Turing’s Thesis

Fall 2004 COMP 335 1

Turing’s Thesis


Turing’s thesis:

Any computation carried outby mechanical meanscan be performed by a Turing Machine

(1930)


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


Definition of Algorithm:

An algorithm for functionis a Turing Machine which computes

)(wf

)(wf


When we say:

There exists an algorithm

Algorithms are Turing Machines

We mean:

There exists a Turing Machinethat executes the algorithm


Variationsof the

Turing Machine


Read-Write Head

Control Unit

a a c b a cb b a a

Deterministic

The Standard Model

Infinite Tape

(Left or Right)


Variations of the Standard Model

• Stay-Option • Semi-Infinite Tape• Off-Line• Multitape• Multidimensional• Nondeterministic

Turing machines with:


We want to prove:

Each Class has the samepower with the Standard Model

The variations form differentTuring Machine Classes

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Same Power of two classes means:

Both classes of Turing machines accept the same languages

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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

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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

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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

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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

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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:

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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:

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1q 2qLba ,

1q 2qLba ,

Stay-Option Machine

Simulation in Standard Machine

Similar for Right moves

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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

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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

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Semi-Infinite Tape

.........# a b a c

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Standard Turing machines simulateSemi-infinite tape machines:

Trivial

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Semi-infinite tape machines simulateStandard Turing machines:

Standard machine

.........

Semi-infinite tape machine

..................

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Standard machine

.........

Semi-infinite tape machine with two tracks

..................

reference point

#

#

Right part

Left part

a b c d e

ac bd e

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1q2q

Rq2Lq1

Lq2 Rq1

Left part Right part

Standard machine


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1q 2qRga ,

Standard machine

Lq1Lq2

Lgxax ),,(),(

Rq1Rq2

Rxgxa ),,(),(


Left part

Right part

For all symbols x

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Standard machine.................. a b c d e

1q

.........


#

#

Right part

Left part ac bd e

Lq1

Time 1

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Time 2

g b c d e

2q

#

#

Right part

Left part gc bd e

Lq2

Standard machine..................

.........


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Lq1Rq1

R),#,(#)#,(#


Left part

At the border:

Rq1Lq1

R),#,(#)#,(# Right part

Fall 2004 COMP 335 34

.........


#

#

Right part

Left part gc bd e

Lq1

.........#

#

Right part

Left part gc bd e

Rq1

Time 1

Time 2

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Theorem: Semi-infinite tape machineshave the same power with Standard Turing machines

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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

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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

##

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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

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Off-line machineshave the same power withStansard machines

Theorem:

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Multitape Turing Machines

a b c e f g

Control unit

Tape 1 Tape 2

Input

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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

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Multitape machines simulate Standard Machines:

Use just one tape

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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:

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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

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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

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Theorem: Multi-tape machineshave the same power withStandard Turing Machines

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Same power doesn’t imply same speed:

Language }{ nnbaL

Acceptance Time

Standard machine

Two-tape machine

2n

n

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}{ 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)

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MultiDimensional Turing Machines

x

y

ab

c

Two-dimensional tape

HEADPosition: +2, -1

MOVES: L,R,U,DU: up D: down

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Multidimensional machines simulate Standard machines:

Use one dimension

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Standard machines simulateMultidimensional machines:

Standard machine:

• Use a two track tape

• Store symbols in track 1• Store coordinates in track 2

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x

y

ab

c

a1

b#

symbols

coordinates

Two-dimensional machine

Standard Machine

1 # 2 # 1c

# 1

1q

1q

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Repeat for each transition

• Update current symbol• Compute coordinates of next position• Go to new position

Standard machine:

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MultiDimensional Machineshave the same powerwith Standard Turing Machines

Theorem:

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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

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Input string is accepted if this a possible computation

w

yqxwq f

0

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 simulateNonDeterministic machines:

Keeps track of all possible computations

Deterministic machine:

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Non-Deterministic Choices

Computation 1

1q

2q

4q

3q

5q

6q 7q

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Non-Deterministic Choices

Computation 2

1q

2q

4q

3q

5q

6q 7q

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• Keeps track of all possible computations

Deterministic machine:

Simulation

• Stores computations in a 2D tape

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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

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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

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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 the NonDeterministic machine

Turing’s Thesis

Documents

languageoriginal machine

examplestayoption machine

standard machines proof

standard machinefor

standard machinesimilar

classes of turing machines

stayoption machines

symbol comp