On Finite Group Automata And Their Extension

Automata theory is a mathematical theory, which essentially studies the functions,the structures, and the relationships between them for discrete mathematics systems, and the purpose is to analyze and synthesize automata. With the development of modern science and technology, and the request of basic theory on computer science , automata theory has already been an important foundation of the theory and application for many subjects.In this paper, we first simply introduced the main achievements on group automata theory in mathematics. Then we studied the structure and the algebra characteristic and the states of finite group automa. And at the same time, we discussed both the construction and the decomposition of finite group automata, we also simply introduced general state machine and finite semigroup automata and binary finite group automata.In this paper we used the methods of rings and modules theory and graph theory to study on finite group automata and got some results.In a view from whole,this paper gave many ways to study on finite group automata in further.There are three parts in this paper, each part as a charpter.In charpter one, we introduced the main achievements on group automata theory in mathematics and gave basic concepts and notation on finite group automa.Charpter two is about both the construction and the decomposition of finite group automata and gave the principles to judge minimal and faithfullness of finite group automata.The following are the main results of Charpter two.Theorem2.2.1 Let M i = (Q i , G ,δi)be finite group automata for i = 1,2 andω: Q2×G2→G1 is a mapping which satisfies withω( q2 , g 2 h2 ) =ω( q2 , g 2 )ω(δ2 ( q2 , g 2 ), h2),where q2∈Q2 , g 2 ,h2∈G2,defineδωas follow:δω(( q1 , q2 ), g 2 ) =δ1 ( q1 ,ω( q2 , g 2 ))×δ2 ( q2 , g 2 ), ?( q1 , q2 )∈Q1×Q2,then M 1ωM 2 = (Q 1×Q2 , G2 ,δω)is a finite group automaton,which is called the cascade product of M 1 and M 2.Theorem2.3.1 Let M = (Q , G ,δ)be a finite group automaton,then M is minimal if and only if q1 e = q2 e,for q1 , q2∈Q,where e is the identity of G , implies q1 = q2.Theorem2.3.3 Let M = (δ1 , G1 , Q , G2 ,δ2)be a binary finite group automaton,then M is Minimal if and only if if e1 ( q1 e 2 ) = e1 ( q2 e2),for q1 , q2∈Q,and ei is the identity of Gi ,for i = 1,2,then q1 = q2.Theorem2.3.5 Let M = (Q , G ,δ)be a finite group automaton,then M is faithful if...