translation if you want to see the current state of automatic on-demand free online translation.

When I tried it I extracted a sentence “In the meantime the reactors found two ( and half ) essential problems with the proof.” but

I have no idea whether that is in any way faithful to the sentiment

actually expressed. ]]>

Although not immediately obvious, the roots of the PvNP problem may lie in how we define the satisfiability of the atomic formulas of the first order arithmetic PA under an interpretation (following Tarski’s inductive definitions of the satisfaction, and truth, of the formulas of a formal language under an interpretation).

For instance, if an n-ary atomic formula [f(x1, x2, …, xn)] of PA interprets under the standard interpretation I_Standard of PA over the domain N of the natural numbers as the arithmetical relation f(x1, x2, …, xn), and the PA numeral [ai] as the natural number ai, then [f(x1, x2, …, xn)] is defined as satisfied under I_Standard for any given sequence of numerals [a1, a2, …, an] if, and only if, f(x1, x2, …, xn) is effectively verifiable instantiationally over N, and f(a1, a2, …, an) holds for the natural number sequence (a1, a2, …, an).

(See Lemmas 16 & 17 in the link below.)

There is, however, an alternative interpretation I_Algorithmic of PA over N where, if an n-ary atomic formula [f(x1, x2, …, xn)] of PA interprets under I_Algorithmic as the arithmetical relation f(x1, x2, …, xn), and the numeral [ai] as the natural number ai, then [f(x1, x2, …, xn)] is defined as satisfied under I_Algorithmic for any given sequence of numerals [a1, a2, …, an] if, and only if, f(x1, x2, …, xn) is effectively computable (decidable) algorithmically over N, and f(a1, a2, …, an) holds for the natural number sequence (a1, a2, …, an).

(See Lemma 18 in the link below.)

Now, it can be argued that the interpretation I_Algorithmic is sound; and that a PA formula [f] is provable if, and only if, f interprets under I_Algorithmic as an arithmetical relation that is effectively computable (decidable) algorithmically as always true over N.

(See Theorems 4 & 6 in the link below.)

This could be the bridge that Deolalikar appears to be implicitly seeking by his argumentation since, in view of Goedel’s construction of a PA formula that is PA-unprovable but effectively verifiable as always true over N, it would immediately follow that P=/=NP (irrespective, however, of the precise definitions of the classes P and NP).

(See Theorem 7 in the link below.)

]]>That was a piece of absent-mindedness on my part. Many thanks for pointing it out — I have now added Rudich’s name to the sentence in question.

]]>Please forgive repetition!

]]>Please forgive pleonastic repetition!

]]>Incidentally, the model of using random reversible logic circuits to build a random polynomial length circuit reminds me of the product replacement algorithm in the theory of random walks:

http://en.wikipedia.org/wiki/Nielsen_transformation

This may only be a superficial similarity though.

]]>That’s a fair summary of what I think. Actually, I’d say it’s a bit closer than a spiritual cousin of the natural proofs barrier — more like an illegitimate sibling. It’s not rigorous, and I’m not sure whether it yields any insights that are not rigorously yielded by the natural proofs, but I find the general principle, that the output of a random circuit is incredibly hard to distinguish from a random function, easier to think about than the more precise natural proofs result. Also, when you say, “according to general belief,” I don’t actually feel that I can speak for the TCS community here. Somebody did once make an interesting comment on this blog, however, which was that Razborov believes more strongly that the output of a random circuit is pseudorandom than that factorizing is hard. (I hope I’m representing the comment and Razborov’s beliefs correctly.)

Another point is that it is conceivable that some very clever simplicity criterion could work, but the phrase polylog-parametrization doesn’t sound sufficiently weird and contorted to feel plausible.

]]>This does look to be getting at the very heart of the matter, and is quite convincing to me at least that the whole “bound complexity using the structure of the solution space” strategy is indeed doomed to failure. I’ll point it out on Lipton’s blog.

]]>The distribution is certainly ample, since about half of all possible inputs belong to it. But the function mixes things up so much that it also seems to me that the various correlations amongst the n random variables (obtained if you take a random n-bit solution) will not be reducible in any sense I can think of.

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