From till at informatik.uni-bremen.de  Tue Aug  9 19:20:20 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Tue Aug  9 19:21:18 2005
Subject: [Flirts] Re: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE62@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE62@goldeneye.ad.parc.com>
Message-ID: <42F8E5D4.4090604@informatik.uni-bremen.de>

Dear Valeria,

thank you for agreeing to continue this interesting discussion on
the FLIRTS mailing list.

The paper "What is a logic?" (for those who do not have it: 
http://www.tzi.de/~till/papers/nel05.pdf) takes the following
perspective: we wanted to clarify what the "identity" of a logic
is, and when two logics are considered to be essentially equal
(or different). While on the model-theoretic side, we have good notions
in the paper (and also some results), the proof-theoretic side is much
weaker developed. The main perspective of the paper is to
distinguish different proof theories by the different connectives
they allow, and the different behaviour of these connectives
(regardless whether these connectives are actually present in
the logic or not - the connectives are defined only in terms of
the vocabulary of an institution with proofs). To do this, we use some
standard tools from categorical logic. You are right, proof term
normalization is completely ignored. This is mainly because we could not
see how it would help us to identify connectives in a logic independent
way, or otherwise contribute to the "identity" of a logic.

However, we say in the conclusion of the paper:

   There are some further proof-theoretic properties that we have not
   treated, like (strong) normal forms for proofs (this would require
   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
   term reductions).  A related topic is cut elimination, which
   would require an even finer structure on $\Sen(\Sigma)$,
   with proof rules of particular format.
   We hope this essay provides a good starting point for
   such investigations.

This finer structure might be seen as inessential for the logic itself
(e.g. FOL with Gentzen style proofs and FOL with natural deduction
proofs are "essentially the same" from this perspective), but of course
this would not preclude a much more fine-grained "identity of prood
system", based on 2-categories of proofs. Does this meet your point?

I must admit that I do not really know much about normal forms of
proofs, and still need to look in your papers more thoroughly. But
before doing that, I want to discuss some point. What me puzzles me a
bit already with the approach of defining connectives in a
proof-theoretic way within 1-categories is the following
(citing from another mail from me written in connection with the paper):

   Note that arrows in proof categories are proofs up to equivalence.
   And we impose certain conditions on this equivalence.
   A simple example: if we infer A/\B from A/\B by conjunction
   elimination and conjunction introduction, then this proof must
   be equivalent to the proof infering A/\B directly from itself,
   because conjunction is product, and <pi_1,pi_2>=id.
   Basically, for propositional logic, our axioms of proof-theoretic
   institutions say that the category of proofs is bicartesian closed
   (ie cartesian closed + finite coproducts, including inital objects).
   Lambek and Scott, "Introduction to categorical higher-order logic",
   show on p.67, that in any bicartesion closed category, for an object
   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
   the initial object). From this it follows that any classical
   bicartesion closed category (i.e. one with A is isomorphic to
   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
   to a thin category, and hence thin itself.

This means that for classical logic, the approach of identifying
different calculi (say, Gentzen or natural deduction) via (2-)categories
of proofs seems to be hopeless, because everything is the same thin
category, namely essentially the usual Lindenbaum-Tarski algebra.
Or am I missing something?

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> 
> Dear Till,
> 
> I noticed that you actually went ahead and did some  of the work we
> discussed a long time ago
> (see below) on adding "categorical logic" to institutions on a joint
> paper with Diaconescu, Goguen and Tarlecki ("What is a Logic?). I was
> invited speaker at the Universal Logic in Montreux, where Diaconescu
> talked about it. 
> 
> While I did feel a bit miffed that my original suggestion of the problem
> wasn't mentioned at all, my problem with the paper is not that. My
> problem is that your approach seems to be the proverbial "throwing the
> baby away with the bathwater". The point of putting real  proofs (as
> opposed to entailment relations) into institutions was to try to use the
> proofs-as-lambda-term-representations to do some real work for us, ie to
> connect to the paradigm of extracting programs from proofs, or to help
> with abstract analysis or to extend type systems in a principled and
> logical way, etc. i.e. any of the usual applications of categorical
> proof theory would do here. 
> 
> You say in page 2 of your joint paper that your new definition of (proof
> theoretic) institution "fully supports proof theory", but the notion of
> proof theoretic institution (or of equivalence of institutions)
> introduced in the paper has nothing much to do with proof theory as
> people normally know it. What you call "proof theoretic institutions" do
> not overcome the suggested limitation of "categorical logic", because
> proof theoretic institutions do not  model the significant aspect of
> proofs, which is their reduction behavior. 
> 
> The point of the Curry_Howard isomorphism is not that you can model
> propositions as objects in a category and proofs as equivalence classes
> of morphisms: the point is that the behaviour of proofs is preserved
> under this modelling. This is why some people think that the
> Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
> The  crucial point is that if proof pi reduces to pi' via alpha-beta-eta
> reductions than the corresponding morphisms are related in the target
> category (either by equality or reduction). Nothing like that happens in
> the proof-theoretic institutions, which is why they are only
> proof-theoretical in name.
> The functor Pr: Sign -> Cat is only about proofs in its name, which you
> presumably realize, as  it is not even spelled out in the definition on
> (page 125 of the book) that Pr stands for proofs. I guess my main
> complaint is that the paper does not define a "proof-theoretical
> institution" in the sense of an institution that preserves proofs, but
> simply as an institution that preserves entailment. But I guess this is
> all right, people will have different perspectives on what is important,
> mathematically speaking.
> 
> Best regards,
> Valeria
> 
> 
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE] 
> Sent: Friday, January 31, 2003 12:14 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Subject: Re: CHI for Institutions
> 
> Dear Valeria,
> 
> agreed, let's devide the work as you suggested.
> 
> For a definition of (some variant of) parchment you might look at p 5ff.
> of "Combingn and representing logical systems using model-theoretic
> parchments", available at my publications page.
> 
> Concerning the Lisbon work: I think it shouldn't be difficult to have a
> meta notion of sequent calculus, like the meta notion of Hilbert
> calculus.
> 
> Concerning the more complex Curry-Howard isos:
> you seem to have one in the paper with Biermann. Then I'll have a look
> on that, before going on with trying to look at Curry-Howard in the
> institutional framework
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com schrieb:
>>Dear Till,
>>Thanks for the very interesting message. Now I have to do some
> reading, I don't even know what a parchment is...
>>But one small thought: your last paragraph about translating the
> Curry-Howard isomorphism in terms of institutions is very interesting
> and seems a more concrete way of pushing forward towards my goal, which
> is different though. My goal is really enriching the whole framework of
> institutions so that it can cope with proofs (and when I say proofs I
> don't mean in a single proof calculus: I usually want a logic to be
> given in different several proof calculi all proved equivalent, like for
> instance for IPL you can give axioms, sequents or Natural deduction and
> you know how to translate proofs from one calculi to the others). So
> another way of pushing forward would be to see if the Lisbon work you've
> mentioned can be "translated" into sequent calculus, for example.
>>Now about your question: no I don't think I know of any reference for
> the diagram in page 26. The first problem  is  that Oyster, PVS and
> NuPRl are computer systems and first of all we would need papers called
> "The essence of Oyster, PVS and NuPRL", or Alf, but I guess these might
> exist. I just haven't had the time or disposition to look for them. This
> would be a different research project altogether it seems to me.
>>Thus a modest proposal: I will read the stuff you've mentioned and try
> to come back with questions/suggestions. Maybe you could try to add some
> details to the suggestion of looking at Curry-Howard in the
> institutional framework?
>>Cheers,
>>Valeria
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From Valeria.dePaiva at parc.com  Tue Aug  9 23:27:34 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Tue Aug  9 23:47:43 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>

Dear Till,
> This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural 
>deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, 
>namely essentially the usual Lindenbaum-Tarski algebra.
>Or am I missing something?
Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.

If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
http://www.cs.bath.ac.uk/~pym/semclasspro.html

Hope this helps,
Best,
Valeria
Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
http://www.ps.uni-sb.de/~lutz/
Dr Valeria de Paiva
PARC
3333 Coyote Hill Road
Palo Alto, CA 94304
USA

-----Original Message-----
From: Till Mossakowski [mailto:till@informatik.uni-bremen.de] 
Sent: Tuesday, August 09, 2005 10:20 AM
To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
Cc: FLIRTS
Subject: Re: Curry-Howard isomorphism for Institutions

Dear Valeria,

thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.

The paper "What is a logic?" (for those who do not have it: 
http://www.tzi.de/~till/papers/nel05.pdf) takes the following
perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.

However, we say in the conclusion of the paper:

   There are some further proof-theoretic properties that we have not
   treated, like (strong) normal forms for proofs (this would require
   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
   term reductions).  A related topic is cut elimination, which
   would require an even finer structure on $\Sen(\Sigma)$,
   with proof rules of particular format.
   We hope this essay provides a good starting point for
   such investigations.

This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?

I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):

   Note that arrows in proof categories are proofs up to equivalence.
   And we impose certain conditions on this equivalence.
   A simple example: if we infer A/\B from A/\B by conjunction
   elimination and conjunction introduction, then this proof must
   be equivalent to the proof infering A/\B directly from itself,
   because conjunction is product, and <pi_1,pi_2>=id.
   Basically, for propositional logic, our axioms of proof-theoretic
   institutions say that the category of proofs is bicartesian closed
   (ie cartesian closed + finite coproducts, including inital objects).
   Lambek and Scott, "Introduction to categorical higher-order logic",
   show on p.67, that in any bicartesion closed category, for an object
   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
   the initial object). From this it follows that any classical
   bicartesion closed category (i.e. one with A is isomorphic to
   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
   to a thin category, and hence thin itself.

This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
Or am I missing something?

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> 
> Dear Till,
> 
> I noticed that you actually went ahead and did some  of the work we 
> discussed a long time ago (see below) on adding "categorical logic" to 
> institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
> ("What is a Logic?). I was invited speaker at the Universal Logic in 
> Montreux, where Diaconescu talked about it.
> 
> While I did feel a bit miffed that my original suggestion of the 
> problem wasn't mentioned at all, my problem with the paper is not 
> that. My problem is that your approach seems to be the proverbial 
> "throwing the baby away with the bathwater". The point of putting real  
> proofs (as opposed to entailment relations) into institutions was to 
> try to use the proofs-as-lambda-term-representations to do some real 
> work for us, ie to connect to the paradigm of extracting programs from 
> proofs, or to help with abstract analysis or to extend type systems in 
> a principled and logical way, etc. i.e. any of the usual applications 
> of categorical proof theory would do here.
> 
> You say in page 2 of your joint paper that your new definition of 
> (proof
> theoretic) institution "fully supports proof theory", but the notion 
> of proof theoretic institution (or of equivalence of institutions) 
> introduced in the paper has nothing much to do with proof theory as 
> people normally know it. What you call "proof theoretic institutions" 
> do not overcome the suggested limitation of "categorical logic", 
> because proof theoretic institutions do not  model the significant 
> aspect of proofs, which is their reduction behavior.
> 
> The point of the Curry_Howard isomorphism is not that you can model 
> propositions as objects in a category and proofs as equivalence 
> classes of morphisms: the point is that the behaviour of proofs is 
> preserved under this modelling. This is why some people think that the 
> Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
> The  crucial point is that if proof pi reduces to pi' via 
> alpha-beta-eta reductions than the corresponding morphisms are related 
> in the target category (either by equality or reduction). Nothing like 
> that happens in the proof-theoretic institutions, which is why they 
> are only proof-theoretical in name.
> The functor Pr: Sign -> Cat is only about proofs in its name, which 
> you presumably realize, as  it is not even spelled out in the 
> definition on (page 125 of the book) that Pr stands for proofs. I 
> guess my main complaint is that the paper does not define a 
> "proof-theoretical institution" in the sense of an institution that 
> preserves proofs, but simply as an institution that preserves 
> entailment. But I guess this is all right, people will have different 
> perspectives on what is important, mathematically speaking.
> 
> Best regards,
> Valeria
> 
> 
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
> Sent: Friday, January 31, 2003 12:14 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Subject: Re: CHI for Institutions
> 
> Dear Valeria,
> 
> agreed, let's devide the work as you suggested.
> 
> For a definition of (some variant of) parchment you might look at p 5ff.
> of "Combingn and representing logical systems using model-theoretic 
> parchments", available at my publications page.
> 
> Concerning the Lisbon work: I think it shouldn't be difficult to have 
> a meta notion of sequent calculus, like the meta notion of Hilbert 
> calculus.
> 
> Concerning the more complex Curry-Howard isos:
> you seem to have one in the paper with Biermann. Then I'll have a look 
> on that, before going on with trying to look at Curry-Howard in the 
> institutional framework
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com schrieb:
>>Dear Till,
>>Thanks for the very interesting message. Now I have to do some
> reading, I don't even know what a parchment is...
>>But one small thought: your last paragraph about translating the
> Curry-Howard isomorphism in terms of institutions is very interesting 
> and seems a more concrete way of pushing forward towards my goal, 
> which is different though. My goal is really enriching the whole 
> framework of institutions so that it can cope with proofs (and when I 
> say proofs I don't mean in a single proof calculus: I usually want a 
> logic to be given in different several proof calculi all proved 
> equivalent, like for instance for IPL you can give axioms, sequents or 
> Natural deduction and you know how to translate proofs from one 
> calculi to the others). So another way of pushing forward would be to 
> see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>Now about your question: no I don't think I know of any reference for
> the diagram in page 26. The first problem  is  that Oyster, PVS and 
> NuPRl are computer systems and first of all we would need papers 
> called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
> these might exist. I just haven't had the time or disposition to look 
> for them. This would be a different research project altogether it seems to me.
>>Thus a modest proposal: I will read the stuff you've mentioned and try
> to come back with questions/suggestions. Maybe you could try to add 
> some details to the suggestion of looking at Curry-Howard in the 
> institutional framework?
>>Cheers,
>>Valeria
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From goguen at cs.ucsd.edu  Wed Aug 10 05:30:38 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Wed Aug 10 05:31:06 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
Message-ID: <42F974DE.6050801@cs.ucsd.edu>

Dear Valeria,

Thanks so much for your remarks!  If you have a nicer way of including
proofs into institutions, such that, for example, classical logic works well
by something other than proof terms for sequents, that would be very
welcome and interesting to us.  Ive heard about the work you mention
in various theses but have never looked at it; your summary is a bit on
the discouraging side, i must admit - too many answers, all likely to be
rather complicated, and no clear tradeoffs.

About the origin of institutions with proofs, it goes back to at least 1980
in conversations between Rod Burstall and i; Rod didnt much like the
idea then but i put it in our JACM paper anyway, where it appeared
after a 7 year delay (submitted 1985, but earlier drafts exist, also with
the proof  idea).

About the formulation in "What is a logic?" it seems a bit extreme to
say that it doesnt really do any proof theory; although it is clear that
it is far from doing everything that proof theorists find interesting, it
does allow defining many basic proof theoretic concepts in a nice
abstract way, especially those that connect with models.  Also, as
far as i know, they idea of using sets of sentences as objects is
new and useful, or did we miss something there also?

I find this dialogue very useful and hope that we can continue it
further - i look forward to learning more!

With all best regards,

    joseph

Valeria.dePaiva@parc.com wrote:

>Dear Till,
>  
>
>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural 
>>deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, 
>>namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
>>    
>>
>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>   (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>
>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>
>Hope this helps,
>Best,
>Valeria
>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>http://www.ps.uni-sb.de/~lutz/
>Dr Valeria de Paiva
>PARC
>3333 Coyote Hill Road
>Palo Alto, CA 94304
>USA
>
>-----Original Message-----
>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de] 
>Sent: Tuesday, August 09, 2005 10:20 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Cc: FLIRTS
>Subject: Re: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>
>The paper "What is a logic?" (for those who do not have it: 
>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>
>However, we say in the conclusion of the paper:
>
>   There are some further proof-theoretic properties that we have not
>   treated, like (strong) normal forms for proofs (this would require
>   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>   term reductions).  A related topic is cut elimination, which
>   would require an even finer structure on $\Sen(\Sigma)$,
>   with proof rules of particular format.
>   We hope this essay provides a good starting point for
>   such investigations.
>
>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>
>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>
>   Note that arrows in proof categories are proofs up to equivalence.
>   And we impose certain conditions on this equivalence.
>   A simple example: if we infer A/\B from A/\B by conjunction
>   elimination and conjunction introduction, then this proof must
>   be equivalent to the proof infering A/\B directly from itself,
>   because conjunction is product, and <pi_1,pi_2>=id.
>   Basically, for propositional logic, our axioms of proof-theoretic
>   institutions say that the category of proofs is bicartesian closed
>   (ie cartesian closed + finite coproducts, including inital objects).
>   Lambek and Scott, "Introduction to categorical higher-order logic",
>   show on p.67, that in any bicartesion closed category, for an object
>   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>   the initial object). From this it follows that any classical
>   bicartesion closed category (i.e. one with A is isomorphic to
>   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>   to a thin category, and hence thin itself.
>
>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>Or am I missing something?
>
>Greetings,
>Till
>
>Valeria.dePaiva@parc.com wrote:
>  
>
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real  
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions" 
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>    
>>
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>>      
>>>
>>reading, I don't even know what a parchment is...
>>    
>>
>>>But one small thought: your last paragraph about translating the
>>>      
>>>
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>    
>>
>>>Now about your question: no I don't think I know of any reference for
>>>      
>>>
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>    
>>
>>>Thus a modest proposal: I will read the stuff you've mentioned and try
>>>      
>>>
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>    
>>
>>>Cheers,
>>>Valeria
>>>      
>>>
>
>
>  
>


From till at informatik.uni-bremen.de  Wed Aug 10 11:32:29 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Wed Aug 10 11:33:47 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
Message-ID: <42F9C9AD.4010202@informatik.uni-bremen.de>

Dear Valeria,

many thanks for the pointers to the literature. Actually, my
last intensive study of categorical proof theory dates back
to the late eighties...
Like Joseph I find it a bit discouraging that there are so many
(unrelated?) proof.theoretic approaches. But maybe institutions
can help, let's see...

Let me recall the programme: trying to identify the essential
properties of a logic by only refering to the vocabulary of an
abstract interface, like institutions.
On the model-theoretic side, Razvan Diaconescu has generalized
much of classical model theory to this setting, including
deep results like the fundamental ultraproduct theorem
(see his forthcoming book "Institutional model theory").
This includes characterizations of abstract
logical connecvtives and quantifiers (i.e. not refering
to some particular syntax, but only to the vocabulary
of institutions) through their "model-theoretic behaviour".

On the proof-theoretic side, this is much less explored.
It seems that there are three levels:

1. entailment systems (= kind of pre-orders)
2. categories of sentences and proofs
3. 2-categories of sentences, proofs and proof reductions.

At level 1., all proofs are identified. The achievment of the
1980's proof theory was to identify good categories of
intuitionistic proofs at level 2, with categorical
characterizations of connectives and quantifiers by their
"proof-theoretic behaviour". The problem was that for
classical logic, these categories collapse to thin categories
= pre-orders, such that we are back at level 1. This
might even not be a problem for defining connectives,
but is just too abstract for proof theorists.
Also practically, a tool should be able to output a proof
tree and not just the unique element of a singleton set...

The work you point out is now on level 3. However, not
just 2-categories are defined, but additional categorical
structure for the connectives is introduced. Thus, the
connectives are no longer definable in terms of the
abstract vocabulary (unless this is extended with this
extra categorical structure, which seems awkward).
Maybe a way out is just to have level 3 for the proof
reductions, but define the connectives at level 2
(which works well even with thin categories). Thus, all
the levels would naturally coexist in parallel (noting
that all the necessary information is contained in the
highest level, because there are "quotienting" constructions
going from a higher to a lower level).

Then, for example, the order-enriched categories of F?hrmann and Pym
should naturally form an institution with proofs. Indeed, the
2-cells here are cut-elimination reductions, which fits nicely
with what we had in mind .However, the two-categorical structure only
captures the order-enrichment, while their categories are also linearly
distributive (i.e. kind of bi-monoidal, where the two monoidal
structures model conjunction and disjunction), plus object-wise monoids
and co-monoids (modeling weakening and contraction). This
richer structure is then ignored at the abstract level,
where conjunction is recoverd as product in the category at level 2
(while the category at level 3 might mot even have products:
quotients do not need to reflect them).

Another point is that our proofs work on sets of sentences, rather
than on sentences. This seems to be related to polycategories,
which, however, only use finite sequences of sentences.

And the next question is of course where this general scheme
also fits for other logics, like modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural 
>>deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, 
>>namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
> Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of researc
h  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
> 
> If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
> http://www.cs.bath.ac.uk/~pym/semclasspro.html
> 
> Hope this helps,
> Best,
> Valeria
> Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
> http://www.ps.uni-sb.de/~lutz/
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@informatik.uni-bremen.de] 
> Sent: Tuesday, August 09, 2005 10:20 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Cc: FLIRTS
> Subject: Re: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
> thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
> 
> The paper "What is a logic?" (for those who do not have it: 
> http://www.tzi.de/~till/papers/nel05.pdf) takes the following
> perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
> 
> However, we say in the conclusion of the paper:
> 
>    There are some further proof-theoretic properties that we have not
>    treated, like (strong) normal forms for proofs (this would require
>    $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>    term reductions).  A related topic is cut elimination, which
>    would require an even finer structure on $\Sen(\Sigma)$,
>    with proof rules of particular format.
>    We hope this essay provides a good starting point for
>    such investigations.
> 
> This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
> 
> I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
> 
>    Note that arrows in proof categories are proofs up to equivalence.
>    And we impose certain conditions on this equivalence.
>    A simple example: if we infer A/\B from A/\B by conjunction
>    elimination and conjunction introduction, then this proof must
>    be equivalent to the proof infering A/\B directly from itself,
>    because conjunction is product, and <pi_1,pi_2>=id.
>    Basically, for propositional logic, our axioms of proof-theoretic
>    institutions say that the category of proofs is bicartesian closed
>    (ie cartesian closed + finite coproducts, including inital objects).
>    Lambek and Scott, "Introduction to categorical higher-order logic",
>    show on p.67, that in any bicartesion closed category, for an object
>    A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>    the initial object). From this it follows that any classical
>    bicartesion closed category (i.e. one with A is isomorphic to
>    (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>    to a thin category, and hence thin itself.
> 
> This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
> Or am I missing something?
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com wrote:
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real  
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions" 
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>reading, I don't even know what a parchment is...
>>>But one small thought: your last paragraph about translating the
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>Now about your question: no I don't think I know of any reference for
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>>Thus a modest proposal: I will read the stuff you've mentioned and try
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>>Cheers,
>>>Valeria
> 
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From Valeria.dePaiva at parc.com  Wed Aug 10 19:15:07 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Wed Aug 10 19:15:30 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>

Dear Till,
>Like Joseph I find it a bit discouraging that there are so many (unrelated?) proof.theoretic approaches.
Well, what I said was that (since I'm no expert on the field) *I* don't know the tradeoffs, maybe one of the guys that is an expert does know the relationships between approaches...

>Let me recall the programme: trying to identify the essential properties of a logic by only refering to the 
>vocabulary of an abstract interface, like institutions.
This is possibly the way you see the programme. For me "to identify the essential properties of a logic" means identifying the essential properties of the *derivations* in this logic. A logic, for me, does not exist without its derivations and proofs.

>On the model-theoretic side, Razvan Diaconescu has generalized much of classical model theory to this setting
This is certainly true.

>It seems that there are three levels:

>1. entailment systems (= kind of pre-orders) 
>2. categories of sentences and proofs 
>3. 2-categories of sentences, proofs and proof reductions.
Not quite. One can and normally does talk about proof reductions using simply categories and morphisms.
You do not need to introduce 2-cells for that.

>The work you point out is now on level 3.
Again not quite. Some of the work I mentioned for *classical* logic is actually at level 2.

Also, of course, there's plenty of work (to be done) on generalized categorical proof-theory of 
*non-classical* logic and relating that to the model-theoretical work on institutions. This is the work I was proposing to do, originally.

>This seems to be related to polycategories, which, however, only use finite sequences of sentences.
Yes, indeed in the kind of proof theory I like, rules mostly have a finite number of hypotheses/assumptions. Girard says somewhere that the infinite is always a potential one, which I think is quite nice.

>And the next question is of course where this general scheme also fits for other logics, like modal logics.
Well, some constructive modal logics fit in already. Others (non-constructive ones) will, once you do your classical logic the way *you* think it seems best. The modularity there is another one of the success criteria, right?

Cheers,
Valeria
-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till Mossakowski
Sent: Wednesday, August 10, 2005 2:32 AM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

many thanks for the pointers to the literature. Actually, my last intensive study of categorical proof theory dates back to the late eighties...
Like Joseph I find it a bit discouraging that there are so many
(unrelated?) proof.theoretic approaches. But maybe institutions can help, let's see...

Let me recall the programme: trying to identify the essential properties of a logic by only refering to the vocabulary of an abstract interface, like institutions.
On the model-theoretic side, Razvan Diaconescu has generalized much of classical model theory to this setting, including deep results like the fundamental ultraproduct theorem (see his forthcoming book "Institutional model theory").
This includes characterizations of abstract logical connecvtives and quantifiers (i.e. not refering to some particular syntax, but only to the vocabulary of institutions) through their "model-theoretic behaviour".

On the proof-theoretic side, this is much less explored.
It seems that there are three levels:

1. entailment systems (= kind of pre-orders) 2. categories of sentences and proofs 3. 2-categories of sentences, proofs and proof reductions.

At level 1., all proofs are identified. The achievment of the 1980's proof theory was to identify good categories of intuitionistic proofs at level 2, with categorical characterizations of connectives and quantifiers by their "proof-theoretic behaviour". The problem was that for classical logic, these categories collapse to thin categories = pre-orders, such that we are back at level 1. This might even not be a problem for defining connectives, but is just too abstract for proof theorists.
Also practically, a tool should be able to output a proof tree and not just the unique element of a singleton set...

The work you point out is now on level 3. However, not just 2-categories are defined, but additional categorical structure for the connectives is introduced. Thus, the connectives are no longer definable in terms of the abstract vocabulary (unless this is extended with this extra categorical structure, which seems awkward).
Maybe a way out is just to have level 3 for the proof reductions, but define the connectives at level 2 (which works well even with thin categories). Thus, all the levels would naturally coexist in parallel (noting that all the necessary information is contained in the highest level, because there are "quotienting" constructions going from a higher to a lower level).

Then, for example, the order-enriched categories of F?hrmann and Pym should naturally form an institution with proofs. Indeed, the 2-cells here are cut-elimination reductions, which fits nicely with what we had in mind .However, the two-categorical structure only captures the order-enrichment, while their categories are also linearly distributive (i.e. kind of bi-monoidal, where the two monoidal structures model conjunction and disjunction), plus object-wise monoids and co-monoids (modeling weakening and contraction). This richer structure is then ignored at the abstract level, where conjunction is recoverd as product in the category at level 2 (while the category at level 3 might mot even have products:
quotients do not need to reflect them).

Another point is that our proofs work on sets of sentences, rather than on sentences. This seems to be related to polycategories, which, however, only use finite sequences of sentences.

And the next question is of course where this general scheme also fits for other logics, like modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
>>This means that for classical logic, the approach of identifying 
>>different calculi (say, Gentzen or natural
>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
> Well, you're missing the gigantic amount of work put into this problem 
> since the early 90's with the theses of Griffin, Murthy, Stewart, 
> Harbelin, Urban, Parigot, etc. It is true that classical logic is 
> harder to model categorically than intuitionistic logic and it's true 
> that despite all this work it's not clear (to me at least) what are 
> the trade-offs between different kinds of classical Curry-Howard 
> systems and their categorical semantics. But I guess by now it's 
> clearly understood by most in the community that the old dictum that 
> "classical logic has no categorical semantics" is dead and buried. 
> Which kind of solution you prefer for the problem (Selinger's control 
> cats or fibrations or order-enriched models or even special kinds of  
> polycategories, etc...I'm sure I'm forgetting half the decent 
> solutions, for which I apologize) is up to you. As I said I don't know 
> of a list of trade-offs or cost/benefits analysis of the several 
> solutions. It's not my main area of researc
h  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
> 
> If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
> http://www.cs.bath.ac.uk/~pym/semclasspro.html
> 
> Hope this helps,
> Best,
> Valeria
> Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
> http://www.ps.uni-sb.de/~lutz/
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
> Sent: Tuesday, August 09, 2005 10:20 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Cc: FLIRTS
> Subject: Re: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
> thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
> 
> The paper "What is a logic?" (for those who do not have it: 
> http://www.tzi.de/~till/papers/nel05.pdf) takes the following
> perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
> 
> However, we say in the conclusion of the paper:
> 
>    There are some further proof-theoretic properties that we have not
>    treated, like (strong) normal forms for proofs (this would require
>    $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>    term reductions).  A related topic is cut elimination, which
>    would require an even finer structure on $\Sen(\Sigma)$,
>    with proof rules of particular format.
>    We hope this essay provides a good starting point for
>    such investigations.
> 
> This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
> 
> I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
> 
>    Note that arrows in proof categories are proofs up to equivalence.
>    And we impose certain conditions on this equivalence.
>    A simple example: if we infer A/\B from A/\B by conjunction
>    elimination and conjunction introduction, then this proof must
>    be equivalent to the proof infering A/\B directly from itself,
>    because conjunction is product, and <pi_1,pi_2>=id.
>    Basically, for propositional logic, our axioms of proof-theoretic
>    institutions say that the category of proofs is bicartesian closed
>    (ie cartesian closed + finite coproducts, including inital objects).
>    Lambek and Scott, "Introduction to categorical higher-order logic",
>    show on p.67, that in any bicartesion closed category, for an object
>    A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>    the initial object). From this it follows that any classical
>    bicartesion closed category (i.e. one with A is isomorphic to
>    (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>    to a thin category, and hence thin itself.
> 
> This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
> Or am I missing something?
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com wrote:
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real 
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions"
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>reading, I don't even know what a parchment is...
>>>But one small thought: your last paragraph about translating the
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>Now about your question: no I don't think I know of any reference for
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>try
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>>Cheers,
>>>Valeria
> 
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till _______________________________________________
Flirts mailing list
Flirts@mail.informatik.uni-bremen.de
http://www.informatik.uni-bremen.de/mailman/listinfo/flirts

From Valeria.dePaiva at parc.com  Wed Aug 10 18:45:16 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Wed Aug 10 20:56:33 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE6F@goldeneye.ad.parc.com>

Dear Joseph,
Thanks for the friendly message.
First, let me make it clear that, no, I do not have "a nicer way of including proofs into institutions", at the moment. I had proposed to Till that we could investigate this problem together, as I had become interested in the idea around 1999. I'm attaching some slides from a talk I gave at NASA Ames then when I was trying to sell them a work proposal along these lines. I thought this was a cool idea, which you probably agree since you've had the same idea some 15 years before me. But I only worked on that for a couple of weeks, preparing the talk, which is just a proposal for some work. Not the work itself.


>they idea of using sets of sentences as objects is new and useful, 
>or did we miss something there also?
Well, I don't think it is useful, as *when* you can do your categorical modelling properly, sets of sentences are modelled for free, either in the case where there's a connective that internalizes the comma (like a categorical tensor) or using the fibration mechanism. So no, I don't see the usefulness at the moment, it doesn't buy me anything new. Maybe you'd like to expand on that?

About:
>it seems a bit extreme to say that it doesn't really do any proof theory;
I'm afraid I don't think it excessive. I went back to the paper "What is a Logic?" to see if I was forgetting any basic interesting proof-theoretic notion, given that you've said
>it does allow defining many basic proof theoretic concepts in a nice abstract way,
But I don't see these many concepts. Would you like to discuss them one by one?

>I find this dialogue very useful and hope that we can continue it further - i look forward to learning more!
So do I, thanks for the discussion!

Best regards,

Valeria
-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph Goguen
Sent: Tuesday, August 09, 2005 8:31 PM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Cc: till@informatik.uni-bremen.de; de Paiva, Valeria <Valeria.dePaiva@parc.com>
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

Thanks so much for your remarks!  If you have a nicer way of including proofs into institutions, such that, for example, classical logic works well by something other than proof terms for sequents, that would be very welcome and interesting to us.  Ive heard about the work you mention in various theses but have never looked at it; your summary is a bit on the discouraging side, i must admit - too many answers, all likely to be rather complicated, and no clear tradeoffs.

About the origin of institutions with proofs, it goes back to at least 1980 in conversations between Rod Burstall and i; Rod didnt much like the idea then but i put it in our JACM paper anyway, where it appeared after a 7 year delay (submitted 1985, but earlier drafts exist, also with the proof  idea).

About the formulation in "What is a logic?" it seems a bit extreme to say that it doesnt really do any proof theory; although it is clear that it is far from doing everything that proof theorists find interesting, it does allow defining many basic proof theoretic concepts in a nice abstract way, especially those that connect with models.  Also, as far as i know, they idea of using sets of sentences as objects is new and useful, or did we miss something there also?


With all best regards,

    joseph

Valeria.dePaiva@parc.com wrote:

>Dear Till,
>  
>
>>This means that for classical logic, the approach of identifying 
>>different calculi (say, Gentzen or natural
>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
>>    
>>
>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>   (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>
>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>
>Hope this helps,
>Best,
>Valeria
>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>http://www.ps.uni-sb.de/~lutz/
>Dr Valeria de Paiva
>PARC
>3333 Coyote Hill Road
>Palo Alto, CA 94304
>USA
>
>-----Original Message-----
>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
>Sent: Tuesday, August 09, 2005 10:20 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Cc: FLIRTS
>Subject: Re: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>
>The paper "What is a logic?" (for those who do not have it: 
>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>
>However, we say in the conclusion of the paper:
>
>   There are some further proof-theoretic properties that we have not
>   treated, like (strong) normal forms for proofs (this would require
>   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>   term reductions).  A related topic is cut elimination, which
>   would require an even finer structure on $\Sen(\Sigma)$,
>   with proof rules of particular format.
>   We hope this essay provides a good starting point for
>   such investigations.
>
>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>
>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>
>   Note that arrows in proof categories are proofs up to equivalence.
>   And we impose certain conditions on this equivalence.
>   A simple example: if we infer A/\B from A/\B by conjunction
>   elimination and conjunction introduction, then this proof must
>   be equivalent to the proof infering A/\B directly from itself,
>   because conjunction is product, and <pi_1,pi_2>=id.
>   Basically, for propositional logic, our axioms of proof-theoretic
>   institutions say that the category of proofs is bicartesian closed
>   (ie cartesian closed + finite coproducts, including inital objects).
>   Lambek and Scott, "Introduction to categorical higher-order logic",
>   show on p.67, that in any bicartesion closed category, for an object
>   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>   the initial object). From this it follows that any classical
>   bicartesion closed category (i.e. one with A is isomorphic to
>   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>   to a thin category, and hence thin itself.
>
>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>Or am I missing something?
>
>Greetings,
>Till
>
>Valeria.dePaiva@parc.com wrote:
>  
>
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real 
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions"
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>    
>>
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>>      
>>>
>>reading, I don't even know what a parchment is...
>>    
>>
>>>But one small thought: your last paragraph about translating the
>>>      
>>>
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>    
>>
>>>Now about your question: no I don't think I know of any reference for
>>>      
>>>
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>    
>>
>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>try
>>>      
>>>
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>    
>>
>>>Cheers,
>>>Valeria
>>>      
>>>
>
>
>  
>


_______________________________________________
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From till at informatik.uni-bremen.de  Wed Aug 10 21:30:32 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Wed Aug 10 21:31:32 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
Message-ID: <42FA55D8.9080805@informatik.uni-bremen.de>

Dear Valeria,

 >>they idea of using sets of sentences as objects is new and useful,
 >>or did we miss something there also?
 > Well, I don't think it is useful, as *when* you can do your
 > categorical modelling properly, sets of sentences are modelled for
 > free, either in the case where there's a connective that internalizes
 > the comma (like a categorical tensor) or using the fibration
 > mechanism. So no, I don't see the usefulness at the moment, it doesn't
 > buy me anything new. Maybe you'd like to expand on that?

For properties like compactness, you need possibly infinite set of
sentences. And many logics do not have infinitary conjunction,
hence you cannot internalize. If then the logic happens to have
infinitary proof rules, I cannot see how to model this by fibrations.
But may be I am missing something again :-).

 >>it does allow defining many basic proof theoretic concepts in a nice 
abstract way,
 > But I don't see these many concepts. Would you like to discuss them 
one by one?

One concept is having not not-elimination, which separates classical
from intuitionistic logic.

> A logic, for me, does not exist without its derivations and proofs.
That's interesting. Strassburger in is paper "What is a logic
and what is a proof" starts with "a logic is a pre-order",
i.e. he omits proofs. Only later proofs come in. And I agree:
there should be a notion of logic with proofs, but a notion
of logic without proofs is useful as well.

>>It seems that there are three levels:
> 
>>1. entailment systems (= kind of pre-orders) 
>>2. categories of sentences and proofs 
>>3. 2-categories of sentences, proofs and proof reductions.
> Not quite. One can and normally does talk about proof reductions using simply categories and morphisms.
> You do not need to introduce 2-cells for that.
I do not understand.
I thought formulas = objects, proofs = 1-cells. So what are
proof reductions other than 2-cells? OK, they might just be
a pre-order on 1-cells. But this just amounts to having
thin Hom-categories.

>>This seems to be related to polycategories, which, however, only use finite sequences of sentences.
> Yes, indeed in the kind of proof theory I like, rules mostly have a finite number of hypotheses/assumptions. Girard says somewhere that the infinite is always a potential one, which I think is quite nice.
I definitely want to include infinite sets of sentences as well.
The whole notion of compactness as studied in traditional logic
only makes sense with infinite sets of sentences.
And ZFC, which is widely used in mathematics, is not finitely
axiomatizable. Neither is first-order Peano arithmetic.
Infinitary rules are less important, but it would be nice not
to exclude them, because they are used occasionally.
Moreover, the framework of institutions with proofs should be
philosophically neutral, and should admit the study of logics
that might be rejected by some people, but not by all.

>  The modularity there is another one of the success criteria, right?
Yes, certainly.

Greetings,
Till

-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From jose at fiadeiro.org  Thu Aug 11 00:50:52 2005
From: jose at fiadeiro.org (=?ISO-8859-1?Q?Jos=E9_Luiz_Fiadeiro?=)
Date: Thu Aug 11 00:51:18 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
Message-ID: <58C13D61-A4E4-4FA1-8444-6AFA8D7AE272@fiadeiro.org>

My two cents...

On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com>  
<Valeria.dePaiva@parc.com> wrote:

>> Let me recall the programme: trying to identify the essential  
>> properties of a logic by only refering to the
>> vocabulary of an abstract interface, like institutions.
>>
> This is possibly the way you see the programme. For me "to identify  
> the essential properties of a logic" means identifying the  
> essential properties of the *derivations* in this logic. A logic,  
> for me, does not exist without its derivations and proofs.


In my opinion, it does not make sense to refer to THE essential  
properties of a logic: it all depends on what you want to do with the  
logic.  Institutions do capture essential properties of a logic in so  
far as the use of a logic for "algebraic specification" is  
concerned.  But one cannot reduce the notion of logic to this  
particular usage.  In fact, even within a generalised notion of  
"algebraic specification", we have found that the structural  
properties of institutions that restrict morphisms to property- 
preserving relationships prevents us from capturing composition as it  
arises in non-deterministic systems.  The so-called "satisfaction  
condition" has also proved to be too restrictive for formalisms that  
work just on subclasses of models that satisfy some closure conditions.

On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
OBJECTS, the structure of logics is in the derivations/proofs.   
However, the "FLIRTS programme" is, as far as I understand, directed  
to Logic as applied to Specification Theory.  In this respect, it is  
not the internal structure of logics that is of interest, but that of  
their applications, namely the structure of specifications.  In other  
words, the ambition of FLIRTS should not be to do Mathematics (or  
Logic with 'L') but Specification Theory (i.e. given usages of logics).

Having said this, I find that it is definitely worth exploring all  
the different constructions that have been mentioned in previous  
messages from the point of view of the applications to Specification  
Theory.  However, I would refrain from trying to go beyond that.


Regards

Jos?



JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Science
University of Leicester
Leicester LE1 7RH
United Kingdom
Tel: +44 116 252 3907
Fax: +44 116 252 3915
http://www.cs.le.ac.uk/



From Valeria.dePaiva at parc.com  Thu Aug 11 01:00:37 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 11 01:00:57 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>

 
Hi,
Of course I agree with Jose that
>it does not make sense to refer to THE essential properties of a logic: 
>it all depends on what you want to do with the logic.

But maybe I missed one referent in this discussion: I thought that Till was referring to the programme
Of adding proofs to institutions in the restricted setting of a proposed research paper that 
we were trying to write. I'm afraid I didn't even know about the FLIRTS programme.
So I read it all in the very restricted setting above.

>In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') 
>but Specification Theory (i.e. given usages of logics).
I know very little about Specification Theory and wouldn't want anyone reading this to think that
I was telling them how to do their job.

Thanks, Jose'.
Best,
Valeria

-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Jos? Luiz Fiadeiro
Sent: Wednesday, August 10, 2005 3:51 PM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

My two cents...

On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com> <Valeria.dePaiva@parc.com> wrote:

>> Let me recall the programme: trying to identify the essential 
>> properties of a logic by only refering to the vocabulary of an 
>> abstract interface, like institutions.
>>
> This is possibly the way you see the programme. For me "to identify 
> the essential properties of a logic" means identifying the essential 
> properties of the *derivations* in this logic. A logic, for me, does 
> not exist without its derivations and proofs.


In my opinion, it does not make sense to refer to THE essential properties of a logic: it all depends on what you want to do with the logic.  Institutions do capture essential properties of a logic in so far as the use of a logic for "algebraic specification" is concerned.  But one cannot reduce the notion of logic to this particular usage.  In fact, even within a generalised notion of "algebraic specification", we have found that the structural properties of institutions that restrict morphisms to property- preserving relationships prevents us from capturing composition as it arises in non-deterministic systems.  The so-called "satisfaction condition" has also proved to be too restrictive for formalisms that work just on subclasses of models that satisfy some closure conditions.

On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
OBJECTS, the structure of logics is in the derivations/proofs.   
However, the "FLIRTS programme" is, as far as I understand, directed to Logic as applied to Specification Theory.  In this respect, it is not the internal structure of logics that is of interest, but that of their applications, namely the structure of specifications.  In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') but Specification Theory (i.e. given usages of logics).

Having said this, I find that it is definitely worth exploring all the different constructions that have been mentioned in previous messages from the point of view of the applications to Specification Theory.  However, I would refrain from trying to go beyond that.


Regards

Jos?



JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Science
University of Leicester
Leicester LE1 7RH
United Kingdom
Tel: +44 116 252 3907
Fax: +44 116 252 3915
http://www.cs.le.ac.uk/



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Flirts mailing list
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From jose at fiadeiro.org  Thu Aug 11 01:09:57 2005
From: jose at fiadeiro.org (=?ISO-8859-1?Q?Jos=E9_Luiz_Fiadeiro?=)
Date: Thu Aug 11 01:10:17 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
Message-ID: <39A7D44B-C1BC-4A62-A185-2FA98F7855C4@fiadeiro.org>


On 11 Aug 2005, at 00:00, <Valeria.dePaiva@parc.com>  
<Valeria.dePaiva@parc.com> wrote:

> But maybe I missed one referent in this discussion: I thought that  
> Till was referring to the programme
> Of adding proofs to institutions in the restricted setting of a  
> proposed research paper that
> we were trying to write. I'm afraid I didn't even know about the  
> FLIRTS programme.
> So I read it all in the very restricted setting above.

I'm afraid that I may be the one who is missing the referent...  One  
the one hand, I don't know about the research paper that you mention;  
on the other hand, my reference to the "FLIRTS programme" is rather  
loose.

I should then rephrase my remarks to mean that I can only understand  
a programme of "adding proofs to institutions" within Specification  
Theory, not within Logic.


Apologies again

Jos?




JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Science
University of Leicester
Leicester LE1 7RH
United Kingdom
Tel: +44 116 252 3907
Fax: +44 116 252 3915
http://www.cs.le.ac.uk/



From Valeria.dePaiva at parc.com  Thu Aug 11 01:49:25 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 11 01:49:55 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE79@goldeneye.ad.parc.com>

Dear Till,

My replies will be somewhat slower, as time is in short supply here.

>For properties like compactness, you need possibly infinite set of
sentences.
All proofs of compactness that I've remembered seen were semantical
proofs,
piggybacking on satisfiability. Hence I have not worried about
compactness much; this goes
back to the notion that the infinities that I'm worried about are all
potential infinities.

And I believe you're right, fibrations won't help if your logic happens
to have infinitary conjunctions,
but the logics I'm interested in, do not have infinitary connectives. I
think I vaguely remember Michael Makkai  being interested in infinitary
conjunctions, so one would have to check what he has to say about it.

>One concept is having not not-elimination, which separates classical
from intuitionistic logic.
But apart from the fact that classical logic has it, while
intuitionistic logic does not, which is really just the definition of
the difference between the logics, I cannot see anything that
institutions with proofs can say about the "concept of
not-not-elimination".

>> A logic, for me, does not exist without its derivations and proofs.
Please don't take the comment without its context: I'm very fond of
algebraic logic and I'm as  keen as any one else is of modelling logics
and forgetting their proofs. But they were there to begin with. I just
don't have to pay attention to them all the time. Modelling is like
that, we allow ourselves to forget stuff when convenient.
The situation would be totally different, if the proofs never existed,
if all that existed was a "soup of symbols, ones unrelated to others".

>But this just amounts to having thin Hom-categories.
Thin Hom-categories is just a different name for  traditional old
categories, so we're in agreement, I take it.

>the framework of institutions with proofs should be philosophically
neutral, 
>and should admit the study of logics that might be rejected by some
people, but not by all.
While I believe that philosophical neutrality is an appealing feature, I
can't see exactly what you're driving at with the second condition. 

Some logics are better-behaved, some are less well-behaved, as far as
proof theory is concerned. 
If one can set up a framework that unifies the "proof-theoretically
well-behaved logics" 
(whatever collection this may end up defining) both model theoretically
and proof-theoretically, then progress would have been made, I reckon.
But of course, the proof is in the pudding, this is all hypothetical
right now.

Best,
Valeria

-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de
[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till
Mossakowski
Sent: Wednesday, August 10, 2005 12:31 PM
To: Formalism, Logic, Institution - Relating, Translating and
Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

 >>they idea of using sets of sentences as objects is new and useful,
>>or did we miss something there also?
 > Well, I don't think it is useful, as *when* you can do your  >
categorical modelling properly, sets of sentences are modelled for  >
free, either in the case where there's a connective that internalizes  >
the comma (like a categorical tensor) or using the fibration  >
mechanism. So no, I don't see the usefulness at the moment, it doesn't
> buy me anything new. Maybe you'd like to expand on that?

For properties like compactness, you need possibly infinite set of
sentences. And many logics do not have infinitary conjunction, hence you
cannot internalize. If then the logic happens to have infinitary proof
rules, I cannot see how to model this by fibrations.
But may be I am missing something again :-).

 >>it does allow defining many basic proof theoretic concepts in a nice
abstract way,  > But I don't see these many concepts. Would you like to
discuss them one by one?

One concept is having not not-elimination, which separates classical
from intuitionistic logic.

> A logic, for me, does not exist without its derivations and proofs.
That's interesting. Strassburger in is paper "What is a logic and what
is a proof" starts with "a logic is a pre-order", i.e. he omits proofs.
Only later proofs come in. And I agree:
there should be a notion of logic with proofs, but a notion of logic
without proofs is useful as well.

>>It seems that there are three levels:
> 
>>1. entailment systems (= kind of pre-orders) 2. categories of 
>>sentences and proofs 3. 2-categories of sentences, proofs and proof 
>>reductions.
> Not quite. One can and normally does talk about proof reductions using
simply categories and morphisms.
> You do not need to introduce 2-cells for that.
I do not understand.
I thought formulas = objects, proofs = 1-cells. So what are proof
reductions other than 2-cells? OK, they might just be a pre-order on
1-cells. But this just amounts to having thin Hom-categories.

>>This seems to be related to polycategories, which, however, only use
finite sequences of sentences.
> Yes, indeed in the kind of proof theory I like, rules mostly have a
finite number of hypotheses/assumptions. Girard says somewhere that the
infinite is always a potential one, which I think is quite nice.
I definitely want to include infinite sets of sentences as well.
The whole notion of compactness as studied in traditional logic only
makes sense with infinite sets of sentences.
And ZFC, which is widely used in mathematics, is not finitely
axiomatizable. Neither is first-order Peano arithmetic.
Infinitary rules are less important, but it would be nice not to exclude
them, because they are used occasionally.
Moreover, the framework of institutions with proofs should be
philosophically neutral, and should admit the study of logics that might
be rejected by some people, but not by all.

>  The modularity there is another one of the success criteria, right?
Yes, certainly.

Greetings,
Till

-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
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From goguen at cs.ucsd.edu  Thu Aug 11 03:25:01 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 11 03:25:25 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE6F@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE6F@goldeneye.ad.parc.com>
Message-ID: <42FAA8ED.90104@cs.ucsd.edu>

Dear Valeria and others,

This is fun, we are getting a lot of different points of view.

Valeria.dePaiva@parc.com wrote:

>Dear Joseph,
>Thanks for the friendly message.
>First, let me make it clear that, no, I do not have "a nicer way of including proofs into institutions", at the moment. I had proposed to Till that we could investigate this problem together, as I had become interested in the idea around 1999. I'm attaching some slides from a talk I gave at NASA Ames then when I was trying to sell them a work proposal along these lines. I thought this was a cool idea, which you probably agree since you've had the same idea some 15 years before me. But I only worked on that for a couple of weeks, preparing the talk, which is just a proposal for some work. Not the work itself.
>
Thanks for the slides.  For some reason page 10 wont print
here, but i get the idea.  It's amusing to see institution
theory referred to as "the mainstream" since that is not a
majority point of view.

>>they idea of using sets of sentences as objects is new and useful, 
>>or did we miss something there also?
>>    
>>
>Well, I don't think it is useful, as *when* you can do your categorical modelling properly, sets of sentences are modelled for free, either in the case where there's a connective that internalizes the comma (like a categorical tensor) or using the fibration mechanism. So no, I don't see the usefulness at the moment, it doesn't buy me anything new. Maybe you'd like to expand on that?
>  
>
I think Till has done a good job on this one.  It's nice to be able to deal
with infinitary logics, compactness, etc. in such a clean way.  By the way,
infinitary logics are sometimes needed for program invariants, despite the
beliefs of people like Hoare and Dijkstra.

>About:
>  
>
>>it seems a bit extreme to say that it doesn't really do any proof theory;
>>    
>>
>I'm afraid I don't think it excessive. I went back to the paper "What is a Logic?" to see if I was forgetting any basic interesting proof-theoretic notion, given that you've said
>  
>
>>it does allow defining many basic proof theoretic concepts in a nice abstract way,
>>    
>>
>But I don't see these many concepts. Would you like to discuss them one by one?
>  
>
Till has mentioned compactness, which i guess is pretty basic.  There has
also been a lot of work on Craig interpolation.  Diaconescu has done
some pretty amazing things, including Beth definability and Robinson
consistency,

>  
>
>>I find this dialogue very useful and hope that we can continue it further - i look forward to learning more!
>>    
>>
>So do I, thanks for the discussion!
>  
>
A perhaps important general point is that institution theory is not trying
to replicate what logicians have done, but to get the same (or better)
results in a much more general setting, so of course, some things are
going to be different.  The bottom line will be what can be done with
the institutional results.

>Best regards,
>
>Valeria
>-----Original Message-----
>From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph Goguen
>Sent: Tuesday, August 09, 2005 8:31 PM
>To: Formalism, Logic, Institution - Relating, Translating and Structuring
>Cc: till@informatik.uni-bremen.de; de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>Thanks so much for your remarks!  If you have a nicer way of including proofs into institutions, such that, for example, classical logic works well by something other than proof terms for sequents, that would be very welcome and interesting to us.  Ive heard about the work you mention in various theses but have never looked at it; your summary is a bit on the discouraging side, i must admit - too many answers, all likely to be rather complicated, and no clear tradeoffs.
>
>About the origin of institutions with proofs, it goes back to at least 1980 in conversations between Rod Burstall and i; Rod didnt much like the idea then but i put it in our JACM paper anyway, where it appeared after a 7 year delay (submitted 1985, but earlier drafts exist, also with the proof  idea).
>
>About the formulation in "What is a logic?" it seems a bit extreme to say that it doesnt really do any proof theory; although it is clear that it is far from doing everything that proof theorists find interesting, it does allow defining many basic proof theoretic concepts in a nice abstract way, especially those that connect with models.  Also, as far as i know, they idea of using sets of sentences as objects is new and useful, or did we miss something there also?
>
>
>With all best regards,
>
>    joseph
>
>Valeria.dePaiva@parc.com wrote:
>
>  
>
>>Dear Till,
>> 
>>
>>    
>>
>>>This means that for classical logic, the approach of identifying 
>>>different calculi (say, Gentzen or natural
>>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>>Or am I missing something?
>>>   
>>>
>>>      
>>>
>>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>>  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>>
>>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>>
>>Hope this helps,
>>Best,
>>Valeria
>>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>>http://www.ps.uni-sb.de/~lutz/
>>Dr Valeria de Paiva
>>PARC
>>3333 Coyote Hill Road
>>Palo Alto, CA 94304
>>USA
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
>>Sent: Tuesday, August 09, 2005 10:20 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Cc: FLIRTS
>>Subject: Re: Curry-Howard isomorphism for Institutions
>>
>>Dear Valeria,
>>
>>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>>
>>The paper "What is a logic?" (for those who do not have it: 
>>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>>
>>However, we say in the conclusion of the paper:
>>
>>  There are some further proof-theoretic properties that we have not
>>  treated, like (strong) normal forms for proofs (this would require
>>  $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>>  term reductions).  A related topic is cut elimination, which
>>  would require an even finer structure on $\Sen(\Sigma)$,
>>  with proof rules of particular format.
>>  We hope this essay provides a good starting point for
>>  such investigations.
>>
>>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>>
>>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>>
>>  Note that arrows in proof categories are proofs up to equivalence.
>>  And we impose certain conditions on this equivalence.
>>  A simple example: if we infer A/\B from A/\B by conjunction
>>  elimination and conjunction introduction, then this proof must
>>  be equivalent to the proof infering A/\B directly from itself,
>>  because conjunction is product, and <pi_1,pi_2>=id.
>>  Basically, for propositional logic, our axioms of proof-theoretic
>>  institutions say that the category of proofs is bicartesian closed
>>  (ie cartesian closed + finite coproducts, including inital objects).
>>  Lambek and Scott, "Introduction to categorical higher-order logic",
>>  show on p.67, that in any bicartesion closed category, for an object
>>  A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>>  the initial object). From this it follows that any classical
>>  bicartesion closed category (i.e. one with A is isomorphic to
>>  (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>>  to a thin category, and hence thin itself.
>>
>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com wrote:
>> 
>>
>>    
>>
>>>Dear Till,
>>>
>>>I noticed that you actually went ahead and did some  of the work we 
>>>discussed a long time ago (see below) on adding "categorical logic" to 
>>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>>Montreux, where Diaconescu talked about it.
>>>
>>>While I did feel a bit miffed that my original suggestion of the 
>>>problem wasn't mentioned at all, my problem with the paper is not 
>>>that. My problem is that your approach seems to be the proverbial 
>>>"throwing the baby away with the bathwater". The point of putting real 
>>>proofs (as opposed to entailment relations) into institutions was to 
>>>try to use the proofs-as-lambda-term-representations to do some real 
>>>work for us, ie to connect to the paradigm of extracting programs from 
>>>proofs, or to help with abstract analysis or to extend type systems in 
>>>a principled and logical way, etc. i.e. any of the usual applications 
>>>of categorical proof theory would do here.
>>>
>>>You say in page 2 of your joint paper that your new definition of 
>>>(proof
>>>theoretic) institution "fully supports proof theory", but the notion 
>>>of proof theoretic institution (or of equivalence of institutions) 
>>>introduced in the paper has nothing much to do with proof theory as 
>>>people normally know it. What you call "proof theoretic institutions"
>>>do not overcome the suggested limitation of "categorical logic", 
>>>because proof theoretic institutions do not  model the significant 
>>>aspect of proofs, which is their reduction behavior.
>>>
>>>The point of the Curry_Howard isomorphism is not that you can model 
>>>propositions as objects in a category and proofs as equivalence 
>>>classes of morphisms: the point is that the behaviour of proofs is 
>>>preserved under this modelling. This is why some people think that the 
>>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>>The  crucial point is that if proof pi reduces to pi' via 
>>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>>in the target category (either by equality or reduction). Nothing like 
>>>that happens in the proof-theoretic institutions, which is why they 
>>>are only proof-theoretical in name.
>>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>>you presumably realize, as  it is not even spelled out in the 
>>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>>guess my main complaint is that the paper does not define a 
>>>"proof-theoretical institution" in the sense of an institution that 
>>>preserves proofs, but simply as an institution that preserves 
>>>entailment. But I guess this is all right, people will have different 
>>>perspectives on what is important, mathematically speaking.
>>>
>>>Best regards,
>>>Valeria
>>>
>>>
>>>
>>>-----Original Message-----
>>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>>Sent: Friday, January 31, 2003 12:14 AM
>>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>>Subject: Re: CHI for Institutions
>>>
>>>Dear Valeria,
>>>
>>>agreed, let's devide the work as you suggested.
>>>
>>>For a definition of (some variant of) parchment you might look at p 5ff.
>>>of "Combingn and representing logical systems using model-theoretic 
>>>parchments", available at my publications page.
>>>
>>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>>calculus.
>>>
>>>Concerning the more complex Curry-Howard isos:
>>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>>on that, before going on with trying to look at Curry-Howard in the 
>>>institutional framework
>>>
>>>Greetings,
>>>Till
>>>
>>>Valeria.dePaiva@parc.com schrieb:
>>>   
>>>
>>>      
>>>
>>>>Dear Till,
>>>>Thanks for the very interesting message. Now I have to do some
>>>>     
>>>>
>>>>        
>>>>
>>>reading, I don't even know what a parchment is...
>>>   
>>>
>>>      
>>>
>>>>But one small thought: your last paragraph about translating the
>>>>     
>>>>
>>>>        
>>>>
>>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>>and seems a more concrete way of pushing forward towards my goal, 
>>>which is different though. My goal is really enriching the whole 
>>>framework of institutions so that it can cope with proofs (and when I 
>>>say proofs I don't mean in a single proof calculus: I usually want a 
>>>logic to be given in different several proof calculi all proved 
>>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>>Natural deduction and you know how to translate proofs from one 
>>>calculi to the others). So another way of pushing forward would be to 
>>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>   
>>>
>>>      
>>>
>>>>Now about your question: no I don't think I know of any reference for
>>>>     
>>>>
>>>>        
>>>>
>>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>>NuPRl are computer systems and first of all we would need papers 
>>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>>these might exist. I just haven't had the time or disposition to look 
>>>for them. This would be a different research project altogether it seems to me.
>>>   
>>>
>>>      
>>>
>>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>>try
>>>>     
>>>>
>>>>        
>>>>
>>>to come back with questions/suggestions. Maybe you could try to add 
>>>some details to the suggestion of looking at Curry-Howard in the 
>>>institutional framework?
>>>   
>>>
>>>      
>>>
>>>>Cheers,
>>>>Valeria
>>>>     
>>>>
>>>>        
>>>>
>> 
>>
>>    
>>
>
>
>_______________________________________________
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>
>------------------------------------------------------------------------
>
>_______________________________________________
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>


From goguen at cs.ucsd.edu  Thu Aug 11 03:46:27 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 11 03:46:34 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <58C13D61-A4E4-4FA1-8444-6AFA8D7AE272@fiadeiro.org>
References: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
	<58C13D61-A4E4-4FA1-8444-6AFA8D7AE272@fiadeiro.org>
Message-ID: <42FAADF3.40902@cs.ucsd.edu>

Dear friends,

A few almost random reactions to recent postings.

Jos? Luiz Fiadeiro wrote:

> My two cents...
>
> On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com>  
> <Valeria.dePaiva@parc.com> wrote:
>
>>> Let me recall the programme: trying to identify the essential  
>>> properties of a logic by only refering to the
>>> vocabulary of an abstract interface, like institutions.
>>>
>> This is possibly the way you see the programme. For me "to identify  
>> the essential properties of a logic" means identifying the  essential 
>> properties of the *derivations* in this logic. A logic,  for me, does 
>> not exist without its derivations and proofs.
>
>
>
> In my opinion, it does not make sense to refer to THE essential  
> properties of a logic: it all depends on what you want to do with the  
> logic.  Institutions do capture essential properties of a logic in so  
> far as the use of a logic for "algebraic specification" is  
> concerned.  But one cannot reduce the notion of logic to this  
> particular usage.  In fact, even within a generalised notion of  
> "algebraic specification", we have found that the structural  
> properties of institutions that restrict morphisms to property- 
> preserving relationships prevents us from capturing composition as it  
> arises in non-deterministic systems.  The so-called "satisfaction  
> condition" has also proved to be too restrictive for formalisms that  
> work just on subclasses of models that satisfy some closure conditions.
>
Yes, one should always be careful about "essences" - they dont seem to 
exist in
the realm that humans inhabit, we always have to think about what we are 
going
to do with something, or at least, that's one version of CS Peirce's 
pragmatism.

About the satisfaction condition: my experience has been that when it 
fails,
there is usually a better to formulate the situation so that it 
succeeds.  For
example, hidden algebra is an institution that can handle nondeterminism and
it does so (in part) by restricting to a subclass of models (those where the
visible part is a fixed given algebra); Ehrig claimed that it wasnt an 
institution
but he had the wrong morphisms.

> On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
> OBJECTS, the structure of logics is in the derivations/proofs.   
> However, the "FLIRTS programme" is, as far as I understand, directed  
> to Logic as applied to Specification Theory.  In this respect, it is  
> not the internal structure of logics that is of interest, but that of  
> their applications, namely the structure of specifications.  In other  
> words, the ambition of FLIRTS should not be to do Mathematics (or  
> Logic with 'L') but Specification Theory (i.e. given usages of logics).

There are many ways to abstract some structures out of logics, including
Lindenbaum algebras, satisfaction relations (advocated by the eminent 
logician
Jon Barwise), proof nets, ...  I dont see why they cannot coexist.

Also, im not sure that FLIRTS should be given such a narrow 
interpretation; it
seems to me it should be up to the participants to decide what they would
like to discuss, and recent work on institutions aimed at logic seems quite
appropriate to me.

>
> Having said this, I find that it is definitely worth exploring all  
> the different constructions that have been mentioned in previous  
> messages from the point of view of the applications to Specification  
> Theory.  However, I would refrain from trying to go beyond that.

It would be a pity if group theory had been restricted to applications 
within
algebra (such as solvability of polynomials), given its wonderful 
applications
in physics.  In fact, Einstein credits Emmy Noether with a key insight 
for his
theory of relativity (that invariants imply conservation principles, the 
socalled
Noether theorem), so we might well not have relativity theory, or quantum
theory, if the original applications were the only ones allowed. 

I dont see why we cant apply institutions wherever we like.  The paper
"What is a logic?" is by no means aimed at specification theory, but was
submitted to a contest asking for the best answer to the question in the
title (rumour says the paper came in second, and might have been first
if the political climate at the conference had been different).

Even worse, i am guilty of applying institutions to cognitive science,
multimedia art, philosophy, user interface design, and programming, none
of which can be called specification theory in any normal sense.

>
>
> Regards
>
> Jos?
>
>
>
> JOSE LUIZ FIADEIRO
> Professor of Software Science and Engineering
>
> http://www.fiadeiro.org/jose
> Mob: +44 779 124 7816
> Skype: jfiadeiro
>
> Department of Computer Science
> University of Leicester
> Leicester LE1 7RH
> United Kingdom
> Tel: +44 116 252 3907
> Fax: +44 116 252 3915
> http://www.cs.le.ac.uk/
>
>
>
> _______________________________________________
> Flirts mailing list
> Flirts@mail.informatik.uni-bremen.de
> http://www.informatik.uni-bremen.de/mailman/listinfo/flirts

From goguen at cs.ucsd.edu  Thu Aug 11 04:06:50 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 11 04:07:34 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
Message-ID: <42FAB2BA.7020005@cs.ucsd.edu>

There seem to be a lot of different programmes going on all
at once, which seems very good to me.  Categorical logic,
for example, has a distinguished history already and many
brilliant practitioners who have done brilliant work, which
has established a fairly clear tradition and direction.  Im
not sure that institution theory can claim as much deep
research, but it can claim some serious applications,
and it does also have a tradition and direction of its
own, which is different from that of categorical logic.

Moreover, even within the institution of institutions
there are different research agendas.  Actually, i
have several different ones just myself!  One that
i forgot to mention in the previous email is semantics
of database systems and ontologies.

Work at Microsoft on data warehousing, for
example (Bernstein & Alagic), and at UCSD on
a system to help ecologists integrate and analyze
their data, use institutions.  If you havent seen it,

    http://research.microsoft.com/db/ModelMgt/

is amusing; it mentions categories not institutions
but institutions are the main feature of ref [1] there.

You could also look at

    http://www.cs.ucsd.edu/~goguen/projs/data.html

especially the first four papers in the biblio there.

Adding proofs to institutions and seeing what that
can be made to do is another programme, but i
think requiring it to conform to categorical logic
or specification theory is not a necessity for
that project, though the many good results in
these areas are certainly highly relevant to it.

In particular, i would like to suggest that the
three levels suggested by Till are a nice and
very institutional way of going about building
an abstraction hierarchy for proofs.  We should
ask if it works, not if it agrees with some other
body of theory, and as far as i can see, it works
rather well.

With very best wishes to all,

   joseph

Valeria.dePaiva@parc.com wrote:

> 
>Hi,
>Of course I agree with Jose that
>  
>
>>it does not make sense to refer to THE essential properties of a logic: 
>>it all depends on what you want to do with the logic.
>>    
>>
>
>But maybe I missed one referent in this discussion: I thought that Till was referring to the programme
>Of adding proofs to institutions in the restricted setting of a proposed research paper that 
>we were trying to write. I'm afraid I didn't even know about the FLIRTS programme.
>So I read it all in the very restricted setting above.
>
>  
>
>>In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') 
>>but Specification Theory (i.e. given usages of logics).
>>    
>>
>I know very little about Specification Theory and wouldn't want anyone reading this to think that
>I was telling them how to do their job.
>
>Thanks, Jose'.
>Best,
>Valeria
>
>-----Original Message-----
>From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Jos? Luiz Fiadeiro
>Sent: Wednesday, August 10, 2005 3:51 PM
>To: Formalism, Logic, Institution - Relating, Translating and Structuring
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>My two cents...
>
>On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com> <Valeria.dePaiva@parc.com> wrote:
>
>  
>
>>>Let me recall the programme: trying to identify the essential 
>>>properties of a logic by only refering to the vocabulary of an 
>>>abstract interface, like institutions.
>>>
>>>      
>>>
>>This is possibly the way you see the programme. For me "to identify 
>>the essential properties of a logic" means identifying the essential 
>>properties of the *derivations* in this logic. A logic, for me, does 
>>not exist without its derivations and proofs.
>>    
>>
>
>
>In my opinion, it does not make sense to refer to THE essential properties of a logic: it all depends on what you want to do with the logic.  Institutions do capture essential properties of a logic in so far as the use of a logic for "algebraic specification" is concerned.  But one cannot reduce the notion of logic to this particular usage.  In fact, even within a generalised notion of "algebraic specification", we have found that the structural properties of institutions that restrict morphisms to property- preserving relationships prevents us from capturing composition as it arises in non-deterministic systems.  The so-called "satisfaction condition" has also proved to be too restrictive for formalisms that work just on subclasses of models that satisfy some closure conditions.
>
>On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
>OBJECTS, the structure of logics is in the derivations/proofs.   
>However, the "FLIRTS programme" is, as far as I understand, directed to Logic as applied to Specification Theory.  In this respect, it is not the internal structure of logics that is of interest, but that of their applications, namely the structure of specifications.  In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') but Specification Theory (i.e. given usages of logics).
>
>Having said this, I find that it is definitely worth exploring all the different constructions that have been mentioned in previous messages from the point of view of the applications to Specification Theory.  However, I would refrain from trying to go beyond that.
>
>
>Regards
>
>Jos?
>
>
>
>JOSE LUIZ FIADEIRO
>Professor of Software Science and Engineering
>
>http://www.fiadeiro.org/jose
>Mob: +44 779 124 7816
>Skype: jfiadeiro
>
>Department of Computer Science
>University of Leicester
>Leicester LE1 7RH
>United Kingdom
>Tel: +44 116 252 3907
>Fax: +44 116 252 3915
>http://www.cs.le.ac.uk/
>
>
>
>_______________________________________________
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>
>_______________________________________________
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>  
>
From jose at fiadeiro.org  Thu Aug 11 10:43:21 2005
From: jose at fiadeiro.org (=?ISO-8859-1?Q?Jos=E9_Luiz_Fiadeiro?=)
Date: Thu Aug 11 10:43:38 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <42FAB2BA.7020005@cs.ucsd.edu>
References: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
	<42FAB2BA.7020005@cs.ucsd.edu>
Message-ID: <CBD4F07F-D706-4C17-9AEF-3A262B967173@fiadeiro.org>

Joseph (and everyone)

I agree with all you say.  Actually, I meant to use the term  
Specification Theory in the widest possible senses to address what is  
perhaps best captured as "modelling", i.e. not just restricted to  
logical formalisms and, indeed, not just to Computer Science.

I guess that what I was trying to say is that it seems more useful to  
look for ways of unifying a number of concepts and techniques that  
people use for "modelling" in a number of domains, than to capture  
the essence of Logic.  At least this is what excites me...

Thanks

Jos?



JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Science
University of Leicester
Leicester LE1 7RH
United Kingdom
Tel: +44 116 252 3907
Fax: +44 116 252 3915
http://www.cs.le.ac.uk/




From goguen at cs.ucsd.edu  Thu Aug 11 17:01:52 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 11 17:02:08 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE7E@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE7E@goldeneye.ad.parc.com>
Message-ID: <42FB6860.1040505@cs.ucsd.edu>

hi again Valeria!

Valeria.dePaiva@parc.com wrote:

> 
>Dear Joseph,
>  
>
>>For some reason page 10 wont print here, but i get the idea. 
>>    
>>
>I am sorry, I think it's eps for pictures that's causing the problem, will see if I can get a pdf version.
>  
>
ps2pdf does the job in unix but i dont really need it, that page is on 
colimits
which i guess i understand well enough.

>>It's amusing to see institution theory referred to as "the mainstream" 
>>since that is not a majority point of view.
>>    
>>
>Indeed, but in the context of categories for specification theory in NASA, it is. Or perhaps it was. I haven't seen the nice guys there for a while...
>
>Craig interpolation I believe we can do totally proof-theoretically in many situations -- without infinite sets of sentences. Compactness, as I said, I have a problem  seeing it as proof theory, but maybe it's just lack of trying.
>  
>
i never said craig interpolation needs infinite sets of sentences, and i 
never said
you couldnt do it proof theoretically!  im just reponding to your 
request for proof
theory stuff that we can do in insitutions - do you remember that 
request?  and
please note that compactness is normally formulated as an assertion 
about proofs
even though it is normally proved model theoretically.

  - joseph

>Best,
>Valeria
>-----Original Message-----
>From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph Goguen
>Sent: Wednesday, August 10, 2005 6:25 PM
>To: Formalism, Logic, Institution - Relating, Translating and Structuring
>Cc: till@informatik.uni-bremen.de
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria and others,
>
>This is fun, we are getting a lot of different points of view.
>
>Valeria.dePaiva@parc.com wrote:
>
>  
>
>>Dear Joseph,
>>Thanks for the friendly message.
>>First, let me make it clear that, no, I do not have "a nicer way of including proofs into institutions", at the moment. I had proposed to Till that we could investigate this problem together, as I had become interested in the idea around 1999. I'm attaching some slides from a talk I gave at NASA Ames then when I was trying to sell them a work proposal along these lines. I thought this was a cool idea, which you probably agree since you've had the same idea some 15 years before me. But I only worked on that for a couple of weeks, preparing the talk, which is just a proposal for some work. Not the work itself.
>>
>>    
>>
>Thanks for the slides.  For some reason page 10 wont print here, but i get the idea.  It's amusing to see institution theory referred to as "the mainstream" since that is not a majority point of view.
>
>  
>
>>>they idea of using sets of sentences as objects is new and useful, or 
>>>did we miss something there also?
>>>   
>>>
>>>      
>>>
>>Well, I don't think it is useful, as *when* you can do your categorical modelling properly, sets of sentences are modelled for free, either in the case where there's a connective that internalizes the comma (like a categorical tensor) or using the fibration mechanism. So no, I don't see the usefulness at the moment, it doesn't buy me anything new. Maybe you'd like to expand on that?
>> 
>>
>>    
>>
>I think Till has done a good job on this one.  It's nice to be able to deal with infinitary logics, compactness, etc. in such a clean way.  By the way, infinitary logics are sometimes needed for program invariants, despite the beliefs of people like Hoare and Dijkstra.
>
>  
>
>>About:
>> 
>>
>>    
>>
>>>it seems a bit extreme to say that it doesn't really do any proof 
>>>theory;
>>>   
>>>
>>>      
>>>
>>I'm afraid I don't think it excessive. I went back to the paper "What 
>>is a Logic?" to see if I was forgetting any basic interesting 
>>proof-theoretic notion, given that you've said
>> 
>>
>>    
>>
>>>it does allow defining many basic proof theoretic concepts in a nice 
>>>abstract way,
>>>   
>>>
>>>      
>>>
>>But I don't see these many concepts. Would you like to discuss them one by one?
>> 
>>
>>    
>>
>Till has mentioned compactness, which i guess is pretty basic.  There has also been a lot of work on Craig interpolation.  Diaconescu has done some pretty amazing things, including Beth definability and Robinson consistency,
>
>  
>
>> 
>>
>>    
>>
>>>I find this dialogue very useful and hope that we can continue it further - i look forward to learning more!
>>>   
>>>
>>>      
>>>
>>So do I, thanks for the discussion!
>> 
>>
>>    
>>
>A perhaps important general point is that institution theory is not trying to replicate what logicians have done, but to get the same (or better) results in a much more general setting, so of course, some things are going to be different.  The bottom line will be what can be done with the institutional results.
>
>  
>
>>Best regards,
>>
>>Valeria
>>-----Original Message-----
>>From: flirts-bounces@informatik.uni-bremen.de 
>>[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph 
>>Goguen
>>Sent: Tuesday, August 09, 2005 8:31 PM
>>To: Formalism, Logic, Institution - Relating, Translating and 
>>Structuring
>>Cc: till@informatik.uni-bremen.de; de Paiva, Valeria 
>><Valeria.dePaiva@parc.com>
>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>
>>Dear Valeria,
>>
>>Thanks so much for your remarks!  If you have a nicer way of including proofs into institutions, such that, for example, classical logic works well by something other than proof terms for sequents, that would be very welcome and interesting to us.  Ive heard about the work you mention in various theses but have never looked at it; your summary is a bit on the discouraging side, i must admit - too many answers, all likely to be rather complicated, and no clear tradeoffs.
>>
>>About the origin of institutions with proofs, it goes back to at least 1980 in conversations between Rod Burstall and i; Rod didnt much like the idea then but i put it in our JACM paper anyway, where it appeared after a 7 year delay (submitted 1985, but earlier drafts exist, also with the proof  idea).
>>
>>About the formulation in "What is a logic?" it seems a bit extreme to say that it doesnt really do any proof theory; although it is clear that it is far from doing everything that proof theorists find interesting, it does allow defining many basic proof theoretic concepts in a nice abstract way, especially those that connect with models.  Also, as far as i know, they idea of using sets of sentences as objects is new and useful, or did we miss something there also?
>>
>>
>>With all best regards,
>>
>>   joseph
>>
>>Valeria.dePaiva@parc.com wrote:
>>
>> 
>>
>>    
>>
>>>Dear Till,
>>>
>>>
>>>   
>>>
>>>      
>>>
>>>>This means that for classical logic, the approach of identifying 
>>>>different calculi (say, Gentzen or natural
>>>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>>>Or am I missing something?
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>>> (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>>>
>>>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>>>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>>>
>>>Hope this helps,
>>>Best,
>>>Valeria
>>>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>>>http://www.ps.uni-sb.de/~lutz/
>>>Dr Valeria de Paiva
>>>PARC
>>>3333 Coyote Hill Road
>>>Palo Alto, CA 94304
>>>USA
>>>
>>>-----Original Message-----
>>>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
>>>Sent: Tuesday, August 09, 2005 10:20 AM
>>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>>Cc: FLIRTS
>>>Subject: Re: Curry-Howard isomorphism for Institutions
>>>
>>>Dear Valeria,
>>>
>>>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>>>
>>>The paper "What is a logic?" (for those who do not have it: 
>>>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>>>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>>>
>>>However, we say in the conclusion of the paper:
>>>
>>> There are some further proof-theoretic properties that we have not  
>>>treated, like (strong) normal forms for proofs (this would require  
>>>$Sen(\Sigma)$ to become 2-category of sentences, proof terms and 
>>>proof  term reductions).  A related topic is cut elimination, which  
>>>would require an even finer structure on $\Sen(\Sigma)$,  with proof 
>>>rules of particular format.
>>> We hope this essay provides a good starting point for  such 
>>>investigations.
>>>
>>>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>>>
>>>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>>>
>>> Note that arrows in proof categories are proofs up to equivalence.
>>> And we impose certain conditions on this equivalence.
>>> A simple example: if we infer A/\B from A/\B by conjunction  
>>>elimination and conjunction introduction, then this proof must  be 
>>>equivalent to the proof infering A/\B directly from itself,  because 
>>>conjunction is product, and <pi_1,pi_2>=id.
>>> Basically, for propositional logic, our axioms of proof-theoretic  
>>>institutions say that the category of proofs is bicartesian closed  
>>>(ie cartesian closed + finite coproducts, including inital objects).
>>> Lambek and Scott, "Introduction to categorical higher-order logic",  
>>>show on p.67, that in any bicartesion closed category, for an object  
>>>A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is  
>>>the initial object). From this it follows that any classical  
>>>bicartesion closed category (i.e. one with A is isomorphic to  
>>>(A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent  to a 
>>>thin category, and hence thin itself.
>>>
>>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>>Or am I missing something?
>>>
>>>Greetings,
>>>Till
>>>
>>>Valeria.dePaiva@parc.com wrote:
>>>
>>>
>>>   
>>>
>>>      
>>>
>>>>Dear Till,
>>>>
>>>>I noticed that you actually went ahead and did some  of the work we 
>>>>discussed a long time ago (see below) on adding "categorical logic" 
>>>>to institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>>>Montreux, where Diaconescu talked about it.
>>>>
>>>>While I did feel a bit miffed that my original suggestion of the 
>>>>problem wasn't mentioned at all, my problem with the paper is not 
>>>>that. My problem is that your approach seems to be the proverbial 
>>>>"throwing the baby away with the bathwater". The point of putting 
>>>>real proofs (as opposed to entailment relations) into institutions 
>>>>was to try to use the proofs-as-lambda-term-representations to do 
>>>>some real work for us, ie to connect to the paradigm of extracting 
>>>>programs from proofs, or to help with abstract analysis or to extend 
>>>>type systems in a principled and logical way, etc. i.e. any of the 
>>>>usual applications of categorical proof theory would do here.
>>>>
>>>>You say in page 2 of your joint paper that your new definition of 
>>>>(proof
>>>>theoretic) institution "fully supports proof theory", but the notion 
>>>>of proof theoretic institution (or of equivalence of institutions) 
>>>>introduced in the paper has nothing much to do with proof theory as 
>>>>people normally know it. What you call "proof theoretic institutions"
>>>>do not overcome the suggested limitation of "categorical logic", 
>>>>because proof theoretic institutions do not  model the significant 
>>>>aspect of proofs, which is their reduction behavior.
>>>>
>>>>The point of the Curry_Howard isomorphism is not that you can model 
>>>>propositions as objects in a category and proofs as equivalence 
>>>>classes of morphisms: the point is that the behaviour of proofs is 
>>>>preserved under this modelling. This is why some people think that 
>>>>the Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>>>The  crucial point is that if proof pi reduces to pi' via 
>>>>alpha-beta-eta reductions than the corresponding morphisms are 
>>>>related in the target category (either by equality or reduction). 
>>>>Nothing like that happens in the proof-theoretic institutions, which 
>>>>is why they are only proof-theoretical in name.
>>>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>>>you presumably realize, as  it is not even spelled out in the 
>>>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>>>guess my main complaint is that the paper does not define a 
>>>>"proof-theoretical institution" in the sense of an institution that 
>>>>preserves proofs, but simply as an institution that preserves 
>>>>entailment. But I guess this is all right, people will have different 
>>>>perspectives on what is important, mathematically speaking.
>>>>
>>>>Best regards,
>>>>Valeria
>>>>
>>>>
>>>>
>>>>-----Original Message-----
>>>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>>>Sent: Friday, January 31, 2003 12:14 AM
>>>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>>>Subject: Re: CHI for Institutions
>>>>
>>>>Dear Valeria,
>>>>
>>>>agreed, let's devide the work as you suggested.
>>>>
>>>>For a definition of (some variant of) parchment you might look at p 5ff.
>>>>of "Combingn and representing logical systems using model-theoretic 
>>>>parchments", available at my publications page.
>>>>
>>>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>>>calculus.
>>>>
>>>>Concerning the more complex Curry-Howard isos:
>>>>you seem to have one in the paper with Biermann. Then I'll have a 
>>>>look on that, before going on with trying to look at Curry-Howard in 
>>>>the institutional framework
>>>>
>>>>Greetings,
>>>>Till
>>>>
>>>>Valeria.dePaiva@parc.com schrieb:
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>Dear Till,
>>>>>Thanks for the very interesting message. Now I have to do some
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>reading, I don't even know what a parchment is...
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>But one small thought: your last paragraph about translating the
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>>>and seems a more concrete way of pushing forward towards my goal, 
>>>>which is different though. My goal is really enriching the whole 
>>>>framework of institutions so that it can cope with proofs (and when I 
>>>>say proofs I don't mean in a single proof calculus: I usually want a 
>>>>logic to be given in different several proof calculi all proved 
>>>>equivalent, like for instance for IPL you can give axioms, sequents 
>>>>or Natural deduction and you know how to translate proofs from one 
>>>>calculi to the others). So another way of pushing forward would be to 
>>>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>Now about your question: no I don't think I know of any reference 
>>>>>for
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>>>NuPRl are computer systems and first of all we would need papers 
>>>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>>>these might exist. I just haven't had the time or disposition to look 
>>>>for them. This would be a different research project altogether it seems to me.
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>>>try
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>to come back with questions/suggestions. Maybe you could try to add 
>>>>some details to the suggestion of looking at Curry-Howard in the 
>>>>institutional framework?
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>Cheers,
>>>>>Valeria
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>   
>>>
>>>      
>>>
>>_______________________________________________
>>Flirts mailing list
>>Flirts@mail.informatik.uni-bremen.de
>>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>>
>>-----------------------------------------------------------------------
>>-
>>
>>_______________________________________________
>>Flirts mailing list
>>Flirts@mail.informatik.uni-bremen.de
>>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>>
>>    
>>
>
>
>_______________________________________________
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>
>  
>


From till at informatik.uni-bremen.de  Thu Aug 11 17:06:05 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Thu Aug 11 17:07:30 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE79@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE79@goldeneye.ad.parc.com>
Message-ID: <42FB695D.3020605@informatik.uni-bremen.de>

Dear Valeria,

after all, it seems that this dicussion converges somewhat.
I think in order to make progress, we should think of looking
at some generalization of the notion of comorphism between institutions
to a new notion of comorphism between "institution with proofs and
reductions". I think that this definition should come out naturally out
of the definition of institutions with 2-categorical sentence structure.
This notion of comorphism between two logics then would consist of
1. translation of signatures
2. translation of sentences
3. translation of proofs
4. translation of proof reductions
5. optionally, translation of models

Of course, the crucial step is to look at some examples, to see if
this notion applies here, or needs to modified. Just to throw in
some ideas: We could take some order-enriched categories in the sense
of Pym and F?hrmann, or some of your modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
> 
> My replies will be somewhat slower, as time is in short supply here.
> 
>>For properties like compactness, you need possibly infinite set of
> sentences.
> All proofs of compactness that I've remembered seen were semantical
> proofs,
> piggybacking on satisfiability. Hence I have not worried about
> compactness much; this goes
> back to the notion that the infinities that I'm worried about are all
> potential infinities.
> 
> And I believe you're right, fibrations won't help if your logic happens
> to have infinitary conjunctions,
> but the logics I'm interested in, do not have infinitary connectives. I
> think I vaguely remember Michael Makkai  being interested in infinitary
> conjunctions, so one would have to check what he has to say about it.
> 
>>One concept is having not not-elimination, which separates classical
> from intuitionistic logic.
> But apart from the fact that classical logic has it, while
> intuitionistic logic does not, which is really just the definition of
> the difference between the logics, I cannot see anything that
> institutions with proofs can say about the "concept of
> not-not-elimination".
> 
>>>A logic, for me, does not exist without its derivations and proofs.
> Please don't take the comment without its context: I'm very fond of
> algebraic logic and I'm as  keen as any one else is of modelling logics
> and forgetting their proofs. But they were there to begin with. I just
> don't have to pay attention to them all the time. Modelling is like
> that, we allow ourselves to forget stuff when convenient.
> The situation would be totally different, if the proofs never existed,
> if all that existed was a "soup of symbols, ones unrelated to others".
> 
>>But this just amounts to having thin Hom-categories.
> Thin Hom-categories is just a different name for  traditional old
> categories, so we're in agreement, I take it.
> 
>>the framework of institutions with proofs should be philosophically
> neutral, 
>>and should admit the study of logics that might be rejected by some
> people, but not by all.
> While I believe that philosophical neutrality is an appealing feature, I
> can't see exactly what you're driving at with the second condition. 
> 
> Some logics are better-behaved, some are less well-behaved, as far as
> proof theory is concerned. 
> If one can set up a framework that unifies the "proof-theoretically
> well-behaved logics" 
> (whatever collection this may end up defining) both model theoretically
> and proof-theoretically, then progress would have been made, I reckon.
> But of course, the proof is in the pudding, this is all hypothetical
> right now.
> 
> Best,
> Valeria
> 
> -----Original Message-----
> From: flirts-bounces@informatik.uni-bremen.de
> [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till
> Mossakowski
> Sent: Wednesday, August 10, 2005 12:31 PM
> To: Formalism, Logic, Institution - Relating, Translating and
> Structuring
> Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
>  >>they idea of using sets of sentences as objects is new and useful,
>>>or did we miss something there also?
>  > Well, I don't think it is useful, as *when* you can do your  >
> categorical modelling properly, sets of sentences are modelled for  >
> free, either in the case where there's a connective that internalizes  >
> the comma (like a categorical tensor) or using the fibration  >
> mechanism. So no, I don't see the usefulness at the moment, it doesn't
>>buy me anything new. Maybe you'd like to expand on that?
> 
> For properties like compactness, you need possibly infinite set of
> sentences. And many logics do not have infinitary conjunction, hence you
> cannot internalize. If then the logic happens to have infinitary proof
> rules, I cannot see how to model this by fibrations.
> But may be I am missing something again :-).
> 
>  >>it does allow defining many basic proof theoretic concepts in a nice
> abstract way,  > But I don't see these many concepts. Would you like to
> discuss them one by one?
> 
> One concept is having not not-elimination, which separates classical
> from intuitionistic logic.
> 
>>A logic, for me, does not exist without its derivations and proofs.
> That's interesting. Strassburger in is paper "What is a logic and what
> is a proof" starts with "a logic is a pre-order", i.e. he omits proofs.
> Only later proofs come in. And I agree:
> there should be a notion of logic with proofs, but a notion of logic
> without proofs is useful as well.
> 
>>>It seems that there are three levels:
>>>1. entailment systems (= kind of pre-orders) 2. categories of 
>>>sentences and proofs 3. 2-categories of sentences, proofs and proof 
>>>reductions.
>>Not quite. One can and normally does talk about proof reductions using
> simply categories and morphisms.
>>You do not need to introduce 2-cells for that.
> I do not understand.
> I thought formulas = objects, proofs = 1-cells. So what are proof
> reductions other than 2-cells? OK, they might just be a pre-order on
> 1-cells. But this just amounts to having thin Hom-categories.
> 
>>>This seems to be related to polycategories, which, however, only use
> finite sequences of sentences.
>>Yes, indeed in the kind of proof theory I like, rules mostly have a
> finite number of hypotheses/assumptions. Girard says somewhere that the
> infinite is always a potential one, which I think is quite nice.
> I definitely want to include infinite sets of sentences as well.
> The whole notion of compactness as studied in traditional logic only
> makes sense with infinite sets of sentences.
> And ZFC, which is widely used in mathematics, is not finitely
> axiomatizable. Neither is first-order Peano arithmetic.
> Infinitary rules are less important, but it would be nice not to exclude
> them, because they are used occasionally.
> Moreover, the framework of institutions with proofs should be
> philosophically neutral, and should admit the study of logics that might
> be rejected by some people, but not by all.
> 
>> The modularity there is another one of the success criteria, right?
> Yes, certainly.
> 
> Greetings,
> Till
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From Valeria.dePaiva at parc.com  Thu Aug 11 18:36:03 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 11 18:36:36 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>

Dear Till,
Yes, I think that a notion like
> This notion of comorphism between two logics then would consist of 
>1. translation of signatures 
>2. translation of sentences 
>3. translation of proofs 
>4. translation of proof reductions 
>5. optionally, translation of models
Would be a good idea. Why not start looking at IL which is so well-understood?
Remind me please, where is the institution for IL described.

Best,

Valeria



-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till Mossakowski
Sent: Thursday, August 11, 2005 8:06 AM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

after all, it seems that this dicussion converges somewhat.
I think in order to make progress, we should think of looking at some generalization of the notion of comorphism between institutions to a new notion of comorphism between "institution with proofs and reductions". I think that this definition should come out naturally out of the definition of institutions with 2-categorical sentence structure.
This notion of comorphism between two logics then would consist of 1. translation of signatures 2. translation of sentences 3. translation of proofs 4. translation of proof reductions 5. optionally, translation of models

Of course, the crucial step is to look at some examples, to see if this notion applies here, or needs to modified. Just to throw in some ideas: We could take some order-enriched categories in the sense of Pym and F?hrmann, or some of your modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
> 
> My replies will be somewhat slower, as time is in short supply here.
> 
>>For properties like compactness, you need possibly infinite set of
> sentences.
> All proofs of compactness that I've remembered seen were semantical 
> proofs, piggybacking on satisfiability. Hence I have not worried about 
> compactness much; this goes back to the notion that the infinities 
> that I'm worried about are all potential infinities.
> 
> And I believe you're right, fibrations won't help if your logic 
> happens to have infinitary conjunctions, but the logics I'm interested 
> in, do not have infinitary connectives. I think I vaguely remember 
> Michael Makkai  being interested in infinitary conjunctions, so one 
> would have to check what he has to say about it.
> 
>>One concept is having not not-elimination, which separates classical
> from intuitionistic logic.
> But apart from the fact that classical logic has it, while 
> intuitionistic logic does not, which is really just the definition of 
> the difference between the logics, I cannot see anything that 
> institutions with proofs can say about the "concept of 
> not-not-elimination".
> 
>>>A logic, for me, does not exist without its derivations and proofs.
> Please don't take the comment without its context: I'm very fond of 
> algebraic logic and I'm as  keen as any one else is of modelling 
> logics and forgetting their proofs. But they were there to begin with. 
> I just don't have to pay attention to them all the time. Modelling is 
> like that, we allow ourselves to forget stuff when convenient.
> The situation would be totally different, if the proofs never existed, 
> if all that existed was a "soup of symbols, ones unrelated to others".
> 
>>But this just amounts to having thin Hom-categories.
> Thin Hom-categories is just a different name for  traditional old 
> categories, so we're in agreement, I take it.
> 
>>the framework of institutions with proofs should be philosophically
> neutral,
>>and should admit the study of logics that might be rejected by some
> people, but not by all.
> While I believe that philosophical neutrality is an appealing feature, 
> I can't see exactly what you're driving at with the second condition.
> 
> Some logics are better-behaved, some are less well-behaved, as far as 
> proof theory is concerned.
> If one can set up a framework that unifies the "proof-theoretically 
> well-behaved logics"
> (whatever collection this may end up defining) both model 
> theoretically and proof-theoretically, then progress would have been made, I reckon.
> But of course, the proof is in the pudding, this is all hypothetical 
> right now.
> 
> Best,
> Valeria
> 
> -----Original Message-----
> From: flirts-bounces@informatik.uni-bremen.de
> [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till 
> Mossakowski
> Sent: Wednesday, August 10, 2005 12:31 PM
> To: Formalism, Logic, Institution - Relating, Translating and 
> Structuring
> Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
>  >>they idea of using sets of sentences as objects is new and useful,
>>>or did we miss something there also?
>  > Well, I don't think it is useful, as *when* you can do your  > 
> categorical modelling properly, sets of sentences are modelled for  > 
> free, either in the case where there's a connective that internalizes  
> > the comma (like a categorical tensor) or using the fibration  > 
> mechanism. So no, I don't see the usefulness at the moment, it doesn't
>>buy me anything new. Maybe you'd like to expand on that?
> 
> For properties like compactness, you need possibly infinite set of 
> sentences. And many logics do not have infinitary conjunction, hence 
> you cannot internalize. If then the logic happens to have infinitary 
> proof rules, I cannot see how to model this by fibrations.
> But may be I am missing something again :-).
> 
>  >>it does allow defining many basic proof theoretic concepts in a 
> nice abstract way,  > But I don't see these many concepts. Would you 
> like to discuss them one by one?
> 
> One concept is having not not-elimination, which separates classical 
> from intuitionistic logic.
> 
>>A logic, for me, does not exist without its derivations and proofs.
> That's interesting. Strassburger in is paper "What is a logic and what 
> is a proof" starts with "a logic is a pre-order", i.e. he omits proofs.
> Only later proofs come in. And I agree:
> there should be a notion of logic with proofs, but a notion of logic 
> without proofs is useful as well.
> 
>>>It seems that there are three levels:
>>>1. entailment systems (= kind of pre-orders) 2. categories of 
>>>sentences and proofs 3. 2-categories of sentences, proofs and proof 
>>>reductions.
>>Not quite. One can and normally does talk about proof reductions using
> simply categories and morphisms.
>>You do not need to introduce 2-cells for that.
> I do not understand.
> I thought formulas = objects, proofs = 1-cells. So what are proof 
> reductions other than 2-cells? OK, they might just be a pre-order on 
> 1-cells. But this just amounts to having thin Hom-categories.
> 
>>>This seems to be related to polycategories, which, however, only use
> finite sequences of sentences.
>>Yes, indeed in the kind of proof theory I like, rules mostly have a
> finite number of hypotheses/assumptions. Girard says somewhere that 
> the infinite is always a potential one, which I think is quite nice.
> I definitely want to include infinite sets of sentences as well.
> The whole notion of compactness as studied in traditional logic only 
> makes sense with infinite sets of sentences.
> And ZFC, which is widely used in mathematics, is not finitely 
> axiomatizable. Neither is first-order Peano arithmetic.
> Infinitary rules are less important, but it would be nice not to 
> exclude them, because they are used occasionally.
> Moreover, the framework of institutions with proofs should be 
> philosophically neutral, and should admit the study of logics that 
> might be rejected by some people, but not by all.
> 
>> The modularity there is another one of the success criteria, right?
> Yes, certainly.
> 
> Greetings,
> Till
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till _______________________________________________
Flirts mailing list
Flirts@mail.informatik.uni-bremen.de
http://www.informatik.uni-bremen.de/mailman/listinfo/flirts

From till at informatik.uni-bremen.de  Thu Aug 11 19:36:09 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Thu Aug 11 19:37:36 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
Message-ID: <42FB8C89.3090802@informatik.uni-bremen.de>

Dear Valeria,

 > Why not start looking at IL which is so well-understood?
> Remind me please, where is the institution for IL described.

Unfortunately, there is not "the" institution for IL. Some
possibilities where I think I now the corresponding institution:
1. first-order intuitionistic logic with Heyting algebra semantics
2. first-order intuitionistic logic with Kripke semantics
3. higher-order intuitionistic logic with with topos semantics
4. higher-order intuitionistic logic with pccc semantics

I think (others: please correct me!) that only 4. is described in the
literature (in connection with the language HasCASL), but I think we
should take 3., because it is better known. Basically, it is
recasting the theory of Lambek-Scott in institutional terms.
I can sketch this more formally next week when I am back to the
office (currently I don't have the Lambek-Scott book at hand).

Greetings,
Till



-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From till at informatik.uni-bremen.de  Tue Aug 16 22:58:57 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Tue Aug 16 23:01:56 2005
Subject: [Flirts] Institutions for propositional categorical logics
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
Message-ID: <43025391.7080409@informatik.uni-bremen.de>

[The Flirts mailing list has been down for a few hours. Sorry for that.
  Should be working now again. Till]

Dear Valeria,

OK, so here is a sketch how to institutionalize propositional
categorical logics a la Curry-Howard (coming out of discussions
with Florian Rabe (a PhD student) and Lutz Schr?der, and of course
being motivated by the triangles on your slides).
I follow Lambek-Scott, Introduction to Categorical Higher-Order Logic
(the first few chapters). Instead of their "deductive systems", let
us be a bit more specific and consider the two-sorted specification
of small categories as a partial equational theory (with sorts
object and morphism), extended by the specification of an
operation 1:object axiomatized to be a terminal object.
Call any extension L of this theory with new operations and (oriented)
equations a "categorical logic".
Moreover, let us take the notion of proof-theoretic institution
from our paper "What is a logic?", but extend it with proof
reductions in the following way: Pr:Sign->PreOrd-Cat now becomes
a functor into the category of pre-order enriched categories
(i.e. categories where Hom-Sets are pre-orders).
I think the full generality of 2-categories is not needed,
because one usually does not distinguish between different reductions
between two given proofs.

Given a categorical logic L, let C be the category of L-algebras.
Since L is an extension of the theory of categories, C is a category
of certain small categories.
Given a set X, let T_L(X) be the (absolutely free) term algebra
over X (i.e. terms with operations in L and variables in X).
Define the proof-theoretic institution I(L) as follows:
Sign = Set
------- Propositions as types (categorically: objects) ---------
Sen(Sigma) = T_L(Sigma)_object,
     i.e. L-terms of sort object with variables in Sigma
Sen(sigma:Sigma1->Sigma2)(t) = sigma^#(t),
     where sigma^#:T_L(Sigma1)->T_L(Sigma2) is the extension
       of sigma:Sigma1->Sigma2 c T_(Sigma2) to terms
------- Categorical models -------------------------------------
Mod(Sigma) = {m:Sigma->|A|, where A \in C}
Model morphisms (F,mu):( m:Sigma->|A|) -> (m':Sigma->|B|) consist
   of functors F:A->B \in C and nat. transformations mu:F o m -> m'
Mod(sigma:Sigma1->Sigma2)(m:Sigma2->|A|) = m o sigma
Mod(sigma:Sigma1->Sigma2)(F,mu) = (F,mu * sigma)
(m:Sigma->|A|) |= phi iff m^#(phi) is a terminal object in A,
   where m^# is again the extension of m to terms
The satisfaction condition follows immediately from simple
   universal algebra: (m o sigma)^# = m^# o Sen(sigma).
------- Proofs as terms ----------------------------------------
Pr(Sigma) has as objects sets of Sigma-sentences
   A morphism from Gamma to Delta consists of a collection
   of terms (t_phi)_{phi\in Delta}, such that
   t_phi involves variables x_i, and
   L \cup {x_i:1->psi_i} |- t_phi:1->phi, with psi_i\in Gamma.
   Identities are variables, and composition is given by substitution.
   The pre-order on morphisms is given by the reduction ordering
   (using the orientend equations in L).
Pr(sigma:Sigma1->Sigma2) extends Sen(sigma) on objects,
   and on morphisms, it replaces symbols according to sigma
   (but the symbols from L are leaved fixed, of course).

The canonical example is L = theory of bicartesian closed
categories; then you get propositional intuitionistic logic,
and proof terms also could be thought as lambda-terms.
Note, however, if L = theory of cartesian categories (i.e.
a logic with just conjunction and true), terms are not
lambda-terms.
I would be interested if and how modal logic is another example,
and how the logic morphisms look.
Perhaps this leads to a joint paper?

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
> Yes, I think that a notion like
>>This notion of comorphism between two logics then would consist of 
>>1. translation of signatures 
>>2. translation of sentences 
>>3. translation of proofs 
>>4. translation of proof reductions 
>>5. optionally, translation of models
> Would be a good idea. Why not start looking at IL which is so well-understood?
> Remind me please, where is the institution for IL described.
> 
> Best,
> 
> Valeria


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From till at informatik.uni-bremen.de  Tue Aug 16 23:02:44 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Tue Aug 16 23:05:41 2005
Subject: [Flirts] An institution of higher-order intuitionistic logic
In-Reply-To: <42FB8C89.3090802@informatik.uni-bremen.de>
References: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
	<42FB8C89.3090802@informatik.uni-bremen.de>
Message-ID: <43025474.7040706@informatik.uni-bremen.de>

Dear Valeria,

here is an institution of higher-order intuitionistic logic.
But I think it is better to start with the propositional case,
also because the present logic does not deal with Curry-Howard.

Signatures are theories L of intuitionistic type theory, signature
morphisms are theory translations (see Lambek/Scott p.197).
L-Sentences are formulas in the symbols of Th. Sentence translation
is obvious (cf. again p.197).
Models of an intuitionistic type theory L consist of a topos T
and a strict logical functor T(L)-> T from the classifying topos T(L)
of L to T; equivalently, a theory translation t:L->L(T) from L to the
internal language L(T) of T. Model reduction is just composition (using
the latter representation of models).
An L-sentence e satisfies an L-model t:L->L(T), if T |= t(e),
or equivalently, L(T) |- t(e).
The satisfaction condition is trivial, because model reduction
is composition (Meseguer has called a similar thing "categorical logic
institutions", but note that he uses a T that is fixed for all models).
Model morphisms from t:L->L(T) to t':L->L(T') are pairs, consisting of
a functor f:T->T' and a natural transformation mu:L(f) o t -> t'.

Proof terms need to be extracted from the calculus on p.134f. of
Lambek/Scott. I do not spell out the details here, because
they do not do so either. Actually, they do not apply Curry-Howard at
all for this logic. Perhaps there is other work that does.

Greetings,
Till

-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From goguen at cs.ucsd.edu  Wed Aug 17 00:18:03 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Wed Aug 17 00:18:32 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <4302661B.8070506@cs.ucsd.edu>

Note: FLIRTS was down, but is now (said to be) up, so i am resending this.

Dear Valeria,

Thanks for the pointer to Phil's paper, i enjoyed reading it!

This paper does not make the claims for Curry-Howard that
you mention, but instead makes the much more modest claim
that systems were "designed by researchers in functional languages
and they depend heavily on logics and type systems whose roots
were traced in this paper" (these roots are work of Church,
Gentzen, Girard, etc.).  On the whole, there are pleasingly few
exaggerated claims in the paper.

My impression (e.g., looking at recent POPL proceedings) is
that there is indeed a lot of work that applies ideas from type
theory to programming languages, but the systems used are
far from beautiful, while the beautiful systems are not directly
useful.  It would be surprising to me if anyone worked out a
Curry-Howard isomorphism for any of these ugly type systems.

 - joseph

Valeria.dePaiva@parc.com wrote:

>Dear Joseph,
>
>While I do agree with 
>
>>The basic Curry-Howard isomorphism ("CHI") is one of the most beautiful
>>pieces of mathematics that is 
>>associated with computer science, and its extension to much more
>>general types is 
>>something that theoretical computer scientists can be proud of.
>
>I beg to differ on
> 
>>As far as practice goes, not much has happened,  
>>
>I guess it all depends on what kind of practice you're thinking of. It
>seems to me that quite a lot of the practical work that goes under the
>rubric of "programming languages" design&implementation (including the
>design of Java and other recent typed programming languages) owes its
>existence to programmers picking up the CHI and using it for their own
>purposes.  Phil Wadler had a nice note on Dr. Dobbs about the CHI called
>Proofs are Programs: 19th Century Logic and 21st Century Computing.
>
>I think the URL is
>
>http://homepages.inf.ed.ac.uk/wadler/papers/frege/frege.pdf
>
>But back to the subject at hand:
>
>>i even have some ideas which i hope to write up a bit later on.
>>Looking forward to seeing you this,
>
>Valeria
>
>-----Original Message-----
>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu] 
>Sent: Friday, August 12, 2005 8:03 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>; Formalism, Logic,
>Institution - Relating, Translating and Structuring
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>It seems to me that your definition of  "proof-theoretical" is too
>narrow to capture what mathematical logicians actually do; it seems to
>me that "proofs about proofs" is a more accurate definition for "proof
>theory", since in practice proofs about proofs are often done with
>respect to models, and experience shows that when you have the right
>models, such proofs are often easier that way.
>
>Im thinking that the origin of these divergences of view may go back to
>some discussions you and Till had about what's in the title of this
>thread, "Curry-Howard isomorphism for Institutions" and your research
>focus on type theory?
>
>The basic Curry-Howard isomorphism ("CHI") is one of the most beautiful
>pieces of mathematics that is associated with computer science, and its
>extension to much more general types is something that theoretical
>computer scientists can be proud of.  As far as practice goes, not much
>has happened, but it seems plausible to me that someday, for some class
>of useful programs, it may be possible for users to specify what they
>want in a "Visual Type Theory" language, which then an automatic theorem
>can constructively prove inhabited, yielding a program that can then be
>optimized by sophisticated transformations into a practical program that
>can be run.  After all, computers continue to follow Moore's law, which
>means still lots of power to come, while theorem proving and compiler
>optimization technologies continue to improve (but not exponentially!).
>So it seems worth continuing research on type theory in pursuit of the
>dream of automatic programming (and other some dreams).
>
>But i dont think this should be allowed to dictate the research
>programmes of institutions, which have always had different goals from
>type theory, one of which is to capture mathematical practice in logic,
>including model theoretic reasoning, not just formal manipulations of
>proofs.  Nevertheless, i think it could be very interesting to see what
>can be done with CHI in an institutional setting, and i even have some
>ideas which i hope to write up a bit later on.
>
>By the way, some time ago i put forward the slogan "types as theories"
>as a view of programming, and i also tried to show that if you have a
>nice module system of the sort supported by institutions, then you do
>not really need higher order logic to reason about typical higher order
>functions.  See
>
>    http://www.cs.ucsd.edu/~goguen/pps/utyop.pdf
>
>But this is not to say that i am against type theory or against higher
>order functions.  In fact, i endorse Till's proposal for research on IL,
>except that i do not think that the model aspect of morphisms should be
>optional.
>
>I look forward to further discussion of all this.
>
>  -- joseph
>
>Valeria.dePaiva@parc.com wrote:
>
>>Dear Joseph,
>>A small clarification:
>>
>>>im just reponding to your request for proof theory stuff that we can 
>>>do in insitutions - do you remember that request?
>>>
>>My request (may be I wasn't clear enough) was for proof-theoretical 
>>stuff that you could do with (the proof-theoretical side of)
>>"institutions with proofs".
>>
>>So Craig interpolation done by model theoretical means doesn't count, 
>>even if I do think it pretty. It's the `method' of proof that's at
>>stake in the request, not the statement itself.
>>
>>Cheers,
>>
>>Valeria
>>
>>-----Original Message-----
>>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
>>Sent: Thursday, August 11, 2005 8:02 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>; Formalism, Logic, 
>>Institution - Relating, Translating and Structuring
>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>
>>hi again Valeria!
>>
>>Valeria.dePaiva@parc.com wrote:
>>
>>>Craig interpolation I believe we can do totally proof-theoretically in
>>>many situations -- without infinite sets of sentences. Compactness, as I
>>>said, I have a problem  seeing it as proof theory, but maybe it's just
>>>lack of trying.
>>>
>>i never said craig interpolation needs infinite sets of sentences, and
>>i never said you couldnt do it proof theoretically!  im just reponding
>>to your request for proof theory stuff that we can do in insitutions -
>>do you remember that request?  and please note that compactness is
>>normally formulated as an assertion about proofs even though it is
>>normally proved model theoretically.
>>
>> - joseph
>>
>>>Best,
>>>Valeria
>>>-----Original Message-----
>>>From: flirts-bounces@informatik.uni-bremen.de
>>>[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph 
>>>Goguen
>>>Sent: Wednesday, August 10, 2005 6:25 PM
>>>To: Formalism, Logic, Institution - Relating, Translating and 
>>>Structuring
>>>Cc: till@informatik.uni-bremen.de
>>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>>
>>>Dear Valeria and others,
>>>
>>>This is fun, we are getting a lot of different points of view.
>>>
>>>Valeria.dePaiva@parc.com wrote:
>>>
>>>>Dear Joseph,
>>>>Thanks for the friendly message.
>>>>First, let me make it clear that, no, I do not have "a nicer way of
>>>>>including proofs into institutions", at the moment. I had proposed to
>>>>>Till that we could investigate this problem together, as I had become
>>>>>interested in the idea around 1999. I'm attaching some slides from a
>>>>>talk I gave at NASA Ames then when I was trying to sell them a work
>>>>>proposal along these lines. I thought this was a cool idea, which you
>>>>>probably agree since you've had the same idea some 15 years before me.
>>>>>But I only worked on that for a couple of weeks, preparing the talk,
>>>>>which is just a proposal for some work. Not the work itself.
>>>>>
>>>>I'm afraid I don't think it excessive. I went back to the paper "What
>>>>is a Logic?" to see if I was forgetting any basic interesting 
>>>>proof-theoretic notion, given that you've said
>>>>>abstract way,     
>>>>>
>>>>But I don't see these many concepts. Would you like to discuss them
>>>>one by one?


From Valeria.dePaiva at parc.com  Wed Aug 17 02:35:59 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Wed Aug 17 02:36:37 2005
Subject: FW: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EEBC@goldeneye.ad.parc.com>

 


Dr Valeria de Paiva
PARC
3333 Coyote Hill Road
Palo Alto, CA 94304
USA


-----Original Message-----
From: de Paiva, Valeria <Valeria.dePaiva@parc.com> 
Sent: Monday, August 15, 2005 10:24 PM
To: 'Joseph Goguen'
Cc: 'flirts@informatik.uni-bremen.de'
Subject: RE: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Joseph,

I'm glad you've enjoyed Phil's paper.

> My impression (e.g., looking at recent POPL proceedings) is that there

>is indeed a lot of work that applies ideas from type theory to 
>programming languages,  but the systems used are far from beautiful, 
>while the beautiful systems are not directly useful.

My point was that the work represented in POPL, etc
> owes its existence to programmers picking up the CHI and using it for 
>their own purposes.
Maybe it is  too much to  ask for  the systems that come from logic,  to
carry-on being pretty. But I guess one may hope and I do. 
I also enjoyed very much David Walker's invited talk at IMLA this year,
when he was asking  the logicians in the audience for help with his type
systems for memory management. But I guess there's no paper written
about it, (yet, perhaps, I don't know). 

I have written some of my own take on these ideas, as far as CHI for
modal logics is concerned, in the preface of the special issue of the
JLC dedicated to the second IMLA (2002 in Copenhagen). You can read it
from http://www.cs.bham.ac.uk/%7Evdp/publications/final-preface.pdf
if interested. But this is of course just a subcase of the general
picture.

Best,
Valeria 

-----Original Message-----
From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
Sent: Monday, August 15, 2005 6:52 PM
To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
Cc: flirts@informatik.uni-bremen.de
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

Thanks for the pointer to Phil's paper, i enjoyed reading it!

This paper does not make the claims for Curry-Howard that you mention,
but instead makes the much more modest claim that systems were "designed
by researchers in functional languages and they depend heavily on logics
and type systems whose roots were traced in this paper" (these roots are
work of Church, Gentzen, Girard, etc.).  On the whole, there are
pleasingly few exaggerated claims in the paper.

My impression (e.g., looking at recent POPL proceedings) is that there
is indeed a lot of work that applies ideas from type theory to
programming languages, but the systems used are far from beautiful,
while the beautiful systems are not directly useful.  It would be
surprising to me if anyone worked out a Curry-Howard isomorphism for any
of these ugly type systems.

  - joseph

Valeria.dePaiva@parc.com wrote:

>Dear Joseph,
>
>While I do agree with
>  
>
>>The basic Curry-Howard isomorphism ("CHI") is one of the most 
>>beautiful
>>    
>>
>pieces of mathematics that is
>  
>
>>associated with computer science, and its extension to much more
>>    
>>
>general types is
>  
>
>>something that theoretical computer scientists can be proud of.
>>    
>>
>
>I beg to differ on
>  
>
>>As far as practice goes, not much has happened,
>>    
>>
>I guess it all depends on what kind of practice you're thinking of. It 
>seems to me that quite a lot of the practical work that goes under the 
>rubric of "programming languages" design&implementation (including the 
>design of Java and other recent typed programming languages) owes its 
>existence to programmers picking up the CHI and using it for their own 
>purposes.  Phil Wadler had a nice note on Dr. Dobbs about the CHI 
>called
>
>Proofs are Programs: 19th Century Logic and 21st Century Computing.
>
>I think the URL is
>
>http://homepages.inf.ed.ac.uk/wadler/papers/frege/frege.pdf
>
>But back to the subject at hand:
>  
>
>>i even have some ideas which i hope to write up a bit later on.
>>    
>>
>Looking forward to seeing you this,
>
>Valeria
>
>-----Original Message-----
>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
>Sent: Friday, August 12, 2005 8:03 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>; Formalism, Logic, 
>Institution - Relating, Translating and Structuring
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>It seems to me that your definition of  "proof-theoretical" is too 
>narrow to capture what mathematical logicians actually do; it seems to 
>me that "proofs about proofs" is a more accurate definition for "proof 
>theory", since in practice proofs about proofs are often done with 
>respect to models, and experience shows that when you have the right 
>models, such proofs are often easier that way.
>
>Im thinking that the origin of these divergences of view may go back to

>some discussions you and Till had about what's in the title of this 
>thread, "Curry-Howard isomorphism for Institutions" and your research 
>focus on type theory?
>
>The basic Curry-Howard isomorphism ("CHI") is one of the most beautiful

>pieces of mathematics that is associated with computer science, and its

>extension to much more general types is something that theoretical 
>computer scientists can be proud of.  As far as practice goes, not much

>has happened, but it seems plausible to me that someday, for some class

>of useful programs, it may be possible for users to specify what they 
>want in a "Visual Type Theory" language, which then an automatic 
>theorem can constructively prove inhabited, yielding a program that can

>then be optimized by sophisticated transformations into a practical 
>program that can be run.  After all, computers continue to follow 
>Moore's law, which means still lots of power to come, while theorem 
>proving and compiler optimization technologies continue to improve (but
not exponentially!).
>So it seems worth continuing research on type theory in pursuit of the 
>dream of automatic programming (and other some dreams).
>
>But i dont think this should be allowed to dictate the research 
>programmes of institutions, which have always had different goals from 
>type theory, one of which is to capture mathematical practice in logic,

>including model theoretic reasoning, not just formal manipulations of 
>proofs.  Nevertheless, i think it could be very interesting to see what

>can be done with CHI in an institutional setting, and i even have some 
>ideas which i hope to write up a bit later on.
>
>By the way, some time ago i put forward the slogan "types as theories"
>as a view of programming, and i also tried to show that if you have a 
>nice module system of the sort supported by institutions, then you do 
>not really need higher order logic to reason about typical higher order

>functions.  See
>
>    http://www.cs.ucsd.edu/~goguen/pps/utyop.pdf
>
>But this is not to say that i am against type theory or against higher 
>order functions.  In fact, i endorse Till's proposal for research on 
>IL, except that i do not think that the model aspect of morphisms 
>should be optional.
>
>I look forward to further discussion of all this.
>
>  -- joseph
>
>Valeria.dePaiva@parc.com wrote:
>
>  
>
>>Dear Joseph,
>>A small clarification:
>> 
>>
>>    
>>
>>>im just reponding to your request for proof theory stuff that we can 
>>>do in insitutions - do you remember that request?
>>>   
>>>
>>>      
>>>
>>My request (may be I wasn't clear enough) was for proof-theoretical 
>>stuff that you could do with (the proof-theoretical side of)
>>    
>>
>"institutions with proofs".
>  
>
>>So Craig interpolation done by model theoretical means doesn't count, 
>>even if I do think it pretty. It's the `method' of proof that's at
>>    
>>
>stake in the request, not the statement itself.
>  
>
>>Cheers,
>>
>>Valeria
>>
>>-----Original Message-----
>>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
>>Sent: Thursday, August 11, 2005 8:02 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>; Formalism, Logic, 
>>Institution - Relating, Translating and Structuring
>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>
>>hi again Valeria!
>>
>>Valeria.dePaiva@parc.com wrote:
>>
>> 
>>
>>    
>>
>>>Craig interpolation I believe we can do totally proof-theoretically 
>>>in
>>>      
>>>
>many situations -- without infinite sets of sentences. Compactness, as 
>I said, I have a problem  seeing it as proof theory, but maybe it's 
>just lack of trying.
>  
>
>>>   
>>>
>>>      
>>>
>>i never said craig interpolation needs infinite sets of sentences, and
>>    
>>
>i never said you couldnt do it proof theoretically!  im just reponding 
>to your request for proof theory stuff that we can do in insitutions - 
>do you remember that request?  and please note that compactness is 
>normally formulated as an assertion about proofs even though it is 
>normally proved model theoretically.
>  
>
>> - joseph
>>
>> 
>>
>>    
>>
>>>Best,
>>>Valeria
>>>-----Original Message-----
>>>From: flirts-bounces@informatik.uni-bremen.de
>>>[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph 
>>>Goguen
>>>Sent: Wednesday, August 10, 2005 6:25 PM
>>>To: Formalism, Logic, Institution - Relating, Translating and 
>>>Structuring
>>>Cc: till@informatik.uni-bremen.de
>>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>>
>>>Dear Valeria and others,
>>>
>>>This is fun, we are getting a lot of different points of view.
>>>
>>>Valeria.dePaiva@parc.com wrote:
>>>
>>>
>>>
>>>   
>>>
>>>      
>>>
>>>>Dear Joseph,
>>>>Thanks for the friendly message.
>>>>First, let me make it clear that, no, I do not have "a nicer way of
>>>>        
>>>>
>including proofs into institutions", at the moment. I had proposed to 
>Till that we could investigate this problem together, as I had become 
>interested in the idea around 1999. I'm attaching some slides from a 
>talk I gave at NASA Ames then when I was trying to sell them a work 
>proposal along these lines. I thought this was a cool idea, which you 
>probably agree since you've had the same idea some 15 years before me.
>But I only worked on that for a couple of weeks, preparing the talk, 
>which is just a proposal for some work. Not the work itself.
>  
>
>>>>     
>>>>
>>>>        
>>>>
>>>>> 
>>>>>
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>I'm afraid I don't think it excessive. I went back to the paper 
>>>>"What
>>>>        
>>>>
>
>  
>
>>>>is a Logic?" to see if I was forgetting any basic interesting 
>>>>proof-theoretic notion, given that you've said
>>>>
>>>>
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>it does allow defining many basic proof theoretic concepts in a 
>>>>>nice
>>>>>          
>>>>>
>
>  
>
>>>>>abstract way,
>>>>> 
>>>>>
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>But I don't see these many concepts. Would you like to discuss them
>>>>        
>>>>
>one by one?
>  
>
>>>>     
>>>>
>>>>        
>>>>
>> 
>>
>>    
>>
>
>  
>

From goguen at cs.ucsd.edu  Wed Aug 17 23:52:22 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Wed Aug 17 23:52:35 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EEBC@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EEBC@goldeneye.ad.parc.com>
Message-ID: <4303B196.10607@cs.ucsd.edu>

Dear Valeria,

It seems that Wadler and I agree with each other, and disagree with you,
in saying that programmers do not use CHI even though they do use types.

Also, there seems to be a bug in the Wadler paper, in that he proves that
the product type constructor is commutative (there are also a few smallish
exaggerated claims, but not this one).

Valeria.dePaiva@parc.com wrote:

>-----Original Message-----
>From: de Paiva, Valeria <Valeria.dePaiva@parc.com> 
>Sent: Monday, August 15, 2005 10:24 PM
>To: 'Joseph Goguen'
>Cc: 'flirts@informatik.uni-bremen.de'
>Subject: RE: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Joseph,
>
>I'm glad you've enjoyed Phil's paper. 
>  
>
>>My impression (e.g., looking at recent POPL proceedings) is that thereis indeed a lot of work that applies ideas from type theory to 
>>programming languages,  but the systems used are far from beautiful, 
>>while the beautiful systems are not directly useful.
>>    
>>
>
>My point was that the work represented in POPL, etc owes its existence to programmers picking up the CHI and using it for their own purposes.
>  
>
>Maybe it is  too much to  ask for  the systems that come from logic,  to
>carry-on being pretty. But I guess one may hope and I do. 
>I also enjoyed very much David Walker's invited talk at IMLA this year,
>when he was asking  the logicians in the audience for help with his type
>systems for memory management. But I guess there's no paper written
>about it, (yet, perhaps, I don't know). 
>
>I have written some of my own take on these ideas, as far as CHI for
>modal logics is concerned, in the preface of the special issue of the
>JLC dedicated to the second IMLA (2002 in Copenhagen). You can read it
>from http://www.cs.bham.ac.uk/%7Evdp/publications/final-preface.pdf
>if interested. But this is of course just a subcase of the general
>picture.
>
>Best,
>Valeria 
>  
>
i will take a look at your paper when i get a chance.

cheers,

joseph

>-----Original Message-----
>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
>Sent: Monday, August 15, 2005 6:52 PM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Cc: flirts@informatik.uni-bremen.de
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>Thanks for the pointer to Phil's paper, i enjoyed reading it!
>
>This paper does not make the claims for Curry-Howard that you mention,
>but instead makes the much more modest claim that systems were "designed
>by researchers in functional languages and they depend heavily on logics
>and type systems whose roots were traced in this paper" (these roots are
>work of Church, Gentzen, Girard, etc.).  On the whole, there are
>pleasingly few exaggerated claims in the paper.
>
>My impression (e.g., looking at recent POPL proceedings) is that there
>is indeed a lot of work that applies ideas from type theory to
>programming languages, but the systems used are far from beautiful,
>while the beautiful systems are not directly useful.  It would be
>surprising to me if anyone worked out a Curry-Howard isomorphism for any of these ugly type systems.
>
>  - joseph
>
>Valeria.dePaiva@parc.com wrote:
>
>  
>
> _______________________________________________
>
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>  
>
From goguen at cs.ucsd.edu  Wed Aug 17 23:59:39 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Wed Aug 17 23:59:45 2005
Subject: [Flirts] institutional formulation of Curry-Howard
Message-ID: <4303B34B.2030909@cs.ucsd.edu>

Example 3 on pp 17-19 of "Information integration in institutions" gives
as institutional formulation of the Curry-Howard isomorphism for a
simple special case, though the approach obviously extends.  The
paper can be fetched from

    http://www.cs.ucsd.edu/~goguen/pps/ifi04.pdf

Probably it still has some bugs, as the notation is a little dense and i
just finished the write up; please let me know if you find any.

I hope this may be a useful contribution to our discussions.

Cheers,

   joseph

From till at informatik.uni-bremen.de  Tue Aug  9 19:20:20 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] Re: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE62@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE62@goldeneye.ad.parc.com>
Message-ID: <42F8E5D4.4090604@informatik.uni-bremen.de>

Dear Valeria,

thank you for agreeing to continue this interesting discussion on
the FLIRTS mailing list.

The paper "What is a logic?" (for those who do not have it: 
http://www.tzi.de/~till/papers/nel05.pdf) takes the following
perspective: we wanted to clarify what the "identity" of a logic
is, and when two logics are considered to be essentially equal
(or different). While on the model-theoretic side, we have good notions
in the paper (and also some results), the proof-theoretic side is much
weaker developed. The main perspective of the paper is to
distinguish different proof theories by the different connectives
they allow, and the different behaviour of these connectives
(regardless whether these connectives are actually present in
the logic or not - the connectives are defined only in terms of
the vocabulary of an institution with proofs). To do this, we use some
standard tools from categorical logic. You are right, proof term
normalization is completely ignored. This is mainly because we could not
see how it would help us to identify connectives in a logic independent
way, or otherwise contribute to the "identity" of a logic.

However, we say in the conclusion of the paper:

   There are some further proof-theoretic properties that we have not
   treated, like (strong) normal forms for proofs (this would require
   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
   term reductions).  A related topic is cut elimination, which
   would require an even finer structure on $\Sen(\Sigma)$,
   with proof rules of particular format.
   We hope this essay provides a good starting point for
   such investigations.

This finer structure might be seen as inessential for the logic itself
(e.g. FOL with Gentzen style proofs and FOL with natural deduction
proofs are "essentially the same" from this perspective), but of course
this would not preclude a much more fine-grained "identity of prood
system", based on 2-categories of proofs. Does this meet your point?

I must admit that I do not really know much about normal forms of
proofs, and still need to look in your papers more thoroughly. But
before doing that, I want to discuss some point. What me puzzles me a
bit already with the approach of defining connectives in a
proof-theoretic way within 1-categories is the following
(citing from another mail from me written in connection with the paper):

   Note that arrows in proof categories are proofs up to equivalence.
   And we impose certain conditions on this equivalence.
   A simple example: if we infer A/\B from A/\B by conjunction
   elimination and conjunction introduction, then this proof must
   be equivalent to the proof infering A/\B directly from itself,
   because conjunction is product, and <pi_1,pi_2>=id.
   Basically, for propositional logic, our axioms of proof-theoretic
   institutions say that the category of proofs is bicartesian closed
   (ie cartesian closed + finite coproducts, including inital objects).
   Lambek and Scott, "Introduction to categorical higher-order logic",
   show on p.67, that in any bicartesion closed category, for an object
   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
   the initial object). From this it follows that any classical
   bicartesion closed category (i.e. one with A is isomorphic to
   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
   to a thin category, and hence thin itself.

This means that for classical logic, the approach of identifying
different calculi (say, Gentzen or natural deduction) via (2-)categories
of proofs seems to be hopeless, because everything is the same thin
category, namely essentially the usual Lindenbaum-Tarski algebra.
Or am I missing something?

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> 
> Dear Till,
> 
> I noticed that you actually went ahead and did some  of the work we
> discussed a long time ago
> (see below) on adding "categorical logic" to institutions on a joint
> paper with Diaconescu, Goguen and Tarlecki ("What is a Logic?). I was
> invited speaker at the Universal Logic in Montreux, where Diaconescu
> talked about it. 
> 
> While I did feel a bit miffed that my original suggestion of the problem
> wasn't mentioned at all, my problem with the paper is not that. My
> problem is that your approach seems to be the proverbial "throwing the
> baby away with the bathwater". The point of putting real  proofs (as
> opposed to entailment relations) into institutions was to try to use the
> proofs-as-lambda-term-representations to do some real work for us, ie to
> connect to the paradigm of extracting programs from proofs, or to help
> with abstract analysis or to extend type systems in a principled and
> logical way, etc. i.e. any of the usual applications of categorical
> proof theory would do here. 
> 
> You say in page 2 of your joint paper that your new definition of (proof
> theoretic) institution "fully supports proof theory", but the notion of
> proof theoretic institution (or of equivalence of institutions)
> introduced in the paper has nothing much to do with proof theory as
> people normally know it. What you call "proof theoretic institutions" do
> not overcome the suggested limitation of "categorical logic", because
> proof theoretic institutions do not  model the significant aspect of
> proofs, which is their reduction behavior. 
> 
> The point of the Curry_Howard isomorphism is not that you can model
> propositions as objects in a category and proofs as equivalence classes
> of morphisms: the point is that the behaviour of proofs is preserved
> under this modelling. This is why some people think that the
> Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
> The  crucial point is that if proof pi reduces to pi' via alpha-beta-eta
> reductions than the corresponding morphisms are related in the target
> category (either by equality or reduction). Nothing like that happens in
> the proof-theoretic institutions, which is why they are only
> proof-theoretical in name.
> The functor Pr: Sign -> Cat is only about proofs in its name, which you
> presumably realize, as  it is not even spelled out in the definition on
> (page 125 of the book) that Pr stands for proofs. I guess my main
> complaint is that the paper does not define a "proof-theoretical
> institution" in the sense of an institution that preserves proofs, but
> simply as an institution that preserves entailment. But I guess this is
> all right, people will have different perspectives on what is important,
> mathematically speaking.
> 
> Best regards,
> Valeria
> 
> 
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE] 
> Sent: Friday, January 31, 2003 12:14 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Subject: Re: CHI for Institutions
> 
> Dear Valeria,
> 
> agreed, let's devide the work as you suggested.
> 
> For a definition of (some variant of) parchment you might look at p 5ff.
> of "Combingn and representing logical systems using model-theoretic
> parchments", available at my publications page.
> 
> Concerning the Lisbon work: I think it shouldn't be difficult to have a
> meta notion of sequent calculus, like the meta notion of Hilbert
> calculus.
> 
> Concerning the more complex Curry-Howard isos:
> you seem to have one in the paper with Biermann. Then I'll have a look
> on that, before going on with trying to look at Curry-Howard in the
> institutional framework
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com schrieb:
>>Dear Till,
>>Thanks for the very interesting message. Now I have to do some
> reading, I don't even know what a parchment is...
>>But one small thought: your last paragraph about translating the
> Curry-Howard isomorphism in terms of institutions is very interesting
> and seems a more concrete way of pushing forward towards my goal, which
> is different though. My goal is really enriching the whole framework of
> institutions so that it can cope with proofs (and when I say proofs I
> don't mean in a single proof calculus: I usually want a logic to be
> given in different several proof calculi all proved equivalent, like for
> instance for IPL you can give axioms, sequents or Natural deduction and
> you know how to translate proofs from one calculi to the others). So
> another way of pushing forward would be to see if the Lisbon work you've
> mentioned can be "translated" into sequent calculus, for example.
>>Now about your question: no I don't think I know of any reference for
> the diagram in page 26. The first problem  is  that Oyster, PVS and
> NuPRl are computer systems and first of all we would need papers called
> "The essence of Oyster, PVS and NuPRL", or Alf, but I guess these might
> exist. I just haven't had the time or disposition to look for them. This
> would be a different research project altogether it seems to me.
>>Thus a modest proposal: I will read the stuff you've mentioned and try
> to come back with questions/suggestions. Maybe you could try to add some
> details to the suggestion of looking at Curry-Howard in the
> institutional framework?
>>Cheers,
>>Valeria
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From Valeria.dePaiva at parc.com  Tue Aug  9 23:27:34 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>

Dear Till,
> This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural 
>deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, 
>namely essentially the usual Lindenbaum-Tarski algebra.
>Or am I missing something?
Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.

If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
http://www.cs.bath.ac.uk/~pym/semclasspro.html

Hope this helps,
Best,
Valeria
Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
http://www.ps.uni-sb.de/~lutz/
Dr Valeria de Paiva
PARC
3333 Coyote Hill Road
Palo Alto, CA 94304
USA

-----Original Message-----
From: Till Mossakowski [mailto:till@informatik.uni-bremen.de] 
Sent: Tuesday, August 09, 2005 10:20 AM
To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
Cc: FLIRTS
Subject: Re: Curry-Howard isomorphism for Institutions

Dear Valeria,

thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.

The paper "What is a logic?" (for those who do not have it: 
http://www.tzi.de/~till/papers/nel05.pdf) takes the following
perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.

However, we say in the conclusion of the paper:

   There are some further proof-theoretic properties that we have not
   treated, like (strong) normal forms for proofs (this would require
   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
   term reductions).  A related topic is cut elimination, which
   would require an even finer structure on $\Sen(\Sigma)$,
   with proof rules of particular format.
   We hope this essay provides a good starting point for
   such investigations.

This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?

I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):

   Note that arrows in proof categories are proofs up to equivalence.
   And we impose certain conditions on this equivalence.
   A simple example: if we infer A/\B from A/\B by conjunction
   elimination and conjunction introduction, then this proof must
   be equivalent to the proof infering A/\B directly from itself,
   because conjunction is product, and <pi_1,pi_2>=id.
   Basically, for propositional logic, our axioms of proof-theoretic
   institutions say that the category of proofs is bicartesian closed
   (ie cartesian closed + finite coproducts, including inital objects).
   Lambek and Scott, "Introduction to categorical higher-order logic",
   show on p.67, that in any bicartesion closed category, for an object
   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
   the initial object). From this it follows that any classical
   bicartesion closed category (i.e. one with A is isomorphic to
   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
   to a thin category, and hence thin itself.

This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
Or am I missing something?

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> 
> Dear Till,
> 
> I noticed that you actually went ahead and did some  of the work we 
> discussed a long time ago (see below) on adding "categorical logic" to 
> institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
> ("What is a Logic?). I was invited speaker at the Universal Logic in 
> Montreux, where Diaconescu talked about it.
> 
> While I did feel a bit miffed that my original suggestion of the 
> problem wasn't mentioned at all, my problem with the paper is not 
> that. My problem is that your approach seems to be the proverbial 
> "throwing the baby away with the bathwater". The point of putting real  
> proofs (as opposed to entailment relations) into institutions was to 
> try to use the proofs-as-lambda-term-representations to do some real 
> work for us, ie to connect to the paradigm of extracting programs from 
> proofs, or to help with abstract analysis or to extend type systems in 
> a principled and logical way, etc. i.e. any of the usual applications 
> of categorical proof theory would do here.
> 
> You say in page 2 of your joint paper that your new definition of 
> (proof
> theoretic) institution "fully supports proof theory", but the notion 
> of proof theoretic institution (or of equivalence of institutions) 
> introduced in the paper has nothing much to do with proof theory as 
> people normally know it. What you call "proof theoretic institutions" 
> do not overcome the suggested limitation of "categorical logic", 
> because proof theoretic institutions do not  model the significant 
> aspect of proofs, which is their reduction behavior.
> 
> The point of the Curry_Howard isomorphism is not that you can model 
> propositions as objects in a category and proofs as equivalence 
> classes of morphisms: the point is that the behaviour of proofs is 
> preserved under this modelling. This is why some people think that the 
> Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
> The  crucial point is that if proof pi reduces to pi' via 
> alpha-beta-eta reductions than the corresponding morphisms are related 
> in the target category (either by equality or reduction). Nothing like 
> that happens in the proof-theoretic institutions, which is why they 
> are only proof-theoretical in name.
> The functor Pr: Sign -> Cat is only about proofs in its name, which 
> you presumably realize, as  it is not even spelled out in the 
> definition on (page 125 of the book) that Pr stands for proofs. I 
> guess my main complaint is that the paper does not define a 
> "proof-theoretical institution" in the sense of an institution that 
> preserves proofs, but simply as an institution that preserves 
> entailment. But I guess this is all right, people will have different 
> perspectives on what is important, mathematically speaking.
> 
> Best regards,
> Valeria
> 
> 
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
> Sent: Friday, January 31, 2003 12:14 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Subject: Re: CHI for Institutions
> 
> Dear Valeria,
> 
> agreed, let's devide the work as you suggested.
> 
> For a definition of (some variant of) parchment you might look at p 5ff.
> of "Combingn and representing logical systems using model-theoretic 
> parchments", available at my publications page.
> 
> Concerning the Lisbon work: I think it shouldn't be difficult to have 
> a meta notion of sequent calculus, like the meta notion of Hilbert 
> calculus.
> 
> Concerning the more complex Curry-Howard isos:
> you seem to have one in the paper with Biermann. Then I'll have a look 
> on that, before going on with trying to look at Curry-Howard in the 
> institutional framework
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com schrieb:
>>Dear Till,
>>Thanks for the very interesting message. Now I have to do some
> reading, I don't even know what a parchment is...
>>But one small thought: your last paragraph about translating the
> Curry-Howard isomorphism in terms of institutions is very interesting 
> and seems a more concrete way of pushing forward towards my goal, 
> which is different though. My goal is really enriching the whole 
> framework of institutions so that it can cope with proofs (and when I 
> say proofs I don't mean in a single proof calculus: I usually want a 
> logic to be given in different several proof calculi all proved 
> equivalent, like for instance for IPL you can give axioms, sequents or 
> Natural deduction and you know how to translate proofs from one 
> calculi to the others). So another way of pushing forward would be to 
> see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>Now about your question: no I don't think I know of any reference for
> the diagram in page 26. The first problem  is  that Oyster, PVS and 
> NuPRl are computer systems and first of all we would need papers 
> called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
> these might exist. I just haven't had the time or disposition to look 
> for them. This would be a different research project altogether it seems to me.
>>Thus a modest proposal: I will read the stuff you've mentioned and try
> to come back with questions/suggestions. Maybe you could try to add 
> some details to the suggestion of looking at Curry-Howard in the 
> institutional framework?
>>Cheers,
>>Valeria
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till


From goguen at cs.ucsd.edu  Wed Aug 10 05:30:38 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
Message-ID: <42F974DE.6050801@cs.ucsd.edu>

Dear Valeria,

Thanks so much for your remarks!  If you have a nicer way of including
proofs into institutions, such that, for example, classical logic works well
by something other than proof terms for sequents, that would be very
welcome and interesting to us.  Ive heard about the work you mention
in various theses but have never looked at it; your summary is a bit on
the discouraging side, i must admit - too many answers, all likely to be
rather complicated, and no clear tradeoffs.

About the origin of institutions with proofs, it goes back to at least 1980
in conversations between Rod Burstall and i; Rod didnt much like the
idea then but i put it in our JACM paper anyway, where it appeared
after a 7 year delay (submitted 1985, but earlier drafts exist, also with
the proof  idea).

About the formulation in "What is a logic?" it seems a bit extreme to
say that it doesnt really do any proof theory; although it is clear that
it is far from doing everything that proof theorists find interesting, it
does allow defining many basic proof theoretic concepts in a nice
abstract way, especially those that connect with models.  Also, as
far as i know, they idea of using sets of sentences as objects is
new and useful, or did we miss something there also?

I find this dialogue very useful and hope that we can continue it
further - i look forward to learning more!

With all best regards,

    joseph

Valeria.dePaiva@parc.com wrote:

>Dear Till,
>  
>
>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural 
>>deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, 
>>namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
>>    
>>
>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>   (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>
>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>
>Hope this helps,
>Best,
>Valeria
>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>http://www.ps.uni-sb.de/~lutz/
>Dr Valeria de Paiva
>PARC
>3333 Coyote Hill Road
>Palo Alto, CA 94304
>USA
>
>-----Original Message-----
>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de] 
>Sent: Tuesday, August 09, 2005 10:20 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Cc: FLIRTS
>Subject: Re: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>
>The paper "What is a logic?" (for those who do not have it: 
>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>
>However, we say in the conclusion of the paper:
>
>   There are some further proof-theoretic properties that we have not
>   treated, like (strong) normal forms for proofs (this would require
>   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>   term reductions).  A related topic is cut elimination, which
>   would require an even finer structure on $\Sen(\Sigma)$,
>   with proof rules of particular format.
>   We hope this essay provides a good starting point for
>   such investigations.
>
>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>
>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>
>   Note that arrows in proof categories are proofs up to equivalence.
>   And we impose certain conditions on this equivalence.
>   A simple example: if we infer A/\B from A/\B by conjunction
>   elimination and conjunction introduction, then this proof must
>   be equivalent to the proof infering A/\B directly from itself,
>   because conjunction is product, and <pi_1,pi_2>=id.
>   Basically, for propositional logic, our axioms of proof-theoretic
>   institutions say that the category of proofs is bicartesian closed
>   (ie cartesian closed + finite coproducts, including inital objects).
>   Lambek and Scott, "Introduction to categorical higher-order logic",
>   show on p.67, that in any bicartesion closed category, for an object
>   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>   the initial object). From this it follows that any classical
>   bicartesion closed category (i.e. one with A is isomorphic to
>   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>   to a thin category, and hence thin itself.
>
>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>Or am I missing something?
>
>Greetings,
>Till
>
>Valeria.dePaiva@parc.com wrote:
>  
>
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real  
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions" 
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>    
>>
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>>      
>>>
>>reading, I don't even know what a parchment is...
>>    
>>
>>>But one small thought: your last paragraph about translating the
>>>      
>>>
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>    
>>
>>>Now about your question: no I don't think I know of any reference for
>>>      
>>>
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>    
>>
>>>Thus a modest proposal: I will read the stuff you've mentioned and try
>>>      
>>>
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>    
>>
>>>Cheers,
>>>Valeria
>>>      
>>>
>
>
>  
>



From till at informatik.uni-bremen.de  Wed Aug 10 11:32:29 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
Message-ID: <42F9C9AD.4010202@informatik.uni-bremen.de>

Dear Valeria,

many thanks for the pointers to the literature. Actually, my
last intensive study of categorical proof theory dates back
to the late eighties...
Like Joseph I find it a bit discouraging that there are so many
(unrelated?) proof.theoretic approaches. But maybe institutions
can help, let's see...

Let me recall the programme: trying to identify the essential
properties of a logic by only refering to the vocabulary of an
abstract interface, like institutions.
On the model-theoretic side, Razvan Diaconescu has generalized
much of classical model theory to this setting, including
deep results like the fundamental ultraproduct theorem
(see his forthcoming book "Institutional model theory").
This includes characterizations of abstract
logical connecvtives and quantifiers (i.e. not refering
to some particular syntax, but only to the vocabulary
of institutions) through their "model-theoretic behaviour".

On the proof-theoretic side, this is much less explored.
It seems that there are three levels:

1. entailment systems (= kind of pre-orders)
2. categories of sentences and proofs
3. 2-categories of sentences, proofs and proof reductions.

At level 1., all proofs are identified. The achievment of the
1980's proof theory was to identify good categories of
intuitionistic proofs at level 2, with categorical
characterizations of connectives and quantifiers by their
"proof-theoretic behaviour". The problem was that for
classical logic, these categories collapse to thin categories
= pre-orders, such that we are back at level 1. This
might even not be a problem for defining connectives,
but is just too abstract for proof theorists.
Also practically, a tool should be able to output a proof
tree and not just the unique element of a singleton set...

The work you point out is now on level 3. However, not
just 2-categories are defined, but additional categorical
structure for the connectives is introduced. Thus, the
connectives are no longer definable in terms of the
abstract vocabulary (unless this is extended with this
extra categorical structure, which seems awkward).
Maybe a way out is just to have level 3 for the proof
reductions, but define the connectives at level 2
(which works well even with thin categories). Thus, all
the levels would naturally coexist in parallel (noting
that all the necessary information is contained in the
highest level, because there are "quotienting" constructions
going from a higher to a lower level).

Then, for example, the order-enriched categories of F?hrmann and Pym
should naturally form an institution with proofs. Indeed, the
2-cells here are cut-elimination reductions, which fits nicely
with what we had in mind .However, the two-categorical structure only
captures the order-enrichment, while their categories are also linearly
distributive (i.e. kind of bi-monoidal, where the two monoidal
structures model conjunction and disjunction), plus object-wise monoids
and co-monoids (modeling weakening and contraction). This
richer structure is then ignored at the abstract level,
where conjunction is recoverd as product in the category at level 2
(while the category at level 3 might mot even have products:
quotients do not need to reflect them).

Another point is that our proofs work on sets of sentences, rather
than on sentences. This seems to be related to polycategories,
which, however, only use finite sequences of sentences.

And the next question is of course where this general scheme
also fits for other logics, like modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural 
>>deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, 
>>namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
> Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of researc
h  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
> 
> If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
> http://www.cs.bath.ac.uk/~pym/semclasspro.html
> 
> Hope this helps,
> Best,
> Valeria
> Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
> http://www.ps.uni-sb.de/~lutz/
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@informatik.uni-bremen.de] 
> Sent: Tuesday, August 09, 2005 10:20 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Cc: FLIRTS
> Subject: Re: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
> thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
> 
> The paper "What is a logic?" (for those who do not have it: 
> http://www.tzi.de/~till/papers/nel05.pdf) takes the following
> perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
> 
> However, we say in the conclusion of the paper:
> 
>    There are some further proof-theoretic properties that we have not
>    treated, like (strong) normal forms for proofs (this would require
>    $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>    term reductions).  A related topic is cut elimination, which
>    would require an even finer structure on $\Sen(\Sigma)$,
>    with proof rules of particular format.
>    We hope this essay provides a good starting point for
>    such investigations.
> 
> This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
> 
> I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
> 
>    Note that arrows in proof categories are proofs up to equivalence.
>    And we impose certain conditions on this equivalence.
>    A simple example: if we infer A/\B from A/\B by conjunction
>    elimination and conjunction introduction, then this proof must
>    be equivalent to the proof infering A/\B directly from itself,
>    because conjunction is product, and <pi_1,pi_2>=id.
>    Basically, for propositional logic, our axioms of proof-theoretic
>    institutions say that the category of proofs is bicartesian closed
>    (ie cartesian closed + finite coproducts, including inital objects).
>    Lambek and Scott, "Introduction to categorical higher-order logic",
>    show on p.67, that in any bicartesion closed category, for an object
>    A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>    the initial object). From this it follows that any classical
>    bicartesion closed category (i.e. one with A is isomorphic to
>    (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>    to a thin category, and hence thin itself.
> 
> This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
> Or am I missing something?
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com wrote:
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real  
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions" 
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>reading, I don't even know what a parchment is...
>>>But one small thought: your last paragraph about translating the
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>Now about your question: no I don't think I know of any reference for
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>>Thus a modest proposal: I will read the stuff you've mentioned and try
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>>Cheers,
>>>Valeria
> 
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From Valeria.dePaiva at parc.com  Wed Aug 10 19:15:07 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>

Dear Till,
>Like Joseph I find it a bit discouraging that there are so many (unrelated?) proof.theoretic approaches.
Well, what I said was that (since I'm no expert on the field) *I* don't know the tradeoffs, maybe one of the guys that is an expert does know the relationships between approaches...

>Let me recall the programme: trying to identify the essential properties of a logic by only refering to the 
>vocabulary of an abstract interface, like institutions.
This is possibly the way you see the programme. For me "to identify the essential properties of a logic" means identifying the essential properties of the *derivations* in this logic. A logic, for me, does not exist without its derivations and proofs.

>On the model-theoretic side, Razvan Diaconescu has generalized much of classical model theory to this setting
This is certainly true.

>It seems that there are three levels:

>1. entailment systems (= kind of pre-orders) 
>2. categories of sentences and proofs 
>3. 2-categories of sentences, proofs and proof reductions.
Not quite. One can and normally does talk about proof reductions using simply categories and morphisms.
You do not need to introduce 2-cells for that.

>The work you point out is now on level 3.
Again not quite. Some of the work I mentioned for *classical* logic is actually at level 2.

Also, of course, there's plenty of work (to be done) on generalized categorical proof-theory of 
*non-classical* logic and relating that to the model-theoretical work on institutions. This is the work I was proposing to do, originally.

>This seems to be related to polycategories, which, however, only use finite sequences of sentences.
Yes, indeed in the kind of proof theory I like, rules mostly have a finite number of hypotheses/assumptions. Girard says somewhere that the infinite is always a potential one, which I think is quite nice.

>And the next question is of course where this general scheme also fits for other logics, like modal logics.
Well, some constructive modal logics fit in already. Others (non-constructive ones) will, once you do your classical logic the way *you* think it seems best. The modularity there is another one of the success criteria, right?

Cheers,
Valeria
-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till Mossakowski
Sent: Wednesday, August 10, 2005 2:32 AM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

many thanks for the pointers to the literature. Actually, my last intensive study of categorical proof theory dates back to the late eighties...
Like Joseph I find it a bit discouraging that there are so many
(unrelated?) proof.theoretic approaches. But maybe institutions can help, let's see...

Let me recall the programme: trying to identify the essential properties of a logic by only refering to the vocabulary of an abstract interface, like institutions.
On the model-theoretic side, Razvan Diaconescu has generalized much of classical model theory to this setting, including deep results like the fundamental ultraproduct theorem (see his forthcoming book "Institutional model theory").
This includes characterizations of abstract logical connecvtives and quantifiers (i.e. not refering to some particular syntax, but only to the vocabulary of institutions) through their "model-theoretic behaviour".

On the proof-theoretic side, this is much less explored.
It seems that there are three levels:

1. entailment systems (= kind of pre-orders) 2. categories of sentences and proofs 3. 2-categories of sentences, proofs and proof reductions.

At level 1., all proofs are identified. The achievment of the 1980's proof theory was to identify good categories of intuitionistic proofs at level 2, with categorical characterizations of connectives and quantifiers by their "proof-theoretic behaviour". The problem was that for classical logic, these categories collapse to thin categories = pre-orders, such that we are back at level 1. This might even not be a problem for defining connectives, but is just too abstract for proof theorists.
Also practically, a tool should be able to output a proof tree and not just the unique element of a singleton set...

The work you point out is now on level 3. However, not just 2-categories are defined, but additional categorical structure for the connectives is introduced. Thus, the connectives are no longer definable in terms of the abstract vocabulary (unless this is extended with this extra categorical structure, which seems awkward).
Maybe a way out is just to have level 3 for the proof reductions, but define the connectives at level 2 (which works well even with thin categories). Thus, all the levels would naturally coexist in parallel (noting that all the necessary information is contained in the highest level, because there are "quotienting" constructions going from a higher to a lower level).

Then, for example, the order-enriched categories of F?hrmann and Pym should naturally form an institution with proofs. Indeed, the 2-cells here are cut-elimination reductions, which fits nicely with what we had in mind .However, the two-categorical structure only captures the order-enrichment, while their categories are also linearly distributive (i.e. kind of bi-monoidal, where the two monoidal structures model conjunction and disjunction), plus object-wise monoids and co-monoids (modeling weakening and contraction). This richer structure is then ignored at the abstract level, where conjunction is recoverd as product in the category at level 2 (while the category at level 3 might mot even have products:
quotients do not need to reflect them).

Another point is that our proofs work on sets of sentences, rather than on sentences. This seems to be related to polycategories, which, however, only use finite sequences of sentences.

And the next question is of course where this general scheme also fits for other logics, like modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
>>This means that for classical logic, the approach of identifying 
>>different calculi (say, Gentzen or natural
>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
> Well, you're missing the gigantic amount of work put into this problem 
> since the early 90's with the theses of Griffin, Murthy, Stewart, 
> Harbelin, Urban, Parigot, etc. It is true that classical logic is 
> harder to model categorically than intuitionistic logic and it's true 
> that despite all this work it's not clear (to me at least) what are 
> the trade-offs between different kinds of classical Curry-Howard 
> systems and their categorical semantics. But I guess by now it's 
> clearly understood by most in the community that the old dictum that 
> "classical logic has no categorical semantics" is dead and buried. 
> Which kind of solution you prefer for the problem (Selinger's control 
> cats or fibrations or order-enriched models or even special kinds of  
> polycategories, etc...I'm sure I'm forgetting half the decent 
> solutions, for which I apologize) is up to you. As I said I don't know 
> of a list of trade-offs or cost/benefits analysis of the several 
> solutions. It's not my main area of researc
h  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
> 
> If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
> http://www.cs.bath.ac.uk/~pym/semclasspro.html
> 
> Hope this helps,
> Best,
> Valeria
> Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
> http://www.ps.uni-sb.de/~lutz/
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
> Sent: Tuesday, August 09, 2005 10:20 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Cc: FLIRTS
> Subject: Re: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
> thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
> 
> The paper "What is a logic?" (for those who do not have it: 
> http://www.tzi.de/~till/papers/nel05.pdf) takes the following
> perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
> 
> However, we say in the conclusion of the paper:
> 
>    There are some further proof-theoretic properties that we have not
>    treated, like (strong) normal forms for proofs (this would require
>    $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>    term reductions).  A related topic is cut elimination, which
>    would require an even finer structure on $\Sen(\Sigma)$,
>    with proof rules of particular format.
>    We hope this essay provides a good starting point for
>    such investigations.
> 
> This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
> 
> I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
> 
>    Note that arrows in proof categories are proofs up to equivalence.
>    And we impose certain conditions on this equivalence.
>    A simple example: if we infer A/\B from A/\B by conjunction
>    elimination and conjunction introduction, then this proof must
>    be equivalent to the proof infering A/\B directly from itself,
>    because conjunction is product, and <pi_1,pi_2>=id.
>    Basically, for propositional logic, our axioms of proof-theoretic
>    institutions say that the category of proofs is bicartesian closed
>    (ie cartesian closed + finite coproducts, including inital objects).
>    Lambek and Scott, "Introduction to categorical higher-order logic",
>    show on p.67, that in any bicartesion closed category, for an object
>    A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>    the initial object). From this it follows that any classical
>    bicartesion closed category (i.e. one with A is isomorphic to
>    (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>    to a thin category, and hence thin itself.
> 
> This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
> Or am I missing something?
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com wrote:
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real 
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions"
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>reading, I don't even know what a parchment is...
>>>But one small thought: your last paragraph about translating the
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>Now about your question: no I don't think I know of any reference for
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>try
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>>Cheers,
>>>Valeria
> 
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till _______________________________________________
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From Valeria.dePaiva at parc.com  Wed Aug 10 18:45:16 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE6F@goldeneye.ad.parc.com>

Dear Joseph,
Thanks for the friendly message.
First, let me make it clear that, no, I do not have "a nicer way of including proofs into institutions", at the moment. I had proposed to Till that we could investigate this problem together, as I had become interested in the idea around 1999. I'm attaching some slides from a talk I gave at NASA Ames then when I was trying to sell them a work proposal along these lines. I thought this was a cool idea, which you probably agree since you've had the same idea some 15 years before me. But I only worked on that for a couple of weeks, preparing the talk, which is just a proposal for some work. Not the work itself.


>they idea of using sets of sentences as objects is new and useful, 
>or did we miss something there also?
Well, I don't think it is useful, as *when* you can do your categorical modelling properly, sets of sentences are modelled for free, either in the case where there's a connective that internalizes the comma (like a categorical tensor) or using the fibration mechanism. So no, I don't see the usefulness at the moment, it doesn't buy me anything new. Maybe you'd like to expand on that?

About:
>it seems a bit extreme to say that it doesn't really do any proof theory;
I'm afraid I don't think it excessive. I went back to the paper "What is a Logic?" to see if I was forgetting any basic interesting proof-theoretic notion, given that you've said
>it does allow defining many basic proof theoretic concepts in a nice abstract way,
But I don't see these many concepts. Would you like to discuss them one by one?

>I find this dialogue very useful and hope that we can continue it further - i look forward to learning more!
So do I, thanks for the discussion!

Best regards,

Valeria
-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph Goguen
Sent: Tuesday, August 09, 2005 8:31 PM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Cc: till@informatik.uni-bremen.de; de Paiva, Valeria <Valeria.dePaiva@parc.com>
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

Thanks so much for your remarks!  If you have a nicer way of including proofs into institutions, such that, for example, classical logic works well by something other than proof terms for sequents, that would be very welcome and interesting to us.  Ive heard about the work you mention in various theses but have never looked at it; your summary is a bit on the discouraging side, i must admit - too many answers, all likely to be rather complicated, and no clear tradeoffs.

About the origin of institutions with proofs, it goes back to at least 1980 in conversations between Rod Burstall and i; Rod didnt much like the idea then but i put it in our JACM paper anyway, where it appeared after a 7 year delay (submitted 1985, but earlier drafts exist, also with the proof  idea).

About the formulation in "What is a logic?" it seems a bit extreme to say that it doesnt really do any proof theory; although it is clear that it is far from doing everything that proof theorists find interesting, it does allow defining many basic proof theoretic concepts in a nice abstract way, especially those that connect with models.  Also, as far as i know, they idea of using sets of sentences as objects is new and useful, or did we miss something there also?


With all best regards,

    joseph

Valeria.dePaiva@parc.com wrote:

>Dear Till,
>  
>
>>This means that for classical logic, the approach of identifying 
>>different calculi (say, Gentzen or natural
>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
>>    
>>
>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>   (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>
>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>
>Hope this helps,
>Best,
>Valeria
>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>http://www.ps.uni-sb.de/~lutz/
>Dr Valeria de Paiva
>PARC
>3333 Coyote Hill Road
>Palo Alto, CA 94304
>USA
>
>-----Original Message-----
>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
>Sent: Tuesday, August 09, 2005 10:20 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Cc: FLIRTS
>Subject: Re: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>
>The paper "What is a logic?" (for those who do not have it: 
>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>
>However, we say in the conclusion of the paper:
>
>   There are some further proof-theoretic properties that we have not
>   treated, like (strong) normal forms for proofs (this would require
>   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>   term reductions).  A related topic is cut elimination, which
>   would require an even finer structure on $\Sen(\Sigma)$,
>   with proof rules of particular format.
>   We hope this essay provides a good starting point for
>   such investigations.
>
>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>
>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>
>   Note that arrows in proof categories are proofs up to equivalence.
>   And we impose certain conditions on this equivalence.
>   A simple example: if we infer A/\B from A/\B by conjunction
>   elimination and conjunction introduction, then this proof must
>   be equivalent to the proof infering A/\B directly from itself,
>   because conjunction is product, and <pi_1,pi_2>=id.
>   Basically, for propositional logic, our axioms of proof-theoretic
>   institutions say that the category of proofs is bicartesian closed
>   (ie cartesian closed + finite coproducts, including inital objects).
>   Lambek and Scott, "Introduction to categorical higher-order logic",
>   show on p.67, that in any bicartesion closed category, for an object
>   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>   the initial object). From this it follows that any classical
>   bicartesion closed category (i.e. one with A is isomorphic to
>   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>   to a thin category, and hence thin itself.
>
>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>Or am I missing something?
>
>Greetings,
>Till
>
>Valeria.dePaiva@parc.com wrote:
>  
>
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real 
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions"
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>    
>>
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>>      
>>>
>>reading, I don't even know what a parchment is...
>>    
>>
>>>But one small thought: your last paragraph about translating the
>>>      
>>>
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>    
>>
>>>Now about your question: no I don't think I know of any reference for
>>>      
>>>
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>    
>>
>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>try
>>>      
>>>
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>    
>>
>>>Cheers,
>>>Valeria
>>>      
>>>
>
>
>  
>


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From till at informatik.uni-bremen.de  Wed Aug 10 21:30:32 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
Message-ID: <42FA55D8.9080805@informatik.uni-bremen.de>

Dear Valeria,

 >>they idea of using sets of sentences as objects is new and useful,
 >>or did we miss something there also?
 > Well, I don't think it is useful, as *when* you can do your
 > categorical modelling properly, sets of sentences are modelled for
 > free, either in the case where there's a connective that internalizes
 > the comma (like a categorical tensor) or using the fibration
 > mechanism. So no, I don't see the usefulness at the moment, it doesn't
 > buy me anything new. Maybe you'd like to expand on that?

For properties like compactness, you need possibly infinite set of
sentences. And many logics do not have infinitary conjunction,
hence you cannot internalize. If then the logic happens to have
infinitary proof rules, I cannot see how to model this by fibrations.
But may be I am missing something again :-).

 >>it does allow defining many basic proof theoretic concepts in a nice 
abstract way,
 > But I don't see these many concepts. Would you like to discuss them 
one by one?

One concept is having not not-elimination, which separates classical
from intuitionistic logic.

> A logic, for me, does not exist without its derivations and proofs.
That's interesting. Strassburger in is paper "What is a logic
and what is a proof" starts with "a logic is a pre-order",
i.e. he omits proofs. Only later proofs come in. And I agree:
there should be a notion of logic with proofs, but a notion
of logic without proofs is useful as well.

>>It seems that there are three levels:
> 
>>1. entailment systems (= kind of pre-orders) 
>>2. categories of sentences and proofs 
>>3. 2-categories of sentences, proofs and proof reductions.
> Not quite. One can and normally does talk about proof reductions using simply categories and morphisms.
> You do not need to introduce 2-cells for that.
I do not understand.
I thought formulas = objects, proofs = 1-cells. So what are
proof reductions other than 2-cells? OK, they might just be
a pre-order on 1-cells. But this just amounts to having
thin Hom-categories.

>>This seems to be related to polycategories, which, however, only use finite sequences of sentences.
> Yes, indeed in the kind of proof theory I like, rules mostly have a finite number of hypotheses/assumptions. Girard says somewhere that the infinite is always a potential one, which I think is quite nice.
I definitely want to include infinite sets of sentences as well.
The whole notion of compactness as studied in traditional logic
only makes sense with infinite sets of sentences.
And ZFC, which is widely used in mathematics, is not finitely
axiomatizable. Neither is first-order Peano arithmetic.
Infinitary rules are less important, but it would be nice not
to exclude them, because they are used occasionally.
Moreover, the framework of institutions with proofs should be
philosophically neutral, and should admit the study of logics
that might be rejected by some people, but not by all.

>  The modularity there is another one of the success criteria, right?
Yes, certainly.

Greetings,
Till

-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From jose at fiadeiro.org  Thu Aug 11 00:50:52 2005
From: jose at fiadeiro.org (=?ISO-8859-1?Q?Jos=E9_Luiz_Fiadeiro?=)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
Message-ID: <58C13D61-A4E4-4FA1-8444-6AFA8D7AE272@fiadeiro.org>

My two cents...

On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com>  
<Valeria.dePaiva@parc.com> wrote:

>> Let me recall the programme: trying to identify the essential  
>> properties of a logic by only refering to the
>> vocabulary of an abstract interface, like institutions.
>>
> This is possibly the way you see the programme. For me "to identify  
> the essential properties of a logic" means identifying the  
> essential properties of the *derivations* in this logic. A logic,  
> for me, does not exist without its derivations and proofs.


In my opinion, it does not make sense to refer to THE essential  
properties of a logic: it all depends on what you want to do with the  
logic.  Institutions do capture essential properties of a logic in so  
far as the use of a logic for "algebraic specification" is  
concerned.  But one cannot reduce the notion of logic to this  
particular usage.  In fact, even within a generalised notion of  
"algebraic specification", we have found that the structural  
properties of institutions that restrict morphisms to property- 
preserving relationships prevents us from capturing composition as it  
arises in non-deterministic systems.  The so-called "satisfaction  
condition" has also proved to be too restrictive for formalisms that  
work just on subclasses of models that satisfy some closure conditions.

On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
OBJECTS, the structure of logics is in the derivations/proofs.   
However, the "FLIRTS programme" is, as far as I understand, directed  
to Logic as applied to Specification Theory.  In this respect, it is  
not the internal structure of logics that is of interest, but that of  
their applications, namely the structure of specifications.  In other  
words, the ambition of FLIRTS should not be to do Mathematics (or  
Logic with 'L') but Specification Theory (i.e. given usages of logics).

Having said this, I find that it is definitely worth exploring all  
the different constructions that have been mentioned in previous  
messages from the point of view of the applications to Specification  
Theory.  However, I would refrain from trying to go beyond that.


Regards

Jos?



JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Science
University of Leicester
Leicester LE1 7RH
United Kingdom
Tel: +44 116 252 3907
Fax: +44 116 252 3915
http://www.cs.le.ac.uk/




From Valeria.dePaiva at parc.com  Thu Aug 11 01:00:37 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>

 
Hi,
Of course I agree with Jose that
>it does not make sense to refer to THE essential properties of a logic: 
>it all depends on what you want to do with the logic.

But maybe I missed one referent in this discussion: I thought that Till was referring to the programme
Of adding proofs to institutions in the restricted setting of a proposed research paper that 
we were trying to write. I'm afraid I didn't even know about the FLIRTS programme.
So I read it all in the very restricted setting above.

>In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') 
>but Specification Theory (i.e. given usages of logics).
I know very little about Specification Theory and wouldn't want anyone reading this to think that
I was telling them how to do their job.

Thanks, Jose'.
Best,
Valeria

-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Jos? Luiz Fiadeiro
Sent: Wednesday, August 10, 2005 3:51 PM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

My two cents...

On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com> <Valeria.dePaiva@parc.com> wrote:

>> Let me recall the programme: trying to identify the essential 
>> properties of a logic by only refering to the vocabulary of an 
>> abstract interface, like institutions.
>>
> This is possibly the way you see the programme. For me "to identify 
> the essential properties of a logic" means identifying the essential 
> properties of the *derivations* in this logic. A logic, for me, does 
> not exist without its derivations and proofs.


In my opinion, it does not make sense to refer to THE essential properties of a logic: it all depends on what you want to do with the logic.  Institutions do capture essential properties of a logic in so far as the use of a logic for "algebraic specification" is concerned.  But one cannot reduce the notion of logic to this particular usage.  In fact, even within a generalised notion of "algebraic specification", we have found that the structural properties of institutions that restrict morphisms to property- preserving relationships prevents us from capturing composition as it arises in non-deterministic systems.  The so-called "satisfaction condition" has also proved to be too restrictive for formalisms that work just on subclasses of models that satisfy some closure conditions.

On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
OBJECTS, the structure of logics is in the derivations/proofs.   
However, the "FLIRTS programme" is, as far as I understand, directed to Logic as applied to Specification Theory.  In this respect, it is not the internal structure of logics that is of interest, but that of their applications, namely the structure of specifications.  In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') but Specification Theory (i.e. given usages of logics).

Having said this, I find that it is definitely worth exploring all the different constructions that have been mentioned in previous messages from the point of view of the applications to Specification Theory.  However, I would refrain from trying to go beyond that.


Regards

Jos?



JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Science
University of Leicester
Leicester LE1 7RH
United Kingdom
Tel: +44 116 252 3907
Fax: +44 116 252 3915
http://www.cs.le.ac.uk/



_______________________________________________
Flirts mailing list
Flirts@mail.informatik.uni-bremen.de
http://www.informatik.uni-bremen.de/mailman/listinfo/flirts


From jose at fiadeiro.org  Thu Aug 11 01:09:57 2005
From: jose at fiadeiro.org (=?ISO-8859-1?Q?Jos=E9_Luiz_Fiadeiro?=)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
Message-ID: <39A7D44B-C1BC-4A62-A185-2FA98F7855C4@fiadeiro.org>


On 11 Aug 2005, at 00:00, <Valeria.dePaiva@parc.com>  
<Valeria.dePaiva@parc.com> wrote:

> But maybe I missed one referent in this discussion: I thought that  
> Till was referring to the programme
> Of adding proofs to institutions in the restricted setting of a  
> proposed research paper that
> we were trying to write. I'm afraid I didn't even know about the  
> FLIRTS programme.
> So I read it all in the very restricted setting above.

I'm afraid that I may be the one who is missing the referent...  One  
the one hand, I don't know about the research paper that you mention;  
on the other hand, my reference to the "FLIRTS programme" is rather  
loose.

I should then rephrase my remarks to mean that I can only understand  
a programme of "adding proofs to institutions" within Specification  
Theory, not within Logic.


Apologies again

Jos?




JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Science
University of Leicester
Leicester LE1 7RH
United Kingdom
Tel: +44 116 252 3907
Fax: +44 116 252 3915
http://www.cs.le.ac.uk/




From Valeria.dePaiva at parc.com  Thu Aug 11 01:49:25 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE79@goldeneye.ad.parc.com>

Dear Till,

My replies will be somewhat slower, as time is in short supply here.

>For properties like compactness, you need possibly infinite set of
sentences.
All proofs of compactness that I've remembered seen were semantical
proofs,
piggybacking on satisfiability. Hence I have not worried about
compactness much; this goes
back to the notion that the infinities that I'm worried about are all
potential infinities.

And I believe you're right, fibrations won't help if your logic happens
to have infinitary conjunctions,
but the logics I'm interested in, do not have infinitary connectives. I
think I vaguely remember Michael Makkai  being interested in infinitary
conjunctions, so one would have to check what he has to say about it.

>One concept is having not not-elimination, which separates classical
from intuitionistic logic.
But apart from the fact that classical logic has it, while
intuitionistic logic does not, which is really just the definition of
the difference between the logics, I cannot see anything that
institutions with proofs can say about the "concept of
not-not-elimination".

>> A logic, for me, does not exist without its derivations and proofs.
Please don't take the comment without its context: I'm very fond of
algebraic logic and I'm as  keen as any one else is of modelling logics
and forgetting their proofs. But they were there to begin with. I just
don't have to pay attention to them all the time. Modelling is like
that, we allow ourselves to forget stuff when convenient.
The situation would be totally different, if the proofs never existed,
if all that existed was a "soup of symbols, ones unrelated to others".

>But this just amounts to having thin Hom-categories.
Thin Hom-categories is just a different name for  traditional old
categories, so we're in agreement, I take it.

>the framework of institutions with proofs should be philosophically
neutral, 
>and should admit the study of logics that might be rejected by some
people, but not by all.
While I believe that philosophical neutrality is an appealing feature, I
can't see exactly what you're driving at with the second condition. 

Some logics are better-behaved, some are less well-behaved, as far as
proof theory is concerned. 
If one can set up a framework that unifies the "proof-theoretically
well-behaved logics" 
(whatever collection this may end up defining) both model theoretically
and proof-theoretically, then progress would have been made, I reckon.
But of course, the proof is in the pudding, this is all hypothetical
right now.

Best,
Valeria

-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de
[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till
Mossakowski
Sent: Wednesday, August 10, 2005 12:31 PM
To: Formalism, Logic, Institution - Relating, Translating and
Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

 >>they idea of using sets of sentences as objects is new and useful,
>>or did we miss something there also?
 > Well, I don't think it is useful, as *when* you can do your  >
categorical modelling properly, sets of sentences are modelled for  >
free, either in the case where there's a connective that internalizes  >
the comma (like a categorical tensor) or using the fibration  >
mechanism. So no, I don't see the usefulness at the moment, it doesn't
> buy me anything new. Maybe you'd like to expand on that?

For properties like compactness, you need possibly infinite set of
sentences. And many logics do not have infinitary conjunction, hence you
cannot internalize. If then the logic happens to have infinitary proof
rules, I cannot see how to model this by fibrations.
But may be I am missing something again :-).

 >>it does allow defining many basic proof theoretic concepts in a nice
abstract way,  > But I don't see these many concepts. Would you like to
discuss them one by one?

One concept is having not not-elimination, which separates classical
from intuitionistic logic.

> A logic, for me, does not exist without its derivations and proofs.
That's interesting. Strassburger in is paper "What is a logic and what
is a proof" starts with "a logic is a pre-order", i.e. he omits proofs.
Only later proofs come in. And I agree:
there should be a notion of logic with proofs, but a notion of logic
without proofs is useful as well.

>>It seems that there are three levels:
> 
>>1. entailment systems (= kind of pre-orders) 2. categories of 
>>sentences and proofs 3. 2-categories of sentences, proofs and proof 
>>reductions.
> Not quite. One can and normally does talk about proof reductions using
simply categories and morphisms.
> You do not need to introduce 2-cells for that.
I do not understand.
I thought formulas = objects, proofs = 1-cells. So what are proof
reductions other than 2-cells? OK, they might just be a pre-order on
1-cells. But this just amounts to having thin Hom-categories.

>>This seems to be related to polycategories, which, however, only use
finite sequences of sentences.
> Yes, indeed in the kind of proof theory I like, rules mostly have a
finite number of hypotheses/assumptions. Girard says somewhere that the
infinite is always a potential one, which I think is quite nice.
I definitely want to include infinite sets of sentences as well.
The whole notion of compactness as studied in traditional logic only
makes sense with infinite sets of sentences.
And ZFC, which is widely used in mathematics, is not finitely
axiomatizable. Neither is first-order Peano arithmetic.
Infinitary rules are less important, but it would be nice not to exclude
them, because they are used occasionally.
Moreover, the framework of institutions with proofs should be
philosophically neutral, and should admit the study of logics that might
be rejected by some people, but not by all.

>  The modularity there is another one of the success criteria, right?
Yes, certainly.

Greetings,
Till

-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
_______________________________________________
Flirts mailing list
Flirts@mail.informatik.uni-bremen.de
http://www.informatik.uni-bremen.de/mailman/listinfo/flirts


From goguen at cs.ucsd.edu  Thu Aug 11 03:25:01 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE6F@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE6F@goldeneye.ad.parc.com>
Message-ID: <42FAA8ED.90104@cs.ucsd.edu>

Dear Valeria and others,

This is fun, we are getting a lot of different points of view.

Valeria.dePaiva@parc.com wrote:

>Dear Joseph,
>Thanks for the friendly message.
>First, let me make it clear that, no, I do not have "a nicer way of including proofs into institutions", at the moment. I had proposed to Till that we could investigate this problem together, as I had become interested in the idea around 1999. I'm attaching some slides from a talk I gave at NASA Ames then when I was trying to sell them a work proposal along these lines. I thought this was a cool idea, which you probably agree since you've had the same idea some 15 years before me. But I only worked on that for a couple of weeks, preparing the talk, which is just a proposal for some work. Not the work itself.
>
Thanks for the slides.  For some reason page 10 wont print
here, but i get the idea.  It's amusing to see institution
theory referred to as "the mainstream" since that is not a
majority point of view.

>>they idea of using sets of sentences as objects is new and useful, 
>>or did we miss something there also?
>>    
>>
>Well, I don't think it is useful, as *when* you can do your categorical modelling properly, sets of sentences are modelled for free, either in the case where there's a connective that internalizes the comma (like a categorical tensor) or using the fibration mechanism. So no, I don't see the usefulness at the moment, it doesn't buy me anything new. Maybe you'd like to expand on that?
>  
>
I think Till has done a good job on this one.  It's nice to be able to deal
with infinitary logics, compactness, etc. in such a clean way.  By the way,
infinitary logics are sometimes needed for program invariants, despite the
beliefs of people like Hoare and Dijkstra.

>About:
>  
>
>>it seems a bit extreme to say that it doesn't really do any proof theory;
>>    
>>
>I'm afraid I don't think it excessive. I went back to the paper "What is a Logic?" to see if I was forgetting any basic interesting proof-theoretic notion, given that you've said
>  
>
>>it does allow defining many basic proof theoretic concepts in a nice abstract way,
>>    
>>
>But I don't see these many concepts. Would you like to discuss them one by one?
>  
>
Till has mentioned compactness, which i guess is pretty basic.  There has
also been a lot of work on Craig interpolation.  Diaconescu has done
some pretty amazing things, including Beth definability and Robinson
consistency,

>  
>
>>I find this dialogue very useful and hope that we can continue it further - i look forward to learning more!
>>    
>>
>So do I, thanks for the discussion!
>  
>
A perhaps important general point is that institution theory is not trying
to replicate what logicians have done, but to get the same (or better)
results in a much more general setting, so of course, some things are
going to be different.  The bottom line will be what can be done with
the institutional results.

>Best regards,
>
>Valeria
>-----Original Message-----
>From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph Goguen
>Sent: Tuesday, August 09, 2005 8:31 PM
>To: Formalism, Logic, Institution - Relating, Translating and Structuring
>Cc: till@informatik.uni-bremen.de; de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>Thanks so much for your remarks!  If you have a nicer way of including proofs into institutions, such that, for example, classical logic works well by something other than proof terms for sequents, that would be very welcome and interesting to us.  Ive heard about the work you mention in various theses but have never looked at it; your summary is a bit on the discouraging side, i must admit - too many answers, all likely to be rather complicated, and no clear tradeoffs.
>
>About the origin of institutions with proofs, it goes back to at least 1980 in conversations between Rod Burstall and i; Rod didnt much like the idea then but i put it in our JACM paper anyway, where it appeared after a 7 year delay (submitted 1985, but earlier drafts exist, also with the proof  idea).
>
>About the formulation in "What is a logic?" it seems a bit extreme to say that it doesnt really do any proof theory; although it is clear that it is far from doing everything that proof theorists find interesting, it does allow defining many basic proof theoretic concepts in a nice abstract way, especially those that connect with models.  Also, as far as i know, they idea of using sets of sentences as objects is new and useful, or did we miss something there also?
>
>
>With all best regards,
>
>    joseph
>
>Valeria.dePaiva@parc.com wrote:
>
>  
>
>>Dear Till,
>> 
>>
>>    
>>
>>>This means that for classical logic, the approach of identifying 
>>>different calculi (say, Gentzen or natural
>>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>>Or am I missing something?
>>>   
>>>
>>>      
>>>
>>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>>  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>>
>>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>>
>>Hope this helps,
>>Best,
>>Valeria
>>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>>http://www.ps.uni-sb.de/~lutz/
>>Dr Valeria de Paiva
>>PARC
>>3333 Coyote Hill Road
>>Palo Alto, CA 94304
>>USA
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
>>Sent: Tuesday, August 09, 2005 10:20 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Cc: FLIRTS
>>Subject: Re: Curry-Howard isomorphism for Institutions
>>
>>Dear Valeria,
>>
>>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>>
>>The paper "What is a logic?" (for those who do not have it: 
>>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>>
>>However, we say in the conclusion of the paper:
>>
>>  There are some further proof-theoretic properties that we have not
>>  treated, like (strong) normal forms for proofs (this would require
>>  $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>>  term reductions).  A related topic is cut elimination, which
>>  would require an even finer structure on $\Sen(\Sigma)$,
>>  with proof rules of particular format.
>>  We hope this essay provides a good starting point for
>>  such investigations.
>>
>>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>>
>>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>>
>>  Note that arrows in proof categories are proofs up to equivalence.
>>  And we impose certain conditions on this equivalence.
>>  A simple example: if we infer A/\B from A/\B by conjunction
>>  elimination and conjunction introduction, then this proof must
>>  be equivalent to the proof infering A/\B directly from itself,
>>  because conjunction is product, and <pi_1,pi_2>=id.
>>  Basically, for propositional logic, our axioms of proof-theoretic
>>  institutions say that the category of proofs is bicartesian closed
>>  (ie cartesian closed + finite coproducts, including inital objects).
>>  Lambek and Scott, "Introduction to categorical higher-order logic",
>>  show on p.67, that in any bicartesion closed category, for an object
>>  A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>>  the initial object). From this it follows that any classical
>>  bicartesion closed category (i.e. one with A is isomorphic to
>>  (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>>  to a thin category, and hence thin itself.
>>
>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com wrote:
>> 
>>
>>    
>>
>>>Dear Till,
>>>
>>>I noticed that you actually went ahead and did some  of the work we 
>>>discussed a long time ago (see below) on adding "categorical logic" to 
>>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>>Montreux, where Diaconescu talked about it.
>>>
>>>While I did feel a bit miffed that my original suggestion of the 
>>>problem wasn't mentioned at all, my problem with the paper is not 
>>>that. My problem is that your approach seems to be the proverbial 
>>>"throwing the baby away with the bathwater". The point of putting real 
>>>proofs (as opposed to entailment relations) into institutions was to 
>>>try to use the proofs-as-lambda-term-representations to do some real 
>>>work for us, ie to connect to the paradigm of extracting programs from 
>>>proofs, or to help with abstract analysis or to extend type systems in 
>>>a principled and logical way, etc. i.e. any of the usual applications 
>>>of categorical proof theory would do here.
>>>
>>>You say in page 2 of your joint paper that your new definition of 
>>>(proof
>>>theoretic) institution "fully supports proof theory", but the notion 
>>>of proof theoretic institution (or of equivalence of institutions) 
>>>introduced in the paper has nothing much to do with proof theory as 
>>>people normally know it. What you call "proof theoretic institutions"
>>>do not overcome the suggested limitation of "categorical logic", 
>>>because proof theoretic institutions do not  model the significant 
>>>aspect of proofs, which is their reduction behavior.
>>>
>>>The point of the Curry_Howard isomorphism is not that you can model 
>>>propositions as objects in a category and proofs as equivalence 
>>>classes of morphisms: the point is that the behaviour of proofs is 
>>>preserved under this modelling. This is why some people think that the 
>>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>>The  crucial point is that if proof pi reduces to pi' via 
>>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>>in the target category (either by equality or reduction). Nothing like 
>>>that happens in the proof-theoretic institutions, which is why they 
>>>are only proof-theoretical in name.
>>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>>you presumably realize, as  it is not even spelled out in the 
>>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>>guess my main complaint is that the paper does not define a 
>>>"proof-theoretical institution" in the sense of an institution that 
>>>preserves proofs, but simply as an institution that preserves 
>>>entailment. But I guess this is all right, people will have different 
>>>perspectives on what is important, mathematically speaking.
>>>
>>>Best regards,
>>>Valeria
>>>
>>>
>>>
>>>-----Original Message-----
>>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>>Sent: Friday, January 31, 2003 12:14 AM
>>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>>Subject: Re: CHI for Institutions
>>>
>>>Dear Valeria,
>>>
>>>agreed, let's devide the work as you suggested.
>>>
>>>For a definition of (some variant of) parchment you might look at p 5ff.
>>>of "Combingn and representing logical systems using model-theoretic 
>>>parchments", available at my publications page.
>>>
>>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>>calculus.
>>>
>>>Concerning the more complex Curry-Howard isos:
>>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>>on that, before going on with trying to look at Curry-Howard in the 
>>>institutional framework
>>>
>>>Greetings,
>>>Till
>>>
>>>Valeria.dePaiva@parc.com schrieb:
>>>   
>>>
>>>      
>>>
>>>>Dear Till,
>>>>Thanks for the very interesting message. Now I have to do some
>>>>     
>>>>
>>>>        
>>>>
>>>reading, I don't even know what a parchment is...
>>>   
>>>
>>>      
>>>
>>>>But one small thought: your last paragraph about translating the
>>>>     
>>>>
>>>>        
>>>>
>>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>>and seems a more concrete way of pushing forward towards my goal, 
>>>which is different though. My goal is really enriching the whole 
>>>framework of institutions so that it can cope with proofs (and when I 
>>>say proofs I don't mean in a single proof calculus: I usually want a 
>>>logic to be given in different several proof calculi all proved 
>>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>>Natural deduction and you know how to translate proofs from one 
>>>calculi to the others). So another way of pushing forward would be to 
>>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>   
>>>
>>>      
>>>
>>>>Now about your question: no I don't think I know of any reference for
>>>>     
>>>>
>>>>        
>>>>
>>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>>NuPRl are computer systems and first of all we would need papers 
>>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>>these might exist. I just haven't had the time or disposition to look 
>>>for them. This would be a different research project altogether it seems to me.
>>>   
>>>
>>>      
>>>
>>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>>try
>>>>     
>>>>
>>>>        
>>>>
>>>to come back with questions/suggestions. Maybe you could try to add 
>>>some details to the suggestion of looking at Curry-Howard in the 
>>>institutional framework?
>>>   
>>>
>>>      
>>>
>>>>Cheers,
>>>>Valeria
>>>>     
>>>>
>>>>        
>>>>
>> 
>>
>>    
>>
>
>
>_______________________________________________
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>
>------------------------------------------------------------------------
>
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>



From goguen at cs.ucsd.edu  Thu Aug 11 03:46:27 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <58C13D61-A4E4-4FA1-8444-6AFA8D7AE272@fiadeiro.org>
References: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
	<58C13D61-A4E4-4FA1-8444-6AFA8D7AE272@fiadeiro.org>
Message-ID: <42FAADF3.40902@cs.ucsd.edu>

Dear friends,

A few almost random reactions to recent postings.

Jos? Luiz Fiadeiro wrote:

> My two cents...
>
> On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com>  
> <Valeria.dePaiva@parc.com> wrote:
>
>>> Let me recall the programme: trying to identify the essential  
>>> properties of a logic by only refering to the
>>> vocabulary of an abstract interface, like institutions.
>>>
>> This is possibly the way you see the programme. For me "to identify  
>> the essential properties of a logic" means identifying the  essential 
>> properties of the *derivations* in this logic. A logic,  for me, does 
>> not exist without its derivations and proofs.
>
>
>
> In my opinion, it does not make sense to refer to THE essential  
> properties of a logic: it all depends on what you want to do with the  
> logic.  Institutions do capture essential properties of a logic in so  
> far as the use of a logic for "algebraic specification" is  
> concerned.  But one cannot reduce the notion of logic to this  
> particular usage.  In fact, even within a generalised notion of  
> "algebraic specification", we have found that the structural  
> properties of institutions that restrict morphisms to property- 
> preserving relationships prevents us from capturing composition as it  
> arises in non-deterministic systems.  The so-called "satisfaction  
> condition" has also proved to be too restrictive for formalisms that  
> work just on subclasses of models that satisfy some closure conditions.
>
Yes, one should always be careful about "essences" - they dont seem to 
exist in
the realm that humans inhabit, we always have to think about what we are 
going
to do with something, or at least, that's one version of CS Peirce's 
pragmatism.

About the satisfaction condition: my experience has been that when it 
fails,
there is usually a better to formulate the situation so that it 
succeeds.  For
example, hidden algebra is an institution that can handle nondeterminism and
it does so (in part) by restricting to a subclass of models (those where the
visible part is a fixed given algebra); Ehrig claimed that it wasnt an 
institution
but he had the wrong morphisms.

> On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
> OBJECTS, the structure of logics is in the derivations/proofs.   
> However, the "FLIRTS programme" is, as far as I understand, directed  
> to Logic as applied to Specification Theory.  In this respect, it is  
> not the internal structure of logics that is of interest, but that of  
> their applications, namely the structure of specifications.  In other  
> words, the ambition of FLIRTS should not be to do Mathematics (or  
> Logic with 'L') but Specification Theory (i.e. given usages of logics).

There are many ways to abstract some structures out of logics, including
Lindenbaum algebras, satisfaction relations (advocated by the eminent 
logician
Jon Barwise), proof nets, ...  I dont see why they cannot coexist.

Also, im not sure that FLIRTS should be given such a narrow 
interpretation; it
seems to me it should be up to the participants to decide what they would
like to discuss, and recent work on institutions aimed at logic seems quite
appropriate to me.

>
> Having said this, I find that it is definitely worth exploring all  
> the different constructions that have been mentioned in previous  
> messages from the point of view of the applications to Specification  
> Theory.  However, I would refrain from trying to go beyond that.

It would be a pity if group theory had been restricted to applications 
within
algebra (such as solvability of polynomials), given its wonderful 
applications
in physics.  In fact, Einstein credits Emmy Noether with a key insight 
for his
theory of relativity (that invariants imply conservation principles, the 
socalled
Noether theorem), so we might well not have relativity theory, or quantum
theory, if the original applications were the only ones allowed. 

I dont see why we cant apply institutions wherever we like.  The paper
"What is a logic?" is by no means aimed at specification theory, but was
submitted to a contest asking for the best answer to the question in the
title (rumour says the paper came in second, and might have been first
if the political climate at the conference had been different).

Even worse, i am guilty of applying institutions to cognitive science,
multimedia art, philosophy, user interface design, and programming, none
of which can be called specification theory in any normal sense.

>
>
> Regards
>
> Jos?
>
>
>
> JOSE LUIZ FIADEIRO
> Professor of Software Science and Engineering
>
> http://www.fiadeiro.org/jose
> Mob: +44 779 124 7816
> Skype: jfiadeiro
>
> Department of Computer Science
> University of Leicester
> Leicester LE1 7RH
> United Kingdom
> Tel: +44 116 252 3907
> Fax: +44 116 252 3915
> http://www.cs.le.ac.uk/
>
>
>
> _______________________________________________
> Flirts mailing list
> Flirts@mail.informatik.uni-bremen.de
> http://www.informatik.uni-bremen.de/mailman/listinfo/flirts


From goguen at cs.ucsd.edu  Thu Aug 11 04:06:50 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
Message-ID: <42FAB2BA.7020005@cs.ucsd.edu>

There seem to be a lot of different programmes going on all
at once, which seems very good to me.  Categorical logic,
for example, has a distinguished history already and many
brilliant practitioners who have done brilliant work, which
has established a fairly clear tradition and direction.  Im
not sure that institution theory can claim as much deep
research, but it can claim some serious applications,
and it does also have a tradition and direction of its
own, which is different from that of categorical logic.

Moreover, even within the institution of institutions
there are different research agendas.  Actually, i
have several different ones just myself!  One that
i forgot to mention in the previous email is semantics
of database systems and ontologies.

Work at Microsoft on data warehousing, for
example (Bernstein & Alagic), and at UCSD on
a system to help ecologists integrate and analyze
their data, use institutions.  If you havent seen it,

    http://research.microsoft.com/db/ModelMgt/

is amusing; it mentions categories not institutions
but institutions are the main feature of ref [1] there.

You could also look at

    http://www.cs.ucsd.edu/~goguen/projs/data.html

especially the first four papers in the biblio there.

Adding proofs to institutions and seeing what that
can be made to do is another programme, but i
think requiring it to conform to categorical logic
or specification theory is not a necessity for
that project, though the many good results in
these areas are certainly highly relevant to it.

In particular, i would like to suggest that the
three levels suggested by Till are a nice and
very institutional way of going about building
an abstraction hierarchy for proofs.  We should
ask if it works, not if it agrees with some other
body of theory, and as far as i can see, it works
rather well.

With very best wishes to all,

   joseph

Valeria.dePaiva@parc.com wrote:

> 
>Hi,
>Of course I agree with Jose that
>  
>
>>it does not make sense to refer to THE essential properties of a logic: 
>>it all depends on what you want to do with the logic.
>>    
>>
>
>But maybe I missed one referent in this discussion: I thought that Till was referring to the programme
>Of adding proofs to institutions in the restricted setting of a proposed research paper that 
>we were trying to write. I'm afraid I didn't even know about the FLIRTS programme.
>So I read it all in the very restricted setting above.
>
>  
>
>>In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') 
>>but Specification Theory (i.e. given usages of logics).
>>    
>>
>I know very little about Specification Theory and wouldn't want anyone reading this to think that
>I was telling them how to do their job.
>
>Thanks, Jose'.
>Best,
>Valeria
>
>-----Original Message-----
>From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Jos? Luiz Fiadeiro
>Sent: Wednesday, August 10, 2005 3:51 PM
>To: Formalism, Logic, Institution - Relating, Translating and Structuring
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>My two cents...
>
>On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com> <Valeria.dePaiva@parc.com> wrote:
>
>  
>
>>>Let me recall the programme: trying to identify the essential 
>>>properties of a logic by only refering to the vocabulary of an 
>>>abstract interface, like institutions.
>>>
>>>      
>>>
>>This is possibly the way you see the programme. For me "to identify 
>>the essential properties of a logic" means identifying the essential 
>>properties of the *derivations* in this logic. A logic, for me, does 
>>not exist without its derivations and proofs.
>>    
>>
>
>
>In my opinion, it does not make sense to refer to THE essential properties of a logic: it all depends on what you want to do with the logic.  Institutions do capture essential properties of a logic in so far as the use of a logic for "algebraic specification" is concerned.  But one cannot reduce the notion of logic to this particular usage.  In fact, even within a generalised notion of "algebraic specification", we have found that the structural properties of institutions that restrict morphisms to property- preserving relationships prevents us from capturing composition as it arises in non-deterministic systems.  The so-called "satisfaction condition" has also proved to be too restrictive for formalisms that work just on subclasses of models that satisfy some closure conditions.
>
>On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
>OBJECTS, the structure of logics is in the derivations/proofs.   
>However, the "FLIRTS programme" is, as far as I understand, directed to Logic as applied to Specification Theory.  In this respect, it is not the internal structure of logics that is of interest, but that of their applications, namely the structure of specifications.  In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') but Specification Theory (i.e. given usages of logics).
>
>Having said this, I find that it is definitely worth exploring all the different constructions that have been mentioned in previous messages from the point of view of the applications to Specification Theory.  However, I would refrain from trying to go beyond that.
>
>
>Regards
>
>Jos?
>
>
>
>JOSE LUIZ FIADEIRO
>Professor of Software Science and Engineering
>
>http://www.fiadeiro.org/jose
>Mob: +44 779 124 7816
>Skype: jfiadeiro
>
>Department of Computer Science
>University of Leicester
>Leicester LE1 7RH
>United Kingdom
>Tel: +44 116 252 3907
>Fax: +44 116 252 3915
>http://www.cs.le.ac.uk/
>
>
>
>_______________________________________________
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>
>_______________________________________________
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>  
>

From jose at fiadeiro.org  Thu Aug 11 10:43:21 2005
From: jose at fiadeiro.org (=?ISO-8859-1?Q?Jos=E9_Luiz_Fiadeiro?=)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <42FAB2BA.7020005@cs.ucsd.edu>
References: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>
	<42FAB2BA.7020005@cs.ucsd.edu>
Message-ID: <CBD4F07F-D706-4C17-9AEF-3A262B967173@fiadeiro.org>

Joseph (and everyone)

I agree with all you say.  Actually, I meant to use the term  
Specification Theory in the widest possible senses to address what is  
perhaps best captured as "modelling", i.e. not just restricted to  
logical formalisms and, indeed, not just to Computer Science.

I guess that what I was trying to say is that it seems more useful to  
look for ways of unifying a number of concepts and techniques that  
people use for "modelling" in a number of domains, than to capture  
the essence of Logic.  At least this is what excites me...

Thanks

Jos?



JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Science
University of Leicester
Leicester LE1 7RH
United Kingdom
Tel: +44 116 252 3907
Fax: +44 116 252 3915
http://www.cs.le.ac.uk/





From goguen at cs.ucsd.edu  Thu Aug 11 17:01:52 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE7E@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE7E@goldeneye.ad.parc.com>
Message-ID: <42FB6860.1040505@cs.ucsd.edu>

hi again Valeria!

Valeria.dePaiva@parc.com wrote:

> 
>Dear Joseph,
>  
>
>>For some reason page 10 wont print here, but i get the idea. 
>>    
>>
>I am sorry, I think it's eps for pictures that's causing the problem, will see if I can get a pdf version.
>  
>
ps2pdf does the job in unix but i dont really need it, that page is on 
colimits
which i guess i understand well enough.

>>It's amusing to see institution theory referred to as "the mainstream" 
>>since that is not a majority point of view.
>>    
>>
>Indeed, but in the context of categories for specification theory in NASA, it is. Or perhaps it was. I haven't seen the nice guys there for a while...
>
>Craig interpolation I believe we can do totally proof-theoretically in many situations -- without infinite sets of sentences. Compactness, as I said, I have a problem  seeing it as proof theory, but maybe it's just lack of trying.
>  
>
i never said craig interpolation needs infinite sets of sentences, and i 
never said
you couldnt do it proof theoretically!  im just reponding to your 
request for proof
theory stuff that we can do in insitutions - do you remember that 
request?  and
please note that compactness is normally formulated as an assertion 
about proofs
even though it is normally proved model theoretically.

  - joseph

>Best,
>Valeria
>-----Original Message-----
>From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph Goguen
>Sent: Wednesday, August 10, 2005 6:25 PM
>To: Formalism, Logic, Institution - Relating, Translating and Structuring
>Cc: till@informatik.uni-bremen.de
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria and others,
>
>This is fun, we are getting a lot of different points of view.
>
>Valeria.dePaiva@parc.com wrote:
>
>  
>
>>Dear Joseph,
>>Thanks for the friendly message.
>>First, let me make it clear that, no, I do not have "a nicer way of including proofs into institutions", at the moment. I had proposed to Till that we could investigate this problem together, as I had become interested in the idea around 1999. I'm attaching some slides from a talk I gave at NASA Ames then when I was trying to sell them a work proposal along these lines. I thought this was a cool idea, which you probably agree since you've had the same idea some 15 years before me. But I only worked on that for a couple of weeks, preparing the talk, which is just a proposal for some work. Not the work itself.
>>
>>    
>>
>Thanks for the slides.  For some reason page 10 wont print here, but i get the idea.  It's amusing to see institution theory referred to as "the mainstream" since that is not a majority point of view.
>
>  
>
>>>they idea of using sets of sentences as objects is new and useful, or 
>>>did we miss something there also?
>>>   
>>>
>>>      
>>>
>>Well, I don't think it is useful, as *when* you can do your categorical modelling properly, sets of sentences are modelled for free, either in the case where there's a connective that internalizes the comma (like a categorical tensor) or using the fibration mechanism. So no, I don't see the usefulness at the moment, it doesn't buy me anything new. Maybe you'd like to expand on that?
>> 
>>
>>    
>>
>I think Till has done a good job on this one.  It's nice to be able to deal with infinitary logics, compactness, etc. in such a clean way.  By the way, infinitary logics are sometimes needed for program invariants, despite the beliefs of people like Hoare and Dijkstra.
>
>  
>
>>About:
>> 
>>
>>    
>>
>>>it seems a bit extreme to say that it doesn't really do any proof 
>>>theory;
>>>   
>>>
>>>      
>>>
>>I'm afraid I don't think it excessive. I went back to the paper "What 
>>is a Logic?" to see if I was forgetting any basic interesting 
>>proof-theoretic notion, given that you've said
>> 
>>
>>    
>>
>>>it does allow defining many basic proof theoretic concepts in a nice 
>>>abstract way,
>>>   
>>>
>>>      
>>>
>>But I don't see these many concepts. Would you like to discuss them one by one?
>> 
>>
>>    
>>
>Till has mentioned compactness, which i guess is pretty basic.  There has also been a lot of work on Craig interpolation.  Diaconescu has done some pretty amazing things, including Beth definability and Robinson consistency,
>
>  
>
>> 
>>
>>    
>>
>>>I find this dialogue very useful and hope that we can continue it further - i look forward to learning more!
>>>   
>>>
>>>      
>>>
>>So do I, thanks for the discussion!
>> 
>>
>>    
>>
>A perhaps important general point is that institution theory is not trying to replicate what logicians have done, but to get the same (or better) results in a much more general setting, so of course, some things are going to be different.  The bottom line will be what can be done with the institutional results.
>
>  
>
>>Best regards,
>>
>>Valeria
>>-----Original Message-----
>>From: flirts-bounces@informatik.uni-bremen.de 
>>[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph 
>>Goguen
>>Sent: Tuesday, August 09, 2005 8:31 PM
>>To: Formalism, Logic, Institution - Relating, Translating and 
>>Structuring
>>Cc: till@informatik.uni-bremen.de; de Paiva, Valeria 
>><Valeria.dePaiva@parc.com>
>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>
>>Dear Valeria,
>>
>>Thanks so much for your remarks!  If you have a nicer way of including proofs into institutions, such that, for example, classical logic works well by something other than proof terms for sequents, that would be very welcome and interesting to us.  Ive heard about the work you mention in various theses but have never looked at it; your summary is a bit on the discouraging side, i must admit - too many answers, all likely to be rather complicated, and no clear tradeoffs.
>>
>>About the origin of institutions with proofs, it goes back to at least 1980 in conversations between Rod Burstall and i; Rod didnt much like the idea then but i put it in our JACM paper anyway, where it appeared after a 7 year delay (submitted 1985, but earlier drafts exist, also with the proof  idea).
>>
>>About the formulation in "What is a logic?" it seems a bit extreme to say that it doesnt really do any proof theory; although it is clear that it is far from doing everything that proof theorists find interesting, it does allow defining many basic proof theoretic concepts in a nice abstract way, especially those that connect with models.  Also, as far as i know, they idea of using sets of sentences as objects is new and useful, or did we miss something there also?
>>
>>
>>With all best regards,
>>
>>   joseph
>>
>>Valeria.dePaiva@parc.com wrote:
>>
>> 
>>
>>    
>>
>>>Dear Till,
>>>
>>>
>>>   
>>>
>>>      
>>>
>>>>This means that for classical logic, the approach of identifying 
>>>>different calculi (say, Gentzen or natural
>>>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>>>Or am I missing something?
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>>> (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>>>
>>>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>>>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>>>
>>>Hope this helps,
>>>Best,
>>>Valeria
>>>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>>>http://www.ps.uni-sb.de/~lutz/
>>>Dr Valeria de Paiva
>>>PARC
>>>3333 Coyote Hill Road
>>>Palo Alto, CA 94304
>>>USA
>>>
>>>-----Original Message-----
>>>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
>>>Sent: Tuesday, August 09, 2005 10:20 AM
>>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>>Cc: FLIRTS
>>>Subject: Re: Curry-Howard isomorphism for Institutions
>>>
>>>Dear Valeria,
>>>
>>>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>>>
>>>The paper "What is a logic?" (for those who do not have it: 
>>>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>>>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>>>
>>>However, we say in the conclusion of the paper:
>>>
>>> There are some further proof-theoretic properties that we have not  
>>>treated, like (strong) normal forms for proofs (this would require  
>>>$Sen(\Sigma)$ to become 2-category of sentences, proof terms and 
>>>proof  term reductions).  A related topic is cut elimination, which  
>>>would require an even finer structure on $\Sen(\Sigma)$,  with proof 
>>>rules of particular format.
>>> We hope this essay provides a good starting point for  such 
>>>investigations.
>>>
>>>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>>>
>>>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>>>
>>> Note that arrows in proof categories are proofs up to equivalence.
>>> And we impose certain conditions on this equivalence.
>>> A simple example: if we infer A/\B from A/\B by conjunction  
>>>elimination and conjunction introduction, then this proof must  be 
>>>equivalent to the proof infering A/\B directly from itself,  because 
>>>conjunction is product, and <pi_1,pi_2>=id.
>>> Basically, for propositional logic, our axioms of proof-theoretic  
>>>institutions say that the category of proofs is bicartesian closed  
>>>(ie cartesian closed + finite coproducts, including inital objects).
>>> Lambek and Scott, "Introduction to categorical higher-order logic",  
>>>show on p.67, that in any bicartesion closed category, for an object  
>>>A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is  
>>>the initial object). From this it follows that any classical  
>>>bicartesion closed category (i.e. one with A is isomorphic to  
>>>(A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent  to a 
>>>thin category, and hence thin itself.
>>>
>>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>>Or am I missing something?
>>>
>>>Greetings,
>>>Till
>>>
>>>Valeria.dePaiva@parc.com wrote:
>>>
>>>
>>>   
>>>
>>>      
>>>
>>>>Dear Till,
>>>>
>>>>I noticed that you actually went ahead and did some  of the work we 
>>>>discussed a long time ago (see below) on adding "categorical logic" 
>>>>to institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>>>Montreux, where Diaconescu talked about it.
>>>>
>>>>While I did feel a bit miffed that my original suggestion of the 
>>>>problem wasn't mentioned at all, my problem with the paper is not 
>>>>that. My problem is that your approach seems to be the proverbial 
>>>>"throwing the baby away with the bathwater". The point of putting 
>>>>real proofs (as opposed to entailment relations) into institutions 
>>>>was to try to use the proofs-as-lambda-term-representations to do 
>>>>some real work for us, ie to connect to the paradigm of extracting 
>>>>programs from proofs, or to help with abstract analysis or to extend 
>>>>type systems in a principled and logical way, etc. i.e. any of the 
>>>>usual applications of categorical proof theory would do here.
>>>>
>>>>You say in page 2 of your joint paper that your new definition of 
>>>>(proof
>>>>theoretic) institution "fully supports proof theory", but the notion 
>>>>of proof theoretic institution (or of equivalence of institutions) 
>>>>introduced in the paper has nothing much to do with proof theory as 
>>>>people normally know it. What you call "proof theoretic institutions"
>>>>do not overcome the suggested limitation of "categorical logic", 
>>>>because proof theoretic institutions do not  model the significant 
>>>>aspect of proofs, which is their reduction behavior.
>>>>
>>>>The point of the Curry_Howard isomorphism is not that you can model 
>>>>propositions as objects in a category and proofs as equivalence 
>>>>classes of morphisms: the point is that the behaviour of proofs is 
>>>>preserved under this modelling. This is why some people think that 
>>>>the Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>>>The  crucial point is that if proof pi reduces to pi' via 
>>>>alpha-beta-eta reductions than the corresponding morphisms are 
>>>>related in the target category (either by equality or reduction). 
>>>>Nothing like that happens in the proof-theoretic institutions, which 
>>>>is why they are only proof-theoretical in name.
>>>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>>>you presumably realize, as  it is not even spelled out in the 
>>>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>>>guess my main complaint is that the paper does not define a 
>>>>"proof-theoretical institution" in the sense of an institution that 
>>>>preserves proofs, but simply as an institution that preserves 
>>>>entailment. But I guess this is all right, people will have different 
>>>>perspectives on what is important, mathematically speaking.
>>>>
>>>>Best regards,
>>>>Valeria
>>>>
>>>>
>>>>
>>>>-----Original Message-----
>>>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>>>Sent: Friday, January 31, 2003 12:14 AM
>>>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>>>Subject: Re: CHI for Institutions
>>>>
>>>>Dear Valeria,
>>>>
>>>>agreed, let's devide the work as you suggested.
>>>>
>>>>For a definition of (some variant of) parchment you might look at p 5ff.
>>>>of "Combingn and representing logical systems using model-theoretic 
>>>>parchments", available at my publications page.
>>>>
>>>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>>>calculus.
>>>>
>>>>Concerning the more complex Curry-Howard isos:
>>>>you seem to have one in the paper with Biermann. Then I'll have a 
>>>>look on that, before going on with trying to look at Curry-Howard in 
>>>>the institutional framework
>>>>
>>>>Greetings,
>>>>Till
>>>>
>>>>Valeria.dePaiva@parc.com schrieb:
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>Dear Till,
>>>>>Thanks for the very interesting message. Now I have to do some
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>reading, I don't even know what a parchment is...
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>But one small thought: your last paragraph about translating the
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>>>and seems a more concrete way of pushing forward towards my goal, 
>>>>which is different though. My goal is really enriching the whole 
>>>>framework of institutions so that it can cope with proofs (and when I 
>>>>say proofs I don't mean in a single proof calculus: I usually want a 
>>>>logic to be given in different several proof calculi all proved 
>>>>equivalent, like for instance for IPL you can give axioms, sequents 
>>>>or Natural deduction and you know how to translate proofs from one 
>>>>calculi to the others). So another way of pushing forward would be to 
>>>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>Now about your question: no I don't think I know of any reference 
>>>>>for
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>>>NuPRl are computer systems and first of all we would need papers 
>>>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>>>these might exist. I just haven't had the time or disposition to look 
>>>>for them. This would be a different research project altogether it seems to me.
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>>>try
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>to come back with questions/suggestions. Maybe you could try to add 
>>>>some details to the suggestion of looking at Curry-Howard in the 
>>>>institutional framework?
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>Cheers,
>>>>>Valeria
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>   
>>>
>>>      
>>>
>>_______________________________________________
>>Flirts mailing list
>>Flirts@mail.informatik.uni-bremen.de
>>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>>
>>-----------------------------------------------------------------------
>>-
>>
>>_______________________________________________
>>Flirts mailing list
>>Flirts@mail.informatik.uni-bremen.de
>>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>>
>>    
>>
>
>
>_______________________________________________
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>
>  
>



From till at informatik.uni-bremen.de  Thu Aug 11 17:06:05 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE79@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE79@goldeneye.ad.parc.com>
Message-ID: <42FB695D.3020605@informatik.uni-bremen.de>

Dear Valeria,

after all, it seems that this dicussion converges somewhat.
I think in order to make progress, we should think of looking
at some generalization of the notion of comorphism between institutions
to a new notion of comorphism between "institution with proofs and
reductions". I think that this definition should come out naturally out
of the definition of institutions with 2-categorical sentence structure.
This notion of comorphism between two logics then would consist of
1. translation of signatures
2. translation of sentences
3. translation of proofs
4. translation of proof reductions
5. optionally, translation of models

Of course, the crucial step is to look at some examples, to see if
this notion applies here, or needs to modified. Just to throw in
some ideas: We could take some order-enriched categories in the sense
of Pym and F?hrmann, or some of your modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
> 
> My replies will be somewhat slower, as time is in short supply here.
> 
>>For properties like compactness, you need possibly infinite set of
> sentences.
> All proofs of compactness that I've remembered seen were semantical
> proofs,
> piggybacking on satisfiability. Hence I have not worried about
> compactness much; this goes
> back to the notion that the infinities that I'm worried about are all
> potential infinities.
> 
> And I believe you're right, fibrations won't help if your logic happens
> to have infinitary conjunctions,
> but the logics I'm interested in, do not have infinitary connectives. I
> think I vaguely remember Michael Makkai  being interested in infinitary
> conjunctions, so one would have to check what he has to say about it.
> 
>>One concept is having not not-elimination, which separates classical
> from intuitionistic logic.
> But apart from the fact that classical logic has it, while
> intuitionistic logic does not, which is really just the definition of
> the difference between the logics, I cannot see anything that
> institutions with proofs can say about the "concept of
> not-not-elimination".
> 
>>>A logic, for me, does not exist without its derivations and proofs.
> Please don't take the comment without its context: I'm very fond of
> algebraic logic and I'm as  keen as any one else is of modelling logics
> and forgetting their proofs. But they were there to begin with. I just
> don't have to pay attention to them all the time. Modelling is like
> that, we allow ourselves to forget stuff when convenient.
> The situation would be totally different, if the proofs never existed,
> if all that existed was a "soup of symbols, ones unrelated to others".
> 
>>But this just amounts to having thin Hom-categories.
> Thin Hom-categories is just a different name for  traditional old
> categories, so we're in agreement, I take it.
> 
>>the framework of institutions with proofs should be philosophically
> neutral, 
>>and should admit the study of logics that might be rejected by some
> people, but not by all.
> While I believe that philosophical neutrality is an appealing feature, I
> can't see exactly what you're driving at with the second condition. 
> 
> Some logics are better-behaved, some are less well-behaved, as far as
> proof theory is concerned. 
> If one can set up a framework that unifies the "proof-theoretically
> well-behaved logics" 
> (whatever collection this may end up defining) both model theoretically
> and proof-theoretically, then progress would have been made, I reckon.
> But of course, the proof is in the pudding, this is all hypothetical
> right now.
> 
> Best,
> Valeria
> 
> -----Original Message-----
> From: flirts-bounces@informatik.uni-bremen.de
> [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till
> Mossakowski
> Sent: Wednesday, August 10, 2005 12:31 PM
> To: Formalism, Logic, Institution - Relating, Translating and
> Structuring
> Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
>  >>they idea of using sets of sentences as objects is new and useful,
>>>or did we miss something there also?
>  > Well, I don't think it is useful, as *when* you can do your  >
> categorical modelling properly, sets of sentences are modelled for  >
> free, either in the case where there's a connective that internalizes  >
> the comma (like a categorical tensor) or using the fibration  >
> mechanism. So no, I don't see the usefulness at the moment, it doesn't
>>buy me anything new. Maybe you'd like to expand on that?
> 
> For properties like compactness, you need possibly infinite set of
> sentences. And many logics do not have infinitary conjunction, hence you
> cannot internalize. If then the logic happens to have infinitary proof
> rules, I cannot see how to model this by fibrations.
> But may be I am missing something again :-).
> 
>  >>it does allow defining many basic proof theoretic concepts in a nice
> abstract way,  > But I don't see these many concepts. Would you like to
> discuss them one by one?
> 
> One concept is having not not-elimination, which separates classical
> from intuitionistic logic.
> 
>>A logic, for me, does not exist without its derivations and proofs.
> That's interesting. Strassburger in is paper "What is a logic and what
> is a proof" starts with "a logic is a pre-order", i.e. he omits proofs.
> Only later proofs come in. And I agree:
> there should be a notion of logic with proofs, but a notion of logic
> without proofs is useful as well.
> 
>>>It seems that there are three levels:
>>>1. entailment systems (= kind of pre-orders) 2. categories of 
>>>sentences and proofs 3. 2-categories of sentences, proofs and proof 
>>>reductions.
>>Not quite. One can and normally does talk about proof reductions using
> simply categories and morphisms.
>>You do not need to introduce 2-cells for that.
> I do not understand.
> I thought formulas = objects, proofs = 1-cells. So what are proof
> reductions other than 2-cells? OK, they might just be a pre-order on
> 1-cells. But this just amounts to having thin Hom-categories.
> 
>>>This seems to be related to polycategories, which, however, only use
> finite sequences of sentences.
>>Yes, indeed in the kind of proof theory I like, rules mostly have a
> finite number of hypotheses/assumptions. Girard says somewhere that the
> infinite is always a potential one, which I think is quite nice.
> I definitely want to include infinite sets of sentences as well.
> The whole notion of compactness as studied in traditional logic only
> makes sense with infinite sets of sentences.
> And ZFC, which is widely used in mathematics, is not finitely
> axiomatizable. Neither is first-order Peano arithmetic.
> Infinitary rules are less important, but it would be nice not to exclude
> them, because they are used occasionally.
> Moreover, the framework of institutions with proofs should be
> philosophically neutral, and should admit the study of logics that might
> be rejected by some people, but not by all.
> 
>> The modularity there is another one of the success criteria, right?
> Yes, certainly.
> 
> Greetings,
> Till
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From Valeria.dePaiva at parc.com  Thu Aug 11 18:36:03 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>

Dear Till,
Yes, I think that a notion like
> This notion of comorphism between two logics then would consist of 
>1. translation of signatures 
>2. translation of sentences 
>3. translation of proofs 
>4. translation of proof reductions 
>5. optionally, translation of models
Would be a good idea. Why not start looking at IL which is so well-understood?
Remind me please, where is the institution for IL described.

Best,

Valeria



-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till Mossakowski
Sent: Thursday, August 11, 2005 8:06 AM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

after all, it seems that this dicussion converges somewhat.
I think in order to make progress, we should think of looking at some generalization of the notion of comorphism between institutions to a new notion of comorphism between "institution with proofs and reductions". I think that this definition should come out naturally out of the definition of institutions with 2-categorical sentence structure.
This notion of comorphism between two logics then would consist of 1. translation of signatures 2. translation of sentences 3. translation of proofs 4. translation of proof reductions 5. optionally, translation of models

Of course, the crucial step is to look at some examples, to see if this notion applies here, or needs to modified. Just to throw in some ideas: We could take some order-enriched categories in the sense of Pym and F?hrmann, or some of your modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
> 
> My replies will be somewhat slower, as time is in short supply here.
> 
>>For properties like compactness, you need possibly infinite set of
> sentences.
> All proofs of compactness that I've remembered seen were semantical 
> proofs, piggybacking on satisfiability. Hence I have not worried about 
> compactness much; this goes back to the notion that the infinities 
> that I'm worried about are all potential infinities.
> 
> And I believe you're right, fibrations won't help if your logic 
> happens to have infinitary conjunctions, but the logics I'm interested 
> in, do not have infinitary connectives. I think I vaguely remember 
> Michael Makkai  being interested in infinitary conjunctions, so one 
> would have to check what he has to say about it.
> 
>>One concept is having not not-elimination, which separates classical
> from intuitionistic logic.
> But apart from the fact that classical logic has it, while 
> intuitionistic logic does not, which is really just the definition of 
> the difference between the logics, I cannot see anything that 
> institutions with proofs can say about the "concept of 
> not-not-elimination".
> 
>>>A logic, for me, does not exist without its derivations and proofs.
> Please don't take the comment without its context: I'm very fond of 
> algebraic logic and I'm as  keen as any one else is of modelling 
> logics and forgetting their proofs. But they were there to begin with. 
> I just don't have to pay attention to them all the time. Modelling is 
> like that, we allow ourselves to forget stuff when convenient.
> The situation would be totally different, if the proofs never existed, 
> if all that existed was a "soup of symbols, ones unrelated to others".
> 
>>But this just amounts to having thin Hom-categories.
> Thin Hom-categories is just a different name for  traditional old 
> categories, so we're in agreement, I take it.
> 
>>the framework of institutions with proofs should be philosophically
> neutral,
>>and should admit the study of logics that might be rejected by some
> people, but not by all.
> While I believe that philosophical neutrality is an appealing feature, 
> I can't see exactly what you're driving at with the second condition.
> 
> Some logics are better-behaved, some are less well-behaved, as far as 
> proof theory is concerned.
> If one can set up a framework that unifies the "proof-theoretically 
> well-behaved logics"
> (whatever collection this may end up defining) both model 
> theoretically and proof-theoretically, then progress would have been made, I reckon.
> But of course, the proof is in the pudding, this is all hypothetical 
> right now.
> 
> Best,
> Valeria
> 
> -----Original Message-----
> From: flirts-bounces@informatik.uni-bremen.de
> [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till 
> Mossakowski
> Sent: Wednesday, August 10, 2005 12:31 PM
> To: Formalism, Logic, Institution - Relating, Translating and 
> Structuring
> Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
>  >>they idea of using sets of sentences as objects is new and useful,
>>>or did we miss something there also?
>  > Well, I don't think it is useful, as *when* you can do your  > 
> categorical modelling properly, sets of sentences are modelled for  > 
> free, either in the case where there's a connective that internalizes  
> > the comma (like a categorical tensor) or using the fibration  > 
> mechanism. So no, I don't see the usefulness at the moment, it doesn't
>>buy me anything new. Maybe you'd like to expand on that?
> 
> For properties like compactness, you need possibly infinite set of 
> sentences. And many logics do not have infinitary conjunction, hence 
> you cannot internalize. If then the logic happens to have infinitary 
> proof rules, I cannot see how to model this by fibrations.
> But may be I am missing something again :-).
> 
>  >>it does allow defining many basic proof theoretic concepts in a 
> nice abstract way,  > But I don't see these many concepts. Would you 
> like to discuss them one by one?
> 
> One concept is having not not-elimination, which separates classical 
> from intuitionistic logic.
> 
>>A logic, for me, does not exist without its derivations and proofs.
> That's interesting. Strassburger in is paper "What is a logic and what 
> is a proof" starts with "a logic is a pre-order", i.e. he omits proofs.
> Only later proofs come in. And I agree:
> there should be a notion of logic with proofs, but a notion of logic 
> without proofs is useful as well.
> 
>>>It seems that there are three levels:
>>>1. entailment systems (= kind of pre-orders) 2. categories of 
>>>sentences and proofs 3. 2-categories of sentences, proofs and proof 
>>>reductions.
>>Not quite. One can and normally does talk about proof reductions using
> simply categories and morphisms.
>>You do not need to introduce 2-cells for that.
> I do not understand.
> I thought formulas = objects, proofs = 1-cells. So what are proof 
> reductions other than 2-cells? OK, they might just be a pre-order on 
> 1-cells. But this just amounts to having thin Hom-categories.
> 
>>>This seems to be related to polycategories, which, however, only use
> finite sequences of sentences.
>>Yes, indeed in the kind of proof theory I like, rules mostly have a
> finite number of hypotheses/assumptions. Girard says somewhere that 
> the infinite is always a potential one, which I think is quite nice.
> I definitely want to include infinite sets of sentences as well.
> The whole notion of compactness as studied in traditional logic only 
> makes sense with infinite sets of sentences.
> And ZFC, which is widely used in mathematics, is not finitely 
> axiomatizable. Neither is first-order Peano arithmetic.
> Infinitary rules are less important, but it would be nice not to 
> exclude them, because they are used occasionally.
> Moreover, the framework of institutions with proofs should be 
> philosophically neutral, and should admit the study of logics that 
> might be rejected by some people, but not by all.
> 
>> The modularity there is another one of the success criteria, right?
> Yes, certainly.
> 
> Greetings,
> Till
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till _______________________________________________
Flirts mailing list
Flirts@mail.informatik.uni-bremen.de
http://www.informatik.uni-bremen.de/mailman/listinfo/flirts


From till at informatik.uni-bremen.de  Thu Aug 11 19:36:09 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Thu Aug 18 10:12:46 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
Message-ID: <42FB8C89.3090802@informatik.uni-bremen.de>

Dear Valeria,

 > Why not start looking at IL which is so well-understood?
> Remind me please, where is the institution for IL described.

Unfortunately, there is not "the" institution for IL. Some
possibilities where I think I now the corresponding institution:
1. first-order intuitionistic logic with Heyting algebra semantics
2. first-order intuitionistic logic with Kripke semantics
3. higher-order intuitionistic logic with with topos semantics
4. higher-order intuitionistic logic with pccc semantics

I think (others: please correct me!) that only 4. is described in the
literature (in connection with the language HasCASL), but I think we
should take 3., because it is better known. Basically, it is
recasting the theory of Lambek-Scott in institutional terms.
I can sketch this more formally next week when I am back to the
office (currently I don't have the Lambek-Scott book at hand).

Greetings,
Till



-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From till at informatik.uni-bremen.de  Tue Aug 16 22:58:57 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Thu Aug 18 10:12:47 2005
Subject: [Flirts] Institutions for propositional categorical logics
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
Message-ID: <43025391.7080409@informatik.uni-bremen.de>

[The Flirts mailing list has been down for a few hours. Sorry for that.
  Should be working now again. Till]

Dear Valeria,

OK, so here is a sketch how to institutionalize propositional
categorical logics a la Curry-Howard (coming out of discussions
with Florian Rabe (a PhD student) and Lutz Schr?der, and of course
being motivated by the triangles on your slides).
I follow Lambek-Scott, Introduction to Categorical Higher-Order Logic
(the first few chapters). Instead of their "deductive systems", let
us be a bit more specific and consider the two-sorted specification
of small categories as a partial equational theory (with sorts
object and morphism), extended by the specification of an
operation 1:object axiomatized to be a terminal object.
Call any extension L of this theory with new operations and (oriented)
equations a "categorical logic".
Moreover, let us take the notion of proof-theoretic institution
from our paper "What is a logic?", but extend it with proof
reductions in the following way: Pr:Sign->PreOrd-Cat now becomes
a functor into the category of pre-order enriched categories
(i.e. categories where Hom-Sets are pre-orders).
I think the full generality of 2-categories is not needed,
because one usually does not distinguish between different reductions
between two given proofs.

Given a categorical logic L, let C be the category of L-algebras.
Since L is an extension of the theory of categories, C is a category
of certain small categories.
Given a set X, let T_L(X) be the (absolutely free) term algebra
over X (i.e. terms with operations in L and variables in X).
Define the proof-theoretic institution I(L) as follows:
Sign = Set
------- Propositions as types (categorically: objects) ---------
Sen(Sigma) = T_L(Sigma)_object,
     i.e. L-terms of sort object with variables in Sigma
Sen(sigma:Sigma1->Sigma2)(t) = sigma^#(t),
     where sigma^#:T_L(Sigma1)->T_L(Sigma2) is the extension
       of sigma:Sigma1->Sigma2 c T_(Sigma2) to terms
------- Categorical models -------------------------------------
Mod(Sigma) = {m:Sigma->|A|, where A \in C}
Model morphisms (F,mu):( m:Sigma->|A|) -> (m':Sigma->|B|) consist
   of functors F:A->B \in C and nat. transformations mu:F o m -> m'
Mod(sigma:Sigma1->Sigma2)(m:Sigma2->|A|) = m o sigma
Mod(sigma:Sigma1->Sigma2)(F,mu) = (F,mu * sigma)
(m:Sigma->|A|) |= phi iff m^#(phi) is a terminal object in A,
   where m^# is again the extension of m to terms
The satisfaction condition follows immediately from simple
   universal algebra: (m o sigma)^# = m^# o Sen(sigma).
------- Proofs as terms ----------------------------------------
Pr(Sigma) has as objects sets of Sigma-sentences
   A morphism from Gamma to Delta consists of a collection
   of terms (t_phi)_{phi\in Delta}, such that
   t_phi involves variables x_i, and
   L \cup {x_i:1->psi_i} |- t_phi:1->phi, with psi_i\in Gamma.
   Identities are variables, and composition is given by substitution.
   The pre-order on morphisms is given by the reduction ordering
   (using the orientend equations in L).
Pr(sigma:Sigma1->Sigma2) extends Sen(sigma) on objects,
   and on morphisms, it replaces symbols according to sigma
   (but the symbols from L are leaved fixed, of course).

The canonical example is L = theory of bicartesian closed
categories; then you get propositional intuitionistic logic,
and proof terms also could be thought as lambda-terms.
Note, however, if L = theory of cartesian categories (i.e.
a logic with just conjunction and true), terms are not
lambda-terms.
I would be interested if and how modal logic is another example,
and how the logic morphisms look.
Perhaps this leads to a joint paper?

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
> Yes, I think that a notion like
>>This notion of comorphism between two logics then would consist of 
>>1. translation of signatures 
>>2. translation of sentences 
>>3. translation of proofs 
>>4. translation of proof reductions 
>>5. optionally, translation of models
> Would be a good idea. Why not start looking at IL which is so well-understood?
> Remind me please, where is the institution for IL described.
> 
> Best,
> 
> Valeria


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From till at informatik.uni-bremen.de  Tue Aug 16 23:02:44 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Thu Aug 18 10:12:47 2005
Subject: [Flirts] An institution of higher-order intuitionistic logic
In-Reply-To: <42FB8C89.3090802@informatik.uni-bremen.de>
References: <1F0E426765530348BB55EB9575AEEBE514EE7F@goldeneye.ad.parc.com>
	<42FB8C89.3090802@informatik.uni-bremen.de>
Message-ID: <43025474.7040706@informatik.uni-bremen.de>

Dear Valeria,

here is an institution of higher-order intuitionistic logic.
But I think it is better to start with the propositional case,
also because the present logic does not deal with Curry-Howard.

Signatures are theories L of intuitionistic type theory, signature
morphisms are theory translations (see Lambek/Scott p.197).
L-Sentences are formulas in the symbols of Th. Sentence translation
is obvious (cf. again p.197).
Models of an intuitionistic type theory L consist of a topos T
and a strict logical functor T(L)-> T from the classifying topos T(L)
of L to T; equivalently, a theory translation t:L->L(T) from L to the
internal language L(T) of T. Model reduction is just composition (using
the latter representation of models).
An L-sentence e satisfies an L-model t:L->L(T), if T |= t(e),
or equivalently, L(T) |- t(e).
The satisfaction condition is trivial, because model reduction
is composition (Meseguer has called a similar thing "categorical logic
institutions", but note that he uses a T that is fixed for all models).
Model morphisms from t:L->L(T) to t':L->L(T') are pairs, consisting of
a functor f:T->T' and a natural transformation mu:L(f) o t -> t'.

Proof terms need to be extracted from the calculus on p.134f. of
Lambek/Scott. I do not spell out the details here, because
they do not do so either. Actually, they do not apply Curry-Howard at
all for this logic. Perhaps there is other work that does.

Greetings,
Till

-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From goguen at cs.ucsd.edu  Wed Aug 17 00:18:03 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 18 10:12:47 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <4302661B.8070506@cs.ucsd.edu>

Note: FLIRTS was down, but is now (said to be) up, so i am resending this.

Dear Valeria,

Thanks for the pointer to Phil's paper, i enjoyed reading it!

This paper does not make the claims for Curry-Howard that
you mention, but instead makes the much more modest claim
that systems were "designed by researchers in functional languages
and they depend heavily on logics and type systems whose roots
were traced in this paper" (these roots are work of Church,
Gentzen, Girard, etc.).  On the whole, there are pleasingly few
exaggerated claims in the paper.

My impression (e.g., looking at recent POPL proceedings) is
that there is indeed a lot of work that applies ideas from type
theory to programming languages, but the systems used are
far from beautiful, while the beautiful systems are not directly
useful.  It would be surprising to me if anyone worked out a
Curry-Howard isomorphism for any of these ugly type systems.

 - joseph

Valeria.dePaiva@parc.com wrote:

>Dear Joseph,
>
>While I do agree with 
>
>>The basic Curry-Howard isomorphism ("CHI") is one of the most beautiful
>>pieces of mathematics that is 
>>associated with computer science, and its extension to much more
>>general types is 
>>something that theoretical computer scientists can be proud of.
>
>I beg to differ on
> 
>>As far as practice goes, not much has happened,  
>>
>I guess it all depends on what kind of practice you're thinking of. It
>seems to me that quite a lot of the practical work that goes under the
>rubric of "programming languages" design&implementation (including the
>design of Java and other recent typed programming languages) owes its
>existence to programmers picking up the CHI and using it for their own
>purposes.  Phil Wadler had a nice note on Dr. Dobbs about the CHI called
>Proofs are Programs: 19th Century Logic and 21st Century Computing.
>
>I think the URL is
>
>http://homepages.inf.ed.ac.uk/wadler/papers/frege/frege.pdf
>
>But back to the subject at hand:
>
>>i even have some ideas which i hope to write up a bit later on.
>>Looking forward to seeing you this,
>
>Valeria
>
>-----Original Message-----
>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu] 
>Sent: Friday, August 12, 2005 8:03 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>; Formalism, Logic,
>Institution - Relating, Translating and Structuring
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>It seems to me that your definition of  "proof-theoretical" is too
>narrow to capture what mathematical logicians actually do; it seems to
>me that "proofs about proofs" is a more accurate definition for "proof
>theory", since in practice proofs about proofs are often done with
>respect to models, and experience shows that when you have the right
>models, such proofs are often easier that way.
>
>Im thinking that the origin of these divergences of view may go back to
>some discussions you and Till had about what's in the title of this
>thread, "Curry-Howard isomorphism for Institutions" and your research
>focus on type theory?
>
>The basic Curry-Howard isomorphism ("CHI") is one of the most beautiful
>pieces of mathematics that is associated with computer science, and its
>extension to much more general types is something that theoretical
>computer scientists can be proud of.  As far as practice goes, not much
>has happened, but it seems plausible to me that someday, for some class
>of useful programs, it may be possible for users to specify what they
>want in a "Visual Type Theory" language, which then an automatic theorem
>can constructively prove inhabited, yielding a program that can then be
>optimized by sophisticated transformations into a practical program that
>can be run.  After all, computers continue to follow Moore's law, which
>means still lots of power to come, while theorem proving and compiler
>optimization technologies continue to improve (but not exponentially!).
>So it seems worth continuing research on type theory in pursuit of the
>dream of automatic programming (and other some dreams).
>
>But i dont think this should be allowed to dictate the research
>programmes of institutions, which have always had different goals from
>type theory, one of which is to capture mathematical practice in logic,
>including model theoretic reasoning, not just formal manipulations of
>proofs.  Nevertheless, i think it could be very interesting to see what
>can be done with CHI in an institutional setting, and i even have some
>ideas which i hope to write up a bit later on.
>
>By the way, some time ago i put forward the slogan "types as theories"
>as a view of programming, and i also tried to show that if you have a
>nice module system of the sort supported by institutions, then you do
>not really need higher order logic to reason about typical higher order
>functions.  See
>
>    http://www.cs.ucsd.edu/~goguen/pps/utyop.pdf
>
>But this is not to say that i am against type theory or against higher
>order functions.  In fact, i endorse Till's proposal for research on IL,
>except that i do not think that the model aspect of morphisms should be
>optional.
>
>I look forward to further discussion of all this.
>
>  -- joseph
>
>Valeria.dePaiva@parc.com wrote:
>
>>Dear Joseph,
>>A small clarification:
>>
>>>im just reponding to your request for proof theory stuff that we can 
>>>do in insitutions - do you remember that request?
>>>
>>My request (may be I wasn't clear enough) was for proof-theoretical 
>>stuff that you could do with (the proof-theoretical side of)
>>"institutions with proofs".
>>
>>So Craig interpolation done by model theoretical means doesn't count, 
>>even if I do think it pretty. It's the `method' of proof that's at
>>stake in the request, not the statement itself.
>>
>>Cheers,
>>
>>Valeria
>>
>>-----Original Message-----
>>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
>>Sent: Thursday, August 11, 2005 8:02 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>; Formalism, Logic, 
>>Institution - Relating, Translating and Structuring
>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>
>>hi again Valeria!
>>
>>Valeria.dePaiva@parc.com wrote:
>>
>>>Craig interpolation I believe we can do totally proof-theoretically in
>>>many situations -- without infinite sets of sentences. Compactness, as I
>>>said, I have a problem  seeing it as proof theory, but maybe it's just
>>>lack of trying.
>>>
>>i never said craig interpolation needs infinite sets of sentences, and
>>i never said you couldnt do it proof theoretically!  im just reponding
>>to your request for proof theory stuff that we can do in insitutions -
>>do you remember that request?  and please note that compactness is
>>normally formulated as an assertion about proofs even though it is
>>normally proved model theoretically.
>>
>> - joseph
>>
>>>Best,
>>>Valeria
>>>-----Original Message-----
>>>From: flirts-bounces@informatik.uni-bremen.de
>>>[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph 
>>>Goguen
>>>Sent: Wednesday, August 10, 2005 6:25 PM
>>>To: Formalism, Logic, Institution - Relating, Translating and 
>>>Structuring
>>>Cc: till@informatik.uni-bremen.de
>>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>>
>>>Dear Valeria and others,
>>>
>>>This is fun, we are getting a lot of different points of view.
>>>
>>>Valeria.dePaiva@parc.com wrote:
>>>
>>>>Dear Joseph,
>>>>Thanks for the friendly message.
>>>>First, let me make it clear that, no, I do not have "a nicer way of
>>>>>including proofs into institutions", at the moment. I had proposed to
>>>>>Till that we could investigate this problem together, as I had become
>>>>>interested in the idea around 1999. I'm attaching some slides from a
>>>>>talk I gave at NASA Ames then when I was trying to sell them a work
>>>>>proposal along these lines. I thought this was a cool idea, which you
>>>>>probably agree since you've had the same idea some 15 years before me.
>>>>>But I only worked on that for a couple of weeks, preparing the talk,
>>>>>which is just a proposal for some work. Not the work itself.
>>>>>
>>>>I'm afraid I don't think it excessive. I went back to the paper "What
>>>>is a Logic?" to see if I was forgetting any basic interesting 
>>>>proof-theoretic notion, given that you've said
>>>>>abstract way,     
>>>>>
>>>>But I don't see these many concepts. Would you like to discuss them
>>>>one by one?



From Valeria.dePaiva at parc.com  Wed Aug 17 02:35:59 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Thu Aug 18 10:12:47 2005
Subject: FW: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EEBC@goldeneye.ad.parc.com>

 


Dr Valeria de Paiva
PARC
3333 Coyote Hill Road
Palo Alto, CA 94304
USA


-----Original Message-----
From: de Paiva, Valeria <Valeria.dePaiva@parc.com> 
Sent: Monday, August 15, 2005 10:24 PM
To: 'Joseph Goguen'
Cc: 'flirts@informatik.uni-bremen.de'
Subject: RE: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Joseph,

I'm glad you've enjoyed Phil's paper.

> My impression (e.g., looking at recent POPL proceedings) is that there

>is indeed a lot of work that applies ideas from type theory to 
>programming languages,  but the systems used are far from beautiful, 
>while the beautiful systems are not directly useful.

My point was that the work represented in POPL, etc
> owes its existence to programmers picking up the CHI and using it for 
>their own purposes.
Maybe it is  too much to  ask for  the systems that come from logic,  to
carry-on being pretty. But I guess one may hope and I do. 
I also enjoyed very much David Walker's invited talk at IMLA this year,
when he was asking  the logicians in the audience for help with his type
systems for memory management. But I guess there's no paper written
about it, (yet, perhaps, I don't know). 

I have written some of my own take on these ideas, as far as CHI for
modal logics is concerned, in the preface of the special issue of the
JLC dedicated to the second IMLA (2002 in Copenhagen). You can read it
from http://www.cs.bham.ac.uk/%7Evdp/publications/final-preface.pdf
if interested. But this is of course just a subcase of the general
picture.

Best,
Valeria 

-----Original Message-----
From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
Sent: Monday, August 15, 2005 6:52 PM
To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
Cc: flirts@informatik.uni-bremen.de
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

Thanks for the pointer to Phil's paper, i enjoyed reading it!

This paper does not make the claims for Curry-Howard that you mention,
but instead makes the much more modest claim that systems were "designed
by researchers in functional languages and they depend heavily on logics
and type systems whose roots were traced in this paper" (these roots are
work of Church, Gentzen, Girard, etc.).  On the whole, there are
pleasingly few exaggerated claims in the paper.

My impression (e.g., looking at recent POPL proceedings) is that there
is indeed a lot of work that applies ideas from type theory to
programming languages, but the systems used are far from beautiful,
while the beautiful systems are not directly useful.  It would be
surprising to me if anyone worked out a Curry-Howard isomorphism for any
of these ugly type systems.

  - joseph

Valeria.dePaiva@parc.com wrote:

>Dear Joseph,
>
>While I do agree with
>  
>
>>The basic Curry-Howard isomorphism ("CHI") is one of the most 
>>beautiful
>>    
>>
>pieces of mathematics that is
>  
>
>>associated with computer science, and its extension to much more
>>    
>>
>general types is
>  
>
>>something that theoretical computer scientists can be proud of.
>>    
>>
>
>I beg to differ on
>  
>
>>As far as practice goes, not much has happened,
>>    
>>
>I guess it all depends on what kind of practice you're thinking of. It 
>seems to me that quite a lot of the practical work that goes under the 
>rubric of "programming languages" design&implementation (including the 
>design of Java and other recent typed programming languages) owes its 
>existence to programmers picking up the CHI and using it for their own 
>purposes.  Phil Wadler had a nice note on Dr. Dobbs about the CHI 
>called
>
>Proofs are Programs: 19th Century Logic and 21st Century Computing.
>
>I think the URL is
>
>http://homepages.inf.ed.ac.uk/wadler/papers/frege/frege.pdf
>
>But back to the subject at hand:
>  
>
>>i even have some ideas which i hope to write up a bit later on.
>>    
>>
>Looking forward to seeing you this,
>
>Valeria
>
>-----Original Message-----
>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
>Sent: Friday, August 12, 2005 8:03 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>; Formalism, Logic, 
>Institution - Relating, Translating and Structuring
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>It seems to me that your definition of  "proof-theoretical" is too 
>narrow to capture what mathematical logicians actually do; it seems to 
>me that "proofs about proofs" is a more accurate definition for "proof 
>theory", since in practice proofs about proofs are often done with 
>respect to models, and experience shows that when you have the right 
>models, such proofs are often easier that way.
>
>Im thinking that the origin of these divergences of view may go back to

>some discussions you and Till had about what's in the title of this 
>thread, "Curry-Howard isomorphism for Institutions" and your research 
>focus on type theory?
>
>The basic Curry-Howard isomorphism ("CHI") is one of the most beautiful

>pieces of mathematics that is associated with computer science, and its

>extension to much more general types is something that theoretical 
>computer scientists can be proud of.  As far as practice goes, not much

>has happened, but it seems plausible to me that someday, for some class

>of useful programs, it may be possible for users to specify what they 
>want in a "Visual Type Theory" language, which then an automatic 
>theorem can constructively prove inhabited, yielding a program that can

>then be optimized by sophisticated transformations into a practical 
>program that can be run.  After all, computers continue to follow 
>Moore's law, which means still lots of power to come, while theorem 
>proving and compiler optimization technologies continue to improve (but
not exponentially!).
>So it seems worth continuing research on type theory in pursuit of the 
>dream of automatic programming (and other some dreams).
>
>But i dont think this should be allowed to dictate the research 
>programmes of institutions, which have always had different goals from 
>type theory, one of which is to capture mathematical practice in logic,

>including model theoretic reasoning, not just formal manipulations of 
>proofs.  Nevertheless, i think it could be very interesting to see what

>can be done with CHI in an institutional setting, and i even have some 
>ideas which i hope to write up a bit later on.
>
>By the way, some time ago i put forward the slogan "types as theories"
>as a view of programming, and i also tried to show that if you have a 
>nice module system of the sort supported by institutions, then you do 
>not really need higher order logic to reason about typical higher order

>functions.  See
>
>    http://www.cs.ucsd.edu/~goguen/pps/utyop.pdf
>
>But this is not to say that i am against type theory or against higher 
>order functions.  In fact, i endorse Till's proposal for research on 
>IL, except that i do not think that the model aspect of morphisms 
>should be optional.
>
>I look forward to further discussion of all this.
>
>  -- joseph
>
>Valeria.dePaiva@parc.com wrote:
>
>  
>
>>Dear Joseph,
>>A small clarification:
>> 
>>
>>    
>>
>>>im just reponding to your request for proof theory stuff that we can 
>>>do in insitutions - do you remember that request?
>>>   
>>>
>>>      
>>>
>>My request (may be I wasn't clear enough) was for proof-theoretical 
>>stuff that you could do with (the proof-theoretical side of)
>>    
>>
>"institutions with proofs".
>  
>
>>So Craig interpolation done by model theoretical means doesn't count, 
>>even if I do think it pretty. It's the `method' of proof that's at
>>    
>>
>stake in the request, not the statement itself.
>  
>
>>Cheers,
>>
>>Valeria
>>
>>-----Original Message-----
>>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
>>Sent: Thursday, August 11, 2005 8:02 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>; Formalism, Logic, 
>>Institution - Relating, Translating and Structuring
>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>
>>hi again Valeria!
>>
>>Valeria.dePaiva@parc.com wrote:
>>
>> 
>>
>>    
>>
>>>Craig interpolation I believe we can do totally proof-theoretically 
>>>in
>>>      
>>>
>many situations -- without infinite sets of sentences. Compactness, as 
>I said, I have a problem  seeing it as proof theory, but maybe it's 
>just lack of trying.
>  
>
>>>   
>>>
>>>      
>>>
>>i never said craig interpolation needs infinite sets of sentences, and
>>    
>>
>i never said you couldnt do it proof theoretically!  im just reponding 
>to your request for proof theory stuff that we can do in insitutions - 
>do you remember that request?  and please note that compactness is 
>normally formulated as an assertion about proofs even though it is 
>normally proved model theoretically.
>  
>
>> - joseph
>>
>> 
>>
>>    
>>
>>>Best,
>>>Valeria
>>>-----Original Message-----
>>>From: flirts-bounces@informatik.uni-bremen.de
>>>[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph 
>>>Goguen
>>>Sent: Wednesday, August 10, 2005 6:25 PM
>>>To: Formalism, Logic, Institution - Relating, Translating and 
>>>Structuring
>>>Cc: till@informatik.uni-bremen.de
>>>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>>>
>>>Dear Valeria and others,
>>>
>>>This is fun, we are getting a lot of different points of view.
>>>
>>>Valeria.dePaiva@parc.com wrote:
>>>
>>>
>>>
>>>   
>>>
>>>      
>>>
>>>>Dear Joseph,
>>>>Thanks for the friendly message.
>>>>First, let me make it clear that, no, I do not have "a nicer way of
>>>>        
>>>>
>including proofs into institutions", at the moment. I had proposed to 
>Till that we could investigate this problem together, as I had become 
>interested in the idea around 1999. I'm attaching some slides from a 
>talk I gave at NASA Ames then when I was trying to sell them a work 
>proposal along these lines. I thought this was a cool idea, which you 
>probably agree since you've had the same idea some 15 years before me.
>But I only worked on that for a couple of weeks, preparing the talk, 
>which is just a proposal for some work. Not the work itself.
>  
>
>>>>     
>>>>
>>>>        
>>>>
>>>>> 
>>>>>
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>I'm afraid I don't think it excessive. I went back to the paper 
>>>>"What
>>>>        
>>>>
>
>  
>
>>>>is a Logic?" to see if I was forgetting any basic interesting 
>>>>proof-theoretic notion, given that you've said
>>>>
>>>>
>>>>  
>>>>
>>>>     
>>>>
>>>>        
>>>>
>>>>>it does allow defining many basic proof theoretic concepts in a 
>>>>>nice
>>>>>          
>>>>>
>
>  
>
>>>>>abstract way,
>>>>> 
>>>>>
>>>>>    
>>>>>
>>>>>       
>>>>>
>>>>>          
>>>>>
>>>>But I don't see these many concepts. Would you like to discuss them
>>>>        
>>>>
>one by one?
>  
>
>>>>     
>>>>
>>>>        
>>>>
>> 
>>
>>    
>>
>
>  
>


From goguen at cs.ucsd.edu  Wed Aug 17 23:52:22 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 18 10:12:47 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EEBC@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EEBC@goldeneye.ad.parc.com>
Message-ID: <4303B196.10607@cs.ucsd.edu>

Dear Valeria,

It seems that Wadler and I agree with each other, and disagree with you,
in saying that programmers do not use CHI even though they do use types.

Also, there seems to be a bug in the Wadler paper, in that he proves that
the product type constructor is commutative (there are also a few smallish
exaggerated claims, but not this one).

Valeria.dePaiva@parc.com wrote:

>-----Original Message-----
>From: de Paiva, Valeria <Valeria.dePaiva@parc.com> 
>Sent: Monday, August 15, 2005 10:24 PM
>To: 'Joseph Goguen'
>Cc: 'flirts@informatik.uni-bremen.de'
>Subject: RE: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Joseph,
>
>I'm glad you've enjoyed Phil's paper. 
>  
>
>>My impression (e.g., looking at recent POPL proceedings) is that thereis indeed a lot of work that applies ideas from type theory to 
>>programming languages,  but the systems used are far from beautiful, 
>>while the beautiful systems are not directly useful.
>>    
>>
>
>My point was that the work represented in POPL, etc owes its existence to programmers picking up the CHI and using it for their own purposes.
>  
>
>Maybe it is  too much to  ask for  the systems that come from logic,  to
>carry-on being pretty. But I guess one may hope and I do. 
>I also enjoyed very much David Walker's invited talk at IMLA this year,
>when he was asking  the logicians in the audience for help with his type
>systems for memory management. But I guess there's no paper written
>about it, (yet, perhaps, I don't know). 
>
>I have written some of my own take on these ideas, as far as CHI for
>modal logics is concerned, in the preface of the special issue of the
>JLC dedicated to the second IMLA (2002 in Copenhagen). You can read it
>from http://www.cs.bham.ac.uk/%7Evdp/publications/final-preface.pdf
>if interested. But this is of course just a subcase of the general
>picture.
>
>Best,
>Valeria 
>  
>
i will take a look at your paper when i get a chance.

cheers,

joseph

>-----Original Message-----
>From: Joseph Goguen [mailto:goguen@cs.ucsd.edu]
>Sent: Monday, August 15, 2005 6:52 PM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Cc: flirts@informatik.uni-bremen.de
>Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>Thanks for the pointer to Phil's paper, i enjoyed reading it!
>
>This paper does not make the claims for Curry-Howard that you mention,
>but instead makes the much more modest claim that systems were "designed
>by researchers in functional languages and they depend heavily on logics
>and type systems whose roots were traced in this paper" (these roots are
>work of Church, Gentzen, Girard, etc.).  On the whole, there are
>pleasingly few exaggerated claims in the paper.
>
>My impression (e.g., looking at recent POPL proceedings) is that there
>is indeed a lot of work that applies ideas from type theory to
>programming languages, but the systems used are far from beautiful,
>while the beautiful systems are not directly useful.  It would be
>surprising to me if anyone worked out a Curry-Howard isomorphism for any of these ugly type systems.
>
>  - joseph
>
>Valeria.dePaiva@parc.com wrote:
>
>  
>
> _______________________________________________
>
>Flirts mailing list
>Flirts@mail.informatik.uni-bremen.de
>http://www.informatik.uni-bremen.de/mailman/listinfo/flirts
>  
>

From goguen at cs.ucsd.edu  Wed Aug 17 23:59:39 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Thu Aug 18 10:12:47 2005
Subject: [Flirts] institutional formulation of Curry-Howard
Message-ID: <4303B34B.2030909@cs.ucsd.edu>

Example 3 on pp 17-19 of "Information integration in institutions" gives
as institutional formulation of the Curry-Howard isomorphism for a
simple special case, though the approach obviously extends.  The
paper can be fetched from

    http://www.cs.ucsd.edu/~goguen/pps/ifi04.pdf

Probably it still has some bugs, as the notation is a little dense and i
just finished the write up; please let me know if you find any.

I hope this may be a useful contribution to our discussions.

Cheers,

   joseph


From goguen at cs.ucsd.edu  Fri Aug 19 20:43:32 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Fri Aug 19 20:43:37 2005
Subject: [Flirts] institutional formulation of Curry-Howard
Message-ID: <43062854.9080905@cs.ucsd.edu>

Thanks to some feedback from Till, the institutional formulation
of the Curry-Howard isomorphism (Example 3 on pp 17-19
of "Information integration in institutions") is now clearer and
easier to read.  As before, it can be fetched from

   http://www.cs.ucsd.edu/~goguen/pps/ifi04.pdf

Some other small things are also improved in
the paper.  Any comments would be appreciated,
as i can still make changes (and not just to
Curry-Howard).

Cheers,

  joseph



From till at informatik.uni-bremen.de  Mon Aug 22 12:21:35 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Mon Aug 22 12:20:49 2005
Subject: [Flirts] Curry-Howard for more complex logics
In-Reply-To: <20050821140210.7852727F7F7@stoilow.imar.ro>
References: <20050821140210.7852727F7F7@stoilow.imar.ro>
Message-ID: <4309A72F.7050803@informatik.uni-bremen.de>

Dear Razvan,

> About Curry-Howard, I still want to ask you whether this is applicable
 > to things like equational logic, or even first order logic.

so far Joseph and I have written proposals for institutionalizing the
propositional part of Curry-Howard. I summarize the relevant features.
Note that we have a formal definition of all these features in the
institutional setting.

propositions       - types
proofs             - terms / functional programs

having true        - having singeltons
having false       - having the empty type
having conjunction - having pairing/projections (products)
having disjunction - having sum injections/case (coproducts)
having implication - having lambda abstraction (function spaces)

Now this can be extended as follows (with a bit of handwaving,
I am not an expert on this, the details need to be studied
in the institutional setting):

having propositional constants - having type constants
having quantification (over propositions) - having polymorphism
having (higher-order) functions (over propositions) -
                         having (higher-order) type constructors
having individuals and relations - having dependent types
   (dependent types also lead to the respetive quantifications etc.)
intuitionistic higher-order logic - having all of above

having not-not-elimination (being classic) - having call/cc
                                   (or \mu-abstraction)
re-use of subproofs - \tilde{\mu}-abstraction


Hence, for equational or first-order logic, you need dependent types.
It seems that then terms for individuals at the left hand side
correspond to terms for individuals at the right hand side,
likewise for function symbols, if one does not want to repesent
them as relations (i.e. dependent types). Equality is then a
dependent type with some inhabitants:

refl a : Eq a a
symm a b : Eq a b -> Eq b a
trans a b c : Eq a b * Eq b c -> Eq a c
cong a b : Eq a b -> Eq (f a) (f b)
or in the relational representation
cong a b : Eq a b * F a a' * F b b' -> Eq a' b'

Greetings,
Till
-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From Valeria.dePaiva at parc.com  Mon Aug 22 19:23:36 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Mon Aug 22 19:24:08 2005
Subject: [Flirts] Curry-Howard for more complex logics
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EEFB@goldeneye.ad.parc.com>

Dear Till and Razvan,
A small clarification:
>Hence, for equational or first-order logic, you need dependent types.
Actually you don't *need* dependent types, it's customary to have
dependent types as they fit better with
the intuitionistic conception of FOL. But Martin Coen, for example,
implemented FOL with proof terms Curry-Howard compliant in Isabelle for
his thesis in 1992. I believe his FOL-with-terms (and no dependent
types)
logic was in the official distribution of Isabelle for several years,
but I don't know if it still is or not. 

Best,
Valeria

Dr Valeria de Paiva
PARC
3333 Coyote Hill Road
Palo Alto, CA 94304
USA


-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de
[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till
Mossakowski
Sent: Monday, August 22, 2005 3:22 AM
To: Razvan Diaconescu
Cc: FLIRTS
Subject: [Flirts] Curry-Howard for more complex logics

Dear Razvan,

> About Curry-Howard, I still want to ask you whether this is applicable
 > to things like equational logic, or even first order logic.

so far Joseph and I have written proposals for institutionalizing the
propositional part of Curry-Howard. I summarize the relevant features.
Note that we have a formal definition of all these features in the
institutional setting.

propositions       - types
proofs             - terms / functional programs

having true        - having singeltons
having false       - having the empty type
having conjunction - having pairing/projections (products) having
disjunction - having sum injections/case (coproducts) having implication
- having lambda abstraction (function spaces)

Now this can be extended as follows (with a bit of handwaving, I am not
an expert on this, the details need to be studied in the institutional
setting):

having propositional constants - having type constants having
quantification (over propositions) - having polymorphism having
(higher-order) functions (over propositions) -
                         having (higher-order) type constructors having
individuals and relations - having dependent types
   (dependent types also lead to the respetive quantifications etc.)
intuitionistic higher-order logic - having all of above

having not-not-elimination (being classic) - having call/cc
                                   (or \mu-abstraction) re-use of
subproofs - \tilde{\mu}-abstraction


Hence, for equational or first-order logic, you need dependent types.
It seems that then terms for individuals at the left hand side
correspond to terms for individuals at the right hand side, likewise for
function symbols, if one does not want to repesent them as relations
(i.e. dependent types). Equality is then a dependent type with some
inhabitants:

refl a : Eq a a
symm a b : Eq a b -> Eq b a
trans a b c : Eq a b * Eq b c -> Eq a c
cong a b : Eq a b -> Eq (f a) (f b)
or in the relational representation
cong a b : Eq a b * F a a' * F b b' -> Eq a' b'

Greetings,
Till
-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
_______________________________________________
Flirts mailing list
Flirts@mail.informatik.uni-bremen.de
http://www.informatik.uni-bremen.de/mailman/listinfo/flirts

From till at informatik.uni-bremen.de  Mon Aug 22 22:44:03 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Mon Aug 22 22:44:10 2005
Subject: [Flirts] Curry-Howard for more complex logics
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EEFB@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EEFB@goldeneye.ad.parc.com>
Message-ID: <430A3913.3010909@informatik.uni-bremen.de>

Dear Valeria,

I have looked into Martin Coen's thesis, available at
http://www.cl.cam.ac.uk/Research/Reports/TR272-mc-interactive-program-derivation.pdf

On p. 29, he defines the type system, and includes Pi-types.
Although he writes "Pi(A,B)", Pi has kind tau => (iota => tau) => tau
(see p.30), which means that B depends on some x, and we arrive at the
usual encoding of quantification as dependent types:

the proposition           \forall x:A . B(x)
corresponds to the type   \Pi x: A . B(x)

Also, in the Isabelle 2004 distribution, you find a logic CCL (by Coen),
and there in Type.thy, the notation PROD x:A. B is used.

It seems that for FOL via Curry-Howard, you *need* dependent types.

Greetings,
Till

On p. Valeria.dePaiva@parc.com wrote:
> Dear Till and Razvan,
> A small clarification:
>>Hence, for equational or first-order logic, you need dependent types.
> Actually you don't *need* dependent types, it's customary to have
> dependent types as they fit better with
> the intuitionistic conception of FOL. But Martin Coen, for example,
> implemented FOL with proof terms Curry-Howard compliant in Isabelle for
> his thesis in 1992. I believe his FOL-with-terms (and no dependent
> types)
> logic was in the official distribution of Isabelle for several years,
> but I don't know if it still is or not. 
> 
> Best,
> Valeria
> 
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From Valeria.dePaiva at parc.com  Mon Aug 22 22:58:08 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Mon Aug 22 22:58:26 2005
Subject: [Flirts] Curry-Howard for more complex logics
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EEFE@goldeneye.ad.parc.com>

Dear Till,
The work of Martin Coen's that I mentioned wasn't his thesis work on CCL
(and application to verification):
 check the Isabelle Library on
 http://isabelle.in.tum.de/library/index.html
Under First-Order Logic, the last item is FOLP (FOL with Proof Terms).
This is the work that I discussed with Martin, and apparently it is
still supported, both in the classical and the intuitionistic versions
(is that so, Larry?).

 Of  course in the classical version the terms are not Curry-Howard
isomorphic to the logic (due to all the problems that we've discussed
before), but in the intuitionistic they are.

Best,
Valeria

Dr Valeria de Paiva
PARC
3333 Coyote Hill Road
Palo Alto, CA 94304
USA


-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de
[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till
Mossakowski
Sent: Monday, August 22, 2005 1:44 PM
To: Formalism, Logic, Institution - Relating, Translating and
Structuring
Cc: Razvan.Diaconescu@imar.ro
Subject: Re: [Flirts] Curry-Howard for more complex logics

Dear Valeria,

I have looked into Martin Coen's thesis, available at
http://www.cl.cam.ac.uk/Research/Reports/TR272-mc-interactive-program-de
rivation.pdf

On p. 29, he defines the type system, and includes Pi-types.
Although he writes "Pi(A,B)", Pi has kind tau => (iota => tau) => tau
(see p.30), which means that B depends on some x, and we arrive at the
usual encoding of quantification as dependent types:

the proposition           \forall x:A . B(x)
corresponds to the type   \Pi x: A . B(x)

Also, in the Isabelle 2004 distribution, you find a logic CCL (by Coen),
and there in Type.thy, the notation PROD x:A. B is used.

It seems that for FOL via Curry-Howard, you *need* dependent types.

Greetings,
Till

On p. Valeria.dePaiva@parc.com wrote:
> Dear Till and Razvan,
> A small clarification:
>>Hence, for equational or first-order logic, you need dependent types.
> Actually you don't *need* dependent types, it's customary to have 
> dependent types as they fit better with the intuitionistic conception 
> of FOL. But Martin Coen, for example, implemented FOL with proof terms

> Curry-Howard compliant in Isabelle for his thesis in 1992. I believe 
> his FOL-with-terms (and no dependent
> types)
> logic was in the official distribution of Isabelle for several years, 
> but I don't know if it still is or not.
> 
> Best,
> Valeria
> 
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
_______________________________________________
Flirts mailing list
Flirts@mail.informatik.uni-bremen.de
http://www.informatik.uni-bremen.de/mailman/listinfo/flirts

From till at informatik.uni-bremen.de  Mon Aug 22 23:56:07 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Mon Aug 22 23:56:08 2005
Subject: [Flirts] Curry-Howard for more complex logics
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EEFE@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EEFE@goldeneye.ad.parc.com>
Message-ID: <430A49F7.5080309@informatik.uni-bremen.de>

Dear Valeria,

interesting. I found this in my Isabelle 2004 distribution, so it is
still supported (although I found only rather elementary things
actually proved).

It seems to be that this is a version of Curry-Howard that just
exploits "proofs = terms", while "propositions = types" is
supported only insofar as the proof terms are typed with their
propositions. Of course, then you do not have dependent types,
but just quantified formulas. But once you try to interpret quantified
formulas as types, then you will see that they behave like dependent
types.

The point seems hence to be that you can avoid complicated
type systems by taking your propositions to immediately be the "types",
without resort to a independently-developed type system.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
> The work of Martin Coen's that I mentioned wasn't his thesis work on CCL
> (and application to verification):
>  check the Isabelle Library on
>  http://isabelle.in.tum.de/library/index.html
> Under First-Order Logic, the last item is FOLP (FOL with Proof Terms).
> This is the work that I discussed with Martin, and apparently it is
> still supported, both in the classical and the intuitionistic versions
> (is that so, Larry?).
> 
>  Of  course in the classical version the terms are not Curry-Howard
> isomorphic to the logic (due to all the problems that we've discussed
> before), but in the intuitionistic they are.
> 
> Best,
> Valeria
> 
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA



-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
From Valeria.dePaiva at parc.com  Tue Aug 23 01:46:22 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Tue Aug 23 01:46:41 2005
Subject: [Flirts] Curry-Howard for more complex logics
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EF05@goldeneye.ad.parc.com>

> although I found only rather elementary things actually proved).
Yeah, I don't think anyone used for anything. But I've asked for it to
be done a long time ago,
On the grounds that traditional logicians might want to have something
like it.

>The point seems hence to be that you can avoid complicated type systems
by taking your propositions to 
>immediately be the "types", without resort to a independently-developed
type system.
Indeed. Of course the point for Martin I believe was different, he
wanted something like a classical calculus of constructions to do  his
verification using it. But I never understood exactly how.

Best,
Valeria


-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de
[mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till
Mossakowski
Sent: Monday, August 22, 2005 2:56 PM
To: Formalism, Logic, Institution - Relating, Translating and
Structuring
Cc: lcp@cl.cam.ac.uk
Subject: Re: [Flirts] Curry-Howard for more complex logics

Dear Valeria,

interesting. I found this in my Isabelle 2004 distribution, so it is
still supported (although I found only rather elementary things actually
proved).

It seems to be that this is a version of Curry-Howard that just exploits
"proofs = terms", while "propositions = types" is supported only insofar
as the proof terms are typed with their propositions. Of course, then
you do not have dependent types, but just quantified formulas. But once
you try to interpret quantified formulas as types, then you will see
that they behave like dependent types.

The point seems hence to be that you can avoid complicated type systems
by taking your propositions to immediately be the "types", without
resort to a independently-developed type system.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
> The work of Martin Coen's that I mentioned wasn't his thesis work on 
> CCL (and application to verification):
>  check the Isabelle Library on
>  http://isabelle.in.tum.de/library/index.html
> Under First-Order Logic, the last item is FOLP (FOL with Proof Terms).
> This is the work that I discussed with Martin, and apparently it is 
> still supported, both in the classical and the intuitionistic versions

> (is that so, Larry?).
> 
>  Of  course in the classical version the terms are not Curry-Howard 
> isomorphic to the logic (due to all the problems that we've discussed 
> before), but in the intuitionistic they are.
> 
> Best,
> Valeria
> 
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA



-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till
_______________________________________________
Flirts mailing list
Flirts@mail.informatik.uni-bremen.de
http://www.informatik.uni-bremen.de/mailman/listinfo/flirts

From lp15 at cam.ac.uk  Tue Aug 23 11:26:53 2005
From: lp15 at cam.ac.uk (Lawrence Paulson)
Date: Sun Aug 28 12:40:55 2005
Subject: [Flirts] Curry-Howard for more complex logics
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EEFE@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EEFE@goldeneye.ad.parc.com>
Message-ID: <333C7635-2841-4F96-A1BD-F2E8353DDF85@cam.ac.uk>

The story is that Martin, after finishing his PhD, started work as an  
RA developing his PhD material. The new work was much better (he  
said), as he had a lazy functional language this time. The idea was  
to support interactive refinement and verification rather than  
deriving programs from proofs, and it was to use classical logic.

But then Martin decided he needed a real job, so he left to join a  
software company and last I heard (some time ago) was writing a  
software system for the Turkish stock exchange.

My impression of CCL was that it had a lot of potential but needed  
further development. By keeping it in the Isabelle distribution, I  
was able to ensure that it could still run, but nothing has happened  
to it for 10 years. If anybody can do something with it now, I'd be  
very happy indeed.

Larry


On 22 Aug 2005, at 21:58, <Valeria.dePaiva@parc.com>  
<Valeria.dePaiva@parc.com> wrote:

> The work of Martin Coen's that I mentioned wasn't his thesis work  
> on CCL
> (and application to verification):
>  check the Isabelle Library on
>  http://isabelle.in.tum.de/library/index.html
> Under First-Order Logic, the last item is FOLP (FOL with Proof Terms).
> This is the work that I discussed with Martin, and apparently it is
> still supported, both in the classical and the intuitionistic versions
> (is that so, Larry?).
>
>  Of  course in the classical version the terms are not Curry-Howard
> isomorphic to the logic (due to all the problems that we've discussed
> before), but in the intuitionistic they are.

From till at informatik.uni-bremen.de  Tue Aug  9 19:20:20 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Tue Aug 30 15:43:12 2005
Subject: [Flirts] Re: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE62@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE62@goldeneye.ad.parc.com>
Message-ID: <42F8E5D4.4090604@informatik.uni-bremen.de>

Dear Valeria,

thank you for agreeing to continue this interesting discussion on
the FLIRTS mailing list.

The paper "What is a logic?" (for those who do not have it: 
http://www.tzi.de/~till/papers/nel05.pdf) takes the following
perspective: we wanted to clarify what the "identity" of a logic
is, and when two logics are considered to be essentially equal
(or different). While on the model-theoretic side, we have good notions
in the paper (and also some results), the proof-theoretic side is much
weaker developed. The main perspective of the paper is to
distinguish different proof theories by the different connectives
they allow, and the different behaviour of these connectives
(regardless whether these connectives are actually present in
the logic or not - the connectives are defined only in terms of
the vocabulary of an institution with proofs). To do this, we use some
standard tools from categorical logic. You are right, proof term
normalization is completely ignored. This is mainly because we could not
see how it would help us to identify connectives in a logic independent
way, or otherwise contribute to the "identity" of a logic.

However, we say in the conclusion of the paper:

   There are some further proof-theoretic properties that we have not
   treated, like (strong) normal forms for proofs (this would require
   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
   term reductions).  A related topic is cut elimination, which
   would require an even finer structure on $\Sen(\Sigma)$,
   with proof rules of particular format.
   We hope this essay provides a good starting point for
   such investigations.

This finer structure might be seen as inessential for the logic itself
(e.g. FOL with Gentzen style proofs and FOL with natural deduction
proofs are "essentially the same" from this perspective), but of course
this would not preclude a much more fine-grained "identity of prood
system", based on 2-categories of proofs. Does this meet your point?

I must admit that I do not really know much about normal forms of
proofs, and still need to look in your papers more thoroughly. But
before doing that, I want to discuss some point. What me puzzles me a
bit already with the approach of defining connectives in a
proof-theoretic way within 1-categories is the following
(citing from another mail from me written in connection with the paper):

   Note that arrows in proof categories are proofs up to equivalence.
   And we impose certain conditions on this equivalence.
   A simple example: if we infer A/\B from A/\B by conjunction
   elimination and conjunction introduction, then this proof must
   be equivalent to the proof infering A/\B directly from itself,
   because conjunction is product, and <pi_1,pi_2>=id.
   Basically, for propositional logic, our axioms of proof-theoretic
   institutions say that the category of proofs is bicartesian closed
   (ie cartesian closed + finite coproducts, including inital objects).
   Lambek and Scott, "Introduction to categorical higher-order logic",
   show on p.67, that in any bicartesion closed category, for an object
   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
   the initial object). From this it follows that any classical
   bicartesion closed category (i.e. one with A is isomorphic to
   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
   to a thin category, and hence thin itself.

This means that for classical logic, the approach of identifying
different calculi (say, Gentzen or natural deduction) via (2-)categories
of proofs seems to be hopeless, because everything is the same thin
category, namely essentially the usual Lindenbaum-Tarski algebra.
Or am I missing something?

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> 
> Dear Till,
> 
> I noticed that you actually went ahead and did some  of the work we
> discussed a long time ago
> (see below) on adding "categorical logic" to institutions on a joint
> paper with Diaconescu, Goguen and Tarlecki ("What is a Logic?). I was
> invited speaker at the Universal Logic in Montreux, where Diaconescu
> talked about it. 
> 
> While I did feel a bit miffed that my original suggestion of the problem
> wasn't mentioned at all, my problem with the paper is not that. My
> problem is that your approach seems to be the proverbial "throwing the
> baby away with the bathwater". The point of putting real  proofs (as
> opposed to entailment relations) into institutions was to try to use the
> proofs-as-lambda-term-representations to do some real work for us, ie to
> connect to the paradigm of extracting programs from proofs, or to help
> with abstract analysis or to extend type systems in a principled and
> logical way, etc. i.e. any of the usual applications of categorical
> proof theory would do here. 
> 
> You say in page 2 of your joint paper that your new definition of (proof
> theoretic) institution "fully supports proof theory", but the notion of
> proof theoretic institution (or of equivalence of institutions)
> introduced in the paper has nothing much to do with proof theory as
> people normally know it. What you call "proof theoretic institutions" do
> not overcome the suggested limitation of "categorical logic", because
> proof theoretic institutions do not  model the significant aspect of
> proofs, which is their reduction behavior. 
> 
> The point of the Curry_Howard isomorphism is not that you can model
> propositions as objects in a category and proofs as equivalence classes
> of morphisms: the point is that the behaviour of proofs is preserved
> under this modelling. This is why some people think that the
> Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
> The  crucial point is that if proof pi reduces to pi' via alpha-beta-eta
> reductions than the corresponding morphisms are related in the target
> category (either by equality or reduction). Nothing like that happens in
> the proof-theoretic institutions, which is why they are only
> proof-theoretical in name.
> The functor Pr: Sign -> Cat is only about proofs in its name, which you
> presumably realize, as  it is not even spelled out in the definition on
> (page 125 of the book) that Pr stands for proofs. I guess my main
> complaint is that the paper does not define a "proof-theoretical
> institution" in the sense of an institution that preserves proofs, but
> simply as an institution that preserves entailment. But I guess this is
> all right, people will have different perspectives on what is important,
> mathematically speaking.
> 
> Best regards,
> Valeria
> 
> 
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE] 
> Sent: Friday, January 31, 2003 12:14 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Subject: Re: CHI for Institutions
> 
> Dear Valeria,
> 
> agreed, let's devide the work as you suggested.
> 
> For a definition of (some variant of) parchment you might look at p 5ff.
> of "Combingn and representing logical systems using model-theoretic
> parchments", available at my publications page.
> 
> Concerning the Lisbon work: I think it shouldn't be difficult to have a
> meta notion of sequent calculus, like the meta notion of Hilbert
> calculus.
> 
> Concerning the more complex Curry-Howard isos:
> you seem to have one in the paper with Biermann. Then I'll have a look
> on that, before going on with trying to look at Curry-Howard in the
> institutional framework
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com schrieb:
>>Dear Till,
>>Thanks for the very interesting message. Now I have to do some
> reading, I don't even know what a parchment is...
>>But one small thought: your last paragraph about translating the
> Curry-Howard isomorphism in terms of institutions is very interesting
> and seems a more concrete way of pushing forward towards my goal, which
> is different though. My goal is really enriching the whole framework of
> institutions so that it can cope with proofs (and when I say proofs I
> don't mean in a single proof calculus: I usually want a logic to be
> given in different several proof calculi all proved equivalent, like for
> instance for IPL you can give axioms, sequents or Natural deduction and
> you know how to translate proofs from one calculi to the others). So
> another way of pushing forward would be to see if the Lisbon work you've
> mentioned can be "translated" into sequent calculus, for example.
>>Now about your question: no I don't think I know of any reference for
> the diagram in page 26. The first problem  is  that Oyster, PVS and
> NuPRl are computer systems and first of all we would need papers called
> "The essence of Oyster, PVS and NuPRL", or Alf, but I guess these might
> exist. I just haven't had the time or disposition to look for them. This
> would be a different research project altogether it seems to me.
>>Thus a modest proposal: I will read the stuff you've mentioned and try
> to come back with questions/suggestions. Maybe you could try to add some
> details to the suggestion of looking at Curry-Howard in the
> institutional framework?
>>Cheers,
>>Valeria
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From Valeria.dePaiva at parc.com  Tue Aug  9 23:27:34 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Tue Aug 30 15:43:12 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>

Dear Till,
> This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural 
>deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, 
>namely essentially the usual Lindenbaum-Tarski algebra.
>Or am I missing something?
Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.

If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
http://www.cs.bath.ac.uk/~pym/semclasspro.html

Hope this helps,
Best,
Valeria
Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
http://www.ps.uni-sb.de/~lutz/
Dr Valeria de Paiva
PARC
3333 Coyote Hill Road
Palo Alto, CA 94304
USA

-----Original Message-----
From: Till Mossakowski [mailto:till@informatik.uni-bremen.de] 
Sent: Tuesday, August 09, 2005 10:20 AM
To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
Cc: FLIRTS
Subject: Re: Curry-Howard isomorphism for Institutions

Dear Valeria,

thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.

The paper "What is a logic?" (for those who do not have it: 
http://www.tzi.de/~till/papers/nel05.pdf) takes the following
perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.

However, we say in the conclusion of the paper:

   There are some further proof-theoretic properties that we have not
   treated, like (strong) normal forms for proofs (this would require
   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
   term reductions).  A related topic is cut elimination, which
   would require an even finer structure on $\Sen(\Sigma)$,
   with proof rules of particular format.
   We hope this essay provides a good starting point for
   such investigations.

This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?

I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):

   Note that arrows in proof categories are proofs up to equivalence.
   And we impose certain conditions on this equivalence.
   A simple example: if we infer A/\B from A/\B by conjunction
   elimination and conjunction introduction, then this proof must
   be equivalent to the proof infering A/\B directly from itself,
   because conjunction is product, and <pi_1,pi_2>=id.
   Basically, for propositional logic, our axioms of proof-theoretic
   institutions say that the category of proofs is bicartesian closed
   (ie cartesian closed + finite coproducts, including inital objects).
   Lambek and Scott, "Introduction to categorical higher-order logic",
   show on p.67, that in any bicartesion closed category, for an object
   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
   the initial object). From this it follows that any classical
   bicartesion closed category (i.e. one with A is isomorphic to
   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
   to a thin category, and hence thin itself.

This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
Or am I missing something?

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> 
> Dear Till,
> 
> I noticed that you actually went ahead and did some  of the work we 
> discussed a long time ago (see below) on adding "categorical logic" to 
> institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
> ("What is a Logic?). I was invited speaker at the Universal Logic in 
> Montreux, where Diaconescu talked about it.
> 
> While I did feel a bit miffed that my original suggestion of the 
> problem wasn't mentioned at all, my problem with the paper is not 
> that. My problem is that your approach seems to be the proverbial 
> "throwing the baby away with the bathwater". The point of putting real  
> proofs (as opposed to entailment relations) into institutions was to 
> try to use the proofs-as-lambda-term-representations to do some real 
> work for us, ie to connect to the paradigm of extracting programs from 
> proofs, or to help with abstract analysis or to extend type systems in 
> a principled and logical way, etc. i.e. any of the usual applications 
> of categorical proof theory would do here.
> 
> You say in page 2 of your joint paper that your new definition of 
> (proof
> theoretic) institution "fully supports proof theory", but the notion 
> of proof theoretic institution (or of equivalence of institutions) 
> introduced in the paper has nothing much to do with proof theory as 
> people normally know it. What you call "proof theoretic institutions" 
> do not overcome the suggested limitation of "categorical logic", 
> because proof theoretic institutions do not  model the significant 
> aspect of proofs, which is their reduction behavior.
> 
> The point of the Curry_Howard isomorphism is not that you can model 
> propositions as objects in a category and proofs as equivalence 
> classes of morphisms: the point is that the behaviour of proofs is 
> preserved under this modelling. This is why some people think that the 
> Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
> The  crucial point is that if proof pi reduces to pi' via 
> alpha-beta-eta reductions than the corresponding morphisms are related 
> in the target category (either by equality or reduction). Nothing like 
> that happens in the proof-theoretic institutions, which is why they 
> are only proof-theoretical in name.
> The functor Pr: Sign -> Cat is only about proofs in its name, which 
> you presumably realize, as  it is not even spelled out in the 
> definition on (page 125 of the book) that Pr stands for proofs. I 
> guess my main complaint is that the paper does not define a 
> "proof-theoretical institution" in the sense of an institution that 
> preserves proofs, but simply as an institution that preserves 
> entailment. But I guess this is all right, people will have different 
> perspectives on what is important, mathematically speaking.
> 
> Best regards,
> Valeria
> 
> 
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
> Sent: Friday, January 31, 2003 12:14 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Subject: Re: CHI for Institutions
> 
> Dear Valeria,
> 
> agreed, let's devide the work as you suggested.
> 
> For a definition of (some variant of) parchment you might look at p 5ff.
> of "Combingn and representing logical systems using model-theoretic 
> parchments", available at my publications page.
> 
> Concerning the Lisbon work: I think it shouldn't be difficult to have 
> a meta notion of sequent calculus, like the meta notion of Hilbert 
> calculus.
> 
> Concerning the more complex Curry-Howard isos:
> you seem to have one in the paper with Biermann. Then I'll have a look 
> on that, before going on with trying to look at Curry-Howard in the 
> institutional framework
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com schrieb:
>>Dear Till,
>>Thanks for the very interesting message. Now I have to do some
> reading, I don't even know what a parchment is...
>>But one small thought: your last paragraph about translating the
> Curry-Howard isomorphism in terms of institutions is very interesting 
> and seems a more concrete way of pushing forward towards my goal, 
> which is different though. My goal is really enriching the whole 
> framework of institutions so that it can cope with proofs (and when I 
> say proofs I don't mean in a single proof calculus: I usually want a 
> logic to be given in different several proof calculi all proved 
> equivalent, like for instance for IPL you can give axioms, sequents or 
> Natural deduction and you know how to translate proofs from one 
> calculi to the others). So another way of pushing forward would be to 
> see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>Now about your question: no I don't think I know of any reference for
> the diagram in page 26. The first problem  is  that Oyster, PVS and 
> NuPRl are computer systems and first of all we would need papers 
> called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
> these might exist. I just haven't had the time or disposition to look 
> for them. This would be a different research project altogether it seems to me.
>>Thus a modest proposal: I will read the stuff you've mentioned and try
> to come back with questions/suggestions. Maybe you could try to add 
> some details to the suggestion of looking at Curry-Howard in the 
> institutional framework?
>>Cheers,
>>Valeria
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till


From goguen at cs.ucsd.edu  Wed Aug 10 05:30:38 2005
From: goguen at cs.ucsd.edu (Joseph Goguen)
Date: Tue Aug 30 15:43:12 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
Message-ID: <42F974DE.6050801@cs.ucsd.edu>

Dear Valeria,

Thanks so much for your remarks!  If you have a nicer way of including
proofs into institutions, such that, for example, classical logic works well
by something other than proof terms for sequents, that would be very
welcome and interesting to us.  Ive heard about the work you mention
in various theses but have never looked at it; your summary is a bit on
the discouraging side, i must admit - too many answers, all likely to be
rather complicated, and no clear tradeoffs.

About the origin of institutions with proofs, it goes back to at least 1980
in conversations between Rod Burstall and i; Rod didnt much like the
idea then but i put it in our JACM paper anyway, where it appeared
after a 7 year delay (submitted 1985, but earlier drafts exist, also with
the proof  idea).

About the formulation in "What is a logic?" it seems a bit extreme to
say that it doesnt really do any proof theory; although it is clear that
it is far from doing everything that proof theorists find interesting, it
does allow defining many basic proof theoretic concepts in a nice
abstract way, especially those that connect with models.  Also, as
far as i know, they idea of using sets of sentences as objects is
new and useful, or did we miss something there also?

I find this dialogue very useful and hope that we can continue it
further - i look forward to learning more!

With all best regards,

    joseph

Valeria.dePaiva@parc.com wrote:

>Dear Till,
>  
>
>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural 
>>deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, 
>>namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
>>    
>>
>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>   (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>
>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>
>Hope this helps,
>Best,
>Valeria
>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>http://www.ps.uni-sb.de/~lutz/
>Dr Valeria de Paiva
>PARC
>3333 Coyote Hill Road
>Palo Alto, CA 94304
>USA
>
>-----Original Message-----
>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de] 
>Sent: Tuesday, August 09, 2005 10:20 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Cc: FLIRTS
>Subject: Re: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>
>The paper "What is a logic?" (for those who do not have it: 
>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>
>However, we say in the conclusion of the paper:
>
>   There are some further proof-theoretic properties that we have not
>   treated, like (strong) normal forms for proofs (this would require
>   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>   term reductions).  A related topic is cut elimination, which
>   would require an even finer structure on $\Sen(\Sigma)$,
>   with proof rules of particular format.
>   We hope this essay provides a good starting point for
>   such investigations.
>
>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>
>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>
>   Note that arrows in proof categories are proofs up to equivalence.
>   And we impose certain conditions on this equivalence.
>   A simple example: if we infer A/\B from A/\B by conjunction
>   elimination and conjunction introduction, then this proof must
>   be equivalent to the proof infering A/\B directly from itself,
>   because conjunction is product, and <pi_1,pi_2>=id.
>   Basically, for propositional logic, our axioms of proof-theoretic
>   institutions say that the category of proofs is bicartesian closed
>   (ie cartesian closed + finite coproducts, including inital objects).
>   Lambek and Scott, "Introduction to categorical higher-order logic",
>   show on p.67, that in any bicartesion closed category, for an object
>   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>   the initial object). From this it follows that any classical
>   bicartesion closed category (i.e. one with A is isomorphic to
>   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>   to a thin category, and hence thin itself.
>
>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>Or am I missing something?
>
>Greetings,
>Till
>
>Valeria.dePaiva@parc.com wrote:
>  
>
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real  
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions" 
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>    
>>
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>>      
>>>
>>reading, I don't even know what a parchment is...
>>    
>>
>>>But one small thought: your last paragraph about translating the
>>>      
>>>
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>    
>>
>>>Now about your question: no I don't think I know of any reference for
>>>      
>>>
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>    
>>
>>>Thus a modest proposal: I will read the stuff you've mentioned and try
>>>      
>>>
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>    
>>
>>>Cheers,
>>>Valeria
>>>      
>>>
>
>
>  
>



From till at informatik.uni-bremen.de  Wed Aug 10 11:32:29 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Tue Aug 30 15:43:12 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE6A@goldeneye.ad.parc.com>
Message-ID: <42F9C9AD.4010202@informatik.uni-bremen.de>

Dear Valeria,

many thanks for the pointers to the literature. Actually, my
last intensive study of categorical proof theory dates back
to the late eighties...
Like Joseph I find it a bit discouraging that there are so many
(unrelated?) proof.theoretic approaches. But maybe institutions
can help, let's see...

Let me recall the programme: trying to identify the essential
properties of a logic by only refering to the vocabulary of an
abstract interface, like institutions.
On the model-theoretic side, Razvan Diaconescu has generalized
much of classical model theory to this setting, including
deep results like the fundamental ultraproduct theorem
(see his forthcoming book "Institutional model theory").
This includes characterizations of abstract
logical connecvtives and quantifiers (i.e. not refering
to some particular syntax, but only to the vocabulary
of institutions) through their "model-theoretic behaviour".

On the proof-theoretic side, this is much less explored.
It seems that there are three levels:

1. entailment systems (= kind of pre-orders)
2. categories of sentences and proofs
3. 2-categories of sentences, proofs and proof reductions.

At level 1., all proofs are identified. The achievment of the
1980's proof theory was to identify good categories of
intuitionistic proofs at level 2, with categorical
characterizations of connectives and quantifiers by their
"proof-theoretic behaviour". The problem was that for
classical logic, these categories collapse to thin categories
= pre-orders, such that we are back at level 1. This
might even not be a problem for defining connectives,
but is just too abstract for proof theorists.
Also practically, a tool should be able to output a proof
tree and not just the unique element of a singleton set...

The work you point out is now on level 3. However, not
just 2-categories are defined, but additional categorical
structure for the connectives is introduced. Thus, the
connectives are no longer definable in terms of the
abstract vocabulary (unless this is extended with this
extra categorical structure, which seems awkward).
Maybe a way out is just to have level 3 for the proof
reductions, but define the connectives at level 2
(which works well even with thin categories). Thus, all
the levels would naturally coexist in parallel (noting
that all the necessary information is contained in the
highest level, because there are "quotienting" constructions
going from a higher to a lower level).

Then, for example, the order-enriched categories of F?hrmann and Pym
should naturally form an institution with proofs. Indeed, the
2-cells here are cut-elimination reductions, which fits nicely
with what we had in mind .However, the two-categorical structure only
captures the order-enrichment, while their categories are also linearly
distributive (i.e. kind of bi-monoidal, where the two monoidal
structures model conjunction and disjunction), plus object-wise monoids
and co-monoids (modeling weakening and contraction). This
richer structure is then ignored at the abstract level,
where conjunction is recoverd as product in the category at level 2
(while the category at level 3 might mot even have products:
quotients do not need to reflect them).

Another point is that our proofs work on sets of sentences, rather
than on sentences. This seems to be related to polycategories,
which, however, only use finite sequences of sentences.

And the next question is of course where this general scheme
also fits for other logics, like modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
>>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural 
>>deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, 
>>namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
> Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of researc
h  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
> 
> If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
> http://www.cs.bath.ac.uk/~pym/semclasspro.html
> 
> Hope this helps,
> Best,
> Valeria
> Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
> http://www.ps.uni-sb.de/~lutz/
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@informatik.uni-bremen.de] 
> Sent: Tuesday, August 09, 2005 10:20 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Cc: FLIRTS
> Subject: Re: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
> thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
> 
> The paper "What is a logic?" (for those who do not have it: 
> http://www.tzi.de/~till/papers/nel05.pdf) takes the following
> perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
> 
> However, we say in the conclusion of the paper:
> 
>    There are some further proof-theoretic properties that we have not
>    treated, like (strong) normal forms for proofs (this would require
>    $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>    term reductions).  A related topic is cut elimination, which
>    would require an even finer structure on $\Sen(\Sigma)$,
>    with proof rules of particular format.
>    We hope this essay provides a good starting point for
>    such investigations.
> 
> This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
> 
> I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
> 
>    Note that arrows in proof categories are proofs up to equivalence.
>    And we impose certain conditions on this equivalence.
>    A simple example: if we infer A/\B from A/\B by conjunction
>    elimination and conjunction introduction, then this proof must
>    be equivalent to the proof infering A/\B directly from itself,
>    because conjunction is product, and <pi_1,pi_2>=id.
>    Basically, for propositional logic, our axioms of proof-theoretic
>    institutions say that the category of proofs is bicartesian closed
>    (ie cartesian closed + finite coproducts, including inital objects).
>    Lambek and Scott, "Introduction to categorical higher-order logic",
>    show on p.67, that in any bicartesion closed category, for an object
>    A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>    the initial object). From this it follows that any classical
>    bicartesion closed category (i.e. one with A is isomorphic to
>    (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>    to a thin category, and hence thin itself.
> 
> This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
> Or am I missing something?
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com wrote:
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real  
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions" 
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>reading, I don't even know what a parchment is...
>>>But one small thought: your last paragraph about translating the
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>Now about your question: no I don't think I know of any reference for
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>>Thus a modest proposal: I will read the stuff you've mentioned and try
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>>Cheers,
>>>Valeria
> 
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From Valeria.dePaiva at parc.com  Wed Aug 10 19:15:07 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Tue Aug 30 15:43:12 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>

Dear Till,
>Like Joseph I find it a bit discouraging that there are so many (unrelated?) proof.theoretic approaches.
Well, what I said was that (since I'm no expert on the field) *I* don't know the tradeoffs, maybe one of the guys that is an expert does know the relationships between approaches...

>Let me recall the programme: trying to identify the essential properties of a logic by only refering to the 
>vocabulary of an abstract interface, like institutions.
This is possibly the way you see the programme. For me "to identify the essential properties of a logic" means identifying the essential properties of the *derivations* in this logic. A logic, for me, does not exist without its derivations and proofs.

>On the model-theoretic side, Razvan Diaconescu has generalized much of classical model theory to this setting
This is certainly true.

>It seems that there are three levels:

>1. entailment systems (= kind of pre-orders) 
>2. categories of sentences and proofs 
>3. 2-categories of sentences, proofs and proof reductions.
Not quite. One can and normally does talk about proof reductions using simply categories and morphisms.
You do not need to introduce 2-cells for that.

>The work you point out is now on level 3.
Again not quite. Some of the work I mentioned for *classical* logic is actually at level 2.

Also, of course, there's plenty of work (to be done) on generalized categorical proof-theory of 
*non-classical* logic and relating that to the model-theoretical work on institutions. This is the work I was proposing to do, originally.

>This seems to be related to polycategories, which, however, only use finite sequences of sentences.
Yes, indeed in the kind of proof theory I like, rules mostly have a finite number of hypotheses/assumptions. Girard says somewhere that the infinite is always a potential one, which I think is quite nice.

>And the next question is of course where this general scheme also fits for other logics, like modal logics.
Well, some constructive modal logics fit in already. Others (non-constructive ones) will, once you do your classical logic the way *you* think it seems best. The modularity there is another one of the success criteria, right?

Cheers,
Valeria
-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Till Mossakowski
Sent: Wednesday, August 10, 2005 2:32 AM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

many thanks for the pointers to the literature. Actually, my last intensive study of categorical proof theory dates back to the late eighties...
Like Joseph I find it a bit discouraging that there are so many
(unrelated?) proof.theoretic approaches. But maybe institutions can help, let's see...

Let me recall the programme: trying to identify the essential properties of a logic by only refering to the vocabulary of an abstract interface, like institutions.
On the model-theoretic side, Razvan Diaconescu has generalized much of classical model theory to this setting, including deep results like the fundamental ultraproduct theorem (see his forthcoming book "Institutional model theory").
This includes characterizations of abstract logical connecvtives and quantifiers (i.e. not refering to some particular syntax, but only to the vocabulary of institutions) through their "model-theoretic behaviour".

On the proof-theoretic side, this is much less explored.
It seems that there are three levels:

1. entailment systems (= kind of pre-orders) 2. categories of sentences and proofs 3. 2-categories of sentences, proofs and proof reductions.

At level 1., all proofs are identified. The achievment of the 1980's proof theory was to identify good categories of intuitionistic proofs at level 2, with categorical characterizations of connectives and quantifiers by their "proof-theoretic behaviour". The problem was that for classical logic, these categories collapse to thin categories = pre-orders, such that we are back at level 1. This might even not be a problem for defining connectives, but is just too abstract for proof theorists.
Also practically, a tool should be able to output a proof tree and not just the unique element of a singleton set...

The work you point out is now on level 3. However, not just 2-categories are defined, but additional categorical structure for the connectives is introduced. Thus, the connectives are no longer definable in terms of the abstract vocabulary (unless this is extended with this extra categorical structure, which seems awkward).
Maybe a way out is just to have level 3 for the proof reductions, but define the connectives at level 2 (which works well even with thin categories). Thus, all the levels would naturally coexist in parallel (noting that all the necessary information is contained in the highest level, because there are "quotienting" constructions going from a higher to a lower level).

Then, for example, the order-enriched categories of F?hrmann and Pym should naturally form an institution with proofs. Indeed, the 2-cells here are cut-elimination reductions, which fits nicely with what we had in mind .However, the two-categorical structure only captures the order-enrichment, while their categories are also linearly distributive (i.e. kind of bi-monoidal, where the two monoidal structures model conjunction and disjunction), plus object-wise monoids and co-monoids (modeling weakening and contraction). This richer structure is then ignored at the abstract level, where conjunction is recoverd as product in the category at level 2 (while the category at level 3 might mot even have products:
quotients do not need to reflect them).

Another point is that our proofs work on sets of sentences, rather than on sentences. This seems to be related to polycategories, which, however, only use finite sequences of sentences.

And the next question is of course where this general scheme also fits for other logics, like modal logics.

Greetings,
Till

Valeria.dePaiva@parc.com wrote:
> Dear Till,
>>This means that for classical logic, the approach of identifying 
>>different calculi (say, Gentzen or natural
>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
> Well, you're missing the gigantic amount of work put into this problem 
> since the early 90's with the theses of Griffin, Murthy, Stewart, 
> Harbelin, Urban, Parigot, etc. It is true that classical logic is 
> harder to model categorically than intuitionistic logic and it's true 
> that despite all this work it's not clear (to me at least) what are 
> the trade-offs between different kinds of classical Curry-Howard 
> systems and their categorical semantics. But I guess by now it's 
> clearly understood by most in the community that the old dictum that 
> "classical logic has no categorical semantics" is dead and buried. 
> Which kind of solution you prefer for the problem (Selinger's control 
> cats or fibrations or order-enriched models or even special kinds of  
> polycategories, etc...I'm sure I'm forgetting half the decent 
> solutions, for which I apologize) is up to you. As I said I don't know 
> of a list of trade-offs or cost/benefits analysis of the several 
> solutions. It's not my main area of researc
h  (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
> 
> If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
> http://www.cs.bath.ac.uk/~pym/semclasspro.html
> 
> Hope this helps,
> Best,
> Valeria
> Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
> http://www.ps.uni-sb.de/~lutz/
> Dr Valeria de Paiva
> PARC
> 3333 Coyote Hill Road
> Palo Alto, CA 94304
> USA
> 
> -----Original Message-----
> From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
> Sent: Tuesday, August 09, 2005 10:20 AM
> To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
> Cc: FLIRTS
> Subject: Re: Curry-Howard isomorphism for Institutions
> 
> Dear Valeria,
> 
> thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
> 
> The paper "What is a logic?" (for those who do not have it: 
> http://www.tzi.de/~till/papers/nel05.pdf) takes the following
> perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
> 
> However, we say in the conclusion of the paper:
> 
>    There are some further proof-theoretic properties that we have not
>    treated, like (strong) normal forms for proofs (this would require
>    $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>    term reductions).  A related topic is cut elimination, which
>    would require an even finer structure on $\Sen(\Sigma)$,
>    with proof rules of particular format.
>    We hope this essay provides a good starting point for
>    such investigations.
> 
> This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
> 
> I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
> 
>    Note that arrows in proof categories are proofs up to equivalence.
>    And we impose certain conditions on this equivalence.
>    A simple example: if we infer A/\B from A/\B by conjunction
>    elimination and conjunction introduction, then this proof must
>    be equivalent to the proof infering A/\B directly from itself,
>    because conjunction is product, and <pi_1,pi_2>=id.
>    Basically, for propositional logic, our axioms of proof-theoretic
>    institutions say that the category of proofs is bicartesian closed
>    (ie cartesian closed + finite coproducts, including inital objects).
>    Lambek and Scott, "Introduction to categorical higher-order logic",
>    show on p.67, that in any bicartesion closed category, for an object
>    A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>    the initial object). From this it follows that any classical
>    bicartesion closed category (i.e. one with A is isomorphic to
>    (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>    to a thin category, and hence thin itself.
> 
> This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
> Or am I missing something?
> 
> Greetings,
> Till
> 
> Valeria.dePaiva@parc.com wrote:
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real 
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions"
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>reading, I don't even know what a parchment is...
>>>But one small thought: your last paragraph about translating the
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>>Now about your question: no I don't think I know of any reference for
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>try
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>>Cheers,
>>>Valeria
> 
> 


-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till _______________________________________________
Flirts mailing list
Flirts@mail.informatik.uni-bremen.de
http://www.informatik.uni-bremen.de/mailman/listinfo/flirts


From Valeria.dePaiva at parc.com  Wed Aug 10 18:45:16 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Tue Aug 30 15:43:12 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE6F@goldeneye.ad.parc.com>

Dear Joseph,
Thanks for the friendly message.
First, let me make it clear that, no, I do not have "a nicer way of including proofs into institutions", at the moment. I had proposed to Till that we could investigate this problem together, as I had become interested in the idea around 1999. I'm attaching some slides from a talk I gave at NASA Ames then when I was trying to sell them a work proposal along these lines. I thought this was a cool idea, which you probably agree since you've had the same idea some 15 years before me. But I only worked on that for a couple of weeks, preparing the talk, which is just a proposal for some work. Not the work itself.


>they idea of using sets of sentences as objects is new and useful, 
>or did we miss something there also?
Well, I don't think it is useful, as *when* you can do your categorical modelling properly, sets of sentences are modelled for free, either in the case where there's a connective that internalizes the comma (like a categorical tensor) or using the fibration mechanism. So no, I don't see the usefulness at the moment, it doesn't buy me anything new. Maybe you'd like to expand on that?

About:
>it seems a bit extreme to say that it doesn't really do any proof theory;
I'm afraid I don't think it excessive. I went back to the paper "What is a Logic?" to see if I was forgetting any basic interesting proof-theoretic notion, given that you've said
>it does allow defining many basic proof theoretic concepts in a nice abstract way,
But I don't see these many concepts. Would you like to discuss them one by one?

>I find this dialogue very useful and hope that we can continue it further - i look forward to learning more!
So do I, thanks for the discussion!

Best regards,

Valeria
-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Joseph Goguen
Sent: Tuesday, August 09, 2005 8:31 PM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Cc: till@informatik.uni-bremen.de; de Paiva, Valeria <Valeria.dePaiva@parc.com>
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

Dear Valeria,

Thanks so much for your remarks!  If you have a nicer way of including proofs into institutions, such that, for example, classical logic works well by something other than proof terms for sequents, that would be very welcome and interesting to us.  Ive heard about the work you mention in various theses but have never looked at it; your summary is a bit on the discouraging side, i must admit - too many answers, all likely to be rather complicated, and no clear tradeoffs.

About the origin of institutions with proofs, it goes back to at least 1980 in conversations between Rod Burstall and i; Rod didnt much like the idea then but i put it in our JACM paper anyway, where it appeared after a 7 year delay (submitted 1985, but earlier drafts exist, also with the proof  idea).

About the formulation in "What is a logic?" it seems a bit extreme to say that it doesnt really do any proof theory; although it is clear that it is far from doing everything that proof theorists find interesting, it does allow defining many basic proof theoretic concepts in a nice abstract way, especially those that connect with models.  Also, as far as i know, they idea of using sets of sentences as objects is new and useful, or did we miss something there also?


With all best regards,

    joseph

Valeria.dePaiva@parc.com wrote:

>Dear Till,
>  
>
>>This means that for classical logic, the approach of identifying 
>>different calculi (say, Gentzen or natural
>>deduction) via (2-)categories of proofs seems to be hopeless, because 
>>everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>>Or am I missing something?
>>    
>>
>Well, you're missing the gigantic amount of work put into this problem since the early 90's with the theses of Griffin, Murthy, Stewart, Harbelin, Urban, Parigot, etc. It is true that classical logic is harder to model categorically than intuitionistic logic and it's true that despite all this work it's not clear (to me at least) what are the trade-offs between different kinds of classical Curry-Howard systems and their categorical semantics. But I guess by now it's clearly understood by most in the community that the old dictum that "classical logic has no categorical semantics" is dead and buried. Which kind of solution you prefer for the problem (Selinger's control cats or fibrations or order-enriched models or even special kinds of  polycategories, etc...I'm sure I'm forgetting half the decent solutions, for which I apologize) is up to you. As I said I don't know of a list of trade-offs or cost/benefits analysis of the several solutions. It's not my main area of research!
>   (I like my logic intuitionistic, by and large), so if you do happen to be able to compile such a list, I'd be very interested in seeing it.
>
>If you'd like more information you could check David Pym's page on Semantics of Classical Proofs, which is mostly devoted to his own solution with Carsten F?hrmann and others, but should, I hope, give pointers to other work:
>http://www.cs.bath.ac.uk/~pym/semclasspro.html
>
>Hope this helps,
>Best,
>Valeria
>Ps I was forgetting the work of Lutz Strassburger and Francois Lamarche, which I shouldn't, check it at:
>http://www.ps.uni-sb.de/~lutz/
>Dr Valeria de Paiva
>PARC
>3333 Coyote Hill Road
>Palo Alto, CA 94304
>USA
>
>-----Original Message-----
>From: Till Mossakowski [mailto:till@informatik.uni-bremen.de]
>Sent: Tuesday, August 09, 2005 10:20 AM
>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>Cc: FLIRTS
>Subject: Re: Curry-Howard isomorphism for Institutions
>
>Dear Valeria,
>
>thank you for agreeing to continue this interesting discussion on the FLIRTS mailing list.
>
>The paper "What is a logic?" (for those who do not have it: 
>http://www.tzi.de/~till/papers/nel05.pdf) takes the following
>perspective: we wanted to clarify what the "identity" of a logic is, and when two logics are considered to be essentially equal (or different). While on the model-theoretic side, we have good notions in the paper (and also some results), the proof-theoretic side is much weaker developed. The main perspective of the paper is to distinguish different proof theories by the different connectives they allow, and the different behaviour of these connectives (regardless whether these connectives are actually present in the logic or not - the connectives are defined only in terms of the vocabulary of an institution with proofs). To do this, we use some standard tools from categorical logic. You are right, proof term normalization is completely ignored. This is mainly because we could not see how it would help us to identify connectives in a logic independent way, or otherwise contribute to the "identity" of a logic.
>
>However, we say in the conclusion of the paper:
>
>   There are some further proof-theoretic properties that we have not
>   treated, like (strong) normal forms for proofs (this would require
>   $Sen(\Sigma)$ to become 2-category of sentences, proof terms and proof
>   term reductions).  A related topic is cut elimination, which
>   would require an even finer structure on $\Sen(\Sigma)$,
>   with proof rules of particular format.
>   We hope this essay provides a good starting point for
>   such investigations.
>
>This finer structure might be seen as inessential for the logic itself (e.g. FOL with Gentzen style proofs and FOL with natural deduction proofs are "essentially the same" from this perspective), but of course this would not preclude a much more fine-grained "identity of prood system", based on 2-categories of proofs. Does this meet your point?
>
>I must admit that I do not really know much about normal forms of proofs, and still need to look in your papers more thoroughly. But before doing that, I want to discuss some point. What me puzzles me a bit already with the approach of defining connectives in a proof-theoretic way within 1-categories is the following (citing from another mail from me written in connection with the paper):
>
>   Note that arrows in proof categories are proofs up to equivalence.
>   And we impose certain conditions on this equivalence.
>   A simple example: if we infer A/\B from A/\B by conjunction
>   elimination and conjunction introduction, then this proof must
>   be equivalent to the proof infering A/\B directly from itself,
>   because conjunction is product, and <pi_1,pi_2>=id.
>   Basically, for propositional logic, our axioms of proof-theoretic
>   institutions say that the category of proofs is bicartesian closed
>   (ie cartesian closed + finite coproducts, including inital objects).
>   Lambek and Scott, "Introduction to categorical higher-order logic",
>   show on p.67, that in any bicartesion closed category, for an object
>   A, either Hom(A,0) is empty, or else A is isomorphic to 0 (where 0 is
>   the initial object). From this it follows that any classical
>   bicartesion closed category (i.e. one with A is isomorphic to
>   (A=>0)=>0 ) is equivalent to a Boolean algebra, i.e. equivalent
>   to a thin category, and hence thin itself.
>
>This means that for classical logic, the approach of identifying different calculi (say, Gentzen or natural deduction) via (2-)categories of proofs seems to be hopeless, because everything is the same thin category, namely essentially the usual Lindenbaum-Tarski algebra.
>Or am I missing something?
>
>Greetings,
>Till
>
>Valeria.dePaiva@parc.com wrote:
>  
>
>>Dear Till,
>>
>>I noticed that you actually went ahead and did some  of the work we 
>>discussed a long time ago (see below) on adding "categorical logic" to 
>>institutions on a joint paper with Diaconescu, Goguen and Tarlecki 
>>("What is a Logic?). I was invited speaker at the Universal Logic in 
>>Montreux, where Diaconescu talked about it.
>>
>>While I did feel a bit miffed that my original suggestion of the 
>>problem wasn't mentioned at all, my problem with the paper is not 
>>that. My problem is that your approach seems to be the proverbial 
>>"throwing the baby away with the bathwater". The point of putting real 
>>proofs (as opposed to entailment relations) into institutions was to 
>>try to use the proofs-as-lambda-term-representations to do some real 
>>work for us, ie to connect to the paradigm of extracting programs from 
>>proofs, or to help with abstract analysis or to extend type systems in 
>>a principled and logical way, etc. i.e. any of the usual applications 
>>of categorical proof theory would do here.
>>
>>You say in page 2 of your joint paper that your new definition of 
>>(proof
>>theoretic) institution "fully supports proof theory", but the notion 
>>of proof theoretic institution (or of equivalence of institutions) 
>>introduced in the paper has nothing much to do with proof theory as 
>>people normally know it. What you call "proof theoretic institutions"
>>do not overcome the suggested limitation of "categorical logic", 
>>because proof theoretic institutions do not  model the significant 
>>aspect of proofs, which is their reduction behavior.
>>
>>The point of the Curry_Howard isomorphism is not that you can model 
>>propositions as objects in a category and proofs as equivalence 
>>classes of morphisms: the point is that the behaviour of proofs is 
>>preserved under this modelling. This is why some people think that the 
>>Curry-Howard isomorphism should be called Curry-Howard-Tait isomorphism.
>>The  crucial point is that if proof pi reduces to pi' via 
>>alpha-beta-eta reductions than the corresponding morphisms are related 
>>in the target category (either by equality or reduction). Nothing like 
>>that happens in the proof-theoretic institutions, which is why they 
>>are only proof-theoretical in name.
>>The functor Pr: Sign -> Cat is only about proofs in its name, which 
>>you presumably realize, as  it is not even spelled out in the 
>>definition on (page 125 of the book) that Pr stands for proofs. I 
>>guess my main complaint is that the paper does not define a 
>>"proof-theoretical institution" in the sense of an institution that 
>>preserves proofs, but simply as an institution that preserves 
>>entailment. But I guess this is all right, people will have different 
>>perspectives on what is important, mathematically speaking.
>>
>>Best regards,
>>Valeria
>>
>>
>>
>>-----Original Message-----
>>From: Till Mossakowski [mailto:till@Informatik.Uni-Bremen.DE]
>>Sent: Friday, January 31, 2003 12:14 AM
>>To: de Paiva, Valeria <Valeria.dePaiva@parc.com>
>>Subject: Re: CHI for Institutions
>>
>>Dear Valeria,
>>
>>agreed, let's devide the work as you suggested.
>>
>>For a definition of (some variant of) parchment you might look at p 5ff.
>>of "Combingn and representing logical systems using model-theoretic 
>>parchments", available at my publications page.
>>
>>Concerning the Lisbon work: I think it shouldn't be difficult to have 
>>a meta notion of sequent calculus, like the meta notion of Hilbert 
>>calculus.
>>
>>Concerning the more complex Curry-Howard isos:
>>you seem to have one in the paper with Biermann. Then I'll have a look 
>>on that, before going on with trying to look at Curry-Howard in the 
>>institutional framework
>>
>>Greetings,
>>Till
>>
>>Valeria.dePaiva@parc.com schrieb:
>>    
>>
>>>Dear Till,
>>>Thanks for the very interesting message. Now I have to do some
>>>      
>>>
>>reading, I don't even know what a parchment is...
>>    
>>
>>>But one small thought: your last paragraph about translating the
>>>      
>>>
>>Curry-Howard isomorphism in terms of institutions is very interesting 
>>and seems a more concrete way of pushing forward towards my goal, 
>>which is different though. My goal is really enriching the whole 
>>framework of institutions so that it can cope with proofs (and when I 
>>say proofs I don't mean in a single proof calculus: I usually want a 
>>logic to be given in different several proof calculi all proved 
>>equivalent, like for instance for IPL you can give axioms, sequents or 
>>Natural deduction and you know how to translate proofs from one 
>>calculi to the others). So another way of pushing forward would be to 
>>see if the Lisbon work you've mentioned can be "translated" into sequent calculus, for example.
>>    
>>
>>>Now about your question: no I don't think I know of any reference for
>>>      
>>>
>>the diagram in page 26. The first problem  is  that Oyster, PVS and 
>>NuPRl are computer systems and first of all we would need papers 
>>called "The essence of Oyster, PVS and NuPRL", or Alf, but I guess 
>>these might exist. I just haven't had the time or disposition to look 
>>for them. This would be a different research project altogether it seems to me.
>>    
>>
>>>Thus a modest proposal: I will read the stuff you've mentioned and 
>>>try
>>>      
>>>
>>to come back with questions/suggestions. Maybe you could try to add 
>>some details to the suggestion of looking at Curry-Howard in the 
>>institutional framework?
>>    
>>
>>>Cheers,
>>>Valeria
>>>      
>>>
>
>
>  
>


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From till at informatik.uni-bremen.de  Wed Aug 10 21:30:32 2005
From: till at informatik.uni-bremen.de (Till Mossakowski)
Date: Tue Aug 30 15:43:12 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
Message-ID: <42FA55D8.9080805@informatik.uni-bremen.de>

Dear Valeria,

 >>they idea of using sets of sentences as objects is new and useful,
 >>or did we miss something there also?
 > Well, I don't think it is useful, as *when* you can do your
 > categorical modelling properly, sets of sentences are modelled for
 > free, either in the case where there's a connective that internalizes
 > the comma (like a categorical tensor) or using the fibration
 > mechanism. So no, I don't see the usefulness at the moment, it doesn't
 > buy me anything new. Maybe you'd like to expand on that?

For properties like compactness, you need possibly infinite set of
sentences. And many logics do not have infinitary conjunction,
hence you cannot internalize. If then the logic happens to have
infinitary proof rules, I cannot see how to model this by fibrations.
But may be I am missing something again :-).

 >>it does allow defining many basic proof theoretic concepts in a nice 
abstract way,
 > But I don't see these many concepts. Would you like to discuss them 
one by one?

One concept is having not not-elimination, which separates classical
from intuitionistic logic.

> A logic, for me, does not exist without its derivations and proofs.
That's interesting. Strassburger in is paper "What is a logic
and what is a proof" starts with "a logic is a pre-order",
i.e. he omits proofs. Only later proofs come in. And I agree:
there should be a notion of logic with proofs, but a notion
of logic without proofs is useful as well.

>>It seems that there are three levels:
> 
>>1. entailment systems (= kind of pre-orders) 
>>2. categories of sentences and proofs 
>>3. 2-categories of sentences, proofs and proof reductions.
> Not quite. One can and normally does talk about proof reductions using simply categories and morphisms.
> You do not need to introduce 2-cells for that.
I do not understand.
I thought formulas = objects, proofs = 1-cells. So what are
proof reductions other than 2-cells? OK, they might just be
a pre-order on 1-cells. But this just amounts to having
thin Hom-categories.

>>This seems to be related to polycategories, which, however, only use finite sequences of sentences.
> Yes, indeed in the kind of proof theory I like, rules mostly have a finite number of hypotheses/assumptions. Girard says somewhere that the infinite is always a potential one, which I think is quite nice.
I definitely want to include infinite sets of sentences as well.
The whole notion of compactness as studied in traditional logic
only makes sense with infinite sets of sentences.
And ZFC, which is widely used in mathematics, is not finitely
axiomatizable. Neither is first-order Peano arithmetic.
Infinitary rules are less important, but it would be nice not
to exclude them, because they are used occasionally.
Moreover, the framework of institutions with proofs should be
philosophically neutral, and should admit the study of logics
that might be rejected by some people, but not by all.

>  The modularity there is another one of the success criteria, right?
Yes, certainly.

Greetings,
Till

-- 
Till Mossakowski               Phone +49-421-218-4683
Dept. of Computer Science      Fax +49-421-218-3054
University of Bremen           till@tzi.de
P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till

From jose at fiadeiro.org  Thu Aug 11 00:50:52 2005
From: jose at fiadeiro.org (=?ISO-8859-1?Q?Jos=E9_Luiz_Fiadeiro?=)
Date: Tue Aug 30 15:43:12 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
In-Reply-To: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
References: <1F0E426765530348BB55EB9575AEEBE514EE70@goldeneye.ad.parc.com>
Message-ID: <58C13D61-A4E4-4FA1-8444-6AFA8D7AE272@fiadeiro.org>

My two cents...

On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com>  
<Valeria.dePaiva@parc.com> wrote:

>> Let me recall the programme: trying to identify the essential  
>> properties of a logic by only refering to the
>> vocabulary of an abstract interface, like institutions.
>>
> This is possibly the way you see the programme. For me "to identify  
> the essential properties of a logic" means identifying the  
> essential properties of the *derivations* in this logic. A logic,  
> for me, does not exist without its derivations and proofs.


In my opinion, it does not make sense to refer to THE essential  
properties of a logic: it all depends on what you want to do with the  
logic.  Institutions do capture essential properties of a logic in so  
far as the use of a logic for "algebraic specification" is  
concerned.  But one cannot reduce the notion of logic to this  
particular usage.  In fact, even within a generalised notion of  
"algebraic specification", we have found that the structural  
properties of institutions that restrict morphisms to property- 
preserving relationships prevents us from capturing composition as it  
arises in non-deterministic systems.  The so-called "satisfaction  
condition" has also proved to be too restrictive for formalisms that  
work just on subclasses of models that satisfy some closure conditions.

On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
OBJECTS, the structure of logics is in the derivations/proofs.   
However, the "FLIRTS programme" is, as far as I understand, directed  
to Logic as applied to Specification Theory.  In this respect, it is  
not the internal structure of logics that is of interest, but that of  
their applications, namely the structure of specifications.  In other  
words, the ambition of FLIRTS should not be to do Mathematics (or  
Logic with 'L') but Specification Theory (i.e. given usages of logics).

Having said this, I find that it is definitely worth exploring all  
the different constructions that have been mentioned in previous  
messages from the point of view of the applications to Specification  
Theory.  However, I would refrain from trying to go beyond that.


Regards

Jos?



JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Science
University of Leicester
Leicester LE1 7RH
United Kingdom
Tel: +44 116 252 3907
Fax: +44 116 252 3915
http://www.cs.le.ac.uk/




From Valeria.dePaiva at parc.com  Thu Aug 11 01:00:37 2005
From: Valeria.dePaiva at parc.com (Valeria.dePaiva@parc.com)
Date: Tue Aug 30 15:43:12 2005
Subject: [Flirts] RE: Curry-Howard isomorphism for Institutions
Message-ID: <1F0E426765530348BB55EB9575AEEBE514EE75@goldeneye.ad.parc.com>

 
Hi,
Of course I agree with Jose that
>it does not make sense to refer to THE essential properties of a logic: 
>it all depends on what you want to do with the logic.

But maybe I missed one referent in this discussion: I thought that Till was referring to the programme
Of adding proofs to institutions in the restricted setting of a proposed research paper that 
we were trying to write. I'm afraid I didn't even know about the FLIRTS programme.
So I read it all in the very restricted setting above.

>In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') 
>but Specification Theory (i.e. given usages of logics).
I know very little about Specification Theory and wouldn't want anyone reading this to think that
I was telling them how to do their job.

Thanks, Jose'.
Best,
Valeria

-----Original Message-----
From: flirts-bounces@informatik.uni-bremen.de [mailto:flirts-bounces@informatik.uni-bremen.de] On Behalf Of Jos? Luiz Fiadeiro
Sent: Wednesday, August 10, 2005 3:51 PM
To: Formalism, Logic, Institution - Relating, Translating and Structuring
Subject: Re: [Flirts] RE: Curry-Howard isomorphism for Institutions

My two cents...

On 10 Aug 2005, at 18:15, <Valeria.dePaiva@parc.com> <Valeria.dePaiva@parc.com> wrote:

>> Let me recall the programme: trying to identify the essential 
>> properties of a logic by only refering to the vocabulary of an 
>> abstract interface, like institutions.
>>
> This is possibly the way you see the programme. For me "to identify 
> the essential properties of a logic" means identifying the essential 
> properties of the *derivations* in this logic. A logic, for me, does 
> not exist without its derivations and proofs.


In my opinion, it does not make sense to refer to THE essential properties of a logic: it all depends on what you want to do with the logic.  Institutions do capture essential properties of a logic in so far as the use of a logic for "algebraic specification" is concerned.  But one cannot reduce the notion of logic to this particular usage.  In fact, even within a generalised notion of "algebraic specification", we have found that the structural properties of institutions that restrict morphisms to property- preserving relationships prevents us from capturing composition as it arises in non-deterministic systems.  The so-called "satisfaction condition" has also proved to be too restrictive for formalisms that work just on subclasses of models that satisfy some closure conditions.

On the other hand, I share Val?ria's view in that, AS MATHEMATICAL  
OBJECTS, the structure of logics is in the derivations/proofs.   
However, the "FLIRTS programme" is, as far as I understand, directed to Logic as applied to Specification Theory.  In this respect, it is not the internal structure of logics that is of interest, but that of their applications, namely the structure of specifications.  In other words, the ambition of FLIRTS should not be to do Mathematics (or Logic with 'L') but Specification Theory (i.e. given usages of logics).

Having said this, I find that it is definitely worth exploring all the different constructions that have been mentioned in previous messages from the point of view of the applications to Specification Theory.  However, I would refrain from trying to go beyond that.


Regards

Jos?



JOSE LUIZ FIADEIRO
Professor of Software Science and Engineering

http://www.fiadeiro.org/jose
Mob: +44 779 124 7816
Skype: jfiadeiro

Department of Computer Scie