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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

  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

the first part of 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 also prove the correctness

of an optimised version of the POSIX matching algorithm.  In the

second part 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. \smallskip

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

Isabelle/HOL

\end{abstract}

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\end{document}

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