Finite Automata Over Finite Fields

Automata theory is a mathematic theory, which essentially researches the functions, the structures, and the relationships between them for discrete mathematics systems, and for the purpose of researching how to analyze and synthesize automata. Along with the development of modern science and technology,the automata theory has already become the important foundations of theory and application for many subject.In this paper, we first summarized the mainly achievements of researches on the three directions: linear automata, circulate automata, and invertible automata, including the internal and the external. Then we mainly researched finite automata over finite fields and got the following achievements on the linear automata ,affine automata, and the periods and the factorization of polynomials :A. Autonomous linear automataFor the first time, we presented the equivalence classes of autonomous linear automaton which have different elementary divisor, and defined all kinds of the autonomous linear automaton for different type distinctly.B. Autonomous affine automata1. It is the first time that we presented the equivalence classes of autonomous affine auotomaton which have different elementary divisor ,and defined all kinds of the autonomous affine automaton for different type distinctly. 2.we used maximal sub-group theory of multiple valued logic theory , discussed the problem of changing the signals of all nodes , and researched the problem of the conjugate classes of autonomous affine automaton.C. Period and factorization of polynomial1. We presented a new method to calculate the periods of different polynomials ,according to it,we debugged successful the procedure to realize on the computer.2. we summarized many different methods which to calculate the polynomial factorization,and studied the factoring of xn-1,and debugged successful a procedure on the computer.