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\documentclass[runningheads]{llncs}
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\usepackage{times}
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\usepackage{isabelle}
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\usepackage{isabellesym}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{mathpartir}
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\usepackage{tikz}
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\usepackage{pgf}
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\usetikzlibrary{positioning}
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\usepackage{pdfsetup}
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%%\usepackage{stmaryrd}
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\usepackage{url}
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\usepackage{color}
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\titlerunning{POSIX Lexing with Derivatives of Regular Expressions}
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\urlstyle{rm}
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\isabellestyle{it}
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\renewcommand{\isastyleminor}{\it}%
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\renewcommand{\isastyle}{\normalsize\it}%
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\def\dn{\,\stackrel{\mbox{\scriptsize def}}{=}\,}
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\renewcommand{\isasymequiv}{$\dn$}
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\renewcommand{\isasymemptyset}{$\varnothing$}
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\renewcommand{\isacharunderscore}{\mbox{$\_\!\_$}}
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\renewcommand{\isasymiota}{\makebox[0mm]{${}^{\prime}$}}
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\def\Brz{Brzozowski}
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\def\der{\backslash}
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\newtheorem{falsehood}{Falsehood}
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\newtheorem{conject}{Conjecture}
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\begin{document}
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\title{POSIX {L}exing with {D}erivatives of {R}egular {E}xpressions (Proof Pearl)}
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\author{Fahad Ausaf\inst{1} \and Roy Dyckhoff\inst{2} \and Christian Urban\inst{3}}
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\institute{King's College London\\
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\email{fahad.ausaf@icloud.com}
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\and University of St Andrews\\
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\email{roy.dyckhoff@st-andrews.ac.uk}
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\and King's College London\\
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\email{christian.urban@kcl.ac.uk}}
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\maketitle
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\begin{abstract}
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Brzozowski introduced the notion of derivatives for regular
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expressions. They can be used for a very simple regular expression
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matching algorithm. Sulzmann and Lu cleverly extended this algorithm
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in order to deal with POSIX matching, which is the underlying
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disambiguation strategy for regular expressions needed in lexers.
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Sulzmann and Lu have made available on-line what they call a
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``rigorous proof'' of the correctness of their algorithm w.r.t.\ their
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specification; regrettably, it appears to us to have unfillable gaps.
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In the first part of this paper we give our inductive definition of
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what a POSIX value is and show $(i)$ that such a value is unique (for
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given regular expression and string being matched) and $(ii)$ that
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Sulzmann and Lu's algorithm always generates such a value (provided
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that the regular expression matches the string). We also prove the
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correctness of an optimised version of the POSIX matching
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algorithm. Our definitions and proof are much simpler than those by
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Sulzmann and Lu and can be easily formalised in Isabelle/HOL. In the
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second part we analyse the correctness argument by Sulzmann and Lu and
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explain why the gaps in this argument cannot be filled easily.\smallskip
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{\bf Keywords:} POSIX matching, Derivatives of Regular Expressions,
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Isabelle/HOL
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\end{abstract}
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\input{session}
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\end{document}
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: t
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%%% End:
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