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\section*{Coursework 2}

\noindent This coursework is worth 8\% and is due on \cwTWO{} 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. Please package everything in a zip-file that creates a
directory with the name \texttt{YournameYourfamilyname} on my end. Thanks!

\subsection*{Disclaimer\alert}

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 eleven syntactic entities:

\begin{enumerate}
\item keywords are

\begin{center}
\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{center} 

\item operators are:
\texttt{+}, 
\texttt{-}, 
\texttt{*}, 
\texttt{\%},
\texttt{/},
\texttt{==}, 
\texttt{!=}, 
\texttt{>}, 
\texttt{<},
\texttt{<=}, 
\texttt{>=},
\texttt{:=},
\texttt{\&\&},
\texttt{||}

\item letters are uppercase and lowercase

\item symbols are letters plus the characters
  \texttt{.},
  \texttt{\_},
  \texttt{>},
  \texttt{<},
  \texttt{=},
  \texttt{;},
  \texttt{,} and
  \texttt{:}

\item strings are enclosed by \texttt{"\ldots"} and consisting of
  symbols, whitespaces and digits
\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}
\item comments start with \texttt{//} and contain symbols, spaces and digits until the end of the line
\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
Figures~\ref{fib} -- \ref{collatz}. You can find the programms also on
KEATS. 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/while-tests/fib.while}}
\caption{Fibonacci program in the WHILE language.\label{fib}}
\end{figure}

\begin{figure}[h]
\mbox{\lstinputlisting[language=While,xleftmargin=10mm]{../progs/while-tests/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/while-tests/factors.while}}
\caption{A program that calculates factors for numbers in the WHILE
  language.\label{factors}}
\end{figure}

\begin{figure}[h]
\mbox{\lstinputlisting[language=While,xleftmargin=10mm]{../progs/while-tests/collatz2.while}}
\caption{A program that calculates the Collatz series for numbers
  between 1 and 100.\label{collatz}}
\end{figure}

\end{document}

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