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. S...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.展开更多
传统的概率转移矩阵(Probabilistic Transfer Matrix,PTM)方法是一种能够比较精确地估计软差错对门级电路可靠度影响的方法,但现有的方法只适用于组合逻辑电路的可靠度估计.本文提出基于PTM的时序电路可靠度估计方法(reliability estima...传统的概率转移矩阵(Probabilistic Transfer Matrix,PTM)方法是一种能够比较精确地估计软差错对门级电路可靠度影响的方法,但现有的方法只适用于组合逻辑电路的可靠度估计.本文提出基于PTM的时序电路可靠度估计方法(reliability estimation of Sequential circuits based on PTM,S-PTM),先把待评估时序电路划分为输出逻辑模块和次态逻辑模块,然后用本文提出的时序电路PTM计算模型得到电路的PTM,最后根据输入信号的概率分布计算出时序电路的可靠度.用ISCAS 89基准电路为对象进行实验和验证,实验表明所提方法是准确和合理的.展开更多
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 ...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）.展开更多
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, usin...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.展开更多
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 giv...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.展开更多
The semi-tensor product(STP)of matrices is generalized to multidimensional arrays,called the compound product of hypermatrices.The product is first defined for three-dimensional hypermatrices with compatible orders an...The semi-tensor product(STP)of matrices is generalized to multidimensional arrays,called the compound product of hypermatrices.The product is first defined for three-dimensional hypermatrices with compatible orders and then extended to general cases.Three different types of hyperdeterminants are introduced and certain properties are revealed.The Lie groups and Lie algebras corresponding to the hypermatrix products are constructed.Finally,these results are applied to dynamical systems.展开更多
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, im...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.展开更多
An equivalent definition of hypermatrices is introduced.The matrix expression of hypermatrices is proposed.Using permu-tation matrices,the conversion between different matrix expressions is revealed.The various kinds ...An equivalent definition of hypermatrices is introduced.The matrix expression of hypermatrices is proposed.Using permu-tation matrices,the conversion between different matrix expressions is revealed.The various kinds of contracted products of hypermatrices are realized by semi-tensor products(STP)of matrices via matrix expressions of hypermatrices.展开更多
In this paper,a criterion for the partially symmetric game(PSG)is derived by using the semitensor product approach.The dimension and the basis of the linear subspace composed of all the PSGs with respect to a given se...In this paper,a criterion for the partially symmetric game(PSG)is derived by using the semitensor product approach.The dimension and the basis of the linear subspace composed of all the PSGs with respect to a given set of partial players are calculated.The testing equations with the minimum number are concretely determined,and the computational complexity is analysed.Finally,two examples are displayed to show the theoretical results.展开更多
基金Supported partly by National Natural Science Foundation of China under Grant No. 60221301 and 60334040 .Dedicated to Academician Han-Fu Chen on the occasion of his 70th birthday.
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.G61074068,G61034007,G61174036)the Research Fund for the Taishan Scholar Project of Shandong Province of Chinathe Natural Science Foundation of Shandong Province(Nos.ZR2010FM013,ZR2010FM018)
文摘传统的概率转移矩阵(Probabilistic Transfer Matrix,PTM)方法是一种能够比较精确地估计软差错对门级电路可靠度影响的方法,但现有的方法只适用于组合逻辑电路的可靠度估计.本文提出基于PTM的时序电路可靠度估计方法(reliability estimation of Sequential circuits based on PTM,S-PTM),先把待评估时序电路划分为输出逻辑模块和次态逻辑模块,然后用本文提出的时序电路PTM计算模型得到电路的PTM,最后根据输入信号的概率分布计算出时序电路的可靠度.用ISCAS 89基准电路为对象进行实验和验证,实验表明所提方法是准确和合理的.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 61174094）, and the Tianjin Natural Science Foundation of China under （14JCYBJC18700 and 13JCY- BJC17400）.
文摘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）.
基金This work was supported by the National Natural Science Foundation of China （Nos. G61374065, G61034007, G61374002） the Fund for the Taishan Scholar Project of Shandong Province, the Natural Science Foundation of Shandong Province （No. ZR2010FM013） the Scientific Research and Development Project of Shandong Provincial Education Department （No. J11LA01 ）
文摘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.
基金supported by the National Natural Science Foundation of China under Grant No.61174071
文摘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.
基金supported partly by the National Natural Science Foundation of China under Grant Nos.62073315 and 62103305Shanghai Pujiang Program under Grant No.21PJ 1413100China Postdoctoral Science Foundation under Grant Nos.2021M703423 and 2022T150686。
文摘The semi-tensor product(STP)of matrices is generalized to multidimensional arrays,called the compound product of hypermatrices.The product is first defined for three-dimensional hypermatrices with compatible orders and then extended to general cases.Three different types of hyperdeterminants are introduced and certain properties are revealed.The Lie groups and Lie algebras corresponding to the hypermatrix products are constructed.Finally,these results are applied to dynamical systems.
基金the National Natural Science Foundation of China (No.60274010, 60343001, 60221301, 60334040)
文摘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.
基金This work was supported partly by the National Natural Science Foundation of China(NSFC)(Nos.62073315,62103305)the Shanghai Pujiang Program(No.21PJ 1413100)China Postdoctoral Science Foundation(Nos.2021M703423,2022T150686).
文摘An equivalent definition of hypermatrices is introduced.The matrix expression of hypermatrices is proposed.Using permu-tation matrices,the conversion between different matrix expressions is revealed.The various kinds of contracted products of hypermatrices are realized by semi-tensor products(STP)of matrices via matrix expressions of hypermatrices.
基金the National Natural Science Foundation of China under Grants 61673012 and 11971240,respectively。
文摘In this paper,a criterion for the partially symmetric game(PSG)is derived by using the semitensor product approach.The dimension and the basis of the linear subspace composed of all the PSGs with respect to a given set of partial players are calculated.The testing equations with the minimum number are concretely determined,and the computational complexity is analysed.Finally,two examples are displayed to show the theoretical results.
