Prefix-Free Regular Languages: Closure Properties, Difference, and Left Quotient

Abstract

We show that the class of prefix-free languages is closed under intersection, difference, concatenation, square, and the k-th power and is not closed under complement, union, symmetric difference, Kleene star, reversal, cyclic shift, shuffle, and left quotient. Then, we study the state complexity of difference and left quotient of prefix-free regular languages. In both cases we get tight bounds. In the case of difference, the tight bound is mn − m − 2n + 4 and is met by binary languages. In the case of left quotient, the tight bound is 2 n − 1. The bound is met by languages over (n − 1)-letter alphabet and cannot be met using smaller alphabets.