Star-free language

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A regular language is said to be star-free if it can be described by a regular expression constructed from the letters of the alphabet, the empty set symbol, all boolean operators – including complementation – and concatenation but no Kleene star.[1] For instance, the language of words over the alphabet \{a,\,b\} that do not have consecutive a's can be defined by (\emptyset^c aa \emptyset^c)^c, where X^c denotes the complement of a subset X of \{a,\,b\}^*. The condition is equivalent to having generalized star height zero.

Marcel-Paul Schützenberger characterized star-free languages as those with aperiodic syntactic monoids.[2][3] They can also be characterized logically as languages definable in FO[<], the monadic first-order logic over the natural numbers with the less-than relation,[4] as the counter-free languages[5] and as languages definable in linear temporal logic.[6]

All star-free languages are in uniform AC0.

See also[edit]


  1. ^ Lawson (2004) p.235
  2. ^ Marcel-Paul Schützenberger (1965). "On finite monoids having only trivial subgroups". Information and Computation 8 (2): 190–194. doi:10.1016/s0019-9958(65)90108-7. 
  3. ^ Lawson (2004) p.262
  4. ^ Straubing, Howard (1994). Finite automata, formal logic, and circuit complexity. Progress in Theoretical Computer Science. Basel: Birkhäuser. p. 79. ISBN 3-7643-3719-2. Zbl 0816.68086. 
  5. ^ McNaughton, Robert; Papert, Seymour (1971). Counter-free Automata. Research Monograph 65. With an appendix by William Henneman. MIT Press. ISBN 0-262-13076-9. Zbl 0232.94024. 
  6. ^ Kamp, Johan Antony Willem (1968). Tense Logic and the Theory of Linear Order. University of California at Los Angeles (UCLA).