Approximate counting and NP search problems

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F22%3A00565597" target="_blank" >RIV/67985840:_____/22:00565597 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1142/S021906132250012X" target="_blank" >https://doi.org/10.1142/S021906132250012X</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1142/S021906132250012X" target="_blank" >10.1142/S021906132250012X</a>

Result language

angličtina

Original language name

Approximate counting and NP search problems

Original language description

We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory APC2 of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829-860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory T22, this shows that APC2 does not prove every ς1b sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log. 79(2) (2014) 496-525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the 'fixing lemma' from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185-239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a PNP oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GA19-05497S" target="_blank" >GA19-05497S: Complexity of mathematical proofs and structures</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2022

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Journal of Mathematical Logic

ISSN

0219-0613

e-ISSN

1793-6691

Volume of the periodical

22

Issue of the periodical within the volume

3

Country of publishing house

SG - SINGAPORE

Number of pages

31

Pages from-to

2250012

UT code for WoS article

000848577100003

EID of the result in the Scopus database

2-s2.0-85133016095