3 Derivative Operations for Classes C N of Boolean Functions
77
Then all k-fold derivatives of f (x) with regard to x 0
g 1 (x) = der
x 0
k f (x)
belong to the class
C N 1
g
re
1 (x), g
id
1 (x)
with the mark function of the k-fold derivative of the class C N with regard to x 0
g
re
1 (x) = der
x 0
k f
re (x),
(3.30)
and the independence function g id
1 (x) that is associated with IDM(g 1 ) satisfies
∀x 0i ∈ x 0 :
MIDC(IDM(g 1 ), x 0i ) = 0;
all k-fold minima of f (x) with regard to x 0
g 2 (x) = min
x 0
k f (x)
belong to the class
C N 2
g
re
2 (x), g
id
2 (x)
with the mark function of the k-fold minimum of the class C N with regard to x 0
g
re
2 (x) = min
x 0
k f
re (x),
(3.31)
and the independence function g id
2 (x) that is associated with IDM(g 2 ) satisfies
∀x 0i ∈ x 0 :
MIDC(IDM(g 2 ), x 0i ) = 0;
all k-fold maxima of f (x) with regard to x 0
g 3 (x) = max
x 0
k f (x)
belong to the class
C N 3
g
re
3 (x), g
id
3 (x)