1. Prof. Busch - LSU1 Turing Machines

2. Prof. Busch - LSU2 The Language Hierarchy Regular Languages Context-Free Languages ? ?

3. Prof. Busch - LSU3 Regular Languages Context-Free Languages Languages accepted by Turing Machines

4. Prof. Busch - LSU4 A Turing Machine...... Tape Read-Write head Control Unit

5. Prof. Busch - LSU5 The Tape...... Read-Write head No boundaries -- infinite length The head moves Left or Right

6. Prof. Busch - LSU6...... Read-Write head The head at each transition (time step): 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right

7. Prof. Busch - LSU7...... Example: Time 0...... Time 1 1. Reads 2. Writes 3. Moves Left

8. Prof. Busch - LSU8...... Time 1...... Time 2 1. Reads 2. Writes 3. Moves Right

9. Prof. Busch - LSU9 The Input String...... Blank symbol head Head starts at the leftmost position of the input string Input string

10. Prof. Busch - LSU10 States & Transitions Read Write Move Left Move Right

11. Prof. Busch - LSU11 Example:...... Time 1 current state

12. Prof. Busch - LSU12...... Time 1...... Time 2

13. Prof. Busch - LSU13...... Time 1...... Time 2 Example:

14. Prof. Busch - LSU14...... Time 1...... Time 2 Example:

15. Prof. Busch - LSU15 Determinism Allowed Not Allowed No lambda transitions allowed Turing Machines are deterministic

16. Prof. Busch - LSU16 Partial Transition Function...... Example: No transition for input symbol Allowed:

17. Prof. Busch - LSU17 Halting The machine halts in a state if there is no transition to follow

18. Prof. Busch - LSU18 Halting Example 1:...... No transition from HALT!!!

19. Prof. Busch - LSU19 Halting Example 2:...... No possible transition from and symbol HALT!!!

20. Prof. Busch - LSU20 Accepting States Allowed Not Allowed Accepting states have no outgoing transitions The machine halts and accepts

21. Prof. Busch - LSU21 Acceptance Accept Input If machine halts in an accept state Reject Input If machine halts in a non-accept state or If machine enters an infinite loop string

22. Prof. Busch - LSU22 Observation: In order to accept an input string, it is not necessary to scan all the symbols in the string

23. Prof. Busch - LSU23 Turing Machine Example Accepts the language: Input alphabet

24. Prof. Busch - LSU24 Time 0

25. Prof. Busch - LSU25 Time 1

26. Prof. Busch - LSU26 Time 2

27. Prof. Busch - LSU27 Time 3

28. Prof. Busch - LSU28 Time 4 Halt & Accept

29. Prof. Busch - LSU29 Rejection Example Time 0

30. Prof. Busch - LSU30 Time 1 No possible Transition Halt & Reject

31. Prof. Busch - LSU31 Accepts the language: but for input alphabet A simpler machine for same language

32. Prof. Busch - LSU32 Time 0 Halt & Accept Not necessary to scan input

33. Prof. Busch - LSU33 Infinite Loop Example A Turing machine for language

34. Prof. Busch - LSU34 Time 0

35. Prof. Busch - LSU35 Time 1

36. Prof. Busch - LSU36 Time 2

37. Prof. Busch - LSU37 Time 2 Time 3 Time 4 Time 5 Infinite loop

38. Prof. Busch - LSU38 Because of the infinite loop: The accepting state cannot be reached The machine never halts The input string is rejected

39. Prof. Busch - LSU39 Another Turing Machine Example Turing machine for the language

40. Prof. Busch - LSU40 Match a’s with b’s: Repeat: replace leftmost a with x find leftmost b and replace it with y Until there are no more a’s or b’s If there is a remaining a or b reject Basic Idea:

41. Prof. Busch - LSU41 Time 0

42. Prof. Busch - LSU42 Time 1

43. Prof. Busch - LSU43 Time 2

44. Prof. Busch - LSU44 Time 3

45. Prof. Busch - LSU45 Time 4

46. Prof. Busch - LSU46 Time 5

47. Prof. Busch - LSU47 Time 6

48. Prof. Busch - LSU48 Time 7

49. Prof. Busch - LSU49 Time 8

50. Prof. Busch - LSU50 Time 9

51. Prof. Busch - LSU51 Time 10

52. Prof. Busch - LSU52 Time 11

53. Prof. Busch - LSU53 Time 12

54. Prof. Busch - LSU54 Halt & Accept Time 13

55. Prof. Busch - LSU55 If we modify the machine for the language we can easily construct a machine for the language Observation:

56. Prof. Busch - LSU56 Formal Definitions for Turing Machines

57. Prof. Busch - LSU57 Transition Function

58. Prof. Busch - LSU58 Transition Function

59. Prof. Busch - LSU59 Turing Machine: States Input alphabet Tape alphabet Transition function Initial state blank Accept states

60. Prof. Busch - LSU60 Configuration Instantaneous description:

61. Prof. Busch - LSU61 Time 4Time 5 A Move: (yields in one mode)

62. Prof. Busch - LSU62 Time 4Time 5 Time 6Time 7 A computation

63. Prof. Busch - LSU63 Equivalent notation:

64. Prof. Busch - LSU64 Initial configuration: Input string

65. Prof. Busch - LSU65 The Accepted Language For any Turing Machine Initial stateAccept state

66. Prof. Busch - LSU66 If a language is accepted by a Turing machine then we say that is: Turing Acceptable Recursively Enumerable Turing Recognizable Other names used: