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  • Slide 1
  • Fall 2006Costas Busch - RPI1 Turings Thesis
  • Slide 2
  • Fall 2006Costas Busch - RPI2 Turings thesis (1930): Any computation carried out by mechanical means can be performed by a Turing Machine
  • Slide 3
  • Fall 2006Costas Busch - RPI3 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
  • Slide 4
  • Fall 2006Costas Busch - RPI4 When we say:There exists an algorithm We mean: There exists a Turing Machine that executes the algorithm
  • Slide 5
  • Fall 2006Costas Busch - RPI5 Variations of the Turing Machine
  • Slide 6
  • Fall 2006Costas Busch - RPI6 Read-Write Head Control Unit Deterministic The Standard Model Infinite Tape (Left or Right)
  • Slide 7
  • Fall 2006Costas Busch - RPI7 Variations of the Standard Model Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional Nondeterministic Turing machines with: Different Turing Machine Classes
  • Slide 8
  • Fall 2006Costas Busch - RPI8 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)
  • Slide 9
  • Fall 2006Costas Busch - RPI9 Same Power of two classes means: for any machine of first class there is a machine of second class such that: and vice-versa
  • Slide 10
  • Fall 2006Costas Busch - RPI10 A technique to prove same power.Simulation: Simulate the machine of one class with a machine of the other class First Class Original Machine Second Class Simulation Machine simulates
  • Slide 11
  • Fall 2006Costas Busch - RPI11 Configurations in the Original Machine have corresponding configurations in the Simulation Machine Original Machine: Simulation Machine:
  • Slide 12
  • Fall 2006Costas Busch - RPI12 the Simulation Machine and the Original Machine accept the same strings Original Machine: Simulation Machine: Accepting Configuration
  • Slide 13
  • Fall 2006Costas Busch - RPI13 Turing Machines with Stay-Option The head can stay in the same position Left, Right, Stay L,R,S: possible head moves
  • Slide 14
  • Fall 2006Costas Busch - RPI14 Example: Time 1 Time 2
  • Slide 15
  • Fall 2006Costas Busch - RPI15 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
  • Slide 16
  • Fall 2006Costas Busch - RPI16 1. Stay-Option Machines simulate Standard Turing machines Trivial: any standard Turing machine is also a Stay-Option machine
  • Slide 17
  • Fall 2006Costas Busch - RPI17 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
  • Slide 18
  • Fall 2006Costas Busch - RPI18 Stay-Option Machine Simulation in Standard Machine For every possible tape symbol
  • Slide 19
  • Fall 2006Costas Busch - RPI19 Stay-Option Machine Simulation in Standard Machine Similar for Right moves For other transitions nothing changes
  • Slide 20
  • Fall 2006Costas Busch - RPI20 example of simulation Stay-Option Machine: 12 Simulation in Standard Machine: 123 END OF PROOF
  • Slide 21
  • Fall 2006Costas Busch - RPI21 Multiple Track Tape track 1 track 2 One symbol One head A useful trick to perform more complicated simulations One Tape
  • Slide 22
  • Fall 2006Costas Busch - RPI22 track 1 track 2 track 1 track 2
  • Slide 23
  • Fall 2006Costas Busch - RPI23 Semi-Infinite Tape......... 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
  • Slide 24
  • Fall 2006Costas Busch - RPI24 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
  • Slide 25
  • Fall 2006Costas Busch - RPI25 1. Standard Turing machines simulate Semi-Infinite machines: a. insert special symbol at left of input string b. Add a self-loop to every state (except states with no outgoing transitions) Standard Turing Machine
  • Slide 26
  • Fall 2006Costas Busch - RPI26 2. Semi-Infinite tape machines simulate Standard Turing machines: Standard machine......... Semi-Infinite tape machine......... Squeeze infinity of both directions in one direction
  • Slide 27
  • Fall 2006Costas Busch - RPI27 Standard machine......... Semi-Infinite tape machine with two tracks......... reference point Right part Left part
  • Slide 28
  • Fall 2006Costas Busch - RPI28 Left partRight part Standard machine Semi-Infinite tape machine
  • Slide 29
  • Fall 2006Costas Busch - RPI29 Standard machine Semi-Infinite tape machine Left part Right part For all tape symbols
  • Slide 30
  • Fall 2006Costas Busch - RPI30 Standard machine......... Semi-Infinite tape machine Right part Left part Time 1
  • Slide 31
  • Fall 2006Costas Busch - RPI31 Time 2 Right part Left part Standard machine......... Semi-Infinite tape machine
  • Slide 32
  • Fall 2006Costas Busch - RPI32 Semi-Infinite tape machine Left part At the border: Right part
  • Slide 33
  • Fall 2006Costas Busch - RPI33......... Semi-Infinite tape machine Right part Left part......... Right part Left part Time 1 Time 2 END OF PROOF
  • Slide 34
  • Fall 2006Costas Busch - RPI34 The Off-Line Machine Control Unit Input File Tape read-only (once) read-write Input string Appears on input file only (state machine) Input string
  • Slide 35
  • Fall 2006Costas Busch - RPI35 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
  • Slide 36
  • Fall 2006Costas Busch - RPI36 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
  • Slide 37
  • Fall 2006Costas Busch - RPI37 1. Copy input file to tape Input File Tape Standard machine Off-line machine
  • Slide 38
  • Fall 2006Costas Busch - RPI38 2. Do computations as in Turing machine Input File Tape Standard machine Off-line machine
  • Slide 39
  • Fall 2006Costas Busch - RPI39 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
  • Slide 40
  • Fall 2006Costas Busch - RPI40 Input File Tape Off-line Machine Standard Machine -- Four track tape Input File head position Tape head position
  • Slide 41
  • Fall 2006Costas Busch - RPI41 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
  • Slide 42
  • Fall 2006Costas Busch - RPI42 Multi-tape Turing Machines Control unit Tape 1Tape 2 Input string Input string appears on Tape 1 (state machine)
  • Slide 43
  • Fall 2006Costas Busch - RPI43 Time 1 Time 2 Tape 1Tape 2 Tape 1 Tape 2
  • Slide 44
  • Fall 2006Costas Busch - RPI44 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
  • Slide 45
  • Fall 2006Costas Busch - RPI45 1. Multi-tape machines simulate Standard Turing Machines: Trivial: Use just one tape
  • Slide 46
  • Fall 2006Costas Busch - RPI46 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:
  • Slide 47
  • Fall 2006Costas Busch - RPI47 Multi-tape Machine Tape 1Tape 2 Standard machine with four track tape Tape 1 head position Tape 2 head position
  • Slide 48
  • Fall 2006Costas Busch - RPI48 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 Tape 1 head position Tape 2 head position Reference point END OF PROOF
  • Slide 49
  • Fall 2006Costas Busch - RPI49 ( steps) Standard Turing machine: Go back and forth times 2-tape machine: 1. Copy to tape 2 2. Compare on tape 1 and tape 2 to match the as with the bs ( steps) time Same power doesnt imply same speed:
  • Slide 50
  • Fall 2006Costas Busch - RPI50 Multidimensional Turing Mach

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