有限/余有限、有穷和反向有穷语言的句法复杂性
Syntactic Complexity of Finite/Cofinite, Definite, and Reverse Definite Languages
摘要：
我们研究了有限/上有限，确定和反向确定语言的句法复杂性。一类语言的句法复杂度被定义为该类语言的句法半群的最大尺寸，作为语言的状态复杂度n的函数。我们证明了（n-1）！是有限/余有限语言的严格上限，只有当字母表大小大于或等于（n-1）时才能达到！(n-2)!.我们证明了这个界也是（n-1）！对于逆定语言，但最小字母表大小是（n-1）！2（n-2）！我们证明了\lfloor e \cdot（n-1）！\rfloor是确定语言的句法复杂度的一个下界，并推测这也是一个上界，满足这个界所需的字母表大小是\floor{e \cdot (n-1)!} - \floor{e \cdot (n-2)!}。我们证明了猜想n \le 4。
英文摘要：
We study the syntactic complexity of finite/cofinite, definite and reverse definite languages. The syntactic complexity of a class of languages is defined as the maximal size of syntactic semigroups of languages from the class, taken as a function of the state complexity n of the languages. We prove that (n-1)! is a tight upper bound for finite/cofinite languages and that it can be reached only if the alphabet size is greater than or equal to (n-1)!-(n-2)!. We prove that the bound is also (n-1)! for reverse definite languages, but the minimal alphabet size is (n-1)!-2(n-2)!. We show that \lfloor e\cdot (n-1)!\rfloor is a lower bound on the syntactic complexity of definite languages, and conjecture that this is also an upper bound, and that the alphabet size required to meet this bound is \floor{e \cdot (n-1)!} - \floor{e \cdot (n-2)!}. We prove the conjecture for n\le 4.
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