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a
b
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/v
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m
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/o
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/o
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1
Fig. 3.2
Illustration of parallel composition M
A
˘
M
B
D
M
A
˘
B
of Example
3.14
-a. (
a
) FSM M
A
;
(
b
) FSM M
B
;(
c
) Automaton of A (
[
-language of M
A
); (
d
) Automaton of B (
[
-language of M
B
/;
(
e
) Automaton of B
*
I
1
[
O
1
;(
f
) FSM M
A
˘
M
B
D
M
A
˘
B
;(
g
) Reduced FSM M
A
˘
M
B
D
M
A
˘
B
.
Double-circled
nodes represent accepting states in (c), (d), (e). In pictures of FSMs, all nodes are
accepting
output signals V and O
2
. The network implements a specification M
C
with input
signals I
1
;I
2
and output signals O
1
;O
2
. Supposing that M
A
and M
C
are known and
M
X
is unknown, we want to define an equation of the type M
A
ˇ
M
X
M
C
,to
capture the FSMs M
B
that in place of M
X
let the network of M
A
and M
B
match the
specification M
C
. Through Definitions
3.1.7
and
3.1.8
we have seen two different
ways to associate an FSM language with a given FSM, and related composition
operators
and
˘
have been introduced in Sect.
3.2
; therefore we introduce two
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