Modal Logics of Topological Relations

Abstract

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 Π 1 1 -hard, 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.

Axioms, Algebras and Topology

“…These have been used to capture both multiple spatial dimensions and the combination of space with time. Yet another approach is to employ modal logics in which spatial relations are associated with the accessibility relation associated with the modal operators [19,71].…”

Towards spatial reasoning in fuzzy description logics

“…It remains to see whether practical reasoning algorithms exists in the case of fuzzy GCI's (it is expected to be more involved as already shown for the crisp case [12]); (ii) our calculus applies to the case where a finite non-empty set of regions is fixed a priori. We still have to work out an algorithm for the case R is not defined a priori (see, e.g., [11], [12], [15]). …”

A new modal logic for reasoning about space: spatial propositional neighborhood logic

“…Unlike Bennett, Cohn, Wolter and Zakharyaschev's work, an important attempt to exploit the whole expressive power of modal logic for reasoning about space (instead of using it for constraint solving) is that of Lutz and Wolter's modal logic for topological relations [LW04]. Lutz and Wolter present a new propositional modal logic, where propositional variables are interpreted in the regions of topological space, and references to other regions are enabled by modal operators interpreted as topological relations.…”

“…As we have recalled in the Introduction, in the literature of spatial reasoning some attention has been given to (existential) theories such as Rectangle Algebra and Region Connection Calculus; nevertheless, for some reason, no representation theorems have been shown for spatial frames (based on directional relations) so far. Some results on this topic can be found in [LW04], and in [BCdC98].…”

“…It is worth noticing that comparing our approach with the previous (modal) ones presents some difficulties, basically because most of the previous work is based on topological relations instead of directional relations. Nevertheless, the expressive power of SpPNL can be compared with Lutz and Wolter's modal logic of topological relations [LW04] over a region structure in the Euclidean space D × D, where regions are limited to rectangles (see next section); to some extent, SpPNL can be considered as a fragment of it. Preliminary results on SpPNL have been presented in [MS06a,MS06b,MNS07].…”