| author | Chengsong |
| Thu, 09 Jan 2020 22:21:04 +0000 | |
| changeset 94 | 2e2dca212fff |
| parent 93 | d486c12deeab |
| child 95 | c969a973fcae |
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/algorithm.aux Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,3 @@ +\relax +\providecommand \oddpage@label [2]{} +\@writefile{loa}{\contentsline {algocf}{\numberline {1}{\ignorespaces ders2forSEQ($r_1$,$r_2$,$s$)}}{1}}
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/algorithm.tex Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,22 @@ +\documentclass{article} +\usepackage[ruled,vlined]{algorithm2e} + +\begin{document} +\begin{algorithm}[H] +\SetAlgoLined +\KwResult{rexp } + Compute list $l_1: [r_1\backslash c_1 \cdots c_n, r_1\backslash c_1 \cdots c_{n-1}, ..., r_1 \backslash c_1, r_1]$\; + Compute list $l_2: [r_2, r_2\backslash c_n, r_2\backslash c_{n-1} c_{n}, ..., r_2 \backslash c_1\cdots c_n]$\; + Compute list $l_3$: + $l_3 \gets []$ \; + \For{$i\gets1$ \KwTo $n$}{ + $\mathit{if} \; \mathit{nullable}(r_1 \backslash c_1 \cdots c_{n-i+1}) \; \mathit{then} \; \mathit{add} \; i \; \mathit{to} \; l_3$ + } + $result \gets r_1 \backslash c_1 \cdots c_n \cdot r_2$\\ + \For{$i\gets0$ \KwTo $l_3.length -1$}{ + $result \gets \mathit{ALT}(result, l2[l3[i]])$ + } + \Return result + \caption{ders2forSEQ($r_1$,$r_2$,$s$)} +\end{algorithm} +\end{document} \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/bad-scala.data Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,34 @@ +%% LaTeX2e file `bad-scala.data' +%% generated by the `filecontents' environment +%% from source `ninems' on 2019/08/18. +%% +1 0.048028 +2 0.126308 +3 0.169670 +4 0.264180 +5 0.384297 +6 0.612785 +7 0.817211 +8 1.089386 +9 1.363222 +10 1.644981 +11 1.987046 +12 2.416012 +13 2.920956 +14 3.398956 +15 3.739649 +16 4.329977 +17 4.932896 +18 5.633909 +19 6.436107 +20 6.927233 +21 7.441163 +22 8.242139 +23 8.901034 +24 9.680487 +25 10.496226 +26 11.385733 +27 12.247613 +28 13.990491 +29 14.245804 +30 15.206283
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/bad_scala.data Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,34 @@ +%% LaTeX2e file `bad_scala.data' +%% generated by the `filecontents' environment +%% from source `ninems' on 2019/08/18. +%% +1000 6.688E-6 +2000 6.28E-6 +3000 8.888E-6 +4000 1.4384E-5 +5000 1.875E-5 +6000 -2.1175E-5 +7000 -1.9611E-5 +8000 -1.8512E-5 +9000 2.6921E-5 +10000 3.2706E-5 +11000 3.2041E-5 +12000 3.8868E-5 +13000 3.9683E-5 +14000 4.0458E-5 +15000 4.5261E-5 +16000 6.0411E-5 +17000 5.2199E-5 +18000 5.443E-5 +19000 5.3428E-5 +20000 5.9657E-5 +21000 6.0396E-5 +22000 6.1441E-5 +23000 6.9136E-5 +24000 7.4773E-5 +25000 8.8255E-5 +26000 7.412E-5 +27000 9.415E-5 +28000 8.3007E-5 +29000 9.6488E-5 +30000 9.8127E-5
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/data.sty Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,123 @@ +% The data files, written on the first run. + + +\begin{filecontents}{re-python2.data} +1 0.033 +5 0.036 +10 0.034 +15 0.036 +18 0.059 +19 0.084 +20 0.141 +21 0.248 +22 0.485 +23 0.878 +24 1.71 +25 3.40 +26 7.08 +27 14.12 +28 26.69 +\end{filecontents} + + +% JavaScript, example (a*)*b +\begin{filecontents}{re-js.data} +5 0.061 +10 0.061 +15 0.061 +20 0.070 +23 0.131 +25 0.308 +26 0.564 +28 1.994 +30 7.648 +31 15.881 +32 32.190 +\end{filecontents} + +% Java 8, example (a*)*b +\begin{filecontents}{re-java.data} +5 0.00298 +10 0.00418 +15 0.00996 +16 0.01710 +17 0.03492 +18 0.03303 +19 0.05084 +20 0.10177 +21 0.19960 +22 0.41159 +23 0.82234 +24 1.70251 +25 3.36112 +26 6.63998 +27 13.35120 +28 29.81185 +\end{filecontents} + +% Java, example (a*)b +\begin{filecontents}{good-java.data} +1 1.5633E-5 +2 1.299E-5 +3 1.1451E-5 +4 1.5846E-5 +5 1.9934E-5 +6 2.174E-5 +7 2.7669E-5 +8 2.8657E-5 +9 2.8161E-5 +10 2.8729E-5 +11 3.5367E-5 +12 3.701E-5 +13 3.84E-5 +14 4.1329E-5 +15 4.8116E-5 +16 5.3597E-5 +17 4.6792E-5 +18 5.8618E-5 +19 6.2078E-5 +20 6.4702E-5 +21 6.1464E-5 +22 6.4693E-5 +23 6.1667E-5 +24 7.1466E-5 +25 7.8089E-5 +26 7.4661E-5 +27 7.5628E-5 +28 8.9169E-5 +29 9.4161E-5 +30 9.8494E-5 +\end{filecontents} + +\begin{filecontents}{bad-scala.data} +1 0.048028 +2 0.126308 +3 0.169670 +4 0.264180 +5 0.384297 +6 0.612785 +7 0.817211 +8 1.089386 +9 1.363222 +10 1.644981 +11 1.987046 +12 2.416012 +13 2.920956 +14 3.398956 +15 3.739649 +16 4.329977 +17 4.932896 +18 5.633909 +19 6.436107 +20 6.927233 +21 7.441163 +22 8.242139 +23 8.901034 +24 9.680487 +25 10.496226 +26 11.385733 +27 12.247613 +28 13.990491 +29 14.245804 +30 15.206283 +\end{filecontents} \ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/etnms.aux Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,61 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} +\citation{AusafDyckhoffUrban2016} +\babel@aux{UKenglish}{} +\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}} +\citation{Davis18} +\citation{17Bir} +\citation{AusafDyckhoffUrban2016} +\citation{OkuiSuzuki2010} +\citation{Vansummeren2006} +\citation{Sulzmann2014} +\citation{AusafDyckhoffUrban2016} +\citation{CrashCourse2014} +\citation{Kuklewicz} +\citation{Sulzmann2014} +\citation{Brzozowski1964} +\@writefile{toc}{\contentsline {section}{\numberline {2}The Algorithm by Brzozowski based on Derivatives of Regular Expressions}{6}{section.2}} +\citation{Sulzmann2014} +\newlabel{graph:*}{{1}{7}{The Algorithm by Brzozowski based on Derivatives of Regular Expressions}{equation.2.1}{}} +\@writefile{toc}{\contentsline {section}{\numberline {3}Values and the Algorithm by Sulzmann and Lu}{7}{section.3}} +\newlabel{graph:2}{{2}{8}{Values and the Algorithm by Sulzmann and Lu}{equation.3.2}{}} +\citation{nielson11bcre} +\citation{AusafDyckhoffUrban2016} +\citation{Antimirov95} +\@writefile{toc}{\contentsline {section}{\numberline {4}Simplification of Regular Expressions}{12}{section.4}} +\citation{Sulzmann2014} +\citation{Sulzmann2014} +\@writefile{toc}{\contentsline {section}{\numberline {5}Current Work}{16}{section.5}} +\bibstyle{plain} +\bibdata{root} +\bibcite{Antimirov95}{1} +\bibcite{AusafDyckhoffUrban2016}{2} +\bibcite{Brzozowski1964}{3} +\bibcite{Davis18}{4} +\bibcite{CrashCourse2014}{5} +\bibcite{Kuklewicz}{6} +\bibcite{nielson11bcre}{7} +\bibcite{OkuiSuzuki2010}{8} +\@writefile{toc}{\contentsline {section}{\numberline {6}Conclusion}{19}{section.6}} +\bibcite{17Bir}{9} +\bibcite{Sulzmann2014}{10} +\bibcite{Vansummeren2006}{11} +\newlabel{LastPage}{{}{20}{}{page.20}{}} +\xdef\lastpage@lastpage{20} +\xdef\lastpage@lastpageHy{20}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/etnms.bbl Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,71 @@ +\begin{thebibliography}{10} + +\bibitem{Antimirov95} +V.~Antimirov. +\newblock {P}artial {D}erivatives of {R}egular {E}xpressions and {F}inite + {A}utomata {C}onstructions. +\newblock {\em Theoretical Computer Science}, 155:291--319, 1995. + +\bibitem{AusafDyckhoffUrban2016} +F.~Ausaf, R.~Dyckhoff, and C.~Urban. +\newblock {POSIX} {L}exing with {D}erivatives of {R}egular {E}xpressions + ({P}roof {P}earl). +\newblock In {\em Proc.~of the 7th International Conference on Interactive + Theorem Proving (ITP)}, volume 9807 of {\em LNCS}, pages 69--86, 2016. + +\bibitem{Brzozowski1964} +J.~A. Brzozowski. +\newblock {D}erivatives of {R}egular {E}xpressions. +\newblock {\em Journal of the {ACM}}, 11(4):481--494, 1964. + +\bibitem{Davis18} +J.~C. Davis, C.~.A. Coghlan, F.~Servant, and D.~Lee. +\newblock {T}he {I}mpact of {R}egular {E}xpression {D}enial of {S}ervice + ({ReDoS}) in {P}ractice: {A}n {E}mpirical {S}tudy at the {E}cosystem {S}cale. +\newblock In {\em Proc.~of the 26th ACM Joint Meeting on European Software + Engineering Conference and Symposium on the Foundations of Software + Engineering (ESEC/FSE)}, pages 246--256, 2018. + +\bibitem{CrashCourse2014} +N.~B.~B. Grathwohl, F.~Henglein, and U.~T. Rasmussen. +\newblock {A} {C}rash-{C}ourse in {R}egular {E}xpression {P}arsing and + {R}egular {E}xpressions as {T}ypes. +\newblock Technical report, University of Copenhagen, 2014. + +\bibitem{Kuklewicz} +C.~Kuklewicz. +\newblock {R}egex {P}osix. +\newblock \url{https://wiki.haskell.org/Regex_Posix}. + +\bibitem{nielson11bcre} +Fritz~Henglein Lasse~Nielsen. +\newblock Bit-coded regular expression parsing. +\newblock {\em LATA}, 2011. + +\bibitem{OkuiSuzuki2010} +S.~Okui and T.~Suzuki. +\newblock {D}isambiguation in {R}egular {E}xpression {M}atching via {P}osition + {A}utomata with {A}ugmented {T}ransitions. +\newblock In {\em Proc.~of the 15th International Conference on Implementation + and Application of Automata (CIAA)}, volume 6482 of {\em LNCS}, pages + 231--240, 2010. + +\bibitem{17Bir} +Asiri Rathnayake and Hayo Thielecke. +\newblock Static analysis for regular expression exponential runtime via + substructural logics. +\newblock {\em arXiv:1405.7058}, 2017. + +\bibitem{Sulzmann2014} +M.~Sulzmann and K.~Lu. +\newblock {POSIX} {R}egular {E}xpression {P}arsing with {D}erivatives. +\newblock In {\em Proc.~of the 12th International Conference on Functional and + Logic Programming (FLOPS)}, volume 8475 of {\em LNCS}, pages 203--220, 2014. + +\bibitem{Vansummeren2006} +S.~Vansummeren. +\newblock {T}ype {I}nference for {U}nique {P}attern {M}atching. +\newblock {\em ACM Transactions on Programming Languages and Systems}, + 28(3):389--428, 2006. + +\end{thebibliography}
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/etnms.out Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,6 @@ +\BOOKMARK [1][-]{section.1}{Introduction}{}% 1 +\BOOKMARK [1][-]{section.2}{The Algorithm by Brzozowski based on Derivatives of Regular Expressions}{}% 2 +\BOOKMARK [1][-]{section.3}{Values and the Algorithm by Sulzmann and Lu}{}% 3 +\BOOKMARK [1][-]{section.4}{Simplification of Regular Expressions}{}% 4 +\BOOKMARK [1][-]{section.5}{Current Work}{}% 5 +\BOOKMARK [1][-]{section.6}{Conclusion}{}% 6
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/etnms.tex Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,1702 @@ +\documentclass[a4paper,UKenglish]{lipics} +\usepackage{graphic} +\usepackage{data} +\usepackage{tikz-cd} +%\usepackage{algorithm} +\usepackage{amsmath} +\usepackage[noend]{algpseudocode} +\usepackage{enumitem} +\usepackage{nccmath} + +\definecolor{darkblue}{rgb}{0,0,0.6} +\hypersetup{colorlinks=true,allcolors=darkblue} +\newcommand{\comment}[1]% +{{\color{red}$\Rightarrow$}\marginpar{\raggedright\small{\bf\color{red}#1}}} + +% \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\bders{\textit{bders}} +\def\lexer{\mathit{lexer}} +\def\blexer{\textit{blexer}} +\def\blexers{\mathit{blexer\_simp}} +\def\mkeps{\mathit{mkeps}} +\def\bmkeps{\textit{bmkeps}} +\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}} +\def\flex{\textit{flex}} +\def\rup{r^\uparrow} +\def\retrieve{\textit{retrieve}} +\def\AALTS{\textit{AALTS}} +\def\AONE{\textit{AONE}} +%\theoremstyle{theorem} +%\newtheorem{theorem}{Theorem} +%\theoremstyle{lemma} +%\newtheorem{lemma}{Lemma} +%\newcommand{\lemmaautorefname}{Lemma} +%\theoremstyle{definition} +%\newtheorem{definition}{Definition} +\algnewcommand\algorithmicswitch{\textbf{switch}} +\algnewcommand\algorithmiccase{\textbf{case}} +\algnewcommand\algorithmicassert{\texttt{assert}} +\algnewcommand\Assert[1]{\State \algorithmicassert(#1)}% +% New "environments" +\algdef{SE}[SWITCH]{Switch}{EndSwitch}[1]{\algorithmicswitch\ #1\ \algorithmicdo}{\algorithmicend\ \algorithmicswitch}% +\algdef{SE}[CASE]{Case}{EndCase}[1]{\algorithmiccase\ #1}{\algorithmicend\ \algorithmiccase}% +\algtext*{EndSwitch}% +\algtext*{EndCase}% + + +\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 does also + answers with \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. We have already developed some + simplification rules for this, but have not yet proved that they + preserve the correctness of the algorithm. We also have not yet + looked at extended regular expressions, such as bounded repetitions, + negation and back-references. +\end{abstract} + +\section{Introduction} + + + +\noindent\rule[0.5ex]{\linewidth}{1pt} +Between the 2 bars are the new materials.\\ +In the past 6 months I was trying to prove that the bit-coded algorithm is correct. +\begin{center} +$\blexers \;r \; s = \blexer \; r \; s$ +\end{center} +\noindent +To prove this, we need to prove these two functions produce the same output +whether or not $r \in L(r)$. +Given the definition of $\blexer$ and $\blexers$: +\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} + + \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 +it boils down to proving the following two propositions(depending on which +branch in the if-else clause is taken): + +\begin{itemize} + +\item{} +When s is a string in the language L(r), \\ +$\textit{bmkeps} (r^\uparrow)\backslash_{simp}\, s = \textit{bmkeps} (r^\uparrow)\backslash s$, \\ +\item{} +when s is not a string of the language L(ar) +ders\_simp(ar, s) is not nullable +\end{itemize} +The second one is relatively straightforward using isabelle to prove. +The first part requires more effort. +It builds on the result that the bit-coded algorithm without simplification +produces the correct result: +\begin{center} +$\blexer \;r^\uparrow s = \lexer \; r\; s$ +\end{center} +\noindent +the definition of lexer and its correctness is +omitted(see \cite{AusafDyckhoffUrban2016}). +if we can prove that the bit-coded algorithm with simplification produces +the same result as the original bit-coded algorithm, +then we are done. +The correctness proof of +\begin{center} +$\blexer \; r^\uparrow s = \lexer \;r \;s$ +\end{center} +\noindent +might provide us insight into proving +\begin{center} +$\blexer \; r^\uparrow \;s = \blexers \; r^\uparrow \;s$ +\end{center} +\noindent +(that is also +why we say the new proof builds on the older one). +The proof defined the function $\flex$ as another way of +expressing the $\lexer$ function: +\begin{center} +$\lexer \;r\; s = \flex \;\textit{id} \; r\;s \;(\mkeps \; r\backslash s)$ +\end{center}. +\noindent +(proof for the above equality will be explained later) +The definition of $flex$ is as follows: +\begin{center} +\begin{tabular}{lcl} +$\textit{flex} \;r\; f\; (c\!::\!s) $ & $\dn$ & $\textit{flex} \; (r\backslash c) \;(\lambda v. f (inj \; r \; c \; v)) \;s$ \\ +$\textit{flex} \;r\; f\; [\,] $ & $\dn$ & $f$ +\end{tabular} +\end{center} +\noindent +here $\flex$ essentially does lexing by +stacking up injection functions while doing derivatives, +explicitly showing the order of characters being +injected back in each step. +With $\flex$ we can write $\lexer$ this way: +\begin{center} +$\lexer \;r\; s = \flex \;id \; r\;s \;(\mkeps r\backslash s)$ +\end{center} +\noindent +$\flex$ focuses on + the injections instead +of the derivatives , +compared +to the original definition of $\lexer$, +which puts equal amount of emphasis on +injection and derivative with respect to each character: +\begin{center} +\begin{tabular}{lcl} +$\textit{lexer} \; r\; (c\!::\!s) $ & $\dn$ & $\textit{case} \; \lexer \; (r\backslash c) \;s \; \textit{of}$ \\ + & & $\textit{None} \; \Longrightarrow \; \textit{None}$\\ + & & $\textbar \; v \; \Longrightarrow \; \inj \; r\;c\;v$\\ +$\textit{lexer} \; r\; [\,] $ & $\dn$ & $\textit{if} \; \nullable (r) \; \textit{then} \; \mkeps (r) \; \textit{else} \;None$ +\end{tabular} +\end{center} +\noindent +Using this feature of $\flex$ we can rewrite the lexing +$w.r.t \; s @ [c]$ in term of lexing +$w.r.t \; s$: +\begin{center} +$\flex \; r \; id \; (s@[c]) \; v = \flex \; r \; id \; s \; (inj \; (r\backslash s) \; c\; v)$. +\end{center} +\noindent +this allows us to use +the inductive hypothesis to get +\begin{center} +$ \flex \; r\; id\; (s@[c])\; v = \textit{decode} \;( \textit{retrieve}\; (\rup \backslash s) \; (\inj \; (r\backslash s) \;c\;v)\;) r$ +\end{center} +\noindent +By using a property of retrieve we have the $\textit{RHS}$ of the above equality is +$decode (retrieve (r^\uparrow \backslash(s @ [c])) v) r$, and this gives the +main lemma result: +\begin{center} +$ \flex \;r\; id \; (s@[c]) \; v =\textit{decode}(\textit{retrieve} (\rup \backslash (s@[c])) \;v) r$ +\end{center} +\noindent +To use this lemma result for our +correctness proof, simply replace the $v$ in the +$\textit{RHS}$ of the above equality with +$\mkeps\;(r\backslash (s@[c]))$, and apply the lemma that + +\begin{center} +$\textit{decode} \; \bmkeps \; \rup \; r = \textit{decode} \; (\textit{retrieve} \; \rup \; \mkeps(r)) \;r$ +\end{center} +\noindent +We get the correctness of our bit-coded algorithm: +\begin{center} +$\flex \;r\; id \; s \; (\mkeps \; r\backslash s) = \textit{decode} \; \bmkeps \; \rup\backslash s \; r$ +\end{center} +\noindent +The bridge between the above chain of equalities +is the use of $\retrieve$, +if we want to use a similar technique for the +simplified version of algorithm, +we face the problem that in the above +equalities, +$\retrieve \; a \; v$ is not always defined. +for example, +$\retrieve \; \AALTS(Z, \AONE(S), \AONE(S)) \; \Left(\Empty)$ +is defined, but not $\retrieve \; \AONE(\Z\S) \; \Left(\Empty)$, +though we can extract the same POSIX +bits from the two annotated regular expressions. +That means, if we +want to prove that +\begin{center} +$\textit{decode} \; \bmkeps \; \rup\backslash s \; r = \textit{decode} \; \bmkeps \; \rup\backslash_{simp} s \; r$ +\end{center} +\noindent +holds by using $\retrieve$, +we probably need to prove an equality like below: +\begin{center} +%$\retrieve \; \rup\backslash_{simp} s \; \mkeps(r\backslash_{simp} s)=\textit{retrieve} \; \rup\backslash s \; \mkeps(r\backslash s)$ +$\retrieve \; \rup\backslash_{simp} s \; \mkeps(r\backslash s)=\textit{retrieve} \; \rup\backslash s \; \mkeps(r\backslash s)$ +\end{center} +\noindent +we would need to rectify the value $\mkeps(r\backslash s)$ into something simpler +to make the retrieve function defined.\\ +%HERE CONSTRUCTION SITE +The vsimp function, defined as follows +tries to simplify the value in lockstep with +regular expression:\\ + + +The problem here is that + +we used retrieve for the key induction: +$decode (retrieve (r\backslash (s @ [c])) v) r $ +$decode (retrieve (r\backslash s) (inj (r\backslash s) c v)) r$ +Here, decode recovers a value that corresponds to a match(possibly partial) +from bits, and the bits are extracted by retrieve, +and the key value $v$ that guides retrieve is +$mkeps r\backslash s$, $inj r c (mkeps r\backslash s)$, $inj (inj (v))$, ...... +if we can +the problem is that +need vsiimp to make a value that is suitable for decoding +$Some(flex rid(s@[c])v) = Some(flex rids(inj (r\backslash s)cv))$ +another way that christian came up with that might circumvent the +prblem of finding suitable value is by not stating the visimp +function but include all possible value in a set that a regex is able to produce, +and proving that both r and sr are able to produce the bits that correspond the POSIX value + +produced by feeding the same initial regular expression $r$ and string $s$ to the + two functions $ders$ and $ders\_simp$. +The reason why +Namely, if $bmkeps( r_1) = bmkeps(r_2)$, then we + + +If we define the equivalence relation $\sim_{m\epsilon}$ between two regular expressions +$r_1$ and $r_2$as follows: +$r_1 \sim_{m\epsilon} r_2 \iff bmkeps(r_1)= bmkeps(r_2)$ +(in other words, they $r1$ and $r2$ produce the same output under the function $bmkeps$.) +Then the first goal +might be restated as +$(r^\uparrow)\backslash_{simp}\, s \sim_{m\epsilon} (r^\uparrow)\backslash s$. +I tried to establish an equivalence relation between the regular experssions +like dddr dddsr,..... +but right now i am only able to establish dsr and dr, using structural induction on r. +Those involve multiple derivative operations are harder to prove. +Two attempts have been made: +(1)induction on the number of der operations(or in other words, the length of the string s), +the inductive hypothesis was initially specified as +"For an arbitrary regular expression r, +For all string s in the language of r whose length do not exceed +the number n, ders s r me derssimp s r" +and the proof goal may be stated as +"For an arbitrary regular expression r, +For all string s in the language of r whose length do not exceed +the number n+1, ders s r me derssimp s r" +the problem here is that although we can easily break down +a string s of length n+1 into s1@list(c), it is not that easy +to use the i.h. as a stepping stone to prove anything because s1 may well be not +in the language L(r). This inhibits us from obtaining the fact that +ders s1 r me derssimps s1 r. +Further exploration is needed to amend this hypothesis so it includes the +situation when s1 is not nullable. +For example, what information(bits? +values?) can be extracted +from the regular expression ders(s1,r) so that we can compute or predict the possible +result of bmkeps after another derivative operation. What function f can used to +carry out the task? The possible way of exploration can be +more directly perceived throught the graph below: +find a function +f +such that +f(bders s1 r) += re1 +f(bderss s1 r) += re2 +bmkeps(bders s r) = g(re1,c) +bmkeps(bderssimp s r) = g(re2,c) +and g(re1,c) = g(re2,c) +The inductive hypothesis would be +"For all strings s1 of length <= n, +f(bders s1 r) += re1 +f(bderss s1 r) += re2" +proving this would be a lemma for the main proof: +the main proof would be +" +bmkeps(bders s r) = g(re1,c) +bmkeps(bderssimp s r) = g(re2,c) +for s = s1@c +" +and f need to be a recursive property for the lemma to be proved: +it needs to store not only the "after one char nullable info", +but also the "after two char nullable info", +and so on so that it is able to predict what f will compute after a derivative operation, +in other words, it needs to be "infinitely recursive"\\ +To prove the lemma, in other words, to get +"For all strings s1 of length <= n+1, +f(bders s1 r) += re3 +f(bderss s1 r) += re4"\\ +from\\ +"For all strings s1 of length <= n, +f(bders s1 r) += re1 +f(bderss s1 r) += re2"\\ +it might be best to construct an auxiliary function h such that\\ +h(re1, c) = re3\\ +h(re2, c) = re4\\ +and re3 = f(bder c (bders s1 r))\\ +re4 = f(simp(bder c (bderss s1 r))) +The key point here is that we are not satisfied with what bders s r will produce under +bmkeps, but also how it will perform after a derivative operation and then bmkeps, and two +derivative operations and so on. In essence, we are preserving the regular expression +itself under the function f, in a less compact way than the regluar expression: we are +not just recording but also interpreting what the regular expression matches. +In other words, we need to prove the properties of bderss s r beyond the bmkeps result, +i.e., not just the nullable ones, but also those containing remaining characters.\\ +(2)we observed the fact that +erase sdddddr= erase sdsdsdsr +that is to say, despite the bits are being moved around on the regular expression +(difference in bits), the structure of the (unannotated)regular expression +after one simplification is exactly the same after the +same sequence of derivative operations +regardless of whether we did simplification +along the way. +However, without erase the above equality does not hold: +for the regular expression +$(a+b)(a+a*)$, +if we do derivative with respect to string $aa$, +we get +%TODO +sdddddr does not equal sdsdsdsr sometimes.\\ +For example, + +This equicalence class method might still have the potential of proving this, +but not yet +i parallelly tried another method of using retrieve\\ + + + +\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} + $b ::= S \mid Z \qquad +bs ::= [] \mid b:bs +$ +\end{center} + +\noindent +The $S$ and $Z$ are arbitrary names for the bits 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} + $\textit{code}(\Empty)$ & $\dn$ & $[]$\\ + $\textit{code}(\Char\,c)$ & $\dn$ & $[]$\\ + $\textit{code}(\Left\,v)$ & $\dn$ & $\Z :: code(v)$\\ + $\textit{code}(\Right\,v)$ & $\dn$ & $\S :: code(v)$\\ + $\textit{code}(\Seq\,v_1\,v_2)$ & $\dn$ & $code(v_1) \,@\, code(v_2)$\\ + $\textit{code}(\Stars\,[])$ & $\dn$ & $[\Z]$\\ + $\textit{code}(\Stars\,(v\!::\!vs))$ & $\dn$ & $\S :: code(v) \;@\; + code(\Stars\,vs)$ +\end{tabular} +\end{center} + +\noindent +Here $\textit{code}$ encodes a value into a bitcodes by converting +$\Left$ into $\Z$, $\Right$ into $\S$, the start point of a non-empty +star iteration into $\S$, and the border where a local star terminates +into $\Z$. 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 $\S$s and $\Z$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@{}} + $\textit{decode}'\,bs\,(\ONE)$ & $\dn$ & $(\Empty, bs)$\\ + $\textit{decode}'\,bs\,(c)$ & $\dn$ & $(\Char\,c, bs)$\\ + $\textit{decode}'\,(\Z\!::\!bs)\;(r_1 + r_2)$ & $\dn$ & + $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r_1\;\textit{in}\; + (\Left\,v, bs_1)$\\ + $\textit{decode}'\,(\S\!::\!bs)\;(r_1 + r_2)$ & $\dn$ & + $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r_2\;\textit{in}\; + (\Right\,v, bs_1)$\\ + $\textit{decode}'\,bs\;(r_1\cdot r_2)$ & $\dn$ & + $\textit{let}\,(v_1, bs_1) = \textit{decode}'\,bs\,r_1\;\textit{in}$\\ + & & $\textit{let}\,(v_2, bs_2) = \textit{decode}'\,bs_1\,r_2$\\ + & & \hspace{35mm}$\textit{in}\;(\Seq\,v_1\,v_2, bs_2)$\\ + $\textit{decode}'\,(\Z\!::\!bs)\,(r^*)$ & $\dn$ & $(\Stars\,[], bs)$\\ + $\textit{decode}'\,(\S\!::\!bs)\,(r^*)$ & $\dn$ & + $\textit{let}\,(v, bs_1) = \textit{decode}'\,bs\,r\;\textit{in}$\\ + & & $\textit{let}\,(\Stars\,vs, bs_2) = \textit{decode}'\,bs_1\,r^*$\\ + & & \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} + $\textit{a}$ & $::=$ & $\textit{ZERO}$\\ + & $\mid$ & $\textit{ONE}\;\;bs$\\ + & $\mid$ & $\textit{CHAR}\;\;bs\,c$\\ + & $\mid$ & $\textit{ALTS}\;\;bs\,as$\\ + & $\mid$ & $\textit{SEQ}\;\;bs\,a_1\,a_2$\\ + & $\mid$ & $\textit{STAR}\;\;bs\,a$ +\end{tabular} +\end{center} +%(in \textit{ALTS}) + +\noindent +where $bs$ stands for bitcodes, $a$ for $\bold{a}$nnotated regular +expressions and $as$ for a list of annotated regular expressions. +The alternative constructor($\textit{ALTS}$) 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} + $(\ZERO)^\uparrow$ & $\dn$ & $\textit{ZERO}$\\ + $(\ONE)^\uparrow$ & $\dn$ & $\textit{ONE}\,[]$\\ + $(c)^\uparrow$ & $\dn$ & $\textit{CHAR}\,[]\,c$\\ + $(r_1 + r_2)^\uparrow$ & $\dn$ & + $\textit{ALTS}\;[]\,List((\textit{fuse}\,[\Z]\,r_1^\uparrow),\, + (\textit{fuse}\,[\S]\,r_2^\uparrow))$\\ + $(r_1\cdot r_2)^\uparrow$ & $\dn$ & + $\textit{SEQ}\;[]\,r_1^\uparrow\,r_2^\uparrow$\\ + $(r^*)^\uparrow$ & $\dn$ & + $\textit{STAR}\;[]\,r^\uparrow$\\ +\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} + $\textit{fuse}\;bs\,(\textit{ZERO})$ & $\dn$ & $\textit{ZERO}$\\ + $\textit{fuse}\;bs\,(\textit{ONE}\,bs')$ & $\dn$ & + $\textit{ONE}\,(bs\,@\,bs')$\\ + $\textit{fuse}\;bs\,(\textit{CHAR}\,bs'\,c)$ & $\dn$ & + $\textit{CHAR}\,(bs\,@\,bs')\,c$\\ + $\textit{fuse}\;bs\,(\textit{ALTS}\,bs'\,as)$ & $\dn$ & + $\textit{ALTS}\,(bs\,@\,bs')\,as$\\ + $\textit{fuse}\;bs\,(\textit{SEQ}\,bs'\,a_1\,a_2)$ & $\dn$ & + $\textit{SEQ}\,(bs\,@\,bs')\,a_1\,a_2$\\ + $\textit{fuse}\;bs\,(\textit{STAR}\,bs'\,a)$ & $\dn$ & + $\textit{STAR}\,(bs\,@\,bs')\,a$ +\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}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/etnms.vtc Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,1 @@ +\contitem\title{POSIX Regular Expression Matching and Lexing}\author{Chengsong Tan}\page{1}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/good-ava.data Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,20 @@ +%% LaTeX2e file `good-ava.data' +%% generated by the `filecontents' environment +%% from source `ninems' on 2019/08/18. +%% +5 1.9934E-5 +10 2.8729E-5 +15 4.8116E-5 +16 5.3597E-5 +17 4.6792E-5 +18 5.8618E-5 +19 6.2078E-5 +20 6.4702E-5 +21 6.1464E-5 +22 6.4693E-5 +23 6.1667E-5 +24 7.1466E-5 +25 7.8089E-5 +26 7.4661E-5 +27 7.5628E-5 +28 8.9169E-5
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/good-java.data Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,20 @@ +%% LaTeX2e file `good-java.data' +%% generated by the `filecontents' environment +%% from source `ninems' on 2019/08/18. +%% +5 1.9934E-5 +10 2.8729E-5 +15 4.8116E-5 +16 5.3597E-5 +17 4.6792E-5 +18 5.8618E-5 +19 6.2078E-5 +20 6.4702E-5 +21 6.1464E-5 +22 6.4693E-5 +23 6.1667E-5 +24 7.1466E-5 +25 7.8089E-5 +26 7.4661E-5 +27 7.5628E-5 +28 8.9169E-5
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/good_java.data Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,34 @@ +%% LaTeX2e file `good_java.data' +%% generated by the `filecontents' environment +%% from source `ninems' on 2019/08/18. +%% +1000 6.688E-6 +2000 6.28E-6 +3000 8.888E-6 +4000 1.4384E-5 +5000 1.875E-5 +6000 -2.1175E-5 +7000 -1.9611E-5 +8000 -1.8512E-5 +9000 2.6921E-5 +10000 3.2706E-5 +11000 3.2041E-5 +12000 3.8868E-5 +13000 3.9683E-5 +14000 4.0458E-5 +15000 4.5261E-5 +16000 6.0411E-5 +17000 5.2199E-5 +18000 5.443E-5 +19000 5.3428E-5 +20000 5.9657E-5 +21000 6.0396E-5 +22000 6.1441E-5 +23000 6.9136E-5 +24000 7.4773E-5 +25000 8.8255E-5 +26000 7.412E-5 +27000 9.415E-5 +28000 8.3007E-5 +29000 9.6488E-5 +30000 9.8127E-5
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/graphic.sty Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,12 @@ +\usepackage{tikz} +%\usepackage{pgf} +%\usetikzlibrary{positioning} +%\usetikzlibrary{calc} +%\usetikzlibrary{automata} +%\usetikzlibrary{arrows} +%\usetikzlibrary{backgrounds} +%\usetikzlibrary{fit} +%\usepackage{tikz-qtree} +\usepackage{pgfplots} + +%\pgfplotsset{compat=1.15}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/lipics.cls Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,646 @@ +%% +%% This is file `lipics.cls', +%% generated with the docstrip utility. +%% +%% The original source files were: +%% +%% lipics.dtx (with options: `class') +%% +%% ----------------------------------------------------------------- +%% Author: le-tex publishing services +%% +%% This file is part of the lipics package for preparing +%% LIPICS articles. +%% +%% Copyright (C) 2010 Schloss Dagstuhl +%% ----------------------------------------------------------------- +\NeedsTeXFormat{LaTeX2e}[2005/12/01] +\ProvidesClass{lipics} + [2010/09/27 v1.1 LIPIcs articles] +\emergencystretch1em +\advance\hoffset-1in +\advance\voffset-1in +\advance\hoffset2.95mm +\newif\if@nobotseplist \@nobotseplistfalse +\def\@endparenv{% + \addpenalty\@endparpenalty\if@nobotseplist\else\addvspace\@topsepadd\fi\@endpetrue} +\def\@doendpe{% + \@endpetrue + \def\par{\@restorepar + \everypar{}% + \par + \if@nobotseplist + \addvspace\topsep + \addvspace\partopsep + \global\@nobotseplistfalse + \fi + \@endpefalse}% + \everypar{{\setbox\z@\lastbox}% + \everypar{}% + \if@nobotseplist\global\@nobotseplistfalse\fi + \@endpefalse}} +\def\enumerate{% + \ifnum \@enumdepth >\thr@@\@toodeep\else + \advance\@enumdepth\@ne + \edef\@enumctr{enum\romannumeral\the\@enumdepth}% + \expandafter + \list + \csname label\@enumctr\endcsname + {\advance\partopsep\topsep + \topsep\z@\@plus\p@ + \ifnum\@listdepth=\@ne + \labelsep0.72em + \else + \ifnum\@listdepth=\tw@ + \labelsep0.3em + \else + \labelsep0.5em + \fi + \fi + \usecounter\@enumctr\def\makelabel##1{\hss\llap{##1}}}% + \fi} +\def\endenumerate{\ifnum\@listdepth=\@ne\global\@nobotseplisttrue\fi\endlist} +\def\itemize{% + \ifnum \@itemdepth >\thr@@\@toodeep\else + \advance\@itemdepth\@ne + \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}% + \expandafter + \list + \csname\@itemitem\endcsname + {\advance\partopsep\topsep + \topsep\z@\@plus\p@ + \ifnum\@listdepth=\@ne + \labelsep0.83em + \else + \ifnum\@listdepth=\tw@ + \labelsep0.75em + \else + \labelsep0.5em + \fi + \fi + \def\makelabel##1{\hss\llap{##1}}}% + \fi} +\def\enditemize{\ifnum\@listdepth=\@ne\global\@nobotseplisttrue\fi\endlist} +\def\@sect#1#2#3#4#5#6[#7]#8{% + \ifnum #2>\c@secnumdepth + \let\@svsec\@empty + \else + \refstepcounter{#1}% + \protected@edef\@svsec{\@seccntformat{#1}\relax}% + \fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup + #6{% + \@hangfrom{\hskip #3\relax + \ifnum #2=1 + \colorbox[rgb]{0.99,0.78,0.07}{\kern0.15em\@svsec\kern0.15em}\quad + \else + \@svsec\quad + \fi}% + \interlinepenalty \@M #8\@@par}% + \endgroup + \csname #1mark\endcsname{#7}% + \addcontentsline{toc}{#1}{% + \ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}% + \fi + #7}% + \else + \def\@svsechd{% + #6{\hskip #3\relax + \@svsec #8}% + \csname #1mark\endcsname{#7}% + \addcontentsline{toc}{#1}{% + \ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}% + \fi + #7}}% + \fi + \@xsect{#5}} +\def\@seccntformat#1{\csname the#1\endcsname} +\def\@biblabel#1{\textcolor{darkgray}{\sffamily\bfseries#1}} +\def\copyrightline{% + \ifx\@serieslogo\@empty + \else + \setbox\@tempboxa\hbox{\includegraphics[height=42\p@]{\@serieslogo}}% + \rlap{\hspace\textwidth\hspace{-\wd\@tempboxa}\hspace{\z@}% + \vtop to\z@{\vskip-0mm\unhbox\@tempboxa\vss}}% + \fi + \scriptsize + \vtop{\hsize\textwidth + \nobreakspace\\ + \@Copyright + \ifx\@Event\@empty\else\@Event.\\\fi + \ifx\@Editors\@empty\else + \@Eds: \@Editors + ; pp. \thepage--\pageref{LastPage}% + \\ + \fi + \setbox\@tempboxa\hbox{\includegraphics[height=14\p@,trim=0 15 0 0]{lipics-logo-bw}}% + \hspace*{\wd\@tempboxa}\enskip + \href{http://www.dagstuhl.de/lipics/}% + {Leibniz International Proceedings in Informatics}\\ + \smash{\unhbox\@tempboxa}\enskip + \href{http://www.dagstuhl.de}% + {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik, Dagstuhl Publishing, Germany}}} +\def\ps@plain{\let\@mkboth\@gobbletwo + \let\@oddhead\@empty + \let\@evenhead\@empty + \let\@evenfoot\copyrightline + \let\@oddfoot\copyrightline} +\def\lipics@opterrshort{Option "\CurrentOption" not supported} +\def\lipics@opterrlong{The option "\CurrentOption" from article.cls is not supported by lipics.cls.} +\DeclareOption{a5paper}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{b5paper}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{legalpaper}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{executivepaper}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{landscape}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{10pt}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{11pt}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{12pt}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{oneside}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{twoside}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{titlepage}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{notitlepage}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{onecolumn}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{twocolumn}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{fleqn}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{openbib}{\@latexerr{\lipics@opterrshort}{\lipics@opterrlong}} +\DeclareOption{a4paper}{\PassOptionsToClass{\CurrentOption}{article} + \advance\hoffset-2.95mm + \advance\voffset8.8mm} +\DeclareOption{numberwithinsect}{\let\numberwithinsect\relax} +\DeclareOption*{\PassOptionsToClass{\CurrentOption}{article}} +\ProcessOptions +\LoadClass[twoside,notitlepage,fleqn]{article} +\renewcommand\normalsize{% + \@setfontsize\normalsize\@xpt{13}% + \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ + \abovedisplayshortskip \z@ \@plus3\p@ + \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ + \belowdisplayskip \abovedisplayskip + \let\@listi\@listI} +\normalsize +\renewcommand\small{% + \@setfontsize\small\@ixpt{11.5}% + \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ + \abovedisplayshortskip \z@ \@plus2\p@ + \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 4\p@ \@plus2\p@ \@minus2\p@ + \parsep 2\p@ \@plus\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\renewcommand\footnotesize{% + \@setfontsize\footnotesize{8.5}{9.5}% + \abovedisplayskip 6\p@ \@plus2\p@ \@minus4\p@ + \abovedisplayshortskip \z@ \@plus\p@ + \belowdisplayshortskip 3\p@ \@plus\p@ \@minus2\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 3\p@ \@plus\p@ \@minus\p@ + \parsep 2\p@ \@plus\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\renewcommand\large{\@setfontsize\large{10.5}{13}} +\renewcommand\Large{\@setfontsize\Large{12}{14}} +\setlength\parindent{1.5em} +\setlength\headheight{3mm} +\setlength\headsep {10mm} +\setlength\footskip{3mm} +\setlength\textwidth{140mm} +\setlength\textheight{222mm} +\setlength\oddsidemargin{32mm} +\setlength\evensidemargin{38mm} +\setlength\marginparwidth{25mm} +\setlength\topmargin{13mm} +\setlength{\skip\footins}{2\baselineskip \@plus 4\p@ \@minus 2\p@} +\def\@listi{\leftmargin\leftmargini + \parsep\z@ \@plus\p@ + \topsep 8\p@ \@plus2\p@ \@minus4\p@ + \itemsep \parsep} +\let\@listI\@listi +\@listi +\def\@listii {\leftmargin\leftmarginii + \labelwidth\leftmarginii + \advance\labelwidth-\labelsep + \topsep 4\p@ \@plus2\p@ \@minus\p@ + \parsep\z@ \@plus\p@ + \itemsep \parsep} +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii + \advance\labelwidth-\labelsep + \topsep 2\p@ \@plus\p@\@minus\p@ + \parsep \z@ + \partopsep \p@ \@plus\z@ \@minus\p@ + \itemsep \z@ \@plus\p@} +\def\ps@headings{% + \def\@evenhead{\large\sffamily\bfseries + \llap{\hbox to0.5\oddsidemargin{\thepage\hss}}\leftmark\hfil}% + \def\@oddhead{\large\sffamily\bfseries\rightmark\hfil + \rlap{\hbox to0.5\oddsidemargin{\hss\thepage}}}% + \def\@oddfoot{\hfil + \rlap{% + \vtop{% + \vskip10mm + \colorbox[rgb]{0.99,0.78,0.07} + {\@tempdima\evensidemargin + \advance\@tempdima1in + \advance\@tempdima\hoffset + \hb@xt@\@tempdima{% + \textcolor{darkgray}{\normalsize\sffamily + \bfseries\quad + \expandafter\textsolittle + \expandafter{\@EventShortName}}% + \strut\hss}}}}} + \let\@evenfoot\@empty + \let\@mkboth\markboth + \let\sectionmark\@gobble + \let\subsectionmark\@gobble} +\pagestyle{headings} +\renewcommand\maketitle{\par + \begingroup + \renewcommand\thefootnote{\@fnsymbol\c@footnote}% + \if@twocolumn + \ifnum \col@number=\@ne + \@maketitle + \else + \twocolumn[\@maketitle]% + \fi + \else + \newpage + \global\@topnum\z@ % Prevents figures from going at top of page. + \@maketitle + \fi + \thispagestyle{plain}\@thanks + \endgroup + \setcounter{footnote}{0}% + \global\let\thanks\relax + \global\let\maketitle\relax + \global\let\@maketitle\relax + \global\let\@thanks\@empty + \global\let\@author\@empty + \global\let\@date\@empty + \global\let\@title\@empty + \global\let\title\relax + \global\let\author\relax + \global\let\date\relax + \global\let\and\relax +} +\newwrite\tocfile +\def\@maketitle{% + \newpage + \null\vskip-\baselineskip + \vskip-\headsep + \@titlerunning + \@authorrunning + \let \footnote \thanks + \parindent\z@ \raggedright + {\LARGE\sffamily\bfseries\mathversion{bold}\@title \par}% + \vskip 1.5em% + \ifnum\c@authors=0 % + \@latexerr{No \noexpand\author given}% + {Provide at least one author. See the LIPIcs class documentation.}% + \else + \@author + \fi + \bgroup + \let\footnote\@gobble + \immediate\openout\tocfile=\jobname.vtc + \protected@write\tocfile{}{% + \string\contitem + \string\title{\@title}% + \string\author{\AB@authfortoc}% + \string\page{\thepage}}% + \closeout\tocfile + \egroup + \par} +\setcounter{secnumdepth}{4} +\renewcommand\section{\@startsection {section}{1}{\z@}% + {-3.5ex \@plus -1ex \@minus -.2ex}% + {2.3ex \@plus.2ex}% + {\sffamily\Large\bfseries\raggedright}} +\renewcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-3.25ex\@plus -1ex \@minus -.2ex}% + {1.5ex \@plus .2ex}% + {\sffamily\Large\bfseries\raggedright}} +\renewcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {-3.25ex\@plus -1ex \@minus -.2ex}% + {1.5ex \@plus .2ex}% + {\sffamily\Large\bfseries\raggedright}} +\renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}% + {-3.25ex \@plus-1ex \@minus-.2ex}% + {1.5ex \@plus .2ex}% + {\sffamily\large\bfseries\raggedright}} +\renewcommand\subparagraph{\@startsection{subparagraph}{5}{\z@}% + {3.25ex \@plus1ex \@minus .2ex}% + {-1em}% + {\sffamily\normalsize\bfseries}} +\setlength\leftmargini \parindent +\setlength\leftmarginii {1.2em} +\setlength\leftmarginiii{1.2em} +\setlength\leftmarginiv {1.2em} +\setlength\leftmarginv {1.2em} +\setlength\leftmarginvi {1.2em} +\renewcommand\labelenumi{% + \textcolor{darkgray}{\sffamily\bfseries\mathversion{bold}\theenumi.}} +\renewcommand\labelenumii{% + \textcolor{darkgray}{\sffamily\bfseries\mathversion{bold}\theenumii.}} +\renewcommand\labelenumiii{% + \textcolor{darkgray}{\sffamily\bfseries\mathversion{bold}\theenumiii.}} +\renewcommand\labelenumiv{% + \textcolor{darkgray}{\sffamily\bfseries\mathversion{bold}\theenumiv.}} +\renewcommand\labelitemi{% + \textcolor[rgb]{0.6,0.6,0.61}{\ifnum\@listdepth=\@ne + \rule{0.67em}{0.33em}% + \else + \rule{0.45em}{0.225em}% + \fi}} +\renewcommand\labelitemii{% + \textcolor[rgb]{0.6,0.6,0.61}{\rule{0.45em}{0.225em}}} +\renewcommand\labelitemiii{% + \textcolor[rgb]{0.6,0.6,0.61}{\sffamily\bfseries\textasteriskcentered}} +\renewcommand\labelitemiv{% + \textcolor[rgb]{0.6,0.6,0.61}{\sffamily\bfseries\textperiodcentered}} +\renewenvironment{description} + {\list{}{\advance\partopsep\topsep\topsep\z@\@plus\p@ + \labelwidth\z@ \itemindent-\leftmargin + \let\makelabel\descriptionlabel}} + {\ifnum\@listdepth=\@ne\global\@nobotseplisttrue\fi\endlist} +\renewcommand*\descriptionlabel[1]{% + \hspace\labelsep\textcolor{darkgray}{\sffamily\bfseries\mathversion{bold}#1}} +\renewenvironment{abstract}{% + \vskip\bigskipamount + \noindent + \rlap{\color[rgb]{0.51,0.50,0.52}\vrule\@width\textwidth\@height1\p@}% + \hspace*{7mm}\fboxsep1.5mm\colorbox[rgb]{1,1,1}{\raisebox{-0.4ex}{% + \large\selectfont\sffamily\bfseries\abstractname}}% + \vskip3\p@ + \fontsize{9.5}{12.5}\selectfont + \noindent\ignorespaces} + {\ifx\@subjclass\@empty\else + \vskip\baselineskip\noindent + \subjclassHeading\@subjclass + \fi + \ifx\@keywords\@empty\else + \vskip\baselineskip\noindent + \keywordsHeading\@keywords + \fi + \ifx\@DOI\@empty\else + \vskip\baselineskip\noindent + \doiHeading\doi{\@DOI}% + \fi} +\renewenvironment{thebibliography}[1] + {\if@noskipsec \leavevmode \fi + \par + \@tempskipa-3.5ex \@plus -1ex \@minus -.2ex\relax + \@afterindenttrue + \@tempskipa -\@tempskipa \@afterindentfalse + \if@nobreak + \everypar{}% + \else + \addpenalty\@secpenalty\addvspace\@tempskipa + \fi + \noindent + \rlap{\color[rgb]{0.51,0.50,0.52}\vrule\@width\textwidth\@height1\p@}% + \hspace*{7mm}\fboxsep1.5mm\colorbox[rgb]{1,1,1}{\raisebox{-0.4ex}{% + \normalsize\sffamily\bfseries\refname}}% + \@xsect{1ex \@plus.2ex}% + \list{\@biblabel{\@arabic\c@enumiv}}% + {\leftmargin8.5mm + \labelsep\leftmargin + \settowidth\labelwidth{\@biblabel{#1}}% + \advance\labelsep-\labelwidth + \usecounter{enumiv}% + \let\p@enumiv\@empty + \renewcommand\theenumiv{\@arabic\c@enumiv}}% + \fontsize{9.5}{12.5}\selectfont + \sloppy + \clubpenalty4000 + \@clubpenalty \clubpenalty + \widowpenalty4000% + \sfcode`\.\@m} + {\def\@noitemerr + {\@latex@warning{Empty `thebibliography' environment}}% + \endlist} +\renewcommand\footnoterule{% + \kern-8\p@ + {\color[rgb]{0.60,0.60,0.61}\hrule\@width40mm\@height1\p@}% + \kern6.6\p@} +\renewcommand\@makefntext[1]{% + \parindent\z@\hangindent1em + \leavevmode + \hb@xt@1em{\@makefnmark\hss}#1} +\usepackage[utf8]{inputenc} +\IfFileExists{lmodern.sty}{\RequirePackage{lmodern}}{} +\RequirePackage[T1]{fontenc} +\RequirePackage{textcomp} +\RequirePackage[mathscr]{eucal} +\RequirePackage{amssymb} +\RequirePackage{soul} +\sodef\textsolittle{}{.12em}{.5em\@plus.08em\@minus.06em}% + {.4em\@plus.275em\@minus.183em} +\RequirePackage{color} +\definecolor{darkgray}{rgb}{0.31,0.31,0.33} +\RequirePackage{babel} +\RequirePackage[tbtags,fleqn]{amsmath} +\RequirePackage{amsthm} +\thm@headfont{% + \textcolor{darkgray}{$\blacktriangleright$}\nobreakspace\sffamily\bfseries} +\def\th@remark{% + \thm@headfont{% + \textcolor{darkgray}{$\blacktriangleright$}\nobreakspace\sffamily}% + \normalfont % body font + \thm@preskip\topsep \divide\thm@preskip\tw@ + \thm@postskip\thm@preskip +} +\def\@endtheorem{\endtrivlist}%\@endpefalse +\renewcommand\qedsymbol{\textcolor{darkgray}{\ensuremath{\blacktriangleleft}}} +\renewenvironment{proof}[1][\proofname]{\par + \pushQED{\qed}% + \normalfont \topsep6\p@\@plus6\p@\relax + \trivlist + \item[\hskip\labelsep + \color{darkgray}\sffamily\bfseries + #1\@addpunct{.}]\ignorespaces +}{% + \popQED\endtrivlist%\@endpefalse +} +\theoremstyle{plain} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}[theorem]{Lemma} +\newtheorem{corollary}[theorem]{Corollary} +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\theoremstyle{remark} +\newtheorem*{remark}{Remark} +\ifx\numberwithinsect\relax + \@addtoreset{theorem}{section} + \edef\thetheorem{\expandafter\noexpand\thesection\@thmcountersep\@thmcounter{theorem}} +\fi +\RequirePackage{graphicx} +\RequirePackage{array} +\let\@classzold\@classz +\def\@classz{% + \expandafter\ifx\d@llarbegin\begingroup + \toks \count@ = + \expandafter{\expandafter\small\the\toks\count@}% + \fi + \@classzold} +\RequirePackage{multirow} +\RequirePackage{tabularx} +\RequirePackage[online]{threeparttable} +\def\TPTtagStyle#1{#1)} +\def\tablenotes{\small\TPT@defaults + \@ifnextchar[\TPT@setuptnotes\TPTdoTablenotes} % ] +\RequirePackage{listings} +\lstset{basicstyle=\small\ttfamily,% + backgroundcolor=\color[rgb]{0.85,0.85,0.86},% + frame=single,framerule=0pt,xleftmargin=\fboxsep,xrightmargin=\fboxsep} +\RequirePackage{lastpage} +\IfFileExists{doi.sty} + {\RequirePackage{doi}% + \renewcommand*{\doitext}{}} + {\RequirePackage{hyperref}% + \def\doi##1{##1}} +\hypersetup{pdfborder={0 0 0}} +\RequirePackage[labelsep=space,singlelinecheck=false,% + font={up,small},labelfont={sf,bf},% + listof=false]{caption}%"listof" instead of "list" for backward compatibility +\@ifpackagelater{hyperref}{2009/12/09} + {\captionsetup{compatibility=false}}%cf. http://groups.google.de/group/comp.text.tex/browse_thread/thread/db9310eb540fbbd8/42e30f3b7b3aa17a?lnk=raot + {} +\DeclareCaptionLabelFormat{boxed}{% + \kern0.05em{\color[rgb]{0.99,0.78,0.07}\rule{0.73em}{0.73em}}% + \hspace*{0.67em}\bothIfFirst{#1}{~}#2} +\captionsetup{labelformat=boxed} +\captionsetup[table]{position=top} +\RequirePackage[figuresright]{rotating} +\RequirePackage{subfig} +\def\titlerunning#1{\gdef\@titlerunning{{\let\footnote\@gobble\markboth{#1}{#1}}}} +\def\authorrunning#1{% + \gdef\@authorrunning{\expandafter\def\expandafter\@tempa\expandafter{#1}% + \ifx\@tempa\@empty\else\markright{#1}\fi}} +\titlerunning{\@title} +\authorrunning{\AB@authrunning} +\newcommand*\volumeinfo[6]{% + {\gdef\@Editors{#1}% + \gdef\@Eds{Editor}\ifnum #2>1 \gdef\@Eds{Editors}\fi + \gdef\@Event{#3}% + \setcounter{page}{#6}}} +\volumeinfo{}{1}{}{}{}{1} +\RequirePackage{authblk} +\renewcommand*\Authand{{ and }} +\renewcommand*\Authfont{\Large\bfseries\mathversion{bold}} +\renewcommand*\AB@authnote[1]{\textsuperscript{#1}} +\renewcommand*\AB@affilnote[1]{\protect\item[#1]} +\renewcommand*\Affilfont{\fontsize{9.5}{12}\selectfont} +\setlength\affilsep{\baselineskip} +\newcommand\AB@authrunning{} +\newcommand\AB@authfortoc{} +\renewcommand\author[2][]% + {\ifnewaffil\addtocounter{affil}{1}% + \edef\AB@thenote{\arabic{affil}}\fi + \if\relax#1\relax\def\AB@note{\AB@thenote}\else\def\AB@note{#1}% + \setcounter{Maxaffil}{0}\fi + \ifnum\value{authors}>1\relax + \@namedef{@sep\number\c@authors}{\Authsep}\fi + \addtocounter{authors}{1}% + \begingroup + \let\protect\@unexpandable@protect \let\and\AB@pand + \def\thanks{\protect\thanks}\def\footnote{\protect\footnote}% + \@temptokena=\expandafter{\AB@authors}% + {\def\\{\protect\\[\@affilsep]\protect\Affilfont + \protect\AB@resetsep}% + \xdef\AB@author{\AB@blk@and#2}% + \ifnewaffil\gdef\AB@las{}\gdef\AB@lasx{\protect\Authand}\gdef\AB@as{}% + \xdef\AB@authors{\the\@temptokena\AB@blk@and}% + \else + \xdef\AB@authors{\the\@temptokena\AB@as\AB@au@str}% + \global\let\AB@las\AB@lasx\gdef\AB@lasx{\protect\Authands}% + \gdef\AB@as{\Authsep}% + \fi + \gdef\AB@au@str{#2}}% + \@temptokena=\expandafter{\AB@authlist}% + \let\\=\authorcr + \xdef\AB@authlist{\the\@temptokena + \protect\@nameuse{@sep\number\c@authors}% + \protect\Authfont#2\AB@authnote{\AB@note}}% + %new + \@temptokena=\expandafter{\AB@authrunning}% + \let\\=\authorcr + \xdef\AB@authrunning{\the\@temptokena + \protect\@nameuse{@sep\number\c@authors}#2}% + % + %new + \@temptokena=\expandafter{\AB@authfortoc}% + \let\\=\authorcr + \xdef\AB@authfortoc{\the\@temptokena + \expandafter\noexpand\csname @sep\number\c@authors\endcsname#2}% + % + \endgroup + \ifnum\value{authors}>2\relax + \@namedef{@sep\number\c@authors}{\Authands}\fi + \newaffilfalse +} +\renewcommand\affil[2][]% + {\newaffiltrue\let\AB@blk@and\AB@pand + \if\relax#1\relax\def\AB@note{\AB@thenote}\else\def\AB@note{#1}% + \setcounter{Maxaffil}{0}\fi + \begingroup + \let\protect\@unexpandable@protect + \def\thanks{\protect\thanks}\def\footnote{\protect\footnote}% + \@temptokena=\expandafter{\AB@authors}% + {\def\\{\protect\\\protect\Affilfont}\xdef\AB@temp{#2}}% + \xdef\AB@authors{\the\@temptokena\AB@las\AB@au@str + \protect\\[\affilsep]\protect\Affilfont\AB@temp}% + \gdef\AB@las{}\gdef\AB@au@str{}% + {\xdef\AB@temp{#2}}% + \@temptokena=\expandafter{\AB@affillist}% + \xdef\AB@affillist{\the\@temptokena \AB@affilsep + \AB@affilnote{\AB@note}\protect\Affilfont\AB@temp}% + \endgroup + \let\AB@affilsep\AB@affilsepx} +\renewcommand\@author{\ifx\AB@affillist\AB@empty\AB@authrunning\else + \ifnum\value{affil}>\value{Maxaffil}\def\rlap##1{##1}% + \AB@authlist\\[\affilsep] + \labelwidth1.5em\labelsep\z@\leftmargini\labelwidth + \edef\@enumctr{enumi}% + \list\theenumi{\usecounter\@enumctr\def\makelabel##1{\rlap{##1}\hss}}% + \AB@affillist + \endlist + \else \AB@authors\fi\fi} +\newcommand*\Copyright[1]{% + \def\@Copyright{% + \setbox\@tempboxa\hbox{\includegraphics[height=14\p@,clip]{cc-by}}% + \hspace*{\wd\@tempboxa}\enskip\ifx#1\@empty \else \textcopyright\ #1;\\\fi + \href{http://creativecommons.org/licenses/by/3.0/}% + {\smash{\unhbox\@tempboxa}}\enskip + licensed under Creative Commons License CC-BY\\ + }} +\Copyright{\@empty} +\def\keywords#1{\def\@keywords{#1}} +\let\@keywords\@empty +\def\keywordsHeading{% + \textcolor{darkgray}{\fontsize{9.5}{12.5}\sffamily\bfseries + Keywords and phrases\enskip}} +\def\subjclass#1{\gdef\@subjclass{#1}} +\let\@subjclass\@empty +\def\subjclassHeading{% + \textcolor{darkgray}{\fontsize{9.5}{12.5}\sffamily\bfseries + 1998 ACM Subject Classification\enskip}} +\def\doiHeading{% + \textcolor{darkgray}{\fontsize{9.5}{12.5}\sffamily\bfseries + Digital Object Identifier\enskip}} +\def\serieslogo#1{\gdef\@serieslogo{#1}} +\serieslogo{} +\def\EventShortName#1{\gdef\@EventShortName{#1}} +\EventShortName{} +\def\DOI#1{\gdef\@DOI{#1}} +\DOI{} +\endinput +%% +%% End of file `lipics.cls'.
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/ninems.bbl Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,71 @@ +\begin{thebibliography}{10} + +\bibitem{Antimirov95} +V.~Antimirov. +\newblock {P}artial {D}erivatives of {R}egular {E}xpressions and {F}inite + {A}utomata {C}onstructions. +\newblock {\em Theoretical Computer Science}, 155:291--319, 1995. + +\bibitem{AusafDyckhoffUrban2016} +F.~Ausaf, R.~Dyckhoff, and C.~Urban. +\newblock {POSIX} {L}exing with {D}erivatives of {R}egular {E}xpressions + ({P}roof {P}earl). +\newblock In {\em Proc.~of the 7th International Conference on Interactive + Theorem Proving (ITP)}, volume 9807 of {\em LNCS}, pages 69--86, 2016. + +\bibitem{Brzozowski1964} +J.~A. Brzozowski. +\newblock {D}erivatives of {R}egular {E}xpressions. +\newblock {\em Journal of the {ACM}}, 11(4):481--494, 1964. + +\bibitem{Davis18} +J.~C. Davis, C.~.A. Coghlan, F.~Servant, and D.~Lee. +\newblock {T}he {I}mpact of {R}egular {E}xpression {D}enial of {S}ervice + ({ReDoS}) in {P}ractice: {A}n {E}mpirical {S}tudy at the {E}cosystem {S}cale. +\newblock In {\em Proc.~of the 26th ACM Joint Meeting on European Software + Engineering Conference and Symposium on the Foundations of Software + Engineering (ESEC/FSE)}, pages 246--256, 2018. + +\bibitem{CrashCourse2014} +N.~B.~B. Grathwohl, F.~Henglein, and U.~T. Rasmussen. +\newblock {A} {C}rash-{C}ourse in {R}egular {E}xpression {P}arsing and + {R}egular {E}xpressions as {T}ypes. +\newblock Technical report, University of Copenhagen, 2014. + +\bibitem{Kuklewicz} +C.~Kuklewicz. +\newblock {R}egex {P}osix. +\newblock \url{https://wiki.haskell.org/Regex_Posix}. + +\bibitem{nielson11bcre} +Fritz~Henglein Lasse~Nielsen. +\newblock Bit-coded regular expression parsing. +\newblock {\em LATA}, 2011. + +\bibitem{OkuiSuzuki2010} +S.~Okui and T.~Suzuki. +\newblock {D}isambiguation in {R}egular {E}xpression {M}atching via {P}osition + {A}utomata with {A}ugmented {T}ransitions. +\newblock In {\em Proc.~of the 15th International Conference on Implementation + and Application of Automata (CIAA)}, volume 6482 of {\em LNCS}, pages + 231--240, 2010. + +\bibitem{17Bir} +Asiri Rathnayake and Hayo Thielecke. +\newblock Static analysis for regular expression exponential runtime via + substructural logics. +\newblock {\em arXiv:1405.7058}, 2017. + +\bibitem{Sulzmann2014} +M.~Sulzmann and K.~Lu. +\newblock {POSIX} {R}egular {E}xpression {P}arsing with {D}erivatives. +\newblock In {\em Proc.~of the 12th International Conference on Functional and + Logic Programming (FLOPS)}, volume 8475 of {\em LNCS}, pages 203--220, 2014. + +\bibitem{Vansummeren2006} +S.~Vansummeren. +\newblock {T}ype {I}nference for {U}nique {P}attern {M}atching. +\newblock {\em ACM Transactions on Programming Languages and Systems}, + 28(3):389--428, 2006. + +\end{thebibliography}
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/ninems.out Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,6 @@ +\BOOKMARK [1][-]{section.1}{Introduction}{}% 1 +\BOOKMARK [1][-]{section.2}{The Algorithm by Brzozowski based on Derivatives of Regular Expressions}{}% 2 +\BOOKMARK [1][-]{section.3}{Values and the Algorithm by Sulzmann and Lu}{}% 3 +\BOOKMARK [1][-]{section.4}{Simplification of Regular Expressions}{}% 4 +\BOOKMARK [1][-]{section.5}{Current Work}{}% 5 +\BOOKMARK [1][-]{section.6}{Conclusion}{}% 6
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/ninems.vtc Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,1 @@ +\contitem\title{POSIX Regular Expression Matching and Lexing}\author{Chengsong Tan}\page{1}
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/etnms/root.bib Thu Jan 09 22:21:04 2020 +0000 @@ -0,0 +1,334 @@ +%% This BibTeX bibliography file was created using BibDesk. +%% https://bibdesk.sourceforge.io/ + +%% Created for CS TAN at 2019-08-18 19:00:13 +0100 + + +%% Saved with string encoding Unicode (UTF-8) + + + +@article{17Bir, + Author = {Asiri Rathnayake and Hayo Thielecke}, + Date-Added = {2019-08-18 17:57:30 +0000}, + Date-Modified = {2019-08-18 18:00:13 +0000}, + Journal = {arXiv:1405.7058}, + Title = {Static Analysis for Regular Expression Exponential Runtime via Substructural Logics}, + Year = {2017}} + +@article{nielson11bcre, + Author = {Lasse Nielsen, Fritz Henglein}, + Date-Added = {2019-07-03 21:09:39 +0000}, + Date-Modified = {2019-07-03 21:17:33 +0000}, + Journal = {LATA}, + Title = {Bit-coded Regular Expression Parsing}, + Year = {2011}, + Bdsk-File-1 = {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}} + +@misc{SE16, + Author = {StackStatus}, + Date-Added = {2019-06-26 11:28:41 +0000}, + Date-Modified = {2019-06-26 16:07:31 +0000}, + Keywords = {ReDos Attack}, + Month = {July}, + Rating = {5}, + Read = {1}, + Title = {Stack Overflow Outage Postmortem}, + Url = {https://stackstatus.net/post/147710624694/outage-postmortem-july-20-2016}, + Year = {2016}, + Bdsk-Url-1 = {https://stackstatus.net/post/147710624694/outage-postmortem-july-20-2016}} + +@article{HosoyaVouillonPierce2005, + Author = {H.~Hosoya and J.~Vouillon and B.~C.~Pierce}, + Journal = {ACM Transactions on Programming Languages and Systems (TOPLAS)}, + Number = 1, + Pages = {46--90}, + Title = {{R}egular {E}xpression {T}ypes for {XML}}, + Volume = 27, + Year = {2005}} + +@misc{POSIX, + Note = {\url{http://pubs.opengroup.org/onlinepubs/009695399/basedefs/xbd_chap09.html}}, + Title = {{T}he {O}pen {G}roup {B}ase {S}pecification {I}ssue 6 {IEEE} {S}td 1003.1 2004 {E}dition}, + Year = {2004}} + +@inproceedings{AusafDyckhoffUrban2016, + Author = {F.~Ausaf and R.~Dyckhoff and C.~Urban}, + Booktitle = {Proc.~of the 7th International Conference on Interactive Theorem Proving (ITP)}, + Pages = {69--86}, + Series = {LNCS}, + Title = {{POSIX} {L}exing with {D}erivatives of {R}egular {E}xpressions ({P}roof {P}earl)}, + Volume = {9807}, + Year = {2016}} + +@article{aduAFP16, + Author = {F.~Ausaf and R.~Dyckhoff and C.~Urban}, + Issn = {2150-914x}, + Journal = {Archive of Formal Proofs}, + Note = {\url{http://www.isa-afp.org/entries/Posix-Lexing.shtml}, Formal proof development}, + Title = {{POSIX} {L}exing with {D}erivatives of {R}egular {E}xpressions}, + Year = 2016} + +@techreport{CrashCourse2014, + Annote = {draft report}, + Author = {N.~B.~B.~Grathwohl and F.~Henglein and U.~T.~Rasmussen}, + Institution = {University of Copenhagen}, + Title = {{A} {C}rash-{C}ourse in {R}egular {E}xpression {P}arsing and {R}egular {E}xpressions as {T}ypes}, + Year = {2014}} + +@inproceedings{Sulzmann2014, + Author = {M.~Sulzmann and K.~Lu}, + Booktitle = {Proc.~of the 12th International Conference on Functional and Logic Programming (FLOPS)}, + Pages = {203--220}, + Series = {LNCS}, + Title = {{POSIX} {R}egular {E}xpression {P}arsing with {D}erivatives}, + Volume = {8475}, + Year = {2014}} + +@inproceedings{Sulzmann2014b, + Author = {M.~Sulzmann and P.~van Steenhoven}, + Booktitle = {Proc.~of the 23rd International Conference on Compiler Construction (CC)}, + Pages = {174--191}, + Series = {LNCS}, + Title = {{A} {F}lexible and {E}fficient {ML} {L}exer {T}ool {B}ased on {E}xtended {R}egular {E}xpression {S}ubmatching}, + Volume = {8409}, + Year = {2014}} + +@book{Pierce2015, + Author = {B.~C.~Pierce and C.~Casinghino and M.~Gaboardi and M.~Greenberg and C.~Hri\c{t}cu and V.~Sj\"{o}berg and B.~Yorgey}, + Note = {\url{http://www.cis.upenn.edu/~bcpierce/sf}}, + Publisher = {Electronic textbook}, + Title = {{S}oftware {F}oundations}, + Year = {2015}} + +@misc{Kuklewicz, + Author = {C.~Kuklewicz}, + Howpublished = {\url{https://wiki.haskell.org/Regex_Posix}}, + Title = {{R}egex {P}osix}} + +@article{Vansummeren2006, + Author = {S.~Vansummeren}, + Journal = {ACM Transactions on Programming Languages and Systems}, + Number = {3}, + Pages = {389--428}, + Title = {{T}ype {I}nference for {U}nique {P}attern {M}atching}, + Volume = {28}, + Year = {2006}} + +@inproceedings{Asperti12, + Author = {A.~Asperti}, + Booktitle = {Proc.~of the 3rd International Conference on Interactive Theorem Proving (ITP)}, + Pages = {283--298}, + Series = {LNCS}, + Title = {{A} {C}ompact {P}roof of {D}ecidability for {R}egular {E}xpression {E}quivalence}, + Volume = {7406}, + Year = {2012}} + +@inproceedings{Frisch2004, + Author = {A.~Frisch and L.~Cardelli}, + Booktitle = {Proc.~of the 31st International Conference on Automata, Languages and Programming (ICALP)}, + Pages = {618--629}, + Series = {LNCS}, + Title = {{G}reedy {R}egular {E}xpression {M}atching}, + Volume = {3142}, + Year = {2004}} + +@article{Antimirov95, + Author = {V.~Antimirov}, + Journal = {Theoretical Computer Science}, + Pages = {291--319}, + Title = {{P}artial {D}erivatives of {R}egular {E}xpressions and {F}inite {A}utomata {C}onstructions}, + Volume = {155}, + Year = {1995}} + +@inproceedings{Nipkow98, + Author = {T.~Nipkow}, + Booktitle = {Proc.~of the 11th International Conference on Theorem Proving in Higher Order Logics (TPHOLs)}, + Pages = {1--15}, + Series = {LNCS}, + Title = {{V}erified {L}exical {A}nalysis}, + Volume = 1479, + Year = 1998} + +@article{Brzozowski1964, + Author = {J.~A.~Brzozowski}, + Journal = {Journal of the {ACM}}, + Number = {4}, + Pages = {481--494}, + Title = {{D}erivatives of {R}egular {E}xpressions}, + Volume = {11}, + Year = {1964}} + +@article{Leroy2009, + Author = {X.~Leroy}, + Journal = {Communications of the ACM}, + Number = 7, + Pages = {107--115}, + Title = {{F}ormal {V}erification of a {R}ealistic {C}ompiler}, + Volume = 52, + Year = 2009} + +@inproceedings{Paulson2015, + Author = 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Number = {2}, + Pages = {346--362}, + Title = {{A} {M}echanisation of {S}ome {C}ontext-{F}ree {L}anguage {T}heory in {HOL4}}, + Volume = {80}, + Year = {2014}} + +@article{Thompson1968, + Author = {K.~Thompson}, + Issue_Date = {June 1968}, + Journal = {Communications of the ACM}, + Number = {6}, + Pages = {419--422}, + Title = {{P}rogramming {T}echniques: {R}egular {E}xpression {S}earch {A}lgorithm}, + Volume = {11}, + Year = {1968}} + +@article{Owens2009, + Author = {S.~Owens and J.~H.~Reppy and A.~Turon}, + Journal = {Journal of Functinal Programming}, + Number = {2}, + Pages = {173--190}, + Title = {{R}egular-{E}xpression {D}erivatives {R}e-{E}xamined}, + Volume = {19}, + Year = {2009}} + +@inproceedings{Sulzmann2015, + Author = {M.~Sulzmann and P.~Thiemann}, + Booktitle = {Proc.~of the 9th International Conference on Language and Automata Theory and Applications (LATA)}, + Pages = {275--286}, + Series = {LNCS}, + Title = {{D}erivatives for {R}egular {S}huffle {E}xpressions}, + 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and Symbolic Computation}, + Number = {4}, + Pages = {377--409}, + Title = {{A}dapting {F}unctional {P}rograms to {H}igher {O}rder {L}ogic}, + Volume = {21}, + Year = {2008}} + +@article{Owens2, + Author = {S.~Owens and K.~Slind}, + Bibsource = {dblp computer science bibliography, http://dblp.org}, + Biburl = {http://dblp.uni-trier.de/rec/bib/journals/lisp/OwensS08}, + Doi = {10.1007/s10990-008-9038-0}, + Journal = {Higher-Order and Symbolic Computation}, + Number = {4}, + Pages = {377--409}, + Timestamp = {Wed, 16 Dec 2009 13:51:02 +0100}, + Title = {Adapting functional programs to higher order logic}, + Url = {http://dx.doi.org/10.1007/s10990-008-9038-0}, + Volume = {21}, + Year = {2008}, + Bdsk-Url-1 = {http://dx.doi.org/10.1007/s10990-008-9038-0}} + +@misc{PCRE, + Title = {{PCRE - Perl Compatible Regular Expressions}}, + Url = {http://www.pcre.org}, + Bdsk-Url-1 = {http://www.pcre.org}} + +@inproceedings{OkuiSuzuki2010, + Author = {S.~Okui and T.~Suzuki}, + Booktitle = {Proc.~of the 15th International Conference on Implementation and Application of Automata (CIAA)}, + Pages = {231--240}, + Series = {LNCS}, + Title = {{D}isambiguation in {R}egular {E}xpression {M}atching via {P}osition {A}utomata with {A}ugmented {T}ransitions}, + Volume = {6482}, + Year = {2010}} + +@techreport{OkuiSuzukiTech, + Author = {S.~Okui and T.~Suzuki}, + Institution = {University of Aizu}, + Title = {{D}isambiguation in {R}egular {E}xpression {M}atching via {P}osition {A}utomata with {A}ugmented {T}ransitions}, + Year = {2013}} + +@inproceedings{Davis18, + Author = {J.~C.~Davis and C.~.A.~Coghlan and F.~Servant and D.~Lee}, + Booktitle = {Proc.~of the 26th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering (ESEC/FSE)}, + Numpages = {11}, + Pages = {246--256}, + Title = {{T}he {I}mpact of {R}egular {E}xpression {D}enial of {S}ervice ({ReDoS}) in {P}ractice: {A}n {E}mpirical {S}tudy at the {E}cosystem {S}cale}, + Year = {2018}}