Generalized star-height problem
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Unsolved problem in computer science: Can all regular languages be expressed using generalized regular expressions with a limited nesting depth of Kleene stars? (more unsolved problems in computer science)
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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 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
- Eggan's theorem and Generalized star height sections of the Star height article
- Star height problem
References
- ↑ Sakarovitch (2009) p.171
- Janusz A. Brzozowski (1980). "Open problems about regular languages". in Ronald V. Book. Formal Language Theory: Perspectives and Open Problems. Academic Press. pp. 23–47.
- Wolfgang Thomas (1981). "Remark on the star-height-problem". Theoretical Computer Science 13 (2): 231–237. doi:10.1016/0304-3975(81)90041-4.
- Jean-Eric Pin; Howard Straubing; Denis Thérien (1992). "Some results on the generalized star-height problem". Information and Computation 101 (2): 219–250. doi:10.1016/0890-5401(92)90063-L. https://www.irif.fr/~jep//PDF/StarHeight.pdf.
- Sakarovitch, Jacques (2009). Elements of automata theory. Translated from the French by Reuben Thomas. Cambridge: Cambridge University Press. ISBN 978-0-521-84425-3.
- Marcel-Paul Schützenberger (1965). "On finite monoids having only trivial subgroups". Information and Control 8 (2): 190–194. doi:10.1016/S0019-9958(65)90108-7.
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