Algorithms for determining relative star height and star height

Abstract

Let C = { R 1 , …, R m } be a finite class of regular languages over a finite alphabet Σ. Let Δ = { b 1 , …, b m } be an alphabet, and δ be the substitution from Δ ∗ into Σ ∗ such that δ ( b i ) = R i for all i (1 ≤ i ≤ m ). Let R be a regular language over Σ which can be defined from C by a finite number of applications of the operators union, concatenation, and star. Then there exist regular languages over Δ which can be transformed onto R by δ. The relative star height of R w.r.t. C is the minimum star height of regular languages over Δ which can be transformed onto R by δ. This paper proves the existence of an algorithm for determining relative star height. This result obviously implies the existence of an algorithm for determining the star height of any regular language.