Collapsing modular counting in bounded arithmetic and constant depth propositional proofs

Jeřabek 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 of bounded arithmetic 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 The first author was supported in part by NSF grant DMS-1101228. In the preliminary stages of this work, the second and third authors were supported by grant no. N N201 382234 of the Polish Ministry of Science and Higher Education. Most of this work was carried out while the second author was visiting the University of California, San Diego, supported by Polish Ministry of Science and Higher Education programme “Mobilnośc Plus” with additional support from a grant from the Simons Foundation (#208717 to Sam Buss).

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