<s>
In	O
computability	O
theory	O
and	O
computational	O
complexity	O
theory	O
,	O
a	O
many-one	B-Algorithm
reduction	I-Algorithm
(	O
also	O
called	O
mapping	B-Algorithm
reduction	I-Algorithm
)	O
is	O
a	O
reduction	B-Algorithm
which	O
converts	O
instances	O
of	O
one	O
decision	O
problem	O
to	O
another	O
decision	O
problems	O
using	O
an	O
effective	O
function	O
.	O
</s>
<s>
Thus	O
if	O
we	O
can	O
decide	O
whether	O
instances	O
are	O
in	O
the	O
language	O
,	O
we	O
can	O
decide	O
whether	O
instances	O
are	O
in	O
its	O
language	O
by	O
applying	O
the	O
reduction	B-Algorithm
and	O
solving	O
.	O
</s>
<s>
Many-one	B-Algorithm
reductions	I-Algorithm
are	O
a	O
special	O
case	O
and	O
stronger	O
form	O
of	O
Turing	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
With	O
many-one	B-Algorithm
reductions	I-Algorithm
,	O
the	O
oracle	O
(	O
that	O
is	O
,	O
our	O
solution	O
for	O
B	O
)	O
can	O
be	O
invoked	O
only	O
once	O
at	O
the	O
end	O
,	O
and	O
the	O
answer	O
cannot	O
be	O
modified	O
.	O
</s>
<s>
This	O
means	O
that	O
if	O
we	O
want	O
to	O
show	O
that	O
problem	O
A	O
can	O
be	O
reduced	O
to	O
problem	O
B	O
,	O
we	O
can	O
use	O
our	O
solution	O
for	O
B	O
only	O
once	O
in	O
our	O
solution	O
for	O
A	O
,	O
unlike	O
in	O
Turing	B-Algorithm
reduction	I-Algorithm
,	O
where	O
we	O
can	O
use	O
our	O
solution	O
for	O
B	O
as	O
many	O
times	O
as	O
needed	O
while	O
solving	O
A	O
.	O
</s>
<s>
This	O
means	O
that	O
many-one	B-Algorithm
reductions	I-Algorithm
map	O
instances	O
of	O
one	O
problem	O
to	O
instances	O
of	O
another	O
,	O
while	O
Turing	B-Algorithm
reductions	I-Algorithm
compute	O
the	O
solution	O
to	O
one	O
problem	O
,	O
assuming	O
the	O
other	O
problem	O
is	O
easy	O
to	O
solve	O
.	O
</s>
<s>
The	O
many-one	B-Algorithm
reduction	I-Algorithm
is	O
more	O
effective	O
at	O
separating	O
problems	O
into	O
distinct	O
complexity	O
classes	O
.	O
</s>
<s>
However	O
,	O
the	O
increased	O
restrictions	O
on	O
many-one	B-Algorithm
reductions	I-Algorithm
make	O
them	O
more	O
difficult	O
to	O
find	O
.	O
</s>
<s>
Many-one	B-Algorithm
reductions	I-Algorithm
were	O
first	O
used	O
by	O
Emil	O
Post	O
in	O
a	O
paper	O
published	O
in	O
1944	O
.	O
</s>
<s>
A	O
many-one	B-Algorithm
reduction	I-Algorithm
from	O
to	O
is	O
a	O
total	O
computable	O
function	O
that	O
has	O
the	O
property	O
that	O
each	O
word	O
is	O
in	O
if	O
and	O
only	O
if	O
is	O
in	O
.	O
</s>
<s>
A	O
set	O
is	O
called	O
many-one	B-Algorithm
complete	O
,	O
or	O
simply	O
m-complete	B-Algorithm
,	O
iff	O
is	O
recursively	O
enumerable	O
and	O
every	O
recursively	O
enumerable	O
set	O
is	O
m-reducible	O
to	O
.	O
</s>
<s>
is	O
an	O
equivalence	O
relation	B-Algorithm
,	O
and	O
its	O
equivalences	O
classes	O
(	O
called	O
the	O
1-degrees	O
)	O
form	O
a	O
poset	O
under	O
.	O
</s>
<s>
Many-one	B-Algorithm
reductions	I-Algorithm
are	O
often	O
subjected	O
to	O
resource	O
restrictions	O
,	O
for	O
example	O
that	O
the	O
reduction	B-Algorithm
function	O
is	O
computable	O
in	O
polynomial	O
time	O
,	O
logarithmic	O
space	O
,	O
by	O
or	O
circuits	O
,	O
or	O
polylogarithmic	O
projections	O
where	O
each	O
subsequent	O
reduction	B-Algorithm
notion	O
is	O
weaker	O
than	O
the	O
prior	O
;	O
see	O
polynomial-time	B-Algorithm
reduction	I-Algorithm
and	O
log-space	B-Algorithm
reduction	I-Algorithm
for	O
details	O
.	O
</s>
<s>
Given	O
decision	O
problems	O
and	O
and	O
an	O
algorithm	O
N	O
which	O
solves	O
instances	O
of	O
,	O
we	O
can	O
use	O
a	O
many-one	B-Algorithm
reduction	I-Algorithm
from	O
to	O
to	O
solve	O
instances	O
of	O
in	O
:	O
</s>
<s>
We	O
say	O
that	O
a	O
class	O
C	O
of	O
languages	O
(	O
or	O
a	O
subset	O
of	O
the	O
power	O
set	O
of	O
the	O
natural	O
numbers	O
)	O
is	O
closed	O
under	O
many-one	B-Algorithm
reducibility	O
if	O
there	O
exists	O
no	O
reduction	B-Algorithm
from	O
a	O
language	O
in	O
C	O
to	O
a	O
language	O
outside	O
C	O
.	O
If	O
a	O
class	O
is	O
closed	O
under	O
many-one	B-Algorithm
reducibility	O
,	O
then	O
many-one	B-Algorithm
reduction	I-Algorithm
can	O
be	O
used	O
to	O
show	O
that	O
a	O
problem	O
is	O
in	O
C	O
by	O
reducing	O
a	O
problem	O
in	O
C	O
to	O
it	O
.	O
</s>
<s>
Many-one	B-Algorithm
reductions	I-Algorithm
are	O
valuable	O
because	O
most	O
well-studied	O
complexity	O
classes	O
are	O
closed	O
under	O
some	O
type	O
of	O
many-one	B-Algorithm
reducibility	O
,	O
including	O
P	O
,	O
NP	O
,	O
L	O
,	O
NL	O
,	O
co-NP	O
,	O
PSPACE	O
,	O
EXP	O
,	O
and	O
many	O
others	O
.	O
</s>
<s>
It	O
is	O
known	O
for	O
example	O
that	O
the	O
first	O
four	O
listed	O
are	O
closed	O
up	O
to	O
the	O
very	O
weak	O
reduction	B-Algorithm
notion	O
of	O
polylogarithmic	O
time	O
projections	O
.	O
</s>
<s>
These	O
classes	O
are	O
not	O
closed	O
under	O
arbitrary	O
many-one	B-Algorithm
reductions	I-Algorithm
,	O
however	O
.	O
</s>
<s>
One	O
may	O
also	O
ask	O
about	O
generalized	O
cases	O
of	O
many-one	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
One	O
such	O
example	O
is	O
e-reduction	O
,	O
where	O
we	O
consider	O
that	O
are	O
recursively	O
enumerable	O
instead	O
of	O
restricting	O
to	O
recursive	O
.	O
</s>
<s>
The	O
resulting	O
reducibility	O
relation	B-Algorithm
is	O
denoted	O
,	O
and	O
its	O
poset	O
has	O
been	O
studied	O
in	O
a	O
similar	O
vein	O
to	O
that	O
of	O
the	O
Turing	O
degrees	O
.	O
</s>
<s>
The	O
relations	O
of	O
many-one	B-Algorithm
reducibility	O
and	O
1-reducibility	O
are	O
transitive	O
and	O
reflexive	O
and	O
thus	O
induce	O
a	O
preorder	O
on	O
the	O
powerset	B-General_Concept
of	O
the	O
natural	O
numbers	O
.	O
</s>
<s>
A	O
set	O
is	O
many-one	B-Algorithm
reducible	I-Algorithm
to	O
the	O
halting	O
problem	O
if	O
and	O
only	O
if	O
it	O
is	O
recursively	O
enumerable	O
.	O
</s>
<s>
This	O
says	O
that	O
with	O
regards	O
to	O
many-one	B-Algorithm
reducibility	O
,	O
the	O
halting	O
problem	O
is	O
the	O
most	O
complicated	O
of	O
all	O
recursively	O
enumerable	O
problems	O
.	O
</s>
<s>
The	O
specialized	O
halting	O
problem	O
for	O
an	O
individual	O
Turing	O
machine	O
T	O
(	O
i.e.	O
,	O
the	O
set	O
of	O
inputs	O
for	O
which	O
T	O
eventually	O
halts	O
)	O
is	O
many-one	B-Algorithm
complete	O
iff	O
T	O
is	O
a	O
universal	O
Turing	O
machine	O
.	O
</s>
<s>
Emil	O
Post	O
showed	O
that	O
there	O
exist	O
recursively	O
enumerable	O
sets	O
that	O
are	O
neither	O
decidable	O
nor	O
m-complete	B-Algorithm
,	O
and	O
hence	O
that	O
there	O
exist	O
nonuniversal	O
Turing	O
machines	O
whose	O
individual	O
halting	O
problems	O
are	O
nevertheless	O
undecidable	O
.	O
</s>
<s>
A	O
polynomial-time	B-Algorithm
many-one	I-Algorithm
reduction	I-Algorithm
from	O
a	O
problem	O
A	O
to	O
a	O
problem	O
B	O
(	O
both	O
of	O
which	O
are	O
usually	O
required	O
to	O
be	O
decision	O
problems	O
)	O
is	O
a	O
polynomial-time	B-Algorithm
algorithm	O
for	O
transforming	O
inputs	O
to	O
problem	O
A	O
into	O
inputs	O
to	O
problem	O
B	O
,	O
such	O
that	O
the	O
transformed	O
problem	O
has	O
the	O
same	O
output	O
as	O
the	O
original	O
problem	O
.	O
</s>
<s>
Polynomial-time	B-Algorithm
many-one	I-Algorithm
reductions	I-Algorithm
may	O
also	O
be	O
known	O
as	O
polynomial	O
transformations	O
or	O
Karp	B-Algorithm
reductions	I-Algorithm
,	O
named	O
after	O
Richard	O
Karp	O
.	O
</s>
<s>
A	O
reduction	B-Algorithm
of	O
this	O
type	O
is	O
denoted	O
by	O
or	O
.	O
</s>
