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CIS 705 --- Programming Languages --- Spring 2009
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Assignment 4
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Due by 2:30 p.m.\ on Thursday, April 2
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The context for this assignment is Chapters~5--7 of \emph{TAPL}
and Chapter~2 of \emph{DAWOC}.
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\section*{Exercise 1 (60 Points)}

The goal of this exercise is to prove the results that we called
Propositions~D--G in class, when discussing Chapter~5 of \emph{TAPL}.
You should prove these propositions
in sequence, and you may use the results about $\fun$, $\fun^n$
and $\fun^*$ that were previously established.

\begin{description}
\item[Proposition D (15 Points)] For all closed terms $t_1$ and
  $t_2$, if $t_1\,t_2$ converges, then $t_1$, $t_2$ and
  $\overline{t_1}\;\overline{t_2}$ converge and
  $\overline{t_1t_2} = \overline{\overline{t_1}\;\overline{t_2}}$.

\item[Proposition E (15 Points)] For all closed terms $t_1$ and $t_2$,
  if $t_1$, $t_2$ and $\overline{t_1}\;\overline{t_2}$ converge, then
  $t_1\,t_2$ converges and $\overline{t_1t_2} =
  \overline{\overline{t_1}\;\overline{t_2}}$.

\item[Proposition F (15 Points)] For all $n\geq 2$ and closed terms
  $t_1$, \ldots, $t_n$, if $t_1\,\cdots\,t_n$ converges, then $t_1$,
  \ldots, $t_n$ and $\overline{t_1}\;\cdots\;\overline{t_n}$ converge
  and $\overline{t_1\,\cdots\,t_n} =
  \overline{\overline{t_1}\;\cdots\;\overline{t_n}}$.

\item[Proposition G (15 Points)] For all $n\geq 2$ and closed terms
  $t_1$, \ldots, $t_n$, if $t_1$, \ldots, $t_n$ and
  $\overline{t_1}\;\cdots\;\overline{t_n}$ converge, then
  $t_1\,\cdots\,t_n$ converges and $\overline{t_1\,\cdots\,t_n} =
  \overline{\overline{t_1}\;\cdots\;\overline{t_n}}$.
\end{description}

\subsection*{Submission}

Submit your solution to this exercise on paper, not electronically.

\section*{Exercise 2 (40 Points)}

The goal of this assignment is to translate the OCaml expression
\begin{verbatim}
let rec gcd n m =
      if m = 0
      then n
      else gcd m (n mod m)
in gcd
\end{verbatim}
into the untyped lambda calculus.  This expression has type
\texttt{int~->~int~->~int} and is the well-known greatest common
divisor function.  Although its type involves integers, it is only
intended to be called with natural numbers.  Of course, the untyped
lambda calculus term that you translate it into won't be typed.

You should write a closed lambda calculus term that, when applied to
closed values representing natural numbers $n$ and $m$,
converges to a closed value representing the greatest common divisor
of $n$ and $m$.

Your answer should reside in a file named \texttt{gcd-untyped.txt}.
To test that your term works correctly on natural numbers $n$ and $m$,
copy your term to a new file, parenthesize it, and follow it with
\begin{mverbatim}
     $c_n$
     $c_m$
     (lambda x. lambda y. y x)
     (lambda x. x);
\end{mverbatim}
Then give the untyped lambda calculus interpreter the new file as
its input.  The output should look like
\begin{mverbatim}
     (lambda y.
        y
        (lambda y'.
           y'
           (lambda y''.
              y''
              $...$ (lambda x.x) $...$)))
\end{mverbatim}
where the number of occurrences of ``\texttt{lambda~y}''---with
some number of primes on \texttt{y}---is the greatest common divisor
of $n$ and $m$.

The Linux/Mac OS X shell script \texttt{gcd-test} automates the
process of creating the input to the untyped lambda calculus
interpreter.  Running
\vspace{-.34cm}
\begin{mverbatim}
     gcd-test gcd-untyped.txt $n$ $m$
\end{mverbatim}
puts the text described above in the file \texttt{gcd-test-output}.
For example, running
\begin{verbatim}
gcd-test gcd-untyped.txt 4 6
\end{verbatim}
will put the text
\begin{mverbatim}
     ( /* beginning of gcd code */
     $...$
     /* end of gcd code */ )

     (lambda s. lambda z. s(s(s(s z)))) /* 4 */
     (lambda s. lambda z. s(s(s(s(s(s z)))))) /* 6 */
     (lambda x. lambda y. y x)
     (lambda x. x);
\end{mverbatim}
in \texttt{gcd-test-output}.  Giving this file to the untyped
lambda calculus interpreter should produce the output (formatted
slightly differently)
\begin{verbatim}
(lambda y.
   y
   (lambda y'.
      y'
      (lambda x.x)))
\end{verbatim}
because the greatest common divisor of $4$ and $6$ is $2$.

Try to make your lambda calculus term as easy to understand as
possible.  Use comments, which begin with ``\texttt{/*}'' and
end with ``\texttt{*/}'', to explain what you are doing.
Format your term carefully.  Although your term should run to
completion on small inputs, you don't have to be concerned with
its efficiency.

\subsection*{Submission}

Your untyped lambda calculus term should begin with a comment
containing your name.  It should reside in an ordinary ASCII file,
\texttt{gcd-untyped.txt}.  Submit your program by emailing it to
me.  I will acknowledge receiving it.  Make sure that you retain an
electronic copy of your program.
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