<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
model	B-Application
checking	I-Application
or	O
property	O
checking	O
is	O
a	O
method	O
for	O
checking	O
whether	O
a	O
finite-state	B-Architecture
model	I-Architecture
of	O
a	O
system	O
meets	O
a	O
given	O
specification	B-Application
(	O
also	O
known	O
as	O
correctness	O
)	O
.	O
</s>
<s>
This	O
is	O
typically	O
associated	O
with	O
hardware	B-Architecture
or	O
software	O
systems	O
,	O
where	O
the	O
specification	B-Application
contains	O
liveness	O
requirements	O
(	O
such	O
as	O
avoidance	O
of	O
livelock	B-Application
)	O
as	O
well	O
as	O
safety	O
requirements	O
(	O
such	O
as	O
avoidance	O
of	O
states	O
representing	O
a	O
system	B-General_Concept
crash	I-General_Concept
)	O
.	O
</s>
<s>
In	O
order	O
to	O
solve	O
such	O
a	O
problem	O
algorithmically	O
,	O
both	O
the	O
model	O
of	O
the	O
system	O
and	O
its	O
specification	B-Application
are	O
formulated	O
in	O
some	O
precise	O
mathematical	O
language	O
.	O
</s>
<s>
A	O
simple	O
model-checking	B-Application
problem	O
consists	O
of	O
verifying	O
whether	O
a	O
formula	O
in	O
the	O
propositional	O
logic	O
is	O
satisfied	O
by	O
a	O
given	O
structure	O
.	O
</s>
<s>
During	O
refinement	O
,	O
the	O
specification	B-Application
is	O
complemented	O
with	O
details	O
that	O
are	O
unnecessary	O
in	O
the	O
higher-level	O
specification	B-Application
.	O
</s>
<s>
There	O
is	O
no	O
need	O
to	O
verify	O
the	O
newly	O
introduced	O
properties	O
against	O
the	O
original	O
specification	B-Application
since	O
this	O
is	O
not	O
possible	O
.	O
</s>
<s>
The	O
implementation	O
or	O
design	O
is	O
regarded	O
as	O
a	O
model	O
of	O
the	O
system	O
,	O
whereas	O
the	O
specifications	B-Application
are	O
properties	O
that	O
the	O
model	O
must	O
satisfy	O
.	O
</s>
<s>
An	O
important	O
class	O
of	O
model-checking	B-Application
methods	O
has	O
been	O
developed	O
for	O
checking	O
models	O
of	O
hardware	B-Architecture
and	O
software	O
designs	O
where	O
the	O
specification	B-Application
is	O
given	O
by	O
a	O
temporal	O
logic	O
formula	O
.	O
</s>
<s>
Pioneering	O
work	O
in	O
temporal	O
logic	O
specification	B-Application
was	O
done	O
by	O
Amir	O
Pnueli	O
,	O
who	O
received	O
the	O
1996	O
Turing	O
award	O
for	O
"	O
seminal	O
work	O
introducing	O
temporal	O
logic	O
into	O
computing	B-General_Concept
science	I-General_Concept
"	O
.	O
</s>
<s>
Model	B-Application
checking	I-Application
began	O
with	O
the	O
pioneering	O
work	O
of	O
E	O
.	O
M	O
.	O
Clarke	O
,	O
E	O
.	O
A	O
.	O
Emerson	O
,	O
by	O
J	O
.	O
P	O
.	O
Queille	O
,	O
and	O
J	O
.	O
Sifakis	O
.	O
</s>
<s>
Clarke	O
,	O
Emerson	O
,	O
and	O
Sifakis	O
shared	O
the	O
2007	O
Turing	O
Award	O
for	O
their	O
seminal	O
work	O
founding	O
and	O
developing	O
the	O
field	O
of	O
model	B-Application
checking	I-Application
.	O
</s>
<s>
Model	B-Application
checking	I-Application
is	O
most	O
often	O
applied	O
to	O
hardware	B-Architecture
designs	O
.	O
</s>
<s>
In	O
embedded-systems	O
hardware	B-Architecture
,	O
it	O
is	O
possible	O
to	O
validate	O
a	O
specification	B-Application
delivered	O
,	O
e.g.	O
,	O
by	O
means	O
of	O
UML	O
activity	O
diagrams	O
or	O
control-interpreted	O
Petri	B-Operating_System
nets	I-Operating_System
.	O
</s>
<s>
The	O
structure	O
is	O
usually	O
given	O
as	O
a	O
source	O
code	O
description	O
in	O
an	O
industrial	O
hardware	B-Architecture
description	O
language	O
or	O
a	O
special-purpose	O
language	O
.	O
</s>
<s>
Such	O
a	O
program	O
corresponds	O
to	O
a	O
finite	B-Architecture
state	I-Architecture
machine	I-Architecture
(	O
FSM	O
)	O
,	O
i.e.	O
,	O
a	O
directed	O
graph	O
consisting	O
of	O
nodes	B-Data_Structure
(	O
or	O
vertices	O
)	O
and	O
edges	O
.	O
</s>
<s>
The	O
nodes	B-Data_Structure
represent	O
states	O
of	O
a	O
system	O
,	O
the	O
edges	O
represent	O
possible	O
transitions	O
that	O
may	O
alter	O
the	O
state	O
,	O
while	O
the	O
atomic	O
propositions	O
represent	O
the	O
basic	O
properties	O
that	O
hold	O
at	O
a	O
point	O
of	O
execution	O
.	O
</s>
<s>
If	O
is	O
finite	O
,	O
as	O
it	O
is	O
in	O
hardware	B-Architecture
,	O
model	B-Application
checking	I-Application
reduces	O
to	O
a	O
graph	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
Instead	O
of	O
enumerating	B-Algorithm
reachable	O
states	O
one	O
at	O
a	O
time	O
,	O
the	O
state	O
space	O
can	O
sometimes	O
be	O
traversed	O
more	O
efficiently	O
by	O
considering	O
large	O
numbers	O
of	O
states	O
at	O
a	O
single	O
step	O
.	O
</s>
<s>
When	O
such	O
state-space	O
traversal	O
is	O
based	O
on	O
representations	O
of	O
a	O
set	O
of	O
states	O
and	O
transition	O
relations	O
as	O
logical	O
formulas	O
,	O
binary	B-Application
decision	I-Application
diagrams	I-Application
(	O
BDD	O
)	O
or	O
other	O
related	O
data	O
structures	O
,	O
the	O
model-checking	B-Application
method	O
is	O
symbolic	O
.	O
</s>
<s>
Historically	O
,	O
the	O
first	O
symbolic	O
methods	O
used	O
BDDs	B-Application
.	O
</s>
<s>
After	O
the	O
success	O
of	O
propositional	B-Algorithm
satisfiability	I-Algorithm
in	O
solving	O
the	O
planning	B-Application
problem	O
in	O
artificial	B-Application
intelligence	I-Application
(	O
see	O
satplan	B-Application
)	O
in	O
1996	O
,	O
the	O
same	O
approach	O
was	O
generalized	O
to	O
model	B-Application
checking	I-Application
for	O
linear	O
temporal	O
logic	O
(	O
LTL	O
)	O
:	O
the	O
planning	B-Application
problem	O
corresponds	O
to	O
model	B-Application
checking	I-Application
for	O
safety	O
properties	O
.	O
</s>
<s>
This	O
method	O
is	O
known	O
as	O
bounded	O
model	B-Application
checking	I-Application
.	O
</s>
<s>
The	O
success	O
of	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
solvers	I-Algorithm
in	O
bounded	O
model	B-Application
checking	I-Application
led	O
to	O
the	O
widespread	O
use	O
of	O
satisfiability	O
solvers	O
in	O
symbolic	O
model	B-Application
checking	I-Application
.	O
</s>
<s>
The	O
authors	O
of	O
"	O
Patterns	O
in	O
Property	O
Specification	B-Application
for	O
Finite-State	O
Verification	O
"	O
translate	O
this	O
requirement	O
into	O
the	O
following	O
LTL	O
formula	O
:	O
</s>
<s>
Model-checking	B-Application
tools	O
face	O
a	O
combinatorial	O
blow	O
up	O
of	O
the	O
state-space	O
,	O
commonly	O
known	O
as	O
the	O
state	O
explosion	O
problem	O
,	O
that	O
must	O
be	O
addressed	O
to	O
solve	O
most	O
real-world	O
problems	O
.	O
</s>
<s>
Symbolic	O
algorithms	O
avoid	O
ever	O
explicitly	O
constructing	O
the	O
graph	O
for	O
the	O
finite	B-Architecture
state	I-Architecture
machines	I-Architecture
(	O
FSM	O
)	O
;	O
instead	O
,	O
they	O
represent	O
the	O
graph	O
implicitly	O
using	O
a	O
formula	O
in	O
quantified	O
propositional	O
logic	O
.	O
</s>
<s>
The	O
use	O
of	O
binary	B-Application
decision	I-Application
diagrams	I-Application
(	O
BDDs	B-Application
)	O
was	O
made	O
popular	O
by	O
the	O
work	O
of	O
Ken	O
McMillan	O
and	O
the	O
development	O
of	O
open-source	O
BDD	O
manipulation	O
libraries	O
such	O
as	O
CUDD	O
and	O
BuDDy	O
.	O
</s>
<s>
Bounded	O
model-checking	B-Application
algorithms	O
unroll	O
the	O
FSM	O
for	O
a	O
fixed	O
number	O
of	O
steps	O
,	O
,	O
and	O
check	O
whether	O
a	O
property	O
violation	O
can	O
occur	O
in	O
or	O
fewer	O
steps	O
.	O
</s>
<s>
This	O
typically	O
involves	O
encoding	O
the	O
restricted	O
model	O
as	O
an	O
instance	O
of	O
SAT	B-Algorithm
.	O
</s>
<s>
Iterative	B-Algorithm
deepening	I-Algorithm
depth-first	I-Algorithm
search	I-Algorithm
)	O
.	O
</s>
<s>
properties	O
of	O
mutual	B-Operating_System
exclusion	I-Operating_System
.	O
</s>
<s>
Model-checking	B-Application
tools	O
were	O
initially	O
developed	O
to	O
reason	O
about	O
the	O
logical	O
correctness	O
of	O
discrete	B-Application
state	I-Application
systems	O
,	O
but	O
have	O
since	O
been	O
extended	O
to	O
deal	O
with	O
real-time	O
and	O
limited	O
forms	O
of	O
hybrid	O
systems	O
.	O
</s>
<s>
Model	B-Application
checking	I-Application
is	O
also	O
studied	O
in	O
the	O
field	O
of	O
computational	O
complexity	O
theory	O
.	O
</s>
<s>
Given	O
a	O
finite	O
interpretation	O
,	O
for	O
instance	O
,	O
one	O
described	O
as	O
a	O
relational	B-Application
database	I-Application
,	O
decide	O
whether	O
the	O
interpretation	O
is	O
a	O
model	O
of	O
the	O
formula	O
.	O
</s>
<s>
It	O
is	O
tractable	O
when	O
imposing	O
some	O
restrictions	O
on	O
the	O
input	O
structure	O
:	O
for	O
instance	O
,	O
requiring	O
that	O
it	O
has	O
treewidth	O
bounded	O
by	O
a	O
constant	O
(	O
which	O
more	O
generally	O
implies	O
the	O
tractability	O
of	O
model	B-Application
checking	I-Application
for	O
monadic	O
second-order	O
logic	O
)	O
,	O
bounding	O
the	O
degree	O
of	O
every	O
domain	O
element	O
,	O
and	O
more	O
general	O
conditions	O
such	O
as	O
bounded	O
expansion	O
,	O
locally	O
bounded	O
expansion	O
,	O
and	O
nowhere-dense	O
structures	O
.	O
</s>
<s>
These	O
results	O
have	O
been	O
extended	O
to	O
the	O
task	O
of	O
enumerating	B-Algorithm
all	O
solutions	O
to	O
a	O
first-order	O
formula	O
with	O
free	O
variables	O
.	O
</s>
<s>
Here	O
is	O
a	O
list	O
of	O
significant	O
model-checking	B-Application
tools	O
:	O
</s>
<s>
Storm	O
:	O
A	O
model	B-Application
checker	I-Application
for	O
probabilistic	O
systems	O
.	O
</s>
<s>
Zing	O
–	O
experimental	O
tool	O
from	O
Microsoft	O
to	O
validate	O
state	O
models	O
of	O
software	O
at	O
various	O
levels	O
:	O
high-level	O
protocol	O
descriptions	O
,	O
work-flow	O
specifications	B-Application
,	O
web	O
services	O
,	O
device	O
drivers	O
,	O
and	O
protocols	O
in	O
the	O
core	O
of	O
the	O
operating	O
system	O
.	O
</s>
