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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
Numbers Treasure Hunt Following each question, click on the answer. If correct, the next page will load with a graphic first – these can be used to check.
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