Dot-Depth of Star-Free Events

A regular event is star-free if it can be denoted by a regular expression involving only 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-depth of 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 of regular noncounting events, and also corresponds to the family of finite automata whose semigroups have no nontrivial subgroups (group-free 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.