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Turing Machines

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The Language Hierarchy

*aRegular Languages

Context-Free Languagesnnba Rww

nnn cba ww?

** ba

?

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A Turing Machine

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

Tape

Read-Write headControl Unit

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The Tape

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

Read-Write head

No boundaries -- infinite length

The head moves Left or Right

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

Read-Write head

The head at each transition (time step):

1. Reads a symbol2. Writes a symbol3. Moves Left or Right

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

Example:Time 0

............Time 1

1. Reads2. Writes

a a cb

a b k c

ak

3. Moves Left

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............Time 1

a b k c

............ Time 2a k c f

1. Reads2. Writes

b f

3. Moves Right

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States & Transitions

1q 2q Lba ,

Read Write Move Left

1q 2q Rba ,

Move Right

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Example:

1q 2q Rba ,

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

Time 1

1qcurrent state

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............ a b caTime 1

1q 2q Rba ,

............ a b cbTime 2

1q

2q

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............ a b caTime 1

1q 2q Lba ,

............ a b cbTime 2

1q

2q

Example:

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............ a b caTime 1

1q 2q R g ,

............ g a b cbTime 2

1q

2q

Example:

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Determinism

1q

2q Rba ,

Allowed Not Allowed

3q Ld b ,

1q

2q Rba ,

3q Ld a ,

No lambda transitions allowed

Turing Machines are deterministic

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Partial Transition Function

1q

2q Rba ,

3q Ld b ,

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

1q

Example:

No transitionfor input symbol c

Allowed:

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Halting

The machine halts in a state if there isno transition to follow

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Halting Example 1:

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

1q

1q No transition fromHALT!!!

1q

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Halting Example 2:

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

1q

1q

2q Rba ,

3q Ld b ,

No possible transitionfrom and symbol

HALT!!!

1q c

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Accepting States

1q 2q Allowed

1q 2q Not Allowed

Accepting states have no outgoing transitionsThe machine halts and accepts

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Acceptance

Accept Input If machine haltsin an accept state

Reject Input

If machine haltsin a non-accept state

orIf machine entersan infinite loop

string

string

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Observation:

In order to accept an input string,it is not necessary to scan all thesymbols in the string

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Turing Machine Example

Accepts the language: *a

0q

Raa ,

L,1q

Input alphabet },{ b a

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aaTime 0

0q

a

0q

Raa ,

L,1q

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aaTime 1

0q

a

0q

Raa ,

L,1q

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aaTime 2

0q

a

0q

Raa ,

L,1q

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aaTime 3

0q

a

0q

Raa ,

L,1q

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aaTime 4

1q

a

0q

Raa ,

L,1q

Halt & Accept

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Rejection Example

0q

Raa ,

L,1q

baTime 0

0q

a

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0q

Raa ,

L,1q

baTime 1

0q

a

No possible TransitionHalt & Reject

A i l hi f l

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Accepts the language: *a

0q

but for input alphabet }{a A simpler machine for same language

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aaTime 0

0q

a

0q

Halt & Accept

Not necessary to scan input

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Infinite Loop Example

0q

Raa ,

L,1q

Lbb ,

A Turing machinefor language *)(* baba

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baTime 0

0q

a

0q

Raa ,

L,1q

Lbb ,

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baTime 1

0q

a

0q

Raa ,

L,1q

Lbb ,

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baTime 2

0q

a

0q

Raa ,

L,1q

Lbb ,

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baTime 2

0q

a

baTime 3

0q

a

baTime 4

0q

a

baTime 5

0q

a

I n f i n

i t e l o

o p

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Because of the infinite loop:

The accepting state cannot be reached

The machine never halts

The input string is rejected

h h l

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Another Turing Machine Example

Turing machine for the language }{ nnba

0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

R x x ,

R y y ,

R yy , 4q

L,

1n

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Match as with bs: Repeat:

replace leftmost a with xfind leftmost b and replace it with yUntil there are no more as or bs

If there is a remaining a or b reject

Basic Idea:

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

R x x ,

R y y ,

R yy , 4q

L,

ba

0q

a bTime 0

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

R x x ,

R y y ,

R yy , 4q

L,

b x

1q

a b Time 1

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

R x x ,

R y y ,

R yy , 4q

L,

b x

1q

a b Time 2

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

R x x ,

R y y ,

R yy , 4q

L,

y x

2q

a b Time 3

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

R x x ,

R y y ,

R yy , 4q

L,

y x

2q

a b Time 4

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

R x x ,

R y y ,

R yy , 4q

L,

y x

0q

a b Time 5

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

R x x ,

R y y ,

R yy , 4q

L,

y x

1q

x b Time 6

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

R x x ,

R y y ,

R yy , 4q

L,

y x

1q

x b Time 7

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

Rxx ,

R y y ,

R yy , 4q

L,

y x x y

2q

Time 8

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

Rxx ,

R y y ,

R yy , 4q

L,

y x x y

2q

Time 9

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

Rxx ,

R y y ,

R yy , 4q

L,

y x

0q

x y Time 10

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

Rxx ,

R y y ,

R yy , 4q

L,

y x

3q

x y Time 11

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

Rxx ,

R y y ,

R yy , 4q

L,

y x

3q

x y Time 12

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0q 1q 2q3q R xa ,

Raa , R y y ,

L yb ,

Laa , L y y ,

Rxx ,

R y y ,

R yy , 4q

L,

y x

4q

x y

Halt & Accept

Time 13


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Turing machine for the languageA = {02n n >= O}


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Turing machine for the languageA = {02n n >= O}

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modify themachine for the language }{ nnba

And constructa machine for the language }{ nnn cba

Home Work:

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Formal Definitionsfor

Turing Machines

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Transition Function

1q 2q Rba ,

),,(),( 21 Rbqaq

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1q 2q Ld c ,

),,(),( 21 Ld qcq

Transition Function

Turing Machine:

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g

),,,,,,( 0 F qQ

States

Inputalphabet

Tapealphabet

TransitionfunctionInitial

stateblank

Acceptstates

Configuration

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Configuration

ba

1q

a

Instantaneous description:

c

baqca 1

Time 4 Time 5

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y x

2q

a b y x

0q

a b

A Move: aybq x xaybq 02

(yields in one mode)

Time 4 Time 5

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y x

2q

a b y x

0q

a b

bq xxy ybq xxaybq x xaybq 1102

y x

1q

x b

Time 6 y x

1q

x b

Time 7

A computation

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bq xxy ybq xxaybq x xaybq 1102

bq xxy xaybq 12Equivalent notation:

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Initial configuration: wq0

ba

0q

a b

w

Input string

The Accepted Language

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The Accepted Language

For any Turing Machine

}:{)( 210 xq xwqw M L f

Initial state Accept state

If l i dL

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If a language is acceptedby a Turing machinethen we say that is:

Turing AcceptableRecursively Enumerable

M L

L Turing Recognizable

Other names used:

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Computing Functionswith

Turing Machines

)(

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A function )(w f

Domain: Result Region:

has:

D

Dw

S

S w f )()(w f

A function may have many parameters:

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A function may have many parameters:

y x y x f ),(

Example: Addition function

Integer Domain

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Integer Domain

Unary:

Binary:

Decimal:

11111

101

5

We prefer unary representation:

easier to manipulate with Turing machines

Definition:

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A function is computable if

there is a Turing Machine such that:

f

Initial configuration Final configuration

Dw Domain

0q

w

f q

)(w f

accept stateinitial state

For all

n other words:

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)(0

w f qwq f

InitialConfiguration

FinalConfiguration

A function is computable if

there is a Turing Machine such that:

f

Dw DomainFor all

Example

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Example

The function y x y x f ),( is computable

Turing Machine:

Input string: y x0 unary

Output string: 0 xy unary

y x, are integers

x y

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0

0q

1 1 1 1

y

1 Start

initial state

The 0 is the delimiter thatseparates the two numbers

x y

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0

0q

1 1 1 11

0

f q

1 1

y x

11

Start

Finish

final state

initial state

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0

f q

1 1

y x

11Finish

final state

The 0 here helps when we use

the result for other operations

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Construct the TM for Unary Addition

Turing machine for function y x y x f ),(

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0q

g

1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

Execution Example: Time 0

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p

11 x

11 y 0

0q

1 1 1 1

Time 0 x y

Final Result

0

4q

1 1 1 1

y x

(=2)(=2)

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0

0q

1 1Time 0

0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

1 1

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0q

0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

01 11 1Time 1

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

0

0q

1 1 1 1Time 2

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

1q

1 11 11Time 3

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

1q

1 1 1 11Time 4

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

1q

1 11 11Time 5

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

2q

1 1 1 11Time 6

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 11 01Time 7

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 1 1 01Time 8

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 11 01Time 9

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 1 1 01Time 10

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 11 01Time 11

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0q 1q 2q

3q L, L,01

L,11

R,

R,10

R,11

4q

R,11

4q

1 1 1 01

HALT & accept

Time 12

Another Example

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The function x x f 2)( is computable

Turing Machine:

Input string: x unary

Output string: xx unary

x is integer

x

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0q

1

1 1

1

fq

1 1

x2

11

Start

Finish

accept state

initial state

Turing Machine Pseudocode for x x f 2)(

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Replace every 1 with $ Repeat:

Find rightmost $ , replace it with 1

Go to right end, insert 1

Until no more $ remain

Turing Machine for xxf 2)(

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0q 1q 2q

3q

R,1$

L,1

L,

R$,1 L,11 R,11

R,

Turing Machine for x x f 2)(

ExampleStart Finish

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0q 1q 2q

3q

R,1$

L,1

L,

R$,1 L,11 R,11

R,

0q

1 1

3q

1 11 1

Finish

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Combining Turing Machines

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Block Diagram

TuringMachineinput output

Example: y x yx if

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),( y x f 0

y y x

y x if

Comparator

Adder

Eraser

y x,

y x,

y x

y x

y x

0

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A Universal Turing Machine

A limitation of Turing Machines:

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Turing Machines are hardwired

they executeonly one program

A limitation of Turing Machines:

Real Computers are re-programmable

Solution: Universal Turing Machine

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Reprogrammable machine

Simulates any other Turing Machine

Attributes:

Universal Turing Machine

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simulates any Turing Machine

Input of Universal Turing Machine:

Description of transitions of

Input string of

Three tapes Tape 1

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Universal

TuringMachine

Description of

Tape Contents of

State of

p

Tape 2

Tape 3

Tape 1

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We describe Turing machineas a string of symbols:

We encode as a string of symbols

Description of

Alphabet Encoding

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Alphabet Encoding

Symbols: a b c d

Encoding:1

11 111 1111

State Encoding

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States: 1q 2q 3q 4q

Encoding: 1 11 111 111

Head Move Encoding

Move:

Encoding:

L R

1 11

T iti E di g

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111

Transition Encoding

Transition: ),,(),( 21 Lbqaq

Encoding: 10110110101

separator

T ring Machine Encoding

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Turing Machine Encoding

Transitions:),,(),( 21 Lbqaq

Encoding:10110110101

),,(),( 32 Rcqbq

110111011110101100

separator

Tape 1 contents of Universal Turing Machine:

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binary encodingof the simulated machine

1100011101111010100110110110101

Tape 1

A Turing Machine is describedf

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with a binary string of 0s and 1s

The set of Turing machinesforms a language:

each string of this language isthe binary encoding of a Turing Machine

Therefore:

Language of Turing Machines

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g g g

L = { 010100101,

00100100101111,

111010011110010101,

}

(Turing Machine 1)

(Turing Machine 2)

9-10-11turing

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