Collapsing modular counting in bounded arithmetic and constant depth propositional proofs
Collapsing modular counting in bounded arithmetic and constant depth propositional proofs
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有界算术和恒定深度命题证明中的折叠模块化计数
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
K. Zdanowski
中科院分区：
文献类型：
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作者：
S. Buss;L. Kolodziejczyk;K. Zdanowski
杰贝克（JeDimabek）引入了有界算术的片段，这些片段用弱的表面上的鸽子孔原理和近似计数的稳健概念进行了支持。多项式层次结构与模块化计数。与模块化的构量构想的界限和模块化计数的界限的证明。 1101228。在这项工作的初步阶段中，第二和第三作者得到了授予的n201 382234。科学和高等教育部。西蒙斯基金会（＃208717到SAM巴士）。
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).