<s>
In	O
the	O
computational	O
complexity	O
theory	O
of	O
counting	O
problems	O
,	O
a	O
polynomial-time	B-Algorithm
counting	I-Algorithm
reduction	I-Algorithm
is	O
a	O
type	O
of	O
reduction	B-Algorithm
(	O
a	O
transformation	O
from	O
one	O
problem	O
to	O
another	O
)	O
used	O
to	O
define	O
the	O
notion	O
of	O
completeness	O
for	O
the	O
complexity	O
class	O
♯P	O
.	O
</s>
<s>
These	O
reductions	O
may	O
also	O
be	O
called	O
polynomial	O
many-one	B-Algorithm
counting	O
reductions	O
or	O
weakly	O
parsimonious	B-Algorithm
reductions	I-Algorithm
;	O
they	O
are	O
analogous	O
to	O
many-one	B-Algorithm
reductions	I-Algorithm
for	O
decision	O
problems	O
and	O
they	O
generalize	O
the	O
parsimonious	B-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
A	O
polynomial-time	B-Algorithm
counting	I-Algorithm
reduction	I-Algorithm
is	O
usually	O
used	O
to	O
transform	O
instances	O
of	O
a	O
known-hard	O
problem	O
into	O
instances	O
of	O
another	O
problem	O
that	O
is	O
to	O
be	O
proven	O
hard	O
.	O
</s>
<s>
That	O
is	O
,	O
if	O
the	O
input-output	O
relations	O
of	O
and	O
are	O
expressed	O
as	O
functions	O
,	O
then	O
their	O
function	B-Application
composition	I-Application
must	O
obey	O
the	O
identity	O
.	O
</s>
<s>
As	O
a	O
special	O
case	O
,	O
a	O
parsimonious	B-Algorithm
reduction	I-Algorithm
is	O
a	O
polynomial-time	O
transformation	O
on	O
the	O
inputs	O
to	O
problems	O
that	O
preserves	O
the	O
exact	O
values	O
of	O
the	O
outputs	O
.	O
</s>
<s>
Such	O
a	O
reduction	B-Algorithm
can	O
be	O
viewed	O
as	O
a	O
polynomial-time	B-Algorithm
counting	I-Algorithm
reduction	I-Algorithm
,	O
by	O
using	O
the	O
identity	O
function	O
as	O
the	O
function	O
.	O
</s>
<s>
A	O
functional	O
problem	O
is	O
said	O
to	O
be	O
♯	O
P-hard	O
if	O
there	O
exists	O
a	O
polynomial-time	B-Algorithm
counting	I-Algorithm
reduction	I-Algorithm
from	O
every	O
problem	O
in	O
♯P	O
to	O
.	O
</s>
<s>
(	O
Sometimes	O
,	O
as	O
in	O
Valiant	O
's	O
original	O
paper	O
proving	O
the	O
completeness	O
of	O
the	O
permanent	O
of	O
0	O
–	O
1	O
matrices	O
,	O
a	O
weaker	O
notion	O
of	O
reduction	B-Algorithm
,	O
Turing	B-Algorithm
reduction	I-Algorithm
,	O
is	O
instead	O
used	O
for	O
defining	O
♯	O
P-completeness	O
.	O
)	O
</s>
<s>
The	O
usual	O
method	O
of	O
proving	O
a	O
problem	O
in	O
♯P	O
to	O
be	O
♯	O
P-complete	O
is	O
to	O
start	O
with	O
a	O
single	O
known	O
♯	O
P-complete	O
problem	O
and	O
find	O
a	O
polynomial-time	B-Algorithm
counting	I-Algorithm
reduction	I-Algorithm
from	O
to	O
.	O
</s>
<s>
If	O
this	O
reduction	B-Algorithm
exists	O
,	O
then	O
there	O
exists	O
a	O
reduction	B-Algorithm
from	O
any	O
other	O
problem	O
in	O
♯P	O
to	O
,	O
obtained	O
by	O
composing	B-Application
a	O
reduction	B-Algorithm
from	O
the	O
other	O
problem	O
to	O
with	O
the	O
reduction	B-Algorithm
from	O
to	O
.	O
</s>
