1. Busch Complexity Lectures: Turing Machines
Prof. Busch - LSU

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

24. Time 0
Prof. Busch - LSU

25. Time 1
Prof. Busch - LSU

26. Time 2
Prof. Busch - LSU

27. Time 3
Prof. Busch - LSU

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

29. Rejection Example
Time 0
Prof. Busch - LSU

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

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

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

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

34. Time 0
Prof. Busch - LSU

35. Time 1
Prof. Busch - LSU

36. Time 2
Prof. Busch - LSU

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

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

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

40. Basic Idea: 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
Prof. Busch - LSU

41. Time 0
Prof. Busch - LSU

42. Time 1
Prof. Busch - LSU

43. Time 2
Prof. Busch - LSU

44. Time 3
Prof. Busch - LSU

45. Time 4
Prof. Busch - LSU

46. Time 5
Prof. Busch - LSU

47. Time 6
Prof. Busch - LSU

48. Time 7
Prof. Busch - LSU

49. Time 8
Prof. Busch - LSU

50. Time 9
Prof. Busch - LSU

51. Time 10
Prof. Busch - LSU

52. Time 11
Prof. Busch - LSU

53. Time 12
Prof. Busch - LSU

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

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

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

57. Transition Function
Prof. Busch - LSU

58. Transition Function
Prof. Busch - LSU

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

60. Configuration
Instantaneous description:
Prof. Busch - LSU

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

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

63. Equivalent notation:
Prof. Busch - LSU

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

65. The Accepted Language For any Turing Machine Initial state
Accept state
Prof. Busch - LSU

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