1 Introduction and main contributions Finite automata are dynamical systems with discrete inputs and outputs, which belong to the domain of logical systems and have a wide range of applications. In engineering, due to...1 Introduction and main contributions Finite automata are dynamical systems with discrete inputs and outputs, which belong to the domain of logical systems and have a wide range of applications. In engineering, due to the excellent hardware qualities of simple structure, low power consumption and low electromagnetic noise, etc., finite automata are used in avionics and nuclear engineering, where the environment is bad and require strict safety. In science, finite automata serve as one of the main molding tools for discrete event dynamic systems (DEDS)(others are Petri nets, Markov chains and queuing networks, etc.). Studying DEDS is one of the major ways to study the cyber physical systems (CPS) which is the core content of Industry 4.0.展开更多
A new modeling tool, algebraic state space approach to logical dynamic systems, which is developed recently based on the theory of semi-tensor product of matrices （STP）, is applied to the automata field. Using the S...A new modeling tool, algebraic state space approach to logical dynamic systems, which is developed recently based on the theory of semi-tensor product of matrices （STP）, is applied to the automata field. Using the STE this paper investigates the modeling and controlling problems of combined automata constructed in the ways of parallel, serial and feedback. By representing the states, input and output symbols in vector forms, the transition and output functions are expressed as algebraic equations of the states and inputs. Based on such algebraic descriptions, the control problems of combined automata, including output control and state control, are considered, and two necessary and sufficient conditions are presented for the controllability, by which two algorithms are established to find out all the control strings that make a combined automaton go to a target state or produce a desired output. The results are quite different from existing methods and provide a new angle and means to understand and analyze the dynamics of combined automata.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
基金This work was supported by the National Natural Science Foundation of China (Grant Nos. U 1804150, 61573199)the 2018 Henan Province Science and Technique Foundation (182102210045).
文摘1 Introduction and main contributions Finite automata are dynamical systems with discrete inputs and outputs, which belong to the domain of logical systems and have a wide range of applications. In engineering, due to the excellent hardware qualities of simple structure, low power consumption and low electromagnetic noise, etc., finite automata are used in avionics and nuclear engineering, where the environment is bad and require strict safety. In science, finite automata serve as one of the main molding tools for discrete event dynamic systems (DEDS)(others are Petri nets, Markov chains and queuing networks, etc.). Studying DEDS is one of the major ways to study the cyber physical systems (CPS) which is the core content of Industry 4.0.
基金Acknowledgements This work was supported by Key Scientific Research Program of the Higher Education Institutions of Henan Educational Committee （15A416005）, the 2015 Science Foundation of Henan University of Science and Technology for Youths （2015QN016）, and the National Natural Science Foundation of China （Grant Nos. 61573199, 61473115, and U1404610）. The authors would like to express their thanks to Prof. Y G Hong for his helpful suggestions.
文摘A new modeling tool, algebraic state space approach to logical dynamic systems, which is developed recently based on the theory of semi-tensor product of matrices （STP）, is applied to the automata field. Using the STE this paper investigates the modeling and controlling problems of combined automata constructed in the ways of parallel, serial and feedback. By representing the states, input and output symbols in vector forms, the transition and output functions are expressed as algebraic equations of the states and inputs. Based on such algebraic descriptions, the control problems of combined automata, including output control and state control, are considered, and two necessary and sufficient conditions are presented for the controllability, by which two algorithms are established to find out all the control strings that make a combined automaton go to a target state or produce a desired output. The results are quite different from existing methods and provide a new angle and means to understand and analyze the dynamics of combined automata.
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.
