Turings Thesis

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Turings thesis (1930):Any computation carried outby mechanical meanscan be performed by a Turing Machine

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Algorithm:An algorithm for a problem is a Turing Machine which solves the problem The algorithm describes the steps of the mechanical meansThis is easily translated to computation stepsof a Turing machine

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When we say:There exists an algorithmWe mean:There exists a Turing Machinethat executes the algorithm

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Variationsof theTuring Machine

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Read-Write HeadControl UnitDeterministicThe Standard ModelInfinite Tape(Left or Right)

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Variations of the Standard Model Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional NondeterministicTuring machines with:Different Turing Machine Classes

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We will prove:each new class has the same power with Standard Turing MachineSame Power of two machine classes:both classes accept the same set of languages(accept Turing-Recognizable Languages)

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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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A technique to prove same power.Simulation:Simulate the machine of one classwith a machine of the other classFirst Class Original Machine Second ClassSimulation Machinesimulates

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Configurations in the Original Machinehave corresponding configurations in the Simulation MachineOriginal Machine:Simulation Machine:

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the Simulation Machineand the Original Machineaccept the same stringsOriginal Machine:Simulation Machine:Accepting Configuration

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Turing Machines with Stay-OptionThe head can stay in the same positionLeft, Right, StayL,R,S: possible head moves

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Example:Time 1Time 2

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Stay-Option machineshave the same power with Standard Turing machinesTheorem:Proof:1. Stay-Option Machines simulate Standard Turing machines 2.Standard Turing machines simulate Stay-Option machines

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

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

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Stay-Option MachineSimulation in Standard MachineFor every possible tape symbol

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Stay-Option MachineSimulation in Standard MachineSimilar for Right movesFor other transitions nothing changes

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example of simulationStay-Option Machine:12Simulation in Standard Machine:123END OF PROOF

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Multiple Track Tapetrack 1track 2One symbolOne headA useful trick to perform morecomplicated simulationsOne Tape

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track 1track 2track 1track 2

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

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Semi-Infinite machineshave the same power with Standard Turing machinesTheorem:Proof:2. Semi-Infinite Machines simulate Standard Turing machines 1.Standard Turing machines simulate Semi-Infinite machines

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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

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

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Standard machine.........Semi-Infinite tape machine with two tracks..................reference pointRight partLeft part

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Left partRight partStandard machineSemi-Infinite tape machine

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Standard machineSemi-Infinite tape machineLeft partRight partFor all tape symbols

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Standard machine...........................Semi-Infinite tape machineRight partLeft partTime 1

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Time 2Right partLeft partStandard machine...........................Semi-Infinite tape machine

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Semi-Infinite tape machineLeft partAt the border:Right part

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.........Semi-Infinite tape machineRight partLeft part.........Right partLeft partTime 1Time 2END OF PROOF

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The Off-Line MachineControl UnitInput FileTaperead-only (once)read-writeInput string Appears on input file only(state machine)Input string

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Off-Line machineshave the same power with Standard Turing machinesTheorem:Proof:1. Off-Line machines simulate Standard Turing machines 2.Standard Turing machines simulate Off-Line machines

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1. Off-line machines simulate Standard Turing MachinesOff-line machine:1. Copy input file to tape

2. Continue computation as in Standard Turing machine

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1. Copy input file to tapeInput FileTapeStandard machineOff-line machine

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2. Do computations as in Turing machineInput FileTapeStandard machineOff-line machine

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

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Input FileTapeOff-line MachineStandard Machine -- Four track tape Input Filehead positionTapehead position

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Input Filehead positionTapehead positionRepeat for each state transition: Return to reference point Find current input file symbol Find current tape symbol Make transitionReference point(uses special symbol # )END OF PROOF

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Multi-tape Turing MachinesControl unitTape 1Tape 2Input stringInput string appears on Tape 1(state machine)

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Time 1Time 2Tape 1Tape 2Tape 1Tape 2

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Multi-tape machineshave the same power with Standard Turing machinesTheorem:Proof:1. Multi-tape machines simulate Standard Turing machines 2.Standard Turing machines simulate Multi-tape machines

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

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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 tracksStandard machine:

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Multi-tape MachineTape 1Tape 2Standard machine with four track tapeTape 1head positionTape 2head position

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Repeat for each state transition:Return to reference pointFind current symbol in Tape 1Find current symbol in Tape 2Make transitionTape 1head positionTape 2head positionReference pointEND OF PROOF

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( steps)Standard Turing machine:Go back and forth times 2-tape machine:1. Copy to tape 2 2. Compare on tape 1 and tape 2to match the as with the bs( steps)timetimeSame power doesnt imply same speed:

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Multidimensional Turing Machines2-dimensional tapeHEADPosition: +2, -1MOVES: L,R,U,DU: up D: down

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Multidimensional machineshave the same power with Standard Turing machinesTheorem:Proof:1. Multidimensional machines simulate Standard Turing machines 2.Standard Turing machines simulate Multi-Dimensional machines

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1. Multidimensional machines simulate Standard Turing machinesTrivial: Use one dimension

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

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symbolscoordinates2-dimensional machineStandard Machine

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Repeat for each transition followedin the 2-dimensional machine: Update current symbol Compute coordinates of next position Go to new positionStandard machine:END OF PROOF

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Nondeterministic Turing MachinesAllows Non Deterministic ChoicesChoice 1Choice 2

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Time 0Time 1Choice 1Choice 2

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Input string is accepted if there is a computation:Initial configurationFinal ConfigurationAny accept stateThere is a computation:

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Nondeterministic machineshave the same power with Standard Turing machinesTheorem:Proof:1. Nondeterministic machines simulate Standard Turing machines 2.Standard Turing machines simulate Nondeterministic machines

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1. Nondeterministic Machines simulate Standard (deterministic) Turing MachinesTrivial: every deterministic machine is also nondeterministic

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2. Standard (deterministic) Turing machines simulate Nondeterministic machines: Stores all possible computations of the non-deterministic machine on the 2-dimensional tapeDeterministic machine: Uses a 2-dimensional tape (which is equivalent to 1-dimensional tape)

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All possible computation pathsInitial stateStep 1Step 2Step iStep i+1acceptrejectinfinitepath

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The Deterministic Turing machinesimulates all possible computation paths:in a breadth-first search fashionsimultaneouslystep-by-step

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Time 0NonDeterministic machineDeterministic machinecurrentconfiguration

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Computation 1Choice 1Choice 2Computation 2NonDeterministic machineDeterministic machineTime 1

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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 PROOFDeterministic Turing machineUntil either the input string is accepted or rejected in all configurations

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The simulation takes in the worst case exponential time compared to the shortest accepting path length of the nondeterministic machineRemark:

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