1. 1 Lecture 36 Attempt to prove that CFL’s are closed under intersection –Review previous constructions –Translate previous constructions to current setting –Prove modified result

2. 2 High Level Overview

3. 3 CFL closed under set intersection Let L 1 and L 2 be arbitrary CFL’s Let M 1 and M 2 be PDA’s s.t. L(M 1 ) = L 1, L(M 2 ) = L 2 –M 1 and M 2 exist by definition of L 1 and L 2 are CFL’s and the fact that for every CFG, there is an equivalent PDA Construct PDA M 3 from PDA’s M 1 and M 2 Argue L(M 3 ) = L 1 intersect L 2 There exists a PDA M 3 s.t. L(M 3 ) = L 1 intersect L 2 L 1 intersect L 2 is a CFL

4. 4 Visualization Let L 1 and L 2 be arbitrary CFL’s Let M 1 and M 2 be PDA’s s.t. L(M 1 ) = L 1, L(M 2 ) = L 2 M 1 and M 2 exist by definition of L 1 and L 2 are CFL’s and the fact that for every CFG, there is an equivalent PDA Construct PDA M 3 from PDA’s M 1 and M 2 Argue L(M 3 ) = L 1 intersect L 2 There exists a PDA M 3 s.t. L(M 3 ) = L 1 intersect L 2 L 1 intersect L 2 is a CFL L 1 intersect L 2 L1L1 L2L2 CFL M3M3 M1M1 M2M2 PDA’s

5. 5 Algorithm Specification Input –Two PDA’s M 1 and M 2 Output –PDA M 3 such that L(M 3 ) = L(M 1 ) intersection L(M 2 ) PDA M 1 PDA M 2 PDA M 3 A

6. 6 Review Previous Results

7. 7 Underlying Idea Previous Results –recursive languages are closed under set intersection –r.e. languages are closed under set intersection –regular languages are closed under set intersection What is the idea underlying the constructions used to prove these previous results?

8. 8 Implementation with FSA’s Given the basic idea underlying these constructions, how was this idea implemented in when dealing with FSA’s? That is, restate the construction used to prove that the regular languages are closed under set intersection. –Specify the output FSA in terms of the input FSA’s

9. 9 Applying previous approach to PDA’s

10. 10 Applying approach to PDA’s Given the basic idea underlying these constructions, try and implement this idea in a construction working with PDA’s rather than FSA’s. That is, give a construction specifying how the output PDA is built out of the input PDA’s

11. 11 Problem Describe what goes wrong when applying this idea to PDA’s instead of FSA’s. Does this prove that CFL’s are NOT closed under set intersection?

12. 12 Modified Result What happens if the inputs are –1 PDA –1 FSA Is the problem still a problem? If not, what modified result does the resulting construction prove?