<s>
In	O
model	B-Application
checking	I-Application
,	O
a	O
branch	O
of	O
computer	B-General_Concept
science	I-General_Concept
,	O
linear	B-Application
time	I-Application
properties	I-Application
are	O
used	O
to	O
describe	O
requirements	O
of	O
a	O
model	O
of	O
a	O
computer	O
system	O
.	O
</s>
<s>
Example	O
properties	O
include	O
"	O
the	O
vending	O
machine	O
does	O
not	O
dispense	O
a	O
drink	O
until	O
money	O
has	O
been	O
entered	O
"	O
(	O
a	O
safety	B-Application
property	I-Application
)	O
or	O
"	O
the	O
computer	O
program	O
eventually	O
terminates	O
"	O
(	O
a	O
liveness	B-Application
property	I-Application
)	O
.	O
</s>
<s>
For	O
instance	O
,	O
in	O
a	O
model	O
of	O
two	O
traffic	O
lights	O
,	O
the	O
liveness	B-Application
property	I-Application
"	O
both	O
traffic	O
lights	O
are	O
green	O
infinitely	O
often	O
"	O
may	O
only	O
be	O
true	O
under	O
the	O
unconditional	O
fairness	O
constraint	O
"	O
each	O
traffic	O
light	O
changes	O
colour	O
infinitely	O
often	O
"	O
(	O
to	O
exclude	O
the	O
case	O
where	O
one	O
traffic	O
light	O
is	O
"	O
infinitely	O
faster	O
"	O
than	O
the	O
other	O
)	O
.	O
</s>
<s>
Formally	O
,	O
a	O
linear	B-Application
time	I-Application
property	I-Application
is	O
an	O
ω-language	O
over	O
the	O
power	O
set	O
of	O
"	O
atomic	O
propositions	O
"	O
.	O
</s>
<s>
Every	O
property	O
can	O
be	O
rewritten	O
as	O
"	O
P	O
and	O
Q	O
both	O
occur	O
"	O
for	O
some	O
safety	B-Application
property	I-Application
P	O
and	O
liveness	B-Application
property	I-Application
Q	O
.	O
</s>
<s>
Temporal	O
logics	O
such	O
as	O
linear	O
temporal	O
logic	O
describe	O
types	O
of	O
linear	B-Application
time	I-Application
properties	I-Application
using	O
formulae	O
.	O
</s>
<s>
The	O
more	O
general	O
formalism	O
used	O
in	O
Safety	B-Application
and	I-Application
liveness	I-Application
properties	I-Application
can	O
handle	O
this	O
.	O
</s>
<s>
An	O
example	O
of	O
an	O
LT	B-Application
property	I-Application
over	O
the	O
set	O
is	O
"	O
the	O
set	O
of	O
words	O
that	O
contain	O
a	O
infinitely	O
often	O
"	O
.	O
</s>
<s>
An	O
LT	B-Application
property	I-Application
is	O
an	O
ω-language	O
over	O
the	O
alphabet	O
(	O
and	O
vice	O
versa	O
)	O
.	O
</s>
<s>
The	O
closure	O
of	O
an	O
LT	B-Application
property	I-Application
P	O
is	O
:	O
</s>
<s>
Using	O
the	O
theory	O
of	O
finite-state	B-Architecture
machines	I-Architecture
,	O
a	O
program	O
or	O
computer	O
system	O
can	O
be	O
modelled	O
by	O
a	O
Kripke	B-Application
structure	I-Application
.	O
</s>
<s>
LT	B-Application
properties	I-Application
then	O
describe	O
restrictions	O
on	O
the	O
traces	O
(	O
outputs	O
)	O
of	O
a	O
Kripke	B-Application
structure	I-Application
.	O
</s>
<s>
For	O
instance	O
,	O
if	O
two	O
traffic	O
lights	O
at	O
an	O
intersection	O
are	O
represented	O
by	O
a	O
Kripke	B-Application
structure	I-Application
then	O
the	O
atomic	O
propositions	O
may	O
be	O
the	O
possible	O
colours	O
of	O
each	O
light	O
and	O
it	O
may	O
be	O
desirable	O
that	O
the	O
traces	O
satisfy	O
the	O
LT	B-Application
property	I-Application
"	O
the	O
traffic	O
lights	O
cannot	O
both	O
be	O
green	O
at	O
the	O
same	O
time	O
"	O
(	O
to	O
avoid	O
car	O
collisions	O
)	O
.	O
</s>
<s>
If	O
every	O
trace	O
of	O
the	O
Kripke	B-Application
structure	I-Application
TS	O
is	O
a	O
trace	O
of	O
TS	O
then	O
every	O
LT	B-Application
property	I-Application
that	O
TS	O
satisfies	O
is	O
satisfied	O
by	O
TS	O
.	O
</s>
<s>
This	O
is	O
useful	O
in	O
model	B-Application
checking	I-Application
to	O
allow	O
abstraction	O
:	O
if	O
a	O
simplified	O
model	O
of	O
the	O
system	O
satisfies	O
an	O
LT	B-Application
property	I-Application
then	O
the	O
actual	O
model	O
of	O
the	O
system	O
will	O
satisfy	O
it	O
as	O
well	O
.	O
</s>
<s>
A	O
safety	B-Application
property	I-Application
is	O
informally	O
of	O
the	O
form	O
"	O
a	O
bad	O
thing	O
does	O
not	O
happen	O
"	O
.	O
</s>
<s>
Formally	O
,	O
a	O
safety	B-Application
property	I-Application
is	O
an	O
LT	B-Application
property	I-Application
such	O
that	O
any	O
word	O
that	O
violates	O
the	O
property	O
has	O
a	O
"	O
bad	O
prefix	O
"	O
,	O
for	O
which	O
no	O
word	O
with	O
that	O
prefix	O
satisfies	O
the	O
property	O
.	O
</s>
<s>
To	O
verify	O
a	O
safety	B-Application
property	I-Application
,	O
it	O
is	O
sufficient	O
to	O
consider	O
only	O
the	O
finite	O
traces	O
of	O
a	O
Kripke	B-Application
structure	I-Application
and	O
check	O
whether	O
any	O
such	O
trace	O
is	O
a	O
bad	O
prefix	O
.	O
</s>
<s>
An	O
LT	B-Application
property	I-Application
P	O
is	O
a	O
safety	B-Application
property	I-Application
if	O
and	O
only	O
if	O
.	O
</s>
<s>
An	O
invariant	O
property	O
is	O
a	O
type	O
of	O
safety	B-Application
property	I-Application
in	O
which	O
the	O
condition	O
only	O
refers	O
to	O
the	O
current	O
state	O
.	O
</s>
<s>
A	O
Kripke	B-Application
structure	I-Application
satisfies	O
an	O
invariant	O
if	O
and	O
only	O
if	O
every	O
reachable	O
state	O
satisfies	O
the	O
invariant	O
,	O
which	O
can	O
be	O
checked	O
by	O
a	O
breadth-first	B-Algorithm
search	I-Algorithm
or	O
depth-first	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
Safety	B-Application
properties	I-Application
can	O
be	O
verified	O
inductively	O
using	O
invariants	O
.	O
</s>
<s>
A	O
liveness	B-Application
property	I-Application
is	O
informally	O
of	O
the	O
form	O
"	O
something	O
good	O
eventually	O
happens	O
"	O
.	O
</s>
<s>
Formally	O
,	O
P	O
is	O
a	O
liveness	B-Application
property	I-Application
if	O
i.e.	O
</s>
<s>
An	O
example	O
of	O
a	O
liveness	B-Application
property	I-Application
is	O
the	O
previous	O
LT	B-Application
property	I-Application
"	O
the	O
set	O
of	O
words	O
that	O
contain	O
a	O
infinitely	O
often	O
"	O
.	O
</s>
<s>
In	O
terms	O
of	O
computer	O
programs	O
,	O
useful	O
liveness	B-Application
properties	I-Application
include	O
"	O
the	O
program	O
eventually	O
terminates	O
"	O
and	O
,	O
in	O
concurrent	B-Architecture
computing	I-Architecture
,	O
"	O
every	O
process	B-Operating_System
must	O
eventually	O
be	O
served	O
"	O
.	O
</s>
<s>
A	O
persistence	O
property	O
is	O
a	O
liveness	B-Application
property	I-Application
of	O
the	O
form	O
"	O
eventually	O
forever	O
"	O
.	O
</s>
<s>
No	O
LT	B-Application
property	I-Application
other	O
than	O
(	O
the	O
set	O
of	O
all	O
words	O
over	O
)	O
is	O
both	O
a	O
safety	O
and	O
a	O
liveness	B-Application
property	I-Application
.	O
</s>
<s>
Though	O
not	O
every	O
property	O
is	O
a	O
safety	B-Application
property	I-Application
or	O
a	O
liveness	B-Application
property	I-Application
(	O
consider	O
"	O
a	O
occurs	O
exactly	O
once	O
"	O
)	O
,	O
every	O
property	O
is	O
the	O
intersection	O
of	O
a	O
safety	O
and	O
a	O
liveness	B-Application
property	I-Application
.	O
</s>
<s>
In	O
topology	B-Architecture
,	O
the	O
set	O
of	O
all	O
words	O
can	O
be	O
equipped	O
with	O
the	O
metric	O
:	O
</s>
<s>
Then	O
a	O
safety	B-Application
property	I-Application
is	O
a	O
closed	O
set	O
and	O
a	O
liveness	B-Application
property	I-Application
is	O
a	O
dense	O
set	O
.	O
</s>
<s>
Unconditional	O
fairness	O
is	O
of	O
the	O
form	O
"	O
every	O
process	B-Operating_System
gets	O
its	O
turn	O
infinitely	O
often	O
"	O
.	O
</s>
<s>
Strong	O
fairness	O
is	O
of	O
the	O
form	O
"	O
every	O
process	B-Operating_System
gets	O
its	O
turn	O
infinitely	O
often	O
if	O
it	O
is	O
enabled	O
infinitely	O
often	O
"	O
.	O
</s>
<s>
Weak	O
fairness	O
is	O
of	O
the	O
form	O
"	O
every	O
process	B-Operating_System
gets	O
its	O
turn	O
infinitely	O
often	O
if	O
it	O
is	O
continuously	O
enabled	O
from	O
a	O
particular	O
point	O
"	O
.	O
</s>
<s>
A	O
program	O
"	O
fairly	O
satisfies	O
"	O
an	O
LT	B-Application
property	I-Application
P	O
with	O
respect	O
to	O
a	O
set	O
of	O
fairness	O
conditions	O
if	O
for	O
every	O
path	O
,	O
either	O
the	O
path	O
fails	O
a	O
fairness	O
condition	O
or	O
it	O
satisfies	O
P	O
.	O
That	O
is	O
,	O
the	O
property	O
P	O
is	O
satisfied	O
for	O
all	O
fair	O
paths	O
.	O
</s>
<s>
A	O
fairness	O
property	O
is	O
realizable	O
for	O
a	O
Kripke	B-Application
structure	I-Application
if	O
every	O
reachable	O
state	O
has	O
a	O
fair	O
path	O
starting	O
from	O
that	O
state	O
.	O
</s>
<s>
So	O
long	O
as	O
a	O
set	O
of	O
fairness	O
conditions	O
are	O
realizable	O
,	O
they	O
are	O
irrelevant	O
to	O
safety	B-Application
properties	I-Application
.	O
</s>
<s>
Temporal	O
logics	O
such	O
as	O
computation	O
tree	O
logic	O
(	O
CTL	O
)	O
can	O
be	O
used	O
to	O
specify	O
some	O
LT	B-Application
properties	I-Application
.	O
</s>
<s>
All	O
linear	O
temporal	O
logic	O
(	O
LTL	O
)	O
formulae	O
are	O
LT	B-Application
properties	I-Application
.	O
</s>
<s>
By	O
a	O
counting	O
argument	O
,	O
we	O
see	O
that	O
any	O
logic	O
in	O
which	O
each	O
formula	O
is	O
a	O
finite	O
string	O
cannot	O
represent	O
all	O
LT	B-Application
properties	I-Application
,	O
as	O
there	O
must	O
be	O
countably	O
many	O
formulae	O
but	O
there	O
are	O
uncountably	O
many	O
LT	B-Application
properties	I-Application
.	O
</s>
