1. 1 Computing Functions with Turing Machines

2. 2 A function Domain Result Region has:

3. 3 Integer Domain: Unary: Binary: Decimal: 11111 101 5 We prefer Unary representation: Easier to manipulate

4. 4 A function may have many parameters: Example: Addition function are integers

5. 5 Definition: A function is computable if there is a Turing Machine such that: Initial Configuration Final configuration Domain final state

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

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

8. 8 Start Finish final state

9. 9 Turing machine for function

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

11. 11 Time 0Time 1

12. 12 Time 2Time 3

13. 13 Time 4Time 5

14. 14 Time 6Time 7

15. 15 Time 8Time 9

16. 16 Time 10Time 11

17. 17 Time 12 HALT & accept

18. 18 Another Example The function is computable Turing Machine: Input string: unary Output string:unary is integer

19. 19 Start Finish final state

20. 20 Turing Machine Pseudocode for 1. Replace every 1 with $ Repeat: 2. Find rightmost $, replace it with 1 3. Go to right end, insert 1 Until no more $ remain

21. 21 Turing Machine for

22. 22 Example Start Finish

23. 23 Another Example The function is computable if

24. 24 Turing Machine for if Input: Output: or

25. 25 Turing Machine Pseudocode: Match a 1 from with a 1 from 1. Repeat Until all or has been matched 2. If a 1 from is not matched erase tape, write 1 else erase tape, write 0

26. 26 Combining Turing Machines

27. 27 Block Diagram Turing Machine inputoutput

28. 28 Example: if Comparer Adder Eraser

29. 29 Turing ’ s Thesis

30. 30 Do Turing machines have the same power with a digital computer?

31. 31 Do Turing machines have the same power with a digital computer? Intuitive answer: Yes There is no formal answer

32. 32 Turing ’ s thesis: Any computation carried out by mechanical means can be performed by Turing Machine (1930)

33. 33 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

34. 34 Definition of Algorithm: An algorithm for function is a Turing Machine which computes

35. 35 When we say: There exists an algorithm Algorithms are Turing Machines We mean: There exists a Turing Machine