1 Pushdown Automata There are context-free languages that are not regular. Finite automata cannot recognize all context-free languages.

2 Pushdown Automata {a, b}* is regular. {a k b k | k is a constant} is regular. {a n b n | n  0} is not regular.

3 Finite Automata Control unit q 0 Input file yes/no

4 Pushdown Automata Control unit q 0 Input file yes/no Stack

5 Non-deterministic Pushdown Automata (NPDA) M = (Q, , , , q 0, z, F) Q: finite set of internal states  : finite set of symbols - input alphabet  : finite set of symbols - stack alphabet  : Q  (  { })    finite subsets of Q   * transition function q 0  Q: initial state z   : stack start symbol F  Q: set of final states

6 Non-deterministic Pushdown Automata (NPDA)  : Q  (  { })    finite subsets of Q   * stack topstack top replacement

7 Example  (q 1, a, b) = {(q 2, cd), (q 3, )} b d c q1q1 q2q2 q3q3

8 Example M = (Q, , , , q 0, z, F) Q = {q 0, q 1, q 2, q 3 }  (q 0, a, 0) = {(q 1, 10), (q 3, )}  = {a, b}  (q 0,, 0) = {(q 3, )}  = {0, 1}  (q 1, a, 1) = {(q 1, 11)} z = 0  (q 1, b, 1) = {(q 2, )} F = {q 3 }  (q 2, b, 1) = {(q 2, )}  (q 2,, 0) = {(q 3, )}

9 Instantaneous Description (q, w, u) current stateunread part ofstack contents input string

10 Instantaneous Description move: (q 1, aw, bx)  (q 2, w, yx) iff (q 2, y)   (q 1, a, b)

11 Instantaneous Description (q 1, x, y)     (q 2, u, v) (q 1, x, y)     (q 2, u, v)

12 Language accepted by NPDA Let M = (Q, , , , q 0, z, F) be an NPDA. L(M) = {w   * | (q 0, w, z)     (q f,, u), q f  F, u   * }

13 Example L = { w  {a, b} * | n a (w) = n b (w) } M = (Q, , , , q 0, z, F) ?

14 Example L = { w  {a, b} * | n a (w) = n b (w) } M = (Q, , , , q 0, z, F) Q = {q 0, q f }  (q 0,, z) = {(q f, z)}  = {a, b}  (q 0, a, z) = {(q 0, 0z)}  = {0, 1, z}  (q 0, b, z) = {(q 0, 1z)} F = {q f }  (q 0, a, 0) = {(q 0, 00)}  (q 0, b, 0) = {(q 0, )}  (q 0, a, 1) = {(q 0, )}  (q 0, b, 1) = {(q 0, 11)}

15 Example L = { ww R | w  {a, b} + } M = (Q, , , , q 0, z, F) ?

16 Example L = { ww R | w  {a, b} + } M = (Q, , , , q 0, z, F) Q = {q 0, q 1, q 2 }  = {a, b}  = {a, b, z} F = {q 2 }  (q 0, a, a) = {(q 0, aa)}  (q 0,, a) = {(q 1, a)}  (q 0, b, a) = {(q 0, ba)}  (q 0,, b) = {(q 1, b)}  (q 0, a, b) = {(q 0, ab)}  (q 0, b, b) = {(q 0, bb)}  (q 1, a, a) = {(q 1, )}  (q 0, a, z) = {(q 0, az)}  (q 1, b, b) = {(q 1, )}  (q 0, b, z) = {(q 0, bz)}  (q 1,, z) = {(q 2, z)}

17 NPDA and Context-Free Languages Greibach NF  (q 0,, z) = {(q 1, Sz)} S  aSA | a  (q 1, a, S) = {(q 1, SA), (q 0, )} A  bB  (q 1, b, A) = {(q 1, B)} B  b  (q 1, b, B) = {(q 1, )}  (q 1,, z) = {(q 2, )}

18 Theorem For any context-free language L not containing, there exists an NPDA M such that L = L(M).

19 Theorem Proof: G = (V, T, S, P) M = ({q 0, q 1, q f }, T, V  {z}, , q 0, z, {q f })z  V  (q 0,, z) = {(q 1, Sz)} (q 1, u)   (q 1, a, A) iff A  au  P  (q 1,, z) = {(q f, z)}

20 Example Greibach NF  (q 0,, z) = {(q 1, Sz)} S  aA  (q 1, a, S) = {(q 1, A)} A  aABC | bB | a  (q 1, a, A) = {(q 1, ABC), (q 1, )}  (q 1, b, A) = {(q 1, B)} B  b  (q 1, b, B) = {(q 1, )} C  c  (q 1, c, C) = {(q 1, )}  (q 1,, z) = {(q 2, )}

21 Context-Free Grammars for NPDA M = (Q, , , , q 0, z, F) G = (V, T, S, P) L(G) = L(M)

22 Context-Free Grammars for NPDA M = (Q, , , , q 0, z, F) M^: Single final state Final state entered iff the stack is empty  (q i, a, A) = {c 1, c 2,..., c n } c i = (q j, ) c i = (q j, BC)

23 Context-Free Grammars for NPDA  (q i, a, A) = {(q j, ),... } ?

24 Context-Free Grammars for NPDA  (q i, a, A) = {(q j, ),... } ? At q i erase A and move to q j if receiving a

25 Context-Free Grammars for NPDA  (q i, a, A) = {(q j, BC),... } ?

26 Context-Free Grammars for NPDA  (q i, a, A) = {(q j, BC),... } ? At q i erase A and move to q k if receiving a and at q j erase BC and move to q k

27 Context-Free Grammars for NPDA  (q i, a, A) = {(q j, BC),... } ? At q i erase A and move to q k if receiving a and at q j erase BC and move to q k At q i erase A and move to q k if receiving a and at q j erase B and move to q m and at q m erase C and move to q k

28 Context-Free Grammars for NPDA  (q i, a, A) = {(q j, ),... } At q i erase A and move to q j if receiving a (q i Aq j )  a

29 Context-Free Grammars for NPDA  (q i, a, A) = {(q j, BC),... } At q i erase A and move to q k if receiving a and at q j erase B and move to q m and at q m erase C and move to q k (q i Aq k )  a(q j Bq m )(q m Cq k )

30 Context-Free Grammars for NPDA Start symbol: (q 0 zq f )

31 Example M = (Q, , , , q 0, z, F) Q = {q 0, q 1, q 2 }  = {a}  = {A, z} F = {q 2 }  (q 0, a, z) = {(q 0, Az)}  (q 0, a, A) = {(q 0, A)}  (q 0, b, A) = {(q 1, )}  (q 1,, z) = {(q 2, )}

32 Example M = (Q, , , , q 0, z, F) Q = {q 0, q 1, q 2, q 3 }  = {a}  = {A, z} F = {q 2 }  (q 0, a, z) = {(q 0, Az)}  (q 0, a, A) = {(q 0, A)}  (q 0, a, A) = {(q 3, )}  (q 0, b, A) = {(q 1, )}  (q 3,, z) = {(q 3, Az)}  (q 1,, z) = {(q 2, )}  (q 0, b, A) = {(q 1, )}  (q 1,, z) = {(q 2, )}

33 Example M = (Q, , , , q 0, z, F) Q = {q 0, q 1, q 2, q 3 }  = {a}  = {A, z} F = {q 2 }  (q 0, a, z) = {(q 0, Az)}  (q 0, a, A) = {(q 3, )} (q 0 Aq 3 )  a  (q 3,, z) = {(q 3, Az)}  (q 0, b, A) = {(q 1, )} (q 0 Aq 1 )  b  (q 1,, z) = {(q 2, )} (q 1 zq 2 ) 

34 Example M = (Q, , , , q 0, z, F) Q = {q 0, q 1, q 2, q 3 }  = {a}  = {A, z} F = {q 2 }  (q 0, a, z) = {(q 0, Az)} (q 0 zq 0 ) ...  (q 0, a, A) = {(q 3, )} (q 0 zq 1 ) ...  (q 3,, z) = {(q 3, Az)} (q 0 zq 2 ) ...  (q 0, b, A) = {(q 1, )} (q 0 zq 3 ) ...  (q 1,, z) = {(q 2, )}

35 Example M = (Q, , , , q 0, z, F) Q = {q 0, q 1, q 2, q 3 }  = {a}  = {A, z} F = {q 2 }  (q 0, a, z) = {(q 0, Az)} (q 0 zq 0 )  a(q 0 Aq 0 )(q 0 zq 0 ) |  (q 0, a, A) = {(q 3, )} a(q 0 Aq 1 )(q 1 zq 0 ) |  (q 3,, z) = {(q 3, Az)} a(q 0 Aq 2 )(q 2 zq 0 )  (q 0, b, A) = {(q 1, )} a(q 0 Aq 3 )(q 3 zq 0 )  (q 1,, z) = {(q 2, )}...

36 Theorem If L = L(M) for some NPDA M, then L is a context-free language.

37 Theorem Proof: M = (Q, , , , q 0, z, {q f }) G = (V, T, S, P) T =  V = {(q i Aq j ) | A   }S = (q 0 zq f ) P: (q i Aq j )  a iff (q j, )   (q i, a, A) (q i Aq m )  a(q i Bq m )(q m Cq j ) iff (q j, BC)   (q i, a, A)

38 Deterministic Pushdown Automata A DPDA is a pushdown automaton that never has a choice in its move: 1.  (q, a, b) contains at most one element. 2.if  (q,, b) is not empty, then  (q, a, b) must be empty for every a  .

39 Deterministic Context-Free Language A language L is said to be a DCFL iff there exists a DPDA M such that L = L(M).

40 Example M = (Q, , , , q 0, z, F) Q = {q 0, q 1, q 2 }  (q 0, a, 0) = {(q 1, 10)}  = {a, b}  (q 1, a, 1) = {(q 1, 11)}  = {0, 1}  (q 1, b, 1) = {(q 2, )} z = 0  (q 2, b, 1) = {(q 2, )} F = {q 0 }  (q 2,, 0) = {(q 0, )} L = {a n b n | n  0}

41 Homework Exercises: 5, 10, 13 of Section Linz’s book. Exercises: 1, 2, 4, 5, 8, 12 of Section Linz’s book. Exercises: 1, 2, 3, 6, 7 of Section Linz’s book. Presentations: Section 6.3 and Section 7.4.