Comments on: Do NOT use this idea to prove P != NP

By: Kurt

Kurt — Fri, 25 Jan 2008 14:16:13 +0000

Hi ambrosiac,Unfortunately, I never did get around to pursuing this idea. In my post I mentioned that I was struck by the fact that there is no explicit time bound in the computational model. However, as I’m sure you’ve noticed if you’ve worked through the problem in Sipser, the time bound is hidden only slightly below the surface. The k heads moving back in forth on the length-n input basically amount to a counter for n^k.However, the model is still interesting in that in some sense it strips computation in P down to a very elementary level. Whether that’s useful for anything, I’m not sure. It possibly might provide a way around relativization for some proof techniques, although I haven’t really thought that through.

By: ambrosiac

ambrosiac — Fri, 25 Jan 2008 05:20:28 +0000

Hi Kurt. The k-DPDA stuff is interesting. No, I don’t have access to Cook’s papers, so I had to prove it equals P. Always a silver lining ! Has anything been done with this idea , as you indeed suggested?Thank you!

By: twidjaja

twidjaja — Sat, 26 Nov 2005 16:51:30 +0000

Hi Kurt, sorry for my belated reply.It’s interesting that deterministic and non-deterministic PDAs coincides for two-way multi-head PDAs.In this case, I’m not so sure how promising this approach is (as any other approaches to P vs. NP). It’s always hard to say for sure when it comes to separating complexity classes Remember that it is still possible, however unlikely, that the two coincide Let me mention one other thing. Taitslin’s “proof” that NL != P has a gap, which he admits in the Russian version of the paper. But it does contain some good ideas, some of which Stephen Cook has independently thought of (and even published) in the 70s.

By: Kurt

Kurt — Tue, 22 Nov 2005 05:04:43 +0000

Anthony,Surprisingly, it turns out that for two-way multihead PDAs, the deterministic and non-deterministic classes are the same.Cook also describes two-way multihead <>stack<> automata. These differ from the pushdown automata in that they can read up and down the stack without having to pop off anything. For stack automata, the non-deterministic class is larger than the deterministic class.(I had to go back and edit my post a tiny bit, because I had inadvertently used the word ’stack’ when I defined a k-PDA, and I want to avoid any possible confusion.)Thanks for the link, I’ll take a look at that when I have a chance.

By: twidjaja

twidjaja — Tue, 22 Nov 2005 02:36:46 +0000

Addendum: you can trust the author of that paper (Frank Neven). He is well-known in finite model theory and database theory.

By: twidjaja

twidjaja — Tue, 22 Nov 2005 02:34:26 +0000

Kurt: Is it possible thatNP = the class of languages recognizable by a nondeterministic k-PDA for some k?This might be an interesting result if this is true.I know, at least, that nondeterministic 1-PDAs are strictly more powerful than deterministic 1-PDAs. In general, I’m not so sure.Here is something else you might find interesting: characterizing P, LOGSPACE, PSPACE, and EXPTIME using deterministic tree-walking automata plus some extra resources. See http://alpha.luc.ac.be/%7Elucg5503/csl2002.ps (end of section 6). BTW: this is actually a survey paper for foundations of XML researchers like myself