7.4 Symbolic Formulation
91
m 1
m 2
f 1
f 2
d 1
d 2
M PU
e (M PU , m 1 )
e (m 1 , M PU )
e (m 1 , m 2 )
e
(m 2
,
m 1
)
e (m 1 , f 1 )
e ( f 1 , m 1 )
......
......
......
.........
. . . . . . . . . . . .
. . . . . . . . . . . .
e (M PU , m 1 ) = 1 e (M PU , f 1 ) = 1
e (M PU , f 1 ) = 1 e ( f 1 , d 1 ) = 1
e ( f 1 , m 2 ) = 1
e (d 1 , m 2 ) = 1
e (m 2 , M PU ) = 1 e (m 2 , d 2 ) = 1
e (d 1 , M PU ) = 1 e (d 2 , M PU ) = 1
e (u i , v j ) = 0
for all remaining variables.
m 1
m 2
f 1
d 1
d 2
M PU
Fig. 7.2 Symbolic formulation of all possible architectures
operation sequence given by φ. To this end, new Boolean variables are introduced,
which represent all possibilities how an experiment can be realized.
Definition 7.4 Let φ ∈ be an experiment and φ[p] ∈ O with 1 ≤ p ≤ |φ| be the
pth operation in it (where |φ| is the length of the experiment). Further, let φ[p] i ∈ ˆ
V
with 1 ≤ i ≤ maxI(φ[p]) be the ith instance of the module to be employed
at position p. Then, for all φ[p] i , new Boolean variables ex φ[p] i are introduced.
These variables represent whether (ex φ[p] i = 1) or not (ex φ[p] i = 0) the module at
position p of experiment φ is realized using the ith instance.
Example 7.5 Consider the experiment φ 1 := (f, m, d) defined in Example 7.3.
The newly introduced notations and variables are summarized in Table 7.1, i.e. the
sequence of modules as specified by φ 1 (first row), the corresponding φ[p]-notation
(second row), as well as the accordingly introduced ex φ[p] i -variables (third row). A
possible assignment to these ex φ[p] i -variables is given in the fourth row. This eventually represents that the experiment φ 1 is realized using the instances (f 1 , m 2 , d 2 ).
91
m 1
m 2
f 1
f 2
d 1
d 2
M PU
e (M PU , m 1 )
e (m 1 , M PU )
e (m 1 , m 2 )
e
(m 2
,
m 1
)
e (m 1 , f 1 )
e ( f 1 , m 1 )
......
......
......
.........
. . . . . . . . . . . .
. . . . . . . . . . . .
e (M PU , m 1 ) = 1 e (M PU , f 1 ) = 1
e (M PU , f 1 ) = 1 e ( f 1 , d 1 ) = 1
e ( f 1 , m 2 ) = 1
e (d 1 , m 2 ) = 1
e (m 2 , M PU ) = 1 e (m 2 , d 2 ) = 1
e (d 1 , M PU ) = 1 e (d 2 , M PU ) = 1
e (u i , v j ) = 0
for all remaining variables.
m 1
m 2
f 1
d 1
d 2
M PU
Fig. 7.2 Symbolic formulation of all possible architectures
operation sequence given by φ. To this end, new Boolean variables are introduced,
which represent all possibilities how an experiment can be realized.
Definition 7.4 Let φ ∈ be an experiment and φ[p] ∈ O with 1 ≤ p ≤ |φ| be the
pth operation in it (where |φ| is the length of the experiment). Further, let φ[p] i ∈ ˆ
V
with 1 ≤ i ≤ maxI(φ[p]) be the ith instance of the module to be employed
at position p. Then, for all φ[p] i , new Boolean variables ex φ[p] i are introduced.
These variables represent whether (ex φ[p] i = 1) or not (ex φ[p] i = 0) the module at
position p of experiment φ is realized using the ith instance.
Example 7.5 Consider the experiment φ 1 := (f, m, d) defined in Example 7.3.
The newly introduced notations and variables are summarized in Table 7.1, i.e. the
sequence of modules as specified by φ 1 (first row), the corresponding φ[p]-notation
(second row), as well as the accordingly introduced ex φ[p] i -variables (third row). A
possible assignment to these ex φ[p] i -variables is given in the fourth row. This eventually represents that the experiment φ 1 is realized using the instances (f 1 , m 2 , d 2 ).