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-% !TEX program = xelatex
-\documentclass{article}
-\usepackage{../style}
-\usepackage{../langs}
-\usepackage{disclaimer}
-\usepackage{tikz}
-\usepackage{pgf}
-\usepackage{pgfplots}
-\usepackage{stackengine}
-%% \usepackage{accents}
-\newcommand\barbelow[1]{\stackunder[1.2pt]{#1}{\raisebox{-4mm}{\boldmath$\uparrow$}}}
-
-%% change Console to scala.io.StdIn.readByte()
-
-
-\begin{document}
-
-\section*{Part 10 (Scala)}
-
-\mbox{}\hfill\textit{``If there's one feature that makes Scala, `Scala',}\\
-\mbox{}\hfill\textit{ I would pick implicits.''}\smallskip\\
-\mbox{}\hfill\textit{ --- Martin Odersky (creator of the Scala language)}\bigskip\bigskip
-
-
-\noindent
-This part is about a small (esoteric) programming language called
-brainf***. Actually, we will implement an interpreter for our own version
-of this language called brainf*ck++.\bigskip
-
-\IMPORTANT{This part is worth 10\% and you need to submit it on \cwTEN{} at 4pm.}
-
-\noindent
-Also note that the running time of each part will be restricted to a
-maximum of 30 seconds on my laptop.
-
-\DISCLAIMER{}
-\newpage
-
-\subsection*{Reference Implementation}
-
-As usual, this Scala assignment comes with a reference implementation in
-form of two \texttt{jar}-files. You can download them from KEATS. They
-allow you to run any test cases on your own computer. For example you
-can call Scala on the command line with the option \texttt{-cp bf.jar}
-and then query any function from the \texttt{bf.scala} template file.
-You have to prefix the calls with \texttt{CW10a} and \texttt{CW10b},
-respectively. For example
-
-
-\begin{lstlisting}[language={},xleftmargin=1mm,numbers=none,basicstyle=\ttfamily\small]
-$ scala -cp bf.jar
-scala> import CW10a._  
-scala> run(load_bff("sierpinski.bf")) ; ()
-                               *
-                              * *
-                             *   *
-                            * * * *
-                           *       *
-                          * *     * *
-                         *   *   *   *
-                        * * * * * * * *
-                       *               *
-                      * *             * *
-                     *   *           *   *
-                    * * * *         * * * *
-                   *       *       *       *
-                  * *     * *     * *     * *
-                 *   *   *   *   *   *   *   *
-                * * * * * * * * * * * * * * * *
-               *                               *
-              * *                             * *
-             *   *                           *   *
-            * * * *                         * * * *
-           *       *                       *       *
-          * *     * *                     * *     * *
-         *   *   *   *                   *   *   *   *
-        * * * * * * * *                 * * * * * * * *
-       *               *               *               *
-      * *             * *             * *             * *
-     *   *           *   *           *   *           *   *
-    * * * *         * * * *         * * * *         * * * *
-   *       *       *       *       *       *       *       *
-  * *     * *     * *     * *     * *     * *     * *     * *
- *   *   *   *   *   *   *   *   *   *   *   *   *   *   *   *
-* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
-\end{lstlisting}%$
-
-\newpage
-
-\subsection*{Part A (6 Marks)}
-
-Coming from Java or C++, you might think Scala is a rather esoteric
-programming language.  But remember, some serious companies have built
-their business on
-Scala.\footnote{\url{https://en.wikipedia.org/wiki/Scala_(programming_language)\#Companies}}
-I claim functional programming is not a fad.  And there are far, far
-more esoteric languages out there. One is called \emph{brainf***}. 
-\here{https://esolangs.org/wiki/Brainfuck}
-You
-are asked in this part to implement an interpreter for
-a slight extension of this language.
-
-Urban M\"uller developed the original version of brainf*** in 1993.  A close
-relative of this language was already introduced in 1964 by Corado
-B\"ohm, an Italian computer pioneer. The main feature of brainf*** is
-its minimalistic set of instructions---just 8 instructions in total
-and all of which are single characters. Despite the minimalism, this
-language has been shown to be Turing complete\ldots{}if this doesn't
-ring any bell with you: it roughly means that every(!) algorithm can,
-in principle, be implemented in brainf***. It just takes a lot of
-determination and quite a lot of memory resources.
-
-Some relatively sophisticated sample programs in brainf*** are given
-in the file \texttt{bf.scala}, including a brainf*** program for the
-Sierpinski triangle and the Mandelbrot set.  There seems to be even a
-dedicated Windows IDE for bf programs, though I am not sure whether
-this is just an elaborate April fools' joke---judge yourself:
-
-\begin{center}
-\url{https://www.microsoft.com/en-us/p/brainf-ck/9nblgggzhvq5}
-\end{center}  \bigskip
-
-
-\noindent
-As mentioned above, the original brainf*** has 8 single-character
-commands. Our version of bf++ will contain the commands \texttt{'>'},
-\texttt{'<'}, \texttt{'+'}, \texttt{'-'}, \texttt{'.'}, \texttt{'['}
-and \texttt{']'} from the original, and in addition the commands
-\texttt{'@'}, \texttt{'*'} and \texttt{'\#'}.  Every other character
-is considered a comment.
-
-Our interpreter for bf++ operates on memory cells containing
-integers. For this it uses a single memory pointer, called
-\texttt{mp}, that points at each stage to one memory cell.
-
-\begin{center}
-\begin{tikzpicture}
-  \draw [line width=1mm, rounded corners] (0,0) rectangle (5, 0.5);
-  \draw (0.5, 0) -- (0.5, 0.5);
-  \draw (1.0, 0) -- (1.0, 0.5);
-
-  \draw (2.5, 0) -- (2.5, 0.5);
-  \draw (2.0, 0) -- (2.0, 0.5);
-
-  \draw (4.5, 0) -- (4.5, 0.5);
-  \draw (4.0, 0) -- (4.0, 0.5);
-
-  \draw (1.5,0.25) node {$\cdots$};
-  \draw (3.0,0.25) node {$\cdots$};
-
-  \draw [->, thick] (2.25, -0.5) -- (2.25, -0.15);
-  \draw (2.25,-0.8) node {\texttt{mp}};
-
-  \draw (0.7,0.7) node {\sf\footnotesize memory};
-\end{tikzpicture}    
-\end{center}  
-
-\noindent
-This pointer can be moved forward by one memory cell by using the
-command \texttt{'>'}, and backward by using \texttt{'<'}. The commands
-\texttt{'+'} and \texttt{'-'} increase, respectively decrease, by 1
-the content of the memory cell to which the memory pointer currently
-points to. The command for output in bf++ is \texttt{'.'} whereby output works
-by reading the content of the memory cell to which the memory pointer
-points to and printing it out as an ASCII character.\footnote{In the
-  original version of bf, there is also a command for input, but we
-  omit it here. All our programs will be ``autonomous''.}  The
-commands \texttt{'['} and \texttt{']'} are looping
-constructs. Everything in between \texttt{'['} and \texttt{']'} is
-repeated until a counter (memory cell) reaches zero.  A typical
-program in brainf*** looks as follows:
-
-\begin{center}
-\begin{verbatim}
-   ++++++++[>++++[>++>+++>+++>+<<<<-]>+>+>->>+[<]<-]>>.>---.++
-   +++++..+++.>>.<-.<.+++.------.--------.>>+.>++.
-\end{verbatim}
-\end{center}  
-
-\noindent
-This one prints out Hello World\ldots{}obviously \texttt{;o)} We also
-add 3 new commands in the bf++-version of the bf-language. The purpose
-of these commands we explain later.
-
-
-\subsubsection*{Tasks (file bf.scala)}
-
-\begin{itemize}
-\item[(1)]  Write a function that takes a filename (a string) as an argument
-  and requests the corresponding file from disk. It returns the
-  content of the file as a string. If the file does not exists,
-  the function should return the empty string.
-  \mbox{}\hfill[1 Mark]
-  
-\item[(2)] Brainf**k++ memory is represented by a \texttt{Map} from
-  integers to integers. The empty memory is represented by
-  \texttt{Map()}, that is nothing is stored in the
-  memory; \texttt{Map(0 -> 1, 2 -> 3)} stores \texttt{1} at
-  memory location \texttt{0}, and at \texttt{2} it stores \texttt{3}. The
-  convention is that if we query the memory at a location that is
-  \emph{not} defined in the \texttt{Map}, we return \texttt{0}. Write
-  a `safe-read' function, \texttt{sread}, that takes a memory (a \texttt{Map}) and
-  a memory pointer (an \texttt{Int}) as arguments, and `safely' reads the
-  corresponding memory location. If the \texttt{Map} is not defined at
-  the memory pointer, \texttt{sread} returns \texttt{0}.
-
-  Write another function \texttt{write}, which takes a memory, a
-  memory pointer and an integer value as arguments and updates the
-  \texttt{Map} with the value at the given memory location. As usual,
-  the \texttt{Map} is not updated `in-place' but a new map is created
-  with the same data, except the new value is stored at the given memory
-  pointer.\hfill[1 Mark]
-
-\item[(3)] Write two functions, \texttt{jumpRight} and
-  \texttt{jumpLeft}, that are needed to implement the loop constructs
-  in brainf**k++. They take a program (a \texttt{String}) and a program
-  counter (an \texttt{Int}) as arguments and move right (respectively
-  left) in the string in order to find the \textbf{matching}
-  opening/closing bracket. For example, given the following program
-  with the program counter indicated by an arrow:
-
-  \begin{center}
-  \texttt{--[\barbelow{.}.+>--].>.++}
-  \end{center}
-
-  then the matching closing bracket is in 9th position (counting from 0) and
-  \texttt{jumpRight} is supposed to return the position just after this
-  
-  \begin{center}
-  \texttt{--[..+>--]\barbelow{.}>.++}
-  \end{center}
-
-  meaning it jumps to after the loop. Similarly, if you are in 8th position,
-  then \texttt{jumpLeft} is supposed to jump to just after the opening
-  bracket (that is jumping to the beginning of the loop):
-
-  \begin{center}
-    \texttt{--[..+>-\barbelow{-}].>.++}
-    \qquad$\stackrel{\texttt{jumpLeft}}{\longrightarrow}$\qquad
-    \texttt{--[\barbelow{.}.+>--].>.++}
-  \end{center}
-
-  Unfortunately we have to take into account that there might be
-  other opening and closing brackets on the `way' to find the
-  matching bracket. For example in the brain*ck++ program
-
-  \begin{center}
-  \texttt{--[\barbelow{.}.[+>]--].>.++}
-  \end{center}
-
-  we do not want to return the index for the \texttt{'-'} in the 9th
-  position, but the program counter for \texttt{'.'} in 12th
-  position. The easiest to find out whether a bracket is matched is by
-  using levels (which are the third argument in \texttt{jumpLeft} and
-  \texttt{jumpLeft}). In case of \texttt{jumpRight} you increase the
-  level by one whenever you find an opening bracket and decrease by
-  one for a closing bracket. Then in \texttt{jumpRight} you are looking
-  for the closing bracket on level \texttt{0}. For \texttt{jumpLeft} you
-  do the opposite. In this way you can find \textbf{matching} brackets
-  in strings such as
-
-  \begin{center}
-  \texttt{--[\barbelow{.}.[[-]+>[.]]--].>.++}
-  \end{center}
-
-  for which \texttt{jumpRight} should produce the position:
-
-  \begin{center}
-  \texttt{--[..[[-]+>[.]]--]\barbelow{.}>.++}
-  \end{center}
-
-  It is also possible that the position returned by \texttt{jumpRight} or
-  \texttt{jumpLeft} is outside the string in cases where there are
-  no matching brackets. For example
-
-  \begin{center}
-  \texttt{--[\barbelow{.}.[[-]+>[.]]--.>.++}
-  \qquad$\stackrel{\texttt{jumpRight}}{\longrightarrow}$\qquad
-  \texttt{--[..[[-]+>[.]]-->.++\barbelow{\;\phantom{+}}}
-  \end{center}
-  \hfill[2 Marks]
-
-
-\item[(4)] Write a recursive function \texttt{compute} that runs a
-  brain*u*k++ program. It takes a program, a program counter, a memory
-  pointer and a memory as arguments. If the program counter is outside
-  the program string, the execution stops and \texttt{compute} returns the
-  memory. If the program counter is inside the string, it reads the
-  corresponding character and updates the program counter \texttt{pc},
-  memory pointer \texttt{mp} and memory \texttt{mem} according to the
-  rules shown in Figure~\ref{comms}. It then calls recursively
-  \texttt{compute} with the updated data. The most convenient way to
-  implement the brainf**k++ rules in Scala is to use pattern-matching
-  and to calculate a triple consisting of the updated \texttt{pc},
-  \texttt{mp} and \texttt{mem}.
-
-  Write another function \texttt{run} that calls \texttt{compute} with a
-  given brainfu*k++ program and memory, and the program counter and memory pointer
-  set to~$0$. Like \texttt{compute}, it returns the memory after the execution
-  of the program finishes. You can test your brainf**k++ interpreter with the
-  Sierpinski triangle or the Hello world programs (they seem to be particularly
-  useful for debugging purposes), or have a look at
-
-  \begin{center}
-  \url{https://esolangs.org/wiki/Brainfuck}
-  \end{center}
-
-  \noindent for more bf/bf++-programs and the test cases given in \texttt{bf.scala}.\\
-  \mbox{}\hfill[2 Marks]
-  
-  \begin{figure}[p]
-  \begin{center}
-    \begin{tabular}{|@{\hspace{0.5mm}}p{0.8cm}|l|}
-      \hline
-      \hfill\texttt{'>'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp} + 1$\\
-                       $\bullet$ & \texttt{mem} unchanged
-                     \end{tabular}\\\hline   
-      \hfill\texttt{'<'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp} - 1$\\
-                       $\bullet$ & \texttt{mem} unchanged
-                     \end{tabular}\\\hline   
-      \hfill\texttt{'+'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp}$ unchanged\\
-                       $\bullet$ & \texttt{mem} updated with \texttt{mp -> mem(mp) + 1}\\
-                     \end{tabular}\\\hline   
-      \hfill\texttt{'-'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp}$ unchanged\\
-                       $\bullet$ & \texttt{mem} updated with \texttt{mp -> mem(mp) - 1}\\
-                     \end{tabular}\\\hline   
-      \hfill\texttt{'.'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\
-                       $\bullet$ & print out \,\texttt{mem(mp)} as a character\\
-                     \end{tabular}\\\hline   
-      %\hfill\texttt{','} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-      %                 $\bullet$ & $\texttt{pc} + 1$\\
-      %                 $\bullet$ & $\texttt{mp}$ unchanged\\
-      %                 $\bullet$ & \texttt{mem} updated with \texttt{mp -> \textrm{input}}\\
-      %                 \multicolumn{2}{@{}l}{the input is given by \texttt{Console.in.read().toByte}}
-      %               \end{tabular}\\\hline   
-      \hfill\texttt{'['} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       \multicolumn{2}{@{}l}{if \texttt{mem(mp) == 0} then}\\
-                       $\bullet$ & $\texttt{pc = jumpRight(prog, pc + 1, 0)}$\\
-                       $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\medskip\\
-                       \multicolumn{2}{@{}l}{otherwise if \texttt{mem(mp) != 0} then}\\
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\
-                     \end{tabular}
-                     \\\hline   
-      \hfill\texttt{']'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       \multicolumn{2}{@{}l}{if \texttt{mem(mp) != 0} then}\\
-                       $\bullet$ & $\texttt{pc = jumpLeft(prog, pc - 1, 0)}$\\
-                       $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\medskip\\
-                       \multicolumn{2}{@{}l}{otherwise if \texttt{mem(mp) == 0} then}\\
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\
-                           \end{tabular}\\\hline
-      \hfill\texttt{'*'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp}$ unchanged\\
-                       $\bullet$ & \texttt{mem} updated with \texttt{mp -> mem(mp) * mem(mp - 1)}\\
-                             \multicolumn{2}{@{}l}{this multiplies the content of the memory cells at
-                             \texttt{mp} and \texttt{mp - 1}}\\
-                             \multicolumn{2}{@{}l}{and stores the result at \texttt{mp}}
-                           \end{tabular}\\\hline
-      \hfill\texttt{'@'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp}$ unchanged\\
-                             $\bullet$ & \texttt{mem} updated with
-                                         \texttt{mem(mp) -> mem(mp - 1)}\\
-                             \multicolumn{2}{@{}l}{this updates the memory cell having the index stored at \texttt{mem(mp)},}\\
-                             \multicolumn{2}{@{}l}{with the value stored at \texttt{mem(mp - 1)},}
-                           \end{tabular}\\\hline
-      \hfill\texttt{'\#'} & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                       $\bullet$ & $\texttt{pc} + 1$\\
-                       $\bullet$ & $\texttt{mp}$ and \texttt{mem} unchanged\\
-                       $\bullet$ & print out \,\texttt{mem(mp)} as a number\\
-                     \end{tabular}\\\hline  
-      any other char & \begin{tabular}[t]{@{}l@{\hspace{2mm}}l@{}}
-                         $\bullet$ & $\texttt{pc} + 1$\\
-                         $\bullet$ & \texttt{mp} and \texttt{mem} unchanged
-                       \end{tabular}\\
-      \hline                 
-    \end{tabular}
-    \\\mbox{}\\[-10mm]\mbox{}
-  \end{center}
-  \caption{The rules for how commands in the brainf***++ language update the
-    program counter \texttt{pc},
-    the memory pointer \texttt{mp} and the memory \texttt{mem}.\label{comms}}
-  \end{figure}
-\end{itemize}\bigskip  
-
-%%\newpage
-
-\subsection*{Part B (4 Marks)}
-
-I am sure you agree while it is fun to marvel at bf++-programs, like the
-Sierpinski triangle or the Mandelbrot program, being interpreted, it
-is much more fun to write a compiler for the bf++-language.
-
-
-\subsubsection*{Tasks (file bfc.scala)}
-
-\begin{itemize}
-\item[(5)] Compilers, in general, attempt to make programs run
-  faster by precomputing as much information as possible
-  before running the program. In our case we can precompute the
-  addresses where we need to jump at the beginning and end of
-  loops. 
-
-  For this write a function \texttt{jtable} that precomputes the ``jump
-  table'' for a bf++-program. This function takes a bf++-program 
-  as an argument and returns a \texttt{Map[Int, Int]}. The 
-  purpose of this Map is to record the information, in cases
-  a pc-position points to a '\texttt{[}' or a '\texttt{]}',
-  to which pc-position do we need to jump next?
- 
-  For example for the program
-    
-  \begin{center}
-    \texttt{+++++[->++++++++++<]>--<+++[->>++++++++++}
-    \texttt{<<]>>++<<----------[+>.>.<+<]}
-  \end{center}
-
-  we obtain the Map (note the precise numbers might differ depending on white
-  spaces etc.~in the bf-program):
-
-  \begin{center}
-  \texttt{Map(69 -> 61, 5 -> 20, 60 -> 70, 27 -> 44, 43 -> 28, 19 -> 6)}
-  \end{center}  
-  
-  This Map states that for the '\texttt{[}' on position 5, we need to
-  jump to position 20, which is just after the corresponding '\texttt{]}'.
-  Similarly, for the '\texttt{]}' on position 19, we need to jump to
-  position 6, which is just after the '\texttt{[}' on position 5, and so
-  on. The idea is to not calculate this information each time
-  we hit a bracket, but just look up this information in the 
-  \texttt{jtable}. 
-
-  Then adapt the \texttt{compute} and \texttt{run} functions
-  from Part 1 in order to take advantage of the information
-  stored in the \texttt{jtable}. This means whenever \texttt{jumpLeft}
-  and \texttt{jumpRight} was called previously, you should look
-  up the jump address in the \texttt{jtable}. Feel free to reuse
-  the function \texttt{jumpLeft} and \texttt{jumpRight} for
-  calculating the \texttt{jtable}.\hfill{[1 Mark]}
-
-\item[(6)] Compilers try to eliminate any ``dead'' code that could
-  slow down programs and also perform what is often called
-  \emph{peephole
-    optimisations}.\footnote{\url{https://en.wikipedia.org/wiki/Peephole_optimization}}
-  For the latter consider that it is difficult for compilers to
-  comprehend what is intended with whole programs, but they are very good
-  at finding out what small snippets of code do, and then try to
-  generate faster code for such snippets.
-
-  In our case, dead code is everything that is not a bf++-command.
-  Therefore write a function \texttt{optimise} which deletes such
-  dead code from a bf++-program. Moreover this function should replace every substring
-  of the form \pcode{[-]} by a new command \texttt{0}. 
-  The idea is that the loop \pcode{[-]} just resets the
-  memory at the current location to 0. It is more efficient
-  to do this in a single step, rather than stepwise in a loop as in
-  the original bf++-programs.
-
-  In the extended \texttt{compute3} and \texttt{run3} functions you should
-  implement this command by writing 0 to \pcode{mem(mp)}, that is use
-  \pcode{write(mem, mp, 0)} as the rule for the command \texttt{0}.
-  The easiest way to modify a string in this way is to use the regular
-  expression \pcode{"""[^<>+-.\\[\\]@\#*]"""}, which recognises everything that is 
-  not a bf++-command. Similarly, the
-  regular expression \pcode{"""\\[-\\]"""} finds all occurrences of \pcode{[-]}.  By using the Scala method \pcode{.replaceAll} you can replace substrings
-  with new strings.\\
-  \mbox{}\hfill{[1 Mark]}
-
-\item[(7)] Finally, real compilers try to take advantage of modern
-  CPUs which often provide complex operations in hardware that can
-  combine many smaller instructions into a single faster instruction.
-
-  In our case we can optimise the several single increments performed at a
-  memory cell, for example \pcode{++++}, by a single ``increment by
-  4''. For this optimisation we just have to make sure these single
-  increments are all next to each other. Similar optimisations should apply
-  for the bf-commands \pcode{-}, \pcode{<} and
-  \pcode{>}, which can all be replaced by extended versions that take
-  the amount of the increment (decrement) into account. We will do
-  this by introducing two-character bf++-commands. For example
-
-  \begin{center}
-    \begin{tabular}{l|l}
-      original bf-cmds & replacement\\
-      \hline
-      \pcode{+} & \pcode{+A}\\
-      \pcode{++} & \pcode{+B}\\
-      \pcode{+++} & \pcode{+C}\\
-      \ldots{} & \ldots{}\\
-      \pcode{+++....++} & \pcode{+Z}\\
-      \hspace{5mm}(these are 26 \pcode{+}'s)\\
-    \end{tabular} 
-  \end{center}  
-
-
-  If there are more
-  than 26 \pcode{+}'s in a row, then more than one ``two-character''
-  bf-commands need to be generated (the idea is that more than
-  26 copies of a single bf++-command in a row is a rare occurrence in
-  actual bf++-programs). Similar replacements apply
-  for \pcode{-}, \pcode{<} and \pcode{>}, but
-  all other bf++-commands should be unaffected by this
-  change. 
-
-  For this write a function \texttt{combine} which replaces sequences
-  of repeated increment and decrement commands by appropriate
-  two-character commands. In the functions \pcode{compute4} and
-  \pcode{run4}, the ``combine'' and the optimisation from (6) should
-  be performed. Make sure that when a two-character bf++-command is
-  encountered you need to increase the \pcode{pc}-counter by two in
-  order to progress to the next command. For example
-
-  \begin{center}
-  \pcode{combine(optimise(load_bff("benchmark.bf")))}  
-  \end{center}  
-
-  generates the improved program
-
-  \begin{center}
-  \pcode{>A+B[<A+M>A-A]<A[[}\hspace{3mm}\ldots{}
-  \end{center}  
-
-  for the original benchmark program
-
-  \begin{center}
-    \pcode{>++[<+++++++++++++>-]<[[}\hspace{3mm}\ldots
-  \end{center}    
-
-  As you can see, the compiler bets on saving a lot of time on the
-  \pcode{+B} and \pcode{+M} steps so that the optimisations is
-  worthwhile overall (of course for the \pcode{>A}'s and so on, the compiler incurs a
-  penalty). Luckily, after you have performed all
-  optimisations in (5) - (7), you can expect that the
-  \pcode{benchmark.bf} program runs four to five times faster.
-  You can also test whether your compiler produces the correct result
-  by  testing for example
-
-  \begin{center}
-  \pcode{run(load_bff("sierpinski.bf")) == run4(load_bff("sierpinski.bf"))}
-  \end{center}
-
-  which should return true for all the different compiler stages. \\ 
-  \mbox{}\hfill{[2 Marks]}
-\end{itemize}  
-
-\end{document}
-
-
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