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In mathematics, an automatic group is a finitely generated group equipped with several 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.〔.〕 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:〔, Section 2.3, "Automatic Groups: Definition", pp. 45–51.〕 * the ''word-acceptor'', which accepts for every element of ''G'' at least one word in representing it *''multipliers'', one for each , which accept a pair (''w''1, ''w''2), for words ''w''''i'' accepted by the word-acceptor, precisely when in ''G''. The property of being automatic does not depend on the set of generators.〔, Section 2.4, "Invariance under Change of Generators", pp. 52–55.〕 The concept of automatic groups generalizes naturally to automatic semigroups.〔, Section 6.1, "Semigroups and Specialized Axioms", pp. 114–116.〕 ==Properties== Automatic groups have 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.〔, Theorem 2.3.10, p. 50.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「automatic group」の詳細全文を読む スポンサード リンク
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