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

Abstract

It is widely accepted that spatial reasoning plays a central role in artificial intelligence, for it has a wide variety of potential applications, e.g., in robotics, geographical information systems, and medical analysis and diagnosis. While spatial reasoning has been extensively studied at the algebraic level, modal logics for spatial reasoning have received less attention in the literature. In this paper we propose a new modal logic, called Spatial Propositional Neighborhood Logic (SpPNL for short) for spatial reasoning through directional relations. We study the expressive power of SpPNL, we show that it is able to express meaningful spatial statements, we prove a representation theorem for abstract spatial frames, and we devise a (non-terminating) sound and complete tableaux-based deduction system for it. Finally, we compare SpPNL with the well-known algebraic spatial reasoning system called Rectangle Algebra.

A Decidable Spatial Logic with Cone-Shaped Cardinal Directions

“…While the satisfiability problem for spatial modal logics with projection modalities turns out to be highly undecidable [7,9], we prove that Cone Logic enjoys a decidable satisfiability problem (in fact, PSPACE-complete) by taking advantage of a suitable filtration technique. We also show that Cone Logic subsumes interesting interval temporal logics such as the temporal logic of subintervals/superintervals, thus generalizing previous results in the literature [3] and basically disproving a conjecture by Lodaya [6].…”

A Decidable Spatial Generalization of Metric Interval Temporal Logic

“…As pointed out in [17], one of the possible measures of the expressive power of a directional-based spatial logic for rectangles is the comparison with Rectangle Algebra (RA) [12]. In RA, one considers a finite set of objects (rectangles) O 1 , .…”

“…In general, given an algebraic constraint network, the main problem is to establish whether or not the network is consistent, that is, if all constraints can be jointly satisfied. In [17], it has been shown that SpPNL is powerful enough to express and to check the consistency of an RA-constraint network. In [19], the authors show that the same can be done with its decidable fragment WSpPNL.…”

“…As for the first one, the most important contributions are those by Güsgen [11] and by Mukerjee and Joe [12], that introduce Rectangle Algebra (RA), later extended by Balbiani et al in [13], [14]. As for the second one, we mention Venema's Compass Logic [15], whose undecidability has been shown by Marx and Reynolds in [16], Spatial Propositional Neighborhood Logic (SpPNL for short) by Morales et al [17], that generalizes the logic of temporal neighborhood [18] to the two-dimensional space, and the fragment of SpPNL, called Weak Spatial Propositional Neighborhood Logic (WSpPNL), presented in [19]. As for quantitative spatial formalisms, the literature is very scarce.…”

A Logic of Directions

“…Some spatial logics, which can encode directions, are undecidable, e.g. the compass logic [Marx and Reynolds, 1999] and SpPNL [Morales et al, 2007]. The satisfiability problem of some spatial logics (e.g.…”