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.[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: Open problems about regular languages, In: Ronald V. Book, editor, Formal language theory—Perspectives and open problems, pp. 23–47. Academic Press, 1980.
- "Remark on the star-height-problem". Theoretical Computer Science. 13 (2): 231–237. 1981. doi:10.1016/0304-3975(81)90041-4. MR 0594062.
{{cite journal}}: Unknown parameter|auhtor=ignored (|author=suggested) (help) - 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
- 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.
- Schützenberger M.P. (1965). "On finite monoids having only trivial subgroups". Information and Control. 8 (2): 190–194. doi:10.1016/S0019-9958(65)90108-7. ISSN 0019-9958. Zbl 0131.02001.
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