Introduction to the
Theory of Computation
Final exam
15 January 2016
Closed-book. Duration: 3h30.
Please answer each question on a separate sheet with your name and
section. Motivate all your answers and give sufficient details.
1. a) Is the set of all closed intervals of Rwith rational bounds denu-
merable?
b) Is the set of all regular expressions over a finite alphabet Σ denu-
merable? What happens when Σ is infinite (but denumerable)?
2. a) Give a DFA that accepts the language
L1
t
w
|
w
P t
a, b
u
, Na
p
w
q
2 mod 4
u
where Nσ
p
w
q
is the number of letters σcontained in the word w.
b) Give a DFA that accepts the language
L2
t
w
|
w
P t
a, b
u
, aa
R
F act
p
w
qu
c) Give a regular grammar that generates L1
Y
L2.
3. a) Is the language
t
ambncmax
p
m,n
q
|
m, n
P
N
u
regular?
b) Is the language of all well-parenthesized expressions regular?
Example:
ppqpqq
.
4. a) State and prove the pumping lemma for context-free languages.
b) Is the language L
t
aibjckdl
|
i
l
¥
j
k
u
context-free?
5. a) State the Turing-Church Thesis. What type of justification can
be given for this thesis?
b) Are two tape Turing Machines more expressive than the standard
definition of a Turing Machine? Explain.
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