1 Introduction

This document contains information concerning the conduct of the mid-term examination, and some sample questions, representative of those that will be set.

2 Information

The mid-term examination for this course will be held, as advertised, on Thursday, April 14th, during the normal lecture period.

• The examination will be held in the normal lecture venue, Ross lecture theatre.
• The examination will last 60 minutes, including five minutes reading time; it will start at 12.00 prompt, and finish at 1.00pm.
• Please deposit all books and bags at the front of the lecture theatre.
• Ensure that you are seated by 11.55am.

3 Sample Questions

The examination will contain one short question (worth 5 marks), and three longer questions. You should attempt the short question, and only two of the three longer questions, each of which will be worth 10 marks.

0. 5 marks
   Give the responses of the ML system to the following sequence of declarations

   val a = 1;     val b = 2;     fun f a = a + b;     val b = 3;     f b;

1. Long Question 10 marks
   The following datatype can be used to represent trees whose nodes can have an arbitrary number of children.
   1. What tree does the following expression denote (i.e draw a picture):
   2. Define a function to calculate the number of leaves in such a tree.
   3. We can assign a level to each node in a tree as follows. The node at the root is at level 1. Its children are at level 2. Their children are at level 3 and so on.

      Suppose we are interested in trees where an internal node at level n always has exactly n children. Define a function check : ’a Tree ->bool that checks whether a given tree has this property.

2. Long Question 10 marks
   The EQueue signature is like the signature Queue, but is extended with an additional operation multiple enqueue, menq:(Item  list * Queue) -> Queue, intended to add a number of items (in an arbitrary order) to the queue in a single operation.

   signature EQueue =   sig       type Item       type Queue         val empty : Queue       val enq : (Item * Queue) -> Queue       val deq : Queue -> (Item * Queue)       val menq: (Item list * Queue) -> Queue   end

   An implementation of a stack, including this operation, uses the type declaration

   type Queue = Item list list

   the operations empty and menq are implemented as follows:

   val empty = []     fun menq(items, q) = items :: q

   1. Complete the following declarations of the functions enq and deq for this implementation

      fun enq(item,       []) =     | enq(item, (h :: t)) =     fun deq((h :: t) :: r)  =     | deq([] :: r)        =     | deq []              =

   2. What is the complexity of the three operations
      1. enq,
      2. deq,
      3. menq

      for this implementation?

3. Long Question 10 marks
   An implementation of sets of integers is designed to represent a set by a list without repetitions, kept in increasing order. Here is the function union : Set*Set -> Set from this implementation

   fun union(a, []) = a     | union([], b) = b     | union(ah :: at, bh :: bt) =             if ah < bh then ah :: union(at, bh :: bt)             else if ah = bh then ah :: union(at, bt)             else bh :: union(ah :: at, bt)

   1. What is the complexity of this implementation of union?
   2. Give an implementation of the operation insert : (int*Set) ->Set compatible with this representation
   3. Give an O(n) implementation of the operation intersect : Set*Set -> Set, compatible with this representation.

©Michael Fourman 1994-2006