Polynomial-time membership comparable sets

Research output: Contribution to journalArticle

31 Citations (Scopus)

Abstract

This paper studies a notion called polynomial-time membership comparable sets. For a function g. a set A is polynomial-time g-membership comparable if there is a polynomial-time computable function f such that for any x1,..., xm with m ≥ g(max{|x1|,..., |xm|}), f outputs b , {0,1}m such that (A(x1),..., A(xm)) = b. The following is a list of major results proven in the paper : 1. Polynomial-time membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomial-time membership comparable sets have polynomial-size circuits. 3. For any function f and any constant c > 0, if a set is ≤p f(n)-tt -reducible to a P-selective set, then the set is polynomial-time (1 + c) log f(n)-membership comparable. 4. For any C chosen from {PSPACE, UP, FewP, NP, C=P, PP, MOD2P, MOD3P,...}, if C ⊃ P-mc(c log n) for some c <1, then C = P. As a corollary of the last two results, it is shown that if there is some constant c <1 such that all C are polynomial-time nc-truth-table reducible to some P-selective sets, then C = P, which resolves a question that has been left open for a long time.

Original languageEnglish (US)
Pages (from-to)1068-1081
Number of pages14
JournalSIAM Journal on Computing
Volume24
Issue number5
StatePublished - 1995
Externally publishedYes

Fingerprint

Polynomial time
Polynomials
Truth table
G-function
Resolve
Corollary
Polynomial
Networks (circuits)
Output

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Applied Mathematics
  • Theoretical Computer Science

Cite this

Polynomial-time membership comparable sets. / Ogihara, Mitsunori.

In: SIAM Journal on Computing, Vol. 24, No. 5, 1995, p. 1068-1081.

Research output: Contribution to journalArticle

@article{bb1d5339ad77448b99514914ec8aafed,
title = "Polynomial-time membership comparable sets",
abstract = "This paper studies a notion called polynomial-time membership comparable sets. For a function g. a set A is polynomial-time g-membership comparable if there is a polynomial-time computable function f such that for any x1,..., xm with m ≥ g(max{|x1|,..., |xm|}), f outputs b , {0,1}m such that (A(x1),..., A(xm)) = b. The following is a list of major results proven in the paper : 1. Polynomial-time membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomial-time membership comparable sets have polynomial-size circuits. 3. For any function f and any constant c > 0, if a set is ≤p f(n)-tt -reducible to a P-selective set, then the set is polynomial-time (1 + c) log f(n)-membership comparable. 4. For any C chosen from {PSPACE, UP, FewP, NP, C=P, PP, MOD2P, MOD3P,...}, if C ⊃ P-mc(c log n) for some c <1, then C = P. As a corollary of the last two results, it is shown that if there is some constant c <1 such that all C are polynomial-time nc-truth-table reducible to some P-selective sets, then C = P, which resolves a question that has been left open for a long time.",
author = "Mitsunori Ogihara",
year = "1995",
language = "English (US)",
volume = "24",
pages = "1068--1081",
journal = "SIAM Journal on Computing",
issn = "0097-5397",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "5",

}

TY - JOUR

T1 - Polynomial-time membership comparable sets

AU - Ogihara, Mitsunori

PY - 1995

Y1 - 1995

N2 - This paper studies a notion called polynomial-time membership comparable sets. For a function g. a set A is polynomial-time g-membership comparable if there is a polynomial-time computable function f such that for any x1,..., xm with m ≥ g(max{|x1|,..., |xm|}), f outputs b , {0,1}m such that (A(x1),..., A(xm)) = b. The following is a list of major results proven in the paper : 1. Polynomial-time membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomial-time membership comparable sets have polynomial-size circuits. 3. For any function f and any constant c > 0, if a set is ≤p f(n)-tt -reducible to a P-selective set, then the set is polynomial-time (1 + c) log f(n)-membership comparable. 4. For any C chosen from {PSPACE, UP, FewP, NP, C=P, PP, MOD2P, MOD3P,...}, if C ⊃ P-mc(c log n) for some c <1, then C = P. As a corollary of the last two results, it is shown that if there is some constant c <1 such that all C are polynomial-time nc-truth-table reducible to some P-selective sets, then C = P, which resolves a question that has been left open for a long time.

AB - This paper studies a notion called polynomial-time membership comparable sets. For a function g. a set A is polynomial-time g-membership comparable if there is a polynomial-time computable function f such that for any x1,..., xm with m ≥ g(max{|x1|,..., |xm|}), f outputs b , {0,1}m such that (A(x1),..., A(xm)) = b. The following is a list of major results proven in the paper : 1. Polynomial-time membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomial-time membership comparable sets have polynomial-size circuits. 3. For any function f and any constant c > 0, if a set is ≤p f(n)-tt -reducible to a P-selective set, then the set is polynomial-time (1 + c) log f(n)-membership comparable. 4. For any C chosen from {PSPACE, UP, FewP, NP, C=P, PP, MOD2P, MOD3P,...}, if C ⊃ P-mc(c log n) for some c <1, then C = P. As a corollary of the last two results, it is shown that if there is some constant c <1 such that all C are polynomial-time nc-truth-table reducible to some P-selective sets, then C = P, which resolves a question that has been left open for a long time.

UR - http://www.scopus.com/inward/record.url?scp=0029538097&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029538097&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0029538097

VL - 24

SP - 1068

EP - 1081

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 5

ER -