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B. Steinbach and C. Posthoff
2. due to the independence of the class C N of certain directions of change the
vectorial derivative operation with regard to several sets x 0 has the same result
then:
– the vectorial derivative operation of f re (x) with regard to x 0 must be
calculated;
– a unique direction of change for the new vectorial derivative operation must
be determined; and
– the independence matrix IDM(C N ) of the resulting class C N must be adjusted.
The binary vector s has been introduced to support the required manipulations of
the directions of change.
Definition 3.6 (Binary Vector (BV)) Let
f (x) = f (x 1 , x 2 , . . . , x n )
be a Boolean function and
x 0 ⊆ x
be a subset of variables, then
s 0 = BV (x 0 )
(3.11)
is a binary vector of n elements, where s 0 [ i ] = 1 indicates that x i ∈ x 0 .
The unique minimal direction of change for a vectorial derivative operation of a
class C N can be determined using Algorithm 1 that has been used in [8] to determine
the minimal direction of change for such derivative operation of a lattice.
Algorithm 1 s min = MIDC(IDM(f ), x 0 )
Minimal independent direction of change
Input : x 0 ⊆ x: evaluated subset of variables,
Input : IDM(f ): unique independence matrix of n rows and n columns of f (x)
Output : s min : minimal direction of change
1: j ← 1
2: s min ← BV (x 0 )
3: while j ≤ n do
4:
if (s min [ j ] = 1) ∧ (IDM(f )[ j, j ] = 1) then
5:
s min ← s min ⊕ IDM(f )[ j ]
6:
end if
7:
j ← j + 1
8: end while