A General Automata Model for First-Order Temporal Logics (Extended Version)
Luca Geatti, Alessandro Gianola, Nicola Gigante
TL;DR
We address the challenge of algorithmically handling first-order temporal logic by introducing first-order automata and first-order regular languages that symbolically represent infinite-state behaviors. We show how FOLTL and related logics can be encoded into these automata, establish closure properties, and derive sufficient semi-decidability results for non-emptiness; we also provide a self-contained, direct proof of the decidability of monodic FOLTL via monadic automata. The pure-past fragment yields deterministic automata, enabling practical applications in reactive synthesis and monitoring. The framework connects to existing automata-theoretic approaches while offering a broad, principled basis for future verification, synthesis, and runtime analysis of infinite-state systems.
Abstract
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, showing their closure under most common language-theoretic operations. We show how they can capture any FOLTL formula 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.