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 n(n-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)(n-3), and (n - 1)(n-3) + (n - 3)2(n-3) + 1, respectively, and exhibit languages with these syntactic complexities. (C) 2012 Elsevier B.V. All rights reserved.