On Decompositions of Regular Events

Authors:
J. A. Brzozowski;Rina Cohen
Affiliations:
University of Waterloo, Department of Applied Analysis and Computer Science, Waterloo, Ontario, Canada;University of Waterloo, Department of Applied Analysis and Computer Science, Waterloo, Ontario, Canada
Venue:
Journal of the ACM (JACM)
Year:
1969

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Abstract

Decompositions of regular events into star events, i.e. events of the form W = V*, are studied. Mathematically, the structure of a star event is that of a monoid. First it is shown that every regular event contains a finite number of maximal star events, which are shown to be regular and can be effectively computed. Necessary and sufficient conditions for a regular event to be the union of its maximal star events are found. Next, star events are factored out from arbitrary events, yielding the form W - V*T. For each W there exists a unique largest V* and a unique smallest T; an algorithm for finding suitable regular expressions for V and T is developed. Finally, an open problem of Paz and Peleg is answered: Every regular event is decomposable as a finite product of star events and prime events.