Lecture 22 Pumping Lemma for Context Free Languages

1. Lecture 22Pumping Lemma forContext Free Languages CSCE 355 Foundations of Computation Topics: • Normal forms • Pumping Lemma for CFLs • Closure properties November 19, 2008

2. Last Time: • Useless symbols: • generating symbols, • useful symbols • Algorithm for generating and reachable symbols • Removal of useless symbols • Removal of epsilon productions; • Removal of unit productions • Chomsky normal form New: • Chomsky normal form • Chomsky Hierarchy • Pumping Lemma for Context Free Languages

3. Useless symbols: • generating symbols, • useful symbols • Algorithm for generating and reachable symbols • Removal of useless symbols • Removal of epsilon productions; • Removal of unit productions • Chomsky normal form

4. Chomsky Normal Form A CFG (Context Free Grammar) is in Chomsky Normal form if productions are one of the following two forms: • A  BC • A  a References http://www.chomsky.info/

5. Conversion to Chomsky Normal Form • Remove: ε-productions, unit productions • A  BCDE • A  abc • In general • For each terminal ‘a’ create a new non-terminal Na with Na a added as a production • A  B1B2…Bk create a new non-terminals C1C2…Ck and replace the production with • A  B1C1 and • Ci  Bi+1Ci+1 for i=1,…k-3 • Ck-2  Bk-1Bk

6. Example

9. Regular Grammars A CFG is regular if all productions are of the form: A  a or A  aB Note sentential forms in a derivation based on a regular grammar have a unique form! What is it ? Grammar  NFA construction • Create a state for each nonterminal. • A  aB means δ(A, a) = B and • A  a means δ(A, a) = Qfinal and

10. Example

11. Chomsky Hierarchy http://en.wikipedia.org/wiki/Chomsky_hierarchy

12. Chomsky Hierarchy Venn Diagram

13. Backus Naur Form (BNF) • Backus Naur Form • N ::= α | … | β (just a CFG) • http://en.wikipedia.org/wiki/Backus-Naur_form • John Backus • Fortran compiler • http://en.wikipedia.org/wiki/John_Backus • Peter Naur • http://en.wikipedia.org/wiki/Peter_Naur

14. Greibach Normal Form • Each production RHS starts with a terminal • A  aα or S ε • http://en.wikipedia.org/wiki/Greibach_normal_form

15. Showing Languages are not CFLs Recursive productions A  a A | b B  B a | b D  aDb | d A * α A β

16. Pumping Lemma for CFLs Let L be a CFL. Then there exists a constant n such that if z is a string in L of length at least n, then we can write z = uvwxy such that • |vwx| =< n • |vx| > 0 • uviwxiy is in L for all i >= 0.

17. Idea behind proof • Assume CNF (or do for L(G)-{ε}) • Consider Parse Tree • Sufficiently long string z, means the parse tree must be sufficiently big.

18. Similarities to Pumping Lemma for Regular Languages • Given an arbitrary n. • Carefully choose z in L (depending on n) with |z| >= n. • Then for any partition z = uvwxy that satisfies • |vx| > 0 • |vwx| <= n • We must be able to “pump”, i.e. • uviwxiy is in L for all i >= 0

19. Example L = {anbncn | n > 0} • Given L as above, suppose we chose n for the Pumping Lemma (for CFLs). • Choose z = • Consider arbitrary partition of z = uvwxy satisfying • | vwx| =< n • |vx| > 0 • Then show …

20. Example

36. Homework • 7.1.4 • 7.1.3 • 7.1.6