62
B. Steinbach and C. Posthoff
In Example 3.6 we noticed that the two functions f 6 (x 1 , x 2 ) and f 9 (x 1 , x 2 ) of
the class C N 4 are independent of the simultaneous change of the variables x 1 and
x 2 , i.e., the vectorial derivatives of these two functions with regard to (x 1 , x 2 ) are
equal to 0. This is not only a special property of the class C N 4 , but the independence
of a certain direction of change satisfies either all functions or no function of a class
C N .
Theorem 3.3 A class C N is uniquely determined by (3.1) when only coefficients
c i are used for which the value 0 occurs in row i of the main diagonal of the
independence matrix IDM(C N ).
Proof A value 0 in the row i of the main diagonal of the independence matrix
IDM(C N ) indicates that the functions of the class C N depend on x i ; hence, the
associated coefficients c i must be used to generate all functions of the class C N .
A value 1 in the row i of the main diagonal of the independence matrix IDM(C N )
indicates that
– either the functions of the class C N are independent of x i ;
– or the functions of the class C N are independent of the simultaneous change of
all variables of the set x 0 with x i ∈ x 0 ;
in both cases all functions of the class C N can be generated using either the
representative function f re (x i = 0, x 1 ) or f re (x i = 1, x 1 ) so that the associated
coefficients c i can be omitted.
The independence matrix of the class C N implicitly determines also the number of directions of change all functions of this class are independent of. The
rank(IDM(C N )) helps to specify the number of functions of a class C N .
Definition 3.4 (Rank) The rank of an independence matrix IDM(C N ) describes
the number of independent directions of change all Boolean functions f (x) of the
class C N do not depend on. The rank(IDM(C N )) is equal to the number of elements
1 in the main diagonal of the unique echelon shape of IDM(C N ).
Theorem 3.4 The number n f (C N ) of Boolean functions f (x 1 , x 2 , . . . , x n ) of the
class C N is equal to
n f (C N ) = 2
n−rank(IDM(C N )) .
(3.5)
Proof If the Boolean functions f (x 1 , x 2 , . . . , x n ) of the class C N depend on all n
variables, then 2 n Boolean functions can be generated by the n coefficients c i . In this
case we have an empty independence matrix IDM(C N ), the rank(IDM(C N )) = 0,
and we get
n f (C N ) = 2
n−0
= 2
n .
B. Steinbach and C. Posthoff
In Example 3.6 we noticed that the two functions f 6 (x 1 , x 2 ) and f 9 (x 1 , x 2 ) of
the class C N 4 are independent of the simultaneous change of the variables x 1 and
x 2 , i.e., the vectorial derivatives of these two functions with regard to (x 1 , x 2 ) are
equal to 0. This is not only a special property of the class C N 4 , but the independence
of a certain direction of change satisfies either all functions or no function of a class
C N .
Theorem 3.3 A class C N is uniquely determined by (3.1) when only coefficients
c i are used for which the value 0 occurs in row i of the main diagonal of the
independence matrix IDM(C N ).
Proof A value 0 in the row i of the main diagonal of the independence matrix
IDM(C N ) indicates that the functions of the class C N depend on x i ; hence, the
associated coefficients c i must be used to generate all functions of the class C N .
A value 1 in the row i of the main diagonal of the independence matrix IDM(C N )
indicates that
– either the functions of the class C N are independent of x i ;
– or the functions of the class C N are independent of the simultaneous change of
all variables of the set x 0 with x i ∈ x 0 ;
in both cases all functions of the class C N can be generated using either the
representative function f re (x i = 0, x 1 ) or f re (x i = 1, x 1 ) so that the associated
coefficients c i can be omitted.
The independence matrix of the class C N implicitly determines also the number of directions of change all functions of this class are independent of. The
rank(IDM(C N )) helps to specify the number of functions of a class C N .
Definition 3.4 (Rank) The rank of an independence matrix IDM(C N ) describes
the number of independent directions of change all Boolean functions f (x) of the
class C N do not depend on. The rank(IDM(C N )) is equal to the number of elements
1 in the main diagonal of the unique echelon shape of IDM(C N ).
Theorem 3.4 The number n f (C N ) of Boolean functions f (x 1 , x 2 , . . . , x n ) of the
class C N is equal to
n f (C N ) = 2
n−rank(IDM(C N )) .
(3.5)
Proof If the Boolean functions f (x 1 , x 2 , . . . , x n ) of the class C N depend on all n
variables, then 2 n Boolean functions can be generated by the n coefficients c i . In this
case we have an empty independence matrix IDM(C N ), the rank(IDM(C N )) = 0,
and we get
n f (C N ) = 2
n−0
= 2
n .