Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
The number of Boolean functions computed by formulas of a given size
proceedings of the eighth international conference on Random structures and algorithms
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Explicit lower bound of 4.5n - o(n) for boolena circuits
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Super-bits, Demi-bits, and NP/qpoly-natural Proofs
RANDOM '97 Proceedings of the International Workshop on Randomization and Approximation Techniques in Computer Science
Pseudorandom generators, measure theory, and natural proofs
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Cracks in the defenses: scouting out approaches on circuit lower bounds
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
Natural proofs versus derandomization
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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Razborov and Rudich have proved that, under a widely-believed hypothesis about pseudorandom number generators, there do not exist P/poly-computable Boolean function properties with density greater than 2^-^p^o^l^y^(^n^) that exclude P/poly. This famous result is widely regarded as a serious barrier to proving strong lower bounds in circuit complexity theory, because virtually all Boolean function properties used in existing lower bound proofs have the stated complexity and density. In this paper, we show that under the same pseudorandomness hypothesis, there do exist nearly-linear-time-computable Boolean function properties with only slightly lower density (namely, 2^-^q^(^n^) for a quasi-polynomial function q) that not only exclude P/poly, but even separate NP from P/poly. Indeed, we introduce a simple, explicit property called discrimination that does so. We also prove unconditionally that there exist non-uniformly nearly-linear-time-computable Boolean function properties with this same density that exclude P/poly. Along the way we also note that by slightly strengthening Razborov and Rudich@?s argument, one can show that their ''naturalization barrier'' is actually a barrier to proving superquadratic circuit lower bounds, not just P/poly circuit lower bounds. It remains open whether there is a naturalization barrier to proving superlinear circuit lower bounds.