Automatic Structures of Bounded Degree

Abstract

The first-order theory of an automatic structure is known to be decidable but there are examples of automatic structures with nonelementary first-order theories. We prove that the first-order theory of an automatic structure of bounded degree (meaning that the corresponding Gaifman-graph has bounded degree) is elementary decidable. More precisely, we prove an upper bound of triply exponential alternating time with a linear number of alternations. We also present an automatic structure of bounded degree such that the corresponding first-order theory has a lower bound of doubly exponential time with a linear number of alternations. We prove similar results also for tree automatic structures.