Lower Bounds for the Weak Pigeonhole Principle 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 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 Krajı́ček. Supported by the CUR, Generalitat de Catalunya, through grant 1999FI 00532. Partially supported by MEC through grant PB98-0937-C04 (FRESCO project) and CICYT TIC 98-0410-C02-01. Partially supported by MEC through grant PB98-0937-C04 (FRESCO project) Partially supported by ALCOM-FT, IST-99-14186.