Streett和奇偶补的紧上界
Tight Upper Bounds for Streett and Parity Complementation
摘要：
有限自动机在无限字上的互补问题不仅是自动机理论中的一个基本问题，而且是解决数理逻辑、模型检验、程序分析和验证等众多决策问题的基石。对于Streett互补，在当前下限$2^{Ω(n\lg nk)}$和上限$2^{O(nk\lg nk)}$之间存在显著的间隙，其中$n$是状态大小，$k$是Streett对的数量，并且$k$可以与$2^{n}$一样大。自80年代末以来，确定Streett互补的复杂性一直是一个悬而未决的问题。本文给出了一个互补结构，其上界$2^{O(n \lg n+nk \lg k)}$用于$k = O(n)$，$2^{O(n^{2} \lg n)}$用于$k = ω(n)$，它与\cite{CZ11a}中得到的下界匹配良好。我们还得到了一个紧的上限$2^{O(n \lg n)}$奇偶互补。
英文摘要：
Complementation of finite automata on infinite words is not only a fundamental problem in automata theory, but also serves as a cornerstone for solving numerous decision problems in mathematical logic, model-checking, program analysis and verification. For Streett complementation, a significant gap exists between the current lower bound $2^{Ω(n\lg nk)}$ and upper bound $2^{O(nk\lg nk)}$, where $n$ is the state size, $k$ is the number of Streett pairs, and $k$ can be as large as $2^{n}$. Determining the complexity of Streett complementation has been an open question since the late '80s. In this paper show a complementation construction with upper bound $2^{O(n \lg n+nk \lg k)}$ for $k = O(n)$ and $2^{O(n^{2} \lg n)}$ for $k = ω(n)$, which matches well the lower bound obtained in \cite{CZ11a}. We also obtain a tight upper bound $2^{O(n \lg n)}$ for parity complementation.
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