<s>
In	O
computability	O
theory	O
,	O
many	O
reducibility	B-Algorithm
relations	I-Algorithm
(	O
also	O
called	O
reductions	O
,	O
reducibilities	O
,	O
and	O
notions	O
of	O
reducibility	O
)	O
are	O
studied	O
.	O
</s>
<s>
The	O
different	O
reducibility	B-Algorithm
relations	I-Algorithm
vary	O
in	O
the	O
methods	O
they	O
permit	O
such	O
a	O
conversion	O
process	O
to	O
use	O
.	O
</s>
<s>
Every	O
reducibility	B-Algorithm
relation	I-Algorithm
(	O
in	O
fact	O
,	O
every	O
preorder	O
)	O
induces	O
an	O
equivalence	O
relation	O
on	O
the	O
powerset	O
of	O
the	O
natural	O
numbers	O
in	O
which	O
two	O
sets	O
are	O
equivalent	O
if	O
and	O
only	O
if	O
each	O
one	O
is	O
reducible	O
to	O
the	O
other	O
.	O
</s>
<s>
In	O
computability	O
theory	O
,	O
these	O
equivalence	O
classes	O
are	O
called	O
the	O
degrees	O
of	O
the	O
reducibility	B-Algorithm
relation	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Turing	O
degrees	O
are	O
the	O
equivalence	O
classes	O
of	O
sets	O
of	O
naturals	O
induced	O
by	O
Turing	B-Algorithm
reducibility	I-Algorithm
.	O
</s>
<s>
The	O
degrees	O
of	O
any	O
reducibility	B-Algorithm
relation	I-Algorithm
are	O
partially	O
ordered	O
by	O
the	O
relation	O
in	O
the	O
following	O
manner	O
.	O
</s>
<s>
Let	O
be	O
a	O
reducibility	B-Algorithm
relation	I-Algorithm
and	O
let	O
and	O
be	O
two	O
of	O
its	O
degrees	O
.	O
</s>
<s>
The	O
most	O
fundamental	O
reducibility	O
notion	O
is	O
Turing	B-Algorithm
reducibility	I-Algorithm
.	O
</s>
<s>
A	O
set	O
of	O
natural	O
numbers	O
is	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
a	O
set	O
if	O
and	O
only	O
if	O
there	O
is	O
an	O
oracle	O
Turing	O
machine	O
that	O
,	O
when	O
run	O
with	O
as	O
its	O
oracle	O
set	O
,	O
will	O
compute	O
the	O
indicator	O
function	O
(	O
characteristic	O
function	O
)	O
of	O
.	O
</s>
<s>
Equivalently	O
,	O
is	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
if	O
and	O
only	O
if	O
there	O
is	O
an	O
algorithm	O
for	O
computing	O
the	O
indicator	O
function	O
for	O
provided	O
that	O
the	O
algorithm	O
is	O
provided	O
with	O
a	O
means	O
to	O
correctly	O
answer	O
questions	O
of	O
the	O
form	O
"	O
Is	O
in	O
?	O
</s>
<s>
Turing	B-Algorithm
reducibility	I-Algorithm
serves	O
as	O
a	O
dividing	O
line	O
for	O
other	O
reducibility	O
notions	O
because	O
,	O
according	O
to	O
the	O
Church-Turing	O
thesis	O
,	O
it	O
is	O
the	O
most	O
general	O
reducibility	B-Algorithm
relation	I-Algorithm
that	O
is	O
effective	O
.	O
</s>
<s>
Reducibility	B-Algorithm
relations	I-Algorithm
that	O
imply	O
Turing	B-Algorithm
reducibility	I-Algorithm
have	O
come	O
to	O
be	O
known	O
as	O
strong	O
reducibilities	O
,	O
while	O
those	O
that	O
are	O
implied	O
by	O
Turing	B-Algorithm
reducibility	I-Algorithm
are	O
weak	O
reducibilities	O
.	O
</s>
<s>
Equivalently	O
,	O
a	O
strong	O
reducibility	B-Algorithm
relation	I-Algorithm
is	O
one	O
whose	O
degrees	O
form	O
a	O
finer	O
equivalence	O
relation	O
than	O
the	O
Turing	O
degrees	O
,	O
while	O
a	O
weak	O
reducibility	B-Algorithm
relation	I-Algorithm
is	O
one	O
whose	O
degrees	O
form	O
a	O
coarser	O
equivalence	O
relation	O
than	O
Turing	O
equivalence	O
.	O
</s>
<s>
One-one	B-Algorithm
reducibility	I-Algorithm
:	O
is	O
one-one	O
reducible	O
to	O
if	O
there	O
is	O
a	O
computable	O
one-to-one	O
function	O
with	O
for	O
all	O
.	O
</s>
<s>
Many-one	B-Algorithm
reducibility	I-Algorithm
:	O
is	O
many-one	B-Algorithm
reducible	I-Algorithm
to	O
if	O
there	O
is	O
a	O
computable	O
function	O
with	O
for	O
all	O
.	O
</s>
<s>
Truth-table	B-Algorithm
reducible	I-Algorithm
:	O
is	O
truth-table	B-Algorithm
reducible	I-Algorithm
to	O
if	O
is	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
via	O
a	O
single	O
(	O
oracle	O
)	O
Turing	O
machine	O
which	O
produces	O
a	O
total	O
function	O
relative	O
to	O
every	O
oracle	O
.	O
</s>
<s>
Weak	B-Algorithm
truth-table	I-Algorithm
reducible	I-Algorithm
:	O
is	O
weak	B-Algorithm
truth-table	I-Algorithm
reducible	I-Algorithm
to	O
if	O
there	O
is	O
a	O
Turing	B-Algorithm
reduction	I-Algorithm
from	O
to	O
and	O
a	O
computable	O
function	O
which	O
bounds	O
the	O
use	O
.	O
</s>
<s>
Whenever	O
is	O
truth-table	B-Algorithm
reducible	I-Algorithm
to	O
,	O
is	O
also	O
weak	B-Algorithm
truth-table	I-Algorithm
reducible	I-Algorithm
to	O
,	O
since	O
one	O
can	O
construct	O
a	O
computable	O
bound	O
on	O
the	O
use	O
by	O
considering	O
the	O
maximum	O
use	O
over	O
the	O
tree	O
of	O
all	O
oracles	O
,	O
which	O
will	O
exist	O
if	O
the	O
reduction	O
is	O
total	O
on	O
all	O
oracles	O
.	O
</s>
<s>
Positive	O
reducible	O
:	O
is	O
positive	O
reducible	O
to	O
if	O
and	O
only	O
if	O
is	O
truth-table	B-Algorithm
reducible	I-Algorithm
to	O
in	O
a	O
way	O
that	O
one	O
can	O
compute	O
for	O
every	O
a	O
formula	O
consisting	O
of	O
atoms	O
of	O
the	O
form	O
such	O
that	O
these	O
atoms	O
are	O
combined	O
by	O
and	O
's	O
and	O
or	O
's	O
,	O
where	O
the	O
and	O
of	O
and	O
is	O
1	O
if	O
and	O
and	O
so	O
on	O
.	O
</s>
<s>
The	O
most	O
famous	O
of	O
these	O
is	O
bounded	O
truth-table	B-Algorithm
reduction	I-Algorithm
,	O
but	O
there	O
are	O
also	O
bounded	O
Turing	O
,	O
bounded	O
weak	O
truth-table	O
,	O
and	O
others	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
set	O
is	O
bounded	O
truth-table	B-Algorithm
reducible	I-Algorithm
to	O
if	O
and	O
only	O
if	O
the	O
Turing	O
machine	O
computing	O
relative	O
to	O
computes	O
a	O
list	O
of	O
up	O
to	O
numbers	O
,	O
queries	O
on	O
these	O
numbers	O
and	O
then	O
terminates	O
for	O
all	O
possible	O
oracle	O
answers	O
;	O
the	O
value	O
is	O
a	O
constant	O
independent	O
of	O
.	O
</s>
<s>
The	O
difference	O
between	O
bounded	O
weak	O
truth-table	O
and	O
bounded	B-Algorithm
Turing	I-Algorithm
reduction	I-Algorithm
is	O
that	O
in	O
the	O
first	O
case	O
,	O
the	O
up	O
to	O
queries	O
have	O
to	O
be	O
made	O
at	O
the	O
same	O
time	O
while	O
in	O
the	O
second	O
case	O
,	O
the	O
queries	O
can	O
be	O
made	O
one	O
after	O
the	O
other	O
.	O
</s>
<s>
For	O
that	O
reason	O
,	O
there	O
are	O
cases	O
where	O
is	O
bounded	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
but	O
not	O
weak	B-Algorithm
truth-table	I-Algorithm
reducible	I-Algorithm
to	O
.	O
</s>
<s>
Thus	O
if	O
a	O
set	O
is	O
decidable	O
then	O
is	O
reducible	O
to	O
any	O
set	O
under	O
any	O
of	O
the	O
strong	O
reducibility	B-Algorithm
relations	I-Algorithm
listed	O
above	O
,	O
even	O
if	O
is	O
not	O
polynomial-time	O
or	O
exponential-time	O
decidable	O
.	O
</s>
<s>
The	O
most	O
common	O
reducibility	O
in	O
computational	O
complexity	O
theory	O
is	O
polynomial-time	B-Algorithm
reducibility	I-Algorithm
;	O
a	O
set	O
A	O
is	O
polynomial-time	O
reducible	O
to	O
a	O
set	O
if	O
there	O
is	O
a	O
polynomial-time	O
function	O
f	O
such	O
that	O
for	O
every	O
,	O
is	O
in	O
if	O
and	O
only	O
if	O
is	O
in	O
.	O
</s>
<s>
This	O
reducibility	O
is	O
,	O
essentially	O
,	O
a	O
resource-bounded	O
version	O
of	O
many-one	B-Algorithm
reducibility	I-Algorithm
.	O
</s>
<s>
Although	O
Turing	B-Algorithm
reducibility	I-Algorithm
is	O
the	O
most	O
general	O
reducibility	O
that	O
is	O
effective	O
,	O
weaker	O
reducibility	B-Algorithm
relations	I-Algorithm
are	O
commonly	O
studied	O
.	O
</s>
<s>
Equivalently	O
,	O
according	O
to	O
Post	O
's	O
theorem	O
,	O
A	O
is	O
arithmetical	O
in	O
if	O
and	O
only	O
if	O
is	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
,	O
the	O
th	O
Turing	O
jump	O
of	O
,	O
for	O
some	O
natural	O
number	O
.	O
</s>
<s>
Equivalently	O
,	O
is	O
hyperarithmetical	O
in	O
if	O
and	O
only	O
if	O
is	O
Turing	B-Algorithm
reducible	I-Algorithm
to	O
,	O
the	O
th	O
Turing	O
jump	O
of	O
,	O
for	O
some	O
-recursive	O
ordinal	O
.	O
</s>
