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Turing’s Thesis

This document explores Turing's Thesis, which states that any computation executable by mechanical means can be performed by a Turing Machine. It delves into the fundamental concepts of computation, defining algorithms in terms of Turing Machines. Variations of the standard Turing model, including Stay-Option, Semi-Infinite, and Off-line machines, are presented. The document provides insights into the concept of simulation to prove the same computational power among different Turing Machine classes, along with detailed proofs and examples highlighting their equivalence in accepting languages.

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Turing’s Thesis

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  1. Turing’s Thesis Courtesy Costas Busch - RPI

  2. Turing’s thesis: Any computation carried out by mechanical means can be performed by a Turing Machine (1930) Courtesy Costas Busch - RPI

  3. 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 Courtesy Costas Busch - RPI

  4. Definition of Algorithm: An algorithm for function is a Turing Machine which computes Courtesy Costas Busch - RPI

  5. Algorithms are Turing Machines When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm Courtesy Costas Busch - RPI

  6. Variationsof theTuring Machine Courtesy Costas Busch - RPI

  7. The Standard Model Infinite Tape Read-Write Head (Left or Right) Control Unit Deterministic Courtesy Costas Busch - RPI

  8. Variations of the Standard Model Turing machines with: • Stay-Option • Semi-Infinite Tape • Off-Line • Multitape • Multidimensional • Nondeterministic Courtesy Costas Busch - RPI

  9. The variations form different Turing MachineClasses We want to prove: Each Class has the same power with the Standard Model Courtesy Costas Busch - RPI

  10. Same Power of two classes means: Both classes of Turing machines accept the same languages Courtesy Costas Busch - RPI

  11. Same Power of two classes means: For any machine of first class there is a machine of second class such that: And vice-versa Courtesy Costas Busch - RPI

  12. Simulation: a technique to prove same power Simulate the machine of one class with a machine of the other class Second Class Simulation Machine First Class Original Machine Courtesy Costas Busch - RPI

  13. Configurations in the Original Machine correspond to configurations in the Simulation Machine Original Machine: Simulation Machine: Courtesy Costas Busch - RPI

  14. Final Configuration Original Machine: Simulation Machine: The Simulation Machine and the Original Machine accept the same language Courtesy Costas Busch - RPI

  15. Turing Machines with Stay-Option The head can stay in the same position Left, Right, Stay L,R,S: moves Courtesy Costas Busch - RPI

  16. Example: Time 1 Time 2 Courtesy Costas Busch - RPI

  17. Theorem: Stay-Option Machines have the same power with Standard Turing machines Courtesy Costas Busch - RPI

  18. Proof: Part 1: Stay-Option Machines are at least as powerful as Standard machines Proof: a Standard machine is also a Stay-Option machine (that never uses the S move) Courtesy Costas Busch - RPI

  19. Proof: Part 2: Standard Machines are at least as powerful as Stay-Option machines Proof: a standard machine can simulate a Stay-Option machine Courtesy Costas Busch - RPI

  20. Stay-Option Machine Simulation in Standard Machine Similar for Right moves Courtesy Costas Busch - RPI

  21. Stay-Option Machine Simulation in Standard Machine For every symbol Courtesy Costas Busch - RPI

  22. Example Stay-Option Machine: 1 2 Simulation in Standard Machine: 1 2 3 Courtesy Costas Busch - RPI

  23. Standard Machine--Multiple Track Tape track 1 track 2 one symbol Courtesy Costas Busch - RPI

  24. track 1 track 2 track 1 track 2 Courtesy Costas Busch - RPI

  25. Semi-Infinite Tape ......... Courtesy Costas Busch - RPI

  26. Standard Turing machines simulate Semi-infinite tape machines: Trivial Courtesy Costas Busch - RPI

  27. Semi-infinite tape machines simulate Standard Turing machines: Standard machine ......... ......... Semi-infinite tape machine ......... Courtesy Costas Busch - RPI

  28. Standard machine ......... ......... reference point Semi-infinite tape machine with two tracks Right part ......... Left part Courtesy Costas Busch - RPI

  29. Standard machine Semi-infinite tape machine Left part Right part Courtesy Costas Busch - RPI

  30. Standard machine Semi-infinite tape machine Right part Left part For all symbols Courtesy Costas Busch - RPI

  31. Time 1 Standard machine ......... ......... Semi-infinite tape machine Right part ......... Left part Courtesy Costas Busch - RPI

  32. Time 2 Standard machine ......... ......... Semi-infinite tape machine Right part ......... Left part Courtesy Costas Busch - RPI

  33. At the border: Semi-infinite tape machine Right part Left part Courtesy Costas Busch - RPI

  34. Semi-infinite tape machine Time 1 Right part ......... Left part Time 2 Right part ......... Left part Courtesy Costas Busch - RPI

  35. Theorem: Semi-infinite tape machines have the same power with Standard Turing machines Courtesy Costas Busch - RPI

  36. The Off-Line Machine Input File read-only Control Unit read-write Tape Courtesy Costas Busch - RPI

  37. Off-line machines simulate Standard Turing Machines: Off-line machine: 1. Copy input file to tape 2. Continue computation as in Standard Turing machine Courtesy Costas Busch - RPI

  38. Standard machine Off-line machine Tape Input File 1. Copy input file to tape Courtesy Costas Busch - RPI

  39. Standard machine Off-line machine Tape Input File 2. Do computations as in Turing machine Courtesy Costas Busch - RPI

  40. Standard Turing machines simulate Off-line machines: Use a Standard machine with four track tape to keep track of the Off-line input file and tape contents Courtesy Costas Busch - RPI

  41. Off-line Machine Tape Input File Four track tape -- Standard Machine Input File head position Tape head position Courtesy Costas Busch - RPI

  42. Reference point Input File head position Tape head position Repeat for each state transition: • Return to reference point • Find current input file symbol • Find current tape symbol • Make transition Courtesy Costas Busch - RPI

  43. Theorem: Off-line machines have the same power with Standard machines Courtesy Costas Busch - RPI

  44. Multitape Turing Machines Control unit Tape 1 Tape 2 Input Courtesy Costas Busch - RPI

  45. Tape 1 Time 1 Tape 2 Time 2 Courtesy Costas Busch - RPI

  46. Multitape machines simulate Standard Machines: Use just one tape Courtesy Costas Busch - RPI

  47. Standard machines simulate Multitape machines: Standard machine: • Use a multi-track tape • A tape of the Multiple tape machine • corresponds to a pair of tracks Courtesy Costas Busch - RPI

  48. Multitape Machine Tape 1 Tape 2 Standard machine with four track tape Tape 1 head position Tape 2 head position Courtesy Costas Busch - RPI

  49. Reference point Tape 1 head position Tape 2 head position Repeat for each state transition: • Return to reference point • Find current symbol in Tape 1 • Find current symbol in Tape 2 • Make transition Courtesy Costas Busch - RPI

  50. Theorem: Multi-tape machines have the same power with Standard Turing Machines Courtesy Costas Busch - RPI

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