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where
σ
A
denotes the value assignment obtained from
σ
by restricting its
domain to the set
A
,and
update
(
σ
r
,σ
)
|
∈
r
→
V
is the value assignment to
registers which is defined by:
update
(
σ
r
,σ
)
r
=
σ
(
r
)if
r
∈
dom
σ
σ
r
(
r
)otherwise,
for
r
∈
r
.
5
Composition of Modules
We shall now describe how modules can be combined in parallel (thereby forming
new modules) using single assignment programs to connect their ports.
5.1
Abstract Syntax for Composition of Modules
Assume that a composition
M
of modules is given by:
-
sets of input ports
i
, output ports
o
, signals
s
,andregisters
r
,
-
a single assignment program
p
r
)
∼
Expr
,and
-
a finite collection of modules
m
1
,... m
k
, where module
m
j
,for1
∈
(
o
∪
s
∪
−→
≤
j
≤
k
,
is given by
m
j
=(
i
j
, o
j
, Conf
j
, conf
0
j
,p
j
, out
j
, next
j
).
In order to express the well-formedness of
M
, we associate the
trivial depen-
dency graph
graph(
m
j
)=
triv
j
with every module
m
j
,where(
i, o
)
∈
triv
j
,for
i
j
and
o
every
i
o
j
, i.e. we consider a module a black-box and assume as
little as possible, i.e. that an output port depends on every input port.
The
composition M is well-formed
, if the following conditions hold:
∈
∈
-
no output port of
m
j
is an assigned variable of
p
, i.e.
o
j
∩
dom
p
=
∅
,for
k
,
-
the modules
m
1
,...,m
k
have disjoint sets of output ports, i.e.
o
j
∩
1
≤
j
≤
o
l
=
∅
,
=
l
,and
-
the dependency graph of
M
, denoted graph(
M
)=
for
i
≺
p
∪
j
=1
graph(
m
j
), is
acyclic, has
i
only as its sources, and
o
is included in the nodes of the graph.
These conditions are sucient to exclude what often is called combinatorial
loops.
5.2
Semantics for Composition of Modules
The semantics for
M
is defined by (a kind) of product automata construction.
Itisthemodule(
i, o, Conf
M
, conf
0
M
,p
M
, out
M
, next
M
), where
-
Conf
M
=
Conf
1
×···×
Conf
k
×
(
r
→
V
),
-
conf
0
M
=(
conf
01
,...,conf
0
k
,r
0
),
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