### 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 x_{1},..., x_{m} with m ≥ g(max{|x_{1}|,..., |x_{m}|}), f outputs b , {0,1}^{m} such that (A(x_{1}),..., A(x_{m})) = 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, MOD_{2}P, MOD_{3}P,...}, 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 n^{c}-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 language | English (US) |
---|---|

Pages (from-to) | 1068-1081 |

Number of pages | 14 |

Journal | SIAM Journal on Computing |

Volume | 24 |

Issue number | 5 |

State | Published - 1995 |

Externally published | Yes |

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### ASJC Scopus subject areas

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

### Cite this

*SIAM Journal on Computing*,

*24*(5), 1068-1081.

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

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 24, no. 5, pp. 1068-1081.

}

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 -