COMS W3261
Computer Science Theory
Lecture 16: November 4, 2009
Decision Properties of CFL's

1. Outline

• Review
• Time complexity of an algorithm
• Decision properties of CFL's
• Undecidable problems for CFL's

2. Review

• Pumping lemma for CFL's
• Closure properties of CFL's
• Nonclosure properties of CFL's

3. Time Compexity of an Algorithm

• Given an algorithm A to solve a problem P, we say A has time complexity T(n), if A can solve any instance of problem P of size n in O(T(n)) steps.
• The time complexity of an algorithm is its worst-case running time on any problem of size n.
• We say T(n) is
• linear if T(n) ≤ cn for some constant c
• quadratic if T(n) ≤ cn² for some constant c
• exponential if T(n) ≤ c2ⁿ for some constant c

4. 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.

5. 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
for row = 2 to n do
  for col = 1 to row - 1 do
    for k = col to row - 1 do {
      i = col
      j = col + row - 1
      if (A → BC is in P) and
         (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). Thus, the running time of the CYK algorithm is O(n³).

6. Undecidable Problems for CFL's

• If a problem cannot be solved by any algorithm, then the problem is said to be undecidable.
• The following problems are undecidable:
• Is a given CFG ambiguous?
• Is a given CFL inherently ambiguous?
• Is the intersection of two CFL's empty?
• Given two CFG's G₁ and G₂, is L(G₁) = L(G₂)?
• Given a CFG G = (V, T, P, S), is L(G) = T*?

7. Practice Problems

1. HMU, Exercise 7.4.1(a).
2. HMU, Exercise 7.4.3(a).
3. Augment the CYK algorithm to construct all parse trees for the given input. What is the time complexity of your algorithm? Show how your algorithm works on Exercise 7.4.3(a).
4. (Hard) HMU, Exercise 7.4.1(c).

8. Reading Assignment

• HMU: Sects. 7.4-7.7