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Variations of the Turing Machine
The Standard Model Infinite Tape Read-Write Head (Left or Right)Control Unit Deterministic
Variations of the Standard ModelTuring machines with: Stay-Option Semi-Infinite Tape Off-Line Multitape Multidimensional Nondeterministic
The variations form differentTuring Machine Classes We want to prove: Each Class has the same power with the Standard Model
Same Power of two classes means:Both classes of Turing machines accept the same languages
Same Power of two classes means:For any machine of first class there is a machine of second class such that: And vice-versa
Simulation: a technique to prove same power Simulate the machine of one class with a machine of the other class Second Class Simulation Machine First Class Original Machine
Configurations in the Original Machinecorrespond to configurations in the Simulation Machine Original Machine: Simulation Machine:
Final Configuration Original Machine: Simulation Machine: The Simulation Machine and the Original Machine accept the same language
Turing Machines with Stay-OptionThe head can stay in the same position Left, Right, Stay L,R,S: moves
Example: Time 1 Time 2
Theorem: Stay-Option Machines have the same power with Standard Turing machines
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)
Proof: Part 2: Standard Machines are at least as powerful as Stay-Option machines Proof: a standard machine can simulate a Stay-Option machine
Stay-Option Machine Simulation in Standard Machine Similar for Right moves
Stay-Option Machine Simulation in Standard Machine For every symbol
Example Stay-Option Machine: 1 2 Simulation in Standard Machine: 1 2 3
Standard Machine--Multiple Track Tapeone symbol
track 1 track 2 track 1 track 2
Standard Turing machines simulateSemi-infinite tape machines: Trivial
Semi-infinite tape machines simulateStandard Turing machines: Standard machine Semi-infinite tape machine
Standard machine reference point Semi-infinite tape machine with two tracks Right part Left part
Theorem: Semi-infinite tape machines have the same power with Standard Turing machines
The Off-Line Machine Input File read-only Control Unit read-write Tape
Off-line machines simulateStandard Turing Machines: Off-line machine: 1. Copy input file to tape 2. Continue computation as in Standard Turing machine
Standard machine Off-line machine Tape Input File 1. Copy input file to tape
Standard machine Off-line machine Tape Input File 2. Do computations as in Turing machine
Standard Turing machines simulateOff-line machines: Use a Standard machine with four track tape to keep track of the Off-line input file and tape contents
Off-line Machine Tape Input File Four track tape -- Standard Machine Input File head position Tape head position
Theorem: Off-line machines have the same power with Standard machines
Multitape Turing MachinesControl unit Tape 1 Tape 2 Input
Tape 1 Time 1 Tape 2 Time 2
Multitape machines simulateStandard Machines: Use just one tape
Standard machines simulateMultitape machines: Standard machine: Use a multi-track tape A tape of the Multiple tape machine corresponds to a pair of tracks
Multitape Machine Tape 1 Tape 2 Standard machine with four track tape Tape 1 head position Tape 2 head position
Reference point Tape 1 head position Tape 2 head position Repeat for each state transition: Return to reference point Find current symbol in Tape 1 Find current symbol in Tape 2 Make transition
Theorem: Multi-tape machines have the same power with Standard Turing Machines
Same power doesn’t imply same speed:Language Acceptance Time Standard machine Two-tape machine
Standard machine: Go back and forth times Two-tape machine: Copy to tape 2 ( steps) ( steps) Leave on tape 1 Compare tape 1 and tape 2 ( steps)
MultiDimensional Turing MachinesTwo-dimensional tape MOVES: L,R,U,D HEAD U: up D: down Position: +2, -1
Multidimensional machines simulateStandard machines: Use one dimension
Standard machines simulateMultidimensional machines: Standard machine: Use a two track tape Store symbols in track 1 Store coordinates in track 2
Two-dimensional machineStandard Machine symbols coordinates
Standard machine: Repeat for each transition Update current symbol Compute coordinates of next position Go to new position
Theorem: MultiDimensional Machines have the same power with Standard Turing Machines
NonDeterministic Turing MachinesNon Deterministic Choice
Time 0 Time 1 Choice 1 Choice 2
Input string is accepted ifthis a possible computation Initial configuration Final Configuration Final state
NonDeterministic Machines simulateStandard (deterministic) Machines: Every deterministic machine is also a nondeterministic machine
Deterministic machines simulateNonDeterministic machines: Deterministic machine: Keeps track of all possible computations
Non-Deterministic ChoicesComputation 1
Non-Deterministic ChoicesComputation 2
Simulation Deterministic machine:Keeps track of all possible computations Stores computations in a two-dimensional tape
NonDeterministic machineTime 0 Deterministic machine Computation 1
NonDeterministic machineTime 1 Choice 1 Choice 2 Deterministic machine Computation 1 Computation 2
Theorem: NonDeterministic Machineshave the same power with Deterministic machines
Remark: The simulation in the Deterministic machine takes time exponential time compared to the NonDeterministic machine
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