1. Non-Context-Free Languages Sections: 2.3 page 123 October 17, 2008
CS310
Non-Context-Free Languages
Sections: 2.3 page 123
October 17, 2008

2. Pumping Lemma (take two)
Theorem: For any CFG there is an equivalent grammar in CNF.
Pumping lemma (CFG): Suppose A is a CFG. There exists a number p such that
if s  A and |s|  p
then s = udxyz where
udixyiz  A, i  0
|dy| > 0
|dxy|  p
For the NFA, P was the number of states in the machine.
Here P is the number of Non-Terminals in the language.
Key, we can pump a string and keep it in the language. By pumping the string we add a level to the parse tree which could produce pumped terminals around the middle.
Note: Your book
ugses uvxyz
But, capital V is the set of variables for the grammar so that can get confusing

3. Pumping a Parse Tree T T | | R R ud2xy2z u d x y z u d y z d x y

4. Proof Suppose A is a CFG in CNF and s  A, |s|  p = 2 |V|+1 2 ?
The height of the parse tree for s is ?

5. Example L = {aibici | i  0} a PDA cannot represent this. Why?
Pumping Lemma:
s = u= d= x= y= z=

6. Example L = {aibjck | k  j  i  0} a PDA cannot represent this. Why?
Pumping Lemma:
s = u= d= x= y= z=

7. Example
L = { ww | w  {0, 1}* }
s =

8. Example
L = { w # x | wR is substring of x; w,x  {0, 1}* }