Deolalikar’s Claim: One Year Later



Still no final chapter, lessons learned



Vinay Deolalikar just over a year ago claimed that he had a proof that .

Today I want to make a short comment on the status of his claim.

I recently received an email asking: Has the anniversary slipped your notice?

No. But there is not a lot to say about the situation—for better or for worse. Here is all that we know publicly:

His web page still claims the result, and explains that it is out to a journal for the standard refereeing process.

A few weeks ago when I inquired “what is up with the proof,” his e-mail reply said the same thing. He still believes that he has a proof, and reiterated that it is being checked by referees.

He has expanded and updated the paper, and claims to have answered all the issues that were raised on the web here and elsewhere.

He will not let the public see his proof, nor will he post it, so it remains an unknown.

Lessons

Ken and I learned several lessons from last year’s event.

There is immense interest in the problem. This surprised us to some degree. We know the problem is important, is intriguing, and is hard. But we had no idea that so many people would be interested in the prospect of a proof.

There is immense power in the web as a method of understanding mathematical claims. The proof attempt was read by many, and this quickly led to insights about it. From Fields Medalists to professional mathematicians to amateurs, all helped with the analysis of the claim. We were amazed at the power of the crowd in this situation.

There is immense difficulty in explaining a new proof, especially for someone who primarily works in another area. We believe that Vinay’s main area of expertise is not computational complexity. Thus, it is to be expected that he might have trouble in quickly explaining the key insight that could make his proof work. Perhaps more immediately, this may cause trouble understanding the objections raised.

The online community desires quick reactions and assessments. However, intricate mathematical arguments by their nature do not allow for them. What happens is that every word acquires a very high “slope.” Change the perceived meaning of a phrase a little from the intended meaning, and you have a wide difference in collective understanding. This happened with the word “serious” and even the word “proof” itself. The same slope applied to our own ethical decisions about presentation, not always to everyone’s liking.

There is joy in grappling with new challenges, and feeling that others in a world community are pulling with you. This is so even if the evaluation of these challenges finds flaws in their ultimate objective. It is a realm where intellectual honesty prevails, and that is the gold standard of our field. This and our second point above were the aspects noted by popular author Steven Landsburg here.

A Worry

My biggest worry about all this is that claims of a major result should be resolved in a timely fashion, since otherwise unpleasant claims of credit and priority may arise.

I worked years ago on an open problem in the area of decision procedures. I proved an lower bound on the so called Vector Addition Reachability Problem. See my earlier discussion for the details. While I proved a lower bound, for quite a while there was no upper bound—the problem could have been undecidable.

Here is what I said happened: (The proofs are of the decidability of the problem.)

In act II, two proofs were found: the first by Ernst Mayr and shortly thereafter a proof by Rao Kosaraju. Many did not believe Mayr’s proof, since it was unclear in many places. Understanding these long complex arguments, with many interlocking definitions and lemmas, is difficult. Kosaraju created his proof because he felt Mayr’s proof was wrong. You can imagine there was an immediate controversy. Which proof was right? Both? Neither? There were claims that Kosaraju’s proof was essentially Mayr’s proof, but written in a clear way. The controversy got messy very quickly. Finally, in act III there is now a new proof by Jérôme Leroux that seems much cleaner. and more likely to be correct. More on this in the future.

I would hate to see this happen with the much more important problem of . I also notice that I still owe a discussion of Leroux’s proof that the problem is decidable. Oh well.

Of course, unlike the vector-addition case, there is no compelling sign of a proof here. However, there may be substantial contributions, perhaps new conditional results, that can possibly come from the work. This may be where the worry most applies. If there is something in Vinay Deolalikar’s long paper, then this is worth attending to, and the first onus is on the author.

Open Problems

How should we remember the event last year? Are there other lessons to be taken away?