<s>
In	O
Boolean	O
algebra	O
,	O
the	O
algebraic	B-Application
normal	I-Application
form	I-Application
(	O
ANF	B-Application
)	O
,	O
ring	B-Application
sum	I-Application
normal	I-Application
form	I-Application
(	O
RSNF	B-Application
or	O
RNF	B-Application
)	O
,	O
Zhegalkin	O
normal	O
form	O
,	O
or	O
Reed	O
–	O
Muller	O
expansion	O
is	O
a	O
way	O
of	O
writing	O
propositional	O
logic	O
formulas	O
in	O
one	O
of	O
three	O
subforms	O
:	O
</s>
<s>
One	O
or	O
more	O
variables	O
are	O
combined	O
into	O
a	O
term	O
by	O
AND	O
(	O
)	O
,	O
then	O
one	O
or	O
more	O
terms	O
are	O
combined	O
by	O
XOR	O
(	O
)	O
together	O
into	O
ANF	B-Application
.	O
</s>
<s>
Formulas	O
written	O
in	O
ANF	B-Application
are	O
also	O
known	O
as	O
Zhegalkin	O
polynomials	O
and	O
Positive	O
Polarity	O
(	O
or	O
Parity	O
)	O
Reed	O
–	O
Muller	O
expressions	O
(	O
PPRM	O
)	O
.	O
</s>
<s>
ANF	B-Application
is	O
a	O
canonical	O
form	O
,	O
which	O
means	O
that	O
two	O
logically	O
equivalent	O
formulas	O
will	O
convert	O
to	O
the	O
same	O
ANF	B-Application
,	O
easily	O
showing	O
whether	O
two	O
formulas	O
are	O
equivalent	O
for	O
automated	B-Application
theorem	I-Application
proving	I-Application
.	O
</s>
<s>
Unlike	O
other	O
normal	O
forms	O
,	O
it	O
can	O
be	O
represented	O
as	O
a	O
simple	O
list	O
of	O
lists	O
of	O
variable	O
names	O
—	O
conjunctive	B-Application
and	O
disjunctive	B-Application
normal	I-Application
forms	I-Application
also	O
require	O
recording	O
whether	O
each	O
variable	O
is	O
negated	O
or	O
not	O
.	O
</s>
<s>
Negation	B-Application
normal	I-Application
form	I-Application
is	O
unsuitable	O
for	O
determining	O
equivalence	O
,	O
since	O
on	O
negation	B-Application
normal	I-Application
forms	I-Application
,	O
equivalence	O
does	O
not	O
imply	O
equality	O
:	O
a	O
¬a	O
is	O
n't	O
reduced	O
to	O
the	O
same	O
thing	O
as	O
1	O
,	O
even	O
though	O
they	O
're	O
logically	O
equivalent	O
.	O
</s>
<s>
Putting	O
a	O
formula	O
into	O
ANF	B-Application
also	O
makes	O
it	O
easy	O
to	O
identify	O
linear	O
functions	O
(	O
used	O
,	O
for	O
example	O
,	O
in	O
linear-feedback	B-Architecture
shift	I-Architecture
registers	I-Architecture
)	O
:	O
a	O
linear	O
function	O
is	O
one	O
that	O
is	O
a	O
sum	O
of	O
single	O
literals	O
.	O
</s>
<s>
Properties	O
of	O
nonlinear-feedback	O
shift	B-General_Concept
registers	I-General_Concept
can	O
also	O
be	O
deduced	O
from	O
certain	O
properties	O
of	O
the	O
feedback	O
function	O
in	O
ANF	B-Application
.	O
</s>
<s>
There	O
are	O
straightforward	O
ways	O
to	O
perform	O
the	O
standard	O
boolean	O
operations	O
on	O
ANF	B-Application
inputs	O
in	O
order	O
to	O
get	O
ANF	B-Application
results	O
.	O
</s>
<s>
Each	O
variable	O
in	O
a	O
formula	O
is	O
already	O
in	O
pure	O
ANF	B-Application
,	O
so	O
you	O
only	O
need	O
to	O
perform	O
the	O
formula	O
's	O
boolean	O
operations	O
as	O
shown	O
above	O
to	O
get	O
the	O
entire	O
formula	O
into	O
ANF	B-Application
.	O
</s>
<s>
ANF	B-Application
is	O
sometimes	O
described	O
in	O
an	O
equivalent	O
way	O
:	O
</s>
<s>
For	O
example	O
,	O
let	O
us	O
construct	O
ANF	B-Application
of	O
(	O
logical	O
or	O
)	O
:	O
</s>
<s>
by	O
distribution	O
,	O
we	O
get	O
the	O
final	O
ANF	B-Application
:	O
</s>
