Dot-depth of star-free events

Abstract

A regular event is star-free if it can be denoted by a regular expression involvingonly Boolean operations and concatenation (dot). The family of star-free events can be constructed by alternately applying Boolean operations and concatenation. This approach leads to a hierarchy of star-free events, and to the definition of “dot-depth” of a star-free event which appears to be useful as a measure of the complexity of the event. Properties of dot-depth are examined; for example, it is shown that the dot-depthof a star-free event cannot be increased by the quotient operation with respect to any language, nor can it be increased by multiplying the event by a finite language. The use of two-sided quotients adds insight to the theory of star-free events and permits the derivation of some new properties of these events; in particular, every star-free event has at least one quotient which is either empty or full. The family of star-free events has been shown to be equivalent to the family ofregular noncounting events, and also corresponds to the family of finite automata whose semigroups have no nontrivial subgroups (group-fee automata). In the final section an algorithm is developed for constructing a star-free expression for the event accepted by a group-free finite automaton. An upper bound for the dot-depth of the event is found. We conjecture that for each n≥0, there exist star-free events of dot depth n. We have been able to show this only for n≤2; the general problem remains open.