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

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 alphabet size is (n−1)!−2(n−2)!. We show that ⌊e · (n−1)!⌋ 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 ⌊e · (n− 1)!⌋− ⌊e · (n− 2)!⌋. We prove the conjecture for n ≤ 4.