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\section*{categorical compositional distributional semantics}

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Categorical compositional distributional semantics, also known as \textbf{DisCoCat} for short, uses category theory to combine the benefits of two very different approaches to [[linguistics]]: [[categorial grammar]] and \href{https://en.wikipedia.org/wiki/Distributional_semantics}{distributional semantics}. Specifically, it uses the fact that [[pregroups]] and the category of finite-dimensional real [[vector space|vector spaces]] are both examples of [[rigid monoidal category|rigid monoidal categories]]. Semantics for [[pregroup grammars]] may then be given as a strong [[monoidal functor]], which allows sentence meanings to be computed from word meanings by linear algebra, compositionally on the grammatical structure of the sentence.

DisCoCat was first introduced in \hyperlink{Coecke10}{Coecke, Sadrzadeh and Clark 2010}.

There is a curious (if probably shallow) analogy between DisCoCat and [[topological quantum field theory]], in that both concern strong monoidal functors from a free ([[rigid monoidal category|rigid monoidal]] or [[compact closed category|compact closed]]) category to a category of vector spaces.

\hypertarget{simple_example}{}\subsection*{{Simple example}}\label{simple_example}

Consider a very simple fragment of English consisting of nouns and verbs. Write $n$ for the type of nouns, and $s$ for the type of sentences. Let $\mathcal{C}$ be the free pregroup generated by $n$ and $s$. In $\mathcal{C}$, a noun will be assigned the type $n$ and a reflexive verb will be assigned the type $n^{r} s n^{l}$, since it will produce a sentence if provided with nouns on the left and right. Now given nouns such as ``Alice'' and ``Bob'' and reflexive verbs such as ``loves'' and ``hates'', the grammaticality of a sentence such as ``Alice hates Bob'' is witnessed by the fact that

\begin{displaymath}
n n^{r} s n^{l} n \leq s
\end{displaymath}
in the free pregroup.

A strong monoidal functor $F : \mathcal{C} \to FVect_{\mathbb{R}}$ is completely determined by where it sends the generators $n$ and $s$. The vector spaces $F (n)$ and $F (s)$ are called the \emph{noun space} and the \emph{sentence space}, and their choice is a matter of linguistic modelling. A common approach is to pick a large set $n$ of \emph{basis words}, say $n = 1000$ (or as many as your sparse matrix implementation can handle), and take $F (n) = \mathbb{R}^{1000}$. The choice of $F (s)$ is not well understood, but $F (s) = \mathbb{R}^{2}$ is a common choice, with the standard basis vectors interpreted as ``true'' and ``false''.

Given a grammatical parsing such as $n n^{r} s n^{l} n \leq s$, we automatically get a linear map

\begin{displaymath}
F (n n^{r} s n^{l} n) : F (n) \otimes F (n) \otimes F (s) \otimes F (n) \otimes F (n) \to F (s)
\end{displaymath}
given by appropriate tensor contractions.

The final ingredient we need are \emph{word vectors}. We need to pick vectors $[\![ Alice ]\!], [\![ Bob ]\!] \in F (n)$ and $[\![ loves ]\!] \in F (n) \otimes F (s) \otimes F (n)$. The choice of these vectors is also a matter for linguistics, but a common idea is to count how often Alice (for example) appears in a corpus (for example the full text of Wikipedia) close to each of the chosen basis words.

Finally, we obtain a sentence-vector

\begin{displaymath}
F (n n^{r} s n^{l} n \leq s) ([\![ Alice ]\!] \otimes [\![ loves ]\!] \otimes [\![ Bob ]\!]) \in F (s)
\end{displaymath}
\hypertarget{relative_pronouns_and_frobenius_algebras}{}\subsection*{{Relative pronouns and Frobenius algebras}}\label{relative_pronouns_and_frobenius_algebras}

\hypertarget{free_pregroups_vs_free_rigid_monoidal_categories}{}\subsection*{{Free pregroups vs free rigid monoidal categories}}\label{free_pregroups_vs_free_rigid_monoidal_categories}

All earlier papers on DisCoCat use free pregroups for the grammar category, building on earlier work by [[Joachim Lambek]] on pregroup grammars. Unfortunately, in \hyperlink{Preller14}{Preller 2014, fact 1} it is proven that any strong monoidal functor from a free pregroup to $FVect$ is necessarily trivial, in the sense that it sends every generator to the monoidal unit. One should instead use a free [[rigid monoidal category]], a kind of [[categorification]]. Morphisms in the free autonomous category can be viewed as [[proof relevance|proof relevant]] grammatical reductions. They may be encoded as [[string diagrams]], see [[pregroup grammar]].

This allows a slightly more elegant reformulation of our basic example. Let $\mathcal{C}$ be the free autonomous category on the objects $n, s$ and the morphisms $Alice : 1 \to n$, $Bob : 1 \to n$ and $loves : 1 \to n^{r} s n^{l}$. Then a sentence such as ``Alice loves Bob'', together with its grammaticality, is witnessed by a morphism $f : 1 \to s$. Now the word-vectors $[\![ Alice ]\!] = F (Alice)$, $[\![ Bob ]\!] = F (Bob)$ and $[\![ loves ]\!] = F (loves)$ become part of the data defining $F : \mathcal{C} \to FVect$, and the resulting sentence vector is simply $F (f)$.

\hypertarget{the_category_of_discocat_models}{}\subsection*{{The Category of DisCoCat Models}}\label{the_category_of_discocat_models}

One criticism of DisCoCat is that a single model is unlikely to be complex enough to capture the intricacies of a natural language. A response to this criticism is that DisCoCat models should only model a given fragment of language with some level of detail. These models can then be combined or updated to represent more complex fragments of natural language. Indeed, in \emph{\href{https://arxiv.org/abs/1811.11041}{Translating and Evolving: Towards a Model of Language Change in DisCoCat}} the authors construct a category $DisCoCat$ of DisCoCat models where

\begin{itemize}%
\item objects are [[strong monoidal functors]]

\begin{displaymath}
F \colon (J,*) \to (FVect,\otimes)
\end{displaymath}
where $(J,*)$ is a category which is freely monoidal on some set of grammatical types. In practice $J$ might be freely equipped with an [[rigid monoidal category|rigid]] structure or a [[Frobenius algebra]] on each object.


\item A morphism from a language model $F \colon J \to FVect$ to a language model $F' \colon J' \to FVect$ is a tuple $(j , \alpha)$ where $j$ is a [[monoidal functor]] $i \colon J \to J'$ and $\alpha \colon F \Rightarrow F' \circ j$ is a [[monoidal natural transformation]] filling the following triangle $\backslash$begin\{centre\} $\backslash$begin\{xymatrix\} J$\backslash$arr{\tt \symbol{94}}F $\backslash$ard\emph{j\& FVect $\backslash$ J' $\backslash$arur{\tt \symbol{94}}\{F'\}$\backslash$ \& $\backslash$end\{xymatrix\} $\backslash$end\{centre\} Morphisms in $DisCoCat$ are called translations because they represent ways of changing your semantic interpretations of grammatical types in a way which is coherent with a change in grammar.}



\end{itemize}
\hypertarget{the_product_space_representation}{}\subsubsection*{{The Product Space Representation}}\label{the_product_space_representation}

In \href{https://arxiv.org/abs/1003.4394}{\emph{Mathematical foundations for a compositional distributional model of meaning}} a different description of a DisCoCat model was described. Let $J$ be a free [[autonomous category]] representing grammar and assume that you have an assignment $V_a$ of each grammatical type $a$ to a finite dimensional vector space. Then grammatical computations of meaning can be performed in the [[product category]] $J \times FVect$. Let $a_1 \cdot \ldots \cdot a_n$ be an object in $J$ and let $v \in V_{a_1} \otimes \ldots \otimes V_{a_n}$ be its semantic meaning. Then a grammatical reduction $f \colon a \to b$ built from the units and counits in $J$ can be paired with the corresponding structure maps in $FVect$. This pairing allows you to apply the structure maps corresponding to $f$ to the vector $v$ to get a new semantic meaning. In essence, this approach identifies an autonomous subcategory of $J \times FVect$ where meaning computations can occur.

This product space representation can be related to monoidal functor perspective via the [[category of elements]]. Using the forgetful functor $i \colon FVect \to Set$ the product space of a language model $F \colon J \to FVect$ can be defined as the category of elements $\int i \circ F$. The product space representation is useful for equipping a language model with a specific set of words. If $W$ is a set of words then a map $W \to \int i \circ F$ describes an assignment of each word to a grammatical type and semantic meaning.

\hypertarget{variations}{}\subsection*{{Variations}}\label{variations}

DisCoCat is relatively easy to modify by replacing the semantic category $FVect$ with some other category with the appropriate structure. Examples include

\begin{itemize}%
\item [[relation|Relations]], which yields a semantics in [[regular logic]], closely related to [[cartesian bicategory|Cartesian bicategories]], see \hyperlink{Felice19}{Felice, Meichanetzidis and Toumi 2019}


\item [[density matrix|Density matrices]] to model semantic ambiguity such as ``financial bank'' vs ``river bank''


\item Convex relations (i.e. [[relation|relations]] in the category of [[convex space|convex spaces]]) to model cognition (\hyperlink{Bolt16}{Bolt et al 2016})


\item [[game theory|Games]] to model dialogue (\hyperlink{Hedges18}{Hedges and Lewis 2018})



\end{itemize}
\hypertarget{references}{}\section*{{References}}\label{references}

\begin{itemize}%
\item \href{https://golem.ph.utexas.edu/category/2018/02/linguistics_using_category_the.html}{Linguistics using category theory} on the n-category café


\item \href{https://golem.ph.utexas.edu/category/2018/03/cognition_convexity_and_catego.html}{Cognition, convexity and category theory} on the n-category café


\item \href{https://golem.ph.utexas.edu/category/2019/06/meeting_the_dialogue_challenge.html}{Meeting the Dialogue Challenge} on the n-category café


\item [[Bob Coecke]], [[Mehrnoosh Sadrzadeh]] and [[Stephen Clark]], \emph{Mathematical foundations for a compositional distributional model of meaning}. Lambek Festschrift, special issue of Linguistic Analysis, 2010. (\href{https://arxiv.org/abs/1003.4394}{arXiv:1003.4394})


\item [[Anne Preller]], \emph{From logical to distributional methods}. QPL 2013. (\href{https://arxiv.org/abs/1412.8527}{arXiv:1412.8527})


\item [[Josef Bolt]], [[Bob Coecke]], [[Fabrizio Genovese]], [[Martha Lewis]], [[Daniel Marsden]] and [[Robin Piedeleu]], \emph{Interacting conceptual spaces I: Grammatical composition of concepts}, SLPCS 2016 (\href{https://arxiv.org/abs/1703.08314}{arXiv:1703.08314}


\item [[Giovanni de Felice]], [[Konstantinos Meichanetzidis]] and [[Alexis Toumi]], \emph{Functorial Question Answering}, ACT 2019. (\href{https://arxiv.org/abs/1905.07408}{arXiv:1905.07408}])


\item [[Jules Hedges]] and [[Martha Lewis]], \emph{Towards functorial language-games}, CAPNS 2018. (\href{https://arxiv.org/abs/1807.07828}{arXiv:1807.07828})



\end{itemize}
The category $DisCoCat$ and it's relationship to the product space representation is discussed in:

\begin{itemize}%
\item [[Tai-Danae Bradley]], [[Martha Lewis]], [[Jade Master]], and [[Brad Theilman]] \emph{Translating and Evolving: Towards a Model of Language Change in DisCoCat} \href{https://arxiv.org/abs/1811.11041}{arXiv:1811.11041}

\end{itemize}
[[!redirects DisCoCat]]



\end{document}