Pseudorandom Generators in Propositional Proof Complexity

We call a pseudorandom generator Gn : {0, 1}n → {0, 1}m hard for a propositional proof system P if P can not efficiently prove the (properly encoded) statement Gn(x1, . . . , xn) 6= b for any string b ∈ {0, 1}m. We consider a variety of “combinatorial” pseudorandom generators inspired by the Nisan-Wigderson generator on the one hand, and by the construction of Tseitin tautologies on the other. We prove that under certain circumstances these generators are hard for such proof systems as Resolution, Polynomial Calculus and Polynomial Calculus with Resolution (PCR). ∗Moscow State University, Moscow, Russia [email protected]. Supported by INTAS grant # 96-753 and by the Russian Basic Research Foundation †Institute of Computer Science, Hebrew University, Jerusalem, Israel. [email protected]. ‡Steklov Mathematical Institute, Moscow, Russia [email protected]. Supported by INTAS grant # 96-753 and by the Russian Basic Research Foundation; part of this work was done while visiting Princeton University and DIMACS §Institute for Advanced Study, Princeton, and Institute of Computer Science, Hebrew University, Jerusalem, [email protected]. This research was supported by grant number 69/96 of the Israel Science Foundation, founded by the Israel Academy for Sciences and Humanities. Support for this research has been provided by The Alfred P. Sloan Foundation