Lower Bounds for the Weak Pigeonhole Principle and Random Formulas beyond Resolution

We work with an extension of Resolution, called Res(2), that allows clauses with conjunctions of two literals. In this system there are rules to introduce and eliminate such conjunctions. We prove that the weak pigeonhole principle PHPncn and random unsatisfiable CNF formulas require exponential-size proofs in this system. This is the strongest system beyond Resolution for which such lower bounds are known. As a consequence to the result about the weak pigeonhole principle, Res(log) is exponentially more powerful than Res(2). Also we prove that Resolution cannot polynomially simulate Res(2) and that Res(2) does not have feasible monotone interpolation solving an open problem posed by Krajicek.

[1]  J. Krajícek.  On the weak pigeonhole principle , 2001 .

[2]  Ran Raz,et al.  Regular resolution lower bounds for the weak pigeonhole principle , 2001, STOC '01.

[3]  Toniann Pitassi,et al.  A new proof of the weak pigeonhole principle , 2000, Symposium on the Theory of Computing.

[4]  J. Kraj.  On the Weak Pigeonhole Principle , 2001 .

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

[6]  Noga Alon,et al.  The monotone circuit complexity of boolean functions , 1987, Comb..

[7]  Samuel R. Buss,et al.  Resolution Proofs of Generalized Pigeonhole Principles , 1988, Theor. Comput. Sci..

[8]  Samuel R. Buss.  Polynomial Size Proofs of the Propositional Pigeonhole Principle , 1987, J. Symb. Log..

[9]  Jan Krajícek,et al.  Exponential Lower Bounds for the Pigeonhole Principle , 1992, STOC.

[10]  Ran Raz,et al.  Lower bounds for cutting planes proofs with small coefficients , 1995, Symposium on the Theory of Computing.

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

[12]  Samuel R. Buss,et al.  Resolution and the Weak Pigeonhole Principle , 1997, CSL.

[13]  Pavel Pudlák,et al.  Lower bounds for resolution and cutting plane proofs and monotone computations , 1997, Journal of Symbolic Logic.

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

[15]  G. Grimmett,et al.  Probability and random processes , 2002 .

[16]  Michael E. Saks,et al.  The Efficiency of Resolution and Davis--Putnam Procedures , 2002, SIAM J. Comput..

[17]  Jeff B. Paris,et al.  Provability of the Pigeonhole Principle and the Existence of Infinitely Many Primes , 1988, J. Symb. Log..

[18]  Alexander A. Razborov,et al.  Improved Resolution Lower Bounds for the Weak Pigeonhole Principle , 2001, Electron. Colloquium Comput. Complex..

[19]  Alexander A. Razborov,et al.  Electronic Colloquium on Computational Complexity, Report No. 75 (2001) Resolution Lower Bounds for the Weak Functional Pigeonhole Principle , 2001 .

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