Mapping spatial relations

Abstract

One of the fundamental issues in cognitive science is the problem of grounding concepts in the perceptual world. In this thesis, I present a computational theory for how spatial relations are grounded in the perceptual world. Three constraints are critical to this theory: abstractness, groundedness and flexibility all of which need to be satisfied in order to explain the structure of spatial concepts. I then show how a formal framework, based on the mathematical notions of category theory can be used to model the grounding problem. The key computational ideas are that of minimal mapping and derivations. A minimal mapping of two categories, A and B, is the "smallest' category, C, that contains A and B. A derivation is a sequence of categories that follow a minimal mapping rule. Derivations and minimal mappings are used to model three domains - the semantics of prepositions, the structure of a toy "Jigsaw World" and the semantics of generic terms and quantifiers. In each case, I show how the computational theory gives rise to insights that are not available upon a purely empirical analysis. In particular, the derivational account shows the importance of stable, non-accidental features and of multiple scales in spatial cognition.