COMS W3261 CS Theory
Lecture 6: Properties of Regular Languages - II

1. Decision Problems for Regular Languages

2. Testing Equivalence of States

3. Testing Equivalence of DFA's

4. Minimizing the Number of States in a DFA

5. Practice Problems

  1. Let L be a regular language over the alphabet Σ. Show that it is decidable whether L = Σ*.
  2. Prove that the two regular expressions (a+b)* and (a*b*)* generate the same language by showing that their minimum-state DFA's are the same (up to renaming of states).
  3. Describe an algorithm to determine whether two regular expressions are equivalent. What is the running time of your algorithm?
  4. Minimize the number of states in the following DFA having the start state A, the set of final states {C, E, F, G }, and the following transition function:

  5. Consider the function on languages noprefix(L) = { w in L | no proper prefix of w is a member of L}. Show that the regular languages are closed under the noprefix function.
  6. [Hard but a fundamental characterization of regular languages] An equivalence relation R on a language L contained in Σ* is right invariant if xRy implies xzRyz for all z in Σ*. R is of finite index if it partitions L into a finite number of equivalence classes. Show that L is regular if and only if it is the union of some of the equivalence classes of a right-invariant equivalence relation on L of finite index.

6. Reading