Module 35

1
Module 35

• Attempt to prove that CFLs are closed under
  intersection
• Review previous constructions
• Translate previous constructions to current
  setting
• Prove modified result

2
High Level Overview
3
CFL closed under set intersection

• Let L1 and L2 be arbitrary CFLs
• Let M1 and M2 be PDAs s.t. L(M1) L1, L(M2)
  L2
• M1 and M2 exist by definition of L1 and L2 are
  CFLs and the fact that for every CFG, there is
  an equivalent PDA
• Construct PDA M3 from PDAs M1 and M2
• Argue L(M3) L1 intersect L2
• There exists a PDA M3 s.t. L(M3) L1 intersect
  L2
• L1 intersect L2 is a CFL

4
Visualization

• Let L1 and L2 be arbitrary CFLs
• Let M1 and M2 be PDAs s.t. L(M1) L1, L(M2)
  L2
• M1 and M2 exist by definition of L1 and L2 are
  CFLs and the fact that for every CFG, there is
  an equivalent PDA
• Construct PDA M3 from PDAs M1 and M2
• Argue L(M3) L1 intersect L2
• There exists a PDA M3 s.t. L(M3) L1 intersect
  L2
• L1 intersect L2 is a CFL

CFL
5
Algorithm Specification

• Input
• Two PDAs M1 and M2
• Output
• PDA M3 such that L(M3) L(M1) intersection L(M2)
  

PDA M1 PDA M2
PDA M3
6
Review Previous Results
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
Implementation with FSAs

• Given the basic idea underlying these
  constructions, how was this idea implemented in
  when dealing with FSAs?
• 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 FSAs

9
Applying previous approach to PDAs
10
Applying approach to PDAs

• Given the basic idea underlying these
  constructions, try and implement this idea in a
  construction working with PDAs rather than
  FSAs.
• That is, give a construction specifying how the
  output PDA is built out of the input PDAs

11
Problem

• Describe what goes wrong when applying this idea
  to PDAs instead of FSAs.
• Does this prove that CFLs are NOT closed under
  set intersection?

12
Modified Result

• What happens if the inputs are 1 FSA and 1 PDA?
• What modified result does the resulting
  construction prove?