Generalized star-height problem 📖 Wikipedia

The generalized star-height problem in formal language theory is the open question whether all regular languages can be expressed using generalized regular expressions with a limited nesting depth of Kleene stars. Here, generalized regular expressions are defined like regular expressions, but they have a built-in complement operator. For a regular language, its generalized star height is defined as the minimum nesting depth of Kleene stars needed in order to describe the language by means of a generalized regular expression, hence the name of the problem.

More specifically, it is an open question whether a nesting depth of more than 1 is required, and if so, whether there is an algorithm to determine the minimum required star height.^[1]

Regular languages of generalized star-height 0 are also known as star-free languages. The theorem of Schützenberger provides an algebraic characterization of star-free languages by means of aperiodic syntactic monoids. In particular star-free languages are a proper decidable subclass of regular languages.

• Eggan's theorem and Generalized star height sections of the Star height article
• Star height problem

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• Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. ISBN 978-0-521-84425-3. Zbl 1188.68177.
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• Jean-Eric Pin: The star-height problem