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CIS 705 --- Programming Languages --- Spring 2009
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Assignment 3
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Due by 2:30 p.m.\ on Thursday, March 12
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The context for this assignment is Chapter~2 of \emph{DAWOC},
the core and standard\\OCaml libraries, and Assignment 2
(including the model answers \texttt{balanced.ml}\\and
\texttt{balanced-inefficient.ml}).
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\section*{Exercise 1 (25 Points)}

Using a few well-chosen examples, explain how the versions of
\begin{verbatim}
val balanced : int list -> int list list
\end{verbatim}
defined by \texttt{balanced-inefficient.ml} and
\texttt{balanced.ml} work, and compare their relative efficiencies.

\section*{Exercise 2 (35 Points)}

The function
\begin{verbatim}
val insert : 'a -> 'a list -> 'a list
\end{verbatim}
of \texttt{balanced.ml} runs out of stack space when used with
lists containing millions of elements, resulting in the
error message
\begin{verbatim}
Stack overflow during evaluation (looping recursion?).
\end{verbatim}

Reimplement \texttt{insert} to meet the same specification, but to be
usable with lists of millions of elements.  Try to minimize the stack
space it uses, as well as the number of \texttt{::}'s and comparisons
that it does; minimizing stack space should have highest priority.
You may use as many auxiliary functions as you wish,
and these functions may be defined either inside or before
\texttt{insert}'s definition.

Program in a strictly functional style, without using the imperative
features of Chapter~3 of \emph{DAWOC}.  Format your program in a way
that makes its structure clear.  Choose meaningful names for
functions, and choose other identifiers with care.  Document your
functions by abstractly explaining their input/output behavior.

\section*{Exercise 3 (40 Points)}

Prove that the functions
\begin{verbatim}
val extend   : int -> int list -> int list option * int list
val balanced : int list -> int list list
\end{verbatim}
of \texttt{balanced.ml} meet their specifications.  You may assume
that the auxiliary and library functions they use are correct, as well as that
the auxiliary function \texttt{bal} of \texttt{balanced}'s definition terminates
on all inputs meeting its precondition.

(A function meets its specification iff, for all inputs to the
function, if those inputs meet the function's precondition, then the
function returns a value meeting the function's postcondition.  When
proving a recursively defined function correct, one may assume that
a recursive call whose arguments meet the precondition returns a
value that meets the postcondition.)

\section*{Submission}

Submit your solutions to Exercises 1 and 3 on paper, not electronically.

Email your solution to Exercise 2 to me, being sure to keep an
electronic copy.  I will acknowledge receiving it.  Your submission
should \emph{not} include any testing or extraneous code.  It should
begin with a comment including your name.
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