Polynomial-time membership comparable sets

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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 ≤pf(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
DOIs
StatePublished - Jan 1 1995
Externally publishedYes

ASJC Scopus subject areas

  • Computer Science(all)
  • Mathematics(all)

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