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\begin{document}

\section*{Part 10 (Scala)}

\mbox{}\hfill\textit{``If there's one feature that makes Scala,}\\

\mbox{}\hfill\textit{`Scala', I would pick implicits.''}\smallskip\\

\mbox{}\hfill\textit{ --- Martin Odersky (creator of the Scala language)}\bigskip\medskip

\noindent

This part is about a small programming

language called brainf***. The part is worth 10\% and you need to 

submit on \cwTEN{} at 4pm.\bigskip

\IMPORTANT{}

\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***}. You

are asked in this part to implement an interpreter for

this language.

Urban M\"uller developed 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, brainf*** has 8 single-character commands, namely

\texttt{'>'}, \texttt{'<'}, \texttt{'+'}, \texttt{'-'}, \texttt{'.'},

\texttt{','}, \texttt{'['} and \texttt{']'}. Every other character is

considered a comment.  Brainf*** operates on memory cells containing

integers. For this it uses a single memory pointer that points at each

stage to one memory cell. 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 commands for input/output

are \texttt{','} and \texttt{'.'}. 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. Input works the other way, taking some

user input and storing it in the cell to which the memory pointer

points to. 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 ;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[1 Mark]

\item[(2)] Brainf*** 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***. 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 brainf*** 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

  brainf*** 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** 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}\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   

      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}

  \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 look 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]}