1. Turing Machines

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

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

4. ......
......
Read-Write head
The head at each time step:
1. Reads a symbol
2. Writes a symbol
3. Moves Left or Right

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

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

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

8. Input string
Blank symbol
......
......
head
Remark: the input string is never empty

9. States & Transitions
Write
Read
Move Left
Move Right

10. Example:
Time 1
......
......
current state

11. Time 1
......
......
Time 2
......
......

12. Example:
Time 1
......
......
Time 2
......
......

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

14. Determinism Not Allowed Turing Machines are deterministic Allowed
No lambda transitions allowed

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

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

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

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

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

20. Turing Machine Example
A Turing machine that accepts the language:

21. Time 0

22. Time 1

23. Time 2

24. Time 3

25. Time 4
Halt & Accept

26. Rejection Example
Time 0

27. Time 1
No possible Transition
Halt & Reject

28. Infinite Loop Example
A Turing machine
for language

29. Time 0

30. Time 1

31. Time 2

32. Time 2
Time 3
Infinite loop
Time 4
Time 5

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

34. Another Turing Machine Example
Turing machine for the language

35. Time 0

36. Time 1

37. Time 2

38. Time 3

39. Time 4

40. Time 5

41. Time 6

42. Time 7

43. Time 8

44. Time 9

45. Time 10

46. Time 11

47. Time 12

48. Time 13
Halt & Accept

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

50. Formal Definitions for Turing Machines

51. Transition Function

52. Transition Function

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

55. Assignment (10%) Form a group consist <= 4 members.
Find the implementation/usage/ function/benefit of:
FA/NFA
Content Free Language/Grammar
Pushdown Automata
Turing Machine
Due date: during next week class