1

Lecture 38

• Showing CFL’s not closed under set intersection and set complement

2

Nonclosure Properties for CFL’s

3

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

4

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 context-free?

5

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

6

Language class hierarchy

All languages over alphabet

RE

REG

HH

EqualCFLREC

Equal-3