First-order Temporal Logic on Finite Traces: Semantic Properties, Decidable Fragments, and Applications

TL;DR

This paper studies first-order temporal logic on finite traces, analyzing semantic properties, decidable fragments, and applications in planning and verification, revealing complexity results and conditions for reasoning equivalences.

Contribution

It provides new semantic and syntactic conditions for reasoning over finite traces and establishes complexity bounds for satisfiability in various fragments.

Findings

Satisfiability on finite traces is ExpSpace-complete for several fragments.

Complexity decreases to NExpTime for bounded finite traces.

Connections between finite trace reasoning and safety properties are established.

Abstract

Formalisms based on temporal logics interpreted over finite strict linear orders, known in the literature as finite traces, have been used for temporal specification in automated planning, process modelling, (runtime) verification and synthesis of programs, as well as in knowledge representation and reasoning. In this paper, we focus on first-order temporal logic on finite traces. We first investigate preservation of equivalences and satisfiability of formulas between finite and infinite traces, by providing a set of semantic and syntactic conditions to guarantee when the distinction between reasoning in the two cases can be blurred. Moreover, we show that the satisfiability problem on finite traces for several decidable fragments of first-order temporal logic is ExpSpace-complete, as in the infinite trace case, while it decreases to NExpTime when finite traces bounded in the number of instants are considered. This leads also to new complexity results for temporal description logics over finite traces. Finally, we investigate applications to planning and verification, in particular by establishing connections with the notions of insensitivity to infiniteness and safety from the literature.