<s>
Properties	O
of	O
an	O
execution	O
of	O
a	O
computer	O
program	O
—	O
particularly	O
for	O
concurrent	B-Operating_System
and	O
distributed	B-Architecture
systems	I-Architecture
—	O
have	O
long	O
been	O
formulated	O
by	O
giving	O
safety	B-Application
properties	I-Application
(	O
"	O
bad	O
things	O
do	O
n't	O
happen	O
"	O
)	O
and	O
liveness	B-Application
properties	I-Application
(	O
"	O
good	O
things	O
do	O
happen	O
"	O
)	O
.	O
</s>
<s>
A	O
simple	O
example	O
will	O
illustrate	O
safety	O
and	O
liveness	B-Application
.	O
</s>
<s>
Total	O
correctness	O
is	O
a	O
conjunction	O
of	O
a	O
safety	B-Application
property	I-Application
and	O
a	O
liveness	B-Application
property	I-Application
:	O
</s>
<s>
The	O
safety	B-Application
property	I-Application
prohibits	O
these	O
"	O
bad	O
things	O
"	O
:	O
executions	O
that	O
start	O
in	O
a	O
state	O
satisfying	O
and	O
terminate	O
in	O
a	O
final	O
state	O
that	O
does	O
not	O
satisfy	O
.	O
</s>
<s>
For	O
a	O
program	O
,	O
this	O
safety	B-Application
property	I-Application
is	O
usually	O
written	O
using	O
the	O
Hoare	O
triple	O
.	O
</s>
<s>
The	O
liveness	B-Application
property	I-Application
,	O
the	O
"	O
good	O
thing	O
"	O
,	O
is	O
that	O
execution	O
that	O
starts	O
in	O
a	O
state	O
satisfying	O
terminates	O
.	O
</s>
<s>
A	O
"	O
good	O
thing	O
"	O
need	O
not	O
be	O
discrete	O
,	O
but	O
the	O
liveness	B-Application
property	I-Application
of	O
termination	O
is	O
discrete	O
.	O
</s>
<s>
Formal	O
definitions	O
that	O
were	O
ultimately	O
proposed	O
for	O
safety	B-Application
properties	I-Application
and	O
liveness	B-Application
properties	I-Application
demonstrated	O
that	O
this	O
decomposition	O
is	O
not	O
only	O
intuitively	O
appealing	O
but	O
is	O
also	O
complete	O
:	O
all	O
properties	O
of	O
an	O
execution	O
are	O
a	O
conjunction	O
of	O
safety	B-Application
and	I-Application
liveness	I-Application
properties	I-Application
.	O
</s>
<s>
Moreover	O
,	O
undertaking	O
the	O
decomposition	O
can	O
be	O
helpful	O
,	O
because	O
the	O
formal	O
definitions	O
enable	O
a	O
proof	O
that	O
different	O
methods	O
must	O
be	O
used	O
for	O
verifying	O
safety	B-Application
properties	I-Application
versus	O
for	O
verifying	O
liveness	B-Application
properties	I-Application
.	O
</s>
<s>
A	O
safety	B-Application
property	I-Application
proscribes	O
discrete	O
bad	O
things	O
from	O
occurring	O
during	O
an	O
execution	O
.	O
</s>
<s>
A	O
safety	B-Application
property	I-Application
thus	O
characterizes	O
what	O
is	O
permitted	O
by	O
stating	O
what	O
is	O
prohibited	O
.	O
</s>
<s>
Examples	O
of	O
a	O
discrete	O
bad	O
thing	O
that	O
could	O
be	O
used	O
to	O
define	O
a	O
safety	B-Application
property	I-Application
include	O
:	O
</s>
<s>
An	O
execution	O
of	O
two	O
concurrent	B-Operating_System
processes	O
,	O
where	O
the	O
program	O
counters	O
for	O
both	O
processes	O
designate	O
statements	O
within	O
a	O
critical	B-Operating_System
section	I-Operating_System
;	O
</s>
<s>
An	O
execution	O
of	O
two	O
concurrent	B-Operating_System
processes	O
where	O
each	O
process	O
is	O
waiting	O
for	O
another	O
to	O
change	O
state	O
(	O
known	O
as	O
deadlock	B-Operating_System
)	O
.	O
</s>
<s>
The	O
essential	O
characteristic	O
of	O
a	O
safety	B-Application
property	I-Application
is	O
:	O
</s>
<s>
for	O
that	O
safety	B-Application
property	I-Application
occurs	O
at	O
some	O
point	O
in	O
.	O
</s>
<s>
We	O
take	O
this	O
inference	O
about	O
the	O
irremediability	O
of	O
bad	O
things	O
to	O
be	O
the	O
defining	O
characteristic	O
for	O
to	O
be	O
a	O
safety	B-Application
property	I-Application
.	O
</s>
<s>
Formalizing	O
this	O
in	O
predicate	O
logic	O
gives	O
a	O
formal	O
definition	O
for	O
being	O
a	O
safety	B-Application
property	I-Application
.	O
</s>
<s>
A	O
liveness	B-Application
property	I-Application
prescribes	O
good	O
things	O
for	O
every	O
execution	O
or	O
,	O
equivalently	O
,	O
describes	O
something	O
that	O
must	O
happen	O
during	O
an	O
execution	O
.	O
</s>
<s>
Examples	O
of	O
a	O
good	O
thing	O
used	O
to	O
define	O
a	O
liveness	B-Application
property	I-Application
include	O
:	O
</s>
<s>
Guaranteed	O
eventual	O
entry	O
to	O
a	O
critical	B-Operating_System
section	I-Operating_System
whenever	O
entry	O
is	O
attempted	O
;	O
</s>
<s>
Producing	O
an	O
answer	O
within	O
a	O
specified	O
real-time	O
bound	O
is	O
a	O
safety	B-Application
property	I-Application
rather	O
than	O
a	O
liveness	B-Application
property	I-Application
.	O
</s>
<s>
Deadlock	B-Operating_System
freedom	O
is	O
a	O
safety	B-Application
property	I-Application
:	O
the	O
"	O
bad	O
thing	O
"	O
is	O
a	O
deadlock	B-Operating_System
(	O
which	O
is	O
discrete	O
)	O
.	O
</s>
<s>
A	O
property	O
that	O
gives	O
a	O
specific	O
bound	O
to	O
the	O
"	O
good	O
thing	O
"	O
is	O
a	O
safety	B-Application
property	I-Application
(	O
as	O
noted	O
above	O
)	O
,	O
whereas	O
the	O
weaker	O
property	O
that	O
merely	O
asserts	O
the	O
bound	O
exists	O
is	O
a	O
liveness	B-Application
property	I-Application
.	O
</s>
<s>
Proving	O
such	O
a	O
liveness	B-Application
property	I-Application
is	O
likely	O
to	O
be	O
easier	O
than	O
proving	O
the	O
tighter	O
safety	B-Application
property	I-Application
because	O
proving	O
the	O
liveness	B-Application
property	I-Application
does	O
n't	O
require	O
the	O
kind	O
of	O
detailed	O
accounting	O
that	O
is	O
required	O
for	O
proving	O
the	O
safety	B-Application
property	I-Application
.	O
</s>
<s>
an	O
would	O
be	O
a	O
"	O
bad	O
thing	O
"	O
and	O
,	O
thus	O
,	O
would	O
be	O
defining	O
a	O
safety	B-Application
property	I-Application
)	O
.	O
</s>
<s>
That	O
leads	O
to	O
defining	O
a	O
liveness	B-Application
property	I-Application
to	O
be	O
a	O
property	O
that	O
does	O
not	O
rule	O
out	O
any	O
finite	O
prefix	O
.	O
</s>
<s>
in	O
his	O
1977	O
paper	O
on	O
proving	O
the	O
correctness	O
of	O
multiprocess	O
(	O
concurrent	B-Operating_System
)	O
programs	O
.	O
</s>
<s>
to	O
its	O
"	O
places	O
"	O
could	O
evolve	O
;	O
Petri	B-Operating_System
net	I-Operating_System
safety	O
was	O
a	O
specific	O
form	O
of	O
boundedness	O
.	O
</s>
<s>
NATO	O
short	O
course	O
on	O
distributed	B-Architecture
systems	I-Architecture
in	O
Munich	O
.	O
</s>
<s>
Alpern	O
and	O
Schneider	O
gives	O
the	O
formal	O
definition	O
for	O
liveness	B-Application
,	O
accompanied	O
by	O
a	O
proof	O
that	O
all	O
properties	O
can	O
be	O
constructed	O
using	O
safety	B-Application
properties	I-Application
and	O
liveness	B-Application
properties	I-Application
.	O
</s>
<s>
That	O
proof	O
was	O
inspired	O
by	O
Gordon	O
Plotkin	O
's	O
insight	O
that	O
safety	B-Application
properties	I-Application
correspond	O
to	O
closed	O
sets	O
and	O
liveness	B-Application
properties	I-Application
correspond	O
to	O
dense	O
sets	O
in	O
a	O
natural	O
topology	B-Architecture
on	O
the	O
set	O
of	O
infinite	O
sequences	O
of	O
program	O
states	O
.	O
</s>
<s>
Subsequently	O
,	O
Alpern	O
and	O
Schneider	O
not	O
only	O
gave	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
characterization	O
for	O
the	O
formal	O
definitions	O
of	O
safety	B-Application
properties	I-Application
and	O
liveness	B-Application
properties	I-Application
but	O
used	O
these	O
automata	O
formulations	O
to	O
show	O
that	O
verification	O
of	O
safety	B-Application
properties	I-Application
would	O
require	O
an	O
invariant	O
and	O
verification	O
of	O
liveness	B-Application
properties	I-Application
would	O
require	O
a	O
well-foundedness	B-Algorithm
argument	I-Algorithm
.	O
</s>
