On groups whose word problem is solved by a~counter automaton

Abstract

We prove that a group G has a word problem that is accepted by a deterministic counter automaton with a weak inverse property if and only if G is virtually abelian. We extend this result to larger classes of groups by considering a generalization of finite state automata, counter automata and pushdown automata. Natural corollaries of our general result include a restricted version of Herbst's classification of groups for which the word problem is a one counter language and a new classification of automata that accept context-free word problems.