Unified mathematics for natural language data and structure

In 1958 Lambek proposed the logic of a residuated monoid to reason about syntax of natural language. This logic was amongst the first substructural logics. It did not have contraction nor weakening, properties that indeed were not necessary for fixed word order languages. Apart from difficulties in generalizing Lambek's logic to free word order languages, such as Sanskrit and Latin, they also failed to model discourse structure, i.e. structures that go beyond sentential boundaries, for instance in "John slept. He snored." In this talk, I will show how endowing Lambek Calculus with controlled modalities overcomes the problem. I will then go through the body of research on Compositional Distributional semantics which defines vector semantics for Lambek Calculi using tensors and tensor contraction. I will show how this semantics extends to modal Lambek Calculi using Fock spaces. This setting unifies structural and statistical approaches to natural language. We are thus able to apply it to large scale corpora and solve mainstream tasks such as disambiguation and semantic similarity.

03B47, 03B65