Examples
While computing the star height of a regular expression is easy, determining the star height of a language can be sometimes tricky. For illustration, the regular expression
over the alphabet A = {a,b} has star height 2. However, the described language is just the set of all words ending in an a: thus the language can also be described by the expression
which is only of star height 1. To prove that this language indeed has star height 1, one still needs to rule out that it could be described by a regular expression of lower star height. For our example, this can be done by an indirect proof: One proves that a language of star height 0 contains only finitely many words. Since the language under consideration is infinite, it cannot be of star height 0.
The star height of a group language is computable: for example, the star height of the language over {a,b} in which the number of occurrences of a and b are congruent modulo 2n is n.
Read more about this topic: Star Height
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