Subject: Is cut semantical?

From: categories <[email-masked]>
Date: Mon, 25 Aug 1997 11:08:44 -0300 (ADT)

Date: Sun, 24 Aug 1997 08:10:45 -0700
From: Vaughan R. Pratt <[email-masked]>

Cut-free proofs correspond more or less to dinatural transformations,
suitably strengthened to require invariance under logical relations
when necessary.  Is there a semantical counterpart in this sense to
proofs with cut?  Or is the chaotic nature of cut incompatible with
this sort of syntax-semantics correspondence?

Vaughan Pratt

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From: categories <[email-masked]>
Date: Mon, 25 Aug 1997 16:47:28 -0300 (ADT)

Date: Mon, 25 Aug 1997 08:46:46 -0700
From: Vaughan R. Pratt <[email-masked]>

Just to clarify, my question was about proofs with cut, not about cut
itself, whose semantics is perfectly clear.  (I received a couple of
replies from people who merely answered the question in the subject
line, apparently without reading the message.)

The difficulty is that the standard semantics of cut as composition is
not injective, in that it erases the information as to the location of
the cut, as a nasty side effect of associativity.  I therefore do not
see how to attach semantical significance to proofs with cut, as
opposed to their cut-free counterparts where the situation is much
clearer.

This issue arises not just for the denotational semantics of proof but
its operational semantics as well.  In theorem-proving, proofs with cut
are in principle attractive because they can be quite short in
comparison to their cut-free counterparts.  In practice the difficulty
when backchaining of deciding where to put cuts, and what formula to
cut on, seems to make it much harder to find proofs with cut than
without.  Good decision methods are much easier to find for cut-free
logics.

Granted that making algorithmic sense of cut (in the context of a
proof) is hard enough, can one even make semantic sense of it *in that
context*?  That is, is there any bijection between proofs with cut and
some reasonable class of mathematical objects?

Presumably the class will not be closed under composition, at least not
of the associative kind.

Far from being the answer to my problem, composition is its root.

Vaughan Pratt

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From: categories <[email-masked]>
Date: Tue, 26 Aug 1997 12:17:06 -0300 (ADT)

Date: Tue, 26 Aug 1997 09:19:07 -0400
From: Michael Barr <[email-masked]>

Maybe I am being naive, but I always thought that cuts were equivalent
to composing arrows.  The fact that in getting an arrow A --> B, you
could take arbitrary paths A --> C --> B corresponds to the unbounded
nature of proofs with cut.  Of course, if you can calculate Hom(A,B)
directly, this is all irrelevant.

Mike

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From: categories <[email-masked]>
Date: Tue, 26 Aug 1997 12:18:12 -0300 (ADT)

Date: Tue, 26 Aug 1997 09:41:49 -0400
From: Michael Barr <[email-masked]>

I should clarify.  Proofs with cuts correspond to composable paths.
If you like, to proofs in the graph (or free category) of a category.

Michael

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From: categories <[email-masked]>
Date: Wed, 27 Aug 1997 10:31:46 -0300 (ADT)

Date: Tue, 26 Aug 1997 09:25:31 -0700
From: Vaughan R. Pratt <[email-masked]>

Yes, the free thing on the syntax you're trying to model usually does
the job, e.g. the Lindenbaum algebra.  But is there any mathematical
object arising in nature that works like proofs with cut, the way
natural transformations do for cut-free proofs?  It is somewhat magical
that natural transformations match proofs in this way when they do, and
reassuring when they don't: they're letting you know they're alive and
require maintenance, unlike free things which are dead and therefore
maintenance-free.

Vaughan