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Pseudorandom Bits for ConstantDepth Circuits with Few Arbitrary Symmetric Gates

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Viola, Emanuele
Copyright Year	2007
Abstract	We exhibit an explicitly computable pseudorandom generator stretching l bits into $m(l) = l^{\Omega(\log l)}$ bits that look random to constantdepth circuits of size $m(l)$ with $\log m(l)$ arbitrary symmetric gates e.g., PARITY, MAJORITY. This improves on a generator by Luby, Velickovic, and Wigderson Proceedings of the Second Israel Symposium on Theory of Computing Systems, 1993, pp. 1824 that achieves the same stretch but fools only circuits of depth 2 with one arbitrary symmetric gate at the top. Our generator fools a strictly richer class of circuits than Nisans generator for constantdepth circuits but Nisans generator has a much bigger stretch Combinatorica, 11 1991, pp. 6370. In particular, we conclude that every function computable by uniform $\poly(n)$size probabilistic constantdepth circuits with $O(\log n)$ arbitrary symmetric gates is in $\mathit{TIME}(2^{n^{o(1)}})$. This seems to be the richest probabilistic circuit class known to admit a subexponential derandomization. Our generator is obtained by constructing an explicit function $f : \zo^n \to \zo$ that is very hard on average for constantdepth circuits of size $s(n) = n^{\Omega(\log n)}$ with $\log s(n)$ arbitrary symmetric gates, and plugging it into the NisanWigderson pseudorandom generator construction J. Comput. System Sci., 49 1994, pp. 149167. The proof of the averagecase hardness of this function is a modification of arguments by Razborov and Wigderson Inform. Process. Lett., 45 1993, pp. 303307 and Hansen and Miltersen Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Comput. Sci. 3153, SpringerVerlag, Berlin, 2004, pp. 334345 and combines Hstads switching lemma Computational Limitations of SmallDepth Circuits, MIT Press, Cambridge, MA, 1987 with a multiparty communication complexity lower bound by Babai, Nisan, and Szegedy J. Comput. System Sci., 45 1992, pp. 204232.
Starting Page	1387
Ending Page	1403
Page Count	17
File Format	PDF
ISSN	00975397
DOI	10.1137/050640941
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	36
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2007-01-22
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems Modes of computation pseudorandom generator symmetric gate constantdepth circuit switching lemma General derandomization averagecase hardness lower bound communication complexity Complexity classes
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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