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\begin{document}

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\section*{context-free grammar}

\hypertarget{contextfree_grammar}{}\section*{{Context-free grammar}}\label{contextfree_grammar}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{early_sources}{Early sources}\dotfill \pageref*{early_sources} \linebreak
\noindent\hyperlink{modern_literature}{Modern literature}\dotfill \pageref*{modern_literature} \linebreak


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

The notion of context-free grammar, now used in [[linguistics]], [[computer science]] and [[mathematics]], was introduced in the works of [[Noam Chomsky]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We write $X + Y$ for the [[disjoint union]] of two sets and use vector notation and the Kleene star $\vec{v} \in X^\star$ to denote sequences in the [[free monoid]].

A context-free grammar is a tuple $(\Sigma, X, R, s)$ where $\Sigma$ and $X$ are finite sets called the \emph{vocabulary} (also called the \emph{terminals}) and the \emph{non-terminals} respectively, $R \subseteq X \times (X + \Sigma)^\star$ is a finite set of \emph{production rules} and $s \in X$ is called the \emph{start symbol}.

The \emph{language} of a context-free grammar is given by $L(G) = \{ \vec{u} \in V^\star \vert s \to_R \vec{u} \}$ where the rewriting relation $(\to_R) \subseteq (V + X)^\star \times (V + X)^\star$ is traditionally defined as the transitive closure of the following directed graph:

\begin{displaymath}
\big\{ (\vec{u} x \vec{w}, \vec{u v w}) \quad \vert \quad \vec{u}, \vec{w} \in (V + X)^\star, (x, \vec{v}) \in R \big\}
\end{displaymath}
One may redefine $L(G) = \big\{ \vec{u} \in \Sigma^\star \vert \C_G(s, \vec{u}) \neq \emptyset \big\}$ where $C_G$ is the free [[monoidal category]] with:

\begin{itemize}%
\item generating objects the [[disjoint union]] $\Sigma + X$,
\item generating arrows the production rules $(x, \vec{v}) \in R$ with $dom(x, \vec{v}) = x$ and $cod(x, \vec{v}) = \vec{v}$.

\end{itemize}
That is, a string $\vec{u} \in \Sigma^\star$ is \emph{grammatical} whenever there exists an arrow from the start symbol $s$ to $\vec{u}$ in $C_G$. Arrows in $C_G$ may be encoded as a \emph{syntax tree}, seen as a special case of a [[string diagram]], e.g.:



Note that context-free grammar is weakly equivalent to [[Lambek|Lambek's]] [[pregroup grammar]] i.e. they generate the same class of languages, see:

\begin{itemize}%
\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})

\end{itemize}
\hypertarget{early_sources}{}\subsection*{{Early sources}}\label{early_sources}

Here is list of some of the original references in English and also of their Russian translations.

\begin{itemize}%
\item N. Chomsky, \emph{Three models for the description of language, I. R. E. Trans. PGIT 2 (1956), 113---124. ( :  .,     ,  , . 2,  1961, 237-266)}
\item N. Chomsky, \emph{On certain formal properties of grammars}, Information and Control \textbf{2} (1959), 137---267; \emph{A note on phrase structure grammars}, Information andControl \textbf{2} 1959\_, 393---395. ( ,  o   .  , . 5, , 1962, 312---315.)
\item N. Chomsky, \emph{On the notion \guillemotleft{}Rule of Grammar\guillemotright{}}, Proc. Symp. Applied Math., 12. Amer. Math. Soc. (1961). ( .,   \guillemotleft{} \guillemotright{}, .   , . 4, \guillemotleft{}\guillemotright{}, 1965, . 34---65.)
\item N. Chomsky, \emph{Context-free grammars and pushdown storage}, Quarterly Progress Reports,  65, Research Laboratory of Electronics, M. I. T., 1962.
\item N. Chomsky, \emph{Formal properties of grammars}, in Handbook of Mathemati- Mathematical Psychology, 2, ch. 12, Wiley, 1963, p. 323---418. ( .,   ,  , . 2, \guillemotleft{}\guillemotright{}, 1966, 121----230.)
\item N. Chomsky, The logical basis for linguistic theory, Proc. IX-th Int. Cong. Linguists (1962). ( ``    , .   , . 4, \guillemotleft{}\guillemotright{}, 1965, 465----575.)
\item N. hmsk, G. A. Miller G. A.. Finite state languages, Information and Control \textbf{1} (1958), 91----112. ( .,  .,     ,  , . 4, , 1962, 231----255.), \emph{Introduction to the formal analysis of natural languages}. Handbook of Mathematical Psychology \textbf{2}, Ch. 12, Wiley, 1963, 269----322 (     ,  , . 1, \guillemotleft{}\guillemotright{}, 1965, . 229----290.)
\item N. Chomsky, M. P. Sch\"u{}tzenberger, \emph{The algebraic theory of context-free languages}, in: Computer programming and formal systems, 118--161 North-Holland 1963, Amsterdam \href{http://www.ams.org/mathscinet-getitem?mr=152391}{MR152391} (  \textbf{3}, 195--242, 1966)

\end{itemize}
\hypertarget{modern_literature}{}\subsection*{{Modern literature}}\label{modern_literature}

\begin{itemize}%
\item wikipedia: \href{http://en.wikipedia.org/wiki/Context-free_language}{context-free language}
\item A. V. Aho, J. D. Ullman, \emph{The theory of parsing, translation, and compiling}, vol 1, Parsing; vol. 2, Compiling. Prentice Hall, 1972.
\item A. V. Aho, J. D. Ullman, \emph{Principles of compiler design}, Addison-Wesley, 1977.

\end{itemize}
Some chapters in ``Handbook of formal language theory'' (3 vols.), G. Rozenberg, A. Salomaa (eds.), Springer 1997:

\begin{itemize}%
\item Jean-Michel Autebert, Jean Berstel, Luc Boasson, \emph{Context-free languages and push-down automata}, vol. 1, ch. 3

\end{itemize}
[[!redirects context-free grammar]] [[!redirects context-free grammars]] [[!redirects context free grammar]] [[!redirects context free grammars]] [[!redirects context-free language]] [[!redirects context-free languages]] [[!redirects context free language]] [[!redirects context free languages]]



\end{document}