The Solutions of Two Star-Height Problems for Regular Trees

Abstract Regular trees can be defined by two types of rational expressions. For these two types we solve the star-height problem, i.e., we show how to construct a rational expression of minimal star-height from the minimal graph of the given tree (i.e., the analogue of the minimal deterministic automation for regular languages). In one case, the minimal starheight is the rank (in the sense of Eggan) of the minimal graph. There corresponds a characterization of the star-height of a prefix-free regular language w.r.t. rational expressions of a special kind (called deterministic) as the rank of its minimal deterministic automaton considered as a graph.

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