Generalized star-height problem

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.

Regular languages of 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.

See also

• star height problem

References

• M.P. Schützenberger, "On finite monoids having only trivial subgroups", Information and Control, 8, 190–194 (1965).

• Jean-Eric Pin, Howard Straubing and Denis Thérien, Some results on the generalized star-height problem, Information and Computation, 101(2):219–250, December 1992. Available from http://www.liafa.jussieu.fr/~jep/Resumes/StarHeight.html