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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 (computer science)|alphabet]], the [[empty set]] symbol, all [[boolean operators]] – including [[Complement (set theory)|complementation]] – and [[concatenation]] but no [[Kleene star]].<ref name=Law235>Lawson (2004) p.235</ref> For instance, the language of words over the alphabet <math>\{a,\,b\}</math> that do not have consecutive a's can be defined by <math>(\emptyset^c aa \emptyset^c)^c</math>, where <math>X^c</math> denotes the complement of a subset <math>X</math> of <math>\{a,\,b\}^*</math>. The condition is equivalent to having [[generalized star height]] zero.
| | Andera is what you can call her but she never truly liked that name. I am really fond of to go to karaoke but I've been using on new issues lately. Since he was 18 he's been working as an info officer but he ideas on changing it. Alaska is where I've always been residing.<br><br>Feel free to visit my web-site :: online psychics; [http://kard.dk/?p=24252 kard.dk], |
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| [[Marcel-Paul Schützenberger]] characterized star-free languages as those with [[Aperiodic monoid|aperiodic]] [[syntactic monoid]]s.<ref>{{cite journal | author=[[Marcel-Paul Schützenberger]] | title=On finite monoids having only trivial subgroups | journal=Information and Computation| year=1965| volume=8 | issue=2 | pages=190–194|url=http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1965-4TrivialSubgroupsIC.pdf}}</ref><ref name=Law262>Lawson (2004) p.262</ref> 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,<ref>{{cite book | last=Straubing | first=Howard | title=Finite automata, formal logic, and circuit complexity | series=Progress in Theoretical Computer Science | location=Basel | publisher=Birkhäuser | year=1994 | isbn=3-7643-3719-2 | zbl=0816.68086 | page=79 }}</ref> as the [[counter-free language]]s<ref>{{cite book | last1=McNaughton | first1=Robert | last2 = Papert | first2=Seymour | author2-link=Seymour Papert | others=With an appendix by William Henneman | series=Research Monograph | volume=65 | year=1971 | title=Counter-free Automata | publisher=MIT Press | isbn=0-262-13076-9 | zbl=0232.94024 }}</ref> and as languages definable in [[linear temporal logic]].<ref>{{cite book | last = Kamp| first=Johan Antony Willem |authorlink=Hans Kamp | title = Tense Logic and the Theory of Linear Order| publisher = University of California at Los Angeles (UCLA) | year=1968}}</ref>
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| All star-free languages are in uniform [[AC0|AC<sup>0</sup>]].
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| ==See also==
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| *[[Star height]]
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| *[[Generalized star height problem]]
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| *[[Star height problem]]
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| ==References==
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| {{reflist}}
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| * {{cite book | last=Lawson | first=Mark V. | title=Finite automata | publisher=Chapman and Hall/CRC | year=2004 | isbn=1-58488-255-7 | zbl=1086.68074 }}
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| * {{cite book|editor=Jörg Flum | editor2=Erich Grädel | editor3=Thomas Wilke | title=Logic and automata: history and perspectives | year=2008 | publisher=Amsterdam University Press | isbn=978-90-5356-576-6 | url=http://www.lsv.ens-cachan.fr/Publis/PAPERS/PDF/DG-WT08.pdf | chapter=First-order definable languages | first1=Volker | last1=Diekert | first2=Paul | last2=Gastin | unused_data=chapter}}
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| {{Formal languages and grammars}}
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| [[Category:Logic in computer science]]
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| [[Category:Formal languages]]
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| [[Category:Automata theory]]
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| {{comp-sci-theory-stub}}
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Latest revision as of 19:50, 9 August 2014
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Feel free to visit my web-site :: online psychics; kard.dk,