The material can be found in the book by Robert [2] and Shih and Dong in [3].

2 Definition and notations

In this section, we will state some definitions and notations about the discrete derivative F ⁰ (x) and the digrath Γ(F ⁰ (x)) when the Boolean function F from {0, 1} ⁿ to {0, 1} ⁿ is given.

The material can be found in the book by Robert [2] and Shih and Dong in [3].

Let {0, 1} be a set with three operations +, ·,¯ defined as follows:

0 + 0 = 0 · 1 = 1 · 0 = 0 · 0 = 0, 1 + 0 = 0 + 1 = 1 + 1 = 1 · 1 = 1,

¯1 = 0, and ¯0 = 1.

For a, b in {0, 1}, we usually repress the dot” · ” of a · b and simply write ab. For each positive integer n, let {0, 1} ⁿ be the set of ordered n-tuples,

x =



 

 

  x ₁

...

x _n



 

 

 

with components x _i ∈ {0, 1} (i = 1, · · · , n). We can consider x as a bit string of length n. On the other hand, we may write x = x ₁ x ₂ · · · x _n . The zero element of {0, 1} ⁿ is the point 0, all of whose components are 0. As usual, let

e 1 =



 

 

 

 

  1 0 ...

0



 

 

 

 

 

, . . . , e n =



 

 

 

 

  0 0 ...

1



 

 

 

 

  .

The order ”≤” in {0, 1} is given by 0 ≤ 0 ≤ 1 ≤ 1. Thus for a, b ∈ {0, 1},

a + b := max{a, b}, ab := min{a, b}.

For x, y ∈ {0, 1} ⁿ , x ≤ y is meant that x _i ≤ y _i (i = 1, · · · , n). For x, y ∈ {0, 1} ⁿ and λ ∈ {0, 1},

define

x + y :=



 

 

 

max{x ₁ , y ₁ } ...

max{x n , y n }



 

 

 

, λx :=



 

 

 

min{λ, x ₁ } ...

min{λ, x n }



 

 

  .

A Boolean matrix is meant to be a matrix over {0, 1}. Boolean matrix addition and Boolean matrix multiplication are the same as in the case of complex matrices but the concerned sums and products of entries are Boolean. Let F : {0, 1} ⁿ → {0, 1} ⁿ be a Boolean function and write

F :=



 

 

  f ₁

...

f _n



 

 

  .

According to Robert [2, p.7], the incidence matrix of F is the n × n Boolean matrix defined by

B(F ) = (b _ij ),

where b _ij := 0 if f _i does not depend on x _j , b _ij := 1 otherwise. More precisely, b _ij = 0 if for any fixed x ₁ , · · · , x _j−1 , x _j+1 , · · · , x _n , f _i (x ₁ , · · · , x _j−1 , x _j+1 , · · · , x _n ) is a constant function of x _j on {0, 1}, b _ij = 1 otherwise.

For x ∈ {0, 1} ⁿ , let

˜ x ^j :=



 

 

 

 

 

 

  x ₁

...

¯ x j

...

x _n



 

 

 

 

 

 

  .

The notation ˜ x ^j _i is the ith component of ˜ x ^j , that is, ˜ x ^j _i = ¯ x i if i = j; ˜ x ^j _i = x i if i 6= j,

and further, we set ˜ x ⁰ = x. Since we can view {0, 1} ⁿ as the vertices of the n-cube, ˜ x ^j may

neighborhood of x [1, p.17] is defined to be the set V _x of vertices of the n-cube formed by x and its n neighbors, that is,

V _x := {x = ˜ x ⁰ , ˜ x ¹ , · · · , ˜ x ⁿ }.

For x ∈ {0, 1} ⁿ and {j 1 , · · · , j k } ⊂ {1, · · · , n}, let us define ˜ x ^j

^,···,j

= y by

y _i :=

 

 

 



x _i if i 6= j ₁ , · · · , j _k ,

˜

x _i if i = j ₁ , · · · , j _k .

The discrete derivative(or the Jacobian Boolean matrix ) of F at x ∈ {0, 1} ⁿ is the Boolean n × n matrix defined by

F ⁰ (x) = (f _ij (x)),

where f _ij (x) := 1 if f _i (x) 6= f _i (˜ x ^j ), f _ij := 0 otherwise. In particular, to say that F ⁰ (x) = 0 (Boolean zero matrix), is the same as to say that F (y) is constant at all points y in the von Neumann neighborhood of x. This is to say that F (y) = F (x) for all y ∈ V _x .

Let A always denote an n × n Boolean matrix. A nonzero element u ∈ {0, 1} ⁿ is called a Boolean eigenvector of A if there exists an λ in {0, 1} such that Au = λu; this λ is called the Boolean eigenvalue associated with the eigenvector u. The symbol σ(A) stands for the set of all Boolean eigenvalues of A, so that σ(A) ⊂ {0, 1}. The Boolean spectral radius of A, denoted by ρ(A), is defined to be the largest Boolean eigenvalues of A. Because of σ(A) 6= ∅ (this fact is not a priori obvious, see [2, p. 48]), we have ρ(A) = 0 or 1. Also ρ(P ^t AP ) = ρ(A) for any permutation matrix P .

The discrete metric on {0, 1} is denoted by δ, that is, δ(x, y) = 1 if x 6= y, δ(x, y) = 0 if x = y. For x, y ∈ {0, 1} ⁿ , the discrete metric δ on {0, 1} induces a metric ρ _H (·, ·) on {0, 1} ⁿ , called the Hamming metric, defined by

ρ _H (x, y) :=

X n i=1

δ(x _i , y _i ),

in other words, ρ _H (x, y) = ]{i; 1 ≤ i ≤ n, x _i 6= y _i }. The Hamming metric ρ _H (·, ·) satisfies:

(i) ρ H (x, y) = ρ H (y, x)(x, y ∈ {0, 1} ⁿ ), (ii) ρ _H (x, y) = 0 ⇔ x = y(x, y ∈ {0, 1} ⁿ ),

(iii) ρ _H (x, y) ≤ ρ _H (x, z) + ρ _H (z, y)(x, y, z ∈ {0, 1} ⁿ ),

For x, y ∈ {0, 1} ⁿ , the Boolean vector distance between x and y, denoted by d(x, y), is defined by

d(x, y) :=



 

 

 

δ(x ₁ , y ₁ ) ...

δ(x _n , y _n )



 

 

  .

Then the Boolean vector distance d also satisfies:

(i) d(x, y) = d(y, x)(x, y ∈ {0, 1} ⁿ ), (ii) d(x, y) = 0 ⇔ x = y(x, y ∈ {0, 1} ⁿ ),

(iii) d(x, y) ≤ d(x, z) + d(z, y)(x, y, z ∈ {0, 1} ⁿ ),

where d(x, z) + d(z, y) is the Boolean sun in {0, 1} ⁿ . Notice that the Boolean vector distance is not a metric, since d(x, y) is not a real valued function.

Let us recall some elementary graph-theoretic notations. The digraph (directed graph) of an n × n Boolean matrix A = (a _ij ), denoted by Γ(A), is the digraph with n nodes P ₁ , · · · , P _n such that there is a directed arc from P _i to P _j if a _ij = 1. A directed path in Γ(A) is a sequence of directed arcs P _i

P _i

, P _i

P _i

, . . . in Γ(A). A cycle in Γ(A) is a directed path that begins and ends at the same node.

Let F is a Boolean mapping from {0, 1} ⁿ to {0, 1} ⁿ . The iteration graph for F is the graph consisting of vertices which are elements of {0, 1} ⁿ and the following arcs: for all x in {0, 1} ⁿ , an arc connects x to F (x). Since {0, 1} ⁿ is finite, it is clear that the iteration defined by x ^r+1 = F (x ^r ) (r = 0, 1, 2, . . .) has only two possible modes of behavior.

It is possible that the sequence generated by this iteration will remain stationary at an

Clearly it follows immediately that ξ is a fixed point for F since F (ξ) = ξ.

If the sequence does not converge in the above case, then it follows that it must repeat itself after a certain number of steps since {0, 1} ⁿ is finite. The element that are repeated in the iteration sequence constitute a cycle {ξ 1 , ξ 2 , . . . , ξ p } and this is defined by

ξ ₂ = F (ξ ₁ ) ...

ξ _p = F (x _p−1 ) ξ ₁ = F (ξ _p ),

where ξ ₁ , . . . , ξ _p are pairwise distinct. This cycle is said to have length p and is should be noted that a fixed point is nothing but a cycle of length 1.

The iteration graph for F is said to be simple if the iteration graph has only one basin and this basin has a unique fixed point for F .

Let ξ = F (ξ) be a fixed point of F in {0, 1} ⁿ . ξ is called an attractor in its von Neumann neighborhood if the following two conditions are valid:

(i) F (V _ξ ) ⊂ V _ξ .

(ii) For all x ⁰ from V ξ , the iteration x ^r+1 = F (x ^r ) (which remains in V ξ according to (i)) reaches ξ in at most n steps (x ⁿ = F ⁿ (x ⁰ ) = ξ).

We say that F is simple if F has a unique fixed point ξ and ξ is a global attractor; i.e.

there exists a positive integer p(≤ 2 ⁿ ) such that F ^p (x ⁰ ) = x ^p = ξ for any initial x ⁰ in {0, 1} ⁿ .

In other words, F is simple if there exists ξ in {0, 1} ⁿ such that for any initial x ⁰ in {0, 1} ⁿ ,

there is a positive integer p(x ⁰ )(≤ 2 ⁿ ) so that F ^p(x

⁾ (x ⁰ ) = ξ.