Specification of Reactive Hardware/Software Systems - Electronic ...

284 Process Part of POOSL
relations. Especially in the context of behaviour-preserving transformations, this is a
very pleasant property. Another appealing property of observation equivalence is that
it benefits from the simple proof technique of finding so-called bisimulations.
A bisimulation establishes a kind of invariant holding between a pair of dynamic systems,
and the technique is to prove two systems equivalent by establishing such an invariant.
Bisimulations are particularly useful if the semantics is given in an operational way (as
in the case of POOSL). The notion of bisimulation was first introduced by Park in [Par81]
and later developed in the context of CCS in [Mil83].
We start defining observation equivalence in terms of weak bisimulations. We then slightly
refine observation equivalence to obtain transformation equivalence.
A binary relation over configurations
(conf p
1
¥ conf p
2 ) implies
(i) the Sys parts of conf p
1 and conf p
2 are syntactically identical
and for all a ¡ Act
(ii) if conf p
1
(iii) if conf p
2
Here,
£
a
conf p
1 ¡ then, for some conf p
2 ¡ , conf p
2
a
conf £ p
2 ¡ then, for some conf p
1 ¡ , conf p
1
¥ Conf p Conf p is a weak bisimulation if
¡
¤
a
conf p
2 ¡ and (conf p
1 ¡ conf ¥ p
2 ¡ ¡ )
¡
a
conf ¤ p
1 ¡ and (conf p
1 ¡ conf ¥ p
2 ¡ ¡ )
¡
¤
a
is a binary transition relation over configurations (often called descendant
relation). Let conf p
1 , conf p
2 be configurations and let l be a communication action. Then
conf p
1
conf p
1
¡
¤
l
2
conf p
2 if conf p
¡
conf ¤ p
p
if conf1 ( £ ) conf p
2
1 £ ( )
l
£ (
£ ) conf p
2
Our notion of weak bisimulation is very similar to that of CCS [Mil80]. The difference is
that CCS does not include (i) in its definition. We, however, need to incorporate (i) to insure
that every data class name referred to by any send or receive action, unambiguously
denotes a single class.
Observation equivalence ( ¢
) on configurations is defined in terms of weak bisimulations.
Configurations conf p
1 and conf p
2 are observation equivalent, written conf p
1
conf p
2 ) ¡
for some weak bisimulation
. So
¢
.
conf p
2 , if (conf p
1 ,
£
¡ is a weak bisimulation ¦
¢
¨
Note that since every POOSL specification is also a configuration, weak bisimulations
and observation equivalence are defined on specifications too.
Although observation equivalence reflects a very strong notion of equivalence, it is
branching equivalence [vGW96].