
diff -r e1e3e259e9b7 -r cfcaf4a5e5b4 cws/cw05.tex
--- a/cws/cw05.tex	Sat Dec 01 15:09:37 2018 +0000
+++ b/cws/cw05.tex	Thu Dec 06 13:15:28 2018 +0000
@@ -9,84 +9,6 @@
 %% \usepackage{accents}
 \newcommand\barbelow[1]{\stackunder[1.2pt]{#1}{\raisebox{-4mm}{\boldmath$\uparrow$}}}
 
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-\end{filecontents}
-
 
 \begin{document}
 
@@ -95,16 +17,10 @@
   
 \section*{Coursework 9 (Scala)}
 
-This coursework is worth 10\%. It is about a regular expression
-matcher and the shunting yard algorithm by Dijkstra. The first part is
-due on 6 December at 11pm; the second, more advanced part, is due on
-21 December at 11pm. In the first part, you are asked to implement a
-regular expression matcher based on derivatives of regular
-expressions. The reason is that regular expression matching in
-languages like Java, JavaScipt and Python can sometimes be extremely
-slow. The advanced part is about the shunting yard algorithm that
-transforms the usual infix notation of arithmetic expressions into the
-postfix notation, which is for example used in compilers.\bigskip
+This coursework is worth 10\%. It is about a small programming
+language called brainf***. The first part is due on 13 December at
+11pm; the second, more advanced part, is due on 20 December at
+11pm.\bigskip
 
 \IMPORTANT{}
 
@@ -115,357 +31,28 @@
 \DISCLAIMER{}
 
 
-\subsection*{Part 1 (6 Marks, Regular Expression Matcher)}
-
-The task is to implement a regular expression matcher that is based on
-derivatives of regular expressions. Most of the functions are defined by
-recursion over regular expressions and can be elegantly implemented
-using Scala's pattern-matching. The implementation should deal with the
-following regular expressions, which have been predefined in the file
-\texttt{re.scala}:
-
-\begin{center}
-\begin{tabular}{lcll}
-  $r$ & $::=$ & $\ZERO$     & cannot match anything\\
-      &   $|$ & $\ONE$      & can only match the empty string\\
-      &   $|$ & $c$         & can match a single character (in this case $c$)\\
-      &   $|$ & $r_1 + r_2$ & can match a string either with $r_1$ or with $r_2$\\
-  &   $|$ & $r_1\cdot r_2$ & can match the first part of a string with $r_1$ and\\
-          &  & & then the second part with $r_2$\\
-      &   $|$ & $r^*$       & can match a string with zero or more copies of $r$\\
-\end{tabular}
-\end{center}
-
-\noindent 
-Why? Knowing how to match regular expressions and strings will let you
-solve a lot of problems that vex other humans. Regular expressions are
-one of the fastest and simplest ways to match patterns in text, and
-are endlessly useful for searching, editing and analysing data in all
-sorts of places (for example analysing network traffic in order to
-detect security breaches). However, you need to be fast, otherwise you
-will stumble over problems such as recently reported at
-
-{\small
-\begin{itemize}
-\item[$\bullet$] \url{http://stackstatus.net/post/147710624694/outage-postmortem-july-20-2016}
-\item[$\bullet$] \url{https://vimeo.com/112065252}
-\item[$\bullet$] \url{http://davidvgalbraith.com/how-i-fixed-atom/}  
-\end{itemize}}
-
-\subsubsection*{Tasks (file re.scala)}
-
-The file \texttt{re.scala} has already a definition for regular
-expressions and also defines some handy shorthand notation for
-regular expressions. The notation in this document matches up
-with the code in the file as follows:
-
-\begin{center}
-  \begin{tabular}{rcl@{\hspace{10mm}}l}
-    & & code: & shorthand:\smallskip \\ 
-  $\ZERO$ & $\mapsto$ & \texttt{ZERO}\\
-  $\ONE$  & $\mapsto$ & \texttt{ONE}\\
-  $c$     & $\mapsto$ & \texttt{CHAR(c)}\\
-  $r_1 + r_2$ & $\mapsto$ & \texttt{ALT(r1, r2)} & \texttt{r1 | r2}\\
-  $r_1 \cdot r_2$ & $\mapsto$ & \texttt{SEQ(r1, r2)} & \texttt{r1 $\sim$ r2}\\
-  $r^*$ & $\mapsto$ &  \texttt{STAR(r)} & \texttt{r.\%}
-\end{tabular}    
-\end{center}  
-
-
-\begin{itemize}
-\item[(1)] Implement a function, called \textit{nullable}, by
-  recursion over regular expressions. This function tests whether a
-  regular expression can match the empty string. This means given a
-  regular expression it either returns true or false. The function
-  \textit{nullable}
-  is defined as follows:
-
-\begin{center}
-\begin{tabular}{lcl}
-$\textit{nullable}(\ZERO)$ & $\dn$ & $\textit{false}$\\
-$\textit{nullable}(\ONE)$  & $\dn$ & $\textit{true}$\\
-$\textit{nullable}(c)$     & $\dn$ & $\textit{false}$\\
-$\textit{nullable}(r_1 + r_2)$ & $\dn$ & $\textit{nullable}(r_1) \vee \textit{nullable}(r_2)$\\
-$\textit{nullable}(r_1 \cdot r_2)$ & $\dn$ & $\textit{nullable}(r_1) \wedge \textit{nullable}(r_2)$\\
-$\textit{nullable}(r^*)$ & $\dn$ & $\textit{true}$\\
-\end{tabular}
-\end{center}~\hfill[1 Mark]
-
-\item[(2)] Implement a function, called \textit{der}, by recursion over
-  regular expressions. It takes a character and a regular expression
-  as arguments and calculates the derivative regular expression according
-  to the rules:
-
-\begin{center}
-\begin{tabular}{lcl}
-$\textit{der}\;c\;(\ZERO)$ & $\dn$ & $\ZERO$\\
-$\textit{der}\;c\;(\ONE)$  & $\dn$ & $\ZERO$\\
-$\textit{der}\;c\;(d)$     & $\dn$ & $\textit{if}\; c = d\;\textit{then} \;\ONE \; \textit{else} \;\ZERO$\\
-$\textit{der}\;c\;(r_1 + r_2)$ & $\dn$ & $(\textit{der}\;c\;r_1) + (\textit{der}\;c\;r_2)$\\
-$\textit{der}\;c\;(r_1 \cdot r_2)$ & $\dn$ & $\textit{if}\;\textit{nullable}(r_1)$\\
-      & & $\textit{then}\;((\textit{der}\;c\;r_1)\cdot r_2) + (\textit{der}\;c\;r_2)$\\
-      & & $\textit{else}\;(\textit{der}\;c\;r_1)\cdot r_2$\\
-$\textit{der}\;c\;(r^*)$ & $\dn$ & $(\textit{der}\;c\;r)\cdot (r^*)$\\
-\end{tabular}
-\end{center}
-
-For example given the regular expression $r = (a \cdot b) \cdot c$, the derivatives
-w.r.t.~the characters $a$, $b$ and $c$ are
-
-\begin{center}
-  \begin{tabular}{lcll}
-    $\textit{der}\;a\;r$ & $=$ & $(\ONE \cdot b)\cdot c$ & \quad($= r'$)\\
-    $\textit{der}\;b\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$\\
-    $\textit{der}\;c\;r$ & $=$ & $(\ZERO \cdot b)\cdot c$
-  \end{tabular}
-\end{center}
-
-Let $r'$ stand for the first derivative, then taking the derivatives of $r'$
-w.r.t.~the characters $a$, $b$ and $c$ gives
-
-\begin{center}
-  \begin{tabular}{lcll}
-    $\textit{der}\;a\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$ \\
-    $\textit{der}\;b\;r'$ & $=$ & $((\ZERO \cdot b) + \ONE)\cdot c$ & \quad($= r''$)\\
-    $\textit{der}\;c\;r'$ & $=$ & $((\ZERO \cdot b) + \ZERO)\cdot c$
-  \end{tabular}
-\end{center}
-
-One more example: Let $r''$ stand for the second derivative above,
-then taking the derivatives of $r''$ w.r.t.~the characters $a$, $b$
-and $c$ gives
-
-\begin{center}
-  \begin{tabular}{lcll}
-    $\textit{der}\;a\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$ \\
-    $\textit{der}\;b\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ZERO$\\
-    $\textit{der}\;c\;r''$ & $=$ & $((\ZERO \cdot b) + \ZERO) \cdot c + \ONE$ &
-    (is $\textit{nullable}$)                      
-  \end{tabular}
-\end{center}
-
-Note, the last derivative can match the empty string, that is it is \textit{nullable}.\\
-\mbox{}\hfill\mbox{[1 Mark]}
-
-\item[(3)] Implement the function \textit{simp}, which recursively
-  traverses a regular expression from the inside to the outside, and
-  on the way simplifies every regular expression on the left (see
-  below) to the regular expression on the right, except it does not
-  simplify inside ${}^*$-regular expressions.
-
-  \begin{center}
-\begin{tabular}{l@{\hspace{4mm}}c@{\hspace{4mm}}ll}
-$r \cdot \ZERO$ & $\mapsto$ & $\ZERO$\\ 
-$\ZERO \cdot r$ & $\mapsto$ & $\ZERO$\\ 
-$r \cdot \ONE$ & $\mapsto$ & $r$\\ 
-$\ONE \cdot r$ & $\mapsto$ & $r$\\ 
-$r + \ZERO$ & $\mapsto$ & $r$\\ 
-$\ZERO + r$ & $\mapsto$ & $r$\\ 
-$r + r$ & $\mapsto$ & $r$\\ 
-\end{tabular}
-  \end{center}
-
-  For example the regular expression
-  \[(r_1 + \ZERO) \cdot \ONE + ((\ONE + r_2) + r_3) \cdot (r_4 \cdot \ZERO)\]
 
-  simplifies to just $r_1$. \textbf{Hint:} Regular expressions can be
-  seen as trees and there are several methods for traversing
-  trees. One of them corresponds to the inside-out traversal, which is
-  sometimes also called post-order traversal'' you traverse inside the
-  tree and on the way up, you apply simplification rules.
-  Furthermore,
-  remember numerical expressions from school times: there you had expressions
-  like $u + \ldots + (1 \cdot x) - \ldots (z + (y \cdot 0)) \ldots$
-  and simplification rules that looked very similar to rules
-  above. You would simplify such numerical expressions by replacing
-  for example the $y \cdot 0$ by $0$, or $1\cdot x$ by $x$, and then
-  look whether more rules are applicable. If you organise the
-  simplification in an inside-out fashion, it is always clear which
-  rule should be applied next.\hfill[1 Mark]
-
-\item[(4)] Implement two functions: The first, called \textit{ders},
-  takes a list of characters and a regular expression as arguments, and
-  builds the derivative w.r.t.~the list as follows:
-
-\begin{center}
-\begin{tabular}{lcl}
-$\textit{ders}\;(Nil)\;r$ & $\dn$ & $r$\\
-  $\textit{ders}\;(c::cs)\;r$  & $\dn$ &
-    $\textit{ders}\;cs\;(\textit{simp}(\textit{der}\;c\;r))$\\
-\end{tabular}
-\end{center}
-
-Note that this function is different from \textit{der}, which only
-takes a single character.
-
-The second function, called \textit{matcher}, takes a string and a
-regular expression as arguments. It builds first the derivatives
-according to \textit{ders} and after that tests whether the resulting
-derivative regular expression can match the empty string (using
-\textit{nullable}).  For example the \textit{matcher} will produce
-true for the regular expression $(a\cdot b)\cdot c$ and the string
-$abc$, but false if you give it the string $ab$. \hfill[1 Mark]
-
-\item[(5)] Implement a function, called \textit{size}, by recursion
-  over regular expressions. If a regular expression is seen as a tree,
-  then \textit{size} should return the number of nodes in such a
-  tree. Therefore this function is defined as follows:
-
-\begin{center}
-\begin{tabular}{lcl}
-$\textit{size}(\ZERO)$ & $\dn$ & $1$\\
-$\textit{size}(\ONE)$  & $\dn$ & $1$\\
-$\textit{size}(c)$     & $\dn$ & $1$\\
-$\textit{size}(r_1 + r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\
-$\textit{size}(r_1 \cdot r_2)$ & $\dn$ & $1 + \textit{size}(r_1) + \textit{size}(r_2)$\\
-$\textit{size}(r^*)$ & $\dn$ & $1 + \textit{size}(r)$\\
-\end{tabular}
-\end{center}
-
-You can use \textit{size} in order to test how much the `evil' regular
-expression $(a^*)^* \cdot b$ grows when taking successive derivatives
-according the letter $a$ without simplification and then compare it to
-taking the derivative, but simplify the result.  The sizes
-are given in \texttt{re.scala}. \hfill[1 Mark]
-
-\item[(6)] You do not have to implement anything specific under this
-  task.  The purpose is that you will be marked for some ``power''
-  test cases. For example can your matcher decide withing 30 seconds
-  whether the regular expression $(a^*)^*\cdot b$ matches strings of the
-  form $aaa\ldots{}aaaa$, for say 1 Million $a$'s. And does simplification
-  simplify the regular expression
-
-  \[
-  \texttt{SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE)}
-  \]  
-
-  \noindent correctly to just \texttt{ONE}, where \texttt{SEQ} is nested
-  50 or more times.\\
-  \mbox{}\hfill[1 Mark]
-\end{itemize}
-
-\subsection*{Background}
-
-Although easily implementable in Scala, the idea behind the derivative
-function might not so easy to be seen. To understand its purpose
-better, assume a regular expression $r$ can match strings of the form
-$c\!::\!cs$ (that means strings which start with a character $c$ and have
-some rest, or tail, $cs$). If you take the derivative of $r$ with
-respect to the character $c$, then you obtain a regular expression
-that can match all the strings $cs$.  In other words, the regular
-expression $\textit{der}\;c\;r$ can match the same strings $c\!::\!cs$
-that can be matched by $r$, except that the $c$ is chopped off.
+\subsection*{Part 1 (6 Marks)}
 
-Assume now $r$ can match the string $abc$. If you take the derivative
-according to $a$ then you obtain a regular expression that can match
-$bc$ (it is $abc$ where the $a$ has been chopped off). If you now
-build the derivative $\textit{der}\;b\;(\textit{der}\;a\;r)$ you
-obtain a regular expression that can match the string $c$ (it is $bc$
-where $b$ is chopped off). If you finally build the derivative of this
-according $c$, that is
-$\textit{der}\;c\;(\textit{der}\;b\;(\textit{der}\;a\;r))$, you obtain
-a regular expression that can match the empty string. You can test
-whether this is indeed the case using the function nullable, which is
-what your matcher is doing.
-
-The purpose of the $\textit{simp}$ function is to keep the regular
-expressions small. Normally the derivative function makes the regular
-expression bigger (see the SEQ case and the example in (2)) and the
-algorithm would be slower and slower over time. The $\textit{simp}$
-function counters this increase in size and the result is that the
-algorithm is fast throughout.  By the way, this algorithm is by Janusz
-Brzozowski who came up with the idea of derivatives in 1964 in his PhD
-thesis.
-
-\begin{center}\small
-\url{https://en.wikipedia.org/wiki/Janusz_Brzozowski_(computer_scientist)}
-\end{center}
-
-
-If you want to see how badly the regular expression matchers do in
-Java\footnote{Version 8 and below; Version 9 and above does not seem to be as
-  catastrophic, but still much worse than the regular expression
-  matcher based on derivatives.}, JavaScript and in Python with the
-`evil' regular expression $(a^*)^*\cdot b$, then have a look at the
-graphs below (you can try it out for yourself: have a look at the file
-\texttt{catastrophic9.java}, \texttt{catastrophic.js} and
-\texttt{catastrophic.py} on KEATS). Compare this with the matcher you
-have implemented. How long can the string of $a$'s be in your matcher
-and still stay within the 30 seconds time limit?
-
-\begin{center}
-\begin{tabular}{@{}cc@{}}
-\multicolumn{2}{c}{Graph: $(a^*)^*\cdot b$ and strings 
-           $\underbrace{a\ldots a}_{n}$}\bigskip\\
-  
-\begin{tikzpicture}
-\begin{axis}[
-    xlabel={$n$},
-    x label style={at={(1.05,0.0)}},
-    ylabel={time in secs},
-    y label style={at={(0.06,0.5)}},
-    enlargelimits=false,
-    xtick={0,5,...,30},
-    xmax=33,
-    ymax=45,
-    ytick={0,5,...,40},
-    scaled ticks=false,
-    axis lines=left,
-    width=6cm,
-    height=5.5cm, 
-    legend entries={Python, Java 8, JavaScript},  
-    legend pos=north west]
-\addplot[blue,mark=*, mark options={fill=white}] table {re-python2.data};
-\addplot[cyan,mark=*, mark options={fill=white}] table {re-java.data};
-\addplot[red,mark=*, mark options={fill=white}] table {re-js.data};
-\end{axis}
-\end{tikzpicture}
-  & 
-\begin{tikzpicture}
-\begin{axis}[
-    xlabel={$n$},
-    x label style={at={(1.05,0.0)}},
-    ylabel={time in secs},
-    y label style={at={(0.06,0.5)}},
-    %enlargelimits=false,
-    %xtick={0,5000,...,30000},
-    xmax=65000,
-    ymax=45,
-    ytick={0,5,...,40},
-    scaled ticks=false,
-    axis lines=left,
-    width=6cm,
-    height=5.5cm, 
-    legend entries={Java 9},  
-    legend pos=north west]
-\addplot[cyan,mark=*, mark options={fill=white}] table {re-java9.data};
-\end{axis}
-\end{tikzpicture}
-\end{tabular}  
-\end{center}
-\newpage
-
-\subsection*{Part 2 (4 Marks)}
-
-Coming from Java or C++, you might think Scala is a quite esoteric
+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}}
 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.
+interpreter and compiler 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, who unfortunately died a few months ago. 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 we know 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}.\bigskip
+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 we know 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 Mandelbot set.\bigskip
 
 \noindent
 As mentioned above, brainf*** has 8 single-character commands, namely
@@ -500,15 +87,21 @@
 \subsubsection*{Tasks (file bf.scala)}
 
 \begin{itemize}
-\item[(2a)] Brainf*** memory is represented by a \texttt{Map} from
+\item[(1)]  Write a function that takes a file name as argument and
+  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)} clearly stores \texttt{1} at
-  memory location \texttt{0}; at \texttt{2} it stores \texttt{3}. 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 function, \texttt{sread}, that takes a memory (a \texttt{Map}) and
-  a memory pointer (an \texttt{Int}) as argument, and safely reads the
+  a memory pointer (an \texttt{Int}) as argument, and `safely' reads the
   corresponding memory location. If the \texttt{Map} is not defined at
   the memory pointer, \texttt{sread} returns \texttt{0}.
 
@@ -519,7 +112,7 @@
   with the same data, except the value is stored at the given memory
   pointer.\hfill[1 Mark]
 
-\item[(2b)] Write two functions, \texttt{jumpRight} and
+\item[(3)] Write two functions, \texttt{jumpRight} and
   \texttt{jumpLeft} that are needed to implement the loop constructs
   of brainf***. They take a program (a \texttt{String}) and a program
   counter (an \texttt{Int}) as argument and move right (respectively
@@ -586,10 +179,10 @@
   \qquad$\stackrel{\texttt{jumpRight}}{\longrightarrow}$\qquad
   \texttt{--[..[[-]+>[.]]-->,++\barbelow{\;\phantom{+}}}
   \end{center}
-  \hfill[1 Mark]
+  \hfill[2 Marks]
 
 
-\item[(2c)] Write a recursive function \texttt{run} that executes a
+\item[(4)] Write a recursive function \texttt{run} that executes 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{run} returns the
@@ -597,13 +190,17 @@
   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{run} with the updated data.
+  \texttt{run} with the updated data. The most convenient way to
+  implement the rules in \texttt{run} is to use pattern-matching
+  and calculating a triple consisting of the new \texttt{pc},
+  \texttt{mp} and \texttt{mem}.
 
   Write another function \texttt{start} that calls \texttt{run} with a
   given brainfu** program and memory, and the program counter and memory pointer
   set to~$0$. Like \texttt{run} 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 or have a look at
+  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}
@@ -673,7 +270,10 @@
   \end{figure}
 \end{itemize}\bigskip  
 
+\subsection*{Part 2 (4 Marks)}
 
+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.
 
 
 \end{document}




