Properties of ContextFree Languages

1
Properties of Context-Free Languages

• Is a certain language context-free?
• Is the family of CFLs closed under a certain
  operation?

2
Pumping Lemma

• Let L be an infinite CFL. Then there exists m ? 0
  such that any w ? L with w ? m can be
  decomposed as w uvxyz where
• vy ? 1
• vxy ? m
• uvixyiz ? L for all i ? 0

3
Pumping Lemma

• Proof
• The RL case S ? xA ? xyA ? xyz
• The CFL case S ? uAz ? uvAyz ? uvxyz

4
Moves in the Game

• The opponent picks m ? 0.
• We choose w ? L with w ? m.
• The opponent chooses the decomposition w uvxyz
  such that vy ? 1 and vxy ? m.
• We pick i such that uvixyiz ? L.

5
Example

• Prove L ww w ? a, b is not a CFL.

6
Moves in the Game

• The opponent picks m ? 0.

7
Moves in the Game

• The opponent picks m ? 0.
• We choose w ambmambm.

8
Moves in the Game

• The opponent picks m ? 0.
• We choose w ambmambm.
• The opponent chooses the decomposition w uvxyz
  such that vy ? 1 and vxy ? m.
• m m m m
• a . . . a b . . . b a . . . a b . . . b
• u v x y z

9
Moves in the Game

• The opponent picks m ? 0.
• We choose w ambmambm.
• The opponent chooses the decomposition w uvxyz
  such that vy ? 1 and vxy ? m.
• m m m m
• a . . . a b . . . b a . . . a b . . . b
• u v x y z
• We pick i such that uvixyiz ? L.

10
Example

• Prove L anbncn n ? 0 is not a CFL.

11
Linear Context-Free Languages

• A CFL L is said to be linear iff there exists a
  linear CFG G such that L L(G).
• (A grammar is linear iff at most 1 variable can
  occur on the right side of any production)

12
Pumping Lemma for Linear CFLs

• Let L be an infinite linear CFL. Then there
  exists m ? 0 such that any w ? L with w ? m can
  be decomposed as w uvxyz where
• vy ? 1
• uvyz ? m
• uvixyiz ? L for all i ? 0

13
Moves in the Game

• The opponent picks m ? 0.
• We choose w ? L with w ? m.
• The opponent chooses the decomposition w uvxyz
  such that vy ? 1 and uvyz ? m.
• We pick i such that uvixyiz ? L.

14
Example

• Prove L w na(w) nb(w) is not linear.

15
Closure Properties of Context-Free Languages

• L1 and L2 are context-free.
• How about L1?L2, L1?L2 , L1L2 , L1, L1 ?

16
Theorem

• If L1 and L2 are context-free, then so are L1?L2
  , L1L2 , L1.
• (The family of context-free languages is closed
  under union, concatenation, and star-closure.)

17
Proof

• G1 (V1, T1, S1, P1) G2 (V2, T2, S2, P2)
• G3 (V1?V2?S3, T1?T2, S3, P1?P2?S3 ? S1
  S2)
• L(G3) L(G1)?L(G2)

18
Proof

• G1 (V1, T1, S1, P1) G2 (V2, T2, S2, P2)
• G4 (V1?V2?S4, T1?T2, S4, P1?P2?S4 ? S1S2)
• L(G4) L(G1).L(G2)

19
Proof

• G1 (V1, T1, S1, P1)
• G5 (V1?S5, T1, S5, P1?S5 ? S1S5 ?)
• L(G5) L(G1)

20
Theorem

• The family of context-free languages is not
  closed under intersection and complement.

21
Proof

• L1 anbncm n ? 0, m ? 0
• L2 anbmcm n ? 0, m ? 0
• L anbncn n ? 0 L1?L2

22
Proof

• L1?L2 L1?L2

23
Homework

• Exercises 2, 7, 8, 9, 14, 15, 16 of Section 8.1.
• Exercises 2, 4, 10, 15 of Section 8.2.
• Presentations
• Section 12.1 Computability and Decidability
  Halting Problem
• Section 13.1 Recursive Functions
• Post Systems Church's Thesis
• Section 13.2 Measures of Complexity
  Complexity Classes