34 

    35 \noindent

    36 Roman numerals are strings consisting of the letters $I$, $V$, $X$,

    37 $L$, $C$, $D$, and $M$. Such strings should be transformed into an

    38 internal representation using the datatypes \texttt{RomanDigit} and

    39 \texttt{RomanNumeral}, and then from this internal representation

    40 converted into an Integer.

    41 

    42 \begin{itemize}

    43 \item[(1)] First write a polymorphic function that recursively

    44   transforms a list of options into an option of a list. For example,

    45   if you have the lists on the left, they should be transformed into

    46   the option on the right:

    47 

    48   \begin{center}

    49   \begin{tabular}{lcl}  

    50     \texttt{List(Some(1), Some(2), Some(3))} & $\Rightarrow$ &

    51     \texttt{Some(List(1, 2, 3))} \\

    52     \texttt{List(Some(1), None, Some(3))} & $\Rightarrow$ &

    53     \texttt{None} \\

    54     \texttt{List()} & $\Rightarrow$ & \texttt{Some(List())}

    55   \end{tabular}  

    56   \end{center}

    57 

    58   This means the function should produce \texttt{None} as soon

    59   as a \texttt{None} is inside the list. Otherwise it produces

    60   a list of all \texttt{Some}s. In case the list is empty, it

    61   produces \texttt{Some} of the empty list. \hfill[1 Mark]

    62 

    63  

    64 \item[(2)] Write a function first a function that converts a character

    65   $I$, $V$, $X$, $L$, $C$, $D$, or $M$ into an option of a \texttt{RomanDigit}.

    66   If it is one of the roman digits, it should produce \texttt{Some};

    67   otherwise \texttt{None}.

    68   

    69   Next write a function that converts a string into a \texttt{RomanNumeral}.

    70   Again, this function should return an \texttt{Option}:

    71   If the string consists of $I$, $V$, $X$, $L$, $C$, $D$, and $M$ only, then

    72   it produces \texttt{Some}; otherwise if there is any other character in

    73   the string, it should produce \texttt{None}. The empty string is just

    74   the empty \texttt{RomanNumeral}, that is empty list of \texttt{RomanDigit}'s.

    75   You should use the function under Task (1) to produce the result.

    76   \hfill[2 Marks]

    77 

    78 \item[(3)] Write a recursive function RomanNumral2Int that converts a

    79   \texttt{RomanNumeral} into an integer. You can assume the generated

    80   integer will be between 0 and 3999.  The argument of the function is

    81   a list of roman digits. It should look how this list starts and then

    82   calculate what the corresponding integer is for this ``start'' and

    83   add it with the integer for the rest of the list. That means if the

    84   argument is of the form shown on the left-hand side, it should do

    85   the calculation on the right-hand side.

    86 

    87   \begin{center}

    88   \begin{tabular}{lcl}

    89     $M::r$    & $\Rightarrow$ & $1000 + \text{roman numeral of rest}\; r$\\

    90     $C::M::r$ & $\Rightarrow$ & $900 + \text{roman numeral of rest}\; r$\\

    91     $D::r$    & $\Rightarrow$ & $500 + \text{roman numeral of rest}\; r$\\

    92     $C::D::r$ & $\Rightarrow$ & $400 + \text{roman numeral of rest}\; r$\\

    93     $C::r$    & $\Rightarrow$ & $100 + \text{roman numeral of rest}\; r$\\

    94     $X::C::r$ & $\Rightarrow$ & $90 + \text{roman numeral of rest}\; r$\\

    95     $L::r$    & $\Rightarrow$ & $50 + \text{roman numeral of rest}\; r$\\

    96     $X::L::r$ & $\Rightarrow$ & $40 + \text{roman numeral of rest}\; r$\\

    97     $X::r$    & $\Rightarrow$ & $10 + \text{roman numeral of rest}\; r$\\

    98     $I::X::r$ & $\Rightarrow$ & $9 + \text{roman numeral of rest}\; r$\\

    99     $V::r$    & $\Rightarrow$ & $5 + \text{roman numeral of rest}\; r$\\

   100     $I::V::r$ & $\Rightarrow$ & $4 + \text{roman numeral of rest}\; r$\\

   101     $I::r$    & $\Rightarrow$ & $1 + \text{roman numeral of rest}\; r$

   102   \end{tabular}  

   103   \end{center}    

   104 

   105   The empty list will be converted into integer $0$.\hfill[1 Mark]

   106   

   107 \item[(4)] Write a function that takes a string and if possible

   108   converts it into the internal representation. If successful, then

   109   calculate the integer (an option of an integer) according to the

   110   function in (3).  If this is not possible, then return

   111   \texttt{None}.\hfill[1 Mark]

   112 

   113 

   114 \item[(5)] The file \texttt{roman.txt} contains a list of roman numerals.

   115   Read in these numerals, convert them into integers and then add them all

   116   up. The function for reading a file is

   117 

   118   \begin{center}

   119   \texttt{Source.fromFile("filename")("ISO-8859-9")}

   120   \end{center}

   121 

   122   Make sure you process the strings correctly by ignoring whitespaces

   123   where neded.\\ \mbox{}\hfill[1 Mark]

   124 \end{itemize}

   125