Complexity of Some Problems from the Theory of Automata

1.1. Given a finite alphabet L', the regular events over X are those accepted by a finite-state automaton. By Kleene's theorem, a subset W of S* is a regular event if and only if it can be constructed from the finiteword sets by boolean operations together with concatenation and *operation. In some sense, the regular events are quite simple because they are accepted by a machine with no storage capacity. Nevertheless, in recent years, much attention has been paid to special subclasses of the class of regular events; in this paper, we shall be concerned with star-free events, events of dot-depth one and piecewise testable events. Star-free events are constructed like regular events from the finite-word sets but with the restriction that the *-operation is not allowed; .events of dot-depth one and piecewise testable events are star free of a very simple form and will be defined below; star-free events have been characterized in the work of Schiitzenberger (1965) in terms of their syntactic monoid; algebraic characterizations of the other two classes have been given (Simon, 1975; Knast, 1983). From an algorithmic point of view, these algebraic characterizations do not yield efficient procedures because computing the syntactic monoid of a regular event given, e.g., by an automaton is obviously time consuming. In this paper, we investigate the complexity of three problems which we now describe in the style of (Garey and Johnson, 1979). We refer the reader to (Aho, Hopcroft, and Ullman, 1983; Garey and Johnson, 1979), for standard concepts of complexity theory.