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Total Search Problems in Bounded Arithmetic and Improved Witnessing

  • Arnold Beckmann
  • Jean-José Razafindrakoto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10388)

Abstract

We define a new class of total search problems as a subclass of Megiddo and Papadimitriou’s class of total \({\mathsf {NP}}\) search problems, in which solutions are verifiable in \({\mathsf {AC}}^0\). We denote this class \(\forall \exists {\mathsf {AC}}^0\). We show that all total \({\mathsf {NP}}\) search problems are equivalent, w.r.t. \({\mathsf {AC}}^0\)-many-one reductions, to search problems in \(\forall \exists {\mathsf {AC}}^0\). Furthermore, we show that \(\forall \exists {\mathsf {AC}}^0\) contains well-known problems such as the Stable Marriage and the Maximal Independent Set problems. We introduce the class of Inflationary Iteration problems in \(\forall \exists {\mathsf {AC}}^0\), and show that it characterizes the provably total \({\mathsf {NP}}\) search problems of the bounded arithmetic theory corresponding to polynomial-time. Cook and Nguyen introduced a generic way of defining a bounded arithmetic theory \({\mathsf {VC}}\) for complexity classes \({\mathsf {C}}\) which can be obtained using a complete problem. For such C we will define a new class \({\mathsf {KPT[C]}}\) of \(\forall \exists {\mathsf {AC}}^0\) search problems based on Student-Teacher games in which the student has computing power limited to \({\mathsf {AC}}^0\). We prove that \({\mathsf {KPT[C]}}\) characterizes the provably total \({\mathsf {NP}}\) search problems of the bounded arithmetic theory corresponding to \({\mathsf {C}}\). All our characterizations are obtained via “new-style” witnessing theorems, where reductions are provable in a theory corresponding to AC \(^0\).

Notes

Acknowledgement

We would like to thank Noahi Eguchi. Our characterisation of \(\forall \varSigma ^B_1({\mathsf {V}}^1)\) using inflationary iteration grew out of discussions with him on his attempt to capture P via a two-sorted theory using axioms on inductive definitions [10].

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Computer Science, College of ScienceSwansea UniversitySwanseaUK

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