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Pushdown Automata

CSC 4170 Theory of Computation. Pushdown Automata. Section 2.2. 2.2.a. Components of a pushdown automaton (PDA). Stack. Input. a a b a c …. x y x z. Push : write a symbol on the top of the stack Pop : delete a symbol from the top of the stack. a,x  y. (Q,  ,  ,  ,s,F). q1.

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Pushdown Automata

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  1. CSC 4170 Theory of Computation Pushdown Automata Section 2.2

  2. 2.2.a Components of a pushdown automaton (PDA) Stack Input a a b a c … x y x z . . . Push: write a symbol on the top of the stack Pop: delete a symbol from the top of the stack a,xy (Q,,,,s,F) q1 q2 If the input symbol is a and the top stack symbol is x, go from q1 to q2, pop x and push y Q is the set of states  is the input alphabet  is the stack alphabet  is the transition function s is the start state FQ is the set of accept states If a=, the read head is not advanced If x=, nothing is popped If y=, nothing is pushed

  3. 2.2.b1 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 0 0 1 1 1 Stack Input

  4. 2.2.b2 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 0 0 1 1 1 $ Stack Input

  5. 2.2.b3 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 $ 0 0 0 1 1 1 Stack Input

  6. 2.2.b4 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 0 $ 0 0 0 1 1 1 Stack Input

  7. 2.2.b5 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 0 0 $ 0 0 0 1 1 1 Stack Input

  8. 2.2.b6 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 0 $ 0 0 0 1 1 1 Stack Input

  9. 2.2.b7 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 $ 0 0 0 1 1 1 Stack Input

  10. 2.2.b8 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 0 0 1 1 1 $ Stack Input

  11. 2.2.b9 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  Accept 0 0 0 1 1 1 Stack Input

  12. 2.2.b10 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  What language does this automaton recognize?

  13. 2.2.b11 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 0 1 Stack Input

  14. 2.2.b12 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 0 1 $ Stack Input

  15. 2.2.b13 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 $ 0 0 1 Stack Input

  16. 2.2.b14 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 0 $ 0 0 1 Stack Input

  17. 2.2.b15 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  Reject 0 $ 0 0 1 Stack Input

  18. 2.2.b16 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 1 1 Stack Input

  19. 2.2.b17 How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  $ 0 1 1 Stack Input

  20. 2.2.b18 Giorgi JaparidzeTheory of Computability How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 $ 0 1 1 Stack Input

  21. 2.2.b19 Giorgi JaparidzeTheory of Computability How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  $ 0 1 1 Stack Input

  22. 2.2.b20 Giorgi JaparidzeTheory of Computability How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  Reject 0 1 1 Stack Input

  23. 2.2.b21 Giorgi JaparidzeTheory of Computability How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 1 0 Stack Input

  24. 2.2.b22 Giorgi JaparidzeTheory of Computability How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  $ 0 1 0 Stack Input

  25. 2.2.b23 Giorgi JaparidzeTheory of Computability How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  0 $ 0 1 0 Stack Input

  26. 2.2.b24 Giorgi JaparidzeTheory of Computability How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  $ 0 1 0 Stack Input

  27. 2.2.b25 Giorgi JaparidzeTheory of Computability How a PDA works ,$ 0,0 q1 q2 1,0   1,0  q3 q4 ,$  Reject 0 1 0 Stack Input

  28. 2.2.c Giorgi JaparidzeTheory of Computability Designing pushdown automata Design a pushdown automaton that recognizes the language {w | w has an equal number of 0s and 1s} 0 = s 1

  29. 2.2.d Giorgi JaparidzeTheory of Computability Converting NFA into PDA Every NFA can be understood as a PDA that never pushes or pops. Just replace every label a of the NFA by a, 1 1 b b,  a , a, a a, a b a, b, 2 3 2 3

  30. 2.2.e Giorgi JaparidzeTheory of Computability Main theorems Theorem 2.20:A language is context-free iff some pushdown automaton recognizes it. Theorem:Not every nondeterministic PDA has an equivalent deterministic PDA. Example 2.18:There is a nondeterministic PDA recognizing {wwR | w{0,1}* } (wR means w reversed), but no deterministic PDA can recognize this language. Proofs omitted.

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