Finite Presentations of Infinite Structures: Automata and Interpretations

Abstract

We study definability problems and algorithmic issues for infinite struc- tures 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 re- lated. 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 express- ible 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.