Variants of Turing Machine
- 1. Fall 2006 Costas Busch - RPI 1 Turing’s Thesis
- 2. Fall 2006 Costas Busch - RPI 2 Turing’s thesis (1930): Any computation carried out by mechanical means can be performed by a Turing Machine
- 3. Fall 2006 Costas Busch - RPI 3 Algorithm: An algorithm for a problem is a Turing Machine which solves the problem The algorithm describes the steps of the mechanical means This is easily translated to computation steps of a Turing machine
- 4. Fall 2006 Costas Busch - RPI 4 When we say: There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm
- 5. Fall 2006 Costas Busch - RPI 5 Variations of the Turing Machine
- 6. Fall 2006 Costas Busch - RPI 6 Read-Write Head Control Unit ◊◊ a a c ◊◊ ◊b a cb b a a Deterministic The Standard Model Infinite Tape (Left or Right)
- 7. Fall 2006 Costas Busch - RPI 7 Variations of the Standard Model • Stay-Option • Semi-Infinite Tape • Off-Line • Multitape • Multidimensional • Nondeterministic Turing machines with: Different Turing Machine Classes
- 8. Fall 2006 Costas Busch - RPI 8 We will prove: each new class has the same power with Standard Turing Machine Same Power of two machine classes: both classes accept the same set of languages (accept Turing-Recognizable Languages)
- 9. Fall 2006 Costas Busch - RPI 9 Same Power of two classes means: for any machine of first class1M there is a machine of second class2M such that: )()( 21 MLML = and vice-versa
- 10. Fall 2006 Costas Busch - RPI 10 A technique to prove same power.Simulation: Simulate the machine of one class with a machine of the other class First Class Original Machine 1M 1M 2M Second Class Simulation Machine simulates 1M
- 11. Fall 2006 Costas Busch - RPI 11 Configurations in the Original Machine have corresponding configurations in the Simulation Machine nddd 10Original Machine: Simulation Machine: nddd ′′′ ∗∗∗ 10 1M 2M 1M 2M
- 12. Fall 2006 Costas Busch - RPI 12 the Simulation Machine and the Original Machine accept the same strings fdOriginal Machine: Simulation Machine: fd′ Accepting Configuration )()( 21 MLML =
- 13. Fall 2006 Costas Busch - RPI 13 Turing Machines with Stay-Option The head can stay in the same position ◊◊ a a c ◊◊ ◊b a cb b a a Left, Right, Stay L,R,S: possible head moves
- 14. Fall 2006 Costas Busch - RPI 14 Example: ◊◊ a a c ◊◊ ◊b a cb b a a Time 1 ◊◊ b a c ◊◊ ◊b a cb b a a Time 2 1q 2q 1q 2q Sba ,→
- 15. Fall 2006 Costas Busch - RPI 15 Stay-Option machines have the same power with Standard Turing machines Theorem: Proof: 1. Stay-Option Machines simulate Standard Turing machines 2. Standard Turing machines simulate Stay-Option machines
- 16. Fall 2006 Costas Busch - RPI 16 1. Stay-Option Machines simulate Standard Turing machines Trivial: any standard Turing machine is also a Stay-Option machine
- 17. Fall 2006 Costas Busch - RPI 17 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
- 18. Fall 2006 Costas Busch - RPI 18 1q 2q Sba ,→ 1q Lba ,→ 2q Rxx ,→ Stay-Option Machine Simulation in Standard Machine For every possible tape symbol x
- 19. Fall 2006 Costas Busch - RPI 19 1q 2q Lba ,→ 1q 2q Lba ,→ Stay-Option Machine Simulation in Standard Machine Similar for Right moves For other transitions nothing changes
- 20. Fall 2006 Costas Busch - RPI 20 example of simulation ◊ ◊a a b a 1q Stay-Option Machine: 1 ◊ ◊b a b a 2q 2 1q 2qSba ,→ Simulation in Standard Machine: ◊ ◊a a b a 1q 1 ◊ ◊b a b a 2q 2 ◊ ◊b a b a 3q 3 END OF PROOF
- 21. Fall 2006 Costas Busch - RPI 21 Multiple Track Tape ◊ ◊ ◊ ◊ ◊ ◊ b d a b b a a c track 1 track 2 One symbol ),( ba One head A useful trick to perform more complicated simulations One Tape
- 22. Fall 2006 Costas Busch - RPI 22 ◊ ◊ ◊ ◊ ◊ ◊ b d a b b a a c track 1 track 2 1q 2q Ldcab ),,(),( → 1q ◊ ◊ ◊ ◊ ◊ ◊ b d a b c d a c track 1 track 2 2q
- 23. Fall 2006 Costas Busch - RPI 23 Semi-Infinite Tape .........a b a c ◊ ◊ • When the head moves left from the border, it returns to the same position The head extends infinitely only to the right • Initial position is the leftmost cell
- 24. Fall 2006 Costas Busch - RPI 24 Semi-Infinite machines have the same power with Standard Turing machines Theorem: Proof: 2. Semi-Infinite Machines simulate Standard Turing machines 1. Standard Turing machines simulate Semi-Infinite machines
- 25. Fall 2006 Costas Busch - RPI 25 1. Standard Turing machines simulate Semi-Infinite machines: a. insert special symbol at left of input string # a b a c ◊ ◊# b. Add a self-loop to every state (except states with no outgoing transitions) R,##→ Standard Turing Machine ◊◊
- 26. Fall 2006 Costas Busch - RPI 26 2. Semi-Infinite tape machines simulate Standard Turing machines: Standard machine ......... Semi-Infinite tape machine .................. Squeeze infinity of both directions in one direction
- 27. Fall 2006 Costas Busch - RPI 27 Standard machine ......... Semi-Infinite tape machine with two tracks .................. reference point # # Right part Left part ◊ ◊ ◊a b c d e ac b d e ◊ ◊ ◊ ◊ ◊
- 28. Fall 2006 Costas Busch - RPI 28 1q 2q R q2 L q1 L q2 R q1 Left part Right part Standard machine Semi-Infinite tape machine
- 29. Fall 2006 Costas Busch - RPI 29 1q 2q Rga ,→ Standard machine L q1 L q2 Lgxax ),,(),( → R q1 R q2 Rxgxa ),,(),( → Semi-Infinite tape machine Left part Right part For all tape symbols x
- 30. Fall 2006 Costas Busch - RPI 30 Standard machine .................. ◊ ◊ ◊a b c d e 1q ......... Semi-Infinite tape machine # # Right part Left part ac b d e ◊ ◊ ◊ ◊ ◊ L q1 Time 1
- 31. Fall 2006 Costas Busch - RPI 31 Time 2 ◊ ◊ ◊g b c d e 2q # # Right part Left part gc b d e ◊ ◊ ◊ ◊ ◊ L q2 Standard machine .................. ......... Semi-Infinite tape machine
- 32. Fall 2006 Costas Busch - RPI 32 L q1 R q1 R),#,(#)#,(# → Semi-Infinite tape machine Left part At the border: R q1 L q1 R),#,(#)#,(# → Right part
- 33. Fall 2006 Costas Busch - RPI 33 ......... Semi-Infinite tape machine # # Right part Left part gc b d e ◊ ◊ ◊ ◊ ◊ L q1 .........# # Right part Left part gc b d e ◊ ◊ ◊ ◊ ◊ R q1 Time 1 Time 2 END OF PROOF
- 34. Fall 2006 Costas Busch - RPI 34 The Off-Line Machine Control Unit Input File Tape read-only (once) a b c d eg ◊ ◊◊◊ read-write Input string Appears on input file only (state machine) Input string
- 35. Fall 2006 Costas Busch - RPI 35 Off-Line machines have the same power with Standard Turing machines Theorem: Proof: 1. Off-Line machines simulate Standard Turing machines 2. Standard Turing machines simulate Off-Line machines
- 36. Fall 2006 Costas Busch - RPI 36 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
- 37. Fall 2006 Costas Busch - RPI 37 1. Copy input file to tape Input File a b c ◊◊ ◊ Tape a b c◊ ◊ ◊ Standard machine Off-line machine a b c
- 38. Fall 2006 Costas Busch - RPI 38 2. Do computations as in Turing machine Input File a b c ◊◊ ◊ Tape a b c◊ ◊ ◊ a b c 1q 1q Standard machine Off-line machine
- 39. Fall 2006 Costas Busch - RPI 39 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
- 40. Fall 2006 Costas Busch - RPI 40 Input File a b c ◊◊ ◊ Tape Off-line Machine e f gd Standard Machine -- Four track tape a b c d e f g 0 0 0 0 0 1 1 Input File head position Tape head position # #
- 41. Fall 2006 Costas Busch - RPI 41 a b c d e f g 0 0 0 0 0 1 1 Input File head position Tape head position # # Repeat for each state transition: 1. Return to reference point 2. Find current input file symbol 3. Find current tape symbol 4. Make transition Reference point (uses special symbol # ) # # END OF PROOF
- 42. Fall 2006 Costas Busch - RPI 42 Multi-tape Turing Machines ◊◊ a b c ◊◊ e f g Control unit Tape 1 Tape 2 Input string Input string appears on Tape 1 (state machine)
- 43. Fall 2006 Costas Busch - RPI 43 ◊◊ a b c ◊◊ e f g 1q 1q ◊◊ a g c ◊◊ e d g 2q 2q Time 1 Time 2 RLdgfb ,),,(),( → 1q 2q Tape 1 Tape 2 Tape 1 Tape 2
- 44. Fall 2006 Costas Busch - RPI 44 Multi-tape machines have the same power with Standard Turing machines Theorem: Proof: 1. Multi-tape machines simulate Standard Turing machines 2. Standard Turing machines simulate Multi-tape machines
- 45. Fall 2006 Costas Busch - RPI 45 1. Multi-tape machines simulate Standard Turing Machines: Trivial: Use just one tape
- 46. Fall 2006 Costas Busch - RPI 46 2. Standard Turing machines simulate Multi-tape machines: • Uses a multi-track tape to simulate the multiple tapes • A tape of the Multi-tape machine corresponds to a pair of tracks Standard machine:
- 47. Fall 2006 Costas Busch - RPI 47 ◊◊ a b c h◊ e f g ◊ Multi-tape Machine Tape 1 Tape 2 Standard machine with four track tape a b c e f g 0 0 0 0 1 1 Tape 1 head position Tape 2 head position h 0
- 48. Fall 2006 Costas Busch - RPI 48 Repeat for each state transition: 1. Return to reference point 2. Find current symbol in Tape 1 3. Find current symbol in Tape 2 4. Make transition a b c e f g 0 0 0 0 1 1 Tape 1 head position Tape 2 head position h 0 # # # # Reference point END OF PROOF
- 49. Fall 2006 Costas Busch - RPI 49 ( steps) }{ nn baL = Standard Turing machine: Go back and forth times)( 2 nO 2-tape machine: 1. Copy to tape 2n b 2. Compare on tape 1 and tape 2 )(nO n b n a to match the a’s with the b’s ( steps))(nO )( 2 nO time )(nO time Same power doesn’t imply same speed:
- 50. Fall 2006 Costas Busch - RPI 50 Multidimensional Turing Machines x y a b c 2-dimensional tape HEAD Position: +2, -1 MOVES: L,R,U,D U: up D: down ◊ ◊ ◊ ◊
- 51. Fall 2006 Costas Busch - RPI 51 Multidimensional machines have the same power with Standard Turing machines Theorem: Proof: 1. Multidimensional machines simulate Standard Turing machines 2. Standard Turing machines simulate Multi-Dimensional machines
- 52. Fall 2006 Costas Busch - RPI 52 1. Multidimensional machines simulate Standard Turing machines Trivial: Use one dimension
- 53. Fall 2006 Costas Busch - RPI 53 2. Standard Turing machines simulate Multidimensional machines Standard machine: • Use a two track tape • Store symbols in track 1 • Store coordinates in track 2
- 54. Fall 2006 Costas Busch - RPI 54 x y a b c◊ ◊ ◊ ◊ a 1 b # symbols coordinates 2-dimensional machine Standard Machine 1 # 2 # 1− c # − 1 1q 1q
- 55. Fall 2006 Costas Busch - RPI 55 Repeat for each transition followed in the 2-dimensional machine: 1. Update current symbol 2. Compute coordinates of next position 3. Go to new position Standard machine: END OF PROOF
- 56. Fall 2006 Costas Busch - RPI 56 Nondeterministic Turing Machines Lba ,→ Rca ,→ 1q 2q 3q Allows Non Deterministic Choices Choice 1 Choice 2
- 57. Fall 2006 Costas Busch - RPI 57 a b c◊ ◊ 1q Lba ,→ Rca ,→ 1q 2q 3q Time 0 Time 1 b b c◊ ◊ 2q c b c◊ ◊ 3q Choice 1 Choice 2
- 58. Fall 2006 Costas Busch - RPI 58 Input string is accepted if there is a computation: w yqxwq f ∗ 0 Initial configuration Final Configuration Any accept state There is a computation:
- 59. Fall 2006 Costas Busch - RPI 59 Nondeterministic machines have the same power with Standard Turing machines Theorem: Proof: 1. Nondeterministic machines simulate Standard Turing machines 2. Standard Turing machines simulate Nondeterministic machines
- 60. Fall 2006 Costas Busch - RPI 60 1. Nondeterministic Machines simulate Standard (deterministic) Turing Machines Trivial: every deterministic machine is also nondeterministic
- 61. Fall 2006 Costas Busch - RPI 61 2. Standard (deterministic) Turing machines simulate Nondeterministic machines: • Stores all possible computations of the non-deterministic machine on the 2-dimensional tape Deterministic machine: • Uses a 2-dimensional tape (which is equivalent to 1-dimensional tape)
- 62. Fall 2006 Costas Busch - RPI 62 All possible computation paths Initial state Step 1 Step 2 Step i Step i+1 acceptreject infinite path
- 63. Fall 2006 Costas Busch - RPI 63 The Deterministic Turing machine simulates all possible computation paths: •in a breadth-first search fashion •simultaneously •step-by-step
- 64. Fall 2006 Costas Busch - RPI 64 a b c◊ ◊ 1q Lba ,→ Rca ,→ 1q 2q 3q Time 0 NonDeterministic machine Deterministic machine a b c 1q # # # # ## # # # # # # # # # current configuration
- 65. Fall 2006 Costas Busch - RPI 65 Lba ,→ Rca ,→ 1q 2q 3q b b c 2q # # # # ## # # # # # # Computation 1 b b c◊ ◊ 2q Choice 1 c b c◊ ◊ 3q Choice 2 c b c 3q ## Computation 2 NonDeterministic machine Deterministic machine Time 1
- 66. Fall 2006 Costas Busch - RPI 66 Repeat For each configuration in current step of non-deterministic machine, if there are two or more choices: 1. Replicate configuration 2. Change the state in the replicas END OF PROOF Deterministic Turing machine Until either the input string is accepted or rejected in all configurations
- 67. Fall 2006 Costas Busch - RPI 67 The simulation takes in the worst case exponential time compared to the shortest accepting path length of the nondeterministic machine Remark: