www.nskinfo.com && www.nsksofttch.com Department of nskinfo-i education

1. www.nskinfo.com && www.nsksofttch.comDepartment of nskinfo-i education CS2303-THEORY OF COMPUTATION TURING MACHINE

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

3. When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm

4. Variationsof theTuring Machine

5. The Standard Model Infinite Tape Read-Write Head (Left or Right) Control Unit Deterministic

6. Variations of the Standard Model Turing machines with: • Stay-Option • Semi-Infinite Tape • Off-Line • Multitape • Multidimensional • Nondeterministic Different Turing MachineClasses

7. Turing Machines with Stay-Option The head can stay in the same position Left, Right, Stay L,R,S: possible head moves

8. Example: Time 1 Time 2

9. Theorem: Stay-Option machines have the same power with Standard Turing machines Proof: 1. Stay-Option Machines simulate Standard Turing machines 2. Standard Turing machines simulate Stay-Option machines

10. 1. Stay-Option Machines simulate Standard Turing machines Trivial: any standard Turing machine is also a Stay-Option machine

11. 2. Standard Turing machines simulate Stay-Option machines We need to simulate the stay head option with two head moves, one left and one right

12. Stay-Option Machine Simulation in Standard Machine For every possible tape symbol

13. For other transitions nothing changes Stay-Option Machine Simulation in Standard Machine Similar for Right moves

14. example of simulation Stay-Option Machine: 1 2 Simulation in Standard Machine: 1 2 3 END OF PROOF

15. Multiple Track Tape A useful trick to perform more complicated simulations One Tape track 1 track 2 One head One symbol

16. track 1 track 2 track 1 track 2

17. Semi-Infinite Tape The head extends infinitely only to the right ......... • Initial position is the leftmost cell • When the head moves left from the border, • it returns to the same position

18. Theorem: Semi-Infinite machines have the same power with Standard Turing machines Proof: 1. Standard Turing machines simulate Semi-Infinite machines 2. Semi-Infinite Machines simulate Standard Turing machines

19. 1. Standard Turing machines simulate Semi-Infinite machines: Standard Turing Machine a. insert special symbol at left of input string b.Add a self-loop to every state (except states with no outgoing transitions)

20. 2. Semi-Infinite tape machines simulate Standard Turing machines: Standard machine ......... ......... Semi-Infinite tape machine ......... Squeeze infinity of both directions in one direction

21. Standard machine ......... ......... reference point Semi-Infinite tape machine with two tracks Right part ......... Left part

22. Standard machine Semi-Infinite tape machine Left part Right part

23. Standard machine Semi-Infinite tape machine Right part Left part For all tape symbols

24. Time 1 Standard machine ......... ......... Semi-Infinite tape machine Right part ......... Left part

25. Time 2 Standard machine ......... ......... Semi-Infinite tape machine Right part ......... Left part

26. At the border: Semi-Infinite tape machine Right part Left part

27. Semi-Infinite tape machine Time 1 Right part ......... Left part Time 2 Right part ......... Left part END OF PROOF

28. The Off-Line Machine Input File read-only (once) Input string Appears on input file only Input string Control Unit (state machine) Tape read-write

29. Theorem: Off-Line machines have the same power with Standard Turing machines Proof: 1. Off-Line machines simulate Standard Turing machines 2. Standard Turing machines simulate Off-Line machines

30. 1. Off-line machines simulate Standard Turing Machines Off-line machine: 1. Copy input file to tape 2. Continue computation as in Standard Turing machine

31. Standard machine Off-line machine Tape Input File 1. Copy input file to tape

32. Standard machine Off-line machine Tape Input File 2. Do computations as in Turing machine

33. 2. Standard Turing machines simulate Off-Line machines: Use a Standard machine with a four-track tape to keep track of the Off-line input file and tape contents

34. Off-line Machine Tape Input File Standard Machine -- Four track tape Input File head position Tape head position

35. Reference point (uses special symbol # ) 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 END OF PROOF

36. Multitape Turing Machines • A multitape Turing machine is like an ordinary TM but it has several tapes instead of one tape. • Initially the input starts on tape 1 and the other tapes are blank. • The transition function is changed to allow for reading, writing, and moving the heads on all the tapes simultaneously. • This means we could read on multiples tape and move in different directions on each tape as well as write a different symbol on each tape, all in one move.

37. Multitape Turing Machine • Theorem: A multitape TM is equivalent in power to an ordinary TM. Recall that two TM’s are equivalent if they recognize the same language. We can show how to convert a multitape TM, M, to a single tape TM, S: • Say that M has k tapes. • Create the TM S to simulate having k tapes by interleaving the information on each of the k tapes on its single tape • Use a new symbol # as a delimiter to separate the contents of each tape • S must also keep track of the location on each of the simulated heads • Write a type symbol with a * to mark the place where the head on the tape would be • The * symbols are new tape symbols that don’t exist with M • The finite control must have the proper logic to distinguish say, x* and x and realize both refer to the same thing, but one is the current tape symbol.

38. Multitape Machine Equivalent Single Tape Machine:

39. Single Tape Equivalent • One final detail • If at any point S moves one of the virtual tape heads onto a #, then this action signifies that M has moved the corresponding head onto the previously unread blank portion of that tape. • To accommodate this situation, S writes a blank symbol on this tape cell and shifts the tape contents to the rightmost # by one, adds a new #, and then continues back where it left off

40. Multi-tape Turing Machines Control unit (state machine) Tape 1 Tape 2 Input string Input string appears on Tape 1

41. Tape 1 Time 1 Tape 2 Tape 1 Time 2 Tape 2

42. Theorem: Multi-tape machines have the same power with Standard Turing machines Proof: 1. Multi-tape machines simulate Standard Turing machines 2. Standard Turing machines simulate Multi-tape machines

43. 1. Multi-tape machines simulate Standard Turing Machines: Trivial: Use just one tape

44. 2. Standard Turing machines simulate Multi-tape machines: Standard machine: • Uses a multi-track tape to simulate • the multiple tapes • A tape of the Multi-tape machine • corresponds to a pair of tracks

45. Multi-tape Machine Tape 1 Tape 2 Standard machine with four track tape Tape 1 head position Tape 2 head position

46. 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 END OF PROOF

47. Same power doesn’t imply same speed: Standard Turing machine: time Go back and forth times to match the a’s with the b’s time 2-tape machine: 1. Copy to tape 2 ( steps) 2. Compare on tape 1 and tape 2 ( steps)

48. Multidimensional Turing Machines 2-dimensional tape MOVES: L,R,U,D HEAD U: up D: down Position: +2, -1

49. Theorem: Multidimensional machines have the same power with Standard Turing machines Proof: 1. Multidimensional machines simulate Standard Turing machines 2. Standard Turing machines simulate Multi-Dimensional machines

50. 1. Multidimensional machines simulate Standard Turing machines Trivial: Use one dimension