Lecture 38
• Showing CFL’s not closed under set
intersection and set complement
1
Nonclosure Properties for CFL’s
2
CFL’s not closed under set
intersection
• How do we prove that CFL’s are not closed
under set intersection?
– State closure property as IF-THEN statement
• If L1 and L2 are CFL’s, then L1 intersect L2 is a CFL
– Proof is by counterexample
• Find 2 CFL’s L1 and L2 such that L1 intersect L2 is
NOT a CFL
3
Counterexample
• What is a possible L1 intersect L2?
– What non-CFL languages do we know?
• What could L1 and L2 be?
– L1 =
– L2 =
– How can we prove that L1 and L2 are contextfree?
4
CFL’s not closed under
complement
• How can we prove that CFL’s are not closed
under complement?
– We could do the same thing, find a
counterexample
– Another way
• Use fact that any language class which is closed
under union and complement must also be closed
under intersection
5
Language class hierarchy
H
H
Equal
Equal-3
REG
CFL
REC
RE
All languages over alphabet S
6