Wednesday, August 11, 2010

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.
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