Polynomial-time Membership Comparable Sets

• Authors: Mitsunori Ogihara
• Venue: SIAM Journal on Computing
• Year: 1995

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 $x_1, \cdots, x_m$ with $m \geq g(\max\{ |x_1|, \cdots, |x_m| \})$, outputs $b \in \{0,1\}^m$ such that $(A(x_1), \cdots, A(x_m)) \neq b$. 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 $c0$, if a set is $\leq^p_{f(n)-tt}$-reducible to a P-selective set, then the set is polynomial-time $(1+c)\log f(n)$-membership comparable. 4. For any $\cal C$ chosen from $\{ {\rm PSPACE, UP, FewP, NP, C_{=}P, PP, MOD_{2}P, MOD_{3}}, \cdots \}$, if $\cal C \subseteq {\rm P-mc}(c\log n)$ for some $c As a corollary of the last two results, it is shown that if there is some constant $c