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\titlerunning{POSIX Lexing with Derivatives of Regular Expressions}

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\begin{document}

\renewcommand{\thefootnote}{$\star$} \footnotetext[1]{This paper is a

  revised and expanded version of \cite{AusafDyckhoffUrban2016}.

  Compared with that paper we give a second definition for POSIX

  values introduced by Okui Suzuki \cite{OkuiSuzuki2010,OkuiSuzukiTech}

  and prove that it is

  equivalent to our original one. This

  second definition is based on an ordering of values and very similar to,

  but not equivalent with, the

  definition given by Sulzmann and Lu~\cite{Sulzmann2014}.

  The advantage of the definition based on the

  ordering is that it implements more directly the informal rules from the

  POSIX standard.

  We also prove Sulzmann \& Lu's conjecture that their bitcoded version

  of the POSIX algorithm is correct. Furthermore we extend our results to additional constructors of regular

  expressions.}  \renewcommand{\thefootnote}{\arabic{footnote}}

\title{POSIX {L}exing with {D}erivatives of {R}egular {E}xpressions}

\author{Fahad Ausaf\inst{1} \and Roy Dyckhoff\inst{2} \and Christian Urban\inst{3}}

\institute{King's College London\\

		\email{fahad.ausaf@icloud.com}

\and University of St Andrews\\

		\email{roy.dyckhoff@st-andrews.ac.uk}

\and King's College London\\

		\email{christian.urban@kcl.ac.uk}}

\maketitle

\begin{abstract}

Brzozowski introduced the notion of derivatives for regular

expressions. They can be used for a very simple regular expression

matching algorithm.  Sulzmann and Lu cleverly extended this algorithm

in order to deal with POSIX matching, which is the underlying

disambiguation strategy for regular expressions needed in lexers.

Their algorithm generates POSIX values which encode the information of

\emph{how} a regular expression matches a string---that is, which part

of the string is matched by which part of the regular expression.  In

this paper we give our inductive definition of what a POSIX value is

and show $(i)$ that such a value is unique (for given regular

expression and string being matched) and $(ii)$ that Sulzmann and Lu's

algorithm always generates such a value (provided that the regular

expression matches the string). We show that $(iii)$ our inductive

definition of a POSIX value is equivalent to an alternative definition

by Okui and Suzuki which identifies POSIX values as least elements

according to an ordering of values.  We also prove the correctness of

Sulzmann's bitcoded version of the POSIX matching algorithm and extend the

results to additional constructors for regular expressions.  \smallskip

{\bf Keywords:} POSIX matching, Derivatives of Regular Expressions,

Isabelle/HOL

\end{abstract}

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\end{document}

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