More formally, the star height of a regular expression E over a finite alphabetA 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 of a regular languageL is defined as the minimum star height among all regular expressions representing L. The intuition is here that if the languageL 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.
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 = 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 in which the number of occurrences of a and b are congruent modulo 2n is n.
Eggan's theorem
In his seminal study of the star height of regular languages, established a relation between the theories of regular expressions, finite automata, and of directed graphs. In subsequent years, this relation became known as Eggan's theorem, cf.. We recall a few concepts from graph theory and automata theory.
a set of labeled edges δ, referred to as transition relation: Q × × Q. Here ε denotes the empty word.
an initial state q0 ∈ Q
a set of states F distinguished as accepting statesF ⊆ Q.
A word w ∈ Σ* is accepted by the ε-NFA if there exists a directed path from the initial state q0 to some final state in F using edges from δ, such that the concatenation of all labels visited along the path yields the wordw. The set of all words over Σ* accepted by the automaton is the language accepted by the automaton A. When speaking of digraph properties of a nondeterministic finite automatonA with state set Q, we naturally address the digraph with vertex set Q induced by its transition relation. Now the theorem is stated as follows. Proofs of this theorem are given by, and more recently by.
Generalized star height
The above definition assumes that regular expressions are built from the elements of the alphabet A using only the standard operators set union, concatenation, and Kleene star. Generalized regular expressions are defined just as regular expressions, but here also the set complement operator is allowed . If we alter the definition such that taking complements does not increase the star height, that is, we can define the generalized star height of a regular language L as the minimum star height among all generalized regular expressions representing L. Note that, whereas it is immediate that a language of star height 0 can contain only finitely many words, there exist infinite languages having generalized star height 0. For instance, the regular expression which we saw in the example above, can be equivalently described by the generalized regular expression since the complement of the empty set is precisely the set of all words over A. Thus the set of all words over the alphabet A ending in the letter a has star height one, while its generalized star height equals zero. Languages of generalized star height zero are also called star-free languages. It can be shown that a language L is star-free if and only if its syntactic monoid is aperiodic.
