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. 

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. 

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.

\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.

%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:

%

One such optimisation 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{HuetZipper}.

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. \ref{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. \ref{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.

Some initial results have been obtained, but more care needs to be taken to make sure 

that the lexer output conforms to the POSIX standard. Formalising correctness of 

such a lexer using zippers will probably require using an imperative

framework with separation logic as it involves

mutable data structures, which is also interesting to look into.  

To further optimise the algorithm, 

I plan to add a ``deduplicate'' function that tells 

whether two regular expressions denote the same 

language using

an efficient and verified equivalence checker.

In this way, the internal data structures can be pruned more aggressively, 

leading to better simplifications and ultimately faster algorithms.

The idea is to put the focus on that subtree, turning other parts

of the tree into a context

One observation about our derivative-based lexing algorithm is that

the derivative operation sometimes traverses the entire regular expression

unnecessarily:

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