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Frontiers of Computer Science 2014年第6期8卷 948-957页

作者： Yongyi YAN Zengqiang CHEN Zhongxin LIU College of Computer and Control Engineering Nankai University Tianjin 300071 China College of Science Civil Aviation University of China Tianjin 300300 China

This paper investigates the transition function and the reachability conditions of finite automata by using a semitensor product of matrices, which is a new powerful matrix analysis tool. The states and input symbols are first expressed in vector forms, then the transition function is described in an algebraic form. Using this algebraic representation, a sufficient and necessary condition of the reachability of any two states is proposed, based on which an algorithm is developed for discovering all the paths from one state to another. Furthermore, a mechanism is established to recognize the language acceptable by a finite automaton. Finally, illustrative examples show that the results/algorithms presented in this paper are suitable for both deterministic finite automata （DFA） and nondeterministic finite automata （NFA）.

关键词： finite automata reachability analysis transition function expression matrix approach semi-tensor product

Nonsingularity of Grain-like cascade FSRs via semi-tensor product

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Science China(Information Sciences) 2018年第1期61卷 58-69页

作者： Jianquan LU Meilin LI Yang LIU Daniel W.C.HO Jrgen KURTHS School of Mathematics Southeast University College of Mathematics Physics and Information Engineering Zhejiang Normal University Department of Mathematics City University of Hong Kong Potsdam Institute for Climate Impact Research

In this paper, Grain-like cascade feedback shift registers(FSRs) are regarded as two Boolean networks(BNs), and the semi-tensor product(STP) of the matrices is used to convert the Grain-like cascade FSRs into an equivalent linear equation. Based on the STP, a novel method is proposed herein to investigate the nonsingularity of Grain-like cascade FSRs. First, we investigate the property of the state transition matrix of Grain-like cascade FSRs. We then propose their sufficient and necessary nonsingularity condition. Next,we regard the Grain-like cascade FSRs as Boolean control networks(BCNs) and further provide a sufficient condition of their nonsingularity. Finally, two examples are provided to illustrate the results obtained in this paper.

关键词： Grain-like cascade FSRs Boolean control networks Boolean networks semi-tensor product nonsingularity

Dynamic output feedback stabilization of deterministic finite automata via the semi-tensor product of matrices approach

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Control Theory and Technology 2021年第2期19卷 170-182页

作者： Roozbeh Abolpour Mohsen Raji Parisa Moradi School of Electrical and Computer Engineering Shiraz UniversityShirazIran

This paper deals with the dynamic output feedback stabilization problem of deterministic finite automata(DFA).The static form of this problem is defined and solved in previous studies via a set of equivalent *** this paper,the dynamic output feedback(DOF)stabilization of DFAs is defined in which the controller is supposed to be another *** DFA controller will be designed to stabilize the equilibrium point of the main DFA through a set of proposed equivalent *** has been proven that the design problem of DOF stabilization is more feasible than the static output feedback(SOF)*** simulation examples are provided to illustrate the results of this paper in more *** first example considers an instance DFA and develops SOF and DOF controllers for *** example explains the concepts of the DOF controller and how it will be implemented in the closed-loop *** the second example,a special DFA is provided in which the DOF stabilization is feasible,whereas the SOF stabilization is *** final example compares the feasibility performance of the SOF and DOF stabilizations through applying them to one hundred random-generated *** results reveal the superiority of the DOF stabilization.

关键词： Discrete event dynamic system Finite automata Dynamic output feedback stabilization semi-tensor product

Robust graph coloring based on the matrix semi-tensor product with application to examination timetabling

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Control Theory and Technology 2014年第2期12卷 187-197页

作者： Meirong XU Yuzhen WANG Airong WEI School of Control Science and Engineering Shandong University Jinan Shandong 250061 China School of Mathematical Sciences University of Jinan Jinan Shandong 250022 China

This paper investigates the robust graph coloring problem with application to a kind of examination timetabling by using the matrix semi-tensor product, and presents a number of new results and algorithms. First, using the matrix semi-tensor product, the robust graph coloring is expressed into a kind of optimization problem taking in an algebraic form of matrices, based on which an algorithm is designed to find all the most robust coloring schemes for any simple graph. Second, an equivalent problem of robust graph coloring is studied, and a necessary and sufficient condition is proposed, from which a new algorithm to find all the most robust coloring schemes is established. Third, a kind of examination timetabling is discussed by using the obtained results, and a method to design a practicable timetabling scheme is presented. Finally, the effectiveness of the results/algorithms presented in this paper is shown by two illustrative examples.

关键词： Robust graph coloring Algorithm Examination timetabling semi-tensor product

Singular Boolean networks:semi-tensor product approach

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Science China(Information Sciences) 2013年第11期56卷 265-278页

作者： FENG JunE YAO Juan CUI Peng School of Mathematics Shandong University School of Control Science and Engineering Shandong University

Singular Boolean networks are introduced in this *** semi-tensor product of matrices and the matrix expression of logical functions,two kinds of the condensed algebraic expressions of singular Boolean networks are *** normalization problem of singular Boolean networks is addressed;that is,under what condition singular Boolean networks can be converted into normal Boolean networks with algebraic *** one sufcient condition and one necessary and sufcient condition are derived for the normalization ***,the solvability of singular Boolean networks is discussed and the concept of admissible initial values of singular Boolean networks is ***,fxed points and cycles of singular Boolean networks are also investigated.

关键词： admissible initial value normalization problem semi-tensor product singular Boolean network

A SURVEY ON semi-tensor product OF MATRICES

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Journal of Systems Science & Complexity 2007年第2期20卷 304-322页

作者： Daizhan CHENG Hongsheng QI Ancheng XUE Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 China

semi-tensor product of matrices is a generalization of conventional matrix product for the case when the two factor matrices do not meet the dimension matching condition. It was firstly proposed about ten years ago. Since then it has been developed and applied to several different fields. In this paper we will first give a brief introduction. Then give a survey on its applications to dynamic systems, to logic, to differential geometry, to abstract algebra, respectively.

关键词： Abstract algebra differential geometry dynamic systems logic semi-tensor product

On semi-tensor product of Matrices and Its Applications

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Acta Mathematicae Applicatae Sinica 2003年第2期19卷 219-228页

作者： Dai-zhan Cheng, Li-jun ZhangInstitute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P.R. China Institute of Systems Science Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100080 P. R. China

Abstract The left semi-tensor product of matrices was proposed in [2]. In this paper the right semi-tensor product is introduced first. Some basic properties are presented and compared with those of the left semi-tensor *** two new applications are investigated. Firstly, its applications to connection, an important concept in differential geometry, is considered. The structure matrix and the Christoffel matrix are introduced. The transfer formulas under coordinate transformation are expressed in matrix form. Certain new results are obtained. Secondly, the structure of finite dimensional Lie algebra, etc. are investigated under the matrix *** applications demonstrate the usefulness of the new matrix products.

关键词： Keywords Matrix semi-tensor product connection Christoffel symbol algebra

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Finding Cut-Edges and the Minimum Spanning Tree via semi-tensor product Approach

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Journal of Systems Science and Information 2018年第5期9卷 459-472页

作者： Xujiao FAN Yong XU Xue SU Jinhuan WANG School of Science Hebei University of Technology

Using the semi-tensor product of matrices, this paper investigates cycles of graphs with application to cut-edges and the minimum spanning tree, and presents a number of new results and algorithms. Firstly, by defining a characteristic logical vector and using the matrix expression of logical functions, an algebraic description is obtained for cycles of graph, based on which a new necessary and sufficient condition is established to find all cycles for any graph. Secondly, using the necessary and sufficient condition of cycles, two algorithms are established to find all cut-edges and the minimum spanning tree, respectively. Finally, the study of an illustrative example shows that the results/algorithms presented in this paper are effective.

关键词： semi-tensor product cycle cut-edge minimum spanning tree

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semi-tensor product Approach to Approximation of the Constrained Joint Spectral Radius

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L（ p, q）-label coloring problem via the semi-tensor product method

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作者： Haoyue Zhang Yang Song Li Deng School of Mechatronic Engineering and Automation Shanghai University

In this paper, a semi-tensor product（STP） approach is proposed to discuss the stability problem of a constrained switching system（CSS）. The switching sequence is subject to the constraint of a nondeterministic finite automaton（NFA）. By the mathematical tool, the constrained joint spectral radius（CJSR） of the CSS can be approximated. We propose the equivalence of a matrix-based expression of the CSS and its lifted l-dimensional switching system by using the STP approach. After that, the approximate relation between the constrained joint spectral radius（CJSR）/the constrained generalized spectral radius（CGSR）and the joint spectral radius（JSR）/generalized spectral radius（GSR） is developed for the switching system. At the same time,there is a special case of NFA, where the equivalence relation between them also is proved.

关键词： Constrained Joint Spectral Radius Constrained Switching System semi-tensor product

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L（ p, q）-label coloring problem via the semi-tensor produc...