Hardware Reference
In-Depth Information
Fig. 3.3
General topology
types of equations over FSMs:
M
A
M
X
M
C
and
M
A
˘
M
X
M
C
;
and solve them by building first the related language equations
L.M
A
/
L.M
X
/
L.M
C
/
and
L.M
A
/
˘
L.M
X
/
L.M
C
/
[
.IO/
?
;
where L.M
A
/ and L.M
C
/ are the FSM languages associated with FSMs M
A
and
M
C
. The latter language equation is justified by the following chain of equivalences
M
A
˘
M
X
M
C
,
L.M
A
˘
M
X
/
L.M
C
/
,
by
Def. 3.2.2
L.M
A
/
˘
L.M
X
/
\
.IO/
?
L.M
C
/
,
L.M
A
/
˘
L.M
X
/
L.M
C
/
[
.IO/
?
:
The last equivalence uses the set-theoretic equality A
\
B
C
,
A
C
C
B.
8
W
hen th
ere is no ambiguity we will denote by A
X
C and A
˘
X
C
[
.IO/
?
the lang
uage eq
uations L.M
A
/
L.M
X
/
L.M
C
/ and L.M
A
/
˘
L.M
X
/
L.M
C
/
[
.IO/
?
,whereL.M
A
/, L.M
X
/ and L.M
C
/ are, respectively,
the
-languages and
[
-languages associated with the FSMs M
A
, M
X
and M
C
.
8
In one direction, A
\
B
C
)
A
\
B
C
A
\
B
C
C
A
\
B
)
A
C
C
A
\
B
)
A
C
C
B.
C
B
)
A
\
B
C
A
\
B
C
B
)
A
\
B
In the oth
er
direction, A
C
C
C , because
A
\
B
6
B if A
\
B
¤;
(if A
\
B
D;
then A
\
B
C ).
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