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B. Steinbach and C. Posthoff
conditional ⊕-operations. The new vector s min is included into the independence
matrix IDM(g) in line 12.
3.3.2 Vectorial Derivative Operations of Classes C N of Boolean
Functions
It follows from the definitions of both the vectorial derivative operations and the
class C N that all vectorial derivative operations of such a class of Boolean functions
f i (x) ∈ C N
f
re (x), f
id (x)
with regard to x 0 result again in a class C N of Boolean functions where all
functions are independent of the simultaneous change of x 0 . Based on the explored
background, the associated mark functions g re (x), and g id (x) can be calculated
using Theorem 3.6.
Theorem 3.6 (Vectorial 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), where f re (x) depends on the simultaneous change of the values of
all variables of x 0 :
MIDC(IDM(f
re (x)), x 0 ) > 0.
(3.13)
Then all vectorial derivatives of f (x) with regard to x 0
g 1 (x) = der
x 0
f (x)
belong to the class
C N 1
g
re
1 (x), g
id
1 (x)
with the mark function of the vectorial derivative of the class C N with regard to x 0
g
re
1 (x) = der
x 0
f
re (x),
(3.14)
and the independence function g id
1 (x) associated with
IDM(g 1 ) = UM(IDM(f
re (x)), x 0 );