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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