Gurevich猜想的相对化
Relativization of Gurevich's Conjectures
摘要：
Gurevich（1988）指出，$\textsf{P}$或$\textsf{NP}\cap \textsf{coNP}$没有逻辑。对于后者的复杂性类，他还表明，存在的逻辑将意味着，$\textsf{NP} \cap \textsf{coNP}$有一个完整的问题下多项式时间减少。我们表明，有一个预言，$\textsf P$确实有一个逻辑和$\textsf P \ne\textsf{NP}$。我们还表明，$\textsf{NP} \cap \textsf{coNP}$的逻辑如下从一个完整的问题的存在和进一步的假设规范标签。对于多项式层次中较高的交集类$Σ^p_n \cap Π^p_n$，逻辑的存在等价于完全问题的存在。
英文摘要：
Gurevich (1988) conjectured that there is no logic for $\textsf{P}$ or for $\textsf{NP}\cap \textsf{coNP}$. For the latter complexity class, he also showed that the existence of a logic would imply that $\textsf{NP} \cap \textsf{coNP}$ has a complete problem under polynomial time reductions. We show that there is an oracle with respect to which $\textsf P$ does have a logic and $\textsf P \ne\textsf{NP}$. We also show that a logic for $\textsf{NP} \cap \textsf{coNP}$ follows from the existence of a complete problem and a further assumption about canonical labelling. For intersection classes $Σ^p_n \cap Π^p_n$ higher in the polynomial hierarchy, the existence of a logic is equivalent to the existence of complete problems.
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