COMS W3261
Computer Science Theory
Lecture 6:September 24, 2012
Properties of Regular Languages


1. Closure Properties of Regular Languages

2. Decision Problems for Regular Languages

3. Testing Equivalence of States

4. Testing Equivalence of DFA's

5. Minimizing the Number of States in a DFA

6. Practice Problems

  1. Prove that the two regular expressions (a+b)* and (a*b*)* generate the same language.
  2. 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.
  3. [Hard] Consider the function on languages remove_middle_third(L) = { xz | for some y, xyz is in L where |x| = |y| = |z|}. Show that the regular languages are not closed under the remove_middle_third function.
  4. [Hard] 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.

7. Reading Assignment