Algorithms for Determining Relative Star height and Star Height

Eggan(l963) introduced the notion of the star height to each regular expression which is a nonnegative integer denoting the nestedness of star operators in this expression. The star height of a regular language is the minimum of the star height of regular expressions denoting this language. We remark here that to each regular language, there exist generally infinitely many regular expressions denoting this language. Eggan(l963) showed that for each nonnegative integer k, there exists a regular language of star height k, and posed the problem of determining the star height of any regular language. Dejean and Schutzenberger (1966) showed that for each nonnegative integer k, there exists a regular language of star height k over the two-letter alphabet. McNaughton (1967) presented an algorithm for determining the loop complexity (i.e., the star height) of any regular language whose syntactic monoid is a group. Cohen (1970, 1971) and Cohen and Brzozowski (1970) investigated many properties of star height, some of which provide algorithms for determining the star height of any regular language of certain reset-free type. Hashiguchi and Honda (1979) presented an algorithm for determining the star height of any reset-free language and any reset language. Hashiguchi (1982B) presented an algorithm for deciding whether or not an arbitrary language is of star height one. To obtain this result, it uses the limitedness theorem on finite automata with distance functions (in short, D-automata) in Hashiguci (1982A, 1983).

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