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\section*{Coursework 2 (Strand 1)}

\noindent This coursework is worth 5\% and is due on 4

November at 18:00. You are asked to implement the Sulzmann \&

Lu lexer for the WHILE language. You can do the

implementation in any programming language you like, but you

need to submit the source code with which you answered the

questions, otherwise a mark of 0\% will be awarded. You can

submit your answers in a txt-file or as pdf. Code submit as 

code.

\subsection*{Disclaimer}

It should be understood that the work you submit represents

your own effort. You have not copied from anyone else. An

exception is the Scala code from KEATS and the code I showed

during the lectures, which you can both freely use. You can

also use your own code from the CW~1.

\subsection*{Question 1}

To implement a lexer for the WHILE language, you first

need to design the appropriate regular expressions for the

following eight syntactic entities:

\begin{enumerate}

\item keywords are

\begin{quote}

\texttt{while}, 

\texttt{if}, 

\texttt{then}, 

\texttt{else}, 

\texttt{do}, 

\texttt{for}, 

\texttt{to}, 

\texttt{true}, 

\texttt{false}, 

\texttt{read}, 

\texttt{write},

\texttt{skip}

\end{quote} 

\item operators are

\begin{quote}

\texttt{+}, 

\texttt{-}, 

\texttt{*}, 

\texttt{\%},

\texttt{/},

\texttt{==}, 

\texttt{!=}, 

\texttt{>}, 

\texttt{<}, 

\texttt{:=},

\texttt{\&\&},

\texttt{||}

\end{quote} 

\item strings are enclosed by \texttt{"\ldots"} 

\item parentheses are \texttt{(}, \texttt{\{}, \texttt{)} and \texttt{\}}

\item there are semicolons \texttt{;}

\item whitespaces are either \texttt{" "} (one or more) or \texttt{$\backslash$n} or

  \texttt{$\backslash$t}

\item identifiers are letters followed by underscores \texttt{\_\!\_}, letters

or digits

\item numbers are \pcode{0}, \pcode{1}, \ldots and so on; give 

a regular expression that can recognise \pcode{0}, but not numbers 

with leading zeroes, such as \pcode{001}

\end{enumerate}

\noindent

You can use the basic regular expressions 

\[

\ZERO,\; \ONE,\; c,\; r_1 + r_2,\; r_1 \cdot r_2,\; r^*

\]

\noindent

but also the following extended regular expressions

\begin{center}

\begin{tabular}{ll}

$[c_1,c_2,\ldots,c_n]$ & a set of characters\\

$r^+$ & one or more times $r$\\

$r^?$ & optional $r$\\

$r^{\{n\}}$ & n-times $r$\\

\end{tabular}

\end{center}

\noindent

Later on you will also need the record regular expression:

\begin{center}

\begin{tabular}{ll}

$REC(x:r)$ & record regular expression\\

\end{tabular}

\end{center}

\noindent Try to design your regular expressions to be as

small as possible. For example you should use character sets

for identifiers and numbers. Feel free to use the general

character constructor \textit{CFUN} introduced in CW 1.

\subsection*{Question 2}

Implement the Sulzmann \& Lu lexer from the lectures. For

this you need to implement the functions $nullable$ and $der$

(you can use your code from CW~1), as well as $mkeps$ and

$inj$. These functions need to be appropriately extended for

the extended regular expressions from Q1. Write down the 

clauses for

\begin{center}

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

$mkeps([c_1,c_2,\ldots,c_n])$  & $\dn$ & $?$\\

$mkeps(r^+)$                   & $\dn$ & $?$\\

$mkeps(r^?)$                   & $\dn$ & $?$\\

$mkeps(r^{\{n\}})$             & $\dn$ & $?$\medskip\\

$inj\, ([c_1,c_2,\ldots,c_n])\,c\,\ldots$  & $\dn$ & $?$\\

$inj\, (r^+)\,c\,\ldots$                   & $\dn$ & $?$\\

$inj\, (r^?)\,c\,\ldots$                   & $\dn$ & $?$\\

$inj\, (r^{\{n\}})\,c\,\ldots$             & $\dn$ & $?$\\

\end{tabular}

\end{center}

\noindent where $inj$ takes three arguments: a regular

expression, a character and a value. Test your lexer code

with at least the two small examples below:

\begin{center}

\begin{tabular}{ll}

regex: & string:\smallskip\\

$a^{\{3\}}$ & $aaa$\\

$(a + \ONE)^{\{3\}}$ & $aa$

\end{tabular}

\end{center}

\noindent Both strings should be successfully lexed by the

respective regular expression, that means the lexer returns 

in both examples a value.

Also add the record regular expression from the

lectures to your lexer and implement a function, say

\pcode{env}, that returns all assignments from a value (such

that you can extract easily the tokens from a value).\medskip 

\noindent

Finally give the tokens for your regular expressions from Q1 and the

string

\begin{center}

\code{"read n;"}

\end{center} 

\noindent

and use your \pcode{env} function to give the token sequence.

\subsection*{Question 3}

Extend your lexer from Q2 to also simplify regular expressions

after each derivation step and rectify the computed values after each

injection. Use this lexer to tokenize the programs in

Figure~\ref{fib}, \ref{loop} and \ref{factors}. Give the tokens of these

programs where whitespaces are filtered out. Make sure you can

tokenise \textbf{exactly} these programs.\bigskip

\begin{figure}[h]

\mbox{\lstinputlisting[language=While,xleftmargin=10mm]{../progs/fib.while}}

\caption{Fibonacci program in the WHILE language.\label{fib}}

\end{figure}

\begin{figure}[h]

\mbox{\lstinputlisting[language=While,xleftmargin=10mm]{../progs/loops.while}}

\caption{The three-nested-loops program in the WHILE language. 

(Usually used for timing measurements.)\label{loop}}

\end{figure}

\begin{figure}[h]

\mbox{\lstinputlisting[language=While,xleftmargin=10mm]{../progs/factors.while}}

\caption{A program that calculates factors for numbers in the WHILE

  language.\label{factors}}

\end{figure}

\end{document}

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