A New Calculation for Boolean Derivative Using Cheng Product

Abstract

The matrix expression and relationships among several definitions of Boolean derivatives are given by using the Cheng product. We introduce several definitions of Boolean derivatives. By using the Cheng product, the matrix expressions of Boolean derivative are given, respectively. Furthermore, the relationships among different definitions are presented. The logical calculation is converted into matrix product. This helps to extend the application of Boolean derivative. At last, an example is given to illustrate the main results.

1. Introduction

Boolean algebra was first proposed by George Bole in 1847. It is basic to many aspects of computing. Boolean calculus was first put forward by Daniell in [1]. From then on, great interest has been drawn, and many papers have come out; see [2–5] and references therein. Some forty years after the Boolean calculus was proposed, Boolean derivative was introduced in [6, 7] and applied to switching theory. From then on, Boolean derivative has developed quickly and has been applied in many fields; see [4, 8–13] and references therein. The application of Boolean derivative contains analysis of random Boolean networks [10], design of discrete event systems [11], and cellular automata [12]. Especially, Boolean derivative is widely used in damage spreading and Lyapunov exponents [13] in cellular automata.

Recently, the semitensor product or Cheng product of matrices is proposed and it is successfully applied to express and analyze the Boolean networks. The biggest advantage using this new tool is that it can convert a logical system into a classic discrete dynamical system [14]. After the conversation, the calculation can be easily achieved by the Matlab toolbox. However, the logical calculation is hard to realize before this new tool comes out. It has been applied to present many properties of Boolean networks, such as the controllability and observability [15, 16], stabilization [17], and realization of Boolean networks [18].

There are several definitions of Boolean derivatives. In this paper, the definitions can be found in [19–21].

The Cheng product is the main tool in the present paper. We considered several definitions of Boolean derivatives. By using the Cheng product, formulas for calculating the Boolean derivatives are provided, and the relationships among different definitions of Boolean derivatives are given.

The rest of the paper is organized as follows. Section 2 introduces some fundamental definitions and some notations used in the paper. In Section 3, several definitions of Boolean derivatives are given, and the relationships among them are presented by using the Cheng product. In Section 4, an example is given to illustrate the main results.

2. Preliminary

The semitensor product, that is, Cheng product is the crucial tool in the present paper. The matrix product is assumed to be the semitensor product in the following discussion. In most cases, the symbol ⋉ is omitted. A review of basic concepts and notations will be given [14] as follows.(1)𝒟₂∶={0,1}. (2)Δ_𝑛∶={𝛿¹_𝑛,…,𝛿^𝑛_𝑛}, where 𝛿^𝑘_𝑛 denotes the 𝑘th column of the identity matrix 𝐼_𝑛.(3)Let ℳ_𝑛×𝑠 denote the set of 𝑛×𝑠 matrices. Assume a matrix 𝑀=[𝛿^𝑗1_𝑛,𝛿^𝑗2_𝑛,…,𝛿^𝑗𝑠_𝑛]∈ℳ_𝑛×𝑠, that is, its columns, Col(𝑀)⊂Δ_𝑛. 𝑀 is called a logical matrix. We denote it as 𝑀=𝛿_𝑛[𝑗₁,𝑗₂,…,𝑗_𝑠] for simplification. The set of 𝑛×𝑚 logical matrices is denoted by ℒ_𝑛×𝑚.(4)Let 𝑋 be a row vector of dimension 𝑛𝑝, and let 𝑌=[𝑦₁,𝑦₂,…,𝑦_𝑝]^𝑇 be a column vector of dimension 𝑝. Then, we split 𝑋 into 𝑝 equal-size blocks as 𝑋¹,…,𝑋^𝑝, which are 1×𝑛 rows. Define the semitensor product, denoted by ⋉, as 𝑋⋉𝑌=^𝑝_𝑖=1𝑋^𝑖𝑦_𝑖∈𝑅^𝑛,𝑌^𝑇⋉𝑋^𝑇=^𝑝_𝑖=1𝑦_𝑖𝑋^𝑖𝑇∈𝑅^𝑛. (2.1)(5)Let 𝑀∈ℳ_𝑚×𝑛 and 𝑁∈ℳ_𝑝×𝑞. If 𝑛 is a factor of 𝑝 or 𝑝 is a factor of 𝑛, then 𝐶=𝑀⋉𝑁 is called the semitensor product of 𝑀 and 𝑁, where 𝐶 consists of 𝑚×𝑞 blocks as 𝐶=(𝐶_𝑖𝑗), and 𝐶_𝑖𝑗=𝑀^𝑖⋉𝑁_𝑗,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑞, (2.2) where 𝑀^𝑖=Row_𝑖(𝑀) denotes the 𝑖th row of the matrix 𝑀, and 𝑁_𝑗=Col_𝑗(𝑁) denotes the 𝑗th column of the matrix 𝑁.(6)To use a matrix expression, we identify 1∼𝛿¹₂, 0∼𝛿²₂. Using this transformation, a logical function 𝑓∶𝒟^𝑘→𝒟 becomes a function 𝑓∶Δ^𝑘₂→Δ₂.(7)Consider a fundamental unary logical function: Negation, ¬𝑃 and five fundamental binary logical functions: Disjunction, 𝑃∨𝑄; Conjunction, 𝑃∧𝑄; Conditional, 𝑃→𝑄; Biconditional, 𝑃↔𝑄; Exclusive OR, 𝑃∨𝑄. Their structure matrices are as follows: 𝑀_¬=𝛿₂[2,1];𝑀_∨=𝛿₂[1,1,1,2];𝑀_∧=𝛿₂[];𝑀1,2,2,2_→=𝛿₂[1,2,1,1];𝑀_↔=𝛿₂[1,2,2,1];𝑀_⊕∶=𝑀_∨=𝛿₂[2,1,1,2]. (2.3)(8)Given a logical function 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛), there exists a unique 2×2^𝑛 matrix 𝑀_𝑓, called the structure matrix, such that 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝑀_𝑓𝑥, (2.4) where 𝑥=⋉^𝑛_𝑖=1𝑥_𝑖∈Δ₂^𝑛.(9)Let 𝑥∈𝑅^𝑡 and 𝐴 is a given matrix, then 𝑥𝐴=(𝐼_𝑡⊗𝐴)𝑥.

3. Main Results

In this section, we will introduce several different definitions of Boolean derivatives. Using the Cheng product, we successfully convert the logical expression into the matrix expression. In addition, the relationships among different definitions of Boolean derivatives are obtained.

In the following, we will introduce some definitions of Boolean derivatives for the logical function 𝑓(𝑥₁,𝑥₂,…,𝑥_𝑛).

Definition 3.1 (see [19]). One gets that 𝜕𝑓𝜕𝑥_𝑗=𝑓𝑥₁,…,𝑥_𝑗,…,𝑥_𝑛⊕𝑓𝑥₁,…,𝑥_𝑗⊕1,…,𝑥_𝑛, (3.1) where 𝑃⊕𝑄 denotes the “Exclusive OR” in logical calculation.
This definition has been proposed for a long time, and it can also be described as 𝜕𝑓 / 𝜕𝑥_𝑗 =𝑓(𝑥₁,…,𝑥_𝑗−1,0,𝑥_𝑗+1,…,𝑥_𝑛)⊕𝑓(𝑥₁,…,𝑥_𝑗−1,1,𝑥_𝑗+1,…,𝑥_𝑛).

Definition 3.2 (see [21]). Let 𝑏 be an 𝑛-variable vector and 𝑏=𝑒_𝑖1⊕𝑒_𝑖2⊕⋯⊕𝑒_𝑖𝑘, 𝑖₁,…,𝑖_𝑘∈{1,2,…,𝑛}, 𝑖₁<𝑖₂<⋯<𝑖_𝑘, 𝑒₁=(1,0,0,…,0), 𝑒₂=(0,1,0,…,0),…,and𝑒_𝑛=(0,0,…,0,1), define 𝜕_𝑏𝑓(𝑥)(𝑥)()𝑥=𝑓⊕𝑓𝑥⊕𝑏=𝑓₁,…,𝑥_𝑖1,…,𝑥_𝑖𝑘,…,𝑥_𝑛⊕𝑓𝑥₁,…,𝑥_𝑖1⊕1,…,𝑥_𝑖𝑘⊕1,…,𝑥_𝑛. (3.2)

Remark 3.3. It is clear that Definition 3.2 is a generalization of Definition 3.1. If 𝑘=1, that is, 𝑏=𝑒_𝑖1, Definition 3.2 then degenerates to Definition 3.1.

Definition 3.4 (see [19]). Therefore, 𝜕𝑓𝜕₊𝑥_𝑗=𝑥_𝑗⊕𝑓(𝑥)𝜕𝑓𝜕𝑥_𝑗,𝜕𝑓𝜕₋𝑥_𝑗=𝑥_𝑗⊕𝑓(𝑥)𝜕𝑓𝜕𝑥_𝑗. (3.3)

Remark 3.5. The present Definition 3.4 is called the directional Boolean derivative sometimes. This is meaningful. It considered the change in the arguments increase, decrease, or remain constant.
Let 𝑓(𝑥)=𝑓(𝑥₁,…,𝑥_𝑛)=𝑀_𝑓𝑥. Then, according to the definitions of the Boolean derivatives, we obtain the following representations: 𝜕𝑓𝜕𝑥_𝑗=𝑓𝑥₁,…,𝑥_𝑗,…,𝑥_𝑛⊕𝑓𝑥₁,…,𝑥_𝑗⊕1,…,𝑥_𝑛=𝑀_𝑓𝑥⊕𝑀_𝑓𝑥₁⋯𝑀_¬𝑥_𝑗⋯𝑥_𝑛=𝑀_⊕𝑀_𝑓𝑥𝑀_𝑓𝑥₁⋯𝑀_¬𝑥_𝑗⋯𝑥_𝑛=𝑀_⊕𝑀_𝑓𝑥𝑀_𝑓𝐼₂^𝑗−1⊗𝑀_¬𝑥=𝑀_⊕𝑀_𝑓𝐼₂^𝑛⊗𝑀_𝑓𝐼₂^𝑗−1⊗𝑀_¬Φ_𝑛𝜕𝑥,_𝑏(𝑥)()𝑥𝑓=𝑓⊕𝑓𝑥⊕𝑏=𝑓₁,…,𝑥_𝑖1,…,𝑥_𝑖𝑘,…,𝑥_𝑛⊕𝑓𝑥₁,…,𝑥_𝑖1⊕1,…,𝑥_𝑖𝑘⊕1,…,𝑥_𝑛=𝑀_𝑓𝑥⊕𝑀_𝑓𝑥₁⋯𝑀_¬𝑥_𝑖1⋯𝑀_¬𝑥_𝑖𝑘⋯𝑥_𝑛=𝑀_⊕𝑀_𝑓𝑥𝑀_𝑓𝐼₂^𝑖₁⁻¹⊗𝑀_¬𝐼₂^𝑖₂⁻¹⊗𝑀_¬⋯𝐼₂^𝑖_𝑘⁻¹⊗𝑀_¬𝑥₁⋯𝑥_𝑛=𝑀_⊕𝑀_𝑓𝑥𝑀_𝑓^𝑖𝑘_𝑖=𝑖1𝐼₂^𝑖−1⊗𝑀_¬𝑥=𝑀_⊕𝑀_𝑓𝐼₂^𝑛⊗𝑀_𝑓^𝑖𝑘_𝑖=𝑖1𝐼₂^𝑖−1⊗𝑀_¬Φ_𝑛𝑥, (3.4) where Φ_𝑛 is the power-reducing matrix defined in [14].
In the following, we will discuss the relationships among the different definitions of Boolean derivatives by using the Cheng product.
First, we give a Lemma that will be used in the proof of Theorem 3.7.

Lemma 3.6. Assume that 𝑎,𝑏∈Δ₂, then (𝑎⊕𝑏)∨(𝑎⊕𝑏)=𝛿¹₂.

Proof. One assumes that (𝑎⊕𝑏)∨(𝑎⊕𝑏)=𝑀_𝑑𝑀_⊕𝑎𝑏𝑀_⊕𝑎𝑏=𝑀_𝑑𝑀_⊕𝑀_¬𝑎𝑏𝑀_⊕𝑎𝑏=𝑀_𝑑𝑀_⊕𝑀_¬𝐼₄⊗𝑀_⊕(𝑎𝑏)²=𝑀_𝑑𝑀_⊕𝑀_¬𝐼₄⊗𝑀_⊕𝑀²²_𝑟𝑎𝑏, (3.5) where 𝑀⁴_𝑟=𝛿¹₄0₄⋯0₄0₄𝛿²₄⋯0₄0⋮⋮⋱⋮₄0₄⋯𝛿⁴₄ (3.6) is the power-reducing matrix defined in [14].
Throughout directional computation, we get that (𝑎⊕𝑏)∨(𝑎⊕𝑏)=𝛿¹₂. (3.7)
This completes the proof.

Next, two Theorems are given to illustrate the relationships among the different definitions. Theorem 3.7 shows the relationships of Definitions 3.1 and 3.4; Theorem 3.8 states the relationships between Definitions 3.1 and 3.2.

Theorem 3.7. One gets that 𝜕𝑓𝜕₊𝑥_𝑗∨𝜕𝑓𝜕₋𝑥_𝑗=𝜕𝑓𝜕𝑥_𝑗. (3.8)

Proof. Therefore, 𝜕𝑓𝜕₊𝑥_𝑗∨𝜕𝑓𝜕₋𝑥_𝑗=𝑥_𝑗⊕𝑓(𝑥)𝜕𝑓𝜕𝑥_𝑗∨𝑥_𝑗⊕𝑓(𝑥)𝜕𝑓𝜕𝑥_𝑗=𝑥_𝑗⊕𝑓(𝑥)∨𝑥_𝑗⊕𝑓(𝑥)𝜕𝑓𝜕𝑥_𝑗. (3.9)
According to Lemma 3.6, one obtains that 𝜕𝑓𝜕₊𝑥_𝑗∨𝜕𝑓𝜕₋𝑥_𝑗=𝜕𝑓𝜕𝑥_𝑗. (3.10)
This completes the proof.

Theorem 3.8. Since, 𝜕𝜕𝑥₁𝜕𝜕𝑥₂⋯𝜕𝑓(𝑥)𝜕𝑥_𝑘=_{𝑗1≤𝑙≤𝑘1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑘}𝜕_{𝑗1,…,𝑗𝑙}𝑓(𝑥)=𝑀^2𝑘_⊕⁻¹_{𝑗1≤𝑙≤𝑘1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑘}𝑀_𝑓𝑥𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬𝑥=𝑀^2𝑘_⊕⁻¹_{𝑗1≤𝑙≤𝑘1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑘}𝑀_𝑓𝐼₂^𝑛⊗𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬Φ_𝑛𝑥. (3.11)

Proof. We prove the statement by induction on 𝑘.(1)The formula is true for 𝑘=1,2.
For 𝑘=1, it is obviously true.
For 𝑘=2: 𝜕𝜕𝑥₁𝜕𝜕𝑥₂=𝜕𝜕𝑥₁𝑓(𝑥)⊕𝑓𝑥⊕𝑒_𝑖2=𝑓(𝑥)⊕𝑓𝑥⊕𝑒_𝑖2⊕𝑓𝑥⊕𝑒_𝑖1⊕𝑓𝑥⊕𝑒_𝑖2⊕𝑒_𝑖1=𝑀³_⊕𝑀_𝑓𝑥𝑀_𝑓𝐼₂^𝑖₂⁻¹⊗𝑀_¬𝑥𝑀_𝑓𝐼₂^𝑖₁⁻¹⊗𝑀_¬𝑥𝑀_𝑓𝐼₂^𝑖₁⁻¹𝑀_¬𝐼₂^𝑖₂⁻¹⊗𝑀_¬𝑥. (3.12)
While, _{𝑗1≤𝑙≤𝑘1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑘}𝜕_{𝑗1,…,𝑗𝑙}𝑓(𝑥)=𝜕_𝑖1𝑓(𝑥)⊕𝜕_𝑖2𝑓(𝑥)⊕𝜕_𝑖1,𝑖2𝑓(𝑥)=𝑓(𝑥)⊕𝑓𝑥⊕𝑒_𝑖1⊕𝑓(𝑥)⊕𝑓𝑥⊕𝑒_𝑖2⊕𝑓(𝑥)⊕𝑓𝑥⊕𝑒_𝑖1⊕𝑒_𝑖2=𝑓(𝑥)⊕𝑓𝑥⊕𝑒_𝑖1⊕𝑓𝑥⊕𝑒_𝑖2⊕𝑓𝑥⊕𝑒_𝑖1⊕𝑒_𝑖2=𝑀³_⊕𝑀_𝑓𝑥𝑀_𝑓𝐼₂^𝑖₁⁻¹⊗𝑀_¬𝑥𝑀_𝑓𝐼₂^𝑖₁⁻¹⊗𝑀_¬𝑥𝑀_𝑓𝐼₂^𝑖₁⁻¹𝑀_¬𝐼₂^𝑖₂⁻¹⊗𝑀_¬𝑥. (3.13) Thus, it’s true for 𝑘=1,2.
Assume that the statement is true for 𝑘=𝑚: 𝜕𝜕𝑥₁𝜕𝜕𝑥₂⋯𝜕𝑓(𝑥)𝜕𝑥_𝑚=_{𝑗1≤𝑙≤𝑚1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚}𝜕_{𝑗1,…,𝑗𝑙}𝑓(𝑥)=𝑀^2𝑚_⊕⁻¹_{𝑗1≤𝑙≤𝑚1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚}𝑀_𝑓𝑥𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬𝑥=𝑀^2𝑚_⊕⁻¹_{𝑗1≤𝑙≤𝑚1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚}𝑀_𝑓𝐼₂^𝑛⊗𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬Φ_𝑛𝑥. (3.14) For 𝑘=𝑚+1, 𝜕𝜕𝑥₁𝜕𝜕𝑥₂⋯𝜕𝑓(𝑥)𝜕𝑥_𝑚+1=𝜕𝜕𝑥_𝑚+1𝜕𝜕𝑥₁⋯𝜕𝑓(𝑥)𝜕𝑥_𝑚=𝜕𝜕𝑥₁⋯𝜕𝑓(𝑥)𝜕𝑥_𝑚⊕𝜕𝜕𝑥₁⋯𝜕𝑓𝑥⊕𝑒_𝑖𝑚+1𝜕𝑥_𝑚. (3.15) In addition, 𝜕𝜕𝑥₁⋯𝜕𝑓𝑥⊕𝑒_𝑖𝑚+1𝜕𝑥_𝑚=_{𝑗1≤𝑙≤𝑚1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚}𝜕_{𝑗1,…,𝑗𝑙}𝑓𝑥⊕𝑒_𝑖𝑚+1=_{𝑗1≤𝑙≤𝑚1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚}𝑓𝑥⊕𝑒_𝑖𝑚+1⊕𝑓𝑥⊕𝑒_𝑗1⊕⋯⊕𝑒_𝑗𝑙⊕𝑒_𝑖𝑚+1=𝑀^2𝑚_⊕⁻¹_{𝑗1≤𝑙≤𝑚1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚}𝑀_𝑓𝐼₂^𝑖_𝑚+1⁻¹⊗𝑀_¬𝑥𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬𝐼₂^𝑖_𝑚+1⁻¹⊗𝑀_¬𝑥. (3.16) Thus, 𝜕𝜕𝑥₁𝜕𝜕𝑥₂⋯𝜕𝑓(𝑥)𝜕𝑥_𝑚+1=𝜕𝜕𝑥_𝑚+1𝜕𝜕𝑥₁⋯𝜕𝑓(𝑥)𝜕𝑥_𝑚=𝜕𝜕𝑥₁⋯𝜕𝑓(𝑥)𝜕𝑥_𝑚⊕𝜕𝜕𝑥₁⋯𝜕𝑓𝑥⊕𝑒_𝑖𝑚+1𝜕𝑥_𝑚=𝑀^2𝑚_⊕⁻¹_{𝑗1≤𝑙≤𝑚1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈{𝑖1,𝑖2,…,𝑖𝑚}}𝑀_𝑓𝑥𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬𝑥⊕𝑀^2𝑚_⊕⁻¹_{𝑗1≤𝑙≤𝑚1<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚}𝑀_𝑓𝐼₂^𝑖_𝑚+1⊗𝑀_¬𝑥𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬𝐼₂^𝑖_𝑚+1⁻¹⊗𝑀_¬𝑥=𝑀^2∗(2𝑚_⊕⁻¹⁾⁺¹_{𝑗1≤𝑙≤𝑚+11<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚+1}𝑀_𝑓𝑥𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬𝑥=𝑀^2𝑚+1_⊕⁻¹_{𝑗1≤𝑙≤𝑚+11<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚+1}𝑀_𝑓𝑥𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬𝑥=𝑀^2𝑚+1_⊕⁻¹_{𝑗1≤𝑙≤𝑚+11<⋯<𝑗𝑙𝑗1,…,𝑗𝑙∈𝑖1,𝑖2,…,𝑖𝑚+1}𝑀_𝑓𝐼₂^𝑛⊗𝑀_𝑓𝐼₂^𝑗₁⁻¹⊗𝑀_¬⋯𝐼₂^𝑗_𝑙⁻¹⊗𝑀_¬Φ_𝑛𝑥. (3.17)
This completes the proof.

Remark 3.9. By converting the logical calculation into matrix product, it is more convenient for the calculation of high-dimensional systems.

4. Example

In this part, we will discuss the application of the main results in the present paper. Rule 150 is one of the eight elementary automata rules. It was introduced by Stephen Wolfram in 1983. Vichniac [12] also investigated its property. In the following, we will discuss the rule 150 using the Cheng product method.

In peripheral one dimensional cellular automata, the XOR rule is 𝑥^𝑡_𝑖=𝑥^𝑡−1_𝑖−1⊕𝑥_𝑖^𝑡−1⊕𝑥^𝑡−1_𝑖+1. (4.1)

If we define a Boolean cellular automata with 𝑁 cells by a global mapping 𝐹∶{0,1}^𝑁↦{0,1}^𝑁. (4.2)

A local transition rule 𝑓 of 𝐹 is given as follows: it consists of its two neighborhood and the considered cell itself 𝑓𝑥₁,𝑥₂,𝑥₃=𝑥₁⊕𝑥₂⊕𝑥₃, (4.3) then 𝑓𝑥₁,𝑥₂,𝑥₃=𝑀²_⊕𝑥₁𝑥₂𝑥₃=𝑀_𝑓𝑥,where𝑀_𝑓=𝑀²_⊕,𝑥=𝑥₁𝑥₂𝑥₃. (4.4)

Define the Boolean derivative of 𝐹 as the 𝑁×𝑁 matrix with 𝐹^′_𝑖𝑗=𝜕𝑥^𝑡_𝑖/𝜕𝑥_𝑗^𝑡−1.

According to the definition of Boolean derivative, it is easy to see that 𝐹^′_𝑖𝑗=0 for 𝑗<𝑖−1and𝑖+1<𝑗.

When 𝑖−1≤𝑗≤𝑖+1, from the definition and the matrix expression of Boolean derivative, one can see that 𝜕𝑓𝜕𝑥_𝑗^𝑡−1=𝑀_⊕𝑀_𝑓𝑥^𝑡−1_𝑖−1𝑥_𝑖^𝑡−1𝑥^𝑡−1_𝑖+1𝑀_𝑓𝐼₂^𝑗−1⊗𝑀_¬𝑥^𝑡−1_𝑖−1𝑥_𝑖^𝑡−1𝑥^𝑡−1_𝑖+1=𝑀³_⊕𝐼₈⊗𝑀²_⊕𝐼_𝑗−1⊗𝑀_¬𝑀⁸_𝑟𝑥^𝑡−1_𝑖−1𝑥_𝑖^𝑡−1𝑥^𝑡−1_𝑖+1. (4.5)

Throughout the directional computation, we can get the following Jacobian matrix: 𝐹^′=1100⋯011110⋯000111⋯00⋮⋮⋱⋱⋱⋮⋮⋯⋯⋯⋯1111⋯⋯⋯⋯11. (4.6)

From the Jacobian matrix, one can see that the new value of the cell is determined by its two neighbors and the cell itself. In the Jacobian matrix, the 1′s shows us that the rule 150 is linearly dependent of the augments, respectively.

5. Discussion

Boolean derivative is an important topic in the research of Boolean function. It has wide applications in different fields. There are several different definitions about the Boolean derivatives. Each definition has its own meaning and application in respective domain. In this paper, we introduce several definitions of Boolean derivatives. Furthermore, the relationships among them are presented. We show the meanings of them, and the matrix expressions of them are also given.

Throughout the paper, the main tool used is the semitensor product, that is, Cheng product. It is a very useful tool. It generalizes the matrix product. In classic matrix product, it is necessary to guarantee that the number of columns of the left matrix matches the number of rows of the right matrix. However, it is not necessary in Cheng product. Cheng product has most of the properties of matrix product, and it has many other new advantages. Cheng product has been now used in many research. In this paper, Cheng product is applied in obtaining the relationship among different definitions. The calculation of logic variables has been converted into matrix product. It is interesting and meaningful. Using the results obtained, the application of Boolean derivative in cryptography and other fields might be extended.

6. Conclusion

In this paper, firstly we have introduced several definitions of Boolean derivatives. Then, we have given formulas to calculate Boolean derivatives by using the semitensor product, that is, Cheng product. We have also investigated the relationship among different definitions of Boolean derivatives. At last, an application is presented to illustrate the main results.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant 61174039 and by the Fundamental Research Funds for the Central Universities of China. The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the paper.