We develop and study the complexity of propositional proof systems of varying strength extending resolution by allowing it to operate with disjunctions of linear equations instead of clauses. We demonstrate polynomial-size refutations for hard tautologies like the pigeonhole principle, Tseitin graph tautologies and the clique-coloring tautologies in these proof systems. Using the (monotone) int...

We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.

We discuss recent results on the propositional proof complexity of Frege proof systems, including some recently discovered quasipolynomial size proofs for the pigeonhole principle and the Kneser-Lovász theorem. These are closely related to formalizability in bounded arithmetic.

We apply the pigeonhole principle to show that there must exist Boolean functions on 7 inputs with a multiplicative complexity of at least 7, i.e., that cannot be computed with only 6 multiplications in the Galois field with two elements.

We give simple new lower bounds on the lengths of Resolution proofs for the pigeonhole principle and for randomly generated formulas. For random formulas, our bounds signiicantly extend the range of formula sizes for which non-trivial lower bounds are known. For example, we show that with probability approaching 1, any Resolution refutation of a randomly chosen 3-CNF formula with at most n 6=5?...

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 PHPcn n and random unsatisfiable CNF formulas require exponential-size proofs in this system. This is the strongest system beyond Resolution for which such lower bou...

The ordering principle states that every finite linear order has a least element. We show that, in the relativized setting, the surjective weak pigeonhole principle for polynomial time functions does not prove a Herbrandized version of the ordering principle over T 2 . This answers an open question raised in [Buss, Ko lodziejczyk and Thapen, 2012] and completes their program to compare the stre...