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

328 Behaviour-Preserving Transformations
transformations. The implementation model will in general act as a blue-print for the
actual implementation. The key idea of transformational design is that it supports the
evaluation of many alternative implementation models, in order to select the most suitable
one. The idea is illustrated in Figure 10.5. Each ellipse denotes a model. A dashed
H
T
T
A
E
T
T
I
B
F
J
C
T T
T
G
T T
K L
T T T T
... ... ... ... ...
Sub-space 1 Sub-space 2
Sub-space 3
Figure 10.5: Illustration of Transformational Design
arrow between two models denotes that these models are equivalent with respect to
some formally defined equivalence relation 10 . A solid arrow between two models indicates
the existence of a transformation with which the one model can be transformed
into the other one (and vice versa).
If the transformation process starts off with (abstract) model A then each one of the models
A, E, H, I, ¡ ¡ is a candidate for being selected as implementation model. According
to Proposition 8 of Appendix B 11 , however, there exist always models that are equivalent
to A, but that can not be reached from A by applying the behaviour-preserving
transformations 12 . Examples of such models are B, F, J, C, G, K, L, ¡ ¡ (see Figure 10.5).
Although model L can be a perfect implementation model, it will never be selected since
there exists no sequence of transformations leading from model A to model L. This result
10In stead of an equivalence relations, one could also use a pre-order. In stead of undirected arrows,
one would then use directed arrows.
11Proposition 8 represents a stronger version of the incompleteness result stated in Proposition 6.
12This statement holds if POOSL is used as specification language and if transformation equivalence is
used as equivalence relation. It is however very likely that the same statement holds for any language
that is expressive enough together with any non-trivial equivalence relation or pre-order.