Syntactic Complexity of Bifix-Free Languages

We study the properties of syntactic monoids of bifix-free regular languages. In particular, we solve an open problem concerning syntactic complexity: We prove that the car-dinality of the syntactic semigroup of a bifix-free language with state complexity n is at most (n−1) n−3 +(n−2) n−3 +(n−3)2 n−3 for n 6. The main proof uses a large construction with the method of injective function. Since this bound is known to be reachable, and the values for n 5 are known, this completely settles the problem. We also prove that (n − 2) n−3 + (n − 3)2 n−3 − 1 is the minimal size of the alphabet required to meet the bound for n 6. Finally, we show that the largest transition semigroups of minimal DFAs which recognize bifix-free languages are unique up to renaming the states.