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+\documentclass[a4paper,UKenglish]{lipics}
+\usepackage{graphic}
+\usepackage{data}
+% \documentclass{article}
+%\usepackage[utf8]{inputenc}
+%\usepackage[english]{babel}
+%\usepackage{listings}
+% \usepackage{amsthm}
+% \usepackage{hyperref}
+% \usepackage[margin=0.5in]{geometry}
+%\usepackage{pmboxdraw}
+ 
+\title{POSIX Regular Expression Matching and Lexing}
+\author{Chengsong Tan}
+\affil{King's College London\\
+London, UK\\
+\texttt{chengsong.tan@kcl.ac.uk}}
+\authorrunning{Chengsong Tan}
+\Copyright{Chengsong Tan}
+
+\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}%
+\newcommand{\ZERO}{\mbox{\bf 0}}
+\newcommand{\ONE}{\mbox{\bf 1}}
+\def\lexer{\mathit{lexer}}
+\def\mkeps{\mathit{mkeps}}
+\def\inj{\mathit{inj}}
+\def\Empty{\mathit{Empty}}
+\def\Left{\mathit{Left}}
+\def\Right{\mathit{Right}}
+\def\Stars{\mathit{Stars}}
+\def\Char{\mathit{Char}}
+\def\Seq{\mathit{Seq}}
+\def\Der{\mathit{Der}}
+\def\nullable{\mathit{nullable}}
+\def\Z{\mathit{Z}}
+\def\S{\mathit{S}}
+
+%\theoremstyle{theorem}
+%\newtheorem{theorem}{Theorem}
+%\theoremstyle{lemma}
+%\newtheorem{lemma}{Lemma}
+%\newcommand{\lemmaautorefname}{Lemma}
+%\theoremstyle{definition}
+%\newtheorem{definition}{Definition}
+
+
+\begin{document}
+
+\maketitle
+
+\begin{abstract}
+  Brzozowski introduced in 1964 a beautifully simple algorithm for
+  regular expression matching based on the notion of derivatives of
+  regular expressions. In 2014, Sulzmann and Lu extended this
+  algorithm to not just give a YES/NO answer for whether or not a regular
+  expression matches a string, but in case it matches also \emph{how}
+  it matches the string.  This is important for applications such as
+  lexing (tokenising a string). The problem is to make the algorithm
+  by Sulzmann and Lu fast on all inputs without breaking its
+  correctness.
+\end{abstract}
+
+\section{Introduction}
+
+This PhD-project is about regular expression matching and
+lexing. Given the maturity of this topic, the reader might wonder:
+Surely, regular expressions must have already been studied to death?
+What could possibly be \emph{not} known in this area? And surely all
+implemented algorithms for regular expression matching are blindingly
+fast?
+
+
+
+Unfortunately these preconceptions are not supported by evidence: Take
+for example the regular expression $(a^*)^*\,b$ and ask whether
+strings of the form $aa..a$ match this regular
+expression. Obviously they do not match---the expected $b$ in the last
+position is missing. One would expect that modern regular expression
+matching engines can find this out very quickly. Alas, if one tries
+this example in JavaScript, Python or Java 8 with strings like 28
+$a$'s, one discovers that this decision takes around 30 seconds and
+takes considerably longer when adding a few more $a$'s, as the graphs
+below show:
+
+\begin{center}
+\begin{tabular}{@{}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{}}
+\begin{tikzpicture}
+\begin{axis}[
+    xlabel={$n$},
+    x label style={at={(1.05,-0.05)}},
+    ylabel={time in secs},
+    enlargelimits=false,
+    xtick={0,5,...,30},
+    xmax=33,
+    ymax=35,
+    ytick={0,5,...,30},
+    scaled ticks=false,
+    axis lines=left,
+    width=5cm,
+    height=4cm, 
+    legend entries={JavaScript},  
+    legend pos=north west,
+    legend cell align=left]
+\addplot[red,mark=*, mark options={fill=white}] table {re-js.data};
+\end{axis}
+\end{tikzpicture}
+  &
+\begin{tikzpicture}
+\begin{axis}[
+    xlabel={$n$},
+    x label style={at={(1.05,-0.05)}},
+    %ylabel={time in secs},
+    enlargelimits=false,
+    xtick={0,5,...,30},
+    xmax=33,
+    ymax=35,
+    ytick={0,5,...,30},
+    scaled ticks=false,
+    axis lines=left,
+    width=5cm,
+    height=4cm, 
+    legend entries={Python},  
+    legend pos=north west,
+    legend cell align=left]
+\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};
+\end{axis}
+\end{tikzpicture}
+  &
+\begin{tikzpicture}
+\begin{axis}[
+    xlabel={$n$},
+    x label style={at={(1.05,-0.05)}},
+    %ylabel={time in secs},
+    enlargelimits=false,
+    xtick={0,5,...,30},
+    xmax=33,
+    ymax=35,
+    ytick={0,5,...,30},
+    scaled ticks=false,
+    axis lines=left,
+    width=5cm,
+    height=4cm, 
+    legend entries={Java 8},  
+    legend pos=north west,
+    legend cell align=left]
+\addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data};
+\end{axis}
+\end{tikzpicture}\\
+\multicolumn{3}{c}{Graphs: Runtime for matching $(a^*)^*\,b$ with strings 
+           of the form $\underbrace{aa..a}_{n}$.}
+\end{tabular}    
+\end{center}  
+
+\noindent These are clearly abysmal and possibly surprising results.
+One would expect these systems doing much better than that---after
+all, given a DFA and a string, whether a string is matched by this DFA
+should be linear.
+
+Admittedly, the regular expression $(a^*)^*\,b$ is carefully chosen to
+exhibit this ``exponential behaviour''.  Unfortunately, such regular
+expressions are not just a few ``outliers'', but actually they are
+frequent enough that a separate name has been created for
+them---\emph{evil regular expressions}. In empiric work, Davis et al
+report that they have found thousands of such evil regular expressions
+in the JavaScript and Python ecosystems \cite{Davis18}.
+
+This exponential blowup sometimes causes real pain in real life:
+for example on 20 July 2016 one evil regular expression brought the
+webpage \href{http://stackexchange.com}{Stack Exchange} to its knees \cite{SE16}.
+In this instance, a regular expression intended to just trim white
+spaces from the beginning and the end of a line actually consumed
+massive amounts of CPU-resources and because of this the web servers
+ground to a halt. This happened when a post with 20,000 white spaces
+was submitted, but importantly the white spaces were neither at the
+beginning nor at the end. As a result, the regular expression matching
+engine needed to backtrack over many choices.
+
+The underlying problem is that many ``real life'' regular expression
+matching engines do not use DFAs for matching. This is because they
+support regular expressions that are not covered by the classical
+automata theory, and in this more general setting there are quite a
+few research questions still unanswered and fast algorithms still need
+to be developed.
+
+There is also another under-researched problem to do with regular
+expressions and lexing, i.e.~the process of breaking up strings into
+sequences of tokens according to some regular expressions. In this
+setting one is not just interested in whether or not a regular
+expression matches a string, but if it matches also in \emph{how} it
+matches the string.  Consider for example a regular expression
+$r_{key}$ for recognising keywords such as \textit{if}, \textit{then}
+and so on; and a regular expression $r_{id}$ for recognising
+identifiers (say, a single character followed by characters or
+numbers). One can then form the compound regular expression
+$(r_{key} + r_{id})^*$ and use it to tokenise strings.  But then how
+should the string \textit{iffoo} be tokenised?  It could be tokenised
+as a keyword followed by an identifier, or the entire string as a
+single identifier.  Similarly, how should the string \textit{if} be
+tokenised? Both regular expressions, $r_{key}$ and $r_{id}$, would
+``fire''---so is it an identifier or a keyword?  While in applications
+there is a well-known strategy to decide these questions, called POSIX
+matching, only relatively recently precise definitions of what POSIX
+matching actually means have been formalised
+\cite{AusafDyckhoffUrban2016,OkuiSuzuki2010,Vansummeren2006}. Roughly,
+POSIX matching means matching the longest initial substring and
+in the case of a tie, the initial submatch is chosen according to some priorities attached to the
+regular expressions (e.g.~keywords have a higher priority than
+identifiers). This sounds rather simple, but according to Grathwohl et
+al \cite[Page 36]{CrashCourse2014} this is not the case. They wrote:
+
+\begin{quote}
+\it{}``The POSIX strategy is more complicated than the greedy because of 
+the dependence on information about the length of matched strings in the 
+various subexpressions.''
+\end{quote}
+
+\noindent
+This is also supported by evidence collected by Kuklewicz
+\cite{Kuklewicz} who noticed that a number of POSIX regular expression
+matchers calculate incorrect results.
+
+Our focus is on an algorithm introduced by Sulzmann and Lu in 2014 for
+regular expression matching according to the POSIX strategy
+\cite{Sulzmann2014}. Their algorithm is based on an older algorithm by
+Brzozowski from 1964 where he introduced the notion of derivatives of
+regular expressions \cite{Brzozowski1964}. We shall briefly explain
+the algorithms next.
+
+\section{The Algorithms by  Brzozowski, and Sulzmann and Lu}
+
+Suppose regular expressions are given by the following grammar (for
+the moment ignore the grammar for values on the right-hand side):
+
+
+\begin{center}
+	\begin{tabular}{c@{\hspace{20mm}}c}
+		\begin{tabular}{@{}rrl@{}}
+			\multicolumn{3}{@{}l}{\textbf{Regular Expressions}}\medskip\\
+			$r$ & $::=$  & $\ZERO$\\
+			& $\mid$ & $\ONE$   \\
+			& $\mid$ & $c$          \\
+			& $\mid$ & $r_1 \cdot r_2$\\
+			& $\mid$ & $r_1 + r_2$   \\
+			\\
+			& $\mid$ & $r^*$         \\
+		\end{tabular}
+		&
+		\begin{tabular}{@{\hspace{0mm}}rrl@{}}
+			\multicolumn{3}{@{}l}{\textbf{Values}}\medskip\\
+			$v$ & $::=$  & \\
+			&        & $\Empty$   \\
+			& $\mid$ & $\Char(c)$          \\
+			& $\mid$ & $\Seq\,v_1\, v_2$\\
+			& $\mid$ & $\Left(v)$   \\
+			& $\mid$ & $\Right(v)$  \\
+			& $\mid$ & $\Stars\,[v_1,\ldots\,v_n]$ \\
+		\end{tabular}
+	\end{tabular}
+\end{center}
+
+\noindent
+The intended meaning of the regular expressions is as usual: $\ZERO$
+cannot match any string, $\ONE$ can match the empty string, the
+character regular expression $c$ can match the character $c$, and so
+on. The brilliant contribution by Brzozowski is the notion of
+\emph{derivatives} of regular expressions.  The idea behind this
+notion is as follows: suppose a regular expression $r$ can match a
+string of the form $c\!::\! s$ (that is a list of characters starting
+with $c$), what does the regular expression look like that can match
+just $s$? Brzozowski gave a neat answer to this question. He defined
+the following operation on regular expressions, written
+$r\backslash c$ (the derivative of $r$ w.r.t.~the character $c$):
+
+\begin{center}
+\begin{tabular}{lcl}
+		$\ZERO \backslash c$ & $\dn$ & $\ZERO$\\  
+		$\ONE \backslash c$  & $\dn$ & $\ZERO$\\
+		$d \backslash c$     & $\dn$ & 
+		$\mathit{if} \;c = d\;\mathit{then}\;\ONE\;\mathit{else}\;\ZERO$\\
+$(r_1 + r_2)\backslash c$     & $\dn$ & $r_1 \backslash c \,+\, r_2 \backslash c$\\
+$(r_1 \cdot r_2)\backslash c$ & $\dn$ & $\mathit{if} \nullable(r_1)$\\
+	&   & $\mathit{then}\;(r_1\backslash c) \cdot r_2 \,+\, r_2\backslash c$\\
+	&   & $\mathit{else}\;(r_1\backslash c) \cdot r_2$\\
+	$(r^*)\backslash c$           & $\dn$ & $(r\backslash c) \cdot r^*$\\
+\end{tabular}
+\end{center}
+
+\noindent
+In this definition $\nullable(\_)$ stands for a simple recursive
+function that tests whether a regular expression can match the empty
+string (its definition is omitted). Assuming the classic notion of a
+\emph{language} of a regular expression, written $L(\_)$, the main
+property of the derivative operation is that
+
+\begin{center}
+$c\!::\!s \in L(r)$ holds
+if and only if $s \in L(r\backslash c)$.
+\end{center}
+
+\noindent
+The beauty of derivatives is that they lead to a really simple regular
+expression matching algorithm: To find out whether a string $s$
+matches with a regular expression $r$, build the derivatives of $r$
+w.r.t.\ (in succession) all the characters of the string $s$. Finally,
+test whether the resulting regular expression can match the empty
+string.  If yes, then $r$ matches $s$, and no in the negative
+case. For us the main advantage is that derivatives can be
+straightforwardly implemented in any functional programming language,
+and are easily definable and reasoned about in theorem provers---the
+definitions just consist of inductive datatypes and simple recursive
+functions. Moreover, the notion of derivatives can be easily
+generalised to cover extended regular expression constructors such as
+the not-regular expression, written $\neg\,r$, or bounded repetitions
+(for example $r^{\{n\}}$ and $r^{\{n..m\}}$), which cannot be so
+straightforwardly realised within the classic automata approach.
+
+
+One limitation, however, of Brzozowski's algorithm is that it only
+produces a YES/NO answer for whether a string is being matched by a
+regular expression.  Sulzmann and Lu~\cite{Sulzmann2014} extended this
+algorithm to allow generation of an actual matching, called a
+\emph{value}---see the grammar above for its definition.  Assuming a
+regular expression matches a string, values encode the information of
+\emph{how} the string is matched by the regular expression---that is,
+which part of the string is matched by which part of the regular
+expression. To illustrate this consider the string $xy$ and the
+regular expression $(x + (y + xy))^*$. We can view this regular
+expression as a tree and if the string $xy$ is matched by two Star
+``iterations'', then the $x$ is matched by the left-most alternative
+in this tree and the $y$ by the right-left alternative. This suggests
+to record this matching as
+
+\begin{center}
+$\Stars\,[\Left\,(\Char\,x), \Right(\Left(\Char\,y))]$
+\end{center}
+
+\noindent
+where $\Stars$ records how many
+iterations were used; and $\Left$, respectively $\Right$, which
+alternative is used. The value for
+matching $xy$ in a single ``iteration'', i.e.~the POSIX value,
+would look as follows
+
+\begin{center}
+$\Stars\,[\Seq\,(\Char\,x)\,(\Char\,y)]$
+\end{center}
+
+\noindent
+where $\Stars$ has only a single-element list for the single iteration
+and $\Seq$ indicates that $xy$ is matched by a sequence regular
+expression.
+
+The contribution of Sulzmann and Lu is an extension of Brzozowski's
+algorithm by a second phase (the first phase being building successive
+derivatives). In this second phase, for every successful match the
+corresponding POSIX value is computed. As mentioned before, from this
+value we can extract the information \emph{how} a regular expression
+matched a string. We omit the details here on how Sulzmann and Lu
+achieved this~(see \cite{Sulzmann2014}). Rather, we shall focus next on the
+process of simplification of regular expressions, which is needed in
+order to obtain \emph{fast} versions of the Brzozowski's, and Sulzmann
+and Lu's algorithms.  This is where the PhD-project hopes to advance
+the state-of-the-art.
+
+
+\section{Simplification of Regular Expressions}
+
+The main drawback of building successive derivatives according to
+Brzozowski's definition is that they can grow very quickly in size.
+This is mainly due to the fact that the derivative operation generates
+often ``useless'' $\ZERO$s and $\ONE$s in derivatives.  As a result,
+if implemented naively both algorithms by Brzozowski and by Sulzmann
+and Lu are excruciatingly slow. For example when starting with the
+regular expression $(a + aa)^*$ and building 12 successive derivatives
+w.r.t.~the character $a$, one obtains a derivative regular expression
+with more than 8000 nodes (when viewed as a tree). Operations like
+derivative and $\nullable$ need to traverse such trees and
+consequently the bigger the size of the derivative the slower the
+algorithm. Fortunately, one can simplify regular expressions after
+each derivative step. Various simplifications of regular expressions
+are possible, such as the simplifications of $\ZERO + r$,
+$r + \ZERO$, $\ONE\cdot r$, $r \cdot \ONE$, and $r + r$ to just
+$r$. These simplifications do not affect the answer for whether a
+regular expression matches a string or not, but fortunately also do
+not affect the POSIX strategy of how regular expressions match
+strings---although the latter is much harder to establish. Some
+initial results in this regard have been obtained in
+\cite{AusafDyckhoffUrban2016}. However, what has not been achieved yet
+is a very tight bound for the size. Such a tight bound is suggested by
+work of Antimirov who proved that (partial) derivatives can be bound
+by the number of characters contained in the initial regular
+expression \cite{Antimirov95}. We believe, and have generated test
+data, that a similar bound can be obtained for the derivatives in
+Sulzmann and Lu's algorithm. Let us give some details about this next.
+
+We first followed Sulzmann and Lu's idea of introducing
+\emph{annotated regular expressions}~\cite{Sulzmann2014}. They are
+defined by the following grammar:
+
+\begin{center}
+\begin{tabular}{lcl}
+  $\textit{a}$ & $::=$  & $\textit{ZERO}$\\
+                  & $\mid$ & $\textit{ONE}\;\;bs$\\
+                  & $\mid$ & $\textit{CHAR}\;\;bs\,c$\\
+                  & $\mid$ & $\textit{ALTS}\;\;bs\,as$\\
+                  & $\mid$ & $\textit{SEQ}\;\;bs\,a_1\,a_2$\\
+                  & $\mid$ & $\textit{STAR}\;\;bs\,a$
+\end{tabular}    
+\end{center}  
+
+\noindent
+where $bs$ stands for bitsequences, and $as$ (in \textit{ALTS}) for a
+list of annotated regular expressions. These bitsequences encode
+information about the (POSIX) value that should be generated by the
+Sulzmann and Lu algorithm. There are operations that can transform the
+usual (un-annotated) regular expressions into annotated regular
+expressions, and there are operations for encoding/decoding values to
+or from bitsequences. For example the encoding function for values is
+defined as follows:
+
+\begin{center}
+\begin{tabular}{lcl}
+  $\textit{code}(\Empty)$ & $\dn$ & $[]$\\
+  $\textit{code}(\Char\,c)$ & $\dn$ & $[]$\\
+  $\textit{code}(\Left\,v)$ & $\dn$ & $\Z :: code(v)$\\
+  $\textit{code}(\Right\,v)$ & $\dn$ & $\S :: code(v)$\\
+  $\textit{code}(\Seq\,v_1\,v_2)$ & $\dn$ & $code(v_1) \,@\, code(v_2)$\\
+  $\textit{code}(\Stars\,[])$ & $\dn$ & $[\S]$\\
+  $\textit{code}(\Stars\,(v\!::\!vs))$ & $\dn$ & $\Z :: code(v) \;@\;
+                                                 code(\Stars\,vs)$
+\end{tabular}    
+\end{center} 
+
+\noindent
+where $\Z$ and $\S$ are arbitrary names for the ``bits'' in the
+bitsequences. Although this encoding is ``lossy'' in the sense of not
+recording all details of a value, Sulzmann and Lu have defined the
+decoding function such that with additional information (namely the
+corresponding regular expression) a value can be precisely extracted
+from a bitsequence.
+
+The main point of the bitsequences and annotated regular expressions
+is that we can apply rather aggressive (in terms of size)
+simplification rules in order to keep derivatives small.  We have
+developed such ``aggressive'' simplification rules and generated test
+data that show that the expected bound can be achieved. Obviously we
+could only partially cover  the search space as there are infinitely
+many regular expressions and strings. One modification we introduced
+is to allow a list of annotated regular expressions in the
+\textit{ALTS} constructor. This allows us to not just delete
+unnecessary $\ZERO$s and $\ONE$s from regular expressions, but also
+unnecessary ``copies'' of regular expressions (very similar to
+simplifying $r + r$ to just $r$, but in a more general
+setting). Another modification is that we use simplification rules
+inspired by Antimirov's work on partial derivatives. They maintain the
+idea that only the first ``copy'' of a regular expression in an
+alternative contributes to the calculation of a POSIX value. All
+subsequent copies can be prunned from the regular expression.
+
+We are currently engaged with proving that our simplification rules
+actually do not affect the POSIX value that should be generated by the
+algorithm according to the specification of a POSIX value and
+furthermore that our derivatives stay small for all derivatives. For
+this proof we use the theorem prover Isabelle. Once completed, this
+result will advance the state-of-the-art: Sulzmann and Lu wrote in
+their paper \cite{Sulzmann2014} about the bitcoded ``incremental
+parsing method'' (that is the matching algorithm outlined in this
+section):
+
+\begin{quote}\it
+  ``Correctness Claim: We further claim that the incremental parsing
+  method in Figure~5 in combination with the simplification steps in
+  Figure 6 yields POSIX parse trees. We have tested this claim
+  extensively by using the method in Figure~3 as a reference but yet
+  have to work out all proof details.''
+\end{quote}  
+
+\noindent
+We would settle the correctness claim and furthermore obtain a much
+tighter bound on the sizes of derivatives. The result is that our
+algorithm should be correct and faster on all inputs.  The original
+blow-up, as observed in JavaScript, Python and Java, would be excluded
+from happening in our algorithm.
+
+\section{Conclusion}
+
+In this PhD-project we are interested in fast algorithms for regular
+expression matching. While this seems to be a ``settled'' area, in
+fact interesting research questions are popping up as soon as one steps
+outside the classic automata theory (for example in terms of what kind
+of regular expressions are supported). The reason why it is
+interesting for us to look at the derivative approach introduced by
+Brzozowski for regular expression matching, and then much further
+developed by Sulzmann and Lu, is that derivatives can elegantly deal
+with some of the regular expressions that are of interest in ``real
+life''. This includes the not-regular expression, written $\neg\,r$
+(that is all strings that are not recognised by $r$), but also bounded
+regular expressions such as $r^{\{n\}}$ and $r^{\{n..m\}}$). There is
+also hope that the derivatives can provide another angle for how to
+deal more efficiently with back-references, which are one of the
+reasons why regular expression engines in JavaScript, Python and Java
+choose to not implement the classic automata approach of transforming
+regular expressions into NFAs and then DFAs---because we simply do not
+know how such back-references can be represented by DFAs.
+
+
+\bibliographystyle{plain}
+\bibliography{root}
+
+
+\end{document}




