Finite Automata

1. Finite Automata Irum Feroz Irum.feroz@superior.edu.pk

2. Why FAs ? • • Recognition • Is string x in language L?

3. Deterministic Finite Automaton (DFA) Input Tape String Output “Accept” or “Reject” Finite Automaton

4. 11 1 0 0,1 1 0111 111 1 0 0 1 The machine accepts a string if the process ends in a double circle Steven Rudich: www.cs.cmu.edu/~rudich rudich0123456789

5. 1 0 0,1 accept states (F) states 1 0 0 1 states Anatomy of a Deterministic Finite Automaton q1 q0 q2 The machine accepts a string if the process ends in a double circle start state (q0) q3 Steven Rudich: www.cs.cmu.edu/~rudich rudich0123456789

6. Steven Rudich: www.cs.cmu.edu/~rudich rudich0123456789

7. 1 0 0,1 1 q1 0 0 1 q0 q2 q3 Anatomy of a Deterministic Finite Automaton The alphabet of a finite automaton is the set where the symbols come from: {0,1} The language of a finite automaton is the set of strings that it accepts Steven Rudich: www.cs.cmu.edu/~rudich rudich0123456789

8. 0,1 q0 All strings of 0s and 1s L(M) =  The Language of Machine M Steven Rudich: www.cs.cmu.edu/~rudich rudich0123456789

9. Transition Graph initial state accepting state transition state Costas Busch - RPI

10. Alphabet For every state, there is a transition for every symbol in the alphabet

11. Initial Configuration head Input Tape Input String Initial state Costas Busch - RPI

12. Scanning the Input Costas Busch - RPI

13. Costas Busch - RPI

14. Costas Busch - RPI

15. Input finished accept Costas Busch - RPI

16. A Rejection Case Input String Costas Busch - RPI

17. Costas Busch - RPI

18. Costas Busch - RPI

19. Input finished reject Costas Busch - RPI

20. Another Rejection Case Tape is empty Input Finished reject Costas Busch - RPI

21. Language Accepted: Costas Busch - RPI

22. To accept a string: all the input string is scanned and the last state is accepting To reject a string: all the input string is scanned and the last state is non-accepting Costas Busch - RPI

23. Another Example Accept state Accept state Accept state Costas Busch - RPI

24. Empty Tape Input Finished accept Costas Busch - RPI

25. Another Example trap state Accept state Costas Busch - RPI

26. Input String Costas Busch - RPI

27. Costas Busch - RPI

28. Costas Busch - RPI

29. Input finished accept Costas Busch - RPI

30. A rejection case Input String Costas Busch - RPI

31. Costas Busch - RPI

32. Costas Busch - RPI

33. Input finished reject

34. Language Accepted:

35. Another Example Alphabet: Language Accepted:

36. Transition Table for symbols states

38. All binary strings containing substring 001

39. Construct DFA to accept (0+1)* 0 1

40. Identify following DFA?? 0 0 0 1 1 1 1 0 00(0+1)*

41. (0+1)*00(0+1)* 1 0 0 0 1 1

42. (0+1)*00 0 1 0 0 1 1

43. (0+1)*(00+01) 0 1 0 0 1 1 0 1

44. (a+b)*abb

45. Exercise • ab(a+b)* • a(a+b)* • (a+b)*a • (a+b)*aa (a+b)* • Exactly two strings baa and ab • (a+b)*(aa+bb) (a+b)* • (a+b) (a+b)b(a+b)* • all words containing aaa or bbb • (a*ba*ba*ba*)*

46. ab(a+b)* accept

47. Find language?