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\section*{pregroup grammar}

[[!redirects pregroup]] [[!redirects pregroups]] [[!redirects pregroup grammars]] [[!redirects Pregroup grammar]] [[!redirects Pregroup grammars]]

\textbf{Pregroup grammar} is a mathematical model of natural language grammar introduced by [[Lambek]], it is part of the [[categorial grammar]] tradition.

\noindent\hyperlink{pregroups}{Pregroups}\dotfill \pageref*{pregroups} \linebreak
\noindent\hyperlink{free_pregroups_and_the_switching_lemma}{Free pregroups and the switching lemma}\dotfill \pageref*{free_pregroups_and_the_switching_lemma} \linebreak
\noindent\hyperlink{from_pregroups_to_compact_2categories}{From pregroups to compact 2-categories}\dotfill \pageref*{from_pregroups_to_compact_2categories} \linebreak
\noindent\hyperlink{PregroupGrammar}{Pregroup grammars as free rigid monoidal categories}\dotfill \pageref*{PregroupGrammar} \linebreak
\noindent\hyperlink{pregroup_semantics_as_strong_monoidal_functors}{Pregroup semantics as strong monoidal functors}\dotfill \pageref*{pregroup_semantics_as_strong_monoidal_functors} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{pregroups}{}\subsection*{{Pregroups}}\label{pregroups}

As defined in \hyperlink{Lambek99}{Lambek 1999}, \hyperlink{Lambek08}{Lambek 2008}, a \textbf{pregroup} is a [[partial order|partially-ordered]] [[monoid]] $(P, \leq, \cdot, 1)$ such that every object $t \in P$ has [[category with duals|left- and right-adjoints]] $t^l, t^r \in P$. Explicitly we have the following four axioms:

\begin{itemize}%
\item $t^l  t \leq 1, \quad t t^r \leq 1 \quad$ (contraction)
\item $1 \leq t t^l, \quad 1 \leq t^r t \quad$ (expansion)

\end{itemize}
In other words, a pregroup is a [[rigid monoidal category]] which is [[thin category|thin]] and [[skeletal category|skeletal]]. Note that a commutative pregroup is simply a [[group]]. Here are a few variants of pregroups that appear in the literature, see \hyperlink{Coecke13}{Coecke 2013}:

\begin{itemize}%
\item If we drop the anti-symmetry axiom for posets, we get a \textbf{quasi-pregroup},
\item If we drop the two expansion axioms, we get a \textbf{protogroup}.

\end{itemize}
\hypertarget{free_pregroups_and_the_switching_lemma}{}\subsubsection*{{Free pregroups and the switching lemma}}\label{free_pregroups_and_the_switching_lemma}

In [[linguistics|linguistic]] applications, one starts with a [[partial order|poset]] $B = \{ s, n, \dots \}$ of \textbf{basic types} (e.g. sentence and noun) and then takes the free pregroup $P_B$ generated by $B$. For a basic type $b \in B$, iterated adjoints $\dots, b^{ll}, b^l, b, b^r, b^{rr}, \dots \in P_B$ are called \textbf{simple types}; an arbitrary type $t \in P_B$ may then be given as a sequence of simple types. The following lemma makes the parsing problem for free pregroups decidable --- i.e. given a sentence type $s \in P_B$ and the types for a sequence of words $t_1, \dots, t_n \in P_B$, is $t_1 \dots  t_n \leq s$?

\textbf{Switching lemma (\hyperlink{Lambek99}{Lambek 1999}):} For any pair of types $t, t' \in P_B$, if $t \leq t'$ then there is a type $t'' \in P_B$ such that $t \leq t''$ without expansions and $t'' \leq t'$ without contractions, i.e. free protogroups recognise the same language as free pregroups.

\hypertarget{from_pregroups_to_compact_2categories}{}\subsubsection*{{From pregroups to compact 2-categories}}\label{from_pregroups_to_compact_2categories}

In \hyperlink{PrellerLambek07}{Preller, Lambek 2007}, Preller and Lambek generalise the free pregroup construction above to free compact 2-categories in order to capture [[proof relevance]]. This allows to distinguish between the distinct parsings of ambiguous phrases such as ``(men and women) who like maths'' vs. ``men and (women who like maths)''.

Note that the compact 2-categories of Lambek and Preller differ from [[compact closed 2-category]] in that they are not required to be symmetric. A non-symmetric compact closed 2-category with one object is simply a [[rigid monoidal category]].

Given a poset of basic types $B$, the objects of the free rigid monoidal category $C_B$ are the same as that of the free (quasi-)pregroup, the arrows may be described as planar [[string diagram|string diagrams]]. Given two types $t, t' \in P_B$, we have that $t \leq t'$ if and only if there is an arrow $r : t \to t'$ in $C_B$.

\hypertarget{PregroupGrammar}{}\subsection*{{Pregroup grammars as free rigid monoidal categories}}\label{PregroupGrammar}

A \textbf{pregroup grammar} is a tuple $G = (B, \Sigma, \Delta, s)$ where $B$ and $\Sigma$ are finite sets called the \emph{basic types} and the \emph{vocabulary} respectively, $\Delta \subseteq \Sigma \times P_B$ is a relation called the \emph{dictionnary} and $s \in P_B$ is a designated \emph{sentence type}. We require that $\Delta$ is finite, i.e. the set of possible types $\Delta(w) \subseteq P_B$ is finite for each word $w \in \Sigma$. The \emph{language} of $G$ is then given by:

\begin{displaymath}
L(G) = \big\{ w_1 \dots w_n \in \Sigma^n \vert n \in \mathbb{N}, \quad \exists t \in \prod_{i \leq n} \Delta(w_i) \quad t \leq s \big\}
\end{displaymath}
i.e. a sequence of words $w_1 \dots w_n \in \Sigma^n$ is said to be grammatical whenever there is a dictionnary entry $(w_i, t_i) \in \Delta$ for each word $w_i$ such that $t_1 \dots t_n \leq s$. One may simplify this by redefining $L(G) = \big\{ w_1 \dots w_n \in \Sigma^n \vert C_G(w_1 \dots w_n, s) \neq \emptyset \big\}$ where $C_G$ is the free [[rigid monoidal category]] with:

\begin{itemize}%
\item generating objects the [[disjoint union]] $B + \Sigma$,
\item generating arrows the dictionnary entries $(w, t) \in \Delta$ with $dom(w, t) = w$ and $cod(w, t) = t$.

\end{itemize}
That is, a sequence of words is grammatical whenever there exists a string diagram going from the sequence of words to the sentence type. Note that traditionally the identity wires for words $w \in \Sigma$ are omitted, hence dictionnary entries are depicted as triangles with no input. Contractions are depicted as cups, e.g. from $(Alice, n), (loves, n^r s n^l), (Bob, n) \in \Delta$ and $n n^r s n^l n \leq s$ we get the following diagram:

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$\backslash$end\{tikzpicture\} $\backslash$end\{centre\}

\textbf{Theorem (\hyperlink{BuszkowskiMoroz08}{Buszkowski, Moroz 2008}):} For each pregroup grammar $G$, there is a [[context-free grammar]] $G'$ such that $L(G) = L(G')$. Furthermore, $G'$ may be computed in time polynomial in the size of $G$.

The opposite direction also holds, hence pregroup grammar and [[context-free grammar]] are said to be \textbf{weakly equivalent}: the translation preserves only the generated languages, it does not preserve the structure of syntax trees.

\hypertarget{pregroup_semantics_as_strong_monoidal_functors}{}\subsection*{{Pregroup semantics as strong monoidal functors}}\label{pregroup_semantics_as_strong_monoidal_functors}

One may give a [[semantics]] to a pregroup grammar $G = (B, \Sigma, \Delta, s)$ by defining a strong [[monoidal functor]] $F : C_G \to S$, where $C_G$ is the free [[rigid monoidal category]] described in \hyperlink{PregroupGrammar}{section 2}. $S$ is a suitable [[rigid monoidal category]], e.g. $\text{FdVect}$ or $\text{Rel}$, depending on the application. Note the similarity with a [[Lawvere theory]] as a monoidal functor from a syntactic category to $\text{Set}$.

We require the image for all words $w \in \Sigma$ to be the monoidal unit $F(w) = I$, hence the image for each dictionnary entry $(w, t) \in \Delta$ is given by a state $F(w, t) : I \to F(t)$. The meaning $F(r) : I \to F(s)$ for a sentence $w_1 \dots w_n \in \Sigma^n$ with grammatical reduction $r : w_1 \dots w_n \to s$ may then be computed from the individual meanings $F(w_i, t_i) : I \to F(t_i)$ of the words, following [[Gottlob Frege|Frege's]] principle of [[compositionality]]. This has been developed in \hyperlink{Preller05}{Preller 2005} as well as in a series of papers by [[Bob Coecke]] and others, see [[categorical compositional distributional semantics]].

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Joachim Lambek, \emph{Type Grammar Revisited}, Logical Aspects of Computational Linguistics (1999)


\item Anne Preller, \emph{Category Theoretical Semantics for Pregroup Grammars}, Logical Aspects of Computational Linguistics, 5th International Conference, LACL 2005 (\href{https://www.researchgate.net/publication/220718685_Category_Theoretical_Semantics_for_Pregroup_Grammars}{pdf})


\item Anne Preller, Joachim Lambek, \emph{Free Compact 2-Categories}, Mathematical Structures in Computer Science, Cambridge University Press (CUP), 2007 (\href{https://hal-lirmm.ccsd.cnrs.fr/lirmm-00137681v2/document}{pdf})


\item Joachim Lambek, \emph{From Word to Sentence: A Computational Algebraic Approach to Grammar}, Polimetrica 2008 (\href{http://www.math.mcgill.ca/barr/lambek/pdffiles/2008lambek.pdf}{pdf})


\item Wojciech Buszkowski, Katarzyna Moroz, \emph{Pregroup Grammars and Context-free Grammars}, Computational Algebraic Approaches to Natural Language, Polimetrica (2008) (\href{https://pdfs.semanticscholar.org/1924/30f2252b6e0a7f982a3ae69a3ccf9c2981c0.pdf}{pdf})


\item Bob Coecke, \emph{An alternative Gospel of structure: order, composition, processes}, Introductory chapter to C. Heunen, M. Sadrzadeh, and E. Grefenstette. Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse. Oxford University Press, 2013 (\href{https://arxiv.org/abs/1307.4038}{arxiv:1307.4038})



\end{itemize}


\end{document}