Pumping Lemma for CFL

1. [Section 8.1] Pumping Lemma for CFL Let G = (V,,S,P) be a CFG. Suppose we have a string u 2 L(G) with the following derivation: S )* vAz )* vwAyz )* vwxyz = u where v,w,x,y,z 2* and A 2 V. Which other strings (besides u) can be generated by G ?

2. [Section 8.1] Pumping Lemma for CFL Let G = (V,,S,P) be a CFG. Suppose we have a string u 2 L(G) with the following derivation: S )* vAz )* vwAyz )* vwxyz = u where v,w,x,y,z 2*. Which other strings (besides u) can be generated by G ? vwmxymz 2 L(G) for every m ¸ 0 Is it true that for every sufficiently long u there exist v,w,x,y,z satisfying the above derivation ?

3. [Section 8.1] Pumping Lemma for CFL Let G = (V,,S,P) be a CFG. Suppose G is in Chomsky normal form, i.e., every rule is of the form A  or A  A1A2 where A,A1,A22 V and 2 (plus we allow the rule S ). Let’s look at a derivation tree for some u 2 L(G).

4. [Section 8.1] Pumping Lemma for CFL Thm (Pumping lemma for CFL) : Let L be a CFL. Then there exists an integer n > 0 such that for every u 2 L with |u| ¸ n there exist strings v,w,x,y,z satisfying: (1) u = vwxyz (2) |wxy| · n (3) |wy| > 0 (4) vwmxymz 2 L for every m ¸ 0

5. [Section 8.1] Pumping Lemma for CFL Example: L = { akbkck | k ¸ 0 }

6. [Section 8.1] Pumping Lemma for CFL Example: L = { aibjck | 0 · i · j · k }

7. [Section 8.2] Closure Properties, Part II Lemma: CFL’s are not closed under intersection.

8. [Section 8.2] Closure Properties, Part II Lemma: CFL’s are not closed under complement.

9. [Section 8.2] Closure Properties, Part II Lemma: CFL’s are not closed under difference.

10. [Section 8.2] Closure Properties, Part II Lemma: CFL’s are closed under intersection with a regular language.

11. [Section 8.2] Closure Properties, Part II Example: L = { w 2 {a,b,c}* | na(w) = nb(w) = nc(w) }

12. [Section 8.3] Decision Problems for CFL’s • A decision problem : answer is YES or NO. • Give algorithms for the following decision problems : • Given a CFL L (by a PDA or a CFG) and a string x, is x in L ?

13. [Section 8.3] Decision Problems for CFL’s • A decision problem : answer is YES or NO. • Give algorithms for the following decision problems : • Given a CFL L (by a PDA or a CFG), is L = ; ?

14. [Section 8.3] Decision Problems for CFL’s • A decision problem : answer is YES or NO. • Give algorithms for the following decision problems : • Given a CFL L (by a PDA or a CFG), is L finite ?