CS 390 Introduction to Theoretical Computer Science

Spring 1994 Final Exam

1(a) Find an NFA- recognizing the language corresponding to the regular expression , by applying literally the algorithm in the textbook(also given in lectures). Do not simplify.

(b) Find an FA that accepts the complement of the language corresponding to the regular expression ab + b.

2(a) Obtain an NFA that accepts the language accepted by the following NFA- :

The accepting states are {3,6}.

(b) Obtain an FA that accepts the language accepted by the following NFA:

The accepting state is {5}.
3. The halting problem is proven to be unsolvable. Explain in what sense it is unsolvable i.e. explain what happens if you want to solve it by a computer.

4. Design a Turing machine that computes the function f(x) = x div 2 for an arbitrary positive integer x. You may use Turing machines given as examples in the textbook or discussed in the lectures. Also describe in English the method that your Turing machine represents.

5(a) What is the final tape configuration for the following Turing machine, if the initial configuration is ?

(b) Explain briefly what the Turing machine of (a) does and how it does.

6. For a language L, let REV(L) denote the language , where REV(x) for a string is defined as follows:

Definition of REV(x):
(1) = .
(2) For any and any ,
REV(xa) = aREV(x).

Answer the following questions:
(a) If r is the regular expression , and is the corresponding language, give a regular expression corresponding to .
(b) Using your answer to (a) as a hint, give a recursive method of finding a regular expression corresponding to the language for a given regular expression s.
(c) Prove that your method of (b) is correct by mathematical induction.

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