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 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.

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