Homomorphism in Intuitionistic Fuzzy Automata

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1 International Journal of Fuzzy Mathematics Systems. ISSN Volume 3, Number 1 (2013), pp Research India Publications Homomorphism in Intuitionistic Fuzzy Automata A. Uma M. Rajasekar Mathematics Section, Faculty of Engineering Technology, Annamalai University, Annamalainagar, Chidambaram, Tamil Nadu, India uma.feat@yahoo.com, mrajdiv@yahoo.com Abstract In this paper we introduce some properties of homomorphism in Intuitionistic Fuzzy Automata. AMS subject classification: 18B20. Keywords: Intuitionistic Fuzzy Automata. 1. Introduction The concept of intuitionistic fuzzy set was introduced by K.T. Atanassov [1], as a generalization of the notion of fuzzy set. Using the notion of intuitionistic fuzzy sets [1], it is possible to obtain intuitionistic fuzzy language [3]. J.B. Jun [2] defined a homomorphism in intuitionistic fuzzy automata. We introduce some properties of an intuitionistic fuzzy automata with homomorphism. 2. Preliminaries 2.1. Fuzzy Set [1] Let a set E be fixed. A Fuzzy set A in E is an object having the form A ={< x, μ A (x) > x E} where, the function μ A (x) :E [0, 1] define the degree of membership of the element x E to the set A for every x E,0 μ A (x) 1.

2 A. Uma M. Rajasekar 2.2. Automata [4] A non-deterministic finite Automaton is a triple A = (Q, X, δ) where Q is a finite set (the set of states), X is an alphabet δ is a subset of Q X Q, called the set of transitions. Two transitions (p, a, q) (p, a, q ) are consecutive if q = p. Consider a word a 0, a 1,..., a n 1 with a i X. A run α in A is sequence of states a 0 a 1 a n 1 q 0 q 1 q 2,..., q n 1 q n 2.3. Fuzzy Automata [3] A Fuzzy Automaton is a triple A = (Q, X, δ), where Q is a nonempty set of states of A, X is a monoid (the input monoid of M), with identity e, δ is a Fuzzy subset of Q X Q, i.e., a map δ : Q X Q [0, 1], such that q, p Q, x, y X. δ(q, e, p) = { 1 if q = p 0 if q = p δ(q, xy, p) = {δ(q, x, r) δ(r, y, p) :r Q} 2.4. Intuitionistic Fuzzy Set [1] Let a set E be fixed. An Intuitionistic Fuzzy set A in E is an object having the form A ={<x, μ A (x), γ A (x) > x E} where, the functions μ A (x) :E [0, 1] γ A (x) :E [0, 1] define the degree of membership the degree of nonmembership of the element x E to the set A, the subset of E respectively, for every x E,0 μ A (x) + γ A (x) Intuitionistic Fuzzy Automata [2] An Intuitionistic Fuzzy Automaton is a triple A = (Q, X, δ), where Q is a set of states of A, X is a monoid (the input monoid of M with identity e), δ is an Intuitionistic Fuzzy subset of Q X Q, such that q, p Q, x, y X. { 1 if q = p δ 1 (q, e, p) = 0 if q = p δ 2 (q, e, p) = { 1 if q = p 0 if q = p δ 1 (q, xy, p) = {δ 1 (q, x, r) δ 1 (r, y, p) :r Q}, δ 2 (q, xy, p) = {δ 2 (q, x, r) δ 2 (r, y, p) :r Q}.

3 Homomorphism in Intuitionistic Fuzzy Automata 2.6. Homomorphism between Automata Let A 1 = (Q 1, X 1, δ 1 ) A 2 = (Q 2, X 2, δ 2 ) be two finite automata. A pair (α, β) of mappings, α : Q 1 Q 2 β : X 1 X 2 is called a homomorphism, written (α, β) :A 1 A 2,if α(δ 1 (q 1, a)) = δ 2 (α(q 1 ), β(a)) q 1 Q 1 a X Homomorphism between Fuzzy Automata [3] Let A 1 = (Q 1, X 1, μ 1 ) A 2 = (Q 2, X 2, μ 2 ) be ffsms. A pair (α, β) of mappings, α : Q 1 Q 2 β : X 1 X 2 is called a homomorphism, written (α, β) :A 1 A 2, if μ 1 (q, x, p) μ 2 (α(q), β(x), α(p)) q, p Q 1 x X 1. The pair (α, β) is called a strong homomorphism if μ 2 (α(q), β(x), α(p)) = {μ 1 (q, x, t) t Q 1, α(t) = α(p)} q, p Q 1 x X Homomorphism between Intuitionistic Fuzzy Automata [2] Let A 1 = (Q 1, X 1, μ 1, γ 1 ) A 2 = (Q 2, X 2, μ 2, γ 2 ) be iffsms. A pair (α, β) of mappings, α : Q 1 Q 2 β : X 1 X 2 is called an intuitionistic fuzzy homomorphism, written (α, β) :A 1 A 2,if μ 1 (q, x, p) μ 2 (α(q), β(x), α(p)) γ 1 (q, x, p) γ 2 (α(q), β(x), α(p)) p, q Q 1 x X 1. The pair (α, β) is called a strong intuitionistic fuzzy homomorphism if μ 2 (α(q), β(x), α(p)) = {μ 1 (q, x, t) t Q, α(t) = α(p)} γ 2 (α(q), β(x), α(p)) = {γ 1 (q, x, t) t Q, α(t) = α(p)} q, p Q 1 x X Some properties of Homomorphism in Inutitionistic Fuzzy Automata Lemma 3.1. Let A 1 = (Q 1, X 1, μ 1, γ 1 ) A 2 = (Q 2, X 2, μ 2, γ 2 ) be two iffsms. Let (α, β) :A 1 A 2 be a strong intuitionistic homomorphism. Then q, r, Q 1, x X 1, if μ 2 (α(q), β(x), α(r)) > 0 γ 2 (α(q), β(x), α(r)) < 1,

4 A. Uma M. Rajasekar then t Q 1 such that μ 1 (q, x, t) > 0, γ 1 (q, x, t) < 1 α(t) = α(r). Furthermore, p Q if α(p) = α(q), then Proof. Let p, q, r Q 1, x X 1, Then μ 1 (q, x, t) μ 1 (p, x, r) γ 1 (q, x, t) γ 1 (p, x, r). μ 2 (α(q), β(x), α(r)) > 0 γ 2 (α(q), β(x), α(r)) < 1. {μ 1 (q, x, s) s Q 1, α(s) = α(r)} > 0 {γ 1 (q, x, s) s Q 1, α(s) = α(r)} < 1. Since Q 1 is finite, t Q 1 such that α(t) = α(r) suppose α(p) = α(q). Then μ 1 (q, x, t) = {μ 1 (q, x, s) s Q 1, α(s) = α(r)} > 0 γ 1 (q, x, t) = {γ 1 (q, x, s) s Q 1, α(s) = α(r)} < 1 μ 1 (q, x, t) = μ 2 (α(q), β(x), α(r)) = μ 2 (α(p), β(x), α(r)) μ 1 (p, x, r) γ 1 (q, x, t) = γ 2 (α(q), β(x), α(r)) = γ 2 (α(p), β(x), α(r)) γ 1 (p, x, r) Definition 3.2. Let A 1 = (Q 1, X 1, μ 1, γ 1 ) A 2 = (Q 2, X 2, μ 2, γ 2 ) be two iffsms. Let (α, β) :A 1 A 2 be an Intuitionistic fuzzy homomorphism. Define β : X 1 X 2 by β (λ) = λ β (ua) = β (u)β(a) u X 1, a X 1. Theorem 3.3. Let A 1 = (Q 1, X 1, μ 1, γ 1 ) A 2 = (Q 2, X 2, μ 2, γ 2 ) be two iffsms. Let (α, β) :A 1 A 2 be an Intuitionistic fuzzy homomorphism. Then μ 1 (q, x, p) μ 2 (α(q), β (x), α(p)) γ 1 (q, x, p) μ 2 (α(q), β (x), α(p)) q, p Q 1 x X 1.

5 Homomorphism in Intuitionistic Fuzzy Automata Proof. Let q, p Q 1 x X1. We prove the result by induction on x =n. If n = 0, then x = λ β (x) = β (λ) = λ. Now if q = p, then If q = p, then μ 1 (q, λ, p) = 1 = μ 2 (α(q), λ, α(p)) γ1 (q, λ, p) = 0 = γ 2 (α(q), λ, α(p)) μ 1 (q, λ, p) = 0 μ 2 (α(q), λ, α(p)) γ1 (q, λ, p) = 1 γ 2 (α(q), λ, α(p)). Suppose now the result is true y X such that y =n 1, n>0. Let x = ya y X 1, a X 1 y =n 1. Now μ 1 (q, x, p) = μ 1 (q, ya, p) = {μ 1 (q, y, r) μ 1 (r, a, p) r Q 1} {μ 2 (α(q), β (y), α(r)) μ 2 (α(r), β(a), α(p)) r Q 1} {μ 2 (α(q), β (y), r ) μ 2 (r, β (a), α(p)) r Q 2 } = μ2 (α(q), β (y)β(a), α(p)) = μ 2 (α(q), β (ya), α(p)) = μ 2 (α(q), β (x), α(p)). γ1 (q, x, p) = μ 1 (q, ya, p) = {γ1 (q, y, r) μ 1 (r, a, p) r Q 1} {γ 2 (α(q), β (y), α(r)) γ 2 (α(r), β(a), α(p)) r Q 1} {γ 2 (α(q), β (y), r ) γ 2 (r, β (a), α(p)) r Q 2 } = γ 2 (α(q), β (y)β(a), α(p)) = γ 2 (α(q), β (ya), α(p)) = γ 2 (α(q), β (x), α(p)).

6 A. Uma M. Rajasekar Theorem 3.4. Let A 1 = (Q 1, X 1, μ 1, γ 1 ) A 2 = (Q 2, X 2, μ 2, γ 2 ) be two iffsms. Let (α, β) :A 1 A 2 be an strong homomorphism. Then α is one-one if only if μ 1 (q, x, p) = μ 2 (α(q), β (x), α(p)) γ 1 (q, x, p) = γ 2 (α(q), β (x), α(p) q, p Q 1 x X 1. Proof. Suppose α is one-one. Let p, q Q 1 x X1. Let x =n. We prove the result by induction on n. Let n = 0. Then x = λ β (λ) = λ. Now α(q) = α(p)iffq = p. Hence μ 1 (q, λ, p) = 1, γ 1 (q, λ, p) = 0iff μ 2 (α(q), β (λ), α(p)) = 1, γ 2 (α(q), β (λ), α(p)) = 0. Suppose the result is true y X1, y =n 1, n>0. Let x = ya, y =n 1, y X1, a X 1. Then μ 2 (α(q), β (x), α(p)) = μ 2 (α(q), β(ya), α(p)) = μ 2 (α(q), β (y)β(a), α(p) = {μ 2 (α(q), β (y), α(r)) μ 2 (α(r), β(a), α(p)) r Q 1} = {μ 1 ((q, y, r) μ 1(r, a, p) r Q 1 } = μ 1 (q, ya, p) = μ 1 (q, x, p). γ2 (α(q), β (x), α(p)) = γ2 (α(q), β(ya), α(p)) = γ2 (α(q), β (y)β(a), α(p)) = {γ 2 (α(q), β (y), α(r)) γ 2 (α(r), β(a), α(p)) r Q 1 } = {γ 1 ((q, y, r) γ 1(r, a, p) r Q 1 } = γ1 (q, ya, p) = γ1 (q, x, p). Conversely, let q, p Q 1 let α(q) = α(p). Then Hence q = p, i.e., α is one-one. 1 = μ 2 (α(q), λ, α(p)) = μ 1 (q, λ, p) 0 = γ2 (α(q), λ, α(p)) = γ 1 (q, λ, p).

7 Homomorphism in Intuitionistic Fuzzy Automata References [1] K. Atanassov, Intuitionistic Fuzzy sets, Fuzzy Sets Systems, 20:87 96, [2] Y.B. Jun, Intuitionistic Fuzzy Finite State machines, J. Appl. Math. Computing, 17: , [3] J.N. Mordeson D.S. Malik, Fuzzy Automata languages, Theory Applications, Chapman Hall/CRC, [4] D. Perrin J.-E.Pin, Semigroups Automata on Infinite words, in semigroups, formal languages groups, Kluwer Acad. Publ., Dordrecht, 49 72, 1995.