COMS W3261
Computer Science Theory
Lecture 8: October 5, 2009
Properties of Regular Languages

1. Outline

• Review
• Closure properties of regular languages
• Midterm review

2. Review

• From regular expressions to ε-NFA's
• Algebraic laws for regular expressions
• The pumping lemma for regular languages

3. Closure Properties of Regular Languages

• A closure property for a family of languages is a theorem that says if we apply a certain operation to the languages in the family, then the resulting language will also be in the family. For example, if we take the union of two regular languages L and M, then the language L ∪ M is also regular. We therefore say the regular languages are closed under the operation of union.
• We can show that the regular languages are closed under the following operations:
• union
• concatenation
• Kleene closure
• intersection
• complement
• difference
• reversal
• homomorphism
• inverse homomorphism

3. Midterm 1

• Wednesday, October 7, 2009, 1:10-2:25pm, 702 Hamilton, closed book
• Covers Sections 1.1-4.2 of HMU
• Know how to design
• DFAs, NFAs, ε-NFAs, regular expressions
• Key algorithms
• subset construction, NFA to DFA, ε-NFA to DFA, DFA to regular expression, regular expression to ε-NFA
• Key tools
• inductive proofs, pumping lemma, properties of regular languages
• Review practice problems from lectures 1-8.

4. Practice Problems

1. Show that the regular languages are closed under complement.
2. Show that the regular languages are closed under intersection.
3. Let A be any language over Σ. Show that A = A* iff AA is contained in A.
4. Let noprefix(A) be the set of strings w in A such that no prefix of w other than w is in A. Show that the regular languages are closed under the noprefix operation.

5. Reading Assignment

• HMU: Sects. 1.1 to 4.2