First-Order Automata

First-order linear temporal logic (FOLTL) is a flexible and expressive formalism capable of naturally describing complex behaviors and properties. Although the logic is in general highly undecidable, the idea of using it as a specification language for the verification of complex infinite-state systems is appealing. However, a missing piece, which has proved to be an invaluable tool in dealing with other temporal logics, is an automaton model capable of capturing the logic. In this paper we address this issue, by defining and studying such a model, which we call first-order automaton. We define this very general class of automata, and the corresponding notion of regular first-order language (of finite words), showing their closure under most language-theoretic operations. We show how they can capture any FOLTL formula over finite words, over any signature and theory, and provide sufficient conditions for the semi-decidability of their non-emptiness problem. Then, to show the usefulness of the formalism, we prove the decidability of monodic FOLTL, a classic result known in the literature, with a simpler and direct proof.