Uniform Interpolation of Basic Tense Logic

2026-06-30

AI Summary

The uniform interpolation property for basic tense logic—namely, bidirectional modal logic K equipped with converse modalities—has long remained unestablished. This work resolves the issue by extending Visser’s semantic argument for modal logic K, introducing a layered (bounded) bisimulation technique based on Kripke models, and incorporating a refined treatment of converse relations. For the first time, a uniform interpolation theorem is thereby established for this logic. The result not only fills a crucial gap in the interpolation theory of basic tense logic but also broadens the applicability of existing semantic proof frameworks for multimodal logics.

Abstract

This paper establishes the uniform interpolation theorem for basic tense logic, which is also known as two-way modal logic or modal logic with converse. First introduced by Arthur Prior, basic tense logic is a syntactic expansion of basic modal logic with a converse modality. Its corresponding accessibility relation is defined as the converse of the standard accessibility relation in a given Kripke model. Although basic tense logic has been widely studied since its introduction, its uniform interpolation property has yet to be fully established. For basic modal logic K, Albert Visser (1996) provided a semantic argument formulated in terms of layered (or bounded) bisimulation, explicitly attributing the uniform interpolation property of K to Silvio Ghilardi. This paper extends Visser's semantic argument to demonstrate that basic tense logic also enjoys the uniform interpolation property.

Problem

• uniform interpolation
• basic tense logic
• two-way modal logic
• converse modality
• Kripke model

Innovation

• uniform interpolation
• basic tense logic
• converse modality
• layered bisimulation
• Kripke semantics