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Fig. 2.2
Composition
topology with three
components
For ease of notation, we will omit the alphabet from the symbol of synchronous
and parallel composition, unless unclear from the context. By definition of the
operations
#
V
,
"
V
,
+
V
,
*
V
,
it follows that
;
L
D
L
; D ;
,
;˘
L
D
*
.V;l/
L
˘;D;
,
;˘
l
L
D
L
˘
l
;D;
.
When l
D1
the definition of l -bounded parallel composition reduces to the
definition of parallel composition of languages, because then .I
[
O/
?
*
.U;l/
becomes
[
O
[
U/
?
, that is the universe over I
.I
[
O
[
U , and so it can be dropped from
the conjunction.
These definitions can be easily extended to more components and more complex
interconnection topologies, e.g., to the topology where U is observable externally or
where U is the cartesian product of two alphabets only one of which is observable.
For instance in Fig.
2.2
we show a composition topology with three components:
A, B and C , which together define the composed system:
.A
"
Z
O
\
B
"
I
O
\
C
"
I
U
V
/
#
I
U
O
:
We notice that I and O are external variables, U is an internal variable that
is observable externally, whereas V and Z are internal variables. The compo-
sition is well-formed because the synchronous composition operator is associa-
tive [150]. Each specific topology dictates the alphabets to which projection and
lifting (restriction and expansion) should be applied. In the most straightforward
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