1. 1 Turing Machines

2. 2 The Language Hierarchy Regular Languages Context-Free Languages ? ?

3. 3 Regular Languages Context-Free Languages Languages accepted by Turing Machines

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

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

6. 6...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right

7. 7...... Example: Time 0...... Time 1 1. Reads 2. Writes 3. Moves Left

8. 8...... Time 1...... Time 2 1. Reads 2. Writes 3. Moves Right

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

10. 10 States & Transitions Read Write Move Left Move Right

11. 11 Example:...... Time 1 current state

12. 12...... Time 1...... Time 2

13. 13...... Time 1...... Time 2 Example:

14. 14...... Time 1...... Time 2 Example:

15. 15 Determinism Allowed Not Allowed No epsilon transitions allowed Turing Machines are deterministic

16. 16 Partial Transition Function...... Example: No transition for input symbol Allowed:

17. 17 Halting The machine halts if there are no possible transitions to follow

18. 18 Example:...... No possible transition HALT!!!

19. 19 Final States Allowed Not Allowed Final states have no outgoing transitions In a final state the machine halts

20. 20 Acceptance Accept Input If machine halts in a final state Reject Input If machine halts in a non-final state or If machine enters an infinite loop

21. 21 Turing Machine Example A Turing machine that accepts language a *

22. 22 Time 0

23. 23 Time 1

24. 24 Time 2

25. 25 Time 3

26. 26 Time 4 Halt & Accept

27. 27 Rejection Example Time 0

28. 28 Time 1 No possible Transition Halt & Reject

29. 29 Infinite Loop Example Another Turing machine for language a * and is this one correct???

30. 30 Time 0

31. 31 Time 1

32. 32 Time 2

33. 33 Time 2 Time 3 Time 4 Time 5... Infinite Loop

34. 34 Because of the infinite loop: The final state cannot be reached The machine never halts The input is not accepted

35. 35 Another Turing Machine Example Turing machine for the language

36. 36 Time 0

37. 37 Time 1

38. 38 Time 2

39. 39 Time 3

40. 40 Time 4

41. 41 Time 5

42. 42 Time 6

43. 43 Time 7

44. 44 Time 8

45. 45 Time 9

46. 46 Time 10

47. 47 Time 11

48. 48 Time 12

49. 49 Halt & Accept Time 13

50. 50 If we modify the machine for the language we can easily construct a machine for the language Observation:

51. 51 Formal Definitions for Turing Machines

52. 52 Transition Function

53. 53 Transition Function

54. 54 Turing Machine: States Input alphabet Tape alphabet Transition function Initial state blank Final states

55. 55 Configuration Instantaneous description:

56. 56 Time 4Time 5 A Move: aybqxxaybq 02 

57. 57 Time 4Time 5 bqxxyybqxxaybqxxaybq 1102 Time 6Time 7 

58. 58 bqxxyybqxxaybqxxaybq 1102 bqxxyxaybq 12  Equivalent notation:  

59. 59 Initial configuration: Input string

60. 60 The Accepted Language For any Turing Machine }:{)( 210 xqxwqwML f   Initial stateFinal state 

61. 61 Standard Turing Machine Deterministic Infinite tape in both directions Tape is the input/output file The machine we described is the standard:

62. 62 Implementation-level descriptions {w|w contains an equal number of 0s and 1s}

63. 63 Implementation-level descriptions {w|w contains an equal number of 0s and 1s} On input string w:

64. 64 Implementation-level descriptions {w|w contains an equal number of 0s and 1s} On input string w: 1)Scan the tape & mark the 1st 0 which is unmarked. If none is found, go to #4. Otherwise, move the head back to the front of the tape.

65. 65 Implementation-level descriptions {w|w contains an equal number of 0s and 1s} On input string w: 1)Scan the tape & mark the 1st 0 which is unmarked. If none is found, go to #4. Otherwise, move the head back to the front of the tape. 2)Scan the tape & mark the 1st 1 which is unmarked. If none is found, reject.

66. 66 Implementation-level descriptions {w|w contains an equal number of 0s and 1s} On input string w: 1)Scan the tape & mark the 1st 0 which is unmarked. If none is found, go to #4. Otherwise, move the head back to the front of the tape. 2)Scan the tape & mark the 1st 1 which is unmarked. If none is found, reject. 3)Move the head back to the front of the tape & go to #1.

67. 67 Implementation-level descriptions {w|w contains an equal number of 0s and 1s} On input string w: 1)Scan the tape & mark the 1st 0 which is unmarked. If none is found, go to #4. Otherwise, move the head back to the front of the tape. 2)Scan the tape & mark the 1st 1 which is unmarked. If none is found, reject. 3)Move the head back to the front of the tape & go to #1. 4)Move the head back to the front of the tape. Scan the tape to see if any unmarked 1s remain. If none are found, accept; otherwise, reject.

68. 68 Try this one: {w|w contains twice as many 0s as 1s} On input string w: 1)Scan the tape &... (Hint, you can “mark” and “double mark”.)

69. 69 Try this one: {w|w = a n b n, n  0} On input string w: 1)Scan the tape &...