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

282 Process Part of POOSL
Rules (s’) and (t’) state that if one of the constituents of a parallel composition is able
to perform an action, the parallel composition itself can also perform that action. By
rule (u’) the constituents of a parallel composition may synchronise on complementary
communication actions. Together rules (s’), (t’) and (u’) determine that POOSL
support asynchronous interleaved concurrency with synchronous communication. The
occurrence of a synchronisation is not observable by the environment. As a consequence
of this, each communication requires precisely two communication partners
(communication is pair-wise). Notice that the applicability of rule (u’) does not imply
the necessity of communication. If rule (u’) is applicable because two extended behaviour
specifications are able to synchronise on complementary actions, then rules (s’)
and (t’) allow their parallel composition to perform these actions in isolation. This leaves
open the possibility to communicate with possible third parties. This possibility can be
ruled out by the use of channel hiding. If the channel of the complementary actions
is hidden in L, then rule (v’) does not allow these actions to be performed in isolation.
Notice that this rule does not forbid the complementary action to synchronise. After all,
the synchronisation of actions results in a silent ¡ action.
9.5.4 Some Properties of the Transition System
In this subsection a number of important properties of the labeled transition system of
Subsection 9.5.3 are stated.
Proposition 2
Let conf p
1 ¡ Conf p and assume that conf p
1
(i) conf p
2
(ii) Reset(conf p
p
1 ) = Reset(conf2 ).
¡ Conf p
(iii) if a ¡ then Abs(a) ¡ AASort(conf p
1 )
(iv) if a ¡ then Chan(a) ¡ ChSort(conf p
1 )
(v) AASort(conf p
2 ) AASort(conf p
1 )
(vi) ChSort(conf p
2 ) ChSort(conf p
1 )
a
conf £ p
2 for some action ¡ a Act. Then we have
The proof is given in Appendix ¨
B.
Item (i) of Proposition 2 states that every labeled relation a
£ is properly defined on
Conf p . Note that, since the definition rules of Conf p are conditional, this is by no means
a trivial property. Item (ii) states that the Reset of a configuration is equal to the Reset of
any of its derivatives. This property is required in the proof of property (i). From (iii)
and (iv) it follows that the abstraction (Abs) of any non-silent action performed by some
configuration is part of the AASort of that configuration, and that the corresponding
channel is part of the configurations’ channel sort. Properties (v) and (vi) state that
the AASort and ChSort of a configuration equal the AASort respectively ChSort of any
derived configuration. Properties (ii), (iii), (iv) and (v) are required in the proofs of
different behaviour-preserving transformations given in Chapter 10.