76
B. Steinbach and C. Posthoff
Fig. 3.4 Karnaugh-map of
the representative function
g re (x) and the independence
matrix IDM(C N (g(x))) that
specify the four Boolean
functions of the calculated
single derivative with regard
to x 2
0 0 1 1
0 0 1 1
0 0 0 0
0 0 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
g
re (x)
1
2
3
4
1 2 3 4
i
j
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 1
IDM(CN (g(x)))
Figure 3.4 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))) which
determine the resulting class C N (g(x)) of the single derivative of the given class
C N (f (x)).
Algorithm 1 found for the given direction of change x 2 (BV (x 2 ) = (0100)) the
vector s min = (0001) of the associated variable x 4 ; hence, the single derivative
of the given class C N (f (x)) with f id = der (x 2 ,x 4 ) f (x) has the same result if it
is calculated with regard to x 2 or with regard to x 4 . All Boolean functions of the
resulting class C N (g(x)) are independent of x 2 , x 4 , and the simultaneous change of
(x 2 , x 4 ):
der
x 2
g(x) = 0,
der
x 4
g(x) = 0,
der
(x 2 ,x 4 )
g(x) = 0;
hence, all four Boolean functions of the resulting class C N (g(x)) contain quadruples
of identical function values. Each pair of these three derivatives specifies this
property. Algorithm 2 selected as unique representation the two single derivatives.
3.3.4 k-Fold Derivative Operations of Classes C N of Boolean
Functions
Repeated derivative operations of the same type with regard to different variables are
summarized to k-fold derivative operations. It is a consequence of Theorem 3.7 that
each k-fold derivative operation of a given class C N
f re (x), f id (x)
results again in
a class C N of Boolean functions.
Theorem 3.8 (k-Fold Derivative Operations of a Class C N of Boolean Functions) Let
f (x) = f (x 0 , x 1 ) = f (x 1 , x 2 , . . . , x n )
be a Boolean function of n variables that belongs to the class C N
f re (x), f id (x)
defined by (3.1), and there is at least one x 0i ∈ x 0 with
der
x 0i
f (x) = 0.
(3.29)
B. Steinbach and C. Posthoff
Fig. 3.4 Karnaugh-map of
the representative function
g re (x) and the independence
matrix IDM(C N (g(x))) that
specify the four Boolean
functions of the calculated
single derivative with regard
to x 2
0 0 1 1
0 0 1 1
0 0 0 0
0 0 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
g
re (x)
1
2
3
4
1 2 3 4
i
j
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 1
IDM(CN (g(x)))
Figure 3.4 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))) which
determine the resulting class C N (g(x)) of the single derivative of the given class
C N (f (x)).
Algorithm 1 found for the given direction of change x 2 (BV (x 2 ) = (0100)) the
vector s min = (0001) of the associated variable x 4 ; hence, the single derivative
of the given class C N (f (x)) with f id = der (x 2 ,x 4 ) f (x) has the same result if it
is calculated with regard to x 2 or with regard to x 4 . All Boolean functions of the
resulting class C N (g(x)) are independent of x 2 , x 4 , and the simultaneous change of
(x 2 , x 4 ):
der
x 2
g(x) = 0,
der
x 4
g(x) = 0,
der
(x 2 ,x 4 )
g(x) = 0;
hence, all four Boolean functions of the resulting class C N (g(x)) contain quadruples
of identical function values. Each pair of these three derivatives specifies this
property. Algorithm 2 selected as unique representation the two single derivatives.
3.3.4 k-Fold Derivative Operations of Classes C N of Boolean
Functions
Repeated derivative operations of the same type with regard to different variables are
summarized to k-fold derivative operations. It is a consequence of Theorem 3.7 that
each k-fold derivative operation of a given class C N
f re (x), f id (x)
results again in
a class C N of Boolean functions.
Theorem 3.8 (k-Fold Derivative Operations of a Class C N of Boolean Functions) Let
f (x) = f (x 0 , x 1 ) = f (x 1 , x 2 , . . . , x n )
be a Boolean function of n variables that belongs to the class C N
f re (x), f id (x)
defined by (3.1), and there is at least one x 0i ∈ x 0 with
der
x 0i
f (x) = 0.
(3.29)