CS 154
Intro. to Automata and Complexity Theory
Handout 23
Autumn 2006
David Dill
November 7, 2006
Problem Set 5
Due: November 14, 2006
Homework:
(Total 100 points) Do the following exercises.
Problem 1.
[10 points]
In this problem you will establish some closure
properties for languages of Turing machines.
(a).
Are recursive languages closed under
intersection
? Justify your answer.
(b).
Are recursively enumerable languages closed under
intersection
? Jus
tify your answer.
Problem 2.
[20 points]
Consider the following problem concerning Turing
machines with a tape alphabet Γ.
Given a Turing machine
M
, input string
w
, and a symbol
X
∈
Γ,
decide whether
M
, when running on input
w
, will ever write the
symbol
X
on its tape.
Show that this problem is undecidable. Is this problem recursively enumer
able? (
Hint:
Can you give a reduction from the universal language
L
u
? It
may help to look at the solution to Exercise 9.3.7(a) in the second edition
of the textbook, although Exercise 9.2.1 is more directly relevant.)
Problem 3.
[20 points]
(Exercise 9.3.6(c) on page 391 of the second edition
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 Winter '08
 Motwani,R
 Halting problem, Turing Machines, Turing, Lall

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