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+\documentclass{article}
+\usepackage{../style}
+\usepackage{../langs}
+
+\begin{document}
+
+\section*{Replacement Coursework 1 (Roman Numerals)}
+
+This coursework is worth 10\%. It is about translating roman numerals
+into integers and also about validating roman numerals.  The coursework
+is due on 2 February at 5pm.  Make sure the files you submit can be
+processed by just calling \texttt{scala <<filename.scala>>}.\bigskip
+
+\noindent
+\textbf{Important:} Do not use any mutable data structures in your
+submission! They are not needed. This menas you cannot use 
+\texttt{ListBuffer}s, for example. Do not use \texttt{return} in your
+code! It has a different meaning in Scala, than in Java.  Do not use
+\texttt{var}! This declares a mutable variable.  Make sure the
+functions you submit are defined on the ``top-level'' of Scala, not
+inside a class or object. Also note that the running time will be
+restricted to a maximum of 360 seconds on my laptop.
+
+
+\subsection*{Disclaimer}
+
+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 I showed during the lectures or uploaded to KEATS, which
+you can freely use.\bigskip
+
+
+\subsection*{Part 1 (Translation)}
+
+\noindent
+Roman numerals are strings consisting of the letters $I$, $V$, $X$,
+$L$, $C$, $D$, and $M$. Such strings should be transformed into an
+internal representation using the datatypes \texttt{RomanDigit} and
+\texttt{RomanNumeral} (defined in \texttt{roman.scala}), and then from
+this internal representation converted into Integers.
+
+\begin{itemize}
+\item[(1)] First write a polymorphic function that recursively
+  transforms a list of options into an option of a list. For example,
+  if you have the lists on the left-hand side, they should be transformed into
+  the options on the right-hand side:
+
+  \begin{center}
+  \begin{tabular}{lcl}  
+    \texttt{List(Some(1), Some(2), Some(3))} & $\Rightarrow$ &
+    \texttt{Some(List(1, 2, 3))} \\
+    \texttt{List(Some(1), None, Some(3))} & $\Rightarrow$ &
+    \texttt{None} \\
+    \texttt{List()} & $\Rightarrow$ & \texttt{Some(List())}
+  \end{tabular}  
+  \end{center}
+
+  This means the function should produce \texttt{None} as soon
+  as a \texttt{None} is inside the list. Otherwise it produces
+  a list of all \texttt{Some}s. In case the list is empty, it
+  produces \texttt{Some} of the empty list. \hfill[1 Mark]
+
+ 
+\item[(2)] Write first a function that converts the characters $I$, $V$,
+  $X$, $L$, $C$, $D$, and $M$ into an option of a \texttt{RomanDigit}.
+  If it is one of the roman digits, it should produce \texttt{Some};
+  otherwise \texttt{None}.
+  
+  Next write a function that converts a string into a
+  \texttt{RomanNumeral}.  Again, this function should return an
+  \texttt{Option}: If the string consists of $I$, $V$, $X$, $L$, $C$,
+  $D$, and $M$ only, then it produces \texttt{Some}; otherwise if
+  there is any other character in the string, it should produce
+  \texttt{None}. The empty string is just the empty
+  \texttt{RomanNumeral}, that is the empty list of
+  \texttt{RomanDigit}'s.  You should use the function under Task (1)
+  to produce the result.  \hfill[2 Marks]
+
+\item[(3)] Write a recursive function \texttt{RomanNumral2Int} that
+  converts a \texttt{RomanNumeral} into an integer. You can assume the
+  generated integer will be between 0 and 3999.  The argument of the
+  function is a list of roman digits. It should look how this list
+  starts and then calculate what the corresponding integer is for this
+  ``start'' and add it with the integer for the rest of the list. That
+  means if the argument is of the form shown on the left-hand side, it
+  should do the calculation on the right-hand side.
+
+  \begin{center}
+  \begin{tabular}{lcl}
+    $M::r$    & $\Rightarrow$ & $1000 + \text{roman numeral of rest}\; r$\\
+    $C::M::r$ & $\Rightarrow$ & $900 + \text{roman numeral of rest}\; r$\\
+    $D::r$    & $\Rightarrow$ & $500 + \text{roman numeral of rest}\; r$\\
+    $C::D::r$ & $\Rightarrow$ & $400 + \text{roman numeral of rest}\; r$\\
+    $C::r$    & $\Rightarrow$ & $100 + \text{roman numeral of rest}\; r$\\
+    $X::C::r$ & $\Rightarrow$ & $90 + \text{roman numeral of rest}\; r$\\
+    $L::r$    & $\Rightarrow$ & $50 + \text{roman numeral of rest}\; r$\\
+    $X::L::r$ & $\Rightarrow$ & $40 + \text{roman numeral of rest}\; r$\\
+    $X::r$    & $\Rightarrow$ & $10 + \text{roman numeral of rest}\; r$\\
+    $I::X::r$ & $\Rightarrow$ & $9 + \text{roman numeral of rest}\; r$\\
+    $V::r$    & $\Rightarrow$ & $5 + \text{roman numeral of rest}\; r$\\
+    $I::V::r$ & $\Rightarrow$ & $4 + \text{roman numeral of rest}\; r$\\
+    $I::r$    & $\Rightarrow$ & $1 + \text{roman numeral of rest}\; r$
+  \end{tabular}  
+  \end{center}    
+
+  The empty list will be converted to integer $0$.\hfill[1 Mark]
+  
+\item[(4)] Write a function that takes a string and if possible
+  converts it into the internal representation. If successful, it then
+  calculates the integer (an option of an integer) according to the
+  function in (3).  If this is not possible, then return
+  \texttt{None}.\hfill[1 Mark]
+
+
+\item[(5)] The file \texttt{roman.txt} contains a list of roman numerals.
+  Read in these numerals, convert them into integers and then add them all
+  up. The Scala function for reading a file is
+
+  \begin{center}
+  \texttt{Source.fromFile("filename")("ISO-8859-9")}
+  \end{center}
+
+  Make sure you process the strings correctly by ignoring whitespaces
+  where needed.\\ \mbox{}\hfill[1 Mark]
+\end{itemize}
+
+
+\subsection*{Part 2 (Validation)}
+
+As you can see the function under Task (3) can produce some unexpected
+results. For example for $XXCIII$ it produces 103. The reason for this
+unexpected result is that $XXCIII$ is actually not a valid roman
+number, neither is $IIII$ for 4 nor $MIM$ for 1999. Although actual
+Romans were not so fussy about this,\footnote{They happily used
+  numbers like $XIIX$ or $IIXX$ for 18.} but modern times declared
+that there are precise rules for what a valid roman number is, namely:
+
+\begin{itemize}
+\item Repeatable roman digits are $I$, $X$, $C$ and $M$. The other ones
+  are non-repeatable. Repeatable digits can be repeated upto 3 times in a
+  number (for example $MMM$ is OK); non-repeatable digits cannot be
+  repeated at all (for example $VV$ is excluded).
+  
+\item If a smaller digits precedes a bigger digit, then $I$ can precede $V$ and $X$; $X$ can preced
+  $L$ and $C$; and $C$ can preced $D$ and $M$. No other combination is permitted in this case.
+
+\item If a smaller digit precedes a bigger digit (for example $IV$), then the smaller number   
+  must be either the first digit in the number, or follow a digit which is at least 10 times its value.
+  So $VIV$ is excluded, because $I$ follows $V$ and $I * 10$ is bigger than $V$; but $XIV$ is
+  allowed, because $I$ follows $X$ and $I * 10$ is equal to $X$.
+
+\item Let us say two digits are called a \emph{compound} roman digit
+  when a smaller digit precedes a bigger digit (so $IV$, $XL$, $CM$
+  for example). If a compound digit is followed by another digit, then
+  this digit must be smaller than the first digit in the compound
+  digit. For example $IXI$ is excluded, but $XLI$ is not.
+
+\item The empty roman numeral is valid.  
+\end{itemize}
+
+\noindent
+The tasks in this part are as follows:
+
+\begin{itemize}
+\item[(6)] Implement a recursive function \texttt{isValidNumeral} that
+  takes a \texttt{RomanNumeral} as argument and produces true if \textbf{all}
+  the rules above are satisfied, and otherwise false.
+
+  Hint: It might be more convenient to test when the rules fail and then return false;
+  return true in all other cases.
+  \mbox{}\hfill[2 Marks]
+
+\item[(7)] Write a recursive function that converts an Integer into a \texttt{RomanNumeral}.
+  You can assume the function will only be called for integers between 0 and 3999.\mbox{}\hfill[1 Mark]
+  
+\item[(8)] Write a function that reads a text file (for example \texttt{roman2.txt})
+  containing valid and invalid roman numerals. Convert all valid roman numerals into
+  integers, add them up and produce the result as a \texttt{RomanNumeral} (using the function
+  from (7)). \hfill[1 Mark]
+\end{itemize}
+  
+
+\end{document}
+
+
+%%% Local Variables: 
+%%% mode: latex
+%%% TeX-master: t
+%%% End: