# Conversion between Logic and Algebraic Expressions of Boolean Control Networks
## 1 Introduction  
Boolean control network (BCN) is a logical system pioneered by [PERSON] in [1] for initially describing genetic regulatory networks, which provides useful modeling tools for dynamical systems whose state-variables take only two values ('True' or 'False'). With the development of information technologies, BCNs have become an effective way to analyze and design circuits by algebraic means in terms of logic gates, and have served as a fundamental approach to digital circuit design, computer programming, and so on (see, e.g., [2; 3; 4; 5; 6; 7]). With the development of digital circuit, the research on logic synthesis becomes particularly important. Recently, [PERSON] and [PERSON] [5] used a novel method for obtaining the parametric general solutions of the 'big' Boolean equation, which cover the most prominent type among basic problems of digital circuit design. [PERSON] et al. [4] analyzed the effects of different graph-based intermediate representations, including binary decision diagram, and-inverter graph and majority inverter graph, and their usability in making efficient circuit realisations. [PERSON] and [PERSON] [3] introduced the Karnaugh maps of logical systems and used in the analysis and design of clocked sequential circuits and in the simplification of multioutput gate circuits. In addition to the traditional logical expression, there are some representations to describe dynamics and estimate performances of BCNs. For example, logic differential calculus [8] represents a mathematical methodology developed for analysis of Boolean functions, which enables us to consider the existing estimates of system importance, as well as some additional properties, of BCNs from the unique methodological standpoint (see, e.g., [9; 10]). The semi-tensor product (STP) of matrices is developed in [11] to convert a Boolean function into an algebraic function, and to model a BCN by a standard discrete-time linear system. A comparison of different representations of Boolean functions can be found in [12]. Describing a logical dynamic system as a linear system is useful for studying BCNs in a control-theoretic framework. Examples include the controllability and observability [13; 14; 15; 16], periodicity and stability [17; 18; 19; 20; 21; 22], optimal control [23; 24],identification [25], etc. The computation complexity is a series problem in directly calculating the network transition matrix based on STP, which frames a limitation on the application of algebraic state-space approach. It has been calculated in [3] that the complexity of STP based method is \(O(2^{6(m+n)}n)\). Recently, [PERSON] et al. [26, 27] proposed a Khatri-Rao product based method to transform the logical form of a BCN into a discrete time bilinear system, which has a complexity of \(O(2^{m+2n})\). In other words, the complexity of the STP based method is \(2^{5m+4n}n\) times that of the Khatri-Rao product based method. This paper will provide a constructive and straightforward method with a complexity of \(O(2^{m+n}n)\) to convert the logic expression to the algebraic expression of a BCN. More concretely, the complexity of the STP based method presented in [11] is \(2^{5(m+n)}\) times that of the method given in this paper, and the Khatri-Rao product based method presented in [26, 27] is \(2^{n}/n\) times that of ours. For a BCN, converting the algebraic expression into the logic expression is an integral part for learning the logical meaning of the algebraic result from the algebraic expression. In [16], a mechanical procedure is provided to retrieve the logical network and logical dynamic equations from the network matrix of a BCN. The conversion formula given in [16] is an implicit equation, which converts the algebraic expression for a Boolean function into an irregular form of the logic expression. This paper will provide an explicit equation to directly convert the algebraic expression to a canonical form of the logic expression. The remainder of this paper is organized as follows.