COMS W3261
Computer Science Theory
Lecture 11: October 10, 2012
Decision and Closure Properties of CFL's

Outline

• Cocke-Younger-Kasami algorithm
• Testing emptiness of a CFG
• Closure properties of CFL's
• Nonclosure properties of CFL's
• Undecidable CFL problems

1. Cocke-Younger-Kasami Algorithm for Testing Membership in a CFL

• Input: a Chomsky normal form CFG G = (V, T, P, S) and a string w = a₁a₂ ... a_n in T*.
• Output: "yes" if w is in L(G), "no" otherwise.
• Method: The CYK algorithm is a dynamic programming algorithm that fills in a triangular table Xij with nonterminals A such that A ⇒* a_ia_i+1 ... a_j.

for i = 1 to n do
  if A → ai is in P then
    add A to Xii
fill in the table, row-by-row, from row 2 to row n
  fill in the cells in each row from left-to-right
    if (A → BC is in P) and for some i ≤ k < j
      (B is in Xik) and (C is in Xk+1,j) then
        add A to Xij
if S is in X1n then
  output "yes"
else
  output "no"

• The algorithm adds nonterminal A to Xij iff there is a production A → BC in P where B ⇒* a_ia_i+1 ... a_k and C ⇒* a_k+1a_k+2 ... a_j.
• To compute entry Xij, we examine at most n pairs of entries: (Xii, Xi+1,j), (Xi,i+1, Xi+2,j), and so on until (Xi,j-1, Xj,j).
• The running time of the CYK algorithm is O(n³).

2. Testing Emptiness of a CFG

• Problem: Given a CFG G, is L(G) empty?
• Emptiness problem is decidable: determine whether the start symbol of G is generating.
• Naive algorithm has O(n²) time complexity where n is the size of G (sum of the lengths of the productions).
• With a more sophisticated list-processing algorithm, emptiness problem can be solved in linear time. See HMU, p. 302.

3. Closure Properties of CFL's

• The context-free languages are closed under
• substitution
• Let Σ be an alphabet and let L_a be a language for each symbol a in Σ. These languages define 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
• homomorphism
• reversal
• intersection with a regular set
• inverse homomorphism

4. 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.

5. Undecidable CFL Problems

• We say a problem that cannot be solved by any Turing machine is undecidable. There is no algorithm that can solve an undecidable problem.
• We shall see that several fundamental questions about context-free grammars and languages are undecidable, such as:
1. Is a given CFG ambiguous?
2. Given a CFG, is there another equivalent CFG that is unambiguous?
3. Do two given CFG's generate the same language?
4. Is the intersection of the languages generated by two CFG's empty?
5. Given a CFG G = (V, T, P, s), is L(G) = T*?

6. Practice Problems

1. Let G be the following grammar:

S → AB | BC
A → BA | a
B → CC | b
C → AB | a

Use the CYK algorithm to determine whether aabab is in L(G).
2. Modify the CYK algorithm to report the number of distinct parse trees there are for a given string w in a CNF grammar G.
3. Let min(L) = { w | w is in L but no proper prefix of w is in L }. Are the CFL's closed under the min operation?
4. Let max(L) = { w | w is in L but for no string x other than ε is wx is in L }. Are the CFL's closed under the max operation?
5. Let init(L) = { w | wx is in L for some string x (possibly the empty string) }. Are the CFL's closed under the init operation?
6. Let cycle(L) = { w | we can write w as xy where yx is in L }. Are the CFL's closed under the cycle operation?
7. Let half(L) = { w | there exists a string x such that |w| = |x| and wx is in L }. Are the CFL's closed under the half operation?

7. Reading Assignment

• HMU: Ch. 7