Syntactic Complexity of Star-Free Languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the maximal syntactic complexity of languages in that subclass, taken as a function of the state complexity of these languages. We study the syntactic complexity of star-free regular languages, that is, languages that can be constructed from finite languages using union, complement and concatenation. We find tight upper bounds on the syntactic complexity of languages accepted by monotonic and partially monotonic automata. We introduce "nearly monotonic" automata, which accept star-free languages, and find a tight upper bound on the syntactic complexity of languages accepted by such automata. We conjecture that this bound is also an upper bound on the syntactic complexity of star-free languages.