Ambiguity and Incomplete Information in Categorical Models of Language
This work addresses a foundational problem in theoretical linguistics and computational semantics for researchers developing categorical models of language, though it appears incremental as it builds on prior enrichment techniques and monadic frameworks.
The paper tackles the lack of a uniform strategy for modeling ambiguity and incomplete information in categorical distributional models of language, particularly in categories like sets and binary relations, by proposing a method to freely enrich arbitrary dagger compact closed categories to capture these phenomena while retaining key structural properties, resulting in a model with real convex combination of binary relations that utilizes scalars non-trivially.
We investigate notions of ambiguity and partial information in categorical distributional models of natural language. Probabilistic ambiguity has previously been studied using Selinger's CPM construction. This construction works well for models built upon vector spaces, as has been shown in quantum computational applications. Unfortunately, it doesn't seem to provide a satisfactory method for introducing mixing in other compact closed categories such as the category of sets and binary relations. We therefore lack a uniform strategy for extending a category to model imprecise linguistic information. In this work we adopt a different approach. We analyze different forms of ambiguous and incomplete information, both with and without quantitative probabilistic data. Each scheme then corresponds to a suitable enrichment of the category in which we model language. We view different monads as encapsulating the informational behaviour of interest, by analogy with their use in modelling side effects in computation. Previous results of Jacobs then allow us to systematically construct suitable bases for enrichment. We show that we can freely enrich arbitrary dagger compact closed categories in order to capture all the phenomena of interest, whilst retaining the important dagger compact closed structure. This allows us to construct a model with real convex combination of binary relations that makes non-trivial use of the scalars. Finally we relate our various different enrichments, showing that finite subconvex algebra enrichment covers all the effects under consideration.