% Chapter Template+
−
+
−
\chapter{Related Work} % Main chapter title+
−
+
−
\label{RelatedWork} +
−
+
−
%In this chapter, we introduce+
−
%work relevant to this thesis.+
−
+
−
\section{Regular Expressions, Derivatives and POSIX Lexing}+
−
+
−
%\subsection{Formalisations of Automata, Regular Expressions, and Matching Algorithms}+
−
Regular expressions were introduced by Kleene in the 1950s \cite{Kleene1956}.+
−
Since then they have become a fundamental concept in +
−
formal languages and automata theory \cite{Sakarovitch2009}.+
−
Brzozowski defined derivatives on regular expressions in his PhD thesis+
−
in 1964 \cite{Brzozowski1964}, in which he proved the finiteness of the numbers+
−
of regular expression derivatives modulo the ACI-axioms.+
−
It is worth pointing out that this result does not directly+
−
translate to our+
−
finiteness proof, +
−
and our proof does not depend on it to be valid.+
−
The key observation is that our version of the Sulzmann and Lu's algorithm+
−
\cite{Sulzmann2014} represents +
−
derivative terms in a way that allows efficient de-duplication,+
−
and we do not make use of an equivalence checker that exploits the ACI-equivalent+
−
terms.+
−
+
−
+
−
Central to this thesis is the work by Sulzmann and Lu \cite{Sulzmann2014}.+
−
They first introduced the elegant and simple idea of injection-based lexing+
−
and bit-coded lexing.+
−
In a follow-up work \cite{Sulzmann2014b}, Sulzmann and Steenhoven+
−
incorporated these ideas+
−
into a tool called \emph{dreml}.+
−
The pencil-and-paper proofs in \cite{Sulzmann2014} based on the ideas+
−
by Frisch and Cardelli \cite{Frisch2004} were later+
−
found to have unfillable gaps by Ausaf \cite{Ausaf},+
−
who came up with an alternative proof inspired by +
−
Vansummeren \cite{Vansummeren2006}.+
−
Sulzmann and Thiemann extended the Brzozowski derivatives to +
−
shuffling regular expressions \cite{Sulzmann2015},+
−
which are a very succinct way of describing basic regular expressions.+
−
+
−
+
−
+
−
Regular expressions and lexers have been a popular topic among+
−
the theorem proving and functional programming community.+
−
In the next section we give a list of lexers+
−
and matchers that come with a machine-checked correctness proof.+
−
+
−
\subsection{Matchers and Lexers with Mechanised Proofs}+
−
We are aware+
−
of a mechanised correctness proof of Brzozowski's derivative-based matcher in HOL4 by+
−
Owens and Slind~\parencite{Owens2008}. Another one in Isabelle/HOL is part+
−
of the work by Krauss and Nipkow \parencite{Krauss2011}.  Another one+
−
in Coq is given by Coquand and Siles \parencite{Coquand2012}.+
−
Also Ribeiro and Du Bois gave one in Agda \parencite{RibeiroAgda2017}.+
−
The most recent works to our best knowledge are the Verbatim \cite{Verbatim}+
−
and Verbatim++ \cite{Verbatim} lexers.+
−
The Verbatim++ lexer adds many correctness-preserving +
−
optimisations to the Verbatim lexer,+
−
and is therefore quite fast on many inputs.+
−
The problem is that they chose to use DFA to speed up things,+
−
for which dealing with bounded repetitions is a bottleneck.+
−
+
−
+
−
This thesis builds on the formal specifications of POSIX+
−
rules and formal proofs by Ausaf et al. \cite{AusafDyckhoffUrban2016}.+
−
The bounded repetitions is a continuation of the work by Ausaf \cite{Ausaf}.+
−
+
−
Automata formalisations are in general harder and more cumbersome to deal+
−
with for theorem provers \cite{Nipkow1998}.+
−
To represent them, one way is to use graphs, but graphs are not inductive datatypes.+
−
Having to set the inductive principle on the number of nodes+
−
in a graph makes formal reasoning non-intuitive and convoluted,+
−
resulting in large formalisations \cite{Lammich2012}. +
−
When combining two graphs, one also needs to make sure that the nodes in+
−
both graphs are distinct. +
−
If they are not distinct, then renaming of the nodes is needed.+
−
Using Coq which provides dependent types can potentially make things slightly easier+
−
\cite{Doczkal2013}+
−
Another representation for automata are matrices. +
−
But the induction for them is not as straightforward either.+
−
Both approaches have been used in the past and resulted in huge formalisations.+
−
There are some more clever representations, for example one by Paulson +
−
using hereditarily finite sets \cite{Paulson2015}. +
−
There the problem with combining graphs can be solved better. +
−
%but we believe that such clever tricks are not very obvious for +
−
%the John-average-Isabelle-user.+
−
+
−
\subsection{Different Definitions of POSIX Rules}+
−
There are different ways to formalise values and POSIX matching.+
−
Cardelli and Frisch \cite{Frisch2004} have developed a notion of+
−
\emph{non-problematic values} which is a slight variation+
−
of the values defined in the inhabitation relation in \ref{fig:inhab}.+
−
They then defined an ordering between values, and showed that+
−
the maximal element of those values correspond to the output+
−
of their GREEDY lexing algorithm.+
−
+
−
Okui and Suzuki \cite{Okui10} allow iterations of values to +
−
flatten to the empty+
−
string for the star inhabitation rule in+
−
\ref{fig:inhab}.+
−
They refer to the more restrictive version as used +
−
in this thesis (which was defined by Ausaf et al.+
−
\cite{AusafDyckhoffUrban2016}) as \emph{canonical values}.+
−
The very interesting link between the work by Ausaf et al.+
−
and Okui and Suzuki is that they have distinct formalisations+
−
of POSIX values, and yet they are provably equivalent! See+
−
\cite{AusafDyckhoffUrban2016} for details of the+
−
alternative definitions given by Okui and Suzuki and the formalisation.+
−
%TODO: EXPLICITLY STATE the OKUI SUZUKI POSIX DEF.+
−
+
−
Sulzmann and Lu themselves have come up with POSIX definitions \cite{Sulzmann2014}.+
−
In their paper they defined an ordering between values with respect+
−
to regular expressions, and tried to establish that their+
−
algorithm outputs the minimal element by a pencil-and-paper proof.+
−
But having the ordering relation taking regular expression as parameters+
−
causes the transitivity of their ordering to not go through.+
−
+
−
When faced with catastrophic backtracking, +
−
sometimes programmers +
−
have to rewrite their regexes in an ad hoc fashion.+
−
Knowing which patterns should be avoided+
−
can be helpful.+
−
\subsection{Static Analysis of Evil Regex Patterns} +
−
or they try to avoid the possibly problematic patterns completely,+
−
for which many false positives exist\parencite{Davis18}.+
−
Animated tools to "debug" regular expressions such as+
−
 \parencite{regexploit2021} \parencite{regex101} are also popular.+
−
We are also aware of static analysis work on regular expressions that+
−
aims to detect potentially expoential regex patterns. Rathnayake and Thielecke +
−
\parencite{Rathnayake2014StaticAF} proposed an algorithm+
−
that detects regular expressions triggering exponential+
−
behavious on backtracking matchers.+
−
Weideman \parencite{Weideman2017Static} came up with +
−
non-linear polynomial worst-time estimates+
−
for regexes, attack string that exploit the worst-time +
−
scenario, and "attack automata" that generates+
−
attack strings.+
−
+
−
+
−
+
−
+
−
\section{Optimisations}+
−
Perhaps the biggest problem that prevents derivative-based lexing +
−
from being more widely adopted+
−
is that they tend to be not blindingly fast in practice, unable to +
−
reach throughputs like gigabytes per second, which is the+
−
application we had in mind when we initially started looking at the topic.+
−
Commercial+
−
regular expression matchers such as Snort \cite{Snort1999} and Bro \cite{Bro}+
−
are capable of inspecting payloads+
−
at line rates (which can be up to a few gigabits per second) against +
−
thousands of rules \cite{communityRules}.+
−
For our algorithm to be more attractive for practical use, we+
−
need more correctness-preserving optimisations.+
−
+
−
FPGA-based implementations such as \cite{Sidhu2001} +
−
have the advantages of being+
−
reconfigurable and parallel, but suffer from lower clock frequency+
−
and scalability.+
−
Traditional automaton approaches that use a DFA instead of NFA+
−
benefit from the fact that only a single transition is needed +
−
for each input character \cite{Becchi08}. Lower memory bandwidth leads+
−
to faster performance.+
−
However, they suffer from exponential blow-ups on bounded repetitions.+
−
Compression techniques are used, such as those in \cite{Kumar2006} and+
−
\cite{Becchi2007}.+
−
Variations of pure NFAs or DFAs like counting-set automata \cite{Turonova2020}+
−
have been+
−
proposed to better deal with bounded repetitions.+
−
+
−
%So far our focus has been mainly on the bit-coded algorithm,+
−
%but the injection-based lexing algorithm +
−
%could also be sped up in various ways:+
−
%+
−
Another direction of optimisation for derivative-based approaches+
−
is defining string derivatives+
−
directly, without recursively decomposing them into +
−
character-by-character derivatives. For example, instead of+
−
replacing $(r_1 + r_2)\backslash (c::cs)$ by+
−
$((r_1 + r_2)\backslash c)\backslash cs$ (as in definition \ref{table:der}), we rather+
−
calculate $(r_1\backslash (c::cs) + r_2\backslash (c::cs))$.+
−
This has the potential to speed up matching because input is +
−
processed in larger granularity.+
−
One interesting thing is to explore whether this can be done+
−
to $\inj$ as well, so that we can generate a lexical value +
−
rather than simply get a matcher.+
−
+
−
+
−
\subsection{Derivatives and Zippers}+
−
Zipper is a data structure designed to focus on +
−
and navigate between local parts of a tree.+
−
The idea is that often operations on a large tree only deal with+
−
local regions each time.+
−
Therefore it would be a waste to +
−
traverse the entire tree if+
−
the operation only+
−
involves a small fraction of it.+
−
It was first formally described by Huet \cite{Huet1997}.+
−
Typical applications of zippers involve text editor buffers+
−
and proof system databases.+
−
In our setting, the idea is to compactify the representation+
−
of derivatives with zippers, thereby making our algorithm faster.+
−
We introduce several works on parsing, derivatives+
−
and zippers.+
−
+
−
Edelmann et al. developed a formalised parser for LL(1) grammars using derivatives+
−
\cite{Zippy2020}.+
−
They adopted zippers to improve the speed, and argued that the runtime+
−
complexity of their algorithm was linear with respect to the input string.+
−
+
−
The idea of using Brzozowski derivatives on general context-free +
−
parsing was first implemented+
−
by Might et al. \cite{Might2011}. +
−
They used memoization and fixpoint construction to eliminate infinite recursion,+
−
which is a major problem for using derivatives on context-free grammars.+
−
The initial version was quite slow----exponential on the size of the input.+
−
Adams et al. \cite{Adams2016} improved the speed and argued that their version+
−
was cubic with respect to the input.+
−
Darragh and Adams \cite{Darragh2020} further improved the performance+
−
by using zippers in an innovative way--their zippers had multiple focuses+
−
instead of just one in a traditional zipper to handle ambiguity.+
−
Their algorithm was not formalised, though.+
−
+
−
+
−
+
−
+
−
\subsection{Back-References}+
−
We introduced regular expressions with back-references+
−
in chapter \ref{Introduction}.+
−
We adopt the common practice of calling them rewbrs (Regular Expressions+
−
With Back References) for brevity.+
−
It has been shown by Aho \cite{Aho1990}+
−
that the k-vertex cover problem can be reduced+
−
to the problem of rewbrs matching, and is therefore NP-complete.+
−
Given the depth of the problem, the progress made at the full generality+
−
of arbitrary rewbrs matching has been slow, with+
−
theoretical work on them being +
−
fairly recent. +
−
+
−
Campaneu et al. studied regexes with back-references+
−
in the context of formal languages theory in +
−
their 2003 work \cite{campeanu2003formal}.+
−
They devised a pumping lemma for Perl-like regexes, +
−
proving that the langugages denoted by them+
−
is  properly  contained in context-sensitive+
−
languages.+
−
More interesting questions such as +
−
whether the language denoted by Perl-like regexes+
−
can express palindromes ($\{w \mid w = w^R\}$)+
−
are discussed in \cite{campeanu2009patterns} +
−
and \cite{CAMPEANU2009Intersect}.+
−
Freydenberger \cite{Frey2013} investigated the +
−
expressive power of back-references. He showed several +
−
undecidability and decriptional complexity results +
−
of back-references, concluding that they add+
−
great power to certain programming tasks but are difficult to harness.+
−
An interesting question would then be+
−
whether we can add restrictions to them, +
−
so that they become expressive enough for practical use such+
−
as html files, but not too powerful.+
−
Freydenberger and Schmid \cite{FREYDENBERGER20191}+
−
introduced the notion of deterministic+
−
regular expressions with back-references to achieve+
−
a better balance between expressiveness and tractability.+
−
+
−
+
−
Fernau and Schmid \cite{FERNAU2015287} and Schmid \cite{Schmid2012}+
−
investigated the time complexity of different variants+
−
of back-references.+
−
We are not aware of any work that uses derivatives on back-references.+
−
+
−
See \cite{BERGLUND2022} for a survey+
−
of these works and comparison between different+
−
flavours  of back-references syntax (e.g. whether references can be circular, +
−
can labels be repeated etc.).+
−
+
−
+
−
 +
−
+
−
+
−
+
−
+
−
+
−
 +
−
+
−
+
−