Growth of Functions

1
Chapter 8 Properties of CFL
2
8.2 Closure Properties for CFL (1)

• A set is closed under some operation if
  performing that operation on members of the set
  always produces a member of that set.
• Assume that
• L1, L2 are context-free languages
• CFG for L1 is G1 (NT1 , ?1, P1, S1
• CFG for L2 is G2 (NT2 , ?2, P2, S2
• R is a regular language
• CFLs are closed under the operations of
• Union, concatenation, Kleene closure
• CFLs are not closed under
• Intersection, complementation

3
8.2 Closure Properties for CFL (2)

• Union L L1 ? L2
• G (S ? NT1 ? NT2, ?1 ? ?2,
• P1 ? P2 ? S ? S1 S2, S)
• Concatenation L L1L2
• G (S ? NT1 ? NT2, ?1 ? ?2,
• P1 ? P2 ? S ? S1S2, S)
• Closure L L1
• G (S ? NT1, ?1, P1 ? S ? ? S1S, S)
• Assumes Non-terminl names distinct in NT1 and
  NT2. Otherwise, can rename consistently.

4
8.2 Closure Properties for CFL (3)

• Intersection L L1 ? L2
• L not necessarily context-free.
• CFLs are not closed under intersection!
• Cant simulate two stacks by one stack.
• L1 an bn cm n, m ? 0
• is CFL
• L2 an bm cm n, m ? 0
• is CFL
• L L1 ? L2 ak bk ck k ? 0
• is NOT CFL

5
8.2 Closure Properties for CFL (4)

• __
• Complementation L L1
• L not necessarily context-free.
• CFLs are not closed under complementation!
• By DeMorgans Law,
• since union closed,
• if complementation closed, so is intersection.
  (contradiction)

6
8.2 Closure Properties for CFL (5)

• Intersection with RL L L1 ? R
• L is context-free.
• Intuition Only one stack to simulate, so OK.
• Example show that L anbn n 0, n ? 100 is
  context-free
• L1 anbn n 0 is context-free
• L2 a100b100 is finite, so it is regular
• __
• L2 is regular (regular languages closed under
  complementation)
• __
• L L1 ? L2 is context-free

7
8.2 Closure Properties for CFL (6)

• For any context free grammar, we can decide
• membership (can the grammar generate a given
  string)
• if it is empty (the grammar does not generate any
  strings)
• if it is infinite (the grammar generates and
  infinite number of strings)
• However, there is no algorithm for deciding
  whether two context-free grammars generate the
  same language!