1. Lecture 11UofH - COSC 3340 - Dr. Verma 1 COSC 3340: Introduction to Theory of Computation University of Houston Dr. Verma Lecture 11

2. UofH - COSC 3340 - Dr. Verma 2 Push Down Automaton (PDA) Language Acceptor Model for CFLs It is an NFA with a stack. Finite State control Input Stack Accept/Reject

3. Lecture 11UofH - COSC 3340 - Dr. Verma 3 PDA (contd.) In one move the PDA can : – change state, – consume a symbol from the input tape or ignore it – pop a symbol from the stack or ignore it – push a symbol onto the stack or not A string is accepted provided the machine when started in the start state consumes the string and reaches a final state.

4. Lecture 11UofH - COSC 3340 - Dr. Verma 4 PDA (contd.) If PDA in state q can consume u, pop x from stack, change state to p, and push w on stack we show it as q0q0 u, x  w q1q1 u, x ; w In JFLAP

5. Lecture 11UofH - COSC 3340 - Dr. Verma 5 Example of a PDA PDA L = {a n b n |n  0} Push S to the stack in the beginning and then pop it at the end before accepting.

6. Lecture 11UofH - COSC 3340 - Dr. Verma 6 JFLAP Simulation

7. Lecture 11UofH - COSC 3340 - Dr. Verma 7 JFLAP Simulation

8. Lecture 11UofH - COSC 3340 - Dr. Verma 8 JFLAP Simulation

9. Lecture 11UofH - COSC 3340 - Dr. Verma 9 JFLAP Simulation

10. Lecture 11UofH - COSC 3340 - Dr. Verma 10 JFLAP Simulation

11. Lecture 11UofH - COSC 3340 - Dr. Verma 11 JFLAP Simulation

12. Lecture 11UofH - COSC 3340 - Dr. Verma 12 JFLAP Simulation

13. Lecture 11UofH - COSC 3340 - Dr. Verma 13 JFLAP Simulation

14. Lecture 11UofH - COSC 3340 - Dr. Verma 14 JFLAP Simulation

15. Lecture 11UofH - COSC 3340 - Dr. Verma 15 JFLAP Simulation

16. Lecture 11UofH - COSC 3340 - Dr. Verma 16 JFLAP Simulation

17. Lecture 11UofH - COSC 3340 - Dr. Verma 17 JFLAP Simulation

18. Lecture 11UofH - COSC 3340 - Dr. Verma 18 JFLAP Simulation

19. Lecture 11UofH - COSC 3340 - Dr. Verma 19 JFLAP Simulation

20. Lecture 11UofH - COSC 3340 - Dr. Verma 20 JFLAP Simulation

21. Lecture 11UofH - COSC 3340 - Dr. Verma 21 JFLAP Simulation

22. Lecture 11UofH - COSC 3340 - Dr. Verma 22 JFLAP Simulation

23. Lecture 11UofH - COSC 3340 - Dr. Verma 23 JFLAP Simulation

24. Lecture 11UofH - COSC 3340 - Dr. Verma 24 JFLAP Simulation

25. Lecture 11UofH - COSC 3340 - Dr. Verma 25 Definition of PDA Formally, a PDA M = (K, , , , s, F), where – K -- finite set of states –  -- is the input alphabet –  -- is the tape alphabet – s  K -- is the start state – F  K -- is the set of final states –   (K X   X   ) X (K X   )

26. Lecture 11UofH - COSC 3340 - Dr. Verma 26 Definition of L(M) Define  * as: (1)  *(q, w, x) = {(q, w, x)}  {(p, w, x) | ((q, ,  ), (p,  ))   } (2)  *(q, uv, xy) = U {  *(p, v, wy) | ((q, u, x), (p, w))   } [at least one of u, x, w is not equal to  ] – i.e., first compute  * for all successor configurations and then take the union of all those sets M accepts w if (f, , x) in  *(s, w,  ) Alternative: if (f, ,  ) in  *(s, w,  ) [we use] L(M) = {w   * | M accepts w}

27. Lecture 11UofH - COSC 3340 - Dr. Verma 27 Example What is L(M)? Push S to the stack in the beginning and then pop it at the end before accepting.