% Chapter Template

\chapter{A Better Bound and Other Extensions} % Main chapter title

\label{Cubic} %In Chapter 5\ref{Chapter5} we discuss stronger simplifications to improve the finite bound
%in Chapter 4 to a polynomial one, and demonstrate how one can extend the
%algorithm to include constructs such as bounded repetitions and negations.
\lstset{style=myScalastyle}


This chapter is a ``work-in-progress''
chapter which records
extensions to our $\blexersimp$.
We intend to formalise this part, which
we have not been able to finish due to time constraints of the PhD.
Nevertheless, we outline the ideas we intend to use for the proof.

We present further improvements
for our lexer algorithm $\blexersimp$.
We devise a stronger simplification algorithm,
called $\bsimpStrong$, which can prune away
similar components in two regular expressions at the same 
alternative level,
even if these regular expressions are not exactly the same.
We call the lexer that uses this stronger simplification function
$\blexerStrong$.
Unfortunately we did not have time to 
work out the proofs, like in
the previous chapters.
We conjecture that both
\begin{center}
	$\blexerStrong \;r \; s = \blexer\; r\;s$
\end{center}
and
\begin{center}
	$\llbracket \bdersStrong{a}{s} \rrbracket = O(\llbracket a \rrbracket^3)$
\end{center}
hold, but a formalisation
is still future work.
We give an informal justification 
why the correctness and cubic size bound proofs
can be achieved
by exploring the connection between the internal
data structure of our $\blexerStrong$ and
Animirov's partial derivatives.\\
%We also present the idempotency property proof
%of $\bsimp$, which leverages the idempotency proof of $\rsimp$.
%This reinforces our claim that the fixpoint construction
%originally required by Sulzmann and Lu can be removed in $\blexersimp$.

%Last but not least, we present our efforts and challenges we met
%in further improving the algorithm by data
%structures such as zippers.



%----------------------------------------------------------------------------------------
%	SECTION strongsimp
%----------------------------------------------------------------------------------------
\section{A Stronger Version of Simplification}
%TODO: search for isabelle proofs of algorithms that check equivalence 
In our bitcoded lexing algorithm, (sub)terms represent (sub)matches.
For example, the regular expression 
\[
	aa \cdot a^*+ a \cdot a^* + aa\cdot a^*
\]
contains three terms, 
expressing three possibilities for how it can match some input.
The first and the third terms are identical, which means we can eliminate
the latter as it will not contribute to a POSIX value.
In $\bsimps$, the $\distinctBy$ function takes care of 
such instances.
The criteria $\distinctBy$ uses for removing a duplicate
$a_2$ in the list
\begin{center}
	$rs_a@[a_1]@rs_b@[a_2]@rs_c$
\end{center}
is that 
\begin{center}
	$\rerase{a_1} = \rerase{a_2}$.
\end{center}
It is characterised as the $LD$ 
rewrite rule in figure \ref{rrewriteRules}.\\
The problem , however, is that identical components
in two slightly different regular expressions cannot be removed.
Consider the simplification
\begin{equation}
	\label{eqn:partialDedup}
	(a+b+d) \cdot r_1 + (a+c+e) \cdot r_1 \stackrel{?}{\rightsquigarrow} (a+b+d) \cdot r_1 + (c+e) \cdot r_1
\end{equation}
where the $(a+\ldots)\cdot r_1$ is deleted in the right alternative.
This is permissible because we have $(a+\ldots)\cdot r_1$ in the left
alternative.
The difficulty is that such  ``buried''
alternatives-sequences are not easily recognised.
But simplification like this actually
cannot be omitted,
as without it the size of derivatives can still
blow up even with our $\textit{bsimp}$ 
function: 
consider again the example
$\protect((a^* + (aa)^* + \ldots + (\underbrace{a\ldots a}_{n a's})^* )^*)^*$,
and set $n$ to a relatively small number,
we get exponential growth:
\begin{figure}[H]
\centering
\begin{tikzpicture}
\begin{axis}[
    %xlabel={$n$},
    myplotstyle,
    xlabel={input length},
    ylabel={size},
    ]
\addplot[blue,mark=*, mark options={fill=white}] table {bsimpExponential.data};
\end{axis}
\end{tikzpicture}
\caption{Size of derivatives of $\blexersimp$ for matching 
	$\protect((a^* + (aa)^* + \ldots + (aaaaa)^* )^*)^*$ 
	with strings 
	of the form $\protect\underbrace{aa..a}_{n}$.}\label{blexerExp}
\end{figure}
\noindent
One possible approach would be to apply the rewriting
rule
\[
	(a+b+d) \cdot r_1  \longrightarrow a \cdot r_1 + b \cdot r_1 + d \cdot r_1
\]
\noindent
in our $\simp$ function,
so that it makes the simplification in \eqref{eqn:partialDedup} possible.
Translating the rule into our $\textit{bsimp}$ function simply
involves adding a new clause to the $\textit{bsimp}_{ASEQ}$ function:
\begin{center}
	\begin{tabular}{@{}lcl@{}}
		$\textit{bsimp}_{ASEQ} \; bs\; a \; b$ & $\dn$ & $ (a,\; b) \textit{match}$\\
						       && $\ldots$\\
   &&$\quad\textit{case} \; (_{bs1}\sum as, a_2') \Rightarrow _{bs1}\sum (
   \map \; (_{[]}\textit{ASEQ} \; \_ \; a_2') \; as)$\\
   &&$\quad\textit{case} \; (a_1', a_2') \Rightarrow   _{bs}a_1' \cdot a_2'$ \\
	\end{tabular}
\end{center}
\noindent
Unfortunately,
if we introduce them in our
setting we would lose the POSIX property of our calculated values. 
For example given the regular expression 
\begin{center}
	$(a + ab)(bc + c)$
\end{center}
and the string $ab$,
then our algorithm generates the following
correct POSIX value
\begin{center}
	$\Seq \; (\Right \; ab) \; (\Right \; c)$.
\end{center}
Essentially it matches the string with the longer Right-alternative
in the first sequence (and
then the 'rest' with the character regular expression $c$ from the second sequence).
If we add the simplification above, then we obtain the following value
\begin{center}
	$\Left \; (\Seq \; a \; (\Left \; bc))$
\end{center}
where the $\Left$-alternatives get priority. 
However this violates the POSIX rules.
The reason for getting this undesired value
is that the new rule splits this regular expression up into 
\begin{center}
	$a\cdot(b c + c) + ab \cdot (bc + c)$,
\end{center}
which becomes a regular expression with a 
quite different structure--the original
was a sequence, and now it becomes an alternative.
With an alternative the maximal munch rule no longer works.\\
A method to reconcile this is to do the 
transformation in \eqref{eqn:partialDedup} ``non-invasively'',
meaning that we traverse the list of regular expressions
\begin{center}
	$rs_a@[a]@rs_c$
\end{center}
in the alternative
\begin{center}
	$\sum ( rs_a@[a]@rs_c)$
\end{center}
using  a function similar to $\distinctBy$,
but this time 
we allow a more general list rewrite:
\begin{figure}[H]
\begin{mathpar}	
	\inferrule * [Right = cubicRule]{\vspace{0mm} }{rs_a@[a]@rs_c 
			\stackrel{s}{\rightsquigarrow }
		rs_a@[\textit{prune} \; a \; rs_a]@rs_c }
\end{mathpar}
\caption{The rule capturing the pruning simplification needed to achieve
a cubic bound}
\label{fig:cubicRule}
\end{figure}
%L \; a_1' = L \; a_1 \setminus (\cup_{a \in rs_a} L \; a)
where $\textit{prune} \;a \; acc$ traverses $a$
without altering the structure of $a$, removing components in $a$
that have appeared in the accumulator $acc$.
For example
\begin{center}
	$\textit{prune} \;\;\; (r_a+r_f+r_g+r_h)r_d \;\; \; [(r_a+r_b+r_c)r_d, (r_e+r_f)r_d] $
\end{center}
should be equal to 
\begin{center}
	$(r_g+r_h)r_d$
\end{center}
because $r_gr_d$ and 
$r_hr_d$ are the only terms
that have not appeared in the accumulator list 
\begin{center}
$[(r_a+r_b+r_c)r_d, (r_e+r_f)r_d]$.
\end{center}
We implemented 
function $\textit{prune}$ in Scala:
\begin{figure}[H]
\begin{lstlisting}
    def prune(r: ARexp, acc: Set[Rexp]) : ARexp = r match{
      case AALTS(bs, rs) => rs.map(r => prune(r, acc)).filter(_ != AZERO) match
      {
        //all components have been removed, meaning this is effectively a duplicate
        //flats will take care of removing this AZERO 
        case Nil => AZERO
        case r::Nil => fuse(bs, r)
        case rs1 => AALTS(bs, rs1)
      }
      case ASEQ(bs, r1, r2) => 
        //remove the r2 in (ra + rb)r2 to identify the duplicate contents of r1
        prune(r1, acc.map(r => removeSeqTail(r, erase(r2)))) match {
          //after pruning, returns 0
          case AZERO => AZERO
          //after pruning, got r1'.r2, where r1' is equal to 1
          case r1p if(isOne(erase(r1p))) => fuse(bs ++ mkepsBC(r1p), r2)
          //assemble the pruned head r1p with r2
          case r1p => ASEQ(bs, r1p, r2)
        }
        //this does the duplicate component removal task
      case r => if(acc(erase(r))) AZERO else r
    }
\end{lstlisting}
\caption{The function $\textit{prune}$ }
\end{figure}
\noindent
The function $\textit{prune}$ 
is a stronger version of $\textit{distinctBy}$.
It does not just walk through a list looking for exact duplicates,
but prunes sub-expressions recursively.
It manages proper contexts by the helper functions
$\textit{removeSeqTail}$, $\textit{isOne}$ and $\textit{atMostEmpty}$.
\begin{figure}[H]
\begin{lstlisting}
    def atMostEmpty(r: Rexp) : Boolean = r match {
      case ZERO => true
      case ONE => true
      case STAR(r) => atMostEmpty(r)
      case SEQ(r1, r2) => atMostEmpty(r1) && atMostEmpty(r2)
      case ALTS(r1, r2) => atMostEmpty(r1) && atMostEmpty(r2)
      case CHAR(_) => false
    }

    def isOne(r: Rexp) : Boolean = r match {
      case ONE => true
      case SEQ(r1, r2) => isOne(r1) && isOne(r2)
      case ALTS(r1, r2) => (isOne(r1) || isOne(r2)) && (atMostEmpty(r1) && atMostEmpty(r2))
      case STAR(r0) => atMostEmpty(r0)
      case CHAR(c) => false
      case ZERO => false
    }

    def removeSeqTail(r: Rexp, tail: Rexp) : Rexp = 
      if (r == tail)
        ONE
      else {
        r match {
          case SEQ(r1, r2) => 
            if(r2 == tail) 
              r1
            else
              ZERO
          case r => ZERO
        }
      }


\end{lstlisting}
\caption{The helper functions of $\textit{prune}$}
\end{figure}
\noindent
Suppose we feed 
\begin{center}
	$r= (\underline{\ONE}+(\underline{f}+b)\cdot g)\cdot (a\cdot(d\cdot e))$
\end{center}
and 
\begin{center}
	$acc = \{a\cdot(d\cdot e),f\cdot (g \cdot (a \cdot (d \cdot e))) \}$
\end{center}
as the input for $\textit{prune}$.
The end result will be
\[
	b\cdot(g\cdot(a\cdot(d\cdot e)))
\]
where the underlined components in $r$ are eliminated.
Looking more closely, at the topmost call 
\[
	\textit{prune} \quad (\ONE+
	(f+b)\cdot g)\cdot (a\cdot(d\cdot e)) \quad
	\{a\cdot(d\cdot e),f\cdot (g \cdot (a \cdot (d \cdot e))) \}
\]
The sequence clause will be called,
where a sub-call
\[
	\textit{prune} \;\; (\ONE+(f+b)\cdot g)\;\; \{\ONE, f\cdot g \}
\]
is made. The terms in the new accumulator $\{\ONE,\; f\cdot g \}$ come from
the two calls to $\textit{removeSeqTail}$:
\[
	\textit{removeSeqTail} \quad\;\; a \cdot(d\cdot e) \quad\;\; a \cdot(d\cdot e) 
\]
and
\[
	\textit{removeSeqTail} \quad \;\; 
	f\cdot(g\cdot (a \cdot(d\cdot e)))\quad  \;\; a \cdot(d\cdot e).
\]
The idea behind $\textit{removeSeqTail}$ is that
when pruning recursively, we need to ``zoom in''
to sub-expressions, and this ``zoom in'' needs to be performed
on the
accumulators as well, otherwise we will be comparing
apples with oranges.
The sub-call 
$\textit{prune} \;\; (\ONE+(f+b)\cdot g)\;\; \{\ONE, f\cdot g \}$
is simpler, which will trigger the alternative clause, causing
a pruning on each element in $(\ONE+(f+b)\cdot g)$,
leaving us $b\cdot g$ only.

Our new lexer with stronger simplification
uses $\textit{prune}$ by making it 
the core component of the deduplicating function
called $\textit{distinctWith}$.
$\textit{DistinctWith}$ ensures that all verbose
parts of a regular expression are pruned away.

\begin{figure}[H]
\begin{lstlisting}
    def turnIntoTerms(r: Rexp): List[Rexp] = r match {
      case SEQ(r1, r2)  => 
        turnIntoTerms(r1).flatMap(r11 => furtherSEQ(r11, r2))
          case ALTS(r1, r2) => turnIntoTerms(r1) ::: turnIntoTerms(r2)
          case ZERO => Nil
          case _ => r :: Nil
    }

    def distinctWith(rs: List[ARexp], 
      pruneFunction: (ARexp, Set[Rexp]) => ARexp, 
      acc: Set[Rexp] = Set()) : List[ARexp] = 
        rs match{
          case Nil => Nil
          case r :: rs => 
            if(acc(erase(r)))
              distinctWith(rs, pruneFunction, acc)
            else {
              val pruned_r = pruneFunction(r, acc)
              pruned_r :: 
              distinctWith(rs, 
                pruneFunction, 
                turnIntoTerms(erase(pruned_r)) ++: acc
                )
            }
        }

\end{lstlisting}
\caption{A Stronger Version of $\textit{distinctBy}$}
\end{figure}
\noindent
Once a regular expression has been pruned,
all its components will be added to the accumulator
to remove any future regular expressions' duplicate components.

The function $\textit{bsimpStrong}$
is very much the same as $\textit{bsimp}$, just with
$\textit{distinctBy}$ replaced 
by $\textit{distinctWith}$.
\begin{figure}[H]
\begin{lstlisting}

    def bsimpStrong(r: ARexp): ARexp =
    {
      r match {
        case ASEQ(bs1, r1, r2) => (bsimpStrong(r1), bsimpStrong(r2)) match {
        case (AZERO, _) => AZERO
        case (_, AZERO) => AZERO
        case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)
        case (r1s, AONE(bs2)) => fuse(bs1, r1s) //assert bs2 == Nil
        case (r1s, r2s) => ASEQ(bs1, r1s, r2s)
        }
        case AALTS(bs1, rs) => {
          distinctWith(flats(rs.map(bsimpStrong(_))), prune) match {
            case Nil => AZERO
            case s :: Nil => fuse(bs1, s)
            case rs => AALTS(bs1, rs)
          }
        }
        case ASTAR(bs, r0) if(atMostEmpty(erase(r0))) => AONE(bs)
        case r => r
      }
    }
    def bdersStrong(s: List[Char], r: ARexp) : ARexp = s match {
        case Nil => r
        case c::s => bdersStrong(s, bsimpStrong(bder(c, r)))
      }

\end{lstlisting}
\caption{The function
$\textit{bsimpStrong}$: a stronger version of $\textit{bsimp}$}
\end{figure}
\noindent
The benefits of using 
$\textit{prune}$ refining the finiteness bound
to a cubic bound has not been formalised yet.
Therefore we choose to use Scala code rather than an Isabelle-style formal 
definition like we did for $\simp$, as the definitions might change
to suit our proof needs.
In the rest of the chapter we will use this convention consistently.

%The function $\textit{prune}$ is used in $\distinctWith$.
%$\distinctWith$ is a stronger version of $\distinctBy$
%which not only removes duplicates as $\distinctBy$ would
%do, but also uses the $\textit{pruneFunction}$
%argument to prune away verbose components in a regular expression.\\
%\begin{figure}[H]
%\begin{lstlisting}
%   //a stronger version of simp
%    def bsimpStrong(r: ARexp): ARexp =
%    {
%      r match {
%        case ASEQ(bs1, r1, r2) => (bsimpStrong(r1), bsimpStrong(r2)) match {
%          //normal clauses same as simp
%        case (AZERO, _) => AZERO
%        case (_, AZERO) => AZERO
%        case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)
%        //bs2 can be discarded
%        case (r1s, AONE(bs2)) => fuse(bs1, r1s) //assert bs2 == Nil
%        case (r1s, r2s) => ASEQ(bs1, r1s, r2s)
%        }
%        case AALTS(bs1, rs) => {
%          //distinctBy(flat_res, erase)
%          distinctWith(flats(rs.map(bsimpStrong(_))), prune) match {
%            case Nil => AZERO
%            case s :: Nil => fuse(bs1, s)
%            case rs => AALTS(bs1, rs)
%          }
%        }
%        //stars that can be treated as 1
%        case ASTAR(bs, r0) if(atMostEmpty(erase(r0))) => AONE(bs)
%        case r => r
%      }
%    }
%    def bdersStrong(s: List[Char], r: ARexp) : ARexp = s match {
%        case Nil => r
%        case c::s => bdersStrong(s, bsimpStrong(bder(c, r)))
%      }
%\end{lstlisting}
%\caption{The function $\bsimpStrong$ and $\bdersStrongs$}
%\end{figure}
%\noindent
%$\distinctWith$, is in turn used in $\bsimpStrong$:
%\begin{figure}[H]
%\begin{lstlisting}
%      //Conjecture: [| bdersStrong(s, r) |] = O([| r |]^3)
%      def bdersStrong(s: List[Char], r: ARexp) : ARexp = s match {
%        case Nil => r
%        case c::s => bdersStrong(s, bsimpStrong(bder(c, r)))
%      }
%\end{lstlisting}
%\caption{The function $\bsimpStrong$ and $\bdersStrongs$}
%\end{figure}
%\noindent
We conjecture that the above Scala function $\bdersStrongs$,
written $\bdersStrong{\_}{\_}$ as an infix notation, 
satisfies the following property:
\begin{conjecture}
	$\llbracket \bdersStrong{a}{s} \rrbracket = O(\llbracket a \rrbracket^3)$
\end{conjecture}
\noindent
The stronger version of $\blexersimp$'s
code in Scala looks like: 
\begin{figure}[H]
\begin{lstlisting}
      def strongBlexer(r: Rexp, s: String) : Option[Val] = {
        Try(Some(decode(r, strong_blex_simp(internalise(r), s.toList)))).getOrElse(None)
      }
      def strong_blex_simp(r: ARexp, s: List[Char]) : Bits = s match {
        case Nil => {
          if (bnullable(r)) {
            mkepsBC(r)
          }
          else
            throw new Exception("Not matched")
        }
        case c::cs => {
          strong_blex_simp(strongBsimp(bder(c, r)), cs)
        }
      }
\end{lstlisting}
\end{figure}
\noindent
We call this lexer $\blexerStrong$.
This version is able to drastically reduce the
internal data structure size which 
otherwise could
trigger exponential behaviours in
$\blexersimp$.
\begin{figure}[H]
\centering
\begin{tabular}{@{}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{}}
\begin{tikzpicture}
\begin{axis}[
    %xlabel={$n$},
    myplotstyle,
    xlabel={input length},
    ylabel={size},
    width = 7cm,
    height = 5cm,
    ]
\addplot[red,mark=*, mark options={fill=white}] table {strongSimpCurve.data};
\end{axis}
\end{tikzpicture}
  &
\begin{tikzpicture}
\begin{axis}[
    %xlabel={$n$},
    myplotstyle,
    xlabel={input length},
    ylabel={size},
    width = 7cm,
    height = 5cm,
    ]
\addplot[blue,mark=*, mark options={fill=white}] table {bsimpExponential.data};
\end{axis}
\end{tikzpicture}\\
\multicolumn{2}{l}{}
\end{tabular}    
\caption{Runtime for matching 
	$\protect((a^* + (aa)^* + \ldots + (aaaaa)^* )^*)^*$ with strings 
	   of the form $\protect\underbrace{aa..a}_{n}$.}\label{fig:aaaaaStarStar}
\end{figure}
\noindent
We would like to preserve the correctness like the one 
we had for $\blexersimp$:
\begin{conjecture}\label{cubicConjecture}
	$\blexerStrong \;r \; s = \blexer\; r\;s$
\end{conjecture}
\noindent
The idea is to maintain key lemmas in
chapter \ref{Bitcoded2} like
$r \stackrel{*}{\rightsquigarrow} \textit{bsimp} \; r$
with the new rewriting rule 
shown in figure \ref{fig:cubicRule} .

In the next sub-section,
we will describe why we 
believe a cubic size bound can be achieved with
the stronger simplification.
For this we give a short introduction to the
partial derivatives,
which were invented by Antimirov \cite{Antimirov95},
and then link them with the result of the function
$\bdersStrongs$.
 
\subsection{Antimirov's partial derivatives}
Partial derivatives were first introduced by 
Antimirov \cite{Antimirov95}.
Partial derivatives are very similar
to Brzozowski derivatives,
but splits children of alternative regular expressions into 
multiple independent terms. This means the output of
partial derivatives become a 
set of regular expressions:
\begin{center}  
 	\begin{tabular}{lcl}
		$\partial_x \; (r_1 \cdot r_2)$ & 
		$\dn$ & $(\partial_x \; r_1) \cdot r_2 \cup
		\partial_x \; r_2 \; \textit{if} \; \; \nullable\; r_1$\\
		      & & $(\partial_x \; r_1)\cdot r_2 \quad\quad 
		      \textit{otherwise}$\\ 
		$\partial_x \; r^*$ & $\dn$ & $(\partial_x \; r) \cdot r^*$\\
		$\partial_x \; c $ & $\dn$ & $\textit{if} \; x = c \; 
		\textit{then} \;
		\{ \ONE\} \;\;\textit{else} \; \varnothing$\\
		$\partial_x(r_1+r_2)$ & $=$ & $\partial_x(r_1) \cup \partial_x(r_2)$\\
		$\partial_x(\ONE)$ & $=$ & $\varnothing$\\
		$\partial_x(\ZERO)$ & $\dn$ & $\varnothing$\\
	\end{tabular}
\end{center}
\noindent
The $\cdot$ between for example 
$(\partial_x \; r_1) \cdot r_2 $ 
is a shorthand notation for the cartesian product 
$(\partial_x \; r_1) \times \{ r_2\}$.
%Each element in the set generated by a partial derivative
%corresponds to a (potentially partial) match
%TODO: define derivatives w.r.t string s
Rather than joining the calculated derivatives $\partial_x r_1$ and $\partial_x r_2$ together
using the $\sum$ constructor, Antimirov put them into
a set.  This means many subterms will be de-duplicated
because they are sets, 
For example, to compute what regular expression $x^*(xx + y)^*$'s 
derivative w.r.t. $x$ is, one can compute a partial derivative
and get two singleton sets $\{x^* \cdot (xx + y)^*\}$ and $\{x \cdot (xx + y) ^* \}$
from $\partial_x(x^*) \cdot (xx + y) ^*$ and $\partial_x((xx + y)^*)$.

The partial derivative w.r.t. a string is defined recursively:
\[
	\partial_{c::cs} r \dn \bigcup_{r'\in (\partial_c r)}
	\partial_{cs} r'
\]
Given an alphabet $\Sigma$, we denote the set of all possible strings
from this alphabet as $\Sigma^*$. 
The set of all possible partial derivatives is defined
as the union of derivatives w.r.t all the strings in the universe:
\begin{center}
	\begin{tabular}{lcl}
		$\textit{PDER}_{\Sigma^*} \; r $ & $\dn $ & $\bigcup_{w \in \Sigma^*}\partial_w \; r$
	\end{tabular}
\end{center}
\noindent
Consider now again our pathological case where the derivatives
grow with a rather aggressive simplification
\begin{center}
	$((a^* + (aa)^* + \ldots + (\underbrace{a\ldots a}_{n a's})^* )^*)^*$
\end{center}
example, if we denote this regular expression as $r$,
we have that
\begin{center}
$\textit{PDER}_{\Sigma^*} \; r = 
\bigcup_{i=1}^{n}\bigcup_{j=0}^{i-1} \{ 
	(\underbrace{a \ldots a}_{\text{j a's}}\cdot
(\underbrace{a \ldots a}_{\text{i a's}})^*)\cdot r \}$, 
\end{center}
with exactly $n * (n + 1) / 2$ terms.
This is in line with our speculation that only $n*(n+1)/2$ terms are
needed. We conjecture that $\bsimpStrong$ is also able to achieve this
upper limit in general
\begin{conjecture}\label{bsimpStrongInclusionPder}
	Using a suitable transformation $f$, we have 
	\begin{center}
		$\forall s.\; f \; (r \bdersStrong  \; s) \subseteq
		 \textit{PDER}_{\Sigma^*} \; r$
	\end{center}
\end{conjecture}
\noindent
because our \ref{fig:cubicRule} will keep only one copy of each term,
where the function $\textit{prune}$ takes care of maintaining
a set like structure similar to partial derivatives.
We might need to adjust $\textit{prune}$
slightly to make sure all duplicate terms are eliminated,
which should be doable.

Antimirov had proven that the sum of all the partial derivative 
terms' sizes is bounded by the cubic of the size of that regular
expression:
\begin{property}\label{pderBound}
	$\llbracket \textit{PDER}_{\Sigma^*} \; r \rrbracket \leq O(\llbracket r \rrbracket^3)$
\end{property}
This property was formalised by Wu et al. \cite{Wu2014}, and the 
details can be found in the archive of formal proofs. \footnote{https://www.isa-afp.org/entries/Myhill-Nerode.html}
Once conjecture \ref{bsimpStrongInclusionPder} is proven, then property \ref{pderBound}
would yield us also a cubic bound for our $\blexerStrong$ algorithm: 
\begin{conjecture}\label{strongCubic}
	$\llbracket r \bdersStrong\; s \rrbracket \leq \llbracket r \rrbracket^3$
\end{conjecture}
\noindent
We leave this as future work.


%To get all the "atomic" components of a regular expression's possible derivatives,
%there is a procedure Antimirov called $\textit{lf}$, short for "linear forms", that takes
%whatever character is available at the head of the string inside the language of a
%regular expression, and gives back the character and the derivative regular expression
%as a pair (which he called "monomial"):
% \begin{center}  
% \begin{tabular}{ccc}
% $\lf(\ONE)$ & $=$ & $\phi$\\
%$\lf(c)$ & $=$ & $\{(c, \ONE) \}$\\
% $\lf(a+b)$ & $=$ & $\lf(a) \cup \lf(b)$\\
% $\lf(r^*)$ & $=$ & $\lf(r) \bigodot \lf(r^*)$\\
%\end{tabular}
%\end{center}
%%TODO: completion
%
%There is a slight difference in the last three clauses compared
%with $\partial$: instead of a dot operator $ \textit{rset} \cdot r$ that attaches the regular 
%expression $r$ with every element inside $\textit{rset}$ to create a set of 
%sequence derivatives, it uses the "circle dot" operator $\bigodot$ which operates 
%on a set of monomials (which Antimirov called "linear form") and a regular 
%expression, and returns a linear form:
% \begin{center}  
% \begin{tabular}{ccc}
% $l \bigodot (\ZERO)$ & $=$ & $\phi$\\
% $l \bigodot (\ONE)$ & $=$ & $l$\\
% $\phi \bigodot t$ & $=$ & $\phi$\\
% $\{ (x, \ZERO) \} \bigodot t$ & $=$ & $\{(x,\ZERO) \}$\\
% $\{ (x, \ONE) \} \bigodot t$ & $=$ & $\{(x,t) \}$\\
%  $\{ (x, p) \} \bigodot t$ & $=$ & $\{(x,p\cdot t) \}$\\
% $\lf(a+b)$ & $=$ & $\lf(a) \cup \lf(b)$\\
% $\lf(r^*)$ & $=$ & $\lf(r) \cdot \lf(r^*)$\\
%\end{tabular}
%\end{center}
%%TODO: completion
%
% Some degree of simplification is applied when doing $\bigodot$, for example,
% $l \bigodot (\ZERO) = \phi$ corresponds to $r \cdot \ZERO \rightsquigarrow \ZERO$,
% and  $l \bigodot (\ONE) = l$ to $l \cdot \ONE \rightsquigarrow l$, and
%  $\{ (x, \ZERO) \} \bigodot t = \{(x,\ZERO) \}$ to $\ZERO \cdot x \rightsquigarrow \ZERO$,
%  and so on.
%  
%  With the function $\lf$ one can compute all possible partial derivatives $\partial_{UNIV}(r)$ of a regular expression $r$ with 
%  an iterative procedure:
%   \begin{center}  
% \begin{tabular}{llll}
%$\textit{while}$ & $(\Delta_i \neq \phi)$  &                &          \\
% 		       &  $\Delta_{i+1}$           &        $ =$ & $\lf(\Delta_i) - \PD_i$ \\
%		       &  $\PD_{i+1}$              &         $ =$ & $\Delta_{i+1} \cup \PD_i$ \\
%$\partial_{UNIV}(r)$ & $=$ & $\PD$ &		     
%\end{tabular}
%\end{center}
%
%
% $(r_1 + r_2) \cdot r_3 \longrightarrow (r_1 \cdot r_3) + (r_2 \cdot r_3)$,




%----------------------------------------------------------------------------------------
%	SECTION 2
%----------------------------------------------------------------------------------------


%The closed form for them looks like:
%%\begin{center}
%%	\begin{tabular}{llrclll}
%%		$r^{\{n+1\}}$ & $ \backslash_{rsimps}$ & $(c::s)$ & $=$ & & \\
%%		$\textit{rsimp}$ & $($ & $
%%		\sum \; ( $ & $\map$ & $(\textit{optermsimp}\;r)$ & $($\\
%%			 & & & & $\textit{nupdates} \;$ & 
%%			 $ s \; r_0 \; [ \textit{Some} \; ([c], n)]$\\
%%			 & & & & $)$ &\\
%%			 & &  $)$ & & &\\
%%			 & $)$ & & & &\\
%%	\end{tabular}
%%\end{center}
%\begin{center}
%	\begin{tabular}{llrcllrllll}
%		$r^{\{n+1\}}$ & $ \backslash_{rsimps}$ & $(c::s)$ & $=$ & & &&&&\\
%			      &&&&$\textit{rsimp}$ & $($ & $
%			      \sum \; ( $ & $\map$ & $(\textit{optermsimp}\;r)$ & $($\\
%					  &&&& & & & & $\;\; \textit{nupdates} \;$ & 
%			 		  $ s \; r_0 \; [ \textit{Some} \; ([c], n)]$\\
%					  &&&& & & & & $)$ &\\
%					  &&&& & &  $)$ & & &\\
%					  &&&& & $)$ & & & &\\
%	\end{tabular}
%\end{center}
%The $\textit{optermsimp}$ function with the argument $r$ 
%chooses from two options: $\ZERO$ or 
%We define for the $r^{\{n\}}$ constructor something similar to $\starupdate$
%and $\starupdates$:
%\begin{center}
%	\begin{tabular}{lcl}
%		$\starupdate \; c \; r \; [] $ & $\dn$ & $[]$\\
%		$\starupdate \; c \; r \; (s :: Ss)$ & $\dn$ & \\
%						     & & $\textit{if} \; 
%						     (\rnullable \; (\rders \; r \; s))$ \\
%						     & & $\textit{then} \;\; (s @ [c]) :: [c] :: (
%						     \starupdate \; c \; r \; Ss)$ \\
%						     & & $\textit{else} \;\; (s @ [c]) :: (
%						     \starupdate \; c \; r \; Ss)$
%	\end{tabular}
%\end{center}
%\noindent
%As a generalisation from characters to strings,
%$\starupdates$ takes a string instead of a character
%as the first input argument, and is otherwise the same
%as $\starupdate$.
%\begin{center}
%	\begin{tabular}{lcl}
%		$\starupdates \; [] \; r \; Ss$ & $=$ & $Ss$\\
%		$\starupdates \; (c :: cs) \; r \; Ss$ &  $=$ &  $\starupdates \; cs \; r \; (
%		\starupdate \; c \; r \; Ss)$
%	\end{tabular}
%\end{center}
%\noindent



%\section{Zippers}
%Zipper is a data structure designed to operate on 
%and navigate between local parts of a tree.
%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.
%Some initial results 
%We first give a brief introduction to what zippers are,
%and other works
%that apply zippers to derivatives
%When dealing with large trees, it would be a waste to 
%traverse the entire tree if
%the operation only
%involves a small fraction of it.
%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:


%----------------------------------------------------------------------------------------
%	SECTION 1
%----------------------------------------------------------------------------------------

%\section{Adding Support for the Negation Construct, and its Correctness Proof}
%We now add support for the negation regular expression:
%\[			r ::=   \ZERO \mid  \ONE
%			 \mid  c  
%			 \mid  r_1 \cdot r_2
%			 \mid  r_1 + r_2   
%			 \mid r^*    
%			 \mid \sim r
%\]
%The $\textit{nullable}$ function's clause for it would be 
%\[
%\textit{nullable}(~r) = \neg \nullable(r)
%\]
%The derivative would be
%\[
%~r \backslash c = ~ (r \backslash c)
%\]
% 
%The most tricky part of lexing for the $~r$ regular expression
% is creating a value for it.
% For other regular expressions, the value aligns with the 
% structure of the regular expression:
% \[
% \vdash \Seq(\Char(a), \Char(b)) : a \cdot b
% \]
%But for the $~r$ regular expression, $s$ is a member of it if and only if
%$s$ does not belong to $L(r)$. 
%That means when there
%is a match for the not regular expression, it is not possible to generate how the string $s$ matched
%with $r$. 
%What we can do is preserve the information of how $s$ was not matched by $r$,
%and there are a number of options to do this.
%
%We could give a partial value when there is a partial match for the regular expression inside
%the $\mathbf{not}$ construct.
%For example, the string $ab$ is not in the language of $(a\cdot b) \cdot c$,
%A value for it could be 
% \[
% \vdash \textit{Not}(\Seq(\Char(a), \Char(b))) : ~((a \cdot b ) \cdot c)
% \]
% The above example demonstrates what value to construct 
% when the string $s$ is at most a real prefix
% of the strings in $L(r)$. When $s$ instead is not a prefix of any strings
% in $L(r)$, it becomes unclear what to return as a value inside the $\textit{Not}$
% constructor.
% 
% Another option would be to either store the string $s$ that resulted in 
% a mis-match for $r$ or a dummy value as a placeholder:
% \[
% \vdash \textit{Not}(abcd) : ~( r_1 )
% \]
%or
% \[
% \vdash \textit{Not}(\textit{Dummy}) : ~( r_1 )
% \] 
% We choose to implement this as it is most straightforward:
% \[
% \mkeps(~(r))  = \textit{if}(\nullable(r)) \; \textit{Error} \; \textit{else} \; \textit{Not}(\textit{Dummy})
% \]
% 
%
%\begin{center}
%	\begin{tabular}{lcl}
%		$\ntset \; r \; (n+1) \; c::cs $ & $\dn$ & $\nupdates \;
%		cs \; r \; [\Some \; ([c], n)]$\\
%		$\ntset \; r\; 0 \; \_$ &  $\dn$ &  $\None$\\
%		$\ntset \; r \; \_ \; [] $ & $ \dn$ & $[]$\\
%	\end{tabular}
%\end{center}

