Extended finite automata over groups

Some results from Dassow and Mitrana (Internat. J. Comput. Algebra (2000)), Griebach (Theoret. Comput. Sci. 7 (1978) 311)and Ibarra et al. (Theoret. Comput. Sci. 2 (1976) 271) are generalized Fur finite automata over arbitrary groups. The closure properties of these automata are poorer and the accepting power is smaller when abelian groups are considered. We prove that the addition of any abelian group to a finite automaton is less powerful than the addition of the multiplicative group of rational numbers. Thus, each language accepted by a finite automaton over an abelian group is actually a unordered vector language, Characterizations of the context-free and recursively enumerable languages classes are set up in the case of non-abelian groups. We investigate also deterministic finite automata over groups, especially over abelian groups. (C) 2001 Elsevier Science B.V. All rights reserved.