Question 1.
[10 marks]
Show whether or not the set of all even integers {…,-4,-2,0,2,4,…} is countable..
Answer 1.
This is countable. To prove this, we show that there is a bijection
between this set and the set of natural numbers. The bijection is as follows:
f(n) = n,
if n is even
= -n-1, otherwise
Question 2.
[10 marks]
Write a regular expression which describes the language of binary numbers from the
alphabet {0,1} which are either odd or a power of 2 (or both).
Answer 2.
(0|1)*1 | 10*
Question 3.
[10 marks]
Construct a deterministic finite-state automaton to recognise the language
described in Question 2.
Answer 3.
0
1
1
1
0
1
0
1
0
2
3
0
Question 4.
[10 marks]
Consider the following grammar with start symbol S:
S  if id then S else S
S  if id then S
S  id
Show that this grammar is ambiguous by giving two distinct parse trees for the
following sentence:
if id then if id then id else id
Answer 4.
S
if
S
id
then
S
if
id then
if
S else
S
id
id
id then
if
S else
id
then
S
S id
id
Question 5.
[10 marks]
Consider the following grammar with start symbol E:
E  E+T | E-T | T
T  T*F | T/F | F
F  (E) | id
Construct a parse tree for the following string:
id*(id-id)/id
Answer 5. E  T  T*F  F*F  id*F  id*T/F  id*F/F  id*(E)/F
 id*(E-T)/F  id*(T-T)/F  id*(F-T)/F  id*(id-T)/F  id*(id-F)/F
 id*(id-id)/F  id*(id-id)/id
Question 6.
[10 marks]
Construct a pushdown automaton that recognises the language
{ajbk: j > k}
Answer 6.
states
input symbols
stack symbols
initial state
final states
transitions
{q, s, f}
{a, b}
{a}
s
{f}
{ ((s, , ), (s, a)),
((s, b, ), (q,)),
((q, b, ), (q,)),
((q, , a), (f, )),
((f, , ), (f, )) }
Question 7.
[10 marks]
Consider a Turing machine with start state 0 and the following transitions:
State
Symbol
(State, Symbol)
0
a
(1, #, R)
0
b
(4, #, R)
0
#
(0, #, Y)
1
a
(1, a, R)
1
b
(1, b, R)
1
#
(2, #, L)
2
a
(3, #, L)
2
#
(2, #, Y)
3
a
(3, a, L)
3
b
(3, b, L)
3
#
(0, #, R)
4
a
(4, a, R)
4
b
(4, b, R)
4
#
(5, #, L)
5
b
(3, #, L)
5
#
(5, #, Y)
Trace the execution of this Turing machine with the string
baaab#
as input.
Answer 7.
0
4
4
4
4
4
5
3
3
3
3
0
1
1
baaab#
#aaab#
#aaab#
#aaab#
#aaab#
#aaab#
#aaab#
#aaa##
#aaa##
#aaa##
#aaa##
#aaa##
##aa##
##aa##
1
2
3
3
0
1
2
##aa##
##aa##
##a###
##a###
##a###
######
######
accepts
Question 8.
[10 marks]
Describe the language which is accepted by the Turing machine in Question 7.
What is meant by the statement f(n) = O(g(n))?
What is the time complexity of the Turing machine in Question 7 (in terms of
O(f(n)) notation) for an input of length n?
Answer 8.
The Turing machine accepts the language {w  {a,b}* | w = wR}
We say that f(n) = O(g(n)) if there are positive constants c and N such that f(n) 
cg(n) for all integers n  N.
The time complexity of the machine is O(n2)
Question 9.
[10 marks]
Is the following problem decidable? Give a proof of your answer.
Given a Turing machine T and a string w, does T loop forever on input w?
Answer 9.
This problem is undecidable. To show this is the case, we reduce the problem of
whether or not a given Turing machine halts over a given string (which is known
to be undecidable) to it. Given a Turing machine T, we construct a Turing
machine T' that accepts the same language but never crashes. Then for any input
string w, T fails to accept w if and only if T' loops forever on w. This means that
the problem of whether or not a given Turing machine halts over a given string is
reducible to this one.
Question 10.
[10 marks]
Describe what is meant by each of the following:
(a) Recursive languages
(b) Recursively enumerable languages
(c) Primitive recursive functions
Show that the function to add together two natural numbers is primitive recursive.
Answer 10.
(a) A language is recursive if there is a decision procedure for every string to
indicate whether or not it belongs to the language.
(b) A language is recursively enumerable if there is a Turing machine which
accepts it.
(c) A function f is primitive recursive if it can be constructed from functions g and
h as follows:
f(x,0) = h(x)
f(x,succ(y)) = g(x,y,f(x,y))
The add function can be defined as a primitive recursive function as follows:
add(m,0) = m
add(m,succ(n)) = succ(add(m,n))