Some Discussions On Intuitionistic Fuzzy Finite State Machines

The theory of fuzzy sets proposed by American cybernetics expert Zadeh in 1962 at first, and the theory has achieved a great success in various fields. Out of several higher or-der fuzzy sets intuitionistic fuzzy sets were introduced by Atanassov in 1983. Intuitionistic fuzzy sets have been found to be more flexible and accurate to deal with problems for some uncertain systems than fuzzy sets.Using the notion of intuitionistic fuzzy sets and fuzzy finite state automata, Young Bae Jun introduces the concepts of intuitionistic fuzzy finite state machines(iffsm) as a generalization of fuzzy finite state automata. Since then Young Bae Jun and some scholar have done a lot of studies, especialy some applications of intuitionistic fuzzy finite state machines have been done in recently. They mainly discuss some properties of associated intuitionistic fuzzy finite state machines, such as intuitionistic successors, intu-itionistic exchange property, intuitionistic connected property and so on. At the same time, they also study intuitionistic subsystems, intuitionistic submachines, intuitionistic q-related, intuitionistic separability,entropy and intuitionistic fuzzy finite switchboard state machines and so on. The results not only have enriched the contents of the automata theory, but also proposed lots of new questions.In this paper, we discuss several properties of intuitionistic fuzzy finite state machines utilizing algebraic techniques. At the same time, we introduced the concept of cascade products, intuitionistic direct product, intuitionistic wreath product, intuitionistic cartesian product of intuitionistic fuzzy finite state machines, give relations between product and covering, Product combination and separable property of intuitionistic fuzzy finite state machines.This paper is composed of four parts, each part of the former three parts is a chapter, and the last part is concluding the remarks.In Chapter 1, we simply introduce the occurrence, development situation of intuitionistic fuzzy finite state machines simply.Some basic concepts and notions of intuitionistic fuzzy finite state machines are given. In Chapter 2, we discuss intuitionistic strongly connected, intuitionistic exchanged, cyclic, commutative properties of intuitionistic fuzzy finite state machines,and study in-tuitionistic subsystems, intuitionistic strongly subsystems and intuitionistic submachine of intuitionistic fuzzy finite state machines. The main results of chapter 2 are as follows:Theorem 2.2.1 Let mA=(Q1,X1,A) and mB=(Q2,X2,B) be two intuitionistic fuzzy finite state machines, let(α,β):mA�'mB be a homomorphism, if pi,q1∈Qi,Pi∈I(q1), then:(1)α(p1)∈I(α(q1)),(2)a(I(q1)) (?) I(α(q1)).Theorem 2.2.6 Let m=(Q,X,A) be a intuitionistic fuzzy finite state machine, Q be a strongly subsystem of m, then Q be a strongly subsystem of m, or theorem is wrong.Theorem 2.2.8 Let mA=(Q1,X,A) and mB=(Q2,X,B) be two intuitionistic fuzzy finite state machines, let (α,β):mA�'mB be a onto homomorphism, andβ:X�'X be a Identical mapping, if Q be a intuitionistic subsystem of mA, then a(Q) be a intuitionistic subsystem of mBTheorem 2.2.10 Let mA=(Qi,X, A) and mB=(Q2,X,B)be two intuitionistic fuzzy finite state machines, let (α,β):mA�'mB be a onto homomorphism, if mA is intuitionistic strongly connected, then mB is also.Theorem 2.2.13 Let mA=(Q1,X1,A) and mB=(Q2,X2,B)be two intuition-istic fuzzy finite state machines, let (α,β):mA�'mB be a onto homomorphism,and a is a bijective, if mA satisfies the intuitionistic exchange property, then mB satisfies the intuitionistic exchange property too. Theorem 2.2.15 Let mA=(Qi, X1,A) and mB=(Q2, x2, B)be two intuitionistic fuzzy finite state machines, let(α,β):mA�'mB be a homomorphism, and a is a surjective, if mA satisfies the intuitionistic cyclic, then mB satisfies the intuitionistic cyclic too.Theorem 2.2.23 Let mA=(Q1,X,A) and mB=(Q2, X, B)be two intuitionistic fuzzy finite state machines, let (α,β):mA�'mB be a onto homomorphism,and a is injective.if mA is commutable, then mB is also.Theorem 2.2.25 m=(Q,X,A) is a intuitionistic fuzzy finite state machine, if m is commutable, then m satisfies the intuitionistic exchange property.In chapter 3, we give the intuitionistic cartesian product, intuitionistic direct prod-uct,intuitionistic Cascade product, and intuitionistic wreath product, the definition of the intuitionistic fuzzy finite state machine, as well as their properties, and we also study the relation between intuitionistic direct product and covering, Cascade Product combination and separable property. The main results of chapter 3 are as follows: Theorem 3.2.4 Let mA=(Q1,X1,A1)andmB=(Q2,X2,A2)be two intuitionistic fuzzy finitc state machines, let X1∩x2=φ,m1·m2=(Q1×Q2,X1∪X2,A1·A2)is the intuitionistic cartesian product of M1 and A2.(1)m1·m2 is complete,if and only if m1 and m2 are complete.(2)m1·m2 is retrieveable,then m1 and m2 are retrieveable.(3)m1与m2 are independent,then m1·m2 is independent.Theorem 3.2.9 Let mA=(Q1,X1,A1)and mB=(Q2,X2,A2)be two intuitionistic fuzzy finite state machines,letm=m1×m2=(Q1×Q2,X1×X2,A1×A2)is a intuitionistic fuzzy finite state machine,The m ain results are as follows:(1)If m is intuitionistic strongly connected,then m1 and m2are intuitionistic strongly connected.(2)If m is cyclic,then m1 and m2 are cyclic.(3)If m is switchable,then m1 and m2 are switchable.(4)If m1 and m2 are exchanged,then m is exchanged.Theorem 3.2.10 Let m1=(Q1,X1,A1)and m2=(Q2,X2,A2)be two intuitionistic fuzzy finite state machines,let m=(Ql×Q2,X2,Aω) is a intuitionistic Cascade product of m1 and m2,The main results are as follows:(1)If m is intuitionistic strongly connected,then m1 and m2 are intuitionistic strongly connected.(2)If m is cyclic,then m1 and m2 are cyclic.Theorem 3.2.11 Let m1=(Ql,X1,A1)and m2=(Q2,X2,A2)be two intuitionistic fuzzy finite state machines,let m=(Q1×Q2,X1Q2.×X2,A°)=m1οm2 is a intuitionistic wreath product of m1 and m2.If m is complete,then m1 and m2 are complete.Theorem 3.2.14 Let m1=(Q1,X1,A1)and m2=(Q2,X2,A2)be two intuitionistic fuzzy finite state machines.(1) Then IFTS(m1Λm2)≥IFTs(m1)ΛIFTS(m2).(2) Then IFTS(m1×m2)≥IFTS(m1)×IFTS(m2).Theorem 3.2.16 Let m1=(Q1,X1,A1)and m2=(Q2,X2,A2)be two intuitionistic fuzzy finite state machines.If m=m1ωm2,(q1,q2),(p1,p2)∈Q1×Q2,(?)y=y1y2…yn∈X2*,Then,μAω*((q1,q2),y,(p1,p2)) =∨{pA1*(q1,ω(q2,y1)ω(q21,y2)…ω(q2n-1,yn),p1)∧μA2(q2,y1,q21)∧μA2(q21,y2,q22)∧…∧μA2(q2n-1,yn,p2)|q2i∈Q2,i=1,2,…,n一1｝γAω*((Q1,q2),y,(p1,p2)) =∧{γA1*(q1,ω(q2,y1)ω(q21,y2)…ω(q2n-1,yn),p1)∨γA2(q2,y1,q21)∨γA2(q21,y2,q22)∨…∨γA2(q2n-1,yn,p2)|q2i∈Q2,i=1,2,…,n一1｝Theorem 3.2.17 Let m1=(Q1,X1,A1)and m2=(Q2,X2,A2)be two intuitionistic fuzzy finite state machines.If Im(A2)={0,1｝,then Aωis Intuitionistic separable. Theorem 3.2.20 Letmi=(Qi,Xi,Ai)be two intuitionistic fuzzy finite state ma-chines.i=1,2,3 Then (ITF(m1)οIFTS(m2))οIFTS(m3)(?)ITFS(m1)ο(IFTS(m2)οIFTS(m3)).Chapter 4 is the conclusion part. It summarizes the main results of this paper and the further investigation of this field is illustrated.