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. That is, they 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 property of being automatic does not depend on the set of generators.
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.