Linear Certificates for Membership Comparability, Quadratic Barriers for Selectors

Abstract

Selectors and comparators supply only partial information about membership: a selector names a member of any pair that meets the language, while a binary membership comparator merely excludes one of the four membership vectors of a pair. We ask how much nonuniform advice turns such information into exact recognition. Our main result extends the optimal nondeterministic advice bound for P-selective sets to every binary membership-comparable language: $2-mc \subseteq NP /(3n+5)\cap\mathrm{coNP}/(3n+5)$, with common fixed advice and certificates of at most $5n+12$ bits. The class is strictly larger; some 2-mc sets are not truth-table reducible to any P-selective set. The proof replaces the tournament king by an independent two-step cover of true signed literals, together with a short-forcing-or-exact-majority dichotomy, and it relativizes. Via an advice-preserving isolation transfer, a deterministic polynomial-time algorithm for promise Unique-Circuit-SAT gives $2-mc \subseteq P/O(n)$. For selectors we determine tight orders of ordinary advice: $Θ(n)$ for errorless average-case computation and $Θ(n^2)$ for worst-case bounded-error computation, the latter independent of the interpreter's coin bound. One oracle realizes both orders on a single language and separates ordinary from coin-dependent advice. The quadratic and linear lower bounds hold for tournament-query procedures and relativized languages, not unconditionally for unrelativized P-selective sets.