arXiv 1408.63342014-08cs.CC

一种构造性证明，表示$Σ_2^P$中的语言不能由大小为$n^k$的电路族决定

A constructive proof presenting languages in $Σ_2^P$ that cannot be decided by circuit families of size $n^k$

Sunny Daniels

备注：: This is a corrected version of my previous article (of the same name) which attracted the attention of Professor Lance Fortnow at Georgia Institute of Technology ("Sixteen Years in the Making" in his Complexity Theory Blog). The original had a missing closing bracket in a footnote and a reference to the wrong step in the machine for $Λ_k$

摘要：

据我所知，在我最初设计这个结果的时候（1998年），这是第一个建设性的证明，对于任何整数$k$，在$Σ_2^P$中有一种语言不能被大小为$n^k$的逻辑电路族模拟。然而，这个结果以前被证明是非建设性的：参见Cai和Watanabe [CW08]以获得关于这个问题历史的更多信息。这个构造性证明是基于构造一个从可满足性问题导出的语言$Γ$和一个由交替图灵机定义的语言$Λ_k$。我们发现，工会的$Γ$和$Λ_k$不能模拟电路的大小$n^k$。

英文摘要：

As far as I know, at the time that I originally devised this result (1998), this was the first constructive proof that, for any integer $k$, there is a language in $Σ_2^P$ that cannot be simulated by a family of logic circuits of size $n^k$. However, this result had previously been proved non-constructively: see Cai and Watanabe [CW08] for more information on the history of this problem. This constructive proof is based upon constructing a language $Γ$ derived from the satisfiabiility problem, and a language $Λ_k$ defined by an alternating Turing machine. We show that the union of $Γ$ and $Λ_k$ cannot be simulated by circuits of size $n^k$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。