Matrix Expression Of Logic Research Articles

Networked Evolutionary Model of Snow-Drift Game Based on Semi-Tensor Product

This paper investigates the networked evolutionary model based on snow-drift game with the strategy of rewards and penalty. Firstly, by using the semi-tensor product of matrices approach, the mathematical model of the networked evolutionary game is built. Secondly, combined with the matrix expression of logic, the mathematical model is expressed as a dynamic logical system and next converted into its evolutionary dynamic algebraic form. Thirdly, the dynamic evolution process is analyzed and the final level of cooperation is discussed. Finally, the effects of the changes in the rewarding and penalty factors on the level of cooperation in the model are studied separately, and the conclusions are verified by examples.

Implementation of Boolean Control Network Based Intelligent System in Smart Home

Smart home is a nondeterministic complex environment. Sensors and actuators network are involved in smart home to collect environmental information and to control the devices used by the user for comfort of life. Generally smart home system is implemented by logical rules which described the relation between each element (sensors and actuators). Controlling using logical rules is difficult. In this paper, we have represented an intelligent system for smart home by using Boolean Control Network. For easy control, we have used matrix expression of logic. The system is controlled in several states and in each state different device is operated through actuator network. Matlab based simulation work is done to show the state changes of the system. The result shows that using matrix expression it is easy to control the state of this system.

Partial variables controllability of Boolean control networks

Partial controllability of system is a special definition of controllability with respect to only a pre-specified subset of a system's state variables. The paper considers problems on the controllability of Boolean control networks with respect to a given part of variables that characterize the Boolean network. Using a semi-tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean control network are converted to discrete time bilinear dynamics. First, the definition of partial controllability of Boolean control networks is introduced. Then, based on the algebraic form, necessary and sufficient conditions for partial state variable controllability with respect to a variables set are obtained. In addition, an illustrative example is provided to construct a proper control sequence with which to steer the partial variables' states to those desired.

Partial state stability and stabilization of Boolean control networks

The paper deals with problems on the stability and stabilization of Boolean networks with respect to a given part of variables characterizing the Boolean network. Using the semi-tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to discrete time linear (bilinear) dynamics called the algebraic form of the Boolean (control) network. The two main results are as follows: (i) From the algebraic form of a Boolean (control) network, two partial stability definitions are introduced. One is partial stability with respect to another variable set, and the other is partial stability with respect to arbitrary initial states. (ii) Based on the algebraic form, necessary and sufficient conditions for the stability of global partial states with respect to the variable set and arbitrary initial states are obtained. Examples are provided to illustrate the effectiveness of our results.

STABILITY AND STABILIZATION OF BOOLEAN NETWORKS

The stability of Boolean networks and stabilization of Boolean control networks are discussed.By using the method of semi-tensor product of matrices and the matrix expression of logic,the dynamics of Boolean network can be represented as a discrete-time dynamic system,then it is converted into an algebraic form.Then the one-to-one correspondence between the structure matrix of the algebraic form and a digital transformation is established.Finally,by using the method of digital transformation,necessary and sufficient conditions for the stability and stabilization of Boolean networks are obtained.

Solvability and control design for synchronization of Boolean networks

This paper addresses the synchronization problem of Boolean networks. Based on the matrix expression of logic, solvability conditions and design procedures of the synchronization of Boolean networks with outputs are given for both open-loop and feedback control. Necessary and sufficient conditions on open-loop control are proposed first with a constructive design procedure. Then sufficient condition for the feedback control case is obtained, and corresponding design procedure is proposed with the help of algorithms to solve logic matrix equations. Numerical examples are also provided to illustrate the proposed control design.

Observability of Boolean Control Networks With State Time Delays

This brief deals with the problem of the observability for the Boolean control networks with time delays in states. First, using semi-tensor product of matrices and the matrix expression of logic, the Boolean control networks with state delays can be converted into discrete time delay dynamics. Then, the observability of the Boolean control networks via two kinds of inputs is investigated by giving necessary and sufficient conditions. Finally, examples are given to illustrate the efficiency of the obtained results.

Stability and stabilization of Boolean networks

AbstractThe stability of Boolean networks and the stabilization of Boolean control networks are investigated. Using semi‐tensor product of matrices and the matrix expression of logic, the dynamics of a Boolean (control) network can be converted to a discrete time linear (bilinear) dynamics, called the algebraic form of the Boolean (control) network. Then the stability can be revealed by analyzing the transition matrix of the corresponding discrete time system. Main results consist of two parts: (i) Using logic coordinate transformation, the known sufficient condition based on incidence matrix has been improved. It can also be used in stabilizer design. (ii) Based on algebraic form, necessary and sufficient conditions for stability and stabilization, respectively, are obtained. Copyright © 2010 John Wiley & Sons, Ltd.

On Boolean Control Networks — An Algebraic Approach

AbstractBoolean network is a powerful tool in describing the genetic regulatory networks. Accompanying the development of systems biology, the analysis and control of Boolean networks have attracted much attention from biologists, physicists, and systems scientists. This paper surveys a series of recent results on Boolean (control) networks, obtained by the algebraic approach proposed by the authors. First, a brief introduction to the semi-tensor product of matrices and the matrix expression of logic is presented. Then the topological structure of Boolean networks is investigated. Finally, the basic control problems of Boolean control networks are explored, which include the controllability, observability, realization, stability and stabilization, disturbance decoupling, identification and optimization.

A Linear Representation of Dynamics of Boolean Networks

A new matrix product, called semi-tensor product of matrices, is reviewed. Using it, a matrix expression of logic is proposed, where a logical variable is expressed as a vector, a logical function is expressed as a multiple linear mapping. Under this framework, a Boolean network equation is converted into an equivalent algebraic form as a conventional discrete-time linear system. Analyzing the transition matrix of the linear system, formulas are obtained to show a) the number of fixed points; b) the numbers of cycles of different lengths; c) transient period, for all points to enter the set of attractors; and d) basin of each attractor. The corresponding algorithms are developed and used to some examples.

State–Space Analysis of Boolean Networks

This paper provides a comprehensive framework for the state-space approach to Boolean networks. First, it surveys the authors' recent work on the topic: Using semitensor product of matrices and the matrix expression of logic, the logical dynamic equations of Boolean (control) networks can be converted into standard discrete-time dynamics. To use the state-space approach, the state space and its subspaces of a Boolean network have been carefully defined. The basis of a subspace has been constructed. Particularly, the regular subspace, Y-friendly subspace, and invariant subspace are precisely defined, and the verifying algorithms are presented. As an application, the indistinct rolling gear structure of a Boolean network is revealed.

Logic and logic-based control

This paper gives a matrix expression of logic. Under the matrix expression, a general description of the logical operators is proposed. Using the semi-tensor product of matrices, the proofs of logical equivalences, implications, etc., can be simplified a lot. Certain general properties are revealed. Then, based on matrix expression, the logical operators are extended to multi-valued logic, which provides a foundation for fuzzy logical inference. Finally, we propose a new type of logic, called mix-valued logic, and a new design technique, called logic-based fuzzy control. They provide a numerically computable framework for the application of fuzzy logic for the control of fuzzy systems.