Oracle without Optimal Proof Systems outside Nondeterministic Subexponential Time

Fabian Egidy, Christian Glaßer
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Abstract

We study the existence of optimal proof systems for sets outside of $\mathrm{NP}$. Currently, no set $L \notin \mathrm{NP}$ is known that has optimal proof systems. Our main result shows that this is not surprising, because we can rule out relativizable proofs of optimality for all sets outside $\mathrm{NTIME}(t)$ where $t$ is slightly superpolynomial. We construct an oracle $O$, such that for any set $L \subseteq \Sigma^*$ at least one of the following two properties holds: $L$ does not have optimal proof systems relative to $O$. $L \in \mathrm{UTIME}^O(2^{2(\log n)^{8+4\log(\log(\log(n)))}})$. The runtime bound is slightly superpolynomial. So there is no relativizable proof showing that a complex set has optimal proof systems. Hence, searching for non-trivial optimal proof systems with relativizable methods can only be successful (if at all) in a narrow range above $\mathrm{NP}$.
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非确定性亚指数时间之外的无最优证明系统的 Oracle
我们研究$\mathrm{NP}$之外的集合是否存在最优证明系统。目前,我们还不知道$L \notin \mathrm{NP}$集合有最优证明系统。我们的主要结果表明,这并不奇怪,因为我们可以排除$t$为略超多项式的$\mathrm{NTIME}(t)$之外所有集合的可相对性最优证明。我们构造了一个算法 $O$,使得对于任何集合 $L \subseteq \Sigma^*$ 至少有以下两个性质之一成立:$L$ 没有相对于 $O$ 的最优证明系统。运行时间约束略超多项式。因此,没有可相对化的证明表明复杂集合具有最优证明系统。因此,用可相对化方法寻找非三维最优证明系统只能在 $\mathrm{NP}$ 以上的狭窄范围内成功(如果有的话)。
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