Polynomial-time membership comparable sets

The paper introduces and studies a notion called polynomial-time membership comparable sets, which is a generalization of P-selective sets. For a function g, a set A is called polynomial-time g-membership comparable if there is a polynomial-time computable function f such that for any x₁,…,x_m with m⩾g(max{|x₁ |,…,|x_m|}), outputs b∈{0,1}^m such that (A(x₁),…A(x_m))≠b. It is shown for each C chosen from {PSPACE, UP, FewP, NP, C=P, PP, MOD₂P, MOD₃P,…}, that if all of C are polynomial-time c(log n)-membership comparable for some fixed constant c<1, then C=P. As a corollary, it is shown that if there is same constant c<1 such that all of 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