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Fall 2004 COMP 335 1
Turing’s Thesis
Fall 2004 COMP 335 2
Turing’s thesis:
Any computation carried outby mechanical meanscan be performed by a Turing Machine
(1930)
Fall 2004 COMP 335 3
Computer Science Law:
A computation is mechanical if and only ifit can be performed by a Turing Machine
There is no known model of computationmore powerful than Turing Machines
Fall 2004 COMP 335 4
Definition of Algorithm:
An algorithm for functionis a Turing Machine which computes
)(wf
)(wf
Fall 2004 COMP 335 5
When we say:
There exists an algorithm
Algorithms are Turing Machines
We mean:
There exists a Turing Machinethat executes the algorithm
Fall 2004 COMP 335 6
Variationsof the
Turing Machine
Fall 2004 COMP 335 7
Read-Write Head
Control Unit
a a c b a cb b a a
Deterministic
The Standard Model
Infinite Tape
(Left or Right)
Fall 2004 COMP 335 8
Variations of the Standard Model
• Stay-Option • Semi-Infinite Tape• Off-Line• Multitape• Multidimensional• Nondeterministic
Turing machines with:
Fall 2004 COMP 335 9
We want to prove:
Each Class has the samepower with the Standard Model
The variations form differentTuring Machine Classes
Fall 2004 COMP 335 10
Same Power of two classes means:
Both classes of Turing machines accept the same languages
Fall 2004 COMP 335 11
Same Power of two classes means:
For any machine of first class 1M
there is a machine of second class 2M
such that: )()( 21 MLML
And vice-versa
Fall 2004 COMP 335 12
a technique to prove same powerSimulation:
Simulate the machine of one classwith a machine of the other class
First ClassOriginal Machine
1M 1M
2M
Second ClassSimulation Machine
Fall 2004 COMP 335 13
Configurations in the Original Machinecorrespond to configurations in the Simulation Machine
nddd 10Original Machine:
Simulation Machine: nddd
10
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The Simulation Machineand the Original Machineaccept the same language
fdOriginal Machine:
Simulation Machine: fd
Final Configuration
Fall 2004 COMP 335 15
Turing Machines with Stay-Option
The head can stay in the same position
a a c b a cb b a a
Left, Right, Stay
L,R,S: moves
Fall 2004 COMP 335 16
Example:
a a c b a cb b a a
Time 1
b a c b a cb b a a
Time 2
1q 2q
1q
2q
Sba ,
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Stay-Option Machineshave the same power with Standard Turing machines
Theorem:
Fall 2004 COMP 335 18
Proof:
Part 1: Stay-Option Machines are at least as powerful as Standard machines
Proof: a Standard machine is alsoa Stay-Option machine(that never uses the S move)
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Part 2: Standard Machines are at least as powerful as Stay-Option machines
Proof: a standard machine can simulatea Stay-Option machine
Proof:
Fall 2004 COMP 335 20
1q 2qLba ,
1q 2qLba ,
Stay-Option Machine
Simulation in Standard Machine
Similar for Right moves
Fall 2004 COMP 335 21
1q 2qSba ,
1q 2qLba ,
3qRxx ,
Stay-Option Machine
Simulation in Standard Machine
For every symbol x
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Example
a a b a
1q
Stay-Option Machine:
1 b a b a
2q
21q 2q
Sba ,
Simulation in Standard Machine:
a a b a
1q
1 b a b a
2q
2 b a b a
3q
3
Fall 2004 COMP 335 23
Standard Machine--Multiple Track Tape
bd
abbaac
track 1
track 2
one symbol
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bd
abbaac
track 1
track 2
1q 2qLdcab ),,(),(
1q
bd
abcdac
track 1
track 2
2q
Fall 2004 COMP 335 25
Semi-Infinite Tape
.........# a b a c
Fall 2004 COMP 335 26
Standard Turing machines simulateSemi-infinite tape machines:
Trivial
Fall 2004 COMP 335 27
Semi-infinite tape machines simulateStandard Turing machines:
Standard machine
.........
Semi-infinite tape machine
..................
Fall 2004 COMP 335 28
Standard machine
.........
Semi-infinite tape machine with two tracks
..................
reference point
#
#
Right part
Left part
a b c d e
ac bd e
Fall 2004 COMP 335 29
1q2q
Rq2Lq1
Lq2 Rq1
Left part Right part
Standard machine
Semi-infinite tape machine
Fall 2004 COMP 335 30
1q 2qRga ,
Standard machine
Lq1Lq2
Lgxax ),,(),(
Rq1Rq2
Rxgxa ),,(),(
Semi-infinite tape machine
Left part
Right part
For all symbols x
Fall 2004 COMP 335 31
Standard machine.................. a b c d e
1q
.........
Semi-infinite tape machine
#
#
Right part
Left part ac bd e
Lq1
Time 1
Fall 2004 COMP 335 32
Time 2
g b c d e
2q
#
#
Right part
Left part gc bd e
Lq2
Standard machine..................
.........
Semi-infinite tape machine
Fall 2004 COMP 335 33
Lq1Rq1
R),#,(#)#,(#
Semi-infinite tape machine
Left part
At the border:
Rq1Lq1
R),#,(#)#,(# Right part
Fall 2004 COMP 335 34
.........
Semi-infinite tape machine
#
#
Right part
Left part gc bd e
Lq1
.........#
#
Right part
Left part gc bd e
Rq1
Time 1
Time 2
Fall 2004 COMP 335 35
Theorem: Semi-infinite tape machineshave the same power with Standard Turing machines
Fall 2004 COMP 335 36
The Off-Line Machine
Control Unit
Input File
Tape
read-only
a b c
d eg
read-write
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Off-line machines simulate Standard Turing Machines:
Off-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 tape
Input Filea b c
Tape
a b c Standard machine
Off-line machine
a b c
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2. Do computations as in Turing machine
Input Filea b c
Tape
a b c
a b c
1q
1q
Standard machine
Off-line machine
Fall 2004 COMP 335 40
Standard Turing machines simulate Off-line machines:
Use a Standard machine with four track tapeto keep track ofthe Off-line input file and tape contents
Fall 2004 COMP 335 41
Input Filea b c
Tape
Off-line Machine
e f gd
Four track tape -- Standard Machine
a b c d
e f g0 0 0
0 0
1
1
Input File
head position
Tapehead position
##
Fall 2004 COMP 335 42
a b c d
e f g0 0 0
0 0
1
1
Input File
head position
Tapehead position
##
Repeat for each state transition:• Return to reference point• Find current input file symbol• Find current tape symbol• Make transition
Reference point
Fall 2004 COMP 335 43
Off-line machineshave the same power withStansard machines
Theorem:
Fall 2004 COMP 335 44
Multitape Turing Machines
a b c e f g
Control unit
Tape 1 Tape 2
Input
Fall 2004 COMP 335 45
a b c e f g
1q 1q
a g c e d g
2q 2q
Time 1
Time 2
RLdgfb ,),,(),( 1q 2q
Tape 1 Tape 2
Fall 2004 COMP 335 46
Multitape machines simulate Standard Machines:
Use just one tape
Fall 2004 COMP 335 47
Standard machines simulate Multitape machines:
• Use a multi-track tape
• A tape of the Multiple tape machine corresponds to a pair of tracks
Standard machine:
Fall 2004 COMP 335 48
a b c h e f g
Multitape MachineTape 1 Tape 2
Standard machine with four track tape
a b c
e f g0 0
0 0
1
1
Tape 1
head position
Tape 2head position
h0
Fall 2004 COMP 335 49
Repeat for each state transition:•Return to reference point•Find current symbol in Tape 1•Find current symbol in Tape 2•Make transition
a b c
e f g0 0
0 0
1
1
Tape 1
head position
Tape 2head position
h0
####
Reference point
Fall 2004 COMP 335 50
Theorem: Multi-tape machineshave the same power withStandard Turing Machines
Fall 2004 COMP 335 51
Same power doesn’t imply same speed:
Language }{ nnbaL
Acceptance Time
Standard machine
Two-tape machine
2n
n
Fall 2004 COMP 335 52
}{ nnbaL
Standard machine:
Go back and forth times 2n
Two-tape machine:
Copy to tape 2 nb
Leave on tape 1 naCompare tape 1 and tape 2
n( steps)
n( steps)
n( steps)
Fall 2004 COMP 335 53
MultiDimensional Turing Machines
x
y
ab
c
Two-dimensional tape
HEADPosition: +2, -1
MOVES: L,R,U,DU: up D: down
Fall 2004 COMP 335 54
Multidimensional machines simulate Standard machines:
Use one dimension
Fall 2004 COMP 335 55
Standard machines simulateMultidimensional machines:
Standard machine:
• Use a two track tape
• Store symbols in track 1• Store coordinates in track 2
Fall 2004 COMP 335 56
x
y
ab
c
a1
b#
symbols
coordinates
Two-dimensional machine
Standard Machine
1 # 2 # 1c
# 1
1q
1q
Fall 2004 COMP 335 57
Repeat for each transition
• Update current symbol• Compute coordinates of next position• Go to new position
Standard machine:
Fall 2004 COMP 335 58
MultiDimensional Machineshave the same powerwith Standard Turing Machines
Theorem:
Fall 2004 COMP 335 59
NonDeterministic Turing Machines
Lba ,
Rca ,
1q
2q
3q
Non Deterministic Choice
Fall 2004 COMP 335 60
a b c
1q
Lba ,
Rca ,
1q
2q
3q
Time 0
Time 1
b b c
2q
c b c
3q
Choice 1 Choice 2
Fall 2004 COMP 335 61
Input string is accepted if this a possible computation
w
yqxwq f
0
Initial configuration Final Configuration
Final state
Fall 2004 COMP 335 62
NonDeterministic Machines simulate Standard (deterministic) Machines:
Every deterministic machine is also a nondeterministic machine
Fall 2004 COMP 335 63
Deterministic machines simulateNonDeterministic machines:
Keeps track of all possible computations
Deterministic machine:
Fall 2004 COMP 335 64
Non-Deterministic Choices
Computation 1
1q
2q
4q
3q
5q
6q 7q
Fall 2004 COMP 335 65
Non-Deterministic Choices
Computation 2
1q
2q
4q
3q
5q
6q 7q
Fall 2004 COMP 335 66
• Keeps track of all possible computations
Deterministic machine:
Simulation
• Stores computations in a 2D tape
Fall 2004 COMP 335 67
a b c
1q
Lba ,
Rca ,
1q
2q
3q
Time 0
NonDeterministic machine
Deterministic machine
a b c1q
# # # # ##### # #
##
# #
Computation 1
Fall 2004 COMP 335 68
Lba ,
Rca ,
1q
2q
3q
b b c2q
# # # # #### #
#
# #
Computation 1
b b c
2q
Choice 1
c b c
3q
Choice 2
c b c3q ## Computation 2
NonDeterministic machine
Deterministic machine
Time 1
Fall 2004 COMP 335 69
Repeat• Execute a step in each computation:
• If there are two or more choices in current computation: 1. Replicate configuration 2. Change the state in the replica
Fall 2004 COMP 335 70
Theorem: NonDeterministic Machines have the same power with Deterministic machines
Fall 2004 COMP 335 71
Remark: The simulation in the Deterministic machine takes time exponential time compared to the NonDeterministic machine
