The question paper for the university of cambridge mathematical tripos part ia paper 5 exam held on june 4, 2007, from 1:30 to 4:30 pm. The paper covers various topics in computer science, operating systems, java programming, numbers and sets, and has six sections (a to e) with questions distributed across them. The paper requires the student to answer two questions from section a, and one question each from sections b, c, d, and e. The paper is divided into multiple pages, with special requirements for stationery and instructions for the examinee on how to attempt the paper.
Download University Mathematical Tripos Paper 5 for Part IA, June 2007 and more Exams Mathematics in PDF only on Docsity!
MATHEMATICAL TRIPOS Part IA
Monday 4 June 2007 1.30 to 4.
PAPER 5
Before you begin, read these instructions carefully.
Answer two questions from Section A, and one question from each of Sections B, C, D and E.
Write on one side of the paper only and begin each answer on a separate sheet.
Write legibly; otherwise you place yourself at a grave disadvantage.
At the end of the examination:
Tie up your answers in six separate bundles, each with its own cover sheet. On each cover sheet, write the numbers of all attempted questions, and circle the number of the question attached.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS
Script paper None Blue cover sheets Tags
You may not start to read the questions
printed on the subsequent pages of this
question paper until instructed that you
may do so by the Invigilator
SECTION A
1 Foundations of Computer Science
(a) Code the ML function merge, which combines two ordered lists to form an ordered list containing the elements of both. [2 marks]
(b) Code a top-down merge sort function in ML. You may assume that basic functions on lists are given, provided you describe them briefly. [3 marks]
(c) State the time complexities of merge and merge sort, justifying your answer carefully. [5 marks]
2 Operating Systems
(a) In relation to scheduling of processes, describe the concept of a working set and briefly outline how it can be used within an operating system. [3 marks]
(b) Briefly explain why context switching between processes is inherently more costly than switching between threads of a process. [3 marks]
(c) Give two reasons why operating system designers often choose to make code in the kernel non-preemptive. [2 marks]
(d) Why would it be bad for a Unix file owned by root with the setuid bit set also to have read, write and execute access permissions granted to all users? [2 marks]
3 Programming in Java
(a) What is meant by a generic in the context of Java? Explain the main purpose of generics and the most important syntax associated with them. [3 marks]
(b) Explain how any program written using generics could be re-written to avoid them. Give a fragment of code illustrating the conversions needed to achieve this. Discuss why many people will view the version that does use generics as displaying better style and robustness. [4 marks]
(c) One of your fellow students puts forward the proposition “String is a sub-class of Object, therefore Vector
SECTION B
5 Foundations of Computer Science
(a) Consider the following piece of ML code:
datatype 'a tree = Lf | Br of 'a * 'a tree * 'a tree; exception Blair;
fun tony p Lf = true | tony p (Br(x,t1,t2)) = if not (p x) then raise Blair else tony p t1 handle Blair => tony p t2;
fun gordon p t = tony p t handle Blair => false;
(i ) Code a function that returns the same results as gordon but makes no use of exceptions. [4 marks]
(ii ) What property of binary trees does gordon express? [3 marks]
(b) Write brief notes on the ML type exn. [3 marks]
(c) Consider the following piece of ML code:
datatype 'a result = Ian of 'a | Cherie of exn;
fun what f x = Ian (f x) handle e => Cherie e;
We ask ML to evaluate the expression
map (what (tony (fn x => x <> 0))) [ta,tb]
and the response is as follows:
val it = [Ian true, Cherie Blair] : bool result list
What is the type of what (tony (fn x => x <> 0)), and what can we infer about the binary trees ta and tb? Justify both answers carefully. [5+5 marks]
6 Foundations of Computer Science
(a) Write brief notes on reference types in ML and on control structures for imperative programming. [6 marks]
Consider the following ML datatype:
datatype 'a meal = Snack of 'a | Lunch of 'a meal * 'a meal | Feast of 'a meal * 'a meal * 'a meal;
(b) Write a function that is equivalent to snacker below but makes no use of references. Briefly explain why the two functions are equivalent.
fun snacker m = let val l = ref [] fun munch (Snack x) = (l := x :: !l) | munch (Lunch (m1,m2)) = (munch m1; munch m2) | munch (Feast (m1,m2,m3)) = (munch m1; munch m2; munch m3) in munch m; !l end;
[5 marks]
(c) Write a function gluttony such that gluttony m1 m2 makes a copy of m1, replacing every Snack node with m2. [3 marks]
(d) Write a function glut such that glut k m1 m2 makes a copy of m1, replacing the kth Snack node with m2. Nodes are counted from left to right, with the leftmost node being number one. [6 marks]
5 (TURN OVER)
SECTION D
9 Programming in Java
(a) Explain how to set up a 2-dimensional array in Java. [2 marks]
(b) A simple spreadsheet is a grid of cells. Each cell can contain one of four possible things:
Nothing – the cell might be empty;
A fixed string, used as a label;
A fixed numerical value, represented as a double;
A formula, as discussed below, which will evaluate to a number.
The sorts of formulae to be supported to start with are very limited, but it is expected that later developments will add more options. For now a formula can indicate that the value in a cell is the sum of two other values whose coordinates are specified relative to the cell being considered. If one of the cells so addressed is empty, contains a string or is off the edge of the grid then it will be treated as if it contains zero.
Somewhere in the spreadsheet program there will need to be methods that make it possible to set the type of content of a cell, and to process the formulae until all values are up to date. They may of course need a number of additional fields and methods not explicitly noted in this specification.
(i ) Design a set of Java classes that you can use to represent this set-up. Explain what fields and methods each will have, and what needs to be public and what can be kept private. At this stage you do not need to implement any elaborate methods, but you should explain what your methods must achieve. [9 marks]
(ii ) Sketch an implementation of the methods involved in causing the spreadsheet to bring all its values up to date after the user alters the value in one cell. [9 marks]
7 (TURN OVER)
10 Programming in Java
(a) A restricted variant on ML has types that are such that a type either is denoted by a type variable, α, β, etc˙, or is a function type of the form τ 1 → τ 2 where τ 1 and τ 2 are simpler types. Design a Java class or set of classes that can be used to represent ML type expressions. Ensure that you provide a static method createNewTypeVar that makes a new ML type variable that is different from all the ones you have had before. Rather than giving your type variables Greek letters for their names you may call them t1, t2, t3,... [7 marks]
(b) Suppose that your types are represented by a class called MLType. Explain all the changes you need to make to your code so that any MLType object has a method with signature
public void mustNotDependOn(TypeVar a) throws ItDoesDependOn;
that will check whether the type variable passed as an argument is present within the type. If it is, then an exception must be thrown. For instance if you passed the mustNotDependOn method of (the representation of) (α → β) → (β → γ) the type-variable δ the method would just return, while if you passed it α, β or γ there would be an exception. [9 marks]
(c) Give an implementation of a method
toString()
which returns a text representation of the MLType object concerned. [4 marks]
