Exploring the Jungle of Intuitionistic Temporal Logics

TL;DR

This paper investigates various axiomatic systems for intuitionistic linear temporal logic, establishing their soundness over different structures and introducing a novel topological semantics for the 'henceforth' modality.

Contribution

It introduces new axiomatic systems for intuitionistic linear temporal logic and a novel topological semantics for the 'henceforth' modality, demonstrating their soundness and distinctness.

Findings

Seven distinct intuitionistic temporal logics identified.

New topological semantics for 'henceforth' modality proposed.

Soundness established for multiple classes of structures.

Abstract

The importance of intuitionistic temporal logics in Computer Science and Artificial Intelligence has become increasingly clear in the last few years. From the proof-theory point of view, intuitionistic temporal logics have made it possible to extend functional languages with new features via type theory, while from its semantical perspective several logics for reasoning about dynamical systems and several semantics for logic programming have their roots in this framework. In this paper we consider several axiomatic systems for intuitionistic linear temporal logic and show that each of these systems is sound for a class of structures based either on Kripke frames or on dynamic topological systems. Our topological semantics features a new interpretation for the `henceforth' modality that is a natural intuitionistic variant of the classical one. Using the soundness results, we show that the seven logics obtained from the axiomatic systems are distinct.