Generalized star-height problem

In formal language theory, the generalized star-height problem 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 with 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.

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. A theorem of Schützenberger provides an algebraic characterization of star-free languages by means of aperiodic syntactic monoids.^[2] In particular, star-free languages are a proper decidable subclass of regular languages.