This paper explores the algebraic essence of universal logic functions(ULFs)from an algebraic perspective.Under the framework of semi-tensor product of matrices,the“sequential nature”of ULFs is revealed.Utilizing th...This paper explores the algebraic essence of universal logic functions(ULFs)from an algebraic perspective.Under the framework of semi-tensor product of matrices,the“sequential nature”of ULFs is revealed.Utilizing the nature,a technique called universal transformation method is proposed,by which any ULF can be transformed into an equivalent expression with desired features that facilitate achieving specific objectives,such as modeling,analyzing and synthesizing universal logical systems.Furthermore,several useful logical operators are constructed in a mixed-dimensional situation,including power-raising operator,power-descending operator,erasure operator,and appending operator.Finally,these results are applied to model and analyze finite state machines and their networks,which demonstrate the practical value of the method and operators.展开更多
Traditional matrix-based approaches in the field of finite state machines construct state transition matrices,and then use the powers of the state transition matrices to represent corresponding dynamic transition proc...Traditional matrix-based approaches in the field of finite state machines construct state transition matrices,and then use the powers of the state transition matrices to represent corresponding dynamic transition processes,which are cornerstones of system analysis.In this study,we propose a static matrix-based approach that revisits a finite state machine from its structure rather than its dynamic transition process,thus avoiding the“explosion of complexity”problem inherent in the existing approaches.Based on the static approach,we reexamine the issues of closed-loop detection and controllability for deterministic finite state machines.In addition,we propose controllable equivalent form and minimal controllable equivalent form concepts and give corresponding algorithms.展开更多
Motivated by the inconvenience or even inability to explain the mathematics of the state space optimization of finite state machines(FSMs)in most existing results,we consider the problem by viewing FSMs as logical dyn...Motivated by the inconvenience or even inability to explain the mathematics of the state space optimization of finite state machines(FSMs)in most existing results,we consider the problem by viewing FSMs as logical dynamic systems.Borrowing ideas from the concept of equilibrium points of dynamic systems in control theory,the concepts of t-equivalent states and t-source equivalent states are introduced.Based on the state transition dynamic equations of FSMs proposed in recent years,several mathematical formulations of t-equivalent states and t-source equivalent states are proposed.These can be analogized to the necessary and sufficient conditions of equilibrium points of dynamic systems in control theory and thus give a mathematical explanation of the optimization problem.Using these mathematical formulations,two methods are designed to find all the t-equivalent states and t-source equivalent states of FSMs.Further,two ways of reducing the state space of FSMs are found.These can be implemented without computers but with only pen and paper in a mathematical manner.In addition,an open question is raised which can further improve these methods into unattended ones.Finally,the correctness and effectiveness of the proposed methods are verified by a practical language model.展开更多
This paper uses the semi-tensor product(STP)of matrices and adopts algebraic methods to study the controllability,reachability,and stabilizability of extended finite state machines(EFSMs).First,we construct the biline...This paper uses the semi-tensor product(STP)of matrices and adopts algebraic methods to study the controllability,reachability,and stabilizability of extended finite state machines(EFSMs).First,we construct the bilinear dynamic system model of the EFSM,laying the foundation for further research.Second,combined with this bilinear dynamic system model,we propose theorems for the controllability,reachability,and stabilizability of the bilinear dynamic system model of the EFSM.Finally,we design an algorithm to determine the controllability and stabilizability of the EFSM.The correctness of the main results is verified through examples.展开更多
基金supported in part by the National Natural Science Foundation of China under Grants 62073124 and U1804150.
文摘This paper explores the algebraic essence of universal logic functions(ULFs)from an algebraic perspective.Under the framework of semi-tensor product of matrices,the“sequential nature”of ULFs is revealed.Utilizing the nature,a technique called universal transformation method is proposed,by which any ULF can be transformed into an equivalent expression with desired features that facilitate achieving specific objectives,such as modeling,analyzing and synthesizing universal logical systems.Furthermore,several useful logical operators are constructed in a mixed-dimensional situation,including power-raising operator,power-descending operator,erasure operator,and appending operator.Finally,these results are applied to model and analyze finite state machines and their networks,which demonstrate the practical value of the method and operators.
基金supported by the National Natural Science Foundation of China(Nos.U1804150,62073124,and 61973175)。
文摘Traditional matrix-based approaches in the field of finite state machines construct state transition matrices,and then use the powers of the state transition matrices to represent corresponding dynamic transition processes,which are cornerstones of system analysis.In this study,we propose a static matrix-based approach that revisits a finite state machine from its structure rather than its dynamic transition process,thus avoiding the“explosion of complexity”problem inherent in the existing approaches.Based on the static approach,we reexamine the issues of closed-loop detection and controllability for deterministic finite state machines.In addition,we propose controllable equivalent form and minimal controllable equivalent form concepts and give corresponding algorithms.
基金Project supported by the National Natural Science Foundation of China(Nos.U1804150,62073124,and 61973175)。
文摘Motivated by the inconvenience or even inability to explain the mathematics of the state space optimization of finite state machines(FSMs)in most existing results,we consider the problem by viewing FSMs as logical dynamic systems.Borrowing ideas from the concept of equilibrium points of dynamic systems in control theory,the concepts of t-equivalent states and t-source equivalent states are introduced.Based on the state transition dynamic equations of FSMs proposed in recent years,several mathematical formulations of t-equivalent states and t-source equivalent states are proposed.These can be analogized to the necessary and sufficient conditions of equilibrium points of dynamic systems in control theory and thus give a mathematical explanation of the optimization problem.Using these mathematical formulations,two methods are designed to find all the t-equivalent states and t-source equivalent states of FSMs.Further,two ways of reducing the state space of FSMs are found.These can be implemented without computers but with only pen and paper in a mathematical manner.In addition,an open question is raised which can further improve these methods into unattended ones.Finally,the correctness and effectiveness of the proposed methods are verified by a practical language model.
基金Project supported by the National Natural Science Foundation of China(Nos.U1804150 and 62073124)。
文摘This paper uses the semi-tensor product(STP)of matrices and adopts algebraic methods to study the controllability,reachability,and stabilizability of extended finite state machines(EFSMs).First,we construct the bilinear dynamic system model of the EFSM,laying the foundation for further research.Second,combined with this bilinear dynamic system model,we propose theorems for the controllability,reachability,and stabilizability of the bilinear dynamic system model of the EFSM.Finally,we design an algorithm to determine the controllability and stabilizability of the EFSM.The correctness of the main results is verified through examples.
