1. Costas Busch - LSU1 Deterministic Finite Automata And Regular Languages

2. Costas Busch - LSU2 Deterministic Finite Automaton (DFA) Input Tape “Accept” or “Reject” String Finite Automaton Output

3. Costas Busch - LSU3 Transition Graph initial state accepting state transition

4. Costas Busch - LSU4 For every state, there is a transition for every symbol in the alphabet Alphabet

5. Costas Busch - LSU5 Initial Configuration Input String Initial state Input Tape head

6. Costas Busch - LSU6 Scanning the Input

7. Costas Busch - LSU7

8. 8

9. 9 accept Input finished Last state determines the outcome

10. Costas Busch - LSU10 A Rejection Case Input String

11. Costas Busch - LSU11

12. Costas Busch - LSU12

13. Costas Busch - LSU13 reject Input finished Last state determines the outcome

14. Costas Busch - LSU14 Another Rejection Case Tape is empty reject Input Finished (no symbol read)

15. Costas Busch - LSU15 Language Accepted: This automaton accepts only one string

16. Costas Busch - LSU16 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. Costas Busch - LSU17 Another Example Accept state Accept state Accept state

18. Costas Busch - LSU18 Empty Tape accept Input Finished

19. Costas Busch - LSU19 Another Example Accept state trap state

20. Costas Busch - LSU20 Input String

21. Costas Busch - LSU21

22. Costas Busch - LSU22

23. Costas Busch - LSU23 accept Input finished

24. Costas Busch - LSU24 A rejection case Input String

25. Costas Busch - LSU25

26. Costas Busch - LSU26

27. Costas Busch - LSU27 reject Input finished

28. Costas Busch - LSU28 Language Accepted:

29. Costas Busch - LSU29 Another Example Alphabet: Language Accepted:

30. Costas Busch - LSU30 Formal Definition Deterministic Finite Automaton (DFA) : set of states : input alphabet : transition function : initial state : set of accepting states

31. Costas Busch - LSU31 Set of States Example

32. Costas Busch - LSU32 Input Alphabet :the input alphabet never contains Example

33. Costas Busch - LSU33 Initial State Example

34. Costas Busch - LSU34 Set of Accepting States Example

35. Costas Busch - LSU35 Transition Function Describes the result of a transition from state with symbol

36. Costas Busch - LSU36 Example:

37. Costas Busch - LSU37

38. Costas Busch - LSU38

39. Costas Busch - LSU39 Transition Table for states symbols

40. Costas Busch - LSU40 Extended Transition Function Describes the resulting state after scanning string from state

41. Costas Busch - LSU41 Example:

42. Costas Busch - LSU42

43. Costas Busch - LSU43

44. Costas Busch - LSU44 Special case: for any state

45. Costas Busch - LSU45 states may be repeated In general: implies that there is a walk of transitions

46. Costas Busch - LSU46 Language accepted by DFA : Language Accepted by DFA it is denoted as and contains all the strings accepted by We also say that recognizes

47. Costas Busch - LSU47 For a DFA Language accepted by :

48. Costas Busch - LSU48 Language rejected by :

49. Costas Busch - LSU49 More DFA Examples Empty languageAll strings

50. Costas Busch - LSU50 Language of the empty string

51. Costas Busch - LSU51 = { all strings with prefix } accept

52. Costas Busch - LSU52 = { all binary strings containing substring }

53. Costas Busch - LSU53 = { all binary strings without substring }

54. Costas Busch - LSU54

55. Costas Busch - LSU55 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. Costas Busch - LSU56 { all strings in {a,b}* with prefix } { all binary strings without substring } Example regular languages: There exist DFAs that accept these languages (see previous slides).

57. Costas Busch - LSU57 There exist languages which are not Regular: There are no DFAs that accept these languages (we will prove this in a later class)