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The work in this thesis offers a number of potential future research directions, both on the level of theory and on the level of experimentation.

On the theoretical side, this thesis gives a theoretical extension of the categorical frame- work of Coecke, Grefenstette, and Sadrzadeh [CGS13] by making use of the Lambek Calcu- lus with the structural control modalities of [Moo96]. A number of issues with our proposal remain open. First of all, we worked with a controlled form of contraction. However, the results of Kanovich, Kuznetsov, and Scedrov [KKS16;KKS17] show that even a controlled form of contraction may lead to undecidability of the employed grammar logic. Although the possibility of undecidability does not directly impede concrete experimentation with the models, it does beg further investigation on the formal properties of the proposed theoretical model. One possible solution is to bound the number of applications of the contraction rule to get a light version of the logic that by its bound becomes decidable. Another theoretical issue was that the direct interpretation of types and proofs as vector spaces and multi-linear maps was not able to distinguish the strict and sloppy readings of verb phrase ellipsis. Al- though we mended this by modelling the vector semantics with a lambda calculus as an intermediary language, a current approach to solve this issue without relying on lambdas, is to replace vector spaces with Fock spaces: these are spaces which offer a choice between any number of tensor products of a given vector space. One can then extract the relevant number of copies of a vector space whenever the grammar logic wills it.

On the experimental side, one of the limits of the type-driven approach to composition of distributional representation is scalability: while neural approaches are able to encode arbitrary sentences, the type-driven approach lacks this flexibility for two reasons. First, unbounded parsing remains a challenge in the typelogical approach; second, the tensor- based approach relies on high quality tensor representations for words, and methods to learn these have been theoretically established — especially in the last chapter of this thesis — but haven’t been implemented on full scale yet. Thus, one potential avenue for research is to make use of systems that have learned how to assign types to words in a typelogical grammar.

Work on extracting categorial lexicons form structured data is abundant: there is the unification algorithm of Buszkowski and Penn [BP90], extensions [BR01], and the thesis of Kanazawa [Kan95]. In addition, there is the CCGBank [HS07], which contains CCG deriva- tions for the dependency trees in the Penn Treebank. On top of the CCGBank, parsers and su- pertaggers have been developed [XAC15;LLZ16]. On the multimodal front, Moot [Moo15] describes a typelogical treebank for French and Moot [Moo18] a language-independent chart parsing method. One promising new direction that interacts well with the approach advo- cated in this thesis, uses self-attention neural networks to create a system that can, after training, map raw text to word types [KMD19].

Next to extracting grammatical types, the other main avenue for research is to expand the implementation that we gave in Chapter 5 of a tensor-based skipgram model to scale up to longer sentences, and be able to evaluate these representations on large datasets such as the Stanford Natural Language Inference datasets (SNLI, [Bow+15]) and it’s multi-genre and crosslingual incarnations (MNLI, [WNB18], XNLI, [Con+18]).

Finally, another potential line of research is to extract a large scale tensor-based model from current state of the art contextualised embeddings like ELMo [Pet+18] and BERT [Dev+19]. Such models assume that the embedding of a word is a vector that is a function of all other words in the sentence the word occurs in. This parameterisation naturally allows one to model some words as a higher order tensor that, by the fact that it encodes a multilinear map, contextualises itself in its arguments. For example, in the adjective-noun case, the vec- tor for the adjectivally modified noun depends on the vector for the noun and the effect of the adjective matrix on it.

Then, to end: although the contributions of this thesis provide support for the relevance of type-driven models in a world that appears to be focused on neural methods, many ques- tions arise. It is my hope that future research in this field may improve our understanding of the relevance of linguistic knowledge in semantic representations.

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