CS 390 Final Exam

April 25, 1996

1. Find a regular expression for the following languages over the alphabet {0,1}.
(a) the set of strings with an odd number of 0's.
(b) the set of strings not containing a substring 000.

2. Find an FA equivalent to the NFA given below. click here for the NFA.

3. Show that the language L = { } is not regular.
Hint: Try simple languages for the infinite set of strings.

4 (a) Inductively define the set of states at distance k (i.e. apply k times) from a state q of an (not ).
(b) Prove by induction that .
for a set of states S of an is defined inductively as follows:
Basis:
Induction: For any , .
Extremal Clause: No state is in unless it can be obtained using Basis and Induction.

5. Design a Turing machine for (a) and (b), and answer (c):
You may use the basic Turing machines, shifter, copier and eraser.
(a) A Turing machine that accepts but not decides the language L = {ab}. (b) A Turing machine that decides the language L = {ab}.
(c) Does the Turing machine of (a) solve the problem "Does a string belong to L ?" ? Explain what the Turing machine does when a string is given to it and explain in what sense it solves or does not solve the problem.

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