64
B. Steinbach and C. Posthoff
The Shannon decomposition of (3.8) with regard to x 01 = (x 02 , x 03 , . . . , x 0k ) results
in
der
x 0
f (x 0 , c 1 ) =
x 02 x 03 . . . x 0k der
x 0
f (x 01 ⊕ c 01 , 0, 0, . . . , 0, c 1 )∨
x 02 x 03 . . . x 0k der
x 0
f (x 01 ⊕ c 01 , 1, 0, . . . , 0, c 1 ) ∨ · · · ∨
x 02 x 03 . . . x 0k der
x 0
f (x 01 ⊕ c 01 , 1, 1, . . . , 1, c 1 ) = 0;
hence, it holds ∀x 01 = c 01 :
f (x 01 ⊕ c 01 , c 01 , c 1 ) ⊕ f (x 01 ⊕ c 01 , c 01 , c 1 ) = 0,
and we get for c 01 = 0:
f (x 01 , c 01 , c 1 ) = f (x 01 , c 01 , c 1 ).
The exchange of both sides of this equation leads to
f (x 01 , c 01 , c 1 ) = f (x 01 , c 01 , c 1 ),
so that (3.9) also holds for c 01 = 1 and ∀x 01 = c 01 which completes the proof.
A class C N can be independent of several directions of change. These directions
of change are uniquely specified by the independence matrix IDM(C N ) and can
alternatively expressed by a disjunction of the associated vectorial derivatives. For
a short notation we define the independence function f id (x).
Definition 3.5 (Independence Function) The independence function f id (x) of a
class C N of Boolean functions of n variables corresponds to the independence matrix
IDM(C N ) such that
f
id (x) =
n
i=1
der
x 0i
f (x),
where
der
x 0i
f (x) = 0
if all elements of the row i in IDM(C N ) are equal to 0, and
x j ∈ x 0i if IDM(C N )[ i, j ] = 1.