1. Chapter 2 Context-Free Language - 02
Original author : Prof. Y-p Chen
Reproduced : ccming 2010/10/20

2. 2.3 Non-Context-Free Language
2.3.1 Pumping Lemma for CFLs
Theorem 2.34 – Pumping lemma for context-free languages: If A is a context-free language, then there is a number p (the pumping length) where, if s is any string in A of length at least p, then s may be divided into five pieces, s = uvxyz satisfying the conditions

3. Proof idea : When the string is long enough, some variable(s) must be repeatedly used in derivations.
Proof :
Let G be a CFG for CFL A. Let b be the maximum number if symbols in the right-hand side of a rule. From the start variable, there are at most b leaves for one step; at most b2 leaves for 2 steps; at most bh leaves for h steps. Conversely if a generated string is at least bh+1 long, each of its parse tree must be at least h+1 high.
Say |V| is the number of variables in G. We set p, the pumping length, to be b|V|+1. Now if s is a string in A and its length is p or more, its parse tree must be at least |V|+1 high.

4. For any such string s, let τ be the parse tree with the smallest number of nodes. We know that τ must be at least |V|+1 high, so it must contain a path from the root to a leaf of length at least |V|+1. That path has least |V|+2 nodes; one at a terminal, the others at variables. Hence that path has at least |V|+1 variables. With G having only |V| variables, some variable R appears more than once on that. So we have :

6. How to use the pumping lemma for CFLs:
Condition 2: Both v and y are not ε . τ is the parse tree with the smallest number of nodes.
Condition 3: vxy has length at most p. Choose R so that both occurrence fall within the bottom |V|+1 variables on the path.
How to use the pumping lemma for CFLs:
Assume the language, say B, is context free in order to obtain a contradiction.
By the pumping lemma for CLFs, a pumping length p exist, and any string w∈B can be pumped if |w|≧p
Find a string s ∈ B , |s|≧p that s cannot be pumped as described in the pumping lemma
The contradiction is obtained, so that B is not context-free

7. Example 2.36 :

8. 2.3.2 Ogden’s lemma for CFLs

9. Example :

11. 2.3.3 Ogden’s lemma for CFGs

12. 2.3.4 Inherently Ambiguous CFLs
Theorem : There exists an inherently ambiguous CFL
Proof : One of such language is

16. 2.4 Properties of CLFs
2.4.1 substitutions

18. 2.4.2 Closure Properties
Let A and B be context-free languages. Let R be a regular language. The result of the following operations are all context-free languages:

19. 2.4.2 Membership Test : CYK Algorithm

20. Example : w = baaba