No abstract

Jeřábek introduced fragments of bounded arithmetic which are axiomatized with weak surjective pigeonhole principles and support a robust notion of approximate counting. We extend these fragments to accommodate modular counting quantifiers. These theories can formalize and prove the relativized versions of Toda’s theorem on the collapse of the polynomial hierarchy with modular counting. We introduce a version of the Paris-Wilkie translation for converting formulas and proofs of bounded arithmetic with modular counting quantifiers into constant depth propositional logic with modular counting gates. We also define Paris-Wilkie translations to Nullstellensatz and polynomial calculus refutations. As an application, we prove that constant depth propositional proofs that use connectives AND, OR, and mod p gates, for p p a prime, can be translated, with quasipolynomial increase in size, into propositional proofs containing only propositional formulas of depth three in which the top level is Boolean, the middle level consists of mod p gates, and the bottom level subformulas are small conjunctions. These results are improved to depth two by using refutations from the weak surjective pigeonhole principles.

The relations between (restrictions of) Hindman's Finite Sums Theorem and (variants of) Ramsey's Theorem give rise to long-standing open problems in combinatorics, computability theory and proof theory. We present some results motivated by these open problems. In particular we investigate the restriction of the Finite Sums Theorem to sums of at most two elements, which is the subject of a long-standing open question by Hindman, Leader and Strauss. We show that this restriction has the same proof-theoretic and computability-theoretic lower bound that is known to hold for the full version of the Finite Sums Theorem. In terms of reverse mathematics it implies ACA 0 . Also, we show that Hindman's Theorem restricted to sums of exactly n elements is equivalent to ACA 0 for each n 3, provided a certain sparsity condition is imposed on the solution set. The same results apply to bounded versions of the Finite Union Theorem, in which such a sparsity condition is already built-in. Further we show that the Finite Sums Theorem for sums of at most two elements is tightly connected to the Increasing Polarized Ramsey's Theorem for pairs introduced by Dzhafarov and Hirst. The latter reduces to the former in the technical sense known as strong computable reducibility.