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| In [[mathematics]], an '''automatic group''' is a [[finitely generated group]] equipped with several [[finite state machine|finite-state automata]]. These automata represent the [[Cayley graph]] of the group, i. e. can tell if a given word representation of a group element is in a "canonical form" and can tell if two elements given in canonical words differ by a generator.<ref>{{citation
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| | last1 = Epstein | first1 = David B. A. | authorlink1 = David B. A. Epstein
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| | last2 = Cannon | first2 = James W.
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| | last3 = Holt | first3 = Derek F.
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| | last4 = Levy | first4 = Silvio V. F.
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| | last5 = Paterson | first5 = Michael S.
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| | last6 = Thurston | first6 = William P. | authorlink6 = William Thurston
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| | title = Word Processing in Groups
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| | publisher = Jones and Bartlett Publishers | location = Boston, MA | year = 1992 | isbn = 0-86720-244-0}}.</ref>
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| More precisely, let ''G'' be a group and ''A'' be a finite set of generators. Then an ''automatic structure'' of ''G'' with respect to ''A'' is a set of finite-state automata:<ref>{{harvtxt|Epstein|Cannon|Holt|Levy|1992}}, Section 2.3, "Automatic Groups: Definition", pp. 45–51.</ref>
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| * the ''word-acceptor'', which accepts for every element of ''G'' at least one word in <math>A^\ast</math> representing it
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| *''multipliers'', one for each <math>a \in A \cup \{1\}</math>, which accept a pair (''w''<sub>1</sub>, ''w''<sub>2</sub>), for words ''w''<sub>''i''</sub> accepted by the word-acceptor, precisely when <math>w_1 a = w_2</math> in ''G''.
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| The property of being automatic does not depend on the set of generators.<ref>{{harvtxt|Epstein|Cannon|Holt|Levy|1992}}, Section 2.4, "Invariance under Change of Generators", pp. 52–55.</ref>
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| The concept of automatic groups generalizes naturally to [[automatic semigroup]]s.<ref>{{harvtxt|Epstein|Cannon|Holt|Levy|1992}}, Section 6.1, "Semigroups and Specialized Axioms", pp. 114–116.</ref> | |
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| ==Properties==
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| Automatic groups have [[word problem for groups|word problem]] solvable in quadratic time. More strongly, a given word can actually be put into canonical form in quadratic time, based on which the word problem may be solved by testing whether the canonical forms of two words are equal.<ref>{{harvtxt|Epstein|Cannon|Holt|Levy|1992}}, Theorem 2.3.10, p. 50.</ref>
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| ==Examples of automatic groups==
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| The automatic groups include:
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| *[[Finite group]]s. To see this take the regular language to be the set of all words in the finite group.
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| *[[Negatively curved group]]s
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| *[[Euclidean group]]s
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| *All finitely generated [[Coxeter group]]s <ref name="BrinkAndHowlett">{{Citation | last = Brink and Howlett | title = A finiteness property and an automatic structure for Coxeter groups | year = 1993 | journal = Mathematische Annalen | publisher = Springer Berlin / Heidelberg | issn= 0025-5831 | postscript = .}}</ref>
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| *[[Braid group]]s
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| *[[Geometrically finite group]]s
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| ==Examples of non-automatic groups==
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| * [[Baumslag-Solitar group]]s
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| * Non-[[Euclidean group|Euclidean]] [[nilpotent group]]s
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| ==Biautomatic groups==
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| A group is '''biautomatic''' if it has two multiplier automata, for left and right multiplication by elements of the generating set respectively. A biautomatic group is clearly automatic.<ref>{{citation | title=Algorithmic problems in groups and semigroups | series=Trends in mathematics | first=Jean-Camille | last=Birget | publisher=Birkhäuser | year=2000 | isbn=0-8176-4130-0 | page=82}}</ref>
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| Examples include:
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| * A [[hyperbolic group]].<ref name=charney>{{citation | last=Charney | first=Ruth | title=Artin groups of finite type are biautomatic | journal=Mathematische Annalen | volume= 292 | year=1992 | doi=10.1007/BF01444642 | pages=671–683}}</ref>
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| * An [[Artin group of finite type]].<ref name=charney/>
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| ==Automatic structures==
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| The idea of describing algebraic structures with finite-automata can be generalized from groups to other structures.<ref>{{cite web | title = Some Thoughts On Automatic Structures | id = {{citeseerx|10.1.1.7.3913}} | first1 = Bakhadyr | last1 = Khoussainov | first2 = Sasha | last2 = Rubin | year = 2002 }}</ref>
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| == References ==
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| {{reflist}}
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| ==Additional reading==
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| *{{citation
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| | last1 = Chiswell | first1 = Ian
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| | title = A Course in Formal Languages, Automata and Groups
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| | publisher = Springer | year = 2008 | isbn = 978-1-84800-939-4}}.
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| [[Category:Computability theory]]
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| [[Category:Properties of groups]]
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| [[Category:Combinatorics on words]]
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| [[Category:Computational group theory]]
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Hello. Let me introduce the author. Her title is Emilia Shroyer but it's not the most female name out there. For many years he's been residing in North Dakota and his family enjoys it. The thing she adores most is physique developing and now she is attempting to earn cash with it. For years he's been working as a receptionist.
my blog post ... Blaze16.com