An Automaton Group with PSPACE-Complete Word Problem

Abstract

We construct an automaton group with a -complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely -complete, compressed word problem and acts over a binary alphabet. Thus, it is optimal in terms of the alphabet size. Our construction directly simulates the computation of a Turing machine in an automaton group and, therefore, seems to be quite versatile. It combines two ideas: the first one is a construction used by D’Angeli, Rodaro and the first author to obtain an inverse automaton semigroup with a -complete word problem and the second one is to utilize a construction used by Barrington to simulate Boolean circuits of bounded degree and logarithmic depth in the group of even permutations over five elements.

The Self-Similarity of Free Semigroups and Groups

“…It also makes sense to consider a slight variation of this problem where one dedicated state of the automaton induces the identity function and we exclude this state as an element of the basis. 10 For a discussion of the word problem with respect to automaton structures see [10,38]. 11 There is an automaton group with an undecidable conjugacy problem [33].…”

On the orbits of automaton semigroups and groups

“…An example for this is the order problem: őrst, it could be shown to be undecidable for automaton semigroups [11] and later this result could be extended to automaton groups [2,12]. Similarly, a result on the complexity of the word problem could be lifted from the (inverse) semigroup case [9] to the groups case [20]. For another important problem, the őniteness problem, the current state is that it has been proven to be undecidable in the semigroup case [11] (and also in a more restrictive setting [8]) but the decidability of the problem in the group case remains unknown.…”