turing machine: structure and operational functionality

Structure and Operational Functionality of

The Turing Machine

11—13—2008

Soren Wellhofer

HistoryStructure and Definition

SamplesVarieties

ComputabilityReferences

Table of Contents

1 History

2 Structure and Definition

3 Samples

4 Varieties

5 Computability

6 References

Soren Wellhofer Structure and Operational Functionality of The Turing Machine

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Time frame — The 20th century

What were mathematicians working on?

• Rediscover Theory of Numbers

• Reduction of all math to fundamental logic

• Arithmetics/computations by means of automatic formalsystem

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What does it mean to be computable?

Turing’s achievements

• Proof of the possibility of a symbol-processing machine

• Simple operations according to rules

• Instruction tables for machine’s moves = formal system

• All computations logically feasible!

→ wrote first programmes

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General StructureFormal definitionAction table

Elements of the Turing Machine

• Read/Write Head

• Infinetly long tape

• Divided into cells

• Cells containing symbols

• Action table = the program

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• Read/Write Head

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• Read/Write Head

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The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

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The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

SamplesVarieties

The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

SamplesVarieties

The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

SamplesVarieties

The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

SamplesVarieties

The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

SamplesVarieties

The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

SamplesVarieties

The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

SamplesVarieties

The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

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The Turing Machine

Infinetly long ...

�� 0 0 00

Read/Write Head

Machine’s State

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The Turing Machine

�� 0 0 00

Read/Write Head

Machine’s State

Now action accordingto state table

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The Turing Machine

�� 0 0 00

Read/Write Head

Machine’s State

State table might say:When in state A reading 0:write 1, move right, changestate to B.

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The Turing Machine

�� 1 0 00

Read/Write Head

Machine’s State

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The Turing Machine

�� 1 0 00

Read/Write Head

Machine’s State

Move right

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The Turing Machine

�� 1 0 00

AMove right

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The Turing Machine

�� 1 0 00

BNew state B

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The Turing Machine

�� 1 0 00

Now read again, etc ...

... until final configuration.

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Formal definition — 7-tupel

M = 〈S , Γ, b, Σ, δ, s0, F 〉

S Finite set of states

Γ Finite set of symbolsb ∈ Γ Blank symbolΣ ⊆ Γ \ {b} Input symbolsδ : S × Γ× {L, R} Finite set of statess0 ∈ S Initial stateF ⊆ S Accepting states

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M = 〈S , Γ, b, Σ, δ, s0, F 〉

S Finite set of statesΓ Finite set of symbols

b ∈ Γ Blank symbolΣ ⊆ Γ \ {b} Input symbolsδ : S × Γ× {L, R} Finite set of statess0 ∈ S Initial stateF ⊆ S Accepting states

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M = 〈S , Γ, b, Σ, δ, s0, F 〉

S Finite set of statesΓ Finite set of symbolsb ∈ Γ Blank symbol

Σ ⊆ Γ \ {b} Input symbolsδ : S × Γ× {L, R} Finite set of statess0 ∈ S Initial stateF ⊆ S Accepting states

SamplesVarieties

M = 〈S , Γ, b, Σ, δ, s0, F 〉

S Finite set of statesΓ Finite set of symbolsb ∈ Γ Blank symbolΣ ⊆ Γ \ {b} Input symbols

δ : S × Γ× {L, R} Finite set of statess0 ∈ S Initial stateF ⊆ S Accepting states

SamplesVarieties

M = 〈S , Γ, b, Σ, δ, s0, F 〉

S Finite set of statesΓ Finite set of symbolsb ∈ Γ Blank symbolΣ ⊆ Γ \ {b} Input symbolsδ : S × Γ× {L, R} Finite set of states

s0 ∈ S Initial stateF ⊆ S Accepting states

SamplesVarieties

M = 〈S , Γ, b, Σ, δ, s0, F 〉

S Finite set of statesΓ Finite set of symbolsb ∈ Γ Blank symbolΣ ⊆ Γ \ {b} Input symbolsδ : S × Γ× {L, R} Finite set of statess0 ∈ S Initial state

F ⊆ S Accepting states

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M = 〈S , Γ, b, Σ, δ, s0, F 〉

S Finite set of statesΓ Finite set of symbolsb ∈ Γ Blank symbolΣ ⊆ Γ \ {b} Input symbolsδ : S × Γ× {L, R} Finite set of statess0 ∈ S Initial stateF ⊆ S Accepting states

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M = 〈S , Γ, b, Σ, δ, s0, F 〉

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M = 〈S , Γ, b, Σ, δ, s0, F 〉

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Action table/Transition function δ — Quintupel

siaj −→ si1aj1dk

When in state si reading symbol aj :

• write symbol aj1

• move into dk , k ∈ {L,R}• change state to si1

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• move into dk , k ∈ {L,R}

• change state to si1

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AdditionClass activityComplement

Unary Numbers

n−→u

1−→X5−→XXXXX

n = mu... number of Xsn... natural number

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Unary Number Addition — Self-performed

Simulating a Turing Machine:

Now it’s your turn!

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Unary Number Addition Machine

S ={0, 1,HALT} Γ={B,X,+}F ={HALT} Σ={X,+}

s0=0 b=B

Action table δ si = 0 si = 1aj = X si1 = 0; aj1 = X; dK = R si1 = HALT; aj1 = B; dK = Raj = + si1 = 0; aj1 = X; dK = R —aj = B si1 = 1; aj1 = B; dK = L —

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S ={0, 1,HALT} Γ={B,X,+}F ={HALT} Σ={X,+}s0=0 b=B

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Action table δ si = 0 si = 1

aj = X si1 = 0; aj1 = X; dK = R si1 = HALT; aj1 = B; dK = Raj = + si1 = 0; aj1 = X; dK = R —aj = B si1 = 1; aj1 = B; dK = L —

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Action table δ si = 0 si = 1aj = X si1 = 0; aj1 = X; dK = R si1 = HALT; aj1 = B; dK = R

aj = + si1 = 0; aj1 = X; dK = R —aj = B si1 = 1; aj1 = B; dK = L —

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Action table δ si = 0 si = 1aj = X si1 = 0; aj1 = X; dK = R si1 = HALT; aj1 = B; dK = Raj = + si1 = 0; aj1 = X; dK = R —

aj = B si1 = 1; aj1 = B; dK = L —

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A Complement Machine

S ={0,HALT} Γ={B, 0, 1}F ={HALT} Σ={0, 1}

s0=0 b=B

Action table δ si = 0aj = 0 si1 = 0; aj1 = 1; dK = Raj = 1 si1 = 0; aj1 = 0; dK = Raj = B si1 = HALT; aj1 = B; dK = R

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S ={0,HALT} Γ={B, 0, 1}F ={HALT} Σ={0, 1}s0=0 b=B

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Action table δ si = 0

aj = 0 si1 = 0; aj1 = 1; dK = Raj = 1 si1 = 0; aj1 = 0; dK = Raj = B si1 = HALT; aj1 = B; dK = R

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Action table δ si = 0aj = 0 si1 = 0; aj1 = 1; dK = R

aj = 1 si1 = 0; aj1 = 0; dK = Raj = B si1 = HALT; aj1 = B; dK = R

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Action table δ si = 0aj = 0 si1 = 0; aj1 = 1; dK = Raj = 1 si1 = 0; aj1 = 0; dK = R

aj = B si1 = HALT; aj1 = B; dK = R

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Variations4-tupelState diagramsInstantaneous descriptionUniversal Turing Machine

Variations of the TMs ...

... provably equivalent

• Two-way infinite tapes

• Arbitrary movement of the head

• Arbitrary numbers of read-write heads

• Arbitrary finite alphabet

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Variations of the TMs ...

... provably equivalent

• Two-way infinite tapes

• Arbitrary movement of the head

• Arbitrary numbers of read-write heads

• Arbitrary finite alphabet

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4-tupel representation

asi −→ si1dk

When in state si reading symbol a:

• Take action d : move right/left orwrite

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asi −→ si1dk

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asi −→ si1dk

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asi −→ si1dk

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State diagram: successor of a unary number

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Instantaneous description

A Turing Machine in an instantaneous state.

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Universal Turing Machine

Features

• Emulates δ of other Turing Machines

• Von Neumann architecture

• Turing-completeness

... 010110101111 00 101011011 00 110110110111 0011010111011110 ...

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Features

... 010110101111 00 101011011 00 110110110111 0011010111011110 ...

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Features

... 010110101111 00 101011011 00 110110110111 0011010111011110 ...

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Features

... 010110101111 00 101011011 00 110110110111 0011010111011110 ...

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Computability

Computable

• Any number ifTM-representable(π, e,

• Numerical functions(+, −,× ,÷)

Incomputable

• Entscheidungsproblem: Willany algorithm A witharbitrary input I halt?h(A, I ) is incomputable

• “Busy beaver” functionΣ(n)

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Computability

Computable

Incomputable

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Computability

Computable

Incomputable

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Computability

Computable

Incomputable

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Church?Turing thesis

“Every effectively calculable function is a computablefunction.”

• Effectively calculable: produced intuitively

• Computable function: computable by a Turing Machine

Solvable by humans = Solvable by machines (algorithm)

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The Universal Turing Machine: A Half-Century SurveyR. HerkenNew York: Oxford University Press, 1988.

[http://plato.stanford.edu/entries/turing-machine/]Turing MachinesStanford Encyclopedia of Philosophy, 04.11.1995

[http://www.intelligentedu.com/turing machines examples.html]Turing Machines: ExamplesJaime Soffer, 2005.

[http://en.wikipedia.org/wiki] Turing Machine, Busy Beaver,Computability, Turing-completeness, Entscheidungsproblem,11-03-2008

turing machine: structure and operational functionality

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