理想语言和闭语言的句法复杂性
Syntactic Complexity of Ideal and Closed Languages
摘要：
正则语言的状态复杂度是接受该语言的最小确定性自动机中的状态数。正则语言的句法复杂度是它的句法半群的基数。正则语言子类的语法复杂度是最坏情况下的语法复杂度，它是该类语言的状态复杂度$n$的函数。我们研究了正则理想语言类及其补集闭语言的句法复杂性。我们证明了$n^{n-1}$是右理想和前缀闭语言复杂度的一个紧上界，并且存在语法复杂度为$n^{n-1}+n-1$的左理想和后缀闭语言，以及语法复杂度为$n^{n-2}+(n-2)2^{n-2}+1$的双边理想和因子闭语言.
英文摘要：
The state complexity of a regular language is the number of states in the minimal deterministic automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity $n$ of languages in that class. We study the syntactic complexity of the class of regular ideal languages and their complements, the closed languages. We prove that $n^{n-1}$ is a tight upper bound on the complexity of right ideals and prefix-closed languages, and that there exist left ideals and suffix-closed languages of syntactic complexity $n^{n-1}+n-1$, and two-sided ideals and factor-closed languages of syntactic complexity $n^{n-2}+(n-2)2^{n-2}+1$.
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