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IDT Open Seminar

ALAN TURING AND HIS LEGACY

100 Years Turing celebration

Gordana Dodig Crnkovic, Computer Science and Network Department

Mälardalen University

March 8th 2012

http://www.mrtc.mdh.se/~gdc/work/TuringCentenary.pdf

http://www.mrtc.mdh.se/~gdc/work/TuringMachine.pdf

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*aFinite Automata

Push-down Automatannba Rww

nnn cba ww

**ba

Turing Machines

Chomsky Language Hyerarchy

TURING MACHINES

“Turing’s "Machines". These machines are humans who calculate.” (Wittgenstein)

“A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine.” (Turing)

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4

............Tape

Read-Write head

Control Unit

Turing Machine

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

Read-Write head

No boundaries -- infinite length

The head moves Left or Right

The Tape

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

Read-Write head

1. Reads a symbol

2. Writes a symbol

3. Moves Left or Right

The head at each time step:

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Head starts at the leftmost position

of the input string

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

Blank symbol

head

a b ca

Input string

The Input String

#####

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Determinism

1q

2qRba ,

Allowed Not Allowed

3qLdb ,

1q

2qRba ,

3qLda ,

No lambda transitions allowed in TM!

Turing Machines are deterministic

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Determinism

Note the difference between state indeterminismwhen not even possible future states are known in advance.

and choice indeterminismwhen possible future states are known,but we do not know which state will be taken.

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Halting

The machine halts if there are

no possible transitions to follow

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Example

............ a b ca

1q

1q

2qRba ,

3qLdb ,

No possible transition

HALT!

# # # # #

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Final States

1q 2q Allowed

1q 2q Not Allowed

• Final states have no outgoing transitions

• In a final state the machine halts

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Acceptance

Accept InputIf machine halts

in a final state

Reject Input

If machine halts

in a non-final state

or

If machine enters

an infinite loop

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Formal Definitions for

Turing Machines

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Transition Function

1q 2qRba ,

),,(),( 21 Rbqaq

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1q 2qLdc ,

),,(),( 21 Ldqcq

Transition Function

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Turing Machine

),#,,,,,( 0 FqQM

Transition

functionInitial

stateblank

Final

states

States

Input

alphabetTape

alphabet

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For any Turing Machine M

}:{)( 210 xqxwqwML f

Initial state Final state

The Accepted Language

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Standard Turing Machine

• Deterministic

• Infinite tape in both directions

•Tape is the input/output file

The machine we described is the standard:

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Computing Functionswith

Turing Machines

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)(0 wfqwq f

Initial

Configuration

Final

Configuration

Dw DomainFor all

A function is computable if

there is a Turing Machine such that

fM

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Example (Addition)

The function yxyxf ),( is computable

Turing Machine:

Input string: yx0 unary

Output string: 0xy unary

yx, are integers

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Start

Finish 0

fq

11

yx

11

final state

0

0q

1 11 1

x y

1

initial state

# #

# #

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0q 1q 2q 3qL,## L,01

L,11

R,##

R,10

R,11

4q

R,11

Turing machine for function yxyxf ),(

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Execution Example:

11x

11y

Time 0

0

0q

1 11 1x y

Final Result

0

4q

1 11 1

yx

(2)

(2)

# #

# #

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Time 0 0

0q

1 11 1

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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0q

01 11 1Time 1

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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0

0q

1 11 1Time 2

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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1q

1 11 11Time 3

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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1q

1 11 11Time 4

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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1q

1 11 11Time 5

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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2q

1 11 11Time 6

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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3q

1 11 01Time 7

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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3q

1 11 01Time 8

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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3q

1 11 01Time 9

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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3q

1 11 01Time 10

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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3q

1 11 01Time 11

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

# #

L,##

R,##

yxyxf ),(

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4q

1 11 01

0q 1q 2q 3qL,01

L,11

R,10

R,11

4q

R,11

HALT & accept

Time 12 # #

L,##

R,##

yxyxf ),(

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Universal Turing Machine

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A limitation of Turing Machines:

Turing Machines are “hardwired”

they execute

only one program

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Solution: Universal Turing Machine

• Reprogrammable machine

• Simulates any other Turing Machine

Characteristics:

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Universal Turing Machine

simulates any other Turing Machine M

Input to Universal Turing Machine:

• Description of transitions ofM

• Initial tape contents of M

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Universal

Turing

Machine

Description of Three tapes

MTape Contents of

Tape 2

State of M

Tape 3

M

Tape 1

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We describe Turing machine

as a string of symbols:

We encode as a string of symbols

M

M

Description of M

Tape 1

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Alphabet Encoding

Symbols: a b c d

Encoding: 1 11 111 1111

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State Encoding

States: 1q 2q 3q 4q

Encoding: 1 11 111 1111

Head Move Encoding

Move:

Encoding:

L R

1 11

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Transition Encoding

Transition: ),,(),( 21 Lbqaq

Encoding: 10110110101

separator

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Machine Encoding

Transitions:

),,(),( 21 Lbqaq

Encoding:

10110110101

),,(),( 32 Rcqbq

110111011110101100

separator

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Tape 1 contents of Universal Turing Machine:

encoding of the simulated machine

as a binary string of 0’s and 1’s

M

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As Turing Machine is described

with a binary string of 0’s and 1’s

the set of Turing machines forms a language:

Each string of the language is

the binary encoding of a Turing Machine.

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Language of Turing Machines

L = { 010100101,

00100100101111,

111010011110010101,

…… }

(Turing Machine 1)

(Turing Machine 2)

……

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Question:

Do Turing machines have

the same power with

a digital computer?

Intuitive answer: Yes

There was no formal proof of Church-Turing thesis until 2008!

CHURCH TURING THESIS

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Dershowitz, N. and Gurevich, Y. A Natural Axiomatization of Computability and Proof of Church's Thesis, Bulletin of Symbolic Logic, v. 14, No. 3, pp. 299-350 (2008)

This formal proof of Church-Turing thesis relies on an axiomatization of computation that excludes randomness, parallelism and quantum computing and thus corresponds to the idea of computing that Church and Turing had.

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Turing’s thesis

Any computation carried out

by algorithmic means

can be performed by a Turing Machine. (1930)

http://www.engr.uconn.edu/~dqg/papers/myth.pdf The Origins of the Turing Thesis Myth Goldin & Wegner