AI 中文总结
本文研究选择器和比较器在非均匀建议下实现精确识别的复杂度，证明二元成员可比语言的最优非确定性建议界，并确定选择器的普通建议紧阶。核心方法包括独立两步覆盖和短强制或精确多数二分法，主要贡献在于扩展P-选择性结果并分离建议类型。
AI 中文摘要
选择器和比较器仅提供关于成员资格的部分信息：选择器从满足语言的任何一对中指定一个成员，而二元成员比较器仅排除一对的四个成员资格向量中的一个。我们询问多少非均匀建议能将此类信息转化为精确识别。我们的主要结果将P-选择性集合的最优非确定性建议界扩展到每个二元成员可比语言：$2-mc \subseteq NP /(3n+5)\cap\mathrm{coNP}/(3n+5)$，具有共同的固定建议和至多$5n+12$位的证书。该类严格更大；一些2-mc集合不能真值表归约到任何P-选择性集合。证明用独立的真签名文字两步覆盖取代锦标赛之王，连同短强制或精确多数二分法，并且可相对化。通过保建议隔离转移，确定性多项式时间算法用于承诺唯一电路SAT给出$2-mc \subseteq P/O(n)$。对于选择器，我们确定普通建议的紧阶：$\u0398(n)$用于无错平均情况计算，$\u0398(n^2)$用于最坏情况有界错误计算，后者独立于解释器的硬币界。一个预言机在单一语言上实现两个阶，并将普通建议与硬币依赖建议分开。二次和线性下界适用于锦标赛查询过程和相对化语言，而非无条件适用于非相对化P-选择性集合。
英文摘要
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.
