<s>
In	O
mathematics	O
,	O
Pfaffian	B-Algorithm
functions	I-Algorithm
are	O
a	O
certain	O
class	O
of	O
functions	O
whose	O
derivative	B-Algorithm
can	O
be	O
written	O
in	O
terms	O
of	O
the	O
original	O
function	O
.	O
</s>
<s>
Other	O
functions	O
may	O
not	O
have	O
the	O
above	O
property	O
,	O
but	O
their	O
derivative	B-Algorithm
may	O
be	O
written	O
in	O
terms	O
of	O
functions	O
like	O
those	O
above	O
.	O
</s>
<s>
A	O
Pfaffian	B-Algorithm
function	I-Algorithm
is	O
then	O
just	O
a	O
polynomial	O
in	O
the	O
functions	O
appearing	O
in	O
a	O
Pfaffian	O
chain	O
and	O
the	O
function	O
argument	O
.	O
</s>
<s>
The	O
numbers	O
r	O
,	O
α	O
,	O
and	O
β	O
are	O
collectively	O
known	O
as	O
the	O
format	O
of	O
the	O
Pfaffian	B-Algorithm
function	I-Algorithm
,	O
and	O
give	O
a	O
useful	O
measure	O
of	O
its	O
complexity	O
.	O
</s>
<s>
The	O
most	O
trivial	O
examples	O
of	O
Pfaffian	B-Algorithm
functions	I-Algorithm
are	O
the	O
polynomials	O
in	O
R[X]	O
.	O
</s>
<s>
Perhaps	O
the	O
simplest	O
nontrivial	O
Pfaffian	B-Algorithm
function	I-Algorithm
is	O
f(x )	O
=	O
ex	O
.	O
</s>
<s>
In	O
the	O
1960s	O
Andrei	O
Gabrielov	O
proved	O
that	O
the	O
structure	O
obtained	O
by	O
starting	O
with	O
R	O
and	O
adding	O
a	O
function	O
symbol	O
for	O
every	O
analytic	B-Language
function	I-Language
restricted	O
to	O
the	O
unit	O
box	O
[	O
0	O
,	O
1 ]m	O
is	O
model	O
complete	O
.	O
</s>
<s>
That	O
is	O
,	O
any	O
set	O
definable	O
in	O
this	O
structure	O
Ran	O
was	O
just	O
the	O
projection	O
of	O
some	O
higher-dimensional	O
set	O
defined	O
by	O
identities	O
and	O
inequalities	O
involving	O
these	O
restricted	O
analytic	B-Language
functions	I-Language
.	O
</s>
<s>
In	O
the	O
1990s	O
,	O
Alex	O
Wilkie	O
showed	O
that	O
one	O
has	O
the	O
same	O
result	O
if	O
instead	O
of	O
adding	O
every	O
restricted	O
analytic	B-Language
function	I-Language
,	O
one	O
just	O
adds	O
the	O
unrestricted	O
exponential	O
function	O
to	O
R	O
to	O
get	O
the	O
ordered	O
real	O
field	O
with	O
exponentiation	O
,	O
Rexp	O
,	O
a	O
result	O
known	O
as	O
Wilkie	O
's	O
theorem	O
.	O
</s>
<s>
Wilkie	O
also	O
tackled	O
the	O
question	O
of	O
which	O
finite	O
sets	O
of	O
analytic	B-Language
functions	I-Language
could	O
be	O
added	O
to	O
R	O
to	O
get	O
a	O
model-completeness	O
result	O
.	O
</s>
<s>
In	O
particular	O
one	O
may	O
add	O
all	O
Pfaffian	B-Algorithm
functions	I-Algorithm
to	O
R	O
to	O
get	O
the	O
structure	O
RPfaff	O
as	O
a	O
variant	O
of	O
Gabrielov	O
's	O
result	O
.	O
</s>
<s>
The	O
equations	O
above	O
that	O
define	O
a	O
Pfaffian	O
chain	O
are	O
said	O
to	O
satisfy	O
a	O
triangular	O
condition	O
,	O
since	O
the	O
derivative	B-Algorithm
of	O
each	O
successive	O
function	O
in	O
the	O
chain	O
is	O
a	O
polynomial	O
in	O
one	O
extra	O
variable	O
.	O
</s>
<s>
If	O
this	O
triangularity	O
condition	O
is	O
relaxed	O
so	O
that	O
the	O
derivative	B-Algorithm
of	O
each	O
function	O
in	O
the	O
chain	O
is	O
a	O
polynomial	O
in	O
all	O
the	O
other	O
functions	O
in	O
the	O
chain	O
,	O
then	O
the	O
chain	O
of	O
functions	O
is	O
known	O
as	O
a	O
Noetherian	O
chain	O
,	O
and	O
a	O
function	O
constructed	O
as	O
a	O
polynomial	O
in	O
this	O
chain	O
is	O
called	O
a	O
Noetherian	O
function	O
.	O
</s>
<s>
and	O
these	O
hold	O
for	O
all	O
real	O
numbers	O
x	O
,	O
so	O
f1	O
,	O
f2	O
is	O
a	O
Noetherian	O
chain	O
on	O
all	O
of	O
R	O
.	O
But	O
there	O
is	O
no	O
polynomial	O
P(x, y )	O
such	O
that	O
the	O
derivative	B-Algorithm
of	O
sin(x )	O
can	O
be	O
written	O
as	O
P( x	O
,	O
sin(x )	O
)	O
,	O
and	O
so	O
this	O
chain	O
is	O
not	O
Pfaffian	O
.	O
</s>
