Comments on: Eliminate Cut Elimination

By: Carnival of Math #18 « JD2718

Carnival of Math #18 « JD2718 — Sat, 06 Oct 2007 16:00:21 +0000

[...] Mathematica Eliminate Cut Elimination Theorems “Is there any major theorem in mathematics drier than Gentzen’s cut elimination [...]

By: JaLisa

JaLisa — Wed, 03 Oct 2007 16:43:23 +0000

I have never used any of the techniques listed but they all sound like good ideas that I will use in my geometry class. They sound so complex but I will get around to it.

By: sigfpe

sigfpe — Wed, 03 Oct 2007 00:16:56 +0000

walt,

Did you read my rant about cut elimination? http://sigfpe.blogspot.com/2006/02/cut-rule-and-reading-seminars.html

I found the responses I got very helpful.

By: Todd Trimble

Todd Trimble — Sat, 29 Sep 2007 14:30:39 +0000

Yes, I too found Gentzen sequent calculus and cut elimination a revelation. My own involvement in this area was through its applications to categorical coherence problems, where sequent deductions can be used to present the morphisms in freely generated categories-with-structure (e.g., free cartesian closed categories or free typed lambda calculi, free symmetric monoidal closed categories, free *-autonomous categories [cf. linear logic]). The basic insight, due to Lambek, is that cut elimination is the sine qua non for developing algorithms that decide equality in such free structures (”word problems” for categorical structures, aka categorical coherence problems).

In fact, this was precisely the subject of my PhD dissertation!

[Note: this comment was submitted a couple of days ago awaiting moderation, because evidently reCaptcha caught me making a mistake in those two words I'm supposed to type below. I took the liberty of making a few edits.]

By: Marc Hamann

Marc Hamann — Thu, 27 Sep 2007 23:41:26 +0000

Maybe I’m just sick, but I was really excited by the sequent calculus when I first learned it. The idea that you could nail down a proof system for logic in such a small number of (once you get the notation) simple rules was a revelation.

The (meta)proofs do tend to get long and tedious sometimes, since they tend to proceed by cases, and you usually get the idea on the first case, but someone had to show that it works for the other cases too. If I’m confident I’ve been convinced, I usually skip over some detail, except for the odd edge case or tricky bit.

If you didn’t enjoy Gentzen, I’d recommend you don’t read Prawitz.

By: hellblazer

hellblazer — Thu, 27 Sep 2007 22:30:20 +0000

How about the Caratheodory extension theorem (in beginners’ measure thory)?

Admittedly I haven’t tried to look at it for over five years now, it might have developed more flavour since… It always seemed to me one of those results you were glad was true, but hoped no one would ask you to prove (or enthuse about).

By: Walt

Walt — Thu, 27 Sep 2007 22:00:21 +0000

I find descriptions of the sequent calculus also dry. Once you have that, the statement of cut elimination is short. I don’t find it to the point because the reason someone would care is not really evident. Pretty much I rely entirely on the analogy with inlining functions to see what it means.

By: Peter

Peter — Thu, 27 Sep 2007 17:58:53 +0000

In the wikipedia entry for Cut-elimination-theorem we read that the cut rule is:

“(1) (A, B, . . .) |- C

and

(2) C |- (D, E, . . .)

allows one to infer

(3) (A, B, . . .) |- (D, E, . . .) ”

In fact, the cut rule is more general than this, since the unstated premisses in (1) (denoted by “. . .”) do not need to be identical to the unstated consequences in (2) (denoted also by “. . .”).

By: Walt

Walt — Thu, 27 Sep 2007 15:49:56 +0000

Sorry about that, Joseph. Sigfpe was right: one of my URLs was malformed, which messed everything else up.

By: sigfpe

sigfpe — Thu, 27 Sep 2007 14:27:29 +0000

I’m confused. Cut elimination has a very simple statement that’s short, simple and to the point. The proof might be on the dry side. I’ve not bothered looking at it as I can imagine it’s essentially the same as ‘inlining’ functions in a compiler and that’s a boringly mechanical process. But the statement seems pretty snappy to me.