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\title{{POSIX} {L}exing with {B}itcoded {D}erivatives}
\titlerunning{POSIX Lexing with Bitcoded Derivatives}
\author{Chengsong Tan}{King's College London}{chengsong.tan@kcl.ac.uk}{}{}
\author{Christian Urban}{King's College London}{christian.urban@kcl.ac.uk}{}{}
\authorrunning{C.~Tan and C.~Urban}
\keywords{POSIX matching, Derivatives of Regular Expressions, Isabelle/HOL}
\category{}
\ccsdesc[100]{Design and analysis of algorithms}
\ccsdesc[100]{Formal languages and automata theory}
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\begin{abstract}
  Sulzmann and Lu described a lexing algorithm that calculates
  Brzozowski derivatives using bit-sequences annotated to regular
  expressions.  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.  The purpose of the bit-sequences in
  Sulzmann and Lu's algorithm is to keep the size of derivatives small
  which is achieved by `aggressively' simplifying regular expressions.
  In this paper we describe a slight variant of Sulzmann and Lu's
  algorithm and \textit{(i)} prove that this algorithm generates
  unique POSIX values; \textit{(ii)} we also establish a cubic bound
  for the size of the derivatives---in earlier works, derivatives can
  grow exponentially even after simplification.
  
%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
\end{abstract}



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