Model Checking Spatial Logics for Closure Spaces

TL;DR

This paper introduces a topology-based logical framework for verifying spatial properties in physical and discrete spaces, extending traditional temporal verification methods to include spatial modalities and collective operators.

Contribution

It develops a novel logic for spatial verification in closure spaces, including new operators and efficient model checking procedures, with a proof-of-concept implementation.

Findings

Defined a spatial logic based on topological and closure space semantics.

Extended the logic with new spatial and collective operators.

Provided efficient model checking algorithms and a proof-of-concept tool.

Abstract

Spatial aspects of computation are becoming increasingly relevant in Computer Science, especially in the field of collective adaptive systems and when dealing with systems distributed in physical space. Traditional formal verification techniques are well suited to analyse the temporal evolution of programs; however, properties of space are typically not taken into account explicitly. We present a topology-based approach to formal verification of spatial properties depending upon physical space. We define an appropriate logic, stemming from the tradition of topological interpretations of modal logics, dating back to earlier logicians such as Tarski, where modalities describe neighbourhood. We lift the topological definitions to the more general setting of closure spaces, also encompassing discrete, graph-based structures. We extend the framework with a spatial surrounded operator, a propagation operator and with some collective operators. The latter are interpreted over arbitrary sets of points instead of individual points in space. We define efficient model checking procedures, both for the individual and the collective spatial fragments of the logic and provide a proof-of-concept tool.