Logics of polyhedral reachability

TL;DR

This paper develops a complete axiom system for a polyhedral spatial logic with reachability modalities, enabling formal reasoning about 3D spatial structures in applications like mesh analysis.

Contribution

It introduces an axiom system for a polyhedral modal logic with reachability, proving its completeness with respect to polyhedral semantics and finite models.

Findings

The logic has the finite model property.

Every finite poset model is realizable as a polyhedral model.

The system is complete for polyhedral semantics.

Abstract

Polyhedral semantics is a recently introduced branch of spatial modal logic, in which modal formulas are interpreted as piecewise linear subsets of an Euclidean space. Polyhedral semantics for the basic modal language has already been well investigated. However, for many practical applications of polyhedral semantics, it is advantageous to enrich the basic modal language with a reachability modality. Recently, a language with an Until-like spatial modality has been introduced, with demonstrated applicability to the analysis of 3D meshes via model checking. In this paper, we exhibit an axiom system for this logic, and show that it is complete with respect to polyhedral semantics. The proof consists of two major steps: First, we show that this logic, which is built over Grzegorczyk's system $\mathsf{Grz}$, has the finite model property. Subsequently, we show that every formula satisfied in a finite poset is also satisfied in a polyhedral model, thereby establishing polyhedral completeness.