<s>
In	O
computability	O
theory	O
,	O
a	O
Turing	B-Algorithm
reduction	I-Algorithm
from	O
a	O
decision	O
problem	O
to	O
a	O
decision	O
problem	O
is	O
an	O
oracle	O
machine	O
which	O
decides	O
problem	O
given	O
an	O
oracle	O
for	O
(	O
Rogers	O
1967	O
,	O
Soare	O
1987	O
)	O
.	O
</s>
<s>
If	O
a	O
Turing	B-Algorithm
reduction	I-Algorithm
from	O
to	O
exists	O
,	O
then	O
every	O
algorithm	O
for	O
can	O
be	O
used	O
to	O
produce	O
an	O
algorithm	O
for	O
,	O
by	O
inserting	O
the	O
algorithm	O
for	O
at	O
each	O
place	O
where	O
the	O
oracle	O
machine	O
computing	O
queries	O
the	O
oracle	O
for	O
.	O
</s>
<s>
A	O
Turing	B-Algorithm
reduction	I-Algorithm
in	O
which	O
the	O
oracle	O
machine	O
runs	O
in	O
polynomial	O
time	O
is	O
known	O
as	O
a	O
Cook	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
The	O
first	O
formal	O
definition	O
of	O
relative	B-Algorithm
computability	I-Algorithm
,	O
then	O
called	O
relative	O
reducibility	O
,	O
was	O
given	O
by	O
Alan	O
Turing	O
in	O
1939	O
in	O
terms	O
of	O
oracle	O
machines	O
.	O
</s>
<s>
In	O
1944	O
Emil	O
Post	O
used	O
the	O
term	O
"	O
Turing	B-Algorithm
reducibility	I-Algorithm
"	O
to	O
refer	O
to	O
the	O
concept	O
.	O
</s>
<s>
If	O
additionally	O
then	O
is	O
called	O
Turing	B-Algorithm
complete	I-Algorithm
for	O
.	O
</s>
<s>
Turing	B-Algorithm
completeness	I-Algorithm
,	O
as	O
just	O
defined	O
above	O
,	O
corresponds	O
only	O
partially	O
to	O
Turing	B-Algorithm
completeness	I-Algorithm
in	O
the	O
sense	O
of	O
computational	O
universality	O
.	O
</s>
<s>
Specifically	O
,	O
a	O
Turing	O
machine	O
is	O
a	O
universal	O
Turing	O
machine	O
if	O
its	O
halting	O
problem	O
(	O
i.e.	O
,	O
the	O
set	O
of	O
inputs	O
for	O
which	O
it	O
eventually	O
halts	O
)	O
is	O
many-one	B-Algorithm
complete	I-Algorithm
.	O
</s>
<s>
The	O
reductions	O
presented	O
here	O
are	O
not	O
only	O
Turing	B-Algorithm
reductions	I-Algorithm
but	O
many-one	B-Algorithm
reductions	I-Algorithm
,	O
discussed	O
below	O
.	O
</s>
<s>
Every	O
computable	O
set	O
is	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
every	O
other	O
set	O
.	O
</s>
<s>
There	O
are	O
pairs	O
of	O
sets	O
such	O
that	O
A	O
is	O
not	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
B	O
and	O
B	O
is	O
not	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
A	O
.	O
</s>
<s>
Thus	O
this	O
relation	O
is	O
not	O
well-founded	B-Algorithm
.	O
</s>
<s>
Every	O
set	O
is	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
its	O
own	O
Turing	O
jump	O
,	O
but	O
the	O
Turing	O
jump	O
of	O
a	O
set	O
is	O
never	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
the	O
original	O
set	O
.	O
</s>
<s>
There	O
are	O
two	O
common	O
ways	O
of	O
producing	O
reductions	O
stronger	O
than	O
Turing	B-Algorithm
reducibility	I-Algorithm
.	O
</s>
<s>
Set	O
is	O
many-one	B-Algorithm
reducible	I-Algorithm
to	O
if	O
there	O
is	O
a	O
total	O
computable	O
function	O
such	O
that	O
an	O
element	O
is	O
in	O
if	O
and	O
only	O
if	O
is	O
in	O
.	O
</s>
<s>
Such	O
a	O
function	O
can	O
be	O
used	O
to	O
generate	O
a	O
Turing	B-Algorithm
reduction	I-Algorithm
(	O
by	O
computing	O
,	O
querying	O
the	O
oracle	O
,	O
and	O
then	O
interpreting	O
the	O
result	O
)	O
.	O
</s>
<s>
A	O
truth	B-Algorithm
table	I-Algorithm
reduction	I-Algorithm
or	O
a	O
weak	B-Algorithm
truth	I-Algorithm
table	I-Algorithm
reduction	I-Algorithm
must	O
present	O
all	O
of	O
its	O
oracle	O
queries	O
at	O
the	O
same	O
time	O
.	O
</s>
<s>
In	O
a	O
truth	B-Algorithm
table	I-Algorithm
reduction	I-Algorithm
,	O
the	O
reduction	O
also	O
gives	O
a	O
boolean	O
function	O
(	O
a	O
truth	O
table	O
)	O
which	O
,	O
when	O
given	O
the	O
answers	O
to	O
the	O
queries	O
,	O
will	O
produce	O
the	O
final	O
answer	O
of	O
the	O
reduction	O
.	O
</s>
<s>
In	O
a	O
weak	B-Algorithm
truth	I-Algorithm
table	I-Algorithm
reduction	I-Algorithm
,	O
the	O
reduction	O
uses	O
the	O
oracle	O
answers	O
as	O
a	O
basis	O
for	O
further	O
computation	O
depending	O
on	O
the	O
given	O
answers	O
(	O
but	O
not	O
using	O
the	O
oracle	O
)	O
.	O
</s>
<s>
Equivalently	O
,	O
a	O
weak	B-Algorithm
truth	I-Algorithm
table	I-Algorithm
reduction	I-Algorithm
is	O
one	O
for	O
which	O
the	O
use	O
of	O
the	O
reduction	O
is	O
bounded	O
by	O
a	O
computable	O
function	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
weak	B-Algorithm
truth	I-Algorithm
table	I-Algorithm
reductions	I-Algorithm
are	O
sometimes	O
called	O
"	O
bounded	O
Turing	O
"	O
reductions	O
.	O
</s>
<s>
The	O
second	O
way	O
to	O
produce	O
a	O
stronger	O
reducibility	O
notion	O
is	O
to	O
limit	O
the	O
computational	O
resources	O
that	O
the	O
program	O
implementing	O
the	O
Turing	B-Algorithm
reduction	I-Algorithm
may	O
use	O
.	O
</s>
<s>
These	O
limits	O
on	O
the	O
computational	O
complexity	O
of	O
the	O
reduction	O
are	O
important	O
when	O
studying	O
subrecursive	O
classes	O
such	O
as	O
P	O
.	O
A	O
set	O
A	O
is	O
polynomial-time	B-Algorithm
reducible	I-Algorithm
to	O
a	O
set	O
if	O
there	O
is	O
a	O
Turing	B-Algorithm
reduction	I-Algorithm
of	O
to	O
that	O
runs	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
The	O
concept	O
of	O
log-space	B-Algorithm
reduction	I-Algorithm
is	O
similar	O
.	O
</s>
<s>
These	O
reductions	O
are	O
stronger	O
in	O
the	O
sense	O
that	O
they	O
provide	O
a	O
finer	O
distinction	O
into	O
equivalence	O
classes	O
,	O
and	O
satisfy	O
more	O
restrictive	O
requirements	O
than	O
Turing	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
There	O
may	O
be	O
no	O
way	O
to	O
build	O
a	O
many-one	B-Algorithm
reduction	I-Algorithm
from	O
one	O
set	O
to	O
another	O
even	O
when	O
a	O
Turing	B-Algorithm
reduction	I-Algorithm
for	O
the	O
same	O
sets	O
exists	O
.	O
</s>
<s>
According	O
to	O
the	O
Church	O
–	O
Turing	O
thesis	O
,	O
a	O
Turing	B-Algorithm
reduction	I-Algorithm
is	O
the	O
most	O
general	O
form	O
of	O
an	O
effectively	O
calculable	O
reduction	O
.	O
</s>
