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

314 Behaviour-Preserving Transformations
c
A B
a
b
c
d
a
¢
¡
c
A B
§ L¥ envs¥
Transformation 9
e BSpec f Sysp
¥ f Sys
¢
¡
e BSpec 1 envs¥ f (L)§ Sys ¥ p Sys ¨
¥
9 allows a certain kind of interchanging of channel hiding and channel renaming.
For instance, configuration £
a¥ a© d¥ BSpec§ © b e¥ ¦ e© ¥ c¥ c© f f Sysp ¥ Sys (where
BSpec is defined in the example of 7) is transformation equivalent with £
(BSpec
c¦ )§ e© c¥ c© a¥ a© d¥ f © b ¥ Sys b¥ p Sys . ¥
Transformation 10
e BSpec
if
e
A B
a
c
b
L¥ envs¥ Sys p ¥ Sys
f
c
¢
¡
a
d
a
BSpec e § f
(1) f ¡ ¡ (ch) f (L) for all ¡ ch ChSort(BSpece Sys ¥ p )
¢
¡
a
c
A B
a
c
f (L)¥ envs¥ Sys p ¥ Sys
¡ (2) f (ch) ch for all ch ChSort(BSpece ¥ Sysp L ¨
)
Here, given sets M and L, M L denotes the difference of the sets, i.e. M L = £
¡ ¡
¡ ¡ L¦
x M
x . The notation f(L) is an abbreviation for £
f ¢
¡ L¦ ¡ ch£ ch .
£
b¦ ¥
By 10 it is allowed to perform an internal renaming of channels that can not be
observed by the environment. For example, the unobservable channel b in specification
BSpec Sysp Sys ¥ (where BSpec is given in the example of 7) can be renamed to,
for example, e, resulting in £
BSpec§ e© b e¦ ¥ Sysp ¥ Sys .
L
b
b
c
c
d
a
a
d
a