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

2. Last Time: New: 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 Hierarchy
Pumping Lemma for Context Free Languages

3. Algorithm for generating and reachable symbols
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

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
Grammar
Languages
Automaton
Production rules (constraints)
Type-0
Recursively enumerable
Turing machine
α  β
no restrictions
Type-1
Context-sensitive
Linear-bounded non-deterministic Turing machine
αAβ  αγβ
Type-2
Context-free
Non-deterministic pushdown automaton
A  α
Type-3
Regular
DFA
A  a or A  aB

12. Chomsky Hierarchy Venn Diagram

13. Backus Naur Form (BNF) Backus Naur Form John Backus Peter Naur
N ::= α | … | β (just a CFG)
John Backus
Fortran compiler
Peter Naur

14. Greibach Normal Form Each production RHS starts with a terminal
A  aα or S ε

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
uviwxi y 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