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 uv i xy i z  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 1.The opponent picks m  0. 2.We choose w  L with |w|  m. 3.The opponent chooses the decomposition w = uvxyz such that |vy|  1 and |vxy|  m. 4.We pick i such that uv i xy i z  L.

5 Example Prove L = {ww | w  {a, b}*} is not a CFL.

6 Moves in the Game 1.The opponent picks m  0.

7 Moves in the Game 1.The opponent picks m  0. 2.We choose w = a m b m a m b m.

8 Moves in the Game 1.The opponent picks m  0. 2.We choose w = a m b m a m b m. 3.The opponent chooses the decomposition w = uvxyz such that |vy|  1 and |vxy|  m. m m m m a... a b... b u v x y z

9 Moves in the Game 1.The opponent picks m  0. 2.We choose w = a m b m a m b m. 3.The opponent chooses the decomposition w = uvxyz such that |vy|  1 and |vxy|  m. m m m m a... a b... b u v x y z 4.We pick i such that uv i xy i z  L.

10 Example Prove L = {a n b n c n | 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 uv i xy i z  L for all i  0

13 Moves in the Game 1.The opponent picks m  0. 2.We choose w  L with |w|  m. 3.The opponent chooses the decomposition w = uvxyz such that |vy|  1 and |uvyz|  m. 4.We pick i such that uv i xy i z  L.

14 Example Prove L = {w | n a (w) = n b (w)} is not linear.

15 Closure Properties of Context-Free Languages L 1 and L 2 are context-free. How about L 1  L 2, L 1  L 2, L 1 L 2, L 1, L 1 * ?

16 Theorem If L 1 and L 2 are context-free, then so are L 1  L 2, L 1 L 2, L 1 *. (The family of context-free languages is closed under union, concatenation, and star-closure.)

17 Proof G 1 = (V 1, T 1, S 1, P 1 ) G 2 = (V 2, T 2, S 2, P 2 ) G 3 = (V 1  V 2  {S 3 }, T 1  T 2, S 3, P 1  P 2  {S 3  S 1 | S 2 }) L(G 3 ) = L(G 1 )  L(G 2 )

18 Proof G 1 = (V 1, T 1, S 1, P 1 ) G 2 = (V 2, T 2, S 2, P 2 ) G 4 = (V 1  V 2  {S 4 }, T 1  T 2, S 4, P 1  P 2  {S 4  S 1 S 2 }) L(G 4 ) = L(G 1 ).L(G 2 )

19 Proof G 1 = (V 1, T 1, S 1, P 1 ) G 5 = (V 1  {S 5 }, T 1, S 5, P 1  {S 5  S 1 S 5 | }) L(G 5 ) = L(G 1 )*

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

21 Proof L 1 = {a n b n c m | n  0, m  0} L 2 = {a n b m c m | n  0, m  0} L = {a n b n c n | n  0} = L 1  L 2

22 Proof L 1  L 2 = L 1  L 2

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