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.)

http://alixcomsi.com/27_Resolving_PvNP_Update.pdf

http://www.panix.com/~jays/Boole.four.page.summary.for.1.August.2006.Cork.conference.Sulzberger.v.5.pdf

Please forgive repetition!

http://www.panix.com/~jays/Boole.four.page.summary.for.1.August.2006.Cork.conference.Sulzberger.pdf

Please forgive pleonastic repetition!

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

This may only be a superficial similarity though.

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.