Syntactic Complexity of Finite/Cofinite, Definite, and Reverse Definite Languages

Details Syntactic Complexity of Finite/Cofinite, Definite, and Reverse Definite Languages Syntactic Complexity of Finite/Cofinite, Definite, and Reverse Definite Languages

2012-03-13

arxiv.org/abs/1203.2873v1

Computer Science

Formal Languages and Automata Theory

12 pages

Scientific paper

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 and reverse definite languages, and that it can be reached only if the alphabet size is greater than or equal to (n-1)! - (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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