Modal Logics of Topological Relations

TL;DR

This paper introduces modal logics for topological spatial relations, analyzing their expressive power and computational complexity, and compares them to existing temporal and interval logics, with implications for spatial reasoning in AI.

Contribution

It develops a family of modal logics based on topological relations, exploring their expressive equivalence to two-variable first-order logic and their high undecidability, extending prior results.

Findings

Modal logics have the same expressive power as two-variable first-order logic.

Complexity of these logics ranges from recursively enumerable to highly undecidable.

Results improve upon previous undecidability findings for interval temporal logics.

Abstract

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity.