Tableau Calculus for the Logic of Comparative Similarity over Arbitrary Distance Spaces

Abstract

The logic $\mathcal{CSL}$ (first introduced by Sheremet, Tishkovsky, Wolter and Zakharyaschev in 2005) allows one to reason about distance comparison and similarity comparison within a modal language. The logic can express assertions of the kind ”A is closer/more similar to B than to C” and has a natural application to spatial reasoning, as well as to reasoning about concept similarity in ontologies. The semantics of $\mathcal{CSL}$ is defined in terms of models based on different classes of distance spaces and it generalizes the logic S4 _u of topological spaces. In this paper we consider $\mathcal{CSL}$ defined over arbitrary distance spaces. The logic comprises a binary modality to represent comparative similarity and a unary modality to express the existence of the minimum of a set of distances. We first show that the semantics of $\mathcal{CSL}$ can be equivalently defined in terms of preferential models. As a consequence we obtain the finite model property of the logic with respect to its preferential semantic, a property that does not hold with respect to the original distance-space semantics. Next we present an analytic tableau calculus based on its preferential semantics. The calculus provides a decision procedure for the logic, its termination is obtained by imposing suitable blocking restrictions.