P vs NP: What I've learnt so far...

(ed: note. it is somewhat embarrassing to be categorized as someone "trying to unravel what is up with the claimed proof". All I've really helped with is getting the wiki set up, and I'm very impressed with the amount of work that everyone is putting into keeping it current)

Whether or not Deolalikar's proof turns out to work, I have to say that in a span of three days, I've learnt a heck of a lot about the individual components in his paper.

• Like everyone else, I learnt my descriptive complexity back in the day, and marvelled at the fact that P, NP and PSPACE could be characterized syntactically. The discussions about the subtleties of the order relation in LFP have been fascinating though, and while the issue remains unresolved, there's a wealth of new(er) material that I've now been exposed to. 
• Similarly, while I was familiar with the idea that hardness could be captured as a phase transition in random k-SAT, the bizarreness of the landscape (clustering! freezing! condensation!) was quite new to me, and it's been illuminating to learn from the experts.
• The relationship between the solution space for k-SAT and error-correcting codes is also something that I was not aware of. And it might be the source of yet another issue with the paper. 
• A point made by Terry Tao is even more intriguing:
  
  

  If nothing else, this whole experience has highlighted a “philosophical” barrier to P != NP which is distinct from the three “hard” barriers of relativisation, natural proofs, and algebraisation, namely the difficulty in using average-case (or “global”) behaviour to separate worst-case complexity, due to the existence of basic problems (e.g. k-SAT and k-XORSAT) which are expected to have similar average case behaviour in many ways, but completely different worst case behaviour. (I guess this difficulty was well known to the experts, but it is probably good to make it more explicit.)

There's been a bit of back-and-forth on whether "too much time" is being spent on perusing this paper, and of course people are entitled to their own opinions on this matter.

My take is slightly different. While none of the "revelations" listed above are new to people in the relevant areas, they're certainly new to me, and given how unlikely it is that I'd be able to pigeon-hole the relevant experts one-on-one to explain these ideas to me, the discussion is serving an important purpose. In effect, it has created a virtual workshop with even more discussion than a real one !

We've spent a lot of time over the past few years bemoaning the fragmentation of theoretical computer science, and have spent many blog-hours arguing over how to make STOC a destination conference for theoreticians again. But what the discussion over this paper has done is bring together experts in very different areas to talk with each other and exchange ideas, results and the crucial "mental frameworks" that are usually impossible to share except in oral discussions.

And that, in my view, is a net plus regardless of what happens with the proof.

A 'polymath' home for analysis of the Deolalikar proof

In collaboration with the Polymath project (and many thanks to Terry Tao for making this happen, and putting in lots of editing work in short order), there's now a new wiki to collect various threads of the discussion on the Deolalikar P != NP paper (which has been updated). With a lot of help from Terry, I've seeded the wiki with the initial discussion points that have come up. Please continue to edit this page as new ideas come up, or there's other discussion of the paper. 



This wiki replaces the Google doc that I had originally set up: it's a lot easier to edit, and Google was struggling under the weight of all the people trying to view the file.

The Lipton/Regan posts will continue to host the main research threads. If you are so inclined, you can use this post for "meta"-discussions as is customary in polymath land. Content will be transferred over to the wiki periodically, and more discussion threads can be forked if things start getting unwieldy (the first Lipton post is pushing 117 comments already).

On the Deolalikar proof: Crowdsourcing the discussion ?

 Update: the google doc has been replaced by a wiki. Please read this post.

By now everyone knows about the new proposed proof that P $\ne$ NP by Vinay Deolalikar. Better minds than mine are on the case right now, and little snippets of questions are coming out in comments (especially on Dick Lipton's original post).

It's a little hard to keep track of all the discussions, so I thought I'd try a crowd-sourcing experiment. At this link is a google doc that contains some of the initial discussion topics being raised around the proof. It's crudely organized, with a section for each major set of issues, and paragraphs for each person's comment, with sourcing via URL.

It's editable, so here's the idea. As you see issues get raised and resolved, enter that into the doc. The following rules apply:

• No discussions: this should be comment/resolution only ! 
• No personal, or nontechnical commentary. This is purely for discussion of the mathematics involved, and nothing else
• Add references (urls, or whatever) whenever possible, and quote all context so as not to cause miscommunication.

The goal is to collect the discussions in one place. Ideally, this would be some kind of polymath setup, and maybe it can be converted over to that (if someone does set it up that way, I'll add a link here).

Let's see if this works !

P vs NP seminar

I'm thinking of topics to discuss over the summer with students here, and decided on introducing people to complexity theory via the question, "Why haven't we solved the P vs NP problem?". The title is a little tongue-in-cheek: the real idea is to use P vs NP to explore some concepts in complexity theory, for an audience that has no real familiarity with the topic.

Thus far, I'm working off the "diagonalization-oracles-boolean-circuits-razborov-rudich" story line, with an addition at the end for algebraization, and P vs NC. But since this isn't my real area of expertise, I'm throwing out a request for suggestions: what topics am I missing when discussing the various attacks on P vs NP ?

p.s I'm using Scott Aaronson's P vs NP survey as a rough starting guide for references.

Ketan Mulmuley at the Center for Intractability

The Center for Intractability recently hosted Ketan Mumuley for a 3-part talk series on his attack on P vs NP via geometric complexity theory. The videos are now online here.

And let me just add here that I think it's fantastic that the center posts video for all the talks. It takes some work to get videos produced for web delivery, and it's so much nicer than reading a paper (or 10).