Language Theory

7/26/2019 Language Theory

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Computing Functions

withTuring Machines


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A function )(wf

Domain:

Result Region:

has:

D Dw

S Swf )(


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A function may have many parameters:

yxyxf +=),(

Example: Addition function


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nteger Domain

!nary:

"inary:

Decimal:

#####

#$#

%

&e prefer unaryrepresentation:

easier to manipulate


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Definition:

A function is computa'le if

there is a Turing Machine such that:

f

M

nitial configuration Final configuration

Dw Domain

0q

w

fq

)(wf

final stateinitial state

For all


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6

)(0 wfqwq f

nitial

Configuration

Final

Configuration

A function is computa'le if

there is a Turing Machine such that:

f

M

n other words:

Dw DomainFor all


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Example

The function yxyxf +=),( is computa'le

Turing Machine:

nput string: yx0

unary

(utput string: 0xy unary

yx, are integers


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0

0q

1 1 1 1

x y

1

0

fq

1 1

yx +

11

)tart

Finish

final state

initial state


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0q

Turing machine for function

1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

yxyxf +=),(


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Execution Example:

11=x

11=y

0

0q

1 1 1 1

Time $

x y

Final Result

0

4q

1 1 1 1yx +

*+,

*+,


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0

0q

1 1Time $

0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

1 1


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0q

0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

01 11 1Time #


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0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

0

0q

1 1 1 1Time +


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0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

1q

1 11 11Time -


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0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

1q

1 11 11Time %


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0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

2q

1 1 1 11Time /


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0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 11 01Time 0


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0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 1 1 01Time 1


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0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 1 1 01Time #$


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0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 11 01Time ##


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0q 1q 2q 3qL,

L,01

L,11

R,

R,10

R,11

4q

R,11

4q

1 1 1 01

3A4T 5 accept

Time #+


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Another Example

The function xxf 2)( =

is computa'le

Turing Machine:

nput string: x unary

(utput string: xx unary

x is integer


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0q

1 1

x

1

1

fq

1 1

x2

11

)tart

Finish

final state

initial state


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Turing Machine 6seudocode for xxf 2)( =

7Replace every #with 8

7Repeat:

7Find rightmost 89 replace it with #

7o to right end9 insert #!ntil no more 8remain


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E l


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0q 1q 2q

3q

R,1$

L,1

L,

R$,1 L,11 R,11

R,

Example

0q

1 1

3q

1 11 1

)tart Finish


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Turing Machine for

nput: yx0

(utput: 1 0or

=),( yxf

0

1 yx >

yx

if

if


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Turing Machine 6seudocode:

Match a #from with a #fromx y

7Repeat

!ntil all of or is matchedx y

7fa #from is not matched erase tape9 write #

else

erase tape9 write $

x


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Com'ining Turing Machines


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Block Diagram

Turing

Machineinput output


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Example:

=),( yxf

0

yx + yx>

yx

if

if

Comparer

Adder

Eraser

yx,

yx,

yx>

yx

yx +

0


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Turing;s Thesis


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Do Turing machines have

the same power with

a digital computer>>

Question:


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Turings thesis:

Any computation carried out

'y mechanical means

can 'e performed 'y a Turing Machine

*#2-$,


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Computer )cience 4aw:

A computation is mechanical

if and only if

it can 'e performed 'y a Turing Machine

There is no ?nown model of computation

more powerful than Turing Machines


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Definition of Algorithm:

An algorithm for functionis a

Turing Machine which computes

)(wf

)(wf


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&hen we say:

There exists an algorithm

Algorithms are Turing Machines

&e mean:

There exists a Turing Machine

Language Theory

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