Algebraic Properties And Tree Language Of Finite Fuzzy Tree Automata

Fuzzy finite automata play an important role in dealing with problems with some dynamic and uncertain systems.Thus fuzzy finite automata was used to be a powerful tool for researching and treatment contains fuzziness natural language in computer field. Due to admissible relations and admissible partitions of finite fuzzy tree au-tomata, we have similar concepts on finite fuzzy tree automata. By used of algebraic methods,this paper study the homomorphism of finite fuzzy tree automata, we also extended admissible partition, admissible relation, covering relations and tree language of finite fuzzy automaton to finite fuzzy tree automata. These research work obtained have extended the previous results of the finite fuzzy automaton and have better dis-cussed some algebraic properties and structure of the finite fuzzy tree automata, finally we got tree language of finite fuzzy tree automata.This paper is composed of five parts, each part of the former four parts is as a chapter, and the last part is concluding remarks.In chapter 1,we introduces basic situation of finite fuzzy tree automata,and re-search background in this paper is set forth, some basic concepts and marks of finite fuzzy tree automata are given.In chapter 2, we discuss algebraic properties of the two finite fuzzy tree automaton under strong homomorphism.The main results of chapter 2 are as follows:Theorem 2.2.1 Let A1=(A1;∑,δ1,β1) and A2=(A2,∑,δ2,β2) be two fuzzy tree automaton in L.α:A1�'A2 be a homomorphism and be a injection, then a is a strong homomorphism if and only ifδ2((α(a1),α(a2),…,α(an)),α(a),δ)=δ1((a1,a2,…,an),a,δ)β2(α(a))=β1(a), (?)ai,a∈A1,i=1,…,n,δ∈∑n.Theorem 2.2.2 Let A1=(A1,∑,δ1,β1) and A2=(A2,∑,δ2,β2) be two fuzzy tree automaton in L,a:A1�'A2 is a epimorphism, if A1 is conjoint, then A2 is also. Theorem 2.2.3 Let A1=(A1,∑,δ1,β1)and A2=(A2,∑,δ2,β2)be two fuzzy tree automaton in L,α:A1�'A2 is a homomorphism,if it is a epimorphism,then A1 is complete iff A2 is also.In chapter 3,we introduce admissible partition,admissible relation and cover-ing relations of the concepts on finite fuzzy tree automata we main study algebraic properties about admissible partition,admissible relation,covering relations and tree language of finite fuzzy tree automata.The main results of chapter 3 are as follows:Theorem 3.2.1.4 Let A1=(A1,∑,δ1,β1)and A2=(A2,∑,δ2,β2)be two fuzzy tree automaton in L,α:A1�'A2 is a natural homomorphism,Then there exists a isomorphismφ:A1/kerα�'A2,such thatα=φ·α'.Whereα':A1�'A1/kerαis a natural homomorphism.Theorem 3.2.2.1 Let A1=(A1,∑,δ1,β1)and A2=(A2,∑,δ2,β2)be two fuzzy tree automaton in L,if A1 and A2 is equivalent,then L(A1)=L(A2).Theorem 3.2.2.2 Let A=(A,∑,δ,β)be a complete fuzzy tree automaton in L, (?)a∈A,β(a)>0,t=(t1,…,tn)∈T∑,δ∈∑n,n≥0,then ti∈L(A),i=1,…,n iff t∈L(A).Theorem 3.2.2.3 Let A1=(A1,∑,δ1,β1)and A2=(A2,∑,δ2,β2)be two fuzzy tree automaton in L,ifα:A1�'A2 is a natural homomorphism,then L(A1)(?)L(A2).Theorem3.3.1.3 Let A=(A,∑,δ,β)be a finite fuzzy tree automatons in L.π={Hi│i=1,…,u,u+1,…,r} andτ={Kj│j=1,…,v,v+1,…,s} be two admissible partitions on A which areδ- orthogonal.Then A≤A/π∧A/τ.Theorem 3.3.1.5 Let A=(A,∑,δ,β) be a finite fuzzy tree automaton in L.π={Hi│i=1,…,n} is an admissible partition of A,thenπis maximal iff A/πis irreducible.Theorem 3.3.1.6 Let A=(A,∑,δ,β) be a finite fuzzy tree automaton in L.│A│=n≥2,there exists an maximal admissible partitionπ,on A.Ifτis an admissible partition of A withπandτareδ-orthogonal,then A≤N1∧…∧m-1Nmwhere N1,…,Nm are irreducible finite fuzzy tree automatons,the state sets Ai of Ni are such that│Ai│<n.Theorem 3.3.1.7 Let A1=(A1,∑,δ1,β1),A2=(A2,∑,δ2,β2)be two finite fuzzy tree automaton in L.if A1≤A2,then L(A1)(?)L(A2). Theorem 3.3.2.2 Let A1=(A1,∑,δ1,β1),A2= (A2,∑,δ2,β2) be two fuzzy tree automaton in L. a:A1�'A2 is a natural homomorphism.ρ1,ρ2 be two admissible relations on fuzzy tree automaton A1,A2 respectively,ρ1(?)kerα. Then there ex-ists a natural homomorphismα':A1/ρ1�'A2/ρ2 such that the following diagram is commutative:whereφ1,φ2 are natural homomorphisms.In chapter 4, we introduce the union and intersection on tree languages of finite fuzzy tree automata.The main results of chapter 3 are as follows:Theorem 4.1 Let A1= (A1,∑,δ1,β1),A2=(A2,∑,δ2,β2) be two complete finite fuzzy tree automaton in L. if A1≤A2, then L(Ai∪42)=L(A1)∪L(A2).Theorem 4.2 Let A1=(A1,∑,δ1,β1), A2=(A2,∑,δ2,β2) be two complete finite fuzzy tree automaton in L. if A1≤A2, then L(A1∩A2)= L(A1)∩L(A2).Chapter 5 is the conclusion part.It summarizes the main results of this paper and the further investigation of this field is illustrated.