# HG changeset patch # User Chengsong # Date 1580903691 0 # Node ID c7825cfacc76910eb75df7fab8cc2551fb031113 # Parent 0acf6b58236e89e00e918c7db7c4716f795cdc0a lualatex is probably the culprit it overwrites the buffer when compiling a document sometimes so some changes you made just now gets lost diff -r 0acf6b58236e -r c7825cfacc76 etnms/etnms.tex --- a/etnms/etnms.tex Wed Feb 05 11:27:51 2020 +0000 +++ b/etnms/etnms.tex Wed Feb 05 11:54:51 2020 +0000 @@ -531,15 +531,15 @@ $(_{bs}\textit{CHAR}\;d)\,\backslash c$ & $\dn$ & $\textit{if}\;c=d\; \;\textit{then}\; _{bs}\textit{ONE}\;\textit{else}\;\textit{ZERO}$\\ - $(_{bs}\textit{ALTS}\;as)\,\backslash c$ & $\dn$ & - $_{bs}\textit{ALTS}\;(as.map(\backslash c))$\\ + $(_{bs}\textit{ALTS}\;\textit{as})\,\backslash c$ & $\dn$ & + $_{bs}\textit{ALTS}\;(\textit{as}.\textit{map}(\backslash c))$\\ $(_{bs}\textit{SEQ}\;a_1\,a_2)\,\backslash c$ & $\dn$ & $\textit{if}\;\textit{bnullable}\,a_1$\\ & &$\textit{then}\;_{bs}\textit{ALTS}\,List((_{[]}\textit{SEQ}\,(a_1\,\backslash c)\,a_2),$\\ & &$\phantom{\textit{then}\;_{bs}\textit{ALTS}\,}(\textit{fuse}\,(\textit{bmkeps}\,a_1)\,(a_2\,\backslash c)))$\\ & &$\textit{else}\;_{bs}\textit{SEQ}\,(a_1\,\backslash c)\,a_2$\\ $(_{bs}\textit{STAR}\,a)\,\backslash c$ & $\dn$ & - $_{bs}\textit{SEQ}\;(\textit{fuse}\, [\Z] (r\,\backslash c))\, + $_{bs}\textit{SEQ}\;(\textit{fuse}\, [0] \; r\,\backslash c )\, (_{bs}\textit{STAR}\,[]\,r)$ \end{tabular} \end{center} @@ -553,36 +553,37 @@ $(_{bs}{\bf d})\,\backslash c$ & $\dn$ & $\textit{if}\;c=d\; \;\textit{then}\; _{bs}\ONE\;\textit{else}\;\ZERO$\\ - $(_{bs}\oplus \;as)\,\backslash c$ & $\dn$ & - $_{bs}\oplus\;(as.map(\backslash c))$\\ + $(_{bs}\oplus \;\textit{as})\,\backslash c$ & $\dn$ & + $_{bs}\oplus\;(\textit{as.map}(\backslash c))$\\ $(_{bs}\;a_1\cdot a_2)\,\backslash c$ & $\dn$ & $\textit{if}\;\textit{bnullable}\,a_1$\\ & &$\textit{then}\;_{bs}\oplus\,[(_{[]}\,(a_1\,\backslash c)\cdot\,a_2),$\\ & &$\phantom{\textit{then},\;_{bs}\oplus\,}(\textit{fuse}\,(\textit{bmkeps}\,a_1)\,(a_2\,\backslash c))]$\\ & &$\textit{else}\;_{bs}\,(a_1\,\backslash c)\cdot a_2$\\ $(_{bs}a^*)\,\backslash c$ & $\dn$ & - $_{bs}\;((\textit{fuse}\, [\Z] (r\,\backslash c))\cdot + $_{bs}\;((\textit{fuse}\, [0] (r\,\backslash c))\cdot (_{[]}r^*))$ \end{tabular} \end{center} %\end{definition} \noindent -For instance, when we unfold $\textit{STAR} \; bs \; a$ into a sequence, -we need to attach an additional bit $Z$ to the front of $r \backslash c$ -to indicate that there is one more star iteration. Also the $SEQ$ clause +For instance, when we unfold $_{bs}a^*$ into a sequence, +we need to attach an additional bit $0$ to the front of $r \backslash c$ +to indicate that there is one more star iteration. Also the sequence clause is more subtle---when $a_1$ is $\textit{bnullable}$ (here \textit{bnullable} is exactly the same as $\textit{nullable}$, except that it is for annotated regular expressions, therefore we omit the -definition). Assume that $bmkeps$ correctly extracts the bitcode for how +definition). Assume that $\textit{bmkeps}$ correctly extracts the bitcode for how $a_1$ matches the string prior to character $c$ (more on this later), -then the right branch of $ALTS$, which is $fuse \; bmkeps \; a_1 (a_2 +then the right branch of alternative, which is $\textit{fuse} \; \bmkeps \; a_1 (a_2 \backslash c)$ will collapse the regular expression $a_1$(as it has already been fully matched) and store the parsing information at the head of the regular expression $a_2 \backslash c$ by fusing to it. The -bitsequence $bs$, which was initially attached to the head of $SEQ$, has -now been elevated to the top-level of $ALTS$, as this information will be -needed whichever way the $SEQ$ is matched---no matter whether $c$ belongs +bitsequence $\textit{bs}$, which was initially attached to the +first element of the sequence $a_1 \cdot a_2$, has +now been elevated to the top-level of $\oplus$, as this information will be +needed whichever way the sequence is matched---no matter whether $c$ belongs to $a_1$ or $ a_2$. After building these derivatives and maintaining all the lexing information, we complete the lexing by collecting the bitcodes using a generalised version of the $\textit{mkeps}$ function @@ -1725,1298 +1726,14 @@ \noindent\rule[0.5ex]{\linewidth}{1pt} -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 this is not the case---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 to do much better than that---after all, -given a DFA and a string, deciding whether a string is matched by this -DFA should be linear in terms of the size of the regular expression and -the string? - -Admittedly, the regular expression $(a^*)^*\,b$ is carefully chosen to -exhibit this super-linear behaviour. But unfortunately, such regular -expressions are not just a few outliers. They are actually -frequent enough to have a separate name 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}. Static analysis -approach that is both sound and complete exists\cite{17Bir}, but the running -time on certain examples in the RegExLib and Snort regular expressions -libraries is unacceptable. Therefore the problem of efficiency still remains. - -This superlinear blowup in matching algorithms sometimes causes -considerable grief 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.\footnote{\url{https://stackstatus.net/post/147710624694/outage-postmortem-july-20-2016}} -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---causing web servers to grind 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. In this example, the time needed to process -the string was $O(n^2)$ with respect to the string length. This -quadratic overhead was enough for the homepage of Stack Exchange to -respond so slowly that the load balancer assumed there must be some -attack and therefore stopped the servers from responding to any -requests. This made the whole site become unavailable. Another very -recent example is a global outage of all Cloudflare servers on 2 July -2019. A poorly written regular expression exhibited exponential -behaviour and exhausted CPUs that serve HTTP traffic. Although the -outage had several causes, at the heart was a regular expression that -was used to monitor network -traffic.\footnote{\url{https://blog.cloudflare.com/details-of-the-cloudflare-outage-on-july-2-2019/}} - -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 (for example how to treat efficiently bounded repetitions, negation and -back-references). -%question: dfa can have exponential states. isn't this the actual reason why they do not use dfas? -%how do they avoid dfas exponential states if they use them for fast matching? - -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 also in \emph{how}. 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}. Such a -definition has also been given by Sulzmann and Lu \cite{Sulzmann2014}, -but the corresponding correctness proof turned out to be faulty -\cite{AusafDyckhoffUrban2016}. Roughly, POSIX matching means matching -the longest initial substring. In the case of a tie, the initial -sub-match 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 in this project 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 this algorithm next. - -\section{The Algorithm by Brzozowski based on Derivatives of Regular -Expressions} - -Suppose (basic) regular expressions are given by the following grammar: -\[ r ::= \ZERO \mid \ONE - \mid c - \mid r_1 \cdot r_2 - \mid r_1 + r_2 - \mid r^* -\] - -\noindent -The intended meaning of the constructors is as follows: $\ZERO$ -cannot match any string, $\ONE$ can match the empty string, the -character regular expression $c$ can match the character $c$, and so -on. - -The ingenious 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 started -with the definition of $nullable$: -\begin{center} - \begin{tabular}{lcl} - $\nullable(\ZERO)$ & $\dn$ & $\mathit{false}$ \\ - $\nullable(\ONE)$ & $\dn$ & $\mathit{true}$ \\ - $\nullable(c)$ & $\dn$ & $\mathit{false}$ \\ - $\nullable(r_1 + r_2)$ & $\dn$ & $\nullable(r_1) \vee \nullable(r_2)$ \\ - $\nullable(r_1\cdot r_2)$ & $\dn$ & $\nullable(r_1) \wedge \nullable(r_2)$ \\ - $\nullable(r^*)$ & $\dn$ & $\mathit{true}$ \\ - \end{tabular} - \end{center} -This function simply tests whether the empty string is in $L(r)$. -He then 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} - -%Assuming the classic notion of a -%\emph{language} of a regular expression, written $L(\_)$, t - -\noindent -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 -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. -For the moment however, we focus only on the usual basic regular expressions. - - -Now if we want to find out whether a string $s$ matches with a regular -expression $r$, we can 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. To implement this idea -we can generalise the derivative operation to strings like this: - -\begin{center} -\begin{tabular}{lcl} -$r \backslash (c\!::\!s) $ & $\dn$ & $(r \backslash c) \backslash s$ \\ -$r \backslash [\,] $ & $\dn$ & $r$ -\end{tabular} -\end{center} - -\noindent -and then define as regular-expression matching algorithm: -\[ -match\;s\;r \;\dn\; nullable(r\backslash s) -\] - -\noindent -This algorithm looks graphically as follows: -\begin{equation}\label{graph:*} -\begin{tikzcd} -r_0 \arrow[r, "\backslash c_0"] & r_1 \arrow[r, "\backslash c_1"] & r_2 \arrow[r, dashed] & r_n \arrow[r,"\textit{nullable}?"] & \;\textrm{YES}/\textrm{NO} -\end{tikzcd} -\end{equation} - -\noindent -where we start with a regular expression $r_0$, build successive -derivatives until we exhaust the string and then use \textit{nullable} -to test whether the result can match the empty string. It can be -relatively easily shown that this matcher is correct (that is given -an $s = c_0...c_{n-1}$ and an $r_0$, it generates YES if and only if $s \in L(r_0)$). - - -\section{Values and the Algorithm by Sulzmann and Lu} - -One limitation 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} or -sometimes also \emph{lexical value}. These values and regular -expressions correspond to each other as illustrated in the following -table: - - -\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 -No value corresponds to $\ZERO$; $\Empty$ corresponds to $\ONE$; -$\Char$ to the character regular expression; $\Seq$ to the sequence -regular expression and so on. The idea of values is to encode a kind of -lexical value for how the sub-parts of a regular expression match the -sub-parts of a string. To see this, suppose a \emph{flatten} operation, -written $|v|$ for values. We can use this function to extract the -underlying string of a value $v$. For example, $|\mathit{Seq} \, -(\textit{Char x}) \, (\textit{Char y})|$ is the string $xy$. Using -flatten, we can describe how values encode lexical values: $\Seq\,v_1\, -v_2$ encodes a tree with two children nodes that tells how the string -$|v_1| @ |v_2|$ matches the regex $r_1 \cdot r_2$ whereby $r_1$ matches -the substring $|v_1|$ and, respectively, $r_2$ matches the substring -$|v_2|$. Exactly how these two are matched is contained in the children -nodes $v_1$ and $v_2$ of parent $\textit{Seq}$. - -To give a concrete example of how values work, 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 $\Stars \; [\ldots]$ records all the -iterations; 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---see \eqref{graph:*}). In this second phase, a POSIX value -is generated in case the regular expression matches the string. -Pictorially, the Sulzmann and Lu algorithm is as follows: - -\begin{ceqn} -\begin{equation}\label{graph:2} -\begin{tikzcd} -r_0 \arrow[r, "\backslash c_0"] \arrow[d] & r_1 \arrow[r, "\backslash c_1"] \arrow[d] & r_2 \arrow[r, dashed] \arrow[d] & r_n \arrow[d, "mkeps" description] \\ -v_0 & v_1 \arrow[l,"inj_{r_0} c_0"] & v_2 \arrow[l, "inj_{r_1} c_1"] & v_n \arrow[l, dashed] -\end{tikzcd} -\end{equation} -\end{ceqn} - -\noindent -For convenience, we shall employ the following notations: the regular -expression we start with is $r_0$, and the given string $s$ is composed -of characters $c_0 c_1 \ldots c_{n-1}$. In the first phase from the -left to right, we build the derivatives $r_1$, $r_2$, \ldots according -to the characters $c_0$, $c_1$ until we exhaust the string and obtain -the derivative $r_n$. We test whether this derivative is -$\textit{nullable}$ or not. If not, we know the string does not match -$r$ and no value needs to be generated. If yes, we start building the -values incrementally by \emph{injecting} back the characters into the -earlier values $v_n, \ldots, v_0$. This is the second phase of the -algorithm from the right to left. For the first value $v_n$, we call the -function $\textit{mkeps}$, which builds the lexical value -for how the empty string has been matched by the (nullable) regular -expression $r_n$. This function is defined as - - \begin{center} - \begin{tabular}{lcl} - $\mkeps(\ONE)$ & $\dn$ & $\Empty$ \\ - $\mkeps(r_{1}+r_{2})$ & $\dn$ - & \textit{if} $\nullable(r_{1})$\\ - & & \textit{then} $\Left(\mkeps(r_{1}))$\\ - & & \textit{else} $\Right(\mkeps(r_{2}))$\\ - $\mkeps(r_1\cdot r_2)$ & $\dn$ & $\Seq\,(\mkeps\,r_1)\,(\mkeps\,r_2)$\\ - $mkeps(r^*)$ & $\dn$ & $\Stars\,[]$ - \end{tabular} - \end{center} - - -\noindent There are no cases for $\ZERO$ and $c$, since -these regular expression cannot match the empty string. Note -also that in case of alternatives we give preference to the -regular expression on the left-hand side. This will become -important later on about what value is calculated. - -After the $\mkeps$-call, we inject back the characters one by one in order to build -the lexical value $v_i$ for how the regex $r_i$ matches the string $s_i$ -($s_i = c_i \ldots c_{n-1}$ ) from the previous lexical value $v_{i+1}$. -After injecting back $n$ characters, we get the lexical value for how $r_0$ -matches $s$. For this Sulzmann and Lu defined a function that reverses -the ``chopping off'' of characters during the derivative phase. The -corresponding function is called \emph{injection}, written -$\textit{inj}$; it takes three arguments: the first one is a regular -expression ${r_{i-1}}$, before the character is chopped off, the second -is a character ${c_{i-1}}$, the character we want to inject and the -third argument is the value ${v_i}$, into which one wants to inject the -character (it corresponds to the regular expression after the character -has been chopped off). The result of this function is a new value. The -definition of $\textit{inj}$ is as follows: - -\begin{center} -\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l} - $\textit{inj}\,(c)\,c\,Empty$ & $\dn$ & $Char\,c$\\ - $\textit{inj}\,(r_1 + r_2)\,c\,\Left(v)$ & $\dn$ & $\Left(\textit{inj}\,r_1\,c\,v)$\\ - $\textit{inj}\,(r_1 + r_2)\,c\,Right(v)$ & $\dn$ & $Right(\textit{inj}\,r_2\,c\,v)$\\ - $\textit{inj}\,(r_1 \cdot r_2)\,c\,Seq(v_1,v_2)$ & $\dn$ & $Seq(\textit{inj}\,r_1\,c\,v_1,v_2)$\\ - $\textit{inj}\,(r_1 \cdot r_2)\,c\,\Left(Seq(v_1,v_2))$ & $\dn$ & $Seq(\textit{inj}\,r_1\,c\,v_1,v_2)$\\ - $\textit{inj}\,(r_1 \cdot r_2)\,c\,Right(v)$ & $\dn$ & $Seq(\textit{mkeps}(r_1),\textit{inj}\,r_2\,c\,v)$\\ - $\textit{inj}\,(r^*)\,c\,Seq(v,Stars\,vs)$ & $\dn$ & $Stars((\textit{inj}\,r\,c\,v)\,::\,vs)$\\ -\end{tabular} -\end{center} - -\noindent This definition is by recursion on the ``shape'' of regular -expressions and values. To understands this definition better consider -the situation when we build the derivative on regular expression $r_{i-1}$. -For this we chop off a character from $r_{i-1}$ to form $r_i$. This leaves a -``hole'' in $r_i$ and its corresponding value $v_i$. -To calculate $v_{i-1}$, we need to -locate where that hole is and fill it. -We can find this location by -comparing $r_{i-1}$ and $v_i$. For instance, if $r_{i-1}$ is of shape -$r_a \cdot r_b$, and $v_i$ is of shape $\Left(Seq(v_1,v_2))$, we know immediately that -% -\[ (r_a \cdot r_b)\backslash c = (r_a\backslash c) \cdot r_b \,+\, r_b\backslash c,\] - -\noindent -otherwise if $r_a$ is not nullable, -\[ (r_a \cdot r_b)\backslash c = (r_a\backslash c) \cdot r_b,\] - -\noindent -the value $v_i$ should be $\Seq(\ldots)$, contradicting the fact that -$v_i$ is actually of shape $\Left(\ldots)$. Furthermore, since $v_i$ is of shape -$\Left(\ldots)$ instead of $\Right(\ldots)$, we know that the left -branch of \[ (r_a \cdot r_b)\backslash c = -\bold{\underline{ (r_a\backslash c) \cdot r_b} }\,+\, r_b\backslash c,\](underlined) - is taken instead of the right one. This means $c$ is chopped off -from $r_a$ rather than $r_b$. -We have therefore found out -that the hole will be on $r_a$. So we recursively call $\inj\, -r_a\,c\,v_a$ to fill that hole in $v_a$. After injection, the value -$v_i$ for $r_i = r_a \cdot r_b$ should be $\Seq\,(\inj\,r_a\,c\,v_a)\,v_b$. -Other clauses can be understood in a similar way. - -%\comment{Other word: insight?} -The following example gives an insight of $\textit{inj}$'s effect and -how Sulzmann and Lu's algorithm works as a whole. Suppose we have a -regular expression $((((a+b)+ab)+c)+abc)^*$, and want to match it -against the string $abc$ (when $abc$ is written as a regular expression, -the standard way of expressing it is $a \cdot (b \cdot c)$. But we -usually omit the parentheses and dots here for better readability. This -algorithm returns a POSIX value, which means it will produce the longest -matching. Consequently, it matches the string $abc$ in one star -iteration, using the longest alternative $abc$ in the sub-expression (we shall use $r$ to denote this -sub-expression for conciseness): - -\[((((a+b)+ab)+c)+\underbrace{abc}_r)\] - -\noindent -Before $\textit{inj}$ is called, our lexer first builds derivative using -string $abc$ (we simplified some regular expressions like $\ZERO \cdot -b$ to $\ZERO$ for conciseness; we also omit parentheses if they are -clear from the context): - -%Similarly, we allow -%$\textit{ALT}$ to take a list of regular expressions as an argument -%instead of just 2 operands to reduce the nested depth of -%$\textit{ALT}$ - -\begin{center} -\begin{tabular}{lcl} -$r^*$ & $\xrightarrow{\backslash a}$ & $r_1 = (\ONE+\ZERO+\ONE \cdot b + \ZERO + \ONE \cdot b \cdot c) \cdot r^*$\\ - & $\xrightarrow{\backslash b}$ & $r_2 = (\ZERO+\ZERO+\ONE \cdot \ONE + \ZERO + \ONE \cdot \ONE \cdot c) \cdot r^* +(\ZERO+\ONE+\ZERO + \ZERO + \ZERO) \cdot r^*$\\ - & $\xrightarrow{\backslash c}$ & $r_3 = ((\ZERO+\ZERO+\ZERO + \ZERO + \ONE \cdot \ONE \cdot \ONE) \cdot r^* + (\ZERO+\ZERO+\ZERO + \ONE + \ZERO) \cdot r^*) + $\\ - & & $\phantom{r_3 = (} ((\ZERO+\ONE+\ZERO + \ZERO + \ZERO) \cdot r^* + (\ZERO+\ZERO+\ZERO + \ONE + \ZERO) \cdot r^* )$ -\end{tabular} -\end{center} - -\noindent -In case $r_3$ is nullable, we can call $\textit{mkeps}$ -to construct a lexical value for how $r_3$ matched the string $abc$. -This function gives the following value $v_3$: - - -\begin{center} -$\Left(\Left(\Seq(\Right(\Seq(\Empty, \Seq(\Empty,\Empty))), \Stars [])))$ -\end{center} -The outer $\Left(\Left(\ldots))$ tells us the leftmost nullable part of $r_3$(underlined): - -\begin{center} - \begin{tabular}{l@{\hspace{2mm}}l} - & $\big(\underline{(\ZERO+\ZERO+\ZERO+ \ZERO+ \ONE \cdot \ONE \cdot \ONE) \cdot r^*} - \;+\; (\ZERO+\ZERO+\ZERO + \ONE + \ZERO) \cdot r^*\big)$ \smallskip\\ - $+$ & $\big((\ZERO+\ONE+\ZERO + \ZERO + \ZERO) \cdot r^* - \;+\; (\ZERO+\ZERO+\ZERO + \ONE + \ZERO) \cdot r^* \big)$ - \end{tabular} - \end{center} - -\noindent - Note that the leftmost location of term $(\ZERO+\ZERO+\ZERO + \ZERO + \ONE \cdot \ONE \cdot - \ONE) \cdot r^*$ (which corresponds to the initial sub-match $abc$) allows - $\textit{mkeps}$ to pick it up because $\textit{mkeps}$ is defined to always choose the - left one when it is nullable. In the case of this example, $abc$ is - preferred over $a$ or $ab$. This $\Left(\Left(\ldots))$ location is - generated by two applications of the splitting clause - -\begin{center} - $(r_1 \cdot r_2)\backslash c \;\;(when \; r_1 \; nullable) \, = (r_1\backslash c) \cdot r_2 \,+\, r_2\backslash c.$ -\end{center} - -\noindent -By this clause, we put $r_1 \backslash c \cdot r_2 $ at the -$\textit{front}$ and $r_2 \backslash c$ at the $\textit{back}$. This -allows $\textit{mkeps}$ to always pick up among two matches the one with a longer -initial sub-match. Removing the outside $\Left(\Left(...))$, the inside -sub-value - -\begin{center} - $\Seq(\Right(\Seq(\Empty, \Seq(\Empty, \Empty))), \Stars [])$ -\end{center} - -\noindent -tells us how the empty string $[]$ is matched with $(\ZERO+\ZERO+\ZERO + \ZERO + \ONE \cdot -\ONE \cdot \ONE) \cdot r^*$. We match $[]$ by a sequence of two nullable regular -expressions. The first one is an alternative, we take the rightmost -alternative---whose language contains the empty string. The second -nullable regular expression is a Kleene star. $\Stars$ tells us how it -generates the nullable regular expression: by 0 iterations to form -$\ONE$. Now $\textit{inj}$ injects characters back and incrementally -builds a lexical value based on $v_3$. Using the value $v_3$, the character -c, and the regular expression $r_2$, we can recover how $r_2$ matched -the string $[c]$ : $\textit{inj} \; r_2 \; c \; v_3$ gives us - \begin{center} - $v_2 = \Left(\Seq(\Right(\Seq(\Empty, \Seq(\Empty, c))), \Stars [])),$ - \end{center} -which tells us how $r_2$ matched $[c]$. After this we inject back the character $b$, and get -\begin{center} -$v_1 = \Seq(\Right(\Seq(\Empty, \Seq(b, c))), \Stars [])$ -\end{center} - for how - \begin{center} - $r_1= (\ONE+\ZERO+\ONE \cdot b + \ZERO + \ONE \cdot b \cdot c) \cdot r*$ - \end{center} - matched the string $bc$ before it split into two substrings. - Finally, after injecting character $a$ back to $v_1$, - we get the lexical value tree - \begin{center} - $v_0= \Stars [\Right(\Seq(a, \Seq(b, c)))]$ - \end{center} - for how $r$ matched $abc$. This completes the algorithm. - -%We omit the details of injection function, which is provided by Sulzmann and Lu's paper \cite{Sulzmann2014}. -Readers might have noticed that the lexical value information is actually -already available when doing derivatives. For example, immediately after -the operation $\backslash a$ we know that if we want to match a string -that starts with $a$, we can either take the initial match to be - - \begin{center} -\begin{enumerate} - \item[1)] just $a$ or - \item[2)] string $ab$ or - \item[3)] string $abc$. -\end{enumerate} -\end{center} - -\noindent -In order to differentiate between these choices, we just need to -remember their positions---$a$ is on the left, $ab$ is in the middle , -and $abc$ is on the right. Which of these alternatives is chosen -later does not affect their relative position because the algorithm does -not change this order. If this parsing information can be determined and -does not change because of later derivatives, there is no point in -traversing this information twice. This leads to an optimisation---if we -store the information for lexical values inside the regular expression, -update it when we do derivative on them, and collect the information -when finished with derivatives and call $\textit{mkeps}$ for deciding which -branch is POSIX, we can generate the lexical value in one pass, instead of -doing the rest $n$ injections. This leads to Sulzmann and Lu's novel -idea of using bitcodes in derivatives. - -In the next section, we shall focus on the bitcoded algorithm and the -process of simplification of regular expressions. This 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 aims to advance the -state-of-the-art. - - -\section{Simplification of Regular Expressions} - -Using bitcodes to guide parsing is not a novel idea. It was applied to -context free grammars and then adapted by Henglein and Nielson for -efficient regular expression lexing using DFAs~\cite{nielson11bcre}. -Sulzmann and Lu took this idea of bitcodes a step further by integrating -bitcodes into derivatives. The reason why we want to use bitcodes in -this project is that we want to introduce more aggressive simplification -rules in order to keep the size of derivatives small throughout. This is -because 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 -$\textit{der}$ 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 simplification 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}. - -Unfortunately, the simplification rules outlined above are not -sufficient to prevent a size explosion in all cases. We -believe a tighter bound can be achieved that prevents an explosion in -\emph{all} cases. Such a tighter 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}. He defined the \emph{partial derivatives} of regular -expressions as follows: - -\begin{center} -\begin{tabular}{lcl} - $\textit{pder} \; c \; \ZERO$ & $\dn$ & $\emptyset$\\ - $\textit{pder} \; c \; \ONE$ & $\dn$ & $\emptyset$ \\ - $\textit{pder} \; c \; d$ & $\dn$ & $\textit{if} \; c \,=\, d \; \{ \ONE \} \; \textit{else} \; \emptyset$ \\ - $\textit{pder} \; c \; r_1+r_2$ & $\dn$ & $pder \; c \; r_1 \cup pder \; c \; r_2$ \\ - $\textit{pder} \; c \; r_1 \cdot r_2$ & $\dn$ & $\textit{if} \; nullable \; r_1 $\\ - & & $\textit{then} \; \{ r \cdot r_2 \mid r \in pder \; c \; r_1 \} \cup pder \; c \; r_2 \;$\\ - & & $\textit{else} \; \{ r \cdot r_2 \mid r \in pder \; c \; r_1 \} $ \\ - $\textit{pder} \; c \; r^*$ & $\dn$ & $ \{ r' \cdot r^* \mid r' \in pder \; c \; r \} $ \\ - \end{tabular} - \end{center} - -\noindent -A partial derivative of a regular expression $r$ is essentially a set of -regular expressions that are either $r$'s children expressions or a -concatenation of them. Antimirov has proved a tight bound of the sum of -the size of \emph{all} partial derivatives no matter what the string -looks like. Roughly speaking the size sum will be at most cubic in the -size of the regular expression. - -If we want the size of derivatives in Sulzmann and Lu's algorithm to -stay below this bound, we would need more aggressive simplifications. -Essentially we need to delete useless $\ZERO$s and $\ONE$s, as well as -deleting duplicates whenever possible. For example, the parentheses in -$(a+b) \cdot c + bc$ can be opened up to get $a\cdot c + b \cdot c + b -\cdot c$, and then simplified to just $a \cdot c + b \cdot c$. Another -example is simplifying $(a^*+a) + (a^*+ \ONE) + (a +\ONE)$ to just -$a^*+a+\ONE$. Adding these more aggressive simplification rules helps us -to achieve the same size bound as that of the partial derivatives. - -In order to implement the idea of ``spilling out alternatives'' and to -make them compatible with the $\text{inj}$-mechanism, we use -\emph{bitcodes}. Bits and bitcodes (lists of bits) are just: - -%This allows us to prove a tight -%bound on the size of regular expression during the running time of the -%algorithm if we can establish the connection between our simplification -%rules and partial derivatives. - - %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. - - -\begin{center} - $b ::= 0 \mid 1 \qquad -bs ::= [] \mid b:bs -$ -\end{center} - -\noindent -The $0$ and $1$ are arbitrary names for the bits and not in bold -in order to avoid -confusion with the regular expressions $\ZERO$ and $\ONE$. Bitcodes (or -bit-lists) can be used to encode values (or incomplete values) in a -compact form. This can be straightforwardly seen in the following -coding function from values to bitcodes: - -\begin{center} -\begin{tabular}{lcl} - $\textit{code}(\Empty)$ & $\dn$ & $[]$\\ - $\textit{code}(\Char\,c)$ & $\dn$ & $[]$\\ - $\textit{code}(\Left\,v)$ & $\dn$ & $0 :: code(v)$\\ - $\textit{code}(\Right\,v)$ & $\dn$ & $1 :: code(v)$\\ - $\textit{code}(\Seq\,v_1\,v_2)$ & $\dn$ & $code(v_1) \,@\, code(v_2)$\\ - $\textit{code}(\Stars\,[])$ & $\dn$ & $[0]$\\ - $\textit{code}(\Stars\,(v\!::\!vs))$ & $\dn$ & $1 :: code(v) \;@\; - code(\Stars\,vs)$ -\end{tabular} -\end{center} - -\noindent -Here $\textit{code}$ encodes a value into a bitcodes by converting -$\Left$ into $0$, $\Right$ into $1$, the start point of a non-empty -star iteration into $\S$, and the border where a local star terminates -into $0$. This coding is lossy, as it throws away the information about -characters, and also does not encode the ``boundary'' between two -sequence values. Moreover, with only the bitcode we cannot even tell -whether the $1$s and $0$s are for $\Left/\Right$ or $\Stars$. The -reason for choosing this compact way of storing information is that the -relatively small size of bits can be easily manipulated and ``moved -around'' in a regular expression. In order to recover values, we will -need the corresponding regular expression as an extra information. This -means the decoding function is defined as: - - -%\begin{definition}[Bitdecoding of Values]\mbox{} -\begin{center} -\begin{tabular}{@{}l@{\hspace{1mm}}c@{\hspace{1mm}}l@{}} - $\textit{decode}'\,bs\,(\ONE)$ & $\dn$ & $(\Empty, bs)$\\ - $\textit{decode}'\,bs\,(c)$ & $\dn$ & $(\Char\,c, bs)$\\ - $\textit{decode}'\,(0\!::\!bs)\;(r_1 + r_2)$ & $\dn$ & - $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r_1\;\textit{in}\; - (\Left\,v, bs_1)$\\ - $\textit{decode}'\,(1\!::\!bs)\;(r_1 + r_2)$ & $\dn$ & - $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r_2\;\textit{in}\; - (\Right\,v, bs_1)$\\ - $\textit{decode}'\,bs\;(r_1\cdot r_2)$ & $\dn$ & - $\textit{let}\,(v_1, bs_1) = \textit{decode}'\,bs\,r_1\;\textit{in}$\\ - & & $\textit{let}\,(v_2, bs_2) = \textit{decode}'\,bs_1\,r_2$\\ - & & \hspace{35mm}$\textit{in}\;(\Seq\,v_1\,v_2, bs_2)$\\ - $\textit{decode}'\,(0\!::\!bs)\,(r^*)$ & $\dn$ & $(\Stars\,[], bs)$\\ - $\textit{decode}'\,(1\!::\!bs)\,(r^*)$ & $\dn$ & - $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r\;\textit{in}$\\ - & & $\textit{let}\,(\Stars\,vs, bs_2) = \textit{decode}'\,bs_1\,r^*$\\ - & & \hspace{35mm}$\textit{in}\;(\Stars\,v\!::\!vs, bs_2)$\bigskip\\ - - $\textit{decode}\,bs\,r$ & $\dn$ & - $\textit{let}\,(v, bs') = \textit{decode}'\,bs\,r\;\textit{in}$\\ - & & $\textit{if}\;bs' = []\;\textit{then}\;\textit{Some}\,v\; - \textit{else}\;\textit{None}$ -\end{tabular} -\end{center} -%\end{definition} - -Sulzmann and Lu's integrated the bitcodes into regular expressions to -create annotated regular expressions \cite{Sulzmann2014}. -\emph{Annotated regular expressions} are defined by the following -grammar:%\comment{ALTS should have an $as$ in the definitions, not just $a_1$ and $a_2$} - -\begin{center} -\begin{tabular}{lcl} - $\textit{a}$ & $::=$ & $\ZERO$\\ - & $\mid$ & $_{bs}\ONE$\\ - & $\mid$ & $_{bs}{\bf c}$\\ - & $\mid$ & $_{bs}\oplus \textit{as}$\\ - & $\mid$ & $_{bs}a_1\cdot a_2$\\ - & $\mid$ & $_{bs}a^*$ -\end{tabular} -\end{center} -%(in \textit{ALTS}) - -\noindent -where $\textit{bs}$ stands for bitcodes, $a$ for $\textit{annotated}$ regular -expressions and $\textit{as}$ for a list of annotated regular expressions. -The alternative constructor($\oplus$) has been generalized to -accept a list of annotated regular expressions rather than just 2. -We will show that these bitcodes encode information about -the (POSIX) value that should be generated by the Sulzmann and Lu -algorithm. - - -To do lexing using annotated regular expressions, we shall first -transform the usual (un-annotated) regular expressions into annotated -regular expressions. This operation is called \emph{internalisation} and -defined as follows: - -%\begin{definition} -\begin{center} -\begin{tabular}{lcl} - $(\ZERO)^\uparrow$ & $\dn$ & $\ZERO$\\ - $(\ONE)^\uparrow$ & $\dn$ & $_{[]}\ONE$\\ - $(c)^\uparrow$ & $\dn$ & $_{bs}\textit{\bf c}$\\ - $(r_1 + r_2)^\uparrow$ & $\dn$ & - $_{[]}\oplus\,[(\textit{fuse}\,[0]\,r_1^\uparrow),\, - (\textit{fuse}\,[1]\,r_2^\uparrow)]$\\ - $(r_1\cdot r_2)^\uparrow$ & $\dn$ & - $_{[]}r_1^\uparrow \cdot r_2^\uparrow$\\ - $(r^*)^\uparrow$ & $\dn$ & - $_{[]}r^\uparrow$\\ -\end{tabular} -\end{center} -%\end{definition} - -\noindent -We use up arrows here to indicate that the basic un-annotated regular -expressions are ``lifted up'' into something slightly more complex. In the -fourth clause, $\textit{fuse}$ is an auxiliary function that helps to -attach bits to the front of an annotated regular expression. Its -definition is as follows: - -\begin{center} -\begin{tabular}{lcl} - $\textit{fuse}\;bs\,(\ZERO)$ & $\dn$ & $\textit{ZERO}$\\ - $\textit{fuse}\;bs\,(\textit{ONE}\,bs')$ & $\dn$ & - $\textit{ONE}\,(bs\,@\,bs')$\\ - $\textit{fuse}\;bs\,(\textit{CHAR}\,bs'\,c)$ & $\dn$ & - $\textit{CHAR}\,(bs\,@\,bs')\,c$\\ - $\textit{fuse}\;bs\,(\textit{ALTS}\,bs'\,as)$ & $\dn$ & - $\textit{ALTS}\,(bs\,@\,bs')\,as$\\ - $\textit{fuse}\;bs\,(\textit{SEQ}\,bs'\,a_1\,a_2)$ & $\dn$ & - $\textit{SEQ}\,(bs\,@\,bs')\,a_1\,a_2$\\ - $\textit{fuse}\;bs\,(\textit{STAR}\,bs'\,a)$ & $\dn$ & - $\textit{STAR}\,(bs\,@\,bs')\,a$ -\end{tabular} -\end{center} - -\noindent -After internalising the regular expression, we perform successive -derivative operations on the annotated regular expressions. This -derivative operation is the same as what we had previously for the -basic regular expressions, except that we beed to take care of -the bitcodes: - - %\begin{definition}{bder} -\begin{center} - \begin{tabular}{@{}lcl@{}} - $(\textit{ZERO})\,\backslash c$ & $\dn$ & $\textit{ZERO}$\\ - $(\textit{ONE}\;bs)\,\backslash c$ & $\dn$ & $\textit{ZERO}$\\ - $(\textit{CHAR}\;bs\,d)\,\backslash c$ & $\dn$ & - $\textit{if}\;c=d\; \;\textit{then}\; - \textit{ONE}\;bs\;\textit{else}\;\textit{ZERO}$\\ - $(\textit{ALTS}\;bs\,as)\,\backslash c$ & $\dn$ & - $\textit{ALTS}\;bs\,(as.map(\backslash c))$\\ - $(\textit{SEQ}\;bs\,a_1\,a_2)\,\backslash c$ & $\dn$ & - $\textit{if}\;\textit{bnullable}\,a_1$\\ - & &$\textit{then}\;\textit{ALTS}\,bs\,List((\textit{SEQ}\,[]\,(a_1\,\backslash c)\,a_2),$\\ - & &$\phantom{\textit{then}\;\textit{ALTS}\,bs\,}(\textit{fuse}\,(\textit{bmkeps}\,a_1)\,(a_2\,\backslash c)))$\\ - & &$\textit{else}\;\textit{SEQ}\,bs\,(a_1\,\backslash c)\,a_2$\\ - $(\textit{STAR}\,bs\,a)\,\backslash c$ & $\dn$ & - $\textit{SEQ}\;bs\,(\textit{fuse}\, [\Z] (r\,\backslash c))\, - (\textit{STAR}\,[]\,r)$ -\end{tabular} -\end{center} -%\end{definition} - -\noindent -For instance, when we unfold $\textit{STAR} \; bs \; a$ into a sequence, -we need to attach an additional bit $Z$ to the front of $r \backslash c$ -to indicate that there is one more star iteration. Also the $SEQ$ clause -is more subtle---when $a_1$ is $\textit{bnullable}$ (here -\textit{bnullable} is exactly the same as $\textit{nullable}$, except -that it is for annotated regular expressions, therefore we omit the -definition). Assume that $bmkeps$ correctly extracts the bitcode for how -$a_1$ matches the string prior to character $c$ (more on this later), -then the right branch of $ALTS$, which is $fuse \; bmkeps \; a_1 (a_2 -\backslash c)$ will collapse the regular expression $a_1$(as it has -already been fully matched) and store the parsing information at the -head of the regular expression $a_2 \backslash c$ by fusing to it. The -bitsequence $bs$, which was initially attached to the head of $SEQ$, has -now been elevated to the top-level of $ALTS$, as this information will be -needed whichever way the $SEQ$ is matched---no matter whether $c$ belongs -to $a_1$ or $ a_2$. After building these derivatives and maintaining all -the lexing information, we complete the lexing by collecting the -bitcodes using a generalised version of the $\textit{mkeps}$ function -for annotated regular expressions, called $\textit{bmkeps}$: - - -%\begin{definition}[\textit{bmkeps}]\mbox{} -\begin{center} -\begin{tabular}{lcl} - $\textit{bmkeps}\,(\textit{ONE}\;bs)$ & $\dn$ & $bs$\\ - $\textit{bmkeps}\,(\textit{ALTS}\;bs\,a::as)$ & $\dn$ & - $\textit{if}\;\textit{bnullable}\,a$\\ - & &$\textit{then}\;bs\,@\,\textit{bmkeps}\,a$\\ - & &$\textit{else}\;bs\,@\,\textit{bmkeps}\,(\textit{ALTS}\;bs\,as)$\\ - $\textit{bmkeps}\,(\textit{SEQ}\;bs\,a_1\,a_2)$ & $\dn$ & - $bs \,@\,\textit{bmkeps}\,a_1\,@\, \textit{bmkeps}\,a_2$\\ - $\textit{bmkeps}\,(\textit{STAR}\;bs\,a)$ & $\dn$ & - $bs \,@\, [\S]$ -\end{tabular} -\end{center} -%\end{definition} - -\noindent -This function completes the value information by travelling along the -path of the regular expression that corresponds to a POSIX value and -collecting all the bitcodes, and using $S$ to indicate the end of star -iterations. If we take the bitcodes produced by $\textit{bmkeps}$ and -decode them, we get the value we expect. The corresponding lexing -algorithm looks as follows: - -\begin{center} -\begin{tabular}{lcl} - $\textit{blexer}\;r\,s$ & $\dn$ & - $\textit{let}\;a = (r^\uparrow)\backslash s\;\textit{in}$\\ - & & $\;\;\textit{if}\; \textit{bnullable}(a)$\\ - & & $\;\;\textit{then}\;\textit{decode}\,(\textit{bmkeps}\,a)\,r$\\ - & & $\;\;\textit{else}\;\textit{None}$ -\end{tabular} -\end{center} - -\noindent -In this definition $\_\backslash s$ is the generalisation of the derivative -operation from characters to strings (just like the derivatives for un-annotated -regular expressions). - -The main point of the bitcodes 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 pruned away from -the regular expression. A recursive definition of our simplification function -that looks somewhat similar to our Scala code is given below: -%\comment{Use $\ZERO$, $\ONE$ and so on. -%Is it $ALTS$ or $ALTS$?}\\ - -\begin{center} - \begin{tabular}{@{}lcl@{}} - - $\textit{simp} \; (\textit{SEQ}\;bs\,a_1\,a_2)$ & $\dn$ & $ (\textit{simp} \; a_1, \textit{simp} \; a_2) \; \textit{match} $ \\ - &&$\quad\textit{case} \; (\ZERO, \_) \Rightarrow \ZERO$ \\ - &&$\quad\textit{case} \; (\_, \ZERO) \Rightarrow \ZERO$ \\ - &&$\quad\textit{case} \; (\ONE, a_2') \Rightarrow \textit{fuse} \; bs \; a_2'$ \\ - &&$\quad\textit{case} \; (a_1', \ONE) \Rightarrow \textit{fuse} \; bs \; a_1'$ \\ - &&$\quad\textit{case} \; (a_1', a_2') \Rightarrow \textit{SEQ} \; bs \; a_1' \; a_2'$ \\ - - $\textit{simp} \; (\textit{ALTS}\;bs\,as)$ & $\dn$ & $\textit{distinct}( \textit{flatten} ( \textit{map simp as})) \; \textit{match} $ \\ - &&$\quad\textit{case} \; [] \Rightarrow \ZERO$ \\ - &&$\quad\textit{case} \; a :: [] \Rightarrow \textit{fuse bs a}$ \\ - &&$\quad\textit{case} \; as' \Rightarrow \textit{ALTS}\;bs\;as'$\\ - - $\textit{simp} \; a$ & $\dn$ & $\textit{a} \qquad \textit{otherwise}$ -\end{tabular} -\end{center} - -\noindent -The simplification does a pattern matching on the regular expression. -When it detected that the regular expression is an alternative or -sequence, it will try to simplify its children regular expressions -recursively and then see if one of the children turn into $\ZERO$ or -$\ONE$, which might trigger further simplification at the current level. -The most involved part is the $\textit{ALTS}$ clause, where we use two -auxiliary functions $\textit{flatten}$ and $\textit{distinct}$ to open up nested -$\textit{ALTS}$ and reduce as many duplicates as possible. Function -$\textit{distinct}$ keeps the first occurring copy only and remove all later ones -when detected duplicates. Function $\textit{flatten}$ opens up nested \textit{ALTS}. -Its recursive definition is given below: - - \begin{center} - \begin{tabular}{@{}lcl@{}} - $\textit{flatten} \; (\textit{ALTS}\;bs\,as) :: as'$ & $\dn$ & $(\textit{map} \; - (\textit{fuse}\;bs)\; \textit{as}) \; @ \; \textit{flatten} \; as' $ \\ - $\textit{flatten} \; \textit{ZERO} :: as'$ & $\dn$ & $ \textit{flatten} \; as' $ \\ - $\textit{flatten} \; a :: as'$ & $\dn$ & $a :: \textit{flatten} \; as'$ \quad(otherwise) -\end{tabular} -\end{center} - -\noindent -Here $\textit{flatten}$ behaves like the traditional functional programming flatten -function, except that it also removes $\ZERO$s. Or in terms of regular expressions, it -removes parentheses, for example changing $a+(b+c)$ into $a+b+c$. - -Suppose we apply simplification after each derivative step, and view -these two operations as an atomic one: $a \backslash_{simp}\,c \dn -\textit{simp}(a \backslash c)$. Then we can use the previous natural -extension from derivative w.r.t.~character to derivative -w.r.t.~string:%\comment{simp in the [] case?} - -\begin{center} -\begin{tabular}{lcl} -$r \backslash_{simp} (c\!::\!s) $ & $\dn$ & $(r \backslash_{simp}\, c) \backslash_{simp}\, s$ \\ -$r \backslash_{simp} [\,] $ & $\dn$ & $r$ -\end{tabular} -\end{center} - -\noindent -we obtain an optimised version of the algorithm: - - \begin{center} -\begin{tabular}{lcl} - $\textit{blexer\_simp}\;r\,s$ & $\dn$ & - $\textit{let}\;a = (r^\uparrow)\backslash_{simp}\, s\;\textit{in}$\\ - & & $\;\;\textit{if}\; \textit{bnullable}(a)$\\ - & & $\;\;\textit{then}\;\textit{decode}\,(\textit{bmkeps}\,a)\,r$\\ - & & $\;\;\textit{else}\;\textit{None}$ -\end{tabular} -\end{center} - -\noindent -This algorithm keeps the regular expression size small, for example, -with this simplification our previous $(a + aa)^*$ example's 8000 nodes -will be reduced to just 6 and stays constant, no matter how long the -input string is. -\section{Current Work} -We are currently engaged in two tasks related to this algorithm. The -first task is 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 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. 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 lexing 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 tree [our lexical values]. 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 like to settle this correctness claim. It is relatively -straightforward to establish that after one simplification step, the part of a -nullable derivative that corresponds to a POSIX value remains intact and can -still be collected, in other words, we can show that -%\comment{Double-check....I -%think this is not the case} -%\comment{If i remember correctly, you have proved this lemma. -%I feel this is indeed not true because you might place arbitrary -%bits on the regex r, however if this is the case, did i remember wrongly that -%you proved something like simplification does not affect $\textit{bmkeps}$ results? -%Anyway, i have amended this a little bit so it does not allow arbitrary bits attached -%to a regex. Maybe it works now.} - -\begin{center} - $\textit{bmkeps} \; a = \textit{bmkeps} \; \textit{bsimp} \; a\;($\textit{provided}$ \; a\; is \; \textit{bnullable} )$ -\end{center} - -\noindent -as this basically comes down to proving actions like removing the -additional $r$ in $r+r$ does not delete important POSIX information in -a regular expression. The hard part of this proof is to establish that - -\begin{center} - $ \textit{blexer}\_{simp}(r, \; s) = \textit{blexer}(r, \; s)$ -\end{center} -%comment{This is not true either...look at the definion blexer/blexer-simp} - -\noindent That is, if we do derivative on regular expression $r$ and then -simplify it, and repeat this process until we exhaust the string, we get a -regular expression $r''$($\textit{LHS}$) that provides the POSIX matching -information, which is exactly the same as the result $r'$($\textit{RHS}$ of the -normal derivative algorithm that only does derivative repeatedly and has no -simplification at all. This might seem at first glance very unintuitive, as -the $r'$ could be exponentially larger than $r''$, but can be explained in the -following way: we are pruning away the possible matches that are not POSIX. -Since there could be exponentially many -non-POSIX matchings and only 1 POSIX matching, it -is understandable that our $r''$ can be a lot smaller. we can still provide -the same POSIX value if there is one. This is not as straightforward as the -previous proposition, as the two regular expressions $r'$ and $r''$ might have -become very different. The crucial point is to find the -$\textit{POSIX}$ information of a regular expression and how it is modified, -augmented and propagated -during simplification in parallel with the regular expression that -has not been simplified in the subsequent derivative operations. To aid this, -we use the helper function retrieve described by Sulzmann and Lu: -\begin{center} -\begin{tabular}{@{}l@{\hspace{2mm}}c@{\hspace{2mm}}l@{}} - $\textit{retrieve}\,(\textit{ONE}\,bs)\,\Empty$ & $\dn$ & $bs$\\ - $\textit{retrieve}\,(\textit{CHAR}\,bs\,c)\,(\Char\,d)$ & $\dn$ & $bs$\\ - $\textit{retrieve}\,(\textit{ALTS}\,bs\,a::as)\,(\Left\,v)$ & $\dn$ & - $bs \,@\, \textit{retrieve}\,a\,v$\\ - $\textit{retrieve}\,(\textit{ALTS}\,bs\,a::as)\,(\Right\,v)$ & $\dn$ & - $bs \,@\, \textit{retrieve}\,(\textit{ALTS}\,bs\,as)\,v$\\ - $\textit{retrieve}\,(\textit{SEQ}\,bs\,a_1\,a_2)\,(\Seq\,v_1\,v_2)$ & $\dn$ & - $bs \,@\,\textit{retrieve}\,a_1\,v_1\,@\, \textit{retrieve}\,a_2\,v_2$\\ - $\textit{retrieve}\,(\textit{STAR}\,bs\,a)\,(\Stars\,[])$ & $\dn$ & - $bs \,@\, [\S]$\\ - $\textit{retrieve}\,(\textit{STAR}\,bs\,a)\,(\Stars\,(v\!::\!vs))$ & $\dn$ &\\ - \multicolumn{3}{l}{ - \hspace{3cm}$bs \,@\, [\Z] \,@\, \textit{retrieve}\,a\,v\,@\, - \textit{retrieve}\,(\textit{STAR}\,[]\,a)\,(\Stars\,vs)$}\\ -\end{tabular} -\end{center} -%\comment{Did not read further}\\ -This function assembles the bitcode -%that corresponds to a lexical value for how -%the current derivative matches the suffix of the string(the characters that -%have not yet appeared, but will appear as the successive derivatives go on. -%How do we get this "future" information? By the value $v$, which is -%computed by a pass of the algorithm that uses -%$inj$ as described in the previous section). -using information from both the derivative regular expression and the -value. Sulzmann and Lu poroposed this function, but did not prove -anything about it. Ausaf and Urban used it to connect the bitcoded -algorithm to the older algorithm by the following equation: - - \begin{center} $inj \;a\; c \; v = \textit{decode} \; (\textit{retrieve}\; - (r^\uparrow)\backslash_{simp} \,c)\,v)$ - \end{center} - -\noindent -whereby $r^\uparrow$ stands for the internalised version of $r$. Ausaf -and Urban also used this fact to prove the correctness of bitcoded -algorithm without simplification. Our purpose of using this, however, -is to establish - -\begin{center} -$ \textit{retrieve} \; -a \; v \;=\; \textit{retrieve} \; (\textit{simp}\,a) \; v'.$ -\end{center} -The idea is that using $v'$, a simplified version of $v$ that had gone -through the same simplification step as $\textit{simp}(a)$, we are able -to extract the bitcode that gives the same parsing information as the -unsimplified one. However, we noticed that constructing such a $v'$ -from $v$ is not so straightforward. The point of this is that we might -be able to finally bridge the gap by proving - -\begin{center} -$\textit{retrieve} \; (r^\uparrow \backslash s) \; v = \;\textit{retrieve} \; -(\textit{simp}(r^\uparrow) \backslash s) \; v'$ -\end{center} - -\noindent -and subsequently - -\begin{center} -$\textit{retrieve} \; (r^\uparrow \backslash s) \; v\; = \; \textit{retrieve} \; -(r^\uparrow \backslash_{simp} \, s) \; v'$. -\end{center} - -\noindent -The $\textit{LHS}$ of the above equation is the bitcode we want. This -would prove that our simplified version of regular expression still -contains all the bitcodes needed. The task here is to find a way to -compute the correct $v'$. - -The second task is to speed up the more aggressive simplification. Currently -it is slower than the original naive simplification by Ausaf and Urban (the -naive version as implemented by Ausaf and Urban of course can ``explode'' in -some cases). It is therefore not surprising that the speed is also much slower -than regular expression engines in popular programming languages such as Java -and Python on most inputs that are linear. For example, just by rewriting the -example regular expression in the beginning of this report $(a^*)^*\,b$ into -$a^*\,b$ would eliminate the ambiguity in the matching and make the time -for matching linear with respect to the input string size. This allows the -DFA approach to become blindingly fast, and dwarf the speed of our current -implementation. For example, here is a comparison of Java regex engine -and our implementation on this example. - -\begin{center} -\begin{tabular}{@{}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{}} -\begin{tikzpicture} -\begin{axis}[ - xlabel={$n*1000$}, - x label style={at={(1.05,-0.05)}}, - ylabel={time in secs}, - enlargelimits=false, - xtick={0,5,...,30}, - xmax=33, - ymax=9, - scaled ticks=true, - axis lines=left, - width=5cm, - height=4cm, - legend entries={Bitcoded Algorithm}, - legend pos=north west, - legend cell align=left] -\addplot[red,mark=*, mark options={fill=white}] table {bad-scala.data}; -\end{axis} -\end{tikzpicture} - & -\begin{tikzpicture} -\begin{axis}[ - xlabel={$n*1000$}, - x label style={at={(1.05,-0.05)}}, - %ylabel={time in secs}, - enlargelimits=false, - xtick={0,5,...,30}, - xmax=33, - ymax=9, - scaled ticks=false, - axis lines=left, - width=5cm, - height=4cm, - legend entries={Java}, - legend pos=north west, - legend cell align=left] -\addplot[cyan,mark=*, mark options={fill=white}] table {good-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} - - -Java regex engine can match string of thousands of characters in a few milliseconds, -whereas our current algorithm gets excruciatingly slow on input of this size. -The running time in theory is linear, however it does not appear to be the -case in an actual implementation. So it needs to be explored how to -make our algorithm faster on all inputs. It could be the recursive calls that are -needed to manipulate bits that are causing the slow down. A possible solution -is to write recursive functions into tail-recusive form. -Another possibility would be to explore -again the connection to DFAs to speed up the algorithm on -subcalls that are small enough. This is very much work in progress. - -\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. -We also plan to implement the bitcoded algorithm -in some imperative language like C to see if the inefficiency of the -Scala implementation -is language specific. To make this research more comprehensive we also plan -to contrast our (faster) version of bitcoded algorithm with the -Symbolic Regex Matcher, the RE2, the Rust Regex Engine, and the static -analysis approach by implementing them in the same language and then compare -their performance. \bibliographystyle{plain} \bibliography{root} \end{document} +