Variations%20of%20the%20Turing%20Machine%20part2

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Variationsof theTuring Machinepart2

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Standard Machine--Multiple Track Tape
track 1
track 2
one symbol
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track 1
track 2
track 1
track 2
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Semi-Infinite Tape
.........
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Standard Turing machines simulate Semi-infinite
tape machines
Trivial
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Semi-infinite tape machines simulate Standard
Turing machines
Turing machine
.........
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Semi-infinite tape machine
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Turing machine
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reference point
Semi-infinite tape with two tracks
Right part
.........
Left part
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Turing machine
Semi-infinite tape machine
Left part
Right part
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Turing machine
Semi-infinite tape machine
Right part
Left part
For all symbols x
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Time 1
Turing machine
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.........
Semi-infinite tape
Right part
.........
Left part
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Time 2
Turing machine
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Semi-infinite tape
Right part
.........
Left part
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At the border
Semi-infinite tape machine
Right part
Left part
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Semi-infinite tape
Time 1
Right part
.........
Left part
Time 2
Right part
.........
Left part
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Theorem
Semi-infinite tape machines have the same power
with Standard Turing machines
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The Off-Line Machine
Input File
read-only
Control Unit
read-write
Tape
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Off-line Machines simulate Turing Machines
Off-line machine
1. Copy input file to tape 2. Continue
computation as in Standard Turing machine
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Turing Machine
Off-line Machine
Tape
Input File
1. Copy input file to tape
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Turing Machine
Off-line Machine
Tape
Input File
2. Do computations as in Turing machine
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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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Off-line Machine
Tape
Input File
Four track tape -- Standard Machine
input file
head position
tape
head position
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Reference point
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
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Theorem
Off-line machines have the same power
with Stansard machines
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Multitape Turing Machines
Control unit
Input
tape 1
tape 2
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Time 1
Time 2
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Multitape machines simulate Standard Machines
Just use one tape
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Standard machines simulate Multitape machines
Standard machine
Use a multi-track tape
A tape of the Multiple tape machine corresponds
to a pair of tracks
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Multitape Machine
Tape 1
Tape 2
Four track tape -- Standard Machine
Tape 1
head position
Tape 2
head position
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Reference point
Tape 1
head position
Tape 2
head position
Repeat for each state transition
Return to reference point Find current symbol on
Tape 1 Find current symbol on Tape 2 make
transition
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Theorem
Multi-tape machines have the same power
with Standard Turing Machines
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Same power doesnt mean same speed
Language
Acceptance Time
Standard machine
Two-tape machine
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Standard machine
Go back and forth times
Two-tape machine
Copy to tape 2
( steps)
( steps)
Leave to tape 1
Compare tape 1 and tape 2
( steps)
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MultiDimensional Turing Machines
Two-dimensional tape
MOVES L,R,U,D
HEAD
Position 2, -1
U up D down
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Multidimensional machines simulate Standard
machines
Just use one dimension
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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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Two-dimensional machine
Standard Machine
symbols
coordinates
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Simulation
Repeat for each transition
Update current symbol Compute coordinates of next
position Go to new position
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Theorem MultiDimensional Machines have
the same power with Turing Machines
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NonDeterministic Turing Machines
Non Deterministic Choice
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TIME 0
TIME 1
Choice 1
Choice 2
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Input is accepted if this a possible
computation
Initial configuration
Final Configuration
Final state
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NonDeterministic Machines simulate Standard
(deterministic) Machines
Every deterministic machine is also a
nondeterministic machine
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Deterministic machines simulate NonDeterministic
machines
Deterministic machine
Keeps track of all possible computations
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Non-Deterministic Choices
Computation 1
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Non-Deterministic Choices
Computation 2
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Simulation
Deterministic machine
Keeps track of all possible computations
Stores computations in a two-dimensional tape
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TIME 0
NonDeterministic
Deterministic
Computation 1
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TIME 1
NonDeterministic
Choice 1
Choice 2
Deterministic
Computation 1
Computation 2
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Repeat Execute a step in each computation
If there is a choice in current computation
replicate configuration change the
state in the replica
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Theorem NonDeterministic Machines
have the same power with deterministic
machines
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Remark The simulation in the
deterministic Machine takes time
exponential time compared to
NonDeterministic machines
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Polynomial Time in NonDeterministic Machine
NP-Time
Polynomial Time in Deterministic Machine
P-Time
Fundamental Problem
P NP ?