### 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) |
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Pages (from-to) | 1068-1081 |

Number of pages | 14 |

Journal | SIAM Journal on Computing |

Volume | 24 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 1995 |

Externally published | Yes |

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

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## Cite this

*SIAM Journal on Computing*,

*24*(5), 1068-1081. https://doi.org/10.1137/S0097539793258131