Intuitionistic Linear Temporal Logics

TL;DR

This paper introduces intuitionistic linear temporal logics based on expanding posets, establishing their decidability, and exploring the definability of temporal operators within these frameworks.

Contribution

It defines new intuitionistic temporal logics on expanding posets and analyzes their properties, including decidability and operator definability.

Findings

$ iltl$ has the effective finite model property and is decidable.

$ extit{itlb}$ lacks the finite model property.

'Until' and 'release' operators are not definable from each other.

Abstract

We consider intuitionistic variants of linear temporal logic with `next', `until' and `release' based on expanding posets: partial orders equipped with an order-preserving transition function. This class of structures gives rise to a logic which we denote $\iltl$, and by imposing additional constraints we obtain the logics $\itlb$ of persistent posets and $\itlht$ of here-and-there temporal logic, both of which have been considered in the literature. We prove that $\iltl$ has the effective finite model property and hence is decidable, while $\itlb$ does not have the finite model property. We also introduce notions of bounded bisimulations for these logics and use them to show that the `until' and `release' operators are not definable in terms of each other, even over the class of persistent posets.