Sergio A. Alvarez web: http://www.cs.wpi.edu/ alvarez/
Department of Computer Science e-mail: alvarez@cs.wpi.edu
Worcester Polytechnic Institute phone: (508) 831-5118
Worcester, MA 01609 USA fax: (508) 831-5776

CS 3133, A Term 1998
Midterm Exam

Instructions.

Read each problem carefully. Write solutions neatly in the spaces provided.
Include a brief description of your solution strategy for each problem.

1. Find a regular expression for each of the following languages over the alphabet :
   1. all strings that start with (at least one) a and contain exactly two b's.

   2. all strings that do not contain the substring ab.

2. Consider the context-free grammar G over defined as follows.

   

   1. Give a leftmost derivation in G for the string eabbab.
   2. Construct a nondeterministic finite automaton (NFA) that accepts L(G).
   3. Find a regular expression for the language L(G).

3. Let M be the NFA over having state set , start state A, accepting state set , and transition function satisfying and for all other pairs .
   1. Draw the state transition diagram of the machine M.
   2. Construct a deterministic finite automaton (DFA) equivalent to M.

4. In this problem L denotes the language over consisting of all strings that contain an equal number of a's and b's. You will show that L is not a regular language.
   1. For every natural number k find a string in L of length k or greater for which pumping is guaranteed to produce some strings that are not in L. Specifically, this means that length and for every decomposition of as uvw, where the strings u, v, w satisfy length , length, there must be some natural number n such that the pumped string is not in L.

   2. Aiming to reach a contradiction, assume tentatively that L actually is a regular language. Explain carefully how the pumping lemma together with part (a) above lead to a contradiction for a certain value of k. Explain the significance of the value of k in this context.