<s>
Input/output	O
automata	O
provide	O
a	O
formal	B-Architecture
model	I-Architecture
,	O
applicable	O
in	O
describing	O
most	O
types	O
of	O
an	O
asynchronous	B-Application
concurrent	B-Architecture
system	I-Architecture
.	O
</s>
<s>
distributed	B-Architecture
systems	I-Architecture
.	O
</s>
<s>
To	O
describe	O
specific	O
types	O
of	O
asynchronous	B-Application
systems	I-Application
,	O
additional	O
structure	O
must	O
be	O
added	O
to	O
this	O
basic	O
model	O
.	O
</s>
<s>
"	O
An	O
I/O	B-Operating_System
automaton	I-Operating_System
models	O
a	O
distributed	B-Architecture
system	I-Architecture
component	O
that	O
can	O
interact	O
with	O
other	O
system	O
components	O
.	O
</s>
<s>
I/O	O
automata	O
can	O
be	O
used	O
to	O
model	O
individual	O
components	O
of	O
a	O
distributed	B-Architecture
system	I-Architecture
,	O
such	O
as	O
a	O
process	O
,	O
a	O
message	O
channel	O
in	O
a	O
message	B-Architecture
passing	I-Architecture
system	O
,	O
or	O
a	O
shared	B-Operating_System
data	I-Operating_System
structure	O
in	O
a	O
shared	B-Operating_System
memory	I-Operating_System
systems	O
.	O
</s>
<s>
Figure	O
1	O
depicts	O
an	O
example	O
of	O
an	O
I/O	B-Operating_System
automaton	I-Operating_System
for	O
a	O
process	O
in	O
an	O
asynchronous	B-Application
message-passing	B-Architecture
distributed	B-Architecture
system	I-Architecture
.	O
</s>
<s>
In	O
this	O
setting	O
,	O
process	O
Pi	O
communicates	O
with	O
other	O
processes	O
by	O
using	O
a	O
message	B-Architecture
passing	I-Architecture
system	O
.	O
</s>
<s>
A	O
message	O
channel	O
can	O
also	O
be	O
modeled	O
by	O
an	O
I/O	B-Operating_System
automaton	I-Operating_System
.	O
</s>
<s>
Figure	O
2	O
illustrates	O
a	O
typical	O
unidirectional	O
FIFO	B-Operating_System
channel	O
automaton	O
,	O
named	O
Ci	O
,	O
j	O
.	O
</s>
<s>
The	O
state	O
of	O
the	O
automaton	O
stores	O
a	O
FIFO	B-Operating_System
queue	O
of	O
all	O
messages	O
that	O
have	O
been	O
sent	O
but	O
not	O
yet	O
received	O
.	O
</s>
<s>
Figure	O
3	O
illustrates	O
an	O
atomic	O
read/write	O
register	O
I/O	B-Operating_System
automaton	I-Operating_System
in	O
a	O
shared	B-Operating_System
memory	I-Operating_System
system	O
with	O
two	O
processes	O
,	O
P1	O
and	O
P2	O
.	O
</s>
<s>
An	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
,	O
or	O
simply	O
an	O
automaton	O
,	O
comprises	O
five	O
components	O
:	O
</s>
<s>
The	O
initial	O
step	O
in	O
formalization	O
of	O
an	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
is	O
the	O
definition	O
of	O
its	O
signature	O
,	O
sig(A )	O
.	O
</s>
<s>
A	O
signature	O
S	O
describes	O
the	O
I/O	B-Operating_System
automaton	I-Operating_System
's	O
actions	O
using	O
three	O
disjoint	B-Algorithm
sets	I-Algorithm
of	O
actions	O
:	O
</s>
<s>
A	O
transition	O
,	O
also	O
known	O
as	O
a	O
step	O
of	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
,	O
is	O
defined	O
as	O
an	O
element	O
(	O
s	O
,	O
π	O
,	O
s	O
 '	O
)	O
of	O
trans(A )	O
.	O
</s>
<s>
For	O
any	O
state	O
s	O
and	O
action	O
π	O
,	O
if	O
the	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
has	O
some	O
transition	O
of	O
the	O
form	O
(	O
s	O
,	O
π	O
,	O
s	O
 '	O
)	O
,	O
then	O
π	O
is	O
said	O
to	O
be	O
enabled	O
in	O
s	O
.	O
I/O	O
automata	O
are	O
described	O
as	O
input-enabled	O
because	O
all	O
input	O
actions	O
are	O
required	O
to	O
be	O
enabled	O
in	O
every	O
state	O
.	O
</s>
<s>
The	O
fifth	O
component	O
of	O
the	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
,	O
tasks	O
(	O
A	O
)	O
,	O
is	O
a	O
task	O
partition	O
,	O
defined	O
as	O
an	O
equivalence	O
relation	O
on	O
locally	O
controlled	O
actions	O
of	O
A	O
,	O
which	O
has	O
at	O
most	O
countably	O
many	O
equivalence	O
classes	O
.	O
</s>
<s>
As	O
described	O
above	O
,	O
Ci	O
,	O
j	O
is	O
an	O
example	O
of	O
I/O	B-Operating_System
automaton	I-Operating_System
that	O
represents	O
a	O
unidirectional	O
FIFO	B-Operating_System
channel	O
from	O
process	O
Pi	O
to	O
process	O
Pj	O
.	O
</s>
<s>
A	O
reachable	O
state	O
in	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
is	O
the	O
final	O
state	O
of	O
a	O
finite	O
execution	O
of	O
A	O
.	O
</s>
<s>
This	O
sequence	O
is	O
also	O
an	O
execution	O
fragment	O
of	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
.	O
</s>
<s>
A	O
trace	O
of	O
an	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
is	O
the	O
sequence	O
of	O
external	O
actions	O
that	O
occurs	O
in	O
some	O
execution	O
α	O
of	O
A	O
.	O
</s>
<s>
Executions	O
a	O
,	O
b	O
and	O
c	O
are	O
three	O
executions	O
of	O
the	O
automaton	O
Ci	O
,	O
j	O
described	O
in	O
Formal	O
Definition	O
of	O
Channel	O
I/O	B-Operating_System
Automaton	I-Operating_System
(	O
where	O
message	O
m	O
∈	O
{	O
0	O
,	O
1}	O
)	O
.	O
</s>
<s>
Figure	O
4	O
depicts	O
a	O
composition	O
of	O
two	O
processes	O
,	O
Pi	O
and	O
Pj	O
and	O
a	O
FIFO	B-Operating_System
message	O
channel	O
Ci	O
,	O
j	O
,	O
matching	O
output	O
actions	O
of	O
one	O
automaton	O
with	O
identically	O
named	O
input	O
actions	O
of	O
other	O
automata	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
hide	O
the	O
output	O
actions	O
of	O
an	O
I/O	B-Operating_System
automaton	I-Operating_System
through	O
"	O
reclassifying	O
them	O
as	O
internal	O
actions.	O
"	O
</s>
<s>
Recall	O
that	O
a	O
task	O
partition	O
is	O
defined	O
as	O
an	O
equivalence	O
relation	O
on	O
an	O
I/O	B-Operating_System
automaton	I-Operating_System
's	O
locally	O
controlled	O
actions	O
,	O
containing	O
at	O
most	O
countably	O
many	O
equivalence	O
classes	O
.	O
</s>
<s>
In	O
an	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
,	O
let	O
C	O
represent	O
a	O
class	O
of	O
tasks(A )	O
.	O
</s>
<s>
Thus	O
,	O
a	O
finite	O
fair	O
execution	O
with	O
final	O
state	O
Sf	O
can	O
be	O
defined	O
as	O
an	O
execution	O
where	O
the	O
automaton	O
continuously	O
gives	O
turns	O
to	O
all	O
tasks	O
in	O
a	O
round-robin	B-Algorithm
fashion	O
,	O
but	O
no	O
actions	O
are	O
executed	O
as	O
none	O
are	O
enabled	O
in	O
Sf	O
.	O
</s>
<s>
The	O
set	O
of	O
fair	O
executions	O
of	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
is	O
represented	O
by	O
fairexecs(A )	O
.	O
</s>
<s>
I/O	O
automata	O
provide	O
an	O
accurate	O
description	O
of	O
asynchronous	B-Application
systems	I-Application
.	O
</s>
<s>
As	O
internal	O
actions	O
of	O
an	O
I/O	B-Operating_System
automaton	I-Operating_System
are	O
invisible	O
to	O
a	O
user	O
;	O
the	O
I/O	B-Operating_System
automaton	I-Operating_System
appears	O
to	O
be	O
a	O
black	B-Device
box	I-Device
and	O
the	O
user	O
merely	O
sees	O
"	O
traces	O
of	O
automaton	O
's	O
executions	O
(	O
or	O
fair	O
executions	O
)	O
.	O
"	O
</s>
<s>
When	O
it	O
is	O
mentioned	O
that	O
I/O	B-Operating_System
automaton	I-Operating_System
A	O
satisfies	O
a	O
trace	O
property	O
P	O
,	O
(	O
at	O
least	O
)	O
two	O
different	O
meanings	O
might	O
be	O
intended	O
:	O
</s>
<s>
Intuitively	O
,	O
a	O
safety	B-Application
property	I-Application
represents	O
the	O
fact	O
that	O
nothing	O
"	O
bad	O
"	O
happens	O
.	O
</s>
<s>
P	O
is	O
a	O
trace	O
safety	B-Application
property	I-Application
,	O
or	O
safety	B-Application
property	I-Application
,	O
if	O
the	O
following	O
criteria	O
hold	O
for	O
traces(P )	O
.	O
</s>
<s>
Informally	O
,	O
a	O
liveness	B-Application
property	I-Application
can	O
be	O
interpreted	O
as	O
something	O
"	O
good	O
"	O
eventually	O
happens	O
.	O
</s>
<s>
P	O
is	O
a	O
trace	O
liveness	B-Application
property	I-Application
,	O
or	O
a	O
liveness	B-Application
property	I-Application
,	O
"	O
if	O
every	O
finite	O
sequence	O
over	O
acts(P )	O
has	O
some	O
extension	O
in	O
traces(P )	O
.	O
"	O
</s>
