COMS W3261
Computer Science Theory
Lecture 15: October 28, 2009
Pumping Lemma and Closure Properties for CFL's

1. Outline

• Midterm 2 will be in class Wednesday, November 11, 2009. It will be closed book and will cover context-free languages.
• Pumping lemma for CFL's
• Closure properties for CFL's

2. Review

• Eliminating useless symbols
• Eliminating ε-productions
• Eliminating unit productions
• Chomsky normal form

3. Pumping Lemma for CFL's

• For every nonfinite context-free language L, there exists a constant n that depends on L such that for all z in L with |z| ≥ n, we can write z as uvwxy where
1. vx ≠ ε,
2. |vwx| ≤ n, and
3. for all i ≥ 0, the string uvⁱwxⁱy is in L.
• Proof: See HMU, pp. 281-282.

4. Applications of the Pumping Lemma

• One important use of the pumping lemma is to prove certain languages are not context free.
• Example: The language L = { ss | s is a string of a's and b's } is not context free.
• The proof will be by contradiction. Assume L is context free. Then by the pumping lemma there is a constant n associated with L such that for all z in L with |z| ≥ n, z can be written as uvwxy such that
1. vx ≠ ε,
2. |vwx| ≤ n, and
3. for all i ≥ 0, the string uvⁱwxⁱy is in L.
• Consider the string z = aⁿbⁿaⁿbⁿ.
• Setting i = 0, condition (3) of the pumping lemma says uwy must also be in L.
• But we can argue uwy cannot be in L, yielding a contradiction. The details can be found in HMU, p. 285.

5. Closure Properties of CFL's

• The context-free languages are closed under
• substitution
• Let Σ be an alphabet and L_a a language for each symbol a in Σ. These languages are called a substitution s on Σ.
• If w = a₁a₂ ... a_n is a string in Σ*, then s(w) = { x₁x₂ ... x_n | x_i is a string in s(a_i) for 1 ≤ i ≤ n }.
• If L is a language, s(L) = { s(w) | w is in L }.
• If L is a CFL over Σ and s(a) is a CFL for each a in Σ, then s(L) is a CFL.
• union
• concatenation
• Kleene star
• reversal
• intersection with a regular set
• inverse homomorphism

6. Nonclosure Properties of CFL's

• The context-free languages are not closed under
• intersection
• L₁ = { aⁿbⁿcⁱ | n, i ≥ 0 } and L₂ = { aⁱbⁿcⁿ | n, i ≥ 0 } are CFL's. But L = L₁ ∩ L₂ = { aⁿbⁿcⁿ | n ≥ 0 } is not a CFL.
• complement
• Suppose comp(L) is context free if L is context free. Since L₁ ∩ L₂ = comp(comp(L₁) ∪ comp(L₂)), this would imply the CFL's are closed under intersection.
• difference
• Suppose L₁ – L₂ is a context free if L₁ and L₂ are context free. If L is a CFL over Σ, then comp(L) = Σ* - L would be context free.

7. Practice Problems

1. Show that { aⁿbⁿcⁿ | n ≥ 0 } is not context free.
2. Show that { aⁿbⁿcⁱ | i ≤ n } is not context free.
3. Show that { ss^Rs | s is a string of a's and b's } is not context free.
4. (Hard) Show that the complement of { ss | ss is a string of a's and b's } is context free.

8. Reading Assignment

• HMU: Sects. 7.2-7.3