Construction and Synchronization Analysis of State Power set Automata Based on Algebraic Methods

To address the cumbersome and inefficient process of constructing state power set automata based on traditional definitions, this paper proposes a new algebraic construction method and performs synchronization analysis of finite state machines (FSMs) using this method. The main objectives of this study are to simplify the construction process of state power set automata, enhance the efficiency of this process, and provide a new method for the synchronization analysis of FSMs. This paper innovatively integrates the semitensor product (STP) theory of matrices with the theory of FSMs. Firstly, by introducing the concept of the power set reachability matrix, the structure matrix of the state power set automaton is calculated. Secondly, in order to effectively analyze the synchronization of FSMs and find the shortest reset word, we propose a synchronization determination theorem and design a corresponding algorithm. The results indicate that this method significantly simplifies the construction process of state power set automata while improving both the accuracy and efficiency of synchronization determination. In summary, this study not only offers a new approach for the construction of state power set automata and the synchronization analysis of FSMs, but also provides significant theoretical support for research in automata theory and related fields.