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1 Variations of the Turing Machine

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

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3 Variations of the Standard Model Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional Nondeterministic Turing machines with:

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4 We want to prove: Each Class has the same power with the Standard Model The variations form different Turing Machine Classes

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5 Same Power of two classes means: Both classes of Turing machines accept the same languages

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6 Same Power of two classes means: For any machine of first class there is a machine of second class such that: And vice-versa

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7 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

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8 Configurations in the Original Machine correspond to configurations in the Simulation Machine Original Machine: Simulation Machine:

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9 The Simulation Machine and the Original Machine accept the same language Original Machine: Simulation Machine: Final Configuration

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

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11 Example: Time 1 Time 2

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12 Stay-Option Machines have the same power with Standard Turing machines Theorem:

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13 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)

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14 Part 2: Standard Machines are at least as powerful as Stay-Option machines Proof:a standard machine can simulate a Stay-Option machine Proof:

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15 Stay-Option Machine Simulation in Standard Machine Similar for Right moves

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16 Stay-Option Machine Simulation in Standard Machine For every symbol

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17 Example Stay-Option Machine: 12 Simulation in Standard Machine: 123

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18 Standard Machine--Multiple Track Tape track 1 track 2 one symbol

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19 track 1 track 2 track 1 track 2

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20 Semi-Infinite Tape

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21 Standard Turing machines simulate Semi-infinite tape machines: Trivial

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22 Semi-infinite tape machines simulate Standard Turing machines: Standard machine Semi-infinite tape machine

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23 Standard machine Semi-infinite tape machine with two tracks reference point Right part Left part

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24 Theorem:Semi-infinite tape machines have the same power with Standard Turing machines

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25 The Off-Line Machine Control Unit Input File Tape read-only read-write

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

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

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

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29 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

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

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31 Off-line machines have the same power with Standard machines Theorem:

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32 Multitape Turing Machines Control unit Tape 1Tape 2 Input

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33 Time 1 Time 2 Tape 1Tape 2

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34 Multitape machines simulate Standard Machines: Use just one tape

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35 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:

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36 Multitape Machine Tape 1Tape 2 Standard machine with four track tape Tape 1 head position Tape 2 head position

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37 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

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38 Theorem:Multi-tape machines have the same power with Standard Turing Machines

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39 Same power doesnt imply same speed: Language Acceptance Time Standard machine Two-tape machine

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40 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)

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41 MultiDimensional Turing Machines Two-dimensional tape HEAD Position: +2, -1 MOVES: L,R,U,D U: up D: down

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42 Multidimensional machines simulate Standard machines: Use one dimension

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43 Standard machines simulate Multidimensional machines: Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2

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44 symbols coordinates Two-dimensional machine Standard Machine

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45 Repeat for each transition Update current symbol Compute coordinates of next position Go to new position Standard machine:

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46 MultiDimensional Machines have the same power with Standard Turing Machines Theorem:

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47 NonDeterministic Turing Machines Non Deterministic Choice

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48 Time 0 Time 1 Choice 1 Choice 2

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49 Input string is accepted if this a possible computation Initial configurationFinal Configuration Final state

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50 NonDeterministic Machines simulate Standard (deterministic) Machines: Every deterministic machine is also a nondeterministic machine

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51 Deterministic machines simulate NonDeterministic machines: Keeps track of all possible computations Deterministic machine:

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52 Non-Deterministic Choices Computation 1

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53 Non-Deterministic Choices Computation 2

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54 Keeps track of all possible computations Deterministic machine: Simulation Stores computations in a two-dimensional tape

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55 Time 0 NonDeterministic machine Deterministic machine Computation 1

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56 Computation 1 Choice 1 Choice 2 Computation 2 NonDeterministic machine Deterministic machine Time 1

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57 Theorem: NonDeterministic Machines have the same power with Deterministic machines

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58 Remark: The simulation in the Deterministic machine takes time exponential time compared to the NonDeterministic machine

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