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\section*{rewriting}

\hypertarget{rewriting}{}\section*{{Rewriting}}\label{rewriting}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Given an alphabet of letters and symbols (perhaps `[[type]]d', so that certain symbols are declared to be `symbols for functions', etc., e.g. `$+$' is a symbol for a binary operation), then one can form [[list|strings]] of letters to give words or more generally (well formed) formulae involving symbols of all types.

In a \emph{rewriting system}, one specifies a set of rules that describe valid replacements of subformulae by other ones. On some formulae of a rewriting system, the rewriting rules may produce conflicts, when two or more rules can be applied.

\begin{uexample}
One type of example of a rewriting system is given by a [[group presentation]], $(X,R)$, where $X$ is a set of generators for a group $G$ and $R$ is a subset of pairs of elements in the free group on $X$. As a definite example, take

\begin{itemize}%
\item $X = \{a,b\}$


\item $R = \{(a a a,1),(b b,1),(a b a b,1)\}$



\end{itemize}
(and the group being presented is the symmetric group, $S_3$ on three symbols). We think of $(a a a,1)$ as a `rewrite rule': `'replace $a a a$ by 1'', often written $a a a \rightsquigarrow 1$ (or just $a a a \to 1$).

Working with the presentation, we can then start generating words in the generators, for instance $a a a b a b$ and then see how that word can be `rewritten' using the rewrite rules to $b a b$ using the first rule and to $a a$ using the last. There is thus a potential conflict between these rewritten forms of the word.

\end{uexample}
It is usual to choose a `[[normal form]]' for each word. In the example, in the symmetric group and the above presentation the elements of the group are often listed as $1, a, a^2, b, a b, a^2 b$. These might be our choice of normal forms for the elements. In other words any element has a unique representative in the form $a^i b^j$, and we \emph{choose} to label them that way. With the two uses of the rules applied to $a a a b a b$, the first did not gave a normal form, the second did.

In order to transform a rewriting system into a computation [[algorithm]], one needs to apply the rules in a deterministic way, using a reduction strategy. We also need to know that there is a unique normal form that can be found for each word and that the `algorithm' will \emph{terminate}, that is it really \emph{is} an algorithm!.

We therefore will need to discuss [[confluence]], [[termination]] and [[reduction strategy|reduction strategies]].

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

\begin{itemize}%
\item wikipedia: \href{http://en.wikipedia.org/wiki/Rewriting}{Rewriting}

\end{itemize}
A classical foundational paper is

\begin{itemize}%
\item Axel Thue, Probleme \"u{}ber Ver\"a{}nderungen von Zeichenreihen nach gegebenen Regeln., Kristiania Vidensk. Selsk, Skr. (1914), no. 10, 493--524.

\end{itemize}
A key lemma is given in the beautiful paper:

\begin{itemize}%
\item [[Max Newman|Maxwell Herman Alexander Newman]], \emph{On theories with a combinatorial definition of ``equivalence''}, Annals of Mathematics \textbf{43} (1942), no. 2, 223--243.

\end{itemize}
For good references on word rewriting see

\begin{itemize}%
\item Ronald V. Book, Friedrich Otto, \emph{String-rewriting systems}, Texts and Monographs in Computer Science, Springer-Verlag, 1993.

\end{itemize}
and on term rewriting

\begin{itemize}%
\item Franz Baader, Tobias Nipkow, \emph{Term rewriting and all that}, Cambridge University Press, 1998.


\item \emph{Terese}, Term rewriting systems, Cambridge Tracts in Theoretical Computer Science, vol. 55, Cambridge University Press, 2003.



\end{itemize}
For the higher dimensional forms of rewriting, one source is in the work of Guiraud and Malbos:

\begin{itemize}%
\item [[Yves Guiraud]], [[Philippe Malbos]], \emph{Identities among relations for higher-dimensional rewriting systems}, S\'e{}minaires et Congr\`e{}s, SMF, (to appear), \href{http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.4538v2.pdf}{arXiv:0910.4538}; \emph{Higher-dimensional categories with finite derivation type}, Theory Appl. Categ. \textbf{22} (2009), No. 18, 420--478, \href{http://www.tac.mta.ca/tac/volumes/22/18/22-18abs.html}{tac}; pdf slides \href{http://ljk.imag.fr/membres/Dominique.Duval/CCS09/CCS09-Malbos.pdf}{Homotopical methods in polygraphic rewriting}, \href{http://www.loria.fr/~guiraudy/exposes/ntdf-luminy.pdf}{n-Cat\'e{}gories de type de d\'e{}rivation fini}.

\end{itemize}
This also includes emerging \emph{homotopical theory of computation} based on [[polygraph]]s (introduced by [[Albert Burroni]]) and polygraphic resolutions (introduced by [[François Métayer]]):

\begin{itemize}%
\item Yves Lafont, \emph{Algebra and geometry of rewriting}, Applied Categorical Structures August \textbf{15}:4, 2007, pp 415-437 \href{http://dx.doi.org/10.1007/s10485-007-9083-6
}{doi}

\end{itemize}
Again within the context of higher dimensional forms of rewriting, [[Tibor Beke]] has given a categorification of certain rewriting procedures of Knuth. This is of relevance here as it contains a strong result on coherence theory:

\begin{itemize}%
\item [[Tibor Beke]], \href{http://faculty.uml.edu/tbeke/knuth.pdf}{Categorification, term rewriting and the Knuth-Bendix procedure}

\end{itemize}
There is a preprint by [[Samuel Mimram]]

\begin{itemize}%
\item \emph{Towards 3-Dimensional Rewriting Theory}, (\href{https://arxiv.org/abs/1403.4094}{arXiv:1403.4094})

\end{itemize}
category: combinatorics, algebra, computer science, logic [[!redirects rewriting]] [[!redirects rewriting system]] [[!redirects rewriting systems]] [[!redirects rewriting procedure]] [[!redirects rewriting procedures]]



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