Introduction to the
Theory of Computation
Final exam
19 August 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 an infinite subset of a set that is non-denumerable necessarily
non-denumerable?
b) Is the union of two denumerable sets necessarily denumerable?
c) Show, using a cardinality argument, that there must exist uncom-
putable functions from Nto N.
2. a) Give a DFA that accepts the language
L1“ tw|wP ta, bu˚, Napwq “ 2 mod 4u
where Nσpwqis the number of letters σcontained in the word w.
b) Give a DFA that accepts the language
L2“ tw|wP ta, bu˚, aa RF actpwqu
c) Give a regular grammar that generates L1YL2.
3. a) Using the pumping lemma, show that the language ta3nbnc˚|nP
Nuis not regular.
b) Let L1and L2be two regular languages over the same alphabet
Σ. Is the language L1‘L2that contains the words that belong
to only one of the two languages regular?
4. a) Is the language L“ taibjck|i“jor j“kwhere i, j, k ě0u
context-free?
b) Show that the intersection of two context-free languages is not
necessarily context-free. Use this to deduce that the complement
of a context-free language is not necessarily context-free. Give
a sufficient criterion for the intersection of two context-free lan-
guages to be context-free.
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