1. 1 Computing Functions with Turing Machines

2. 2 A function Domain: Result Region: has:

3. 3 A function may have many parameters: Example: Addition function

4. 4 Integer Domain Unary: Binary: Decimal: 11111 101 5 We prefer unary representation: easier to manipulate

5. 5 Definition: A function is computable if there is a Turing Machine such that: Initial configurationFinal configuration Domain final stateinitial state For all

6. 6 Initial Configuration Final Configuration A function is computable if there is a Turing Machine such that: In other words: Domain For all

7. 7 Example The function is computable Turing Machine: Input string: unary Output string: unary are integers

8. 8 Start Finish final state initial state

9. 9 Turing machine for function

10. 10 Execution Example: Time 0 Final Result (2)

11. 11 Time 0

12. 12 Time 1

13. 13 Time 2

14. 14 Time 3

15. 15 Time 4

16. 16 Time 5

17. 17 Time 6

18. 18 Time 7

19. 19 Time 8

20. 20 Time 9

21. 21 Time 10

22. 22 Time 11

23. 23 HALT & accept Time 12

24. 24 Another Example The function is computable Turing Machine: Input string: unary Output string:unary is integer

25. 25 Start Finish final state initial state

26. 26 Turing Machine Pseudocode for Replace every 1 with $ Repeat: Find rightmost $, replace it with 1 Go to right end, insert 1 Until no more $ remain

27. 27 Turing Machine for

28. 28 Example Start Finish

29. 29 Another Example The function is computable if

30. 30 Turing Machine for Input: Output: or if

31. 31 Turing Machine Pseudocode: Match a 1 from with a 1 from Repeat Until all of or is matched If a 1 from is not matched erase tape, write 1 else erase tape, write 0

32. 32 Combining Turing Machines

33. 33 Block Diagram Turing Machine inputoutput

34. 34 Example: if Comparer Adder Eraser

35. 35 Turing’s Thesis

36. 36 Do Turing machines have the same power with a digital computer? Intuitive answer: Yes There is no formal answer!!! Question:

37. 37 Turing’s thesis: Any computation carried out by mechanical means can be performed by a Turing Machine (1930)

38. 38 Computer Science Law: A computation is mechanical if and only if it can be performed by a Turing Machine There is no known model of computation more powerful than Turing Machines

39. 39 Definition of Algorithm: An algorithm for function is a Turing Machine which computes

40. 40 When we say: There exists an algorithm Algorithms are Turing Machines We mean: There exists a Turing Machine