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Prikl. Diskr. Mat., 2018, Number 40, Pages 100–104 (Mi pdm622)  

This article is cited in 1 scientific paper (total in 1 paper)

Mathematical Backgrounds of Informatics and Programming

Relativized generic classes $\mathrm P$ and $\mathrm{NP}$

A. N. Rybalov

Omsk State Technical University, Omsk, Russia

Abstract: Classical theorem of Baker, Gill and Solovay states that there exist two oracles $A$ and $B$ such that $\mathrm P^A=\mathrm{NP}^A$, but $\mathrm P^B\neq\mathrm{NP}^B$. This result indicates that the classical tools of computability theory (such as diagonalization) are inapplicable to prove the inequality $\mathrm P\neq\mathrm{NP}$. Generic-case approach to algorithmic problems was suggested by Miasnikov, Kapovich, Schupp and Shpilrain in 2003. This approach studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. Many classical undecidable or hard algorithmic problems become feasible in the generic case. In this paper we introduce generic analogs $\mathrm{genP}$ and $\mathrm{genNP}$ of the classical computational complexity classes $\mathrm P$ and $\mathrm{NP}$. We prove a generic analog of the Baker–Gill–Solovay theorem: there exist two oracles $A$ and $B$ such that $\mathrm{genP}^A=\mathrm{genNP}^A$, but $\mathrm{genP}^B\neq\mathrm{genNP}^B$. Therefore the diagonalization arguments cannot be applied to prove the inequality $\mathrm P\neq\mathrm{NP}$ in the generic case too.

Keywords: generic complexity, $\mathrm P$ vs $\mathrm{NP}$ problem, oracles.

Funding Agency Grant Number
Russian Science Foundation 17-11-01117


DOI: https://doi.org/10.17223/20710410/40/8

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Bibliographic databases:

UDC: 510.52

Citation: A. N. Rybalov, “Relativized generic classes $\mathrm P$ and $\mathrm{NP}$”, Prikl. Diskr. Mat., 2018, no. 40, 100–104

Citation in format AMSBIB
\Bibitem{Ryb18}
\by A.~N.~Rybalov
\paper Relativized generic classes~$\mathrm P$ and~$\mathrm{NP}$
\jour Prikl. Diskr. Mat.
\yr 2018
\issue 40
\pages 100--104
\mathnet{http://mi.mathnet.ru/pdm622}
\crossref{https://doi.org/10.17223/20710410/40/8}
\elib{https://elibrary.ru/item.asp?id=35155727}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Seliverstov, “O dvoichnykh resheniyakh sistem uravnenii”, PDM, 2019, no. 45, 26–32  mathnet  crossref
  • Прикладная дискретная математика
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