Pseudorandom generators in propositional proof complexity

We call a pseudorandom generator G/sub n/:{0,1}/sup n//spl rarr/{0,1}/sup m/ hard for a propositional proof system P if P can not efficiently prove the (properly encoded) statement G/sub n/(x/sub 1/,...,x/sub n/)/spl ne/b for any string b/spl epsiv/{0,1}/sup m/. We consider a variety of "combinatorial" pseudorandom generators inspired by the Nisan-Wigderson generator on 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).

[1]  Endre Szemerédi,et al.  Many hard examples for resolution , 1988, JACM.

[2]  Alexander A. Razborov,et al.  Lower bounds for the polynomial calculus , 1998, computational complexity.

[3]  Alexander A. Razborov,et al.  Natural Proofs , 1997, J. Comput. Syst. Sci..

[4]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[5]  Eli Ben-Sasson,et al.  Short proofs are narrow—resolution made simple , 2001, JACM.

[6]  Alexander A. Razborov,et al.  Lower Bounds for Propositional Proofs and Independence Results in Bounded Arithmetic , 1996, ICALP.

[7]  Andrew Chi-Chih Yao,et al.  Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version) , 1985, FOCS.

[8]  Armin Haken,et al.  The Intractability of Resolution , 1985, Theor. Comput. Sci..

[9]  Eli Ben-Sasson,et al.  Random Cnf’s are Hard for the Polynomial Calculus , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[10]  Noga Alon,et al.  Spectral Techniques in Graph Algorithms , 1998, LATIN.

[11]  Noam Nisan,et al.  Hardness vs. randomness , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[12]  Michael Alekhnovich,et al.  Space complexity in propositional calculus , 2000, STOC '00.

[13]  Noam Nisan,et al.  Pseudorandom bits for constant depth circuits , 1991, Comb..

[14]  Alasdair Urquhart,et al.  Hard examples for resolution , 1987, JACM.

[15]  Russell Impagliazzo,et al.  A lower bound for DLL algorithms for k-SAT (preliminary version) , 2000, SODA '00.

[16]  Samuel R. Buss,et al.  Linear gaps between degrees for the polynomial calculus modulo distinct primes , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[17]  Russell Impagliazzo,et al.  Using the Groebner basis algorithm to find proofs of unsatisfiability , 1996, STOC '96.

[18]  Andrew Chi-Chih Yao,et al.  Theory and application of trapdoor functions , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[19]  A. Razborov Unprovability of lower bounds on circuit size in certain fragments of bounded arithmetic , 1995 .

[20]  Russell Impagliazzo,et al.  Lower bounds for the polynomial calculus and the Gröbner basis algorithm , 1999, computational complexity.

[21]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[22]  Dima Grigoriev,et al.  Tseitin's tautologies and lower bounds for Nullstellensatz proofs , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[23]  Toniann Pitassi,et al.  Simplified and improved resolution lower bounds , 1996, Proceedings of 37th Conference on Foundations of Computer Science.