COT4420
Theory of Computation
April 7, 2016
Problem Set 6
Due: April 19, 2016
This assignment will not be graded.
1. Find a phrase structure grammar (it will basically be a context sensitive grammar with
one rule that goes to λ) for the following language:
L = {an**2 : n ≥ 1}.
2. What is a universal Turing machine? What are the inputs? What is the output or the
result of running the universal Turing machine?
3. What is the halting problem for Turing machines? Is it decidable? Explain.
4. How are recursive enumerable languages and recursive languages related to Turing
machines? Using Turing machines, show that recursively enumerable languages are
closed under union and intersection. Are they closed under complementation?
5. Design a Turing machine with no more than three states that accepts the language
a(a+b)*. Assume Σ = {a,b}. Can you find a two-state machine to do this? If so what
is it?
6. Outline a Turing machine that accepts the following language:
L = {anbncn : n ≥ 1}.
7. Outline a Turing machine that converts binary numbers to their unary representations.
8. Define the complexity classes P and NP.
9. Can a Turing machine be simulated by a pda with two stacks? Explain why or why
not and its ramification.
10. Read up on how Len Adleman solved an instance of the Hamilton path problem by
using DNA molecules. Briefly describe the novelty of this work.
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