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Star-free language

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Title: Star-free language  
Author: World Heritage Encyclopedia
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Subject: Aperiodic finite state automaton, Automata theory, Regular language, Generalized star height problem, Aperiodic semigroup
Collection: Automata Theory, Formal Languages, Logic in Computer Science
Publisher: World Heritage Encyclopedia

Star-free language

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.

An example of a regular language which is not star-free is \{(aa)^n \mid n \geq 0\}.[2]

Marcel-Paul Schützenberger characterized star-free languages as those with aperiodic syntactic monoids.[3][4] They can also be characterized logically as languages definable in FO[<], the first-order logic over the natural numbers with the less-than relation,[5] as the counter-free languages[6] and as languages definable in linear temporal logic.[7]

All star-free languages are in uniform AC0.

See also


  1. ^ Lawson (2004) p.235
  2. ^ Arto Salomaa (1981). Jewels of Formal Language Theory. Computer Science Press. p. 53.  
  3. ^  
  4. ^ Lawson (2004) p.262
  5. ^ Straubing, Howard (1994). Finite automata, formal logic, and circuit complexity. Progress in Theoretical Computer Science. Basel: Birkhäuser. p. 79.  
  6. ^ McNaughton, Robert;  
  7. ^  
  • Lawson, Mark V. (2004). Finite automata. Chapman and Hall/CRC.  
  • Diekert, Volker; Gastin, Paul (2008). "First-order definable languages". In Jörg Flum; Erich Grädel; Thomas Wilke. Logic and automata: history and perspectives (PDF). Amsterdam University Press.  

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