Towards Linguistically-Grounded Spatial Logics

Spatial Logics and Spatial Language. The system of Euclidean geometry is defined by geometrical axioms with points and lines as primitive entities [Hilbert, 1899, Tarski, 1959]. It provides the ground for spatial logics, a formal language used for describing geometrical entities and configurations [Aiello et al., 2007], interpreted over a class of geometric structures. These structures can be any kind of geometrical spaces, such as topological spaces, affine spaces, or metric spaces. While spatial logics can specify spatial (geometry-based) entities, relations, and their axioms in a broad way, a more specific subclass are logics for qualitative spatial reasoning, often called spatial calculi. Their motivation builds on the idea that reasoning over qualitative descriptions resemble more closely human reasoning, and it may thus lead to more efficient and effective reasoning strategies than reasoning over numerical descriptions [Cohn and Hazarika, 2001]. Spatial calculi often focus on one particular qualitative spatial aspect, such as distance, shape, orientation, or topology, and on primitive types of objects in the Euclidean plane, such as points or regions. They define possible spatial relations among spatial entities and their axioms accordingly. Some spatial calculi use linguistic terms to define spatial relations (e.g., [Kurata and Shi, 2008]), aiming at being cognitively adequate and suitable particularly for natural language interpretation. These calculi, however, not precisely reflect natural language semantics but meanings that are determined by the calculus’ own axioms. Specifically, the diverse use of language [Bateman, 2010] requires more complex as well as more flexible logical formalizations. Such formalisms have to take into account contributing aspects for spatial language interpretations, such as context or world knowledge, in order to achieve a link or mapping from spatial language to logics [Hois and Kutz, 2008b], i.e., from a natural language sentence to an abstract spatial formalization [Kordjamshidi et al., 2010]. Even though spatial calculi might not be able to exactly reflect spatial language, they can formalize certain aspects of it, e.g., useful for spatial dialogue systems [Ross, 2008]. We propose a method to analyze the amount of coverage and adequacy a spatial calculus provides by relating the calculus to a linguistic ontology for space using similarities and linguistic corpus data. It allows evaluating whether and where a spatial calculus can be used for natural language interpretation. It can also lead to “more appropriate” spatial logics for spatial language.