<s>
This	O
article	O
describes	O
Kripke	B-Application
structures	I-Application
as	O
used	O
in	O
model	B-Application
checking	I-Application
.	O
</s>
<s>
A	O
Kripke	B-Application
structure	I-Application
is	O
a	O
variation	O
of	O
the	O
transition	B-Application
system	I-Application
,	O
originally	O
proposed	O
by	O
Saul	O
Kripke	O
,	O
used	O
in	O
model	B-Application
checking	I-Application
to	O
represent	O
the	O
behavior	O
of	O
a	O
system	O
.	O
</s>
<s>
Temporal	O
logics	O
are	O
traditionally	O
interpreted	O
in	O
terms	O
of	O
Kripke	B-Application
structures	I-Application
.	O
</s>
<s>
Since	O
is	O
left-total	O
,	O
it	O
is	O
always	O
possible	O
to	O
construct	O
an	O
infinite	O
path	O
through	O
the	O
Kripke	B-Application
structure	I-Application
.	O
</s>
<s>
A	O
deadlock	B-Operating_System
state	O
can	O
be	O
modeled	O
by	O
a	O
single	O
outgoing	O
edge	O
back	O
to	O
itself	O
.	O
</s>
<s>
With	O
this	O
definition	O
,	O
a	O
Kripke	B-Application
structure	I-Application
(	O
say	O
,	O
having	O
only	O
one	O
initial	O
state	O
may	O
be	O
identified	O
with	O
a	O
Moore	B-General_Concept
machine	I-General_Concept
with	O
a	O
singleton	O
input	O
alphabet	O
,	O
and	O
with	O
the	O
output	O
function	O
being	O
its	O
labeling	O
function	O
.	O
</s>
<s>
The	O
figure	O
at	O
right	O
illustrates	O
a	O
Kripke	B-Application
structure	I-Application
,	O
</s>
<s>
Although	O
this	O
terminology	O
is	O
widespread	O
in	O
the	O
model	B-Application
checking	I-Application
community	O
,	O
some	O
textbooks	O
on	O
model	B-Application
checking	I-Application
do	O
not	O
define	O
"	O
Kripke	B-Application
structure	I-Application
"	O
in	O
this	O
extended	O
way	O
(	O
or	O
at	O
all	O
in	O
fact	O
)	O
,	O
but	O
simply	O
use	O
the	O
concept	O
of	O
a	O
(	O
labelled	O
)	O
transition	B-Application
system	I-Application
,	O
which	O
additionally	O
has	O
a	O
set	O
of	O
actions	O
,	O
and	O
the	O
transition	O
relation	O
is	O
defined	O
as	O
a	O
subset	O
of	O
,	O
which	O
they	O
additionally	O
extend	O
to	O
include	O
a	O
set	O
of	O
atomic	O
propositions	O
and	O
a	O
labeling	O
function	O
for	O
the	O
states	O
as	O
well	O
(	O
as	O
defined	O
above	O
.	O
)	O
</s>
<s>
redefine	O
a	O
Kripke	B-Application
structure	I-Application
as	O
a	O
set	O
of	O
transitions	O
(	O
instead	O
of	O
just	O
one	O
)	O
,	O
which	O
is	O
equivalent	O
to	O
the	O
labeled	O
transitions	O
above	O
,	O
when	O
they	O
define	O
the	O
semantics	O
of	O
modal	B-Application
μ-calculus	I-Application
.	O
</s>
