Syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that nn−2 is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be (n−1)n−2+(n−2), (n−1)n−3+(n−2)n−3+(n−3)2n−3, and (n−1)n−3+(n−3)2n−3+1, respectively, and exhibit languages with these syntactic complexities.