Implementation of Automaton Product Networks: From Formal Language to Algebraic Models

This paper investigates the construction of automaton product networks based on the algebraic model of automata. By defining the vector forms of states and events in automaton product networks, we establish algebraic models for these networks. Based on this algebraic model, we derive the state transition function in vector form for automaton product networks. Utilizing the equivalence relationship between the established state transition function and the dynamic structure matrix, we design synthesis formulas for automaton product networks in two scenarios: the shared event set scenario and the general scenario. The synthesis formulas we designed can handle complex network operations, improving computational efficiency and accuracy, while also avoiding the common storage complexity explosion problem found in algebraic methods.