Finite Presentations of Infinite Structures: Automata and Interpretations

Abstract We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by model-theoretic interpretations. These two ways of presenting a structure are related. Indeed, a structure is automatic if, and only if, it is first-order interpretable in an appropriate expansion of Presburger arithmetic or, equivalently, in the infinite binary tree with prefix order and equal length predicate. Similar results hold for ω-automatic structures and appropriate expansions of the real ordered group. We also discuss the relationship to automatic groups. The model checking problem for FO(∃ω), first-order logic extended by the quantifier “there are infinitely many”, is proved to be decidable for automatic and ω-automatic structures. Further, the complexity for various fragments of first-order logic is determined. On the other hand, several important properties not expressible in FO, such as isomorphism or connectedness, turn out to be undecidable for automatic structures. Finally, we investigate methods for proving that a structure does not admit an automatic presentation, and we establish that the class of automatic structures is closed under Feferman–Vaught-like products.