Star height of certain families of regular events

Abstract

This paper studies the relationship between the apparent star height of a given regular expression and the structure of its reduced deterministic state graph. Sufficient conditions for the star height of a regular event R to equal the cycle rank of its reduced state graph G R are derived. The cycle rank of G R is also shown to constitute a lower bound to the star height of certain subsets of R . These results are then applied to fully characterize the star height of events consisting of ℰ sets of paths in finite digraphs and two open problems posed by Eggan are answered.