Formal Definition
More formally, the star height of a regular expression E over a finite alphabet A is inductively defined as follows:
- , and for all alphabet symbols a in A.
Here, is the special regular expression denoting the empty set and ε the special one denoting the empty word; E and F are arbitrary regular expressions.
The star height h(L) of a regular language L is defined as the minimum star height among all regular expressions representing L. The intuition is here that if the language L has large star height, then it is in some sense inherently complex, since it cannot be described by means of an "easy" regular expression, of low star height.
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