<s>
An	O
and-inverter	B-Application
graph	I-Application
(	O
AIG	O
)	O
is	O
a	O
directed	O
,	O
acyclic	O
graph	O
that	O
represents	O
a	O
structural	O
implementation	O
of	O
the	O
logical	O
functionality	O
of	O
a	O
circuit	O
or	O
network	O
.	O
</s>
<s>
Typically	O
,	O
the	O
abstract	O
graph	O
is	O
represented	O
as	O
a	O
data	B-General_Concept
structure	I-General_Concept
in	O
software	O
.	O
</s>
<s>
This	O
makes	O
the	O
AIG	O
an	O
efficient	O
representation	O
in	O
comparison	O
with	O
either	O
the	O
binary	B-Application
decision	I-Application
diagram	I-Application
(	O
BDD	O
)	O
or	O
the	O
"	O
sum-of-product	B-Application
"	O
( ΣoΠ	O
)	O
form	O
,	O
that	O
is	O
,	O
the	O
canonical	O
form	O
in	O
Boolean	O
algebra	O
known	O
as	O
the	O
disjunctive	B-Application
normal	I-Application
form	I-Application
(	O
DNF	O
)	O
.	O
</s>
<s>
and	O
Smith	O
et	O
al.	O
,	O
which	O
reduce	O
circuits	O
to	O
improve	O
area	O
and	O
delay	O
during	O
synthesis	O
,	O
or	O
to	O
speed	O
up	O
formal	B-Application
equivalence	I-Application
checking	I-Application
.	O
</s>
<s>
Another	O
important	O
development	O
was	O
the	O
recent	O
emergence	O
of	O
much	O
more	O
efficient	O
boolean	B-Algorithm
satisfiability	I-Algorithm
(	O
SAT	O
)	O
solvers	O
.	O
</s>
<s>
A	O
well-tuned	O
combination	O
of	O
AIGs	O
and	O
boolean	B-Algorithm
satisfiability	I-Algorithm
made	O
an	O
impact	O
on	O
formal	O
verification	O
,	O
including	O
both	O
model	B-Application
checking	I-Application
and	O
equivalence	B-Application
checking	I-Application
.	O
</s>
<s>
There	O
is	O
a	O
growing	O
understanding	O
that	O
logic	O
and	O
physical	O
synthesis	O
problems	O
can	O
be	O
solved	O
using	O
simulation	O
and	O
boolean	B-Algorithm
satisfiability	I-Algorithm
to	O
compute	O
functional	O
properties	O
(	O
such	O
as	O
symmetries	O
)	O
and	O
node	O
flexibilities	O
(	O
such	O
as	O
don't-care	O
terms	O
,	O
resubstitutions	O
,	O
and	O
SPFDs	O
)	O
.	O
</s>
<s>
This	O
is	O
,	O
to	O
a	O
large	O
extent	O
,	O
due	O
to	O
the	O
simple	O
and	O
uniform	O
structure	O
of	O
AIGs	O
,	O
which	O
allow	O
rewriting	O
,	O
simulation	O
,	O
mapping	O
,	O
placement	O
,	O
and	O
verification	O
to	O
share	O
the	O
same	O
data	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
which	O
,	O
in	O
general	O
,	O
can	O
be	O
unknown	O
)	O
resulting	O
in	O
a	O
data	B-General_Concept
structure	I-General_Concept
that	O
is	O
specifically	O
tailored	O
for	O
applications	O
related	O
to	O
retiming	B-Application
.	O
</s>
<s>
One	O
such	O
technique	O
combines	O
technology	O
mapping	O
and	O
retiming	B-Application
in	O
a	O
single	O
optimization	O
step	O
.	O
</s>
