Automatic group
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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 , which accept a pair (w1, w2), for words wi accepted by the word-acceptor, precisely when w1a = w2 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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[edit] Properties
- Automatic groups have word problem solvable in quadratic time. A given word can actually be put into canonical form in quadratic time.
[edit] Examples of automatic groups
- Finite groups
- Negatively curved groups
- Euclidean groups
- Coxeter groups
- Braid groups
- Geometrically finite groups
[edit] Examples of non-automatic groups
[edit] References
- Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S. & Thurston, William P. (1992), Word Processing in Groups, Boston, MA: Jones and Bartlett Publishers, ISBN 0-86720-244-0.