3 Derivative Operations for Classes C N of Boolean Functions
73
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 change of the value of the variable
x i :
MIDC(IDM(f
re (x)), x i ) > 0.
(3.21)
Then all single derivatives of f (x) with regard to x i
g 1 (x) = der
x i
f (x)
belong to the class
C N 1
g
re
1 (x), g
id
1 (x)
with the mark function of the single derivative of the class C N with regard to x i
g
re
1 (x) = der
x i
f
re (x),
(3.22)
and the independence function g id
1 (x) associated with
IDM(g 1 ) = UM(IDM(f
re (x)), x i );
all single minima of f (x) with regard to x i
g 2 (x) = min
x i
f (x)
belong to the class
C N 2
g
re
2 (x), g
id
2 (x)
with the mark function of the single minimum of the class C N with regard to x i
g
re
2 (x) = min
x i
f
re (x),
(3.23)
and the independence function g id
2 (x) associated with
IDM(g 2 ) = UM(IDM(f
re (x)), x i );
and all single maxima of f (x) with regard to x i
g 3 (x) = max
x i
f (x)