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\begin{document}

\section*{Main Part 5 (Scala, 10 Marks)}

\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***. We will implement an interpreter and compiler for

this language.\bigskip

%\IMPORTANT{This part is worth 10\% and you need to submit it on \cwTEN{} at 5pm.

%Any 1\% you achieve counts as your ``weekly engagement''.}

\IMPORTANTNONE{}

\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{M5a} and \texttt{M5b},

respectively. For example

\begin{lstlisting}[language={},xleftmargin=1mm,numbers=none,basicstyle=\ttfamily\small]

$ scala -cp bf.jar

scala> import M5a._  

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 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 and time.

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{']'}.  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)} 

\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[0.5 Marks]

\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[0.5 Marks]

\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-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}

\item[(5)] Let us also generate some BF programs ourselves: 

  Write a function

  \texttt{generate} which takes a list of characters as input and generates

  a corresponding BF program that prints out this list of characters. One

  way to generate such a program is to consider each character in the list,

  add as many \texttt{"+"} given by the ASCII code of the character, then add the

  \texttt{"."} command for printing and add the loop \texttt{"[-]"} for ``zero-ing'' the memory

  cell; then go to the next character in the list. For example

  the list \texttt{"ABC".toString} produces (as a single string)

  \begin{center}

    \texttt{+++++++++++++++++++++++++++++++++++++++++++++++++++++}

    \texttt{++++++++++++.[-]+++++++++++++++++++++++++++++++++++++}

    \texttt{+++++++++++++++++++++++++++++.[-]++++++++++++++++++++}

    \texttt{+++++++++++++++++++++++++++++++++++++++++++++++.[-]}

  \end{center}\mbox{}\hfill[1 Mark]

\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[(6)] 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[(7)] 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