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# TURING MACHINE STRUCTURE AND OPERATIONAL FUNCTIONALITY

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```									Structure and Operational Functionality of

The Turing Machine

11 — 13 — 2008

¨         ¨
Soren Wellhofer
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

1 History

2 Structure and Deﬁnition

3 Samples

4 Varieties

5 Computability

6 References

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

Time frame — The 20th century

What were mathematicians working on?

• Rediscover Theory of Numbers
• Reduction of all math to fundamental logic

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

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 formal
system

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

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 formal
system

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

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 formal
system

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

What does it mean to be computable?

Turing’s achievements

• Proof of the possibility of a symbol-processing machine
• Simple operations according to rules

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

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

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

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!

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

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

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

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

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Elements of the Turing Machine

• Inﬁnetly long tape
• Divided into cells
• Cells containing symbols

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Elements of the Turing Machine

• Inﬁnetly long tape
• Divided into cells
• Cells containing symbols
• Action table = the program

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Elements of the Turing Machine

• Inﬁnetly long tape
• Divided into cells
• Cells containing symbols
• Action table = the program

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

0
1   0
1   1
0    0              0           0                  0               1
0     1
0      0
1

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Tape

0
1   0
1   1
0    0              0           0                  0               1
0     1
0      0
1

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Tape

0
1   0
1   1
0    0              0           0                  0               1
0     1
0      0
1

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Tape

0
1   0
1   1
0    0              0           0                  0               1
0     1
0      0
1

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Tape

0
1   0
1   1
0    0              0           0                  0               1
0     1
0      0
1

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Tape

0
1   1
0   0
1    0              0           0                  0               1
0     0
1      1
0

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Tape

0
1   1
0   0
1    0              0           0                  0               1
0     0
1      1
0

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Tape

A

0
1   1
0   0
1    0              0           0                  0               1
0     0
1      1
0

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Tape
Machine’s State

A

0
1   1
0   0
1    0              0           0                  0               1
0     0
1      1
0

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Tape
Machine’s State

A

0
1   1
0   0
1    0              0           0                  0               1
0     0
1      1
0

Inﬁnetly long ...
o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

Now action according
to state table                                                            Machine’s State

A

1
0   0
1   0
1    0              0           0                  0               0
1     1
0      1
0

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine
When in state A reading 0:
write 1, move right, change
state to B.
Machine’s State

A

1
0   0
1   0
1    0              0           0                  0               0
1     1
0      1
0

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine
When in state A reading 0:
write 1, move right, change
state to B.
Machine’s State

A

1
0   0
1   0
1    0              1           0                  0               0
1     1
0      1
0

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine
When in state A reading 0:
write 1, move right, change
state to B.
Machine’s State

Move right                  A

1
0    0
1    0
1       0              1           0                  0               0
1     1
0      1
0

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine
State table might say:
When in state A reading 0:
write 1, move right, change
state to B.

Move right                   A

1
0   0
1   0
1    0              1           0                  0               0
1     1
0      1
0

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine
State table might say:
When in state A reading 0:
write 1, move right, change
state to B.

New state B
B

1
0   0
1    0
1     0              1           0                  0               0
1     1
0      1
0

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

The Turing Machine

... until ﬁnal conﬁguration.

B

0
1   1
0   0
1     0              1           0                  0               1
0     0
1      1
0

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Formal deﬁnition — 7-tupel

M = S, Γ, b, Σ, δ, s0 , F

S                                      Finite set of states

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Formal deﬁnition — 7-tupel

M = S, Γ, b, Σ, δ, s0 , F

S                                      Finite set of states
Γ                                      Finite set of symbols

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Formal deﬁnition — 7-tupel

M = S, Γ, b, Σ, δ, s0 , F

S                                      Finite set of states
Γ                                      Finite set of symbols
b∈Γ                                    Blank symbol

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Formal deﬁnition — 7-tupel

M = S, Γ, b, Σ, δ, s0 , F

S                                      Finite set of states
Γ                                      Finite set of symbols
b∈Γ                                    Blank symbol
Σ ⊆ Γ \ {b}                            Input symbols

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Formal deﬁnition — 7-tupel

M = S, Γ, b, Σ, δ, s0 , F

S                                       Finite set of states
Γ                                       Finite set of symbols
b∈Γ                                     Blank symbol
Σ ⊆ Γ \ {b}                             Input symbols
δ : S × Γ × {L, R}                      Finite set of states

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Formal deﬁnition — 7-tupel

M = S, Γ, b, Σ, δ, s0 , F

S                                       Finite set of states
Γ                                       Finite set of symbols
b∈Γ                                     Blank symbol
Σ ⊆ Γ \ {b}                             Input symbols
δ : S × Γ × {L, R}                      Finite set of states
s0 ∈ S                                  Initial state

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Formal deﬁnition — 7-tupel

M = S, Γ, b, Σ, δ, s0 , F

S                                       Finite set of states
Γ                                       Finite set of symbols
b∈Γ                                     Blank symbol
Σ ⊆ Γ \ {b}                             Input symbols
δ : S × Γ × {L, R}                      Finite set of states
s0 ∈ S                                  Initial state
F ⊆S                                    Accepting states

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Formal deﬁnition — 7-tupel

M = S, Γ, b, Σ, δ, s0 , F

S                                       Finite set of states
Γ                                       Finite set of symbols
b∈Γ                                     Blank symbol
Σ ⊆ Γ \ {b}                             Input symbols
δ : S × Γ × {L, R}                      Finite set of states
s0 ∈ S                                  Initial state
F ⊆S                                    Accepting states

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Formal deﬁnition — 7-tupel

M = S, Γ, b, Σ, δ, s0 , F

S                                       Finite set of states
Γ                                       Finite set of symbols
b∈Γ                                     Blank symbol
Σ ⊆ Γ \ {b}                             Input symbols
δ : S × Γ × {L, R}                      Finite set of states
s0 ∈ S                                  Initial state
F ⊆S                                    Accepting states

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Action table/Transition function δ — Quintupel

si aj −→ si1 aj1 dk

When in state si reading symbol aj :

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Action table/Transition function δ — Quintupel

si aj −→ si1 aj1 dk

When in state si reading symbol aj :

• write symbol aj1

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Action table/Transition function δ — Quintupel

si aj −→ si1 aj1 dk

When in state si reading symbol aj :

• write symbol aj1
• move into dk , k ∈ {L, R}

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Action table/Transition function δ — Quintupel

si aj −→ si1 aj1 dk

When in state si reading symbol aj :

• write symbol aj1
• move into dk , k ∈ {L, R}
• change state to si1

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Action table/Transition function δ — Quintupel

si aj −→ si1 aj1 dk

When in state si reading symbol aj :

• write symbol aj1
• move into dk , k ∈ {L, R}
• change state to si1

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
General Structure
Samples
Formal deﬁnition
Varieties
Action table
Computability
References

Action table/Transition function δ — Quintupel

si aj −→ si1 aj1 dk

When in state si reading symbol aj :

• write symbol aj1
• move into dk , k ∈ {L, R}
• change state to si1

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

Unary Numbers

n−→u                                   n=m
1−→X                                   u... number of Xs
5−→XXXXX                               n... natural number

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

Simulating a Turing Machine:

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

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

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

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

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

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

Action table δ            si = 0                                        si = 1

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

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

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

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

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

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

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

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

Action table δ             si = 0                                  si = 1
aj = 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                             —

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

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

Action table δ             si = 0                                  si = 1
aj = 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                             —

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

A Complement Machine

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

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

A Complement Machine

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

o         o
S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

A Complement Machine

S ={0, HALT}               Γ={B, 0, 1}
F ={HALT}                  Σ={0, 1}
s0 =0                      b=B
Action table δ                       si = 0

o         o
S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

A Complement Machine

S ={0, HALT}             Γ={B, 0, 1}
F ={HALT}                Σ={0, 1}
s0 =0                    b=B
Action table δ                     si = 0
aj = 0                  si1 = 0; aj1 = 1; dK = R

o         o
S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

A Complement Machine

S ={0, HALT}             Γ={B, 0, 1}
F ={HALT}                Σ={0, 1}
s0 =0                    b=B
Action table δ                     si = 0
aj = 0                  si1 = 0; aj1 = 1; dK = R
aj = 1                  si1 = 0; aj1 = 0; dK = R

o         o
S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

A Complement Machine

S ={0, HALT}            Γ={B, 0, 1}
F ={HALT}               Σ={0, 1}
s0 =0                   b=B
Action table δ                    si = 0
aj = 0                 si1 = 0; aj1 = 1; dK = R
aj = 1                 si1 = 0; aj1 = 0; dK = R
aj = B              si1 = HALT; aj1 = B; dK = R

o         o
S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Class activity
Varieties
Complement
Computability
References

A Complement Machine

S ={0, HALT}            Γ={B, 0, 1}
F ={HALT}               Σ={0, 1}
s0 =0                   b=B
Action table δ                    si = 0
aj = 0                 si1 = 0; aj1 = 1; dK = R
aj = 1                 si1 = 0; aj1 = 0; dK = R
aj = B              si1 = HALT; aj1 = B; dK = R

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S¨ren Wellh¨fer   Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

Variations of the TMs ...

... provably equivalent

• Two-way inﬁnite tapes
• Arbitrary movement of the head
• Arbitrary ﬁnite alphabet

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

Variations of the TMs ...

... provably equivalent

• Two-way inﬁnite tapes
• Arbitrary movement of the head
• Arbitrary ﬁnite alphabet

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

4-tupel representation

asi −→ si1 dk

When in state si reading symbol a:

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

4-tupel representation

asi −→ si1 dk

When in state si reading symbol a:

• change state to si1

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

4-tupel representation

asi −→ si1 dk

When in state si reading symbol a:

• change state to si1
• Take action d: move right/left or
write

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

4-tupel representation

asi −→ si1 dk

When in state si reading symbol a:

• change state to si1
• Take action d: move right/left or
write

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

State diagram: successor of a unary number

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

Instantaneous description

A Turing Machine in an instantaneous state.

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

Universal Turing Machine

Features
• Emulates δ of other Turing Machines
• Von Neumann architecture

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

Universal Turing Machine

Features
• Emulates δ of other Turing Machines
• Von Neumann architecture
• Turing-completeness

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

Universal Turing Machine

Features
• Emulates δ of other Turing Machines
• Von Neumann architecture
• Turing-completeness

... 010110101111 00 101011011 00 110110110111 00
11010111011110 ...

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Variations
Structure and Deﬁnition
4-tupel
Samples
State diagrams
Varieties
Instantaneous description
Computability
Universal Turing Machine
References

Universal Turing Machine

Features
• Emulates δ of other Turing Machines
• Von Neumann architecture
• Turing-completeness

... 010110101111 00 101011011 00 110110110111 00
11010111011110 ...

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

Computability

Computable
• Any number if
TM-representable
√
(π, e,   )
• Numerical functions
(+, −, × , ÷)

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

Computability

Computable                                   Incomputable
• Any number if                                • Entscheidungsproblem: Will
TM-representable                                 any algorithm A with
√
(π, e,   )                                       arbitrary input I halt?
• Numerical functions                              h(A, I ) is incomputable
(+, −, × , ÷)

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

Computability

Computable                                   Incomputable
• Any number if                                • Entscheidungsproblem: Will
TM-representable                                 any algorithm A with
√
(π, e,   )                                       arbitrary input I halt?
• Numerical functions                              h(A, I ) is incomputable
(+, −, × , ÷)                                • “Busy beaver” function
Σ(n)

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

Computability

Computable                                   Incomputable
• Any number if                                • Entscheidungsproblem: Will
TM-representable                                 any algorithm A with
√
(π, e,   )                                       arbitrary input I halt?
• Numerical functions                              h(A, I ) is incomputable
(+, −, × , ÷)                                • “Busy beaver” function
Σ(n)

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

Church?Turing thesis

“Every eﬀectively calculable function is a computable
function.”

• Eﬀectively calculable: produced intuitively
• Computable function: computable by a Turing Machine

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

Church?Turing thesis

“Every eﬀectively calculable function is a computable
function.”

• Eﬀectively calculable: produced intuitively
• Computable function: computable by a Turing Machine

Solvable by humans = Solvable by machines (algorithm)

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

Church?Turing thesis

“Every eﬀectively calculable function is a computable
function.”

• Eﬀectively calculable: produced intuitively
• Computable function: computable by a Turing Machine

Solvable by humans = Solvable by machines (algorithm)

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine
History
Structure and Deﬁnition
Samples
Varieties
Computability
References

The Universal Turing Machine: A Half-Century Survey
R. Herken
New York: Oxford University Press, 1988.
[http://plato.stanford.edu/entries/turing-machine/]
Turing Machines
Stanford Encyclopedia of Philosophy, 04.11.1995

[http://www.intelligentedu.com/turing machines examples.html]
Turing Machines: Examples
Jaime Soﬀer, 2005.
[http://en.wikipedia.org/wiki] Turing Machine, Busy Beaver,
Computability, Turing-completeness, Entscheidungsproblem,
11-03-2008

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S¨ren Wellh¨fer    Structure and Operational Functionality of The Turing Machine

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