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Computing Functions with Turing Machines

Learn about computable functions, domains, parameters, and Turing Machines. Explore examples using unary representation and pseudocode. Discuss the power of Turing Machines in computer science.

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

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

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

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

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

  8. Start initial state Finish final state

  9. Turing machine for function

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

  11. Time 0

  12. Time 1

  13. Time 2

  14. Time 3

  15. Time 4

  16. Time 5

  17. Time 6

  18. Time 7

  19. Time 8

  20. Time 9

  21. Time 10

  22. Time 11

  23. Time 12 HALT & accept

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

  25. Start initial state Finish final state

  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. Turing Machine for

  28. Example Start Finish

  29. Another Example if The function if is computable

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

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

  32. Combining Turing Machines

  33. Block Diagram Turing Machine input output

  34. Example: if if Adder Comparer Eraser

  35. Turing’s Thesis

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

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

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

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

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