Decidability and complexity in automatic monoids

Abstract

We prove several complexity and decidability results for automatic monoids: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the first-order theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Moreover, we show that for every hyperbolic group the word problem belongs to LOGCFL, which improves a result of Cai [4]. © Springer-Verlag Berlin Heidelberg 2004.