NDLI: Almost-Natural Proofs

Content Provider	IEEE Xplore Digital Library
Author	Chow, T.Y.
Copyright Year	2008
Description	Author affiliation: Center for Commun. Res., Princeton, NJ (Chow, T.Y.)
Abstract	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.
Starting Page	86
Ending Page	91
File Size	224384
Page Count	6
File Format	PDF
ISBN	9780769534367
ISSN	02725428
DOI	10.1109/FOCS.2008.16
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2008-10-25
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Boolean functions Circuits Complexity theory Computer science Drives naturalization barrier natural proofs circuit complexity pseudorandom number generators
Content Type	Text
Resource Type	Article

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