Автоматтық топтар және олардың автоматтық құрылымы

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 word acceptor, which accepts for every element of G at least one word in representing it;
multipliers, one for each , which accept a pair (w1, w2), for words wi accepted by the word acceptor, precisely when in G.

The property of being automatic does not depend on the set of generators.

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 represent the same element (using the multiplier for ). Automatic groups are characterized by the fellow traveler property. Let denote the distance between in the Cayley graph of Then, G is automatic with respect to a word acceptor L if and only if there is a constant such that for all words which differ by at most one generator, the distance between the respective prefixes of u and v is bounded by C. In other words, where for the k th prefix of (or itself if ). This means that when reading the words synchronously, it is possible to keep track of the difference between both elements with a finite number of states (the neighborhood of the identity with diameter C in the Cayley graph).