Semantic Unification - A Sheaf Theoretic Approach to Natural Language

Language is contextual and sheaf theory provides a high level mathematical framework to model contextuality. We show how sheaf theory can model the contextual nature of natural language and how gluing can be used to provide a global semantics for a discourse by putting together the local logical semantics of each sentence within the discourse. We introduce a presheaf structure corresponding to a basic form of Discourse Representation Structures. Within this setting, we formulate a notion of semantic unification — gluing meanings of parts of a discourse into a coherent whole — as a form of sheaf-theoretic gluing. We illustrate this idea with a number of examples where it can used to represent resolutions of anaphoric references. We also discuss multivalued gluing, described using a distributions functor, which can be used to represent situations where multiple gluings are possible, and where we may need to rank them using quantitative measures.