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  • Slide 1
  • Courtesy Costas Busch - RPI1 Turings Thesis
  • Slide 2
  • Courtesy Costas Busch - RPI2 Turings thesis: Any computation carried out by mechanical means can be performed by a Turing Machine (1930)
  • Slide 3
  • Courtesy Costas Busch - RPI3 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
  • Slide 4
  • Courtesy Costas Busch - RPI4 Definition of Algorithm: An algorithm for function is a Turing Machine which computes
  • Slide 5
  • Courtesy Costas Busch - RPI5 When we say: There exists an algorithm Algorithms are Turing Machines We mean: There exists a Turing Machine that executes the algorithm
  • Slide 6
  • Courtesy Costas Busch - RPI6 Variations of the Turing Machine
  • Slide 7
  • Courtesy Costas Busch - RPI7 Read-Write Head Control Unit Deterministic The Standard Model Infinite Tape (Left or Right)
  • Slide 8
  • Courtesy Costas Busch - RPI8 Variations of the Standard Model Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional Nondeterministic Turing machines with:
  • Slide 9
  • Courtesy Costas Busch - RPI9 We want to prove: Each Class has the same power with the Standard Model The variations form different Turing Machine Classes
  • Slide 10
  • Courtesy Costas Busch - RPI10 Same Power of two classes means: Both classes of Turing machines accept the same languages
  • Slide 11
  • Courtesy Costas Busch - RPI11 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 12
  • Courtesy Costas Busch - RPI12 a technique to prove same powerSimulation: Simulate the machine of one class with a machine of the other class First Class Original Machine Second Class Simulation Machine
  • Slide 13
  • Courtesy Costas Busch - RPI13 Configurations in the Original Machine correspond to configurations in the Simulation Machine Original Machine: Simulation Machine:
  • Slide 14
  • Courtesy Costas Busch - RPI14 The Simulation Machine and the Original Machine accept the same language Original Machine: Simulation Machine: Final Configuration
  • Slide 15
  • Courtesy Costas Busch - RPI15 Turing Machines with Stay-Option The head can stay in the same position Left, Right, Stay L,R,S: moves
  • Slide 16
  • Courtesy Costas Busch - RPI16 Example: Time 1 Time 2
  • Slide 17
  • Courtesy Costas Busch - RPI17 Stay-Option Machines have the same power with Standard Turing machines Theorem:
  • Slide 18
  • Courtesy Costas Busch - RPI18 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)
  • Slide 19
  • Courtesy Costas Busch - RPI19 Part 2: Standard Machines are at least as powerful as Stay-Option machines Proof:a standard machine can simulate a Stay-Option machine Proof:
  • Slide 20
  • Courtesy Costas Busch - RPI20 Stay-Option Machine Simulation in Standard Machine Similar for Right moves
  • Slide 21
  • Courtesy Costas Busch - RPI21 Stay-Option Machine Simulation in Standard Machine For every symbol
  • Slide 22
  • Courtesy Costas Busch - RPI22 Example Stay-Option Machine: 12 Simulation in Standard Machine: 123
  • Slide 23
  • Courtesy Costas Busch - RPI23 Standard Machine--Multiple Track Tape track 1 track 2 one symbol
  • Slide 24
  • Courtesy Costas Busch - RPI24 track 1 track 2 track 1 track 2
  • Slide 25
  • Courtesy Costas Busch - RPI25 Semi-Infinite Tape.........
  • Slide 26
  • Courtesy Costas Busch - RPI26 Standard Turing machines simulate Semi-infinite tape machines: Trivial
  • Slide 27
  • Courtesy Costas Busch - RPI27 Semi-infinite tape machines simulate Standard Turing machines: Standard machine......... Semi-infinite tape machine.........
  • Slide 28
  • Courtesy Costas Busch - RPI28 Standard machine......... Semi-infinite tape machine with two tracks......... reference point Right part Left part
  • Slide 29
  • Courtesy Costas Busch - RPI29 Left partRight part Standard machine Semi-infinite tape machine
  • Slide 30
  • Courtesy Costas Busch - RPI30 Standard machine Semi-infinite tape machine Left part Right part For all symbols
  • Slide 31
  • Courtesy Costas Busch - RPI31 Standard machine......... Semi-infinite tape machine Right part Left part Time 1
  • Slide 32
  • Courtesy Costas Busch - RPI32 Time 2 Right part Left part Standard machine......... Semi-infinite tape machine
  • Slide 33
  • Courtesy Costas Busch - RPI33 Semi-infinite tape machine Left part At the border: Right part
  • Slide 34
  • Courtesy Costas Busch - RPI34......... Semi-infinite tape machine Right part Left part......... Right part Left part Time 1 Time 2
  • Slide 35
  • Courtesy Costas Busch - RPI35 Theorem:Semi-infinite tape machines have the same power with Standard Turing machines
  • Slide 36
  • Courtesy Costas Busch - RPI36 The Off-Line Machine Control Unit Input File Tape read-only read-write
  • Slide 37
  • Courtesy Costas Busch - RPI37 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 38
  • Courtesy Costas Busch - RPI38 1. Copy input file to tape Input File Tape Standard machine Off-line machine
  • Slide 39
  • Courtesy Costas Busch - RPI39 2. Do computations as in Turing machine Input File Tape Standard machine Off-line machine
  • Slide 40
  • Courtesy Costas Busch - RPI40 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
  • Slide 41
  • Courtesy Costas Busch - RPI41 Input File Tape Off-line Machine Four track tape -- Standard Machine Input File head position Tape head position
  • Slide 42
  • Courtesy Costas Busch - RPI42 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 Reference point
  • Slide 43
  • Courtesy Costas Busch - RPI43 Off-line machines have the same power with Standard machines Theorem:
  • Slide 44
  • Courtesy Costas Busch - RPI44 Multitape Turing Machines Control unit Tape 1Tape 2 Input
  • Slide 45
  • Courtesy Costas Busch - RPI45 Time 1 Time 2 Tape 1Tape 2
  • Slide 46
  • Courtesy Costas Busch - RPI46 Multitape machines simulate Standard Machines: Use just one tape
  • Slide 47
  • Courtesy Costas Busch - RPI47 Standard machines simulate Multitape machines: Use a multi-track tape A tape of the Multiple tape machine corresponds to a pair of tracks Standard machine:
  • Slide 48
  • Courtesy Costas Busch - RPI48 Multitape Machine Tape 1Tape 2 Standard machine with four track tape Tape 1 head position Tape 2 head position
  • Slide 49
  • Courtesy Costas Busch - RPI49 Repeat for each state transition: Return to reference point Find current symbol in Tape 1 Find current symbol in Tape 2 Make transition Tape 1 head position Tape 2 head position Reference point
  • Slide 50
  • Courtesy Costas Busch - RPI50 Theorem:Multi-tape machines have the same power with Standard Turing Machines
  • Slide 51
  • Courtesy Costas Busch - RPI51 Same power doesnt imply same speed: Language Acceptance Time Standard machine Two-tape machine
  • Slide 52
  • Courtesy Costas Busch - RPI52 Standard machine: Go back and forth times Two-tape machine: Copy to tape 2 Leave on tape 1 Compare tape 1 and tape 2 ( steps)
  • Slide 53
  • Courtesy Costas Busch - RPI53 MultiDimensional Turing Machines Two-dimensional tape HEAD Position: +2, -1 MOVES: L,R,U,D U: up D: down
  • Slide 54
  • Courtesy Costas Busch - RPI54 Multidimensional machines simulate Standard machines: Use one dimension
  • Slide 55
  • Courtesy Costas Busch - RPI55 Standard machines simulate Multidimensional machines: Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2
  • Slide 56
  • Courtesy Costas Busch - RPI56 symbols coordinates Two-dimensional machine Standard Machine
  • Slide 57
  • Courtesy Costas Busch - RPI57 Repeat for each transition Update current symbol Compute coordinates of next position Go to new position Standard machine:
  • Slide 58
  • Courtesy Costas Busch - RPI58 MultiDimensional Machines have the same power with Standard Turing Machines Theorem:
  • Slide 59

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