1. Fall 2006Costas Busch - RPI1 Deterministic Finite Automata And Regular Languages

2. Fall 2006Costas Busch - RPI2 Deterministic Finite Automaton (DFA) Input Tape “Accept” or “Reject” String Finite Automaton Output

3. Fall 2006Costas Busch - RPI3 Transition Graph initial state accepting state transition

4. Fall 2006Costas Busch - RPI4 For every state, there is a transition for every symbol in the alphabet Alphabet

5. Fall 2006Costas Busch - RPI5 Initial Configuration Input String Initial state Input Tape head

6. Fall 2006Costas Busch - RPI6 Scanning the Input

7. Fall 2006Costas Busch - RPI7

8. Fall 2006Costas Busch - RPI8

9. Fall 2006Costas Busch - RPI9 accept Input finished

10. Fall 2006Costas Busch - RPI10 A Rejection Case Input String

11. Fall 2006Costas Busch - RPI11

12. Fall 2006Costas Busch - RPI12

13. Fall 2006Costas Busch - RPI13 reject Input finished

14. Fall 2006Costas Busch - RPI14 Another Rejection Case Tape is empty reject Input Finished

15. Fall 2006Costas Busch - RPI15 Language Accepted:

16. Fall 2006Costas Busch - RPI16 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

17. Fall 2006Costas Busch - RPI17 Another Example Accept state Accept state Accept state

18. Fall 2006Costas Busch - RPI18 Empty Tape accept Input Finished

19. Fall 2006Costas Busch - RPI19 Another Example Accept state trap state

20. Fall 2006Costas Busch - RPI20 Input String

21. Fall 2006Costas Busch - RPI21

22. Fall 2006Costas Busch - RPI22

23. Fall 2006Costas Busch - RPI23 accept Input finished

24. Fall 2006Costas Busch - RPI24 A rejection case Input String

25. Fall 2006Costas Busch - RPI25

26. Fall 2006Costas Busch - RPI26

27. Fall 2006Costas Busch - RPI27 reject Input finished

28. Fall 2006Costas Busch - RPI28 Language Accepted:

29. Fall 2006Costas Busch - RPI29 Another Example Alphabet: Language Accepted:

30. Fall 2006Costas Busch - RPI30 Formal Definition Deterministic Finite Automaton (DFA) : set of states : input alphabet : transition function : initial state : set of accepting states

31. Fall 2006Costas Busch - RPI31 Set of States Example

32. Fall 2006Costas Busch - RPI32 Input Alphabet :the input alphabet never contains Example

33. Fall 2006Costas Busch - RPI33 Initial State Example

34. Fall 2006Costas Busch - RPI34 Set of Accepting States Example

35. Fall 2006Costas Busch - RPI35 Transition Function Describes the result of a transition from state with symbol

36. Fall 2006Costas Busch - RPI36 Example:

37. Fall 2006Costas Busch - RPI37

38. Fall 2006Costas Busch - RPI38

39. Fall 2006Costas Busch - RPI39 Transition Table for states symbols

40. Fall 2006Costas Busch - RPI40 Extended Transition Function Describes the resulting state after scanning string from state

41. Fall 2006Costas Busch - RPI41 Example:

42. Fall 2006Costas Busch - RPI42

43. Fall 2006Costas Busch - RPI43

44. Fall 2006Costas Busch - RPI44 Special case: for any state

45. Fall 2006Costas Busch - RPI45 states may be repeated In general: implies that there is a walk of transitions

46. Fall 2006Costas Busch - RPI46 Language Accepted by DFA it is denoted as and contains all the strings accepted by Language of DFA : We say that a language is accepted (or recognized) by DFA if

47. Fall 2006Costas Busch - RPI47 For a DFA Language accepted by :

48. Fall 2006Costas Busch - RPI48 Language rejected by :

49. Fall 2006Costas Busch - RPI49 More DFA Examples Empty languageAll strings

50. Fall 2006Costas Busch - RPI50 Language of the empty string

51. Fall 2006Costas Busch - RPI51 = { all strings with prefix } accept

52. Fall 2006Costas Busch - RPI52 = { all binary strings containing substring }

53. Fall 2006Costas Busch - RPI53 = { all binary strings without substring }

54. Fall 2006Costas Busch - RPI54

55. Fall 2006Costas Busch - RPI55 Regular Languages Definition: A language is regular if there is a DFA that accepts it ( ) The languages accepted by all DFAs form the family of regular languages

56. Fall 2006Costas Busch - RPI56 { all strings in {a,b}* with prefix } { all binary strings without substring } Example regular languages: There exist automata that accept these languages (see previous slides).

57. Fall 2006Costas Busch - RPI57 There exist languages which are not Regular: There is no DFA that accepts these languages (we will prove this in a later class)