In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata 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.

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  • In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata 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: the word-acceptor, which accepts for every element of G at least one word in A representing it multipliers, one for each <math>a \in A \cup \{1\}</math>, which accept a pair (w1, w2), for words wi accepted by the word-acceptor, precisely when <math>w_1 a = w_2</math> in G. The property of being automatic does not depend on the set of generators. The concept of automatic groups generalizes naturally to automatic semigroups.
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  • In mathematics, an automatic group is a finitely generated group equipped with several finite-state automata. These automata 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.
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  • Automatic group
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