By Topic

Almost-Natural Proofs

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$33 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

1 Author(s)
Chow, T.Y. ; Center for Commun. Res., Princeton, NJ

Razborov and Rudich have shown that so-called "natural proofs" are not useful for separating P from NP unless hard pseudorandom number generators do not exist. This famous result is widely regarded as a serious barrier to proving strong lower bounds in circuit complexity theory. By definition, a natural combinatorial property satisfies two conditions, constructivity and largeness. Our main result is that if the largeness condition is weakened slightly, then not only does the Razborov-Rudich proof break down, but such "almost-natural" (and useful) properties provably exist. Specifically, under the same pseudorandomness assumption that Razborov and Rudich make, a simple, explicit property that we call "discrimination" suffices to separate P/poly from NP; discrimination is nearly linear-time computable and almost large, having density 2-q(n) where q grows slightly faster than a quasi-polynomial function. For those who hope to separate P from NP using random function properties in some sense, discrimination is interesting, because it is constructive, yet may be thought of as a minor alteration of a property of a random function. The proof relies heavily on the self-defeating character of natural proofs. Our proof technique also yields an unconditional result, namely that there exist almost-large and useful properties that are constructive, if we are allowed to call non-uniform low-complexity classes "constructive." We note, though, that this unconditional result can also be proved by a more conventional counting argument.

Published in:

Foundations of Computer Science, 2008. FOCS '08. IEEE 49th Annual IEEE Symposium on

Date of Conference:

25-28 Oct. 2008