Topological Logics of Path-Reachability

2026-06-30

AI Summary

This study investigates a topological modal logic combining path reachability and Cantor derivative modalities, aiming to establish a complete and decidable axiomatization. To this end, the authors introduce a neighborhood-style semantics that facilitates finite model property analysis and construct the first complete axiom systems for both T₁ spaces and the class of all metric spaces. The main contributions include the first proofs of completeness and decidability for this hybrid modal logic over T₁ and metric spaces, as well as an axiomatization of the full topological class in a weaker language. These results provide a foundational theoretical framework for understanding the semantic and computational properties of related logical systems.

Abstract

The topological semantics of modal logic has been an active area of research ever since their introduction in the 1940s, with attention shifting in recent years from standard unimodal logic to more expressive frameworks. In particular, an Until-like path-reachability modality has recently been studied in Bezhanishvili et al. (2024) in polyhedral semantics; this paper investigates its topological counterpart. Focusing on the language combining said modality with the classical Cantor derivative modality, we exhibit an axiomatic system sound and complete both for the class of T1 topologies and for the class of all metric spaces, and establish its decidability. We also axiomatize the logic of all topological models in a weaker language obtained by substituting the closure modality for the Cantor derivative. To prove our results, we introduce an equivalent neighborhood-like semantics allowing for the finite model property.

Problem

• topological logic
• path-reachability
• Cantor derivative
• metric spaces
• axiomatization

Innovation

• topological modal logic
• path-reachability modality
• Cantor derivative
• finite model property
• decidability