Polynomial-Time Membership Comparable Sets

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 polynomialtime computable function f such that for any x1; ; xm with m g(maxfjx1j; ; jxmjg), outputs b 2 f0; 1gm such that (A(x1); ; A(xm)) 6= 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 for any constant c > 0, if a set is pf(n)-tt-reducible to a Pselective set, then the set is polynomial-time (1+ c) logf(n)-membership comparable. 4. For any C chosen from fPSPACE;UP;FewP;NP;C=P;PP;MOD2P;MOD3P; g, if C P-mc(c logn) 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 of 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.