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B. Steinbach and C. Posthoff
x 4 ; hence, the twofold derivative of the C N (f (x)) can be calculated by (3.30)
g
re (x) = der
x 0
k f
re (x) = x 1 x 3 .
The independence matrix IDM(C N (g(x))) must be iteratively calculated for x 2
and x 4 using (3.35):
– the execution of Algorithm 2 (UM) for x 2 includes the binary vector
BV (x 2 ) = (0100)
into the second row of the copied independence matrix IDM(C N (g(x)));
– the execution of Algorithm 2 (UM) for x 4 removes rightmost the value 1 in the
first row (IDM(C N (g(x)))[1, 4] = 1) and includes the binary vector
BV (x 4 ) = (0001)
into the fourth row of the intermediate independence matrix IDM(C N (g(x)));
hence, the resulting independence function is
g
id (x) = der
(x 1 ,x 3 )
g(x) ∨ der
x 2
g(x) ∨ der
x 4
g(x).
This independence function contains one vectorial derivative and two single derivatives; hence, the associated independence matrix contains three values 1 in the main
diagonal, the rank of this independence matrix is
rank(IDM(C N (g(x)))) = 3
so that the class
C N
x 1 x 3 , der
(x 1 ,x 3 )
g(x) ∨ der
x 2
g(x) ∨ der
x 4
g(x)
contains 2 n−rank(IDM(C N (g))) = 2 4−3 = 2 1 = 2 functions g(x) of four (or reduced
only 2) variables. This confirms that due to the two used directions of change for
which given class is depending on the number of the eight given functions has been
reduced by a factor of 2 2 = 4 so that the resulting class contains only the two
functions: C N (g(x)) = {x 1 x 3 , x 1 ⊕ x 3 }. Figure 3.6 shows the Karnaugh-map
of the representative function g re (x) using the original Boolean space B 4 and the
independence matrix IDM(C N (g(x))) that specify the two Boolean functions of the
calculated twofold derivative with regard to (x 2 , x 4 ).