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\fnote{\copyright{} Christian Urban, King's College London, 2014, 2015, 2016, 2017, 2019}

\section*{Handout 4 (Sulzmann \& Lu Algorithm)}

expression or not. Often a more interesting question is to

find out \emph{how} a regular expression matched a string?

Answering this question will also help us with the problem we

are after, namely tokenising an input string. 

The algorithm we will be looking at in this lecture was designed by Sulzmann

\& Lu in a rather recent research paper (from 2014). A link to it is

provided on KEATS, in case you are interested.\footnote{In my

humble opinion this is an interesting instance of the research

literature: it contains a very neat idea, but its presentation

is rather sloppy. In earlier versions of this paper, a King's

undergraduate student and I found several rather annoying typos in the

examples and definitions.} My former PhD student Fahad Ausaf and I even more recently 

wrote a paper where we build on their result: we provided a 

mathematical proof that their algorithm is really correct---the proof 

Sulzmann \& Lu had originally given contained major flaws. Such correctness

proofs are important: Kuklewicz maintains a unit-test library

for the kind of algorithms we are interested in here and he showed 

that many implementations in the ``wild'' are buggy, that is not

satisfy his unit tests:

\begin{center}

\url{http://www.haskell.org/haskellwiki/Regex_Posix}

\end{center}

Coming back to the algorithm by Sulzmann \& Lu, their idea is to

introduce \emph{values} for producing an answer for \emph{how} a regular

expression matches a string. So rather than a boolean like in the

Brzozowski algorithm, a value will be the output of the Sulzman \& Lu

algorithm whenever the regular expression matches the string. If the

string does not match the string, an error will be raised. The

definition for values is given below. They are shown together with the

regular expressions $r$ to which they correspond:

\begin{center}

\begin{tabular}{cc}

\begin{tabular}{@{}rrl@{}}

\multicolumn{3}{c}{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}{c}{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 There is no value for the

$\ZERO$ regular expression, since it does not match any

string. Otherwise there is exactly one value corresponding to

each regular expression with the exception of $r_1 + r_2$

where there are two values, namely $\Left(v)$ and $Right(v)$

corresponding to the two alternatives. Note that $r^*$ is

associated with a list of values, one for each copy of $r$

that was needed to match the string. This means we might also

return the empty list $Stars\,[]$, if no copy was needed

for $r^*$. For sequence, there is exactly one value, composed 

of two component values ($v_1$ and $v_2$).

My implementation of regular expressions and values in Scala is

regular expressions are written entirely with upper-case

character and the rest are lower-case letters.

{\small\lstinputlisting[language=Scala,numbers=none,linebackgroundcolor=

                  {\ifodd\value{lstnumber}\color{capri!3}\fi}]

{../progs/app01.scala}}

{\small\lstinputlisting[language=Scala,numbers=none,linebackgroundcolor=

                  {\ifodd\value{lstnumber}\color{capri!3}\fi}]

{../progs/app02.scala}}

Graphically the algorithm by Sulzmann \& Lu can be illustrated

by the picture in Figure~\ref{Sulz} where the path from the

left to the right involving $\der/\nullable$ is the first phase

of the algorithm and $\textit{mkeps}/\inj$, the path from right to left,

the second phase. This picture shows the steps required when a

regular expression, say $r_1$, matches the string $abc$. We

first build the three derivatives (according to $a$, $b$ and

$c$). We then use $\nullable$ to find out whether the resulting

regular expression can match the empty string. If yes, we call

the function $\textit{mkeps}$. The $\textit{mkeps}$ function calculates a value for how a regular

expression has matched the empty string. Its definition

is as follows:

\begin{figure}[t]

\begin{center}

\begin{tikzpicture}[scale=2,node distance=1.2cm,

                    every node/.style={minimum size=7mm}]

\node (r1)  {$r_1$};

\node (r2) [right=of r1]{$r_2$};

\draw[->,line width=1mm](r1)--(r2) node[above,midway] {$\der\,a$};

\node (r3) [right=of r2]{$r_3$};

\draw[->,line width=1mm](r2)--(r3) node[above,midway] {$\der\,b$};

\node (r4) [right=of r3]{$r_4$};

\draw[->,line width=1mm](r3)--(r4) node[above,midway] {$\der\,c$};

\draw (r4) node[anchor=west] {\;\raisebox{3mm}{$\nullable$}};

\node (v4) [below=of r4]{$v_4$};

\draw[->,line width=1mm](r4) -- (v4);

\node (v3) [left=of v4] {$v_3$};

\draw[->,line width=1mm](v4)--(v3) node[below,midway] {$\inj\,c$};

\node (v2) [left=of v3]{$v_2$};

\draw[->,line width=1mm](v3)--(v2) node[below,midway] {$\inj\,b$};

\node (v1) [left=of v2] {$v_1$};

\draw[->,line width=1mm](v2)--(v1) node[below,midway] {$\inj\,a$};

\draw (r4) node[anchor=north west] {\;\raisebox{-8mm}{$\textit{mkeps}$}};

\end{tikzpicture}

\end{center}

\caption{The two phases of the algorithm by Sulzmann \& Lu.

\label{Sulz}}

\end{figure}

\begin{center}

\begin{tabular}{lcl}

  $\textit{mkeps}(\ONE)$       & $\dn$ & $Empty$\\

  $\textit{mkeps}(r_1 + r_2)$      & $\dn$ & if $\nullable(r_1)$  \\

                          &       & then $\Left(\textit{mkeps}(r_1))$\\

                          &       & else $Right(\textit{mkeps}(r_2))$\\

  $\textit{mkeps}(r_1 \cdot r_2)$  & $\dn$ & $Seq(\textit{mkeps}(r_1),\textit{mkeps}(r_2))$\\

  $\textit{mkeps}(r^*)$            & $\dn$ & $Stars\,[]$  \\

\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.

The second phase of the algorithm is organised so that it will

calculate a value for how the derivative regular expression

has matched a string. For this we need a function that

reverses this ``chopping off'' for values which we did in the

first phase for derivatives. The corresponding function is

called $\inj$ (short for injection). This function takes three

arguments: the first one is a regular expression for which we

want to calculate the value, the second is the character we

want to inject and the third argument is the value where we

will inject the character into. The result of this function is a

new value. The definition of $\inj$ is as follows: 

\begin{center}

\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}

  $\inj\,(c)\,c\,Empty$            & $\dn$ & $Char\,c$\\

  $\inj\,(r_1 + r_2)\,c\,\Left(v)$ & $\dn$ & $\Left(\inj\,r_1\,c\,v)$\\

  $\inj\,(r_1 + r_2)\,c\,Right(v)$ & $\dn$ & $Right(\inj\,r_2\,c\,v)$\\

  $\inj\,(r_1 \cdot r_2)\,c\,Seq(v_1,v_2)$ & $\dn$  & $Seq(\inj\,r_1\,c\,v_1,v_2)$\\

  $\inj\,(r_1 \cdot r_2)\,c\,\Left(Seq(v_1,v_2))$ & $\dn$  & $Seq(\inj\,r_1\,c\,v_1,v_2)$\\

  $\inj\,(r_1 \cdot r_2)\,c\,Right(v)$ & $\dn$  & $Seq(\textit{mkeps}(r_1),\inj\,r_2\,c\,v)$\\

  $\inj\,(r^*)\,c\,Seq(v,Stars\,vs)$         & $\dn$  & $Stars(\inj\,r\,c\,v\,::\,vs)$\\

\end{tabular}

\end{center}

\noindent This definition is by recursion on the regular

expression and by analysing the shape of the values. Therefore

there are three cases for sequence regular

expressions (for all possible shapes of the value we can encounter). The last

clause for the star regular expression returns a list where

the first element is $\inj\,r\,c\,v$ and the other elements are

$vs$. That means $\_\,::\,\_$ should be read as list cons.

To understand what is going on, it might be best to do some

example calculations and compare them with Figure~\ref{Sulz}.

For this note that we have not yet dealt with the need of

simplifying regular expressions (this will be a topic on its

own later). Suppose the regular expression is $a \cdot (b

\cdot c)$ and the input string is $abc$. The derivatives from

the first phase are as follows:

\begin{center}

\begin{tabular}{ll}

$r_1$: & $a \cdot (b \cdot c)$\\

$r_2$: & $\ONE \cdot (b \cdot c)$\\

$r_3$: & $(\ZERO \cdot (b \cdot c)) + (\ONE \cdot c)$\\

$r_4$: & $(\ZERO \cdot (b \cdot c)) + ((\ZERO \cdot c) + \ONE)$\\

\end{tabular}

\end{center}

\noindent According to the simple algorithm, we would test

whether $r_4$ is nullable, which in this case it indeed is.

This means we can use the function $\textit{mkeps}$ to calculate a

value for how $r_4$ was able to match the empty string.

Remember that this function gives preference for alternatives

on the left-hand side. However there is only $\ONE$ on the

very right-hand side of $r_4$ (underlined)

\begin{center}

$r_4$:\;$(\ZERO \cdot (b \cdot c)) + 

         ((\ZERO \cdot c) + \underline{\ONE})$\\

\end{center}

\noindent

that matches the empty string.

Therefore $\textit{mkeps}$ returns the value

\begin{center}

$v_4:\;Right(Right(Empty))$

\end{center}

\noindent If there had been a $\ONE$ on the left, then

$\textit{mkeps}$ would have returned something of the form

$\Left(\ldots)$. The point is that from this value we can

directly read off which part of $r_4$ matched the empty

string: take the right-alternative first, and then the

right-alternative again, then you got to the $\ONE$.

Next we have to ``inject'' the last character, that is $c$ in

the running example, into this value $v_4$ in order to

calculate how $r_3$ could have matched the string $c$.

For this we call injection with $\inj(r_3, c, v_4)$.

According to the definition of $\inj$ we obtain

\begin{center}

$v_3:\;Right(Seq(Empty, Char(c)))$

\end{center}

\noindent This is the correct result, because $r_3$ needs

to use the right-hand alternative, and then $\ONE$ needs

to match the empty string and $c$ needs to match $c$.

Next we need to inject back the letter $b$ into $v_3$.

For this we call injection with $\inj(r_2, b, v_3)$.

This gives

\begin{center}

$v_2:\;Seq(Empty, Seq(Char(b), Char(c)))$

\end{center}

\noindent which is again the correct result for how $r_2$

matched the string $bc$. Finally we need to inject back the

letter $a$ into $v_2$ giving the final result.

For this we call injection with $\inj(r_1, a, v_2)$

and obtain

\begin{center}

$v_1:\;Seq(Char(a), Seq(Char(b), Char(c)))$

\end{center}

\noindent This value corresponds to how the regular expression $r_1$,

namely $a \cdot (b \cdot c)$, matched the string $abc$.

There are a few auxiliary functions that are of interest

when analysing this algorithm. One is called \emph{flatten},

written $|\_|$, which extracts the string ``underlying'' a 

value. It is defined recursively as

\begin{center}

\begin{tabular}{lcl}

  $|Empty|$     & $\dn$ & $[]$\\

  $|Char(c)|$   & $\dn$ & $[c]$\\

  $|\Left(v)|$   & $\dn$ & $|v|$\\

  $|Right(v)|$  & $\dn$ & $|v|$\\

  $|Seq(v_1,v_2)|$& $\dn$ & $|v_1| \,@\, |v_2|$\\

  $|Stars\,[v_1,\ldots ,v_n]|$ & $\dn$ & $|v_1| \,@\ldots @\, |v_n|$\\

\end{tabular}

\end{center}

\noindent Using flatten we can see what are the strings behind 

the values calculated by $\textit{mkeps}$ and $\inj$. In our running 

example:

\begin{center}

\begin{tabular}{ll}

$|v_4|$: & $[]$\\

$|v_3|$: & $c$\\

$|v_2|$: & $bc$\\

$|v_1|$: & $abc$

\end{tabular}

\end{center}

\noindent This indicates that $\inj$ indeed is injecting, or

adding, back a character into the value. 

The definition of $\inj$ might still be very puzzling and each clause

might look arbitrary, but there is in fact a relatively simple idea

behind it. Ultimately the $\inj$-functions is determined by the

derivative functions. For this consider one of the ``squares'' from

Figure~\ref{Sulz}:

  \begin{center}

  \begin{tikzpicture}[scale=2,node distance=1.2cm,

                      every node/.style={minimum size=7mm}]

  \node (r)  {$r$};

  \node (rd) [right=of r]{$r_{der}$};

  \draw[->,line width=1mm](r)--(rd) node[above,midway] {$\der\,c$};

  \node (vd) [below=of r2]{$v_{der}$};

  \draw[->,line width=1mm](rd) -- (vd);

  \node (v) [left=of vd] {$v$};

  \draw[->,line width=1mm](vd)--(v) node[below,midway] {$\inj\,c$};

  \draw[->,line width=0.5mm,dotted](r) -- (v) node[left,midway,red] {\bf ?};

  \end{tikzpicture}

  \end{center}

\noindent

The input to the $\inj$-function is $r$ (on the top left), $c$ (the

character to be injected) and $v_{der}$ (bottom right). The output is

the value $v$ for how the regular expression $r$ matched the

corresponding string. How does $\inj$ calculate this value? Well, it has

to analyse the value $v_{der}$ and transform it into the value $v$ for

the regular expression $r$. The value $v_{der}$ corresponds to the

$r_{der}$-regular expression which is the derivative of $r$. Remember

that $v_{der}$ is the value for how $r_{der}$ matches the corresponding string

where $c$ has been chopped off.

Let $r$ be $r_1 + r_2$. Then $r_{der}$

is of the form $(\der\,c\,r_1) + (\der\,c\,r_2)$. What are the possible

values corresponding to $r_{der}$? Well, they can be only of the form

$\Left(\ldots)$ and $\Right(\ldots)$. Therefore you have two

cases in the $\inj$ function -- one for $\Left$ and one for $\Right$.

\begin{center}

\begin{tabular}{l@{\hspace{1mm}}c@{\hspace{2mm}}l}

    $\inj\,(r_1 + r_2)\,c\,\,\Left(v)$ & $\dn$ & $\ldots$\\

    $\inj\,(r_1 + r_2)\,c\,\,\Right(v)$ & $\dn$ & $\ldots$\\

\end{tabular}

\end{center}

\noindent

Let's look next at the sequence case where $r = r_1 \cdot r_2$. What does the

derivative of $r$ look like? According to the definition it is: 

\begin{center}

  \begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}

    $\der\, c\, (r_1 \cdot r_2)$  & $\dn$  &  if $\nullable (r_1)$\\

    & & then $(\der\,c\,r_1) \cdot r_2 + \der\, c\, r_2$\\ 

    & & else $(\der\, c\, r_1) \cdot r_2$\\

  \end{tabular}  

\end{center}  

\noindent As you can see there is a derivative in the if-branch and another

in the else-branch. In the if-branch we have an alternative of the form 

$\_ + \_$. We already know what the possible values are for such a regular 

expression, namely $\Left$ or $\Right$. Therefore we have in $\inj$ the

two cases:

\begin{center}

  \begin{tabular}{l@{\hspace{1mm}}c@{\hspace{2mm}}l}

    $\inj\,(r_1 \cdot r_2)\,c\,\,\Left(Seq(v_1,v_2))$ & $\dn$  & $\ldots$\\

    $\inj\,(r_1 \cdot r_2)\,c\,\,Right(v)$ & $\dn$  & $\ldots$\\

\end{tabular}

\end{center}

\noindent

In the first case we even know that it is not just a $\Left$-value, but 

$\Left(Seq(\ldots))$, because the corresponding regular expression 

in the derivation is \mbox{$(\der\,c\,r_1) \cdot r_2$}. Hence we know 

it must be a $Seq$-value enclosed inside $\Left$. 

The third clause for $r_1\cdot r_2$ in the $\inj$-function

\begin{center}

  \begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}

    $\inj\,(r_1 \cdot r_2)\,c\,\,Seq(v_1,v_2)$ & $\dn$  & $\ldots$\\

  \end{tabular}

  \end{center}

\noindent corresponds to the else-branch in the derivative. In this

case we know the derivative is of the form $(\der\,c\,r_1) \cdot r_2$ and

therefore the value must be of the form $Seq(\ldots)$.

Hopefully the $inj$-function makes now more sense. I let you explain

the $r^*$ case. What do the derivative of $r^*$  and

the corresponding value look like? Does this explain the shape of 

the clause?

\begin{center}

  \begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l}

    $\inj\,(r^*)\,c\,\,Seq(v,Stars\,vs)$ & $\dn$  & $Stars(\inj\,r\,c\,v\,::\,vs)$\\

  \end{tabular}

\end{center}

\noindent If yes, you made sense of the left-hand sides of

the $\inj$-definition. 

How about the right-hand sides? Well, in the

$r^*$ case for example we have according to the square in the picture 

above a value $v_{der}$ which says how the derivative $r_{der}$

matched the string. Since the derivative is of the form

$(\der\,c\,r) \cdot (r^*)$ the corresponding value is of the

form $Seq(v, Stars\,vs)$. But for $r^*$ we are looking for a value

for the original (top left) regular expression --- so it cannot

be a value of the shape $Seq(\ldots, Stars\ldots)$ because that is the

shape for sequence regular expressions. What we need is a value

of the form $Stars \ldots$ (remember the correspondence between 

values and regular expressions). This suggests to define the right hand 

side as

\begin{center}

$\ldots \dn Stars(\inj\,r\,c\,v\,::\,vs)$

\end{center}  

\noindent It is a value of the right shape, namely $Stars$. It injected 

$c$ into the first-value, which is in fact the value where we need to 

undo the derivative. Remember again the shape of the derivative of $r^*$.

In place where we chopped off the $c$, we now need to do the $\inj$ of $c$.

Therefore $\inj\,r\,c\,v$ in the definition above. That is the same with

the other clauses in $\inj$. 

Phew\ldots{}Sweat\ldots!\#@$\skull$\%\ldots Unfortunately, there is

a gigantic problem with the described algorithm so far: it is very

slow. We need to include in all this the simplification from Lecture

2. And what rotten luck: simplification messes things up and we need

to rectify the mess. This is what we shall do next.

\subsubsection*{Simplification}

Generally the matching algorithms based on derivatives do

poorly unless the regular expressions are simplified after

each derivative step. But this is a bit more involved in the

algorithm of Sulzmann \& Lu. So what follows might require you

to read several times before it makes sense and also might

require that you do some example calculations yourself. As a

first example consider the last derivation step in our earlier

example:

\begin{equation}\label{regexfour}

r_4 = \der\,c\,r_3 = 

(\ZERO \cdot (b \cdot c)) + ((\ZERO \cdot c) + \ONE)

\end{equation}

\noindent Simplifying this regular expression would just give

us $\ONE$. Running $\textit{mkeps}$ with this $\ONE$ as

input, however, would give us with the value $Empty$ instead of

\[Right(Right(Empty))\]

\noindent

that was obtained  without the simplification. The problem  is we need

to  recreate this  more  complicated value,  rather  than just  return

$Empty$. This will require what I call \emph{rectification functions}.

They  need  to  be  calculated  whenever  a  regular  expression  gets

simplified.

Rectification functions take a value as argument and return a

(rectified) value. In the example above the argument would be $Empty$

and the output $Right(Right(Empty))$.  Before we define these

rectification functions in general, let us first take a look again at

our simplification rules:

\begin{center}

\begin{tabular}{l@{\hspace{2mm}}c@{\hspace{2mm}}l}

$r \cdot \ZERO$ & $\mapsto$ & $\ZERO$\\ 

$\ZERO \cdot r$ & $\mapsto$ & $\ZERO$\\ 

$r \cdot \ONE$ & $\mapsto$ & $r$\\ 

$\ONE \cdot r$ & $\mapsto$ & $r$\\ 

$r + \ZERO$ & $\mapsto$ & $r$\\ 

$\ZERO + r$ & $\mapsto$ & $r$\\ 

$r + r$ & $\mapsto$ & $r$\\ 

\end{tabular}

\end{center}