期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2012
卷号：2012
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer a quantitative loss in parameters, and hence do not give nontrivial implications for models where we only know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can indeed construct PRGs which are essentially best possible without in turn improving the lower bounds.
More specifically, say that a circuit family has shrinkage exponent Gamma if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of p. Our PRG uses a seed of length roughly s1(+1) to fool circuits in the family of size s. By instantiating this generic construction, we get PRGs for the following classes:
1) de Morgan formulas of size s, seed length s13+o(1) .
2) Formulas over an arbitrary basis of size s, seed length s12+o(1) .
3) Read-once formulas, seed length s234.
4) Branching programs of size s, seed length s12+o(1) .
The previous best PRGs known for these classes used seeds of length bigger than n2 to output n bits, and worked only when the size s=O(n).
关键词：branching programs, formulas, Pseudorandomness, random restrictions, shrinkage
