Transcript powerpoint

Variations
of the
Turing Machine
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The Standard Model
Infinite Tape
 aababbcac a
Read-Write Head
(Left or Right)
Control Unit
Deterministic
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Variations of the Standard Model
• Turing Machines with Stay-Option
• Turing Machines with Semi-Infinite Tape
• The Off-Line Turing Machine
• Multitape Turing Machines
• Multidimensional Turing Machines
• Nondeterministic Turing Machines
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The variations form different
Turing Machine Classes
We want to prove:
Each Class has the same
power with the Standard Model
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Same Power of two classes means:
Take machine
M1 of first class
There is a machine
such that:
M 2 of second class
L( M1)  L( M 2 )
(second class is at least as powerful
as first class)
And vice-versa
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A technique to prove same power: Simulation
Simulate the machine of one class
with a machine of the other class
First Class
Original Machine
M1
Second Class
Simulation Machine
M2
M1
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Configurations in the Original Machine
correspond to configurations
in the Simulation Machine
Original Machine:
Simulation Machine:
d0  d1    d n



d 0  d1    d n
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Final Configuration
Original Machine:
df
Simulation Machine:
d f
The Simulation Machine
and the Original Machine
will accept the same language
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Turing Machines with Stay-Option
The head can stay in the same position
 aababbcac a
Left, Right, Stay
L,R,S: moves
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Example:
Time 1
 aababbcac a
q1
Time 2
 b ab abb c ac a
q2
q1
a  b, S
q2
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Theorem:
Machines with Stay-Option
have the same power
with the standard Turing machines
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Proof:
Part1 -- Stay-Option Machines
are at least as powerful as
standard machines
Trivial proof:
A standard machine is also
a Stay-Option machine
(that never uses the S move)
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Part2 -- Standard Machines
are at least as powerful as
Stay-Option machines
Proof:
a standard machine
can simulate
a Stay-Option machine
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Stay-Option Machine
q1
a  b, L
q2
Simulation in Standard Machine
q1
a  b, L
Similar for Right moves
q2
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Stay-Option Machine
q1
a  b, S
q2
Simulation in Standard Machine
q1
a  b, L
q2
x  x, R
q3
For every symbol x
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Example
Stay-Option Machine:
a

b
,
S
q1
q2
1
aaba 
q1
2
baba 
q2
Simulation in Standard Machine:
1
aaba 
q1
2
baba 
q2
3
baba 
q3
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