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10.3 Some Properties of Transformation Equivalence 309
By property (ii) of Proposition 3 we know that observation equivalence, which was
defined in terms of weak bisimulations, is itself a weak bisimulation. This property is
required in the proof of Proposition 4 given later in this section.
For transformation equivalence to be of practical use, it should be a (partial) congruence.
If a component of a specification is replaced by a transformation equivalent one, a specification
should be obtained that is transformation equivalent to the original specification.
For example, consider specification ¢ B§ b© c¥ d© a (A ) £
c¦ ¥ Sys b¥ p Sys ¥ 2represented as
A
a
c
and suppose we would like to replace instance B by a different instance C (composed of
D and E) which class is defined in Sysp . The transformed specification ¢ C§ b© c¥ d© a (A £
)
c¦ ¥ Sys b¥ p Sys has the following Instance Structure Diagram:
¥
c
A C a
D E
a
It is then natural to require that B,Sysp
¥
¢
Sys ¢ B§ ¡
b© c¥
d©
a
,Sys p C,Sys implies (A
) £
c¦ b¥ ,Sysp £
¢ C§ b© c¥ d© a Sys
b¥ c¦
¢
¡ ¥ (A ) ,Sysp ¥ Sys .
Proposition 4 guarantees that this requirement indeed holds.
b
b
Proposition 4
Transformation equivalence and observation equivalence are partial congruences.
substitutive under parallel composition, channel hiding, and channel renaming, i.e., if
They are
e BSpec1,envs1,Sysp ¡ ,Sys
¢
( ¢
) BSpece 1 ¡ , envs¡1, Sysp ¡
,Sys
e BSpec2,envs2,Sys and
p ¡ ,Sys
¢
( ¢
) BSpece 2 ¡ ,envs¡2,Sysp ¡ ,Sys then
(1) BSpec e 1 ¢ BSpece 2 ¥ envs1 envs2 ¥ Sys p ¥ Sys
BSpec e 1 ¡ ¢ BSpec e 2 ¡ ¥ envs¡1 envs¡2 ¥ Sysp ¡ ¥ Sys
2 For simplicity we let all transformation examples in this chapter apply to system specifications. It is
however straightforward to change them to apply to configurations too.
c
¢
B
c
¡ ( ¢
)
d
a
d