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Computing Functions with Turing Machines

Computing Functions with Turing Machines. A function. has:. Domain. Result Region. Integer Domain:. Unary:. 11111. 101. Binary:. Decimal:. 5. We prefer Unary representation:. Easier to manipulate. A function may have many parameters:. Example:. Addition function. are integers.

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Computing Functions with Turing Machines

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  1. Computing FunctionswithTuring Machines

  2. A function has: Domain Result Region

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

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

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

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

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

  8. Start Finish final state

  9. Turing machine for function

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

  11. Time 0 Time 1

  12. Time 2 Time 3

  13. Time 4 Time 5

  14. Time 6 Time 7

  15. Time 8 Time 9

  16. Time 10 Time 11

  17. Time 12 HALT & accept

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

  19. Start Finish final state

  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. Turing Machine for

  22. Example Start Finish

  23. Another Example if The function if is computable

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

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

  26. Combining Turing Machines

  27. Block Diagram Turing Machine input output

  28. Example: if if Adder Comparer Eraser

  29. Turing’s Thesis

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

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

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

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

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

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