APPLICATIONS OF A GROUP IN GENERAL FUZZY AUTOMATA. Communicated by S. Alikhani
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1 Algebraic Structures and Their Applications Vol. 4 No. 2 ( 2017 ) pp APPLICATIONS OF A GROUP IN GENERAL FUZZY AUTOMATA M. HORRY Communicated by S. Alikhani Abstract. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2) be a general fuzzy automaton and the set of its states be a group. The aim of this paper is the study of applications of a group in a general fuzzy automaton. For this purpose, we define the concepts of fuzzy normal kernel of a general fuzzy automaton, fuzzy kernel of a general fuzzy automaton, adjustable, multiplicative. Then we obtain the relationships between them. 1. Introduction The concept of fuzzy automata was introduced by Wee in 1967 [11]. Let Σ be a set. A word in Σ is the product of a finite sequence of elements in Σ. Λ will denote the empty word and Σ the set of all words on Σ. The length l(x) of the word x Σ is the number of its letters, so l(λ) = 0. For a nonempty set X, P (X) will denote the set of all fuzzy sets on X and P (X) will denote the set of all subsets on X. A deterministic finite-state automaton is a five-tuple denoted as A = (Q, Σ, f, T, s), where Q is a finite set of states, Σ is a finite set of input symbols, the total function f from Q Σ into DOI: MSC(2010): Primary:18B20 Keywords: (General) Fuzzy automata, group, normal subgroup, fuzzy subgroup, fuzzy normal subgroup. Received: 10 November 2017, Accepted: 17 July 2018 Corresponding author c 2017 Yazd University. 57
2 58 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Q is the state transition, T is a subset of Q of accepting states and s Q is the initial state. A word x = x 1 x 2... x n Σ is said to be accepted by A if there exist states q 0, q 1,..., q n satisfying (1) q 0 = s (2) f(q i 1, x i ) = q i for i = 1, 2,..., n, (3)q n T. The empty word is accepted by A if and only if s T. A nondeterministic finite-state automaton is a five-tuple denoted as A = (Q, Σ, f, T, s), where Q is a finite set of states, Σ is a finite set of input symbols, the partial function f from Q Σ into P (Q) is the state transition, T is a subset of Q of accepting states and s Q is the initial state. A fuzzy finite-state automaton (FFA) is a six-tuple F = (Q, Σ, R, Z, δ, ω), where Q is a finite set of states, Σ is a finite set of input symbols, R is the initial state of F, Z is a finite set of output symbols, δ : Q Σ Q [0, 1] is the fuzzy transition function which is used to map a state (current state) into another state (next state) upon an input symbol, attributing a value in the interval [0, 1] and ω : Q Z is the output function. Associated with each fuzzy transition, there is a membership value in [0, 1] called the weight of the transition. The transition from state q i (current state) to state q j (next state) upon input a k is denoted by δ(q i, a k, q j ). We use this notation to refer both to a transition and its weight. Whenever δ(q i, a k, q j ) is used as a value, it refers to the weight of the transition. Otherwise, it specifies the transition itself. The set of all transitions of F will be denoted by. The above definition is generally accepted as a formal definition of a fuzzy finite-state automaton [4, 5, 6, 7, 8, 9]. In 2004, M. Doostfatemeh and S.C. Kremer extended the notion of fuzzy automata and introduced the notion of general fuzzy automata [1]. In this paper, we define the concepts of fuzzy normal kernel of a general fuzzy automaton, fuzzy kernel of a general fuzzy automaton, adjustable, multiplicative and obtain the relationships between them. Definition 1.1. [1] A general fuzzy automaton (GFA) is an eight-tuple machine F = (Q, Σ, R, Z, δ, ω, F 1, F 2 ), where (i) Q is a finite set of states, Q = {q 1, q 2,..., q n }, (ii) Σ is a finite set of input symbols, Σ = {a 1, a 2,..., a m }, (iii) R is the set of fuzzy start states, R P (Q), (iv) Z is a finite set of output symbols, Z = {b 1, b 2,..., b k }, (v) ω : Q Z is the output function, (vi) δ : (Q [0, 1]) Σ Q [0, 1] is the augmented transition function,
3 Alg. Struc. Appl. Vol. 4 No. 2 (2017) (vii) F 1 : [0, 1] [0, 1] [0, 1] is the membership assignment function, (viii) F 2 : [0, 1] [0, 1] is called the multi-membership resolution function. We note that the function F 1 (µ, δ) has two parameters µ and δ, where µ is the membership value of a predecessor and δ is the weight of a transition and δ : Q Σ Q [0, 1] is the fuzzy transition function which is used to map a state (current state) into another state (next state) upon an input symbol, attributing a value in the interval [0, 1]. In this definition, the process that takes place upon the transition from state q i to q j on input a k is represented as: µ t+1 (q j ) = δ((q i, µ t (q i )), a k, q j ) = F 1 (µ t (q i ), δ(q i, a k, q j )). This means that t {0, 1, 2,, n} and the membership value (mv) of the state q j at time t + 1 is computed by function F 1 using both the membership value of q i at time t and the weight of the transition. The usual options for the function F 1 (µ, δ) are max{µ, δ}, min{µ, δ} and (µ + δ)/2. The multi-membership resolution function resolves the multi-membership active states and assigns a single membership value to them. and Let Q act (t i ) be the fuzzy set of all active states at time t i, i 0. We have Q act (t 0 ) = R Q act (t i ) = {(q, µ t i (q)) : q Q act (t i 1 ), a Σ, δ(q, a, q) }, i 1. Since Q act (t i ) is a fuzzy set, in order to show that a state q belongs to Q act (t i ) and T is a subset of Q act (t i ), we should write: q Domain(Q act (t i )) and T Domain(Q act (t i )). Hereafter, we simply denote them as: q Q act (t i ) and T Q act (t i ). The combination of the operations of functions F 1 and F 2 on a multi-membership state q j leads to the multi-membership resolution algorithm. Algorithm 1. [1] (Multi-membership resolution) If there are several simultaneous transitions to the active state q j at time t + 1, the following algorithm will assign a unified membership value to it: (1) Each transition weight δ(q i, a k, q j ) together with µ t (q i ), will be processed by the membership assignment function F 1, and will produce a membership value. Call this v i. v i = δ((q i, µ t (q i )), a k, q j ) = F 1 (µ t (q i ), δ(q i, a k, q j )). (2) These membership values are not necessarily equal. Hence, they need to be processed by the multi-membership resolution function F 2. (3) The result produced by F 2 will be assigned as the instantaneous membership value of the active state q j, µ t+1 (q j ) = F n 2 [v i ] = F n 2 [F 1 (µ t (q i ), δ(q i, a k, q j ))]. i=1 i=1
4 60 Alg. Struc. Appl. Vol. 4 No. 2 (2017) where n is the number of simultaneous transitions to the active state q j at time t + 1. δ(q i, a k, q j ) is the weight of a transition from q i to q j upon input a k. µ t (q i ) is the membership value of q i at time t. µ t+1 (q j ) is the final membership value of q j at time t + 1. Definition 1.2. [12] Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton. define max-min general fuzzy automata as F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) such that : We δ : Q act Σ Q [0, 1] where Q act = {Q act (t 0 ), Q act (t 1 ), Q act (t 2 ),... } and for all i 0, δ ((q, µ t i 1, q = p, (q)), Λ, p) = 0, otherwise Also, if the input at time t i be u i, where u i Σ, 1 i n, then δ ((q, µ t i 1 (q)), u i, p) = δ((q, µ t i 1 (q)), u i, p), δ ((q, µ t i 1 (q)), u i u i+1, p) = ( δ((q, µ t i 1 (q)), u i, q ) δ((q, µ t i (q )), u i+1, p)), q Q act(t i ) and recursively δ ((q, µ t 0 (q)), u 1 u 2... u n, p) = { δ((q, µ t 0 (q)), u 1, p 1 ) δ((p 1, µ t 1 (p 1 )), u 2, p 2 )... δ((p n 1, µ t n 1 (p n 1 )), u n, p) p 1 Q act (t 1 ), p 2 Q act (t 2 ),..., p n 1 Q act (t n 1 )}. If q Q act (t i ), we should write q belongs to an element of Q act. Hereafter, we simply denote it as: q Q act. Definition 1.3. [5] A fuzzy subset µ of X is a function of X into the closed interval [0,1]. The support of µ is defined to be the set, Supp(µ) = {x X µ(x) > 0} Definition 1.4. [5] Let λ and µ be fuzzy subsets of G. The product λ µ of λ and µ is defined by (λ µ)(x) = {λ(y) µ(z) y, z G, x = y z} for all x G. We let F P (X) denote the fuzzy power set of X. So F P (X) is the set of all fuzzy subsets of X. F P (X) is a semigroup with respect to the product of fuzzy subsets of X.
5 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Definition 1.5. [5] Let (G, ) be a group. A fuzzy subset λ of G is called a fuzzy subgroup of G if the following properties hold: i) λ(x y) λ(x) λ(y) ii) λ(x) = λ(x 1 ) for all x, y G. Definition 1.6. [5] A fuzzy subgroup λ of G is called a fuzzy normal subgroup of G if λ(x y x 1 ) λ(y) for all x, y G. Definition 1.7. [5] Let λ and µ be fuzzy subsets of G and λ µ. Then λ is called a fuzzy normal subgroup of µ if λ(x y x 1 ) λ(x) λ(y) for all x, y G. 2. Applications of a group in general fuzzy automata Definition 2.1. Let F 1 = (Q 1, Σ 1, R 1, Z, ω, δ 1, F 1, F 2 ) and F 2 = (Q 2, Σ 2, R 2, Z, ω, δ 2, F 1, F 2 ) be general fuzzy automata and (Q 1, ) and (Q 2, ) be groups. A pair of functions (f, g), where f : Q 1 Q 2, g : Σ 1 Σ 2, is called a homomorphism from F 1 into F 2 if the following conditions hold: i) f is a group homomorphism, ii) δ 1 ((p, µ t (p)), x, q) δ 2 ((f(p), µ t (f(p))), g(x), f(q)), p, q Q, x Σ 1. Definition 2.2. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and let λ be a fuzzy subset of Q and (Q, ) be a group. λ is called fuzzy normal kernel of F if the following conditions hold: i) λ is a fuzzy normal subgroup of Q, ii) λ(p r 1 ) δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) λ(k) for all p, q, k, r Q, x Σ. Theorem 2.3. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a max-min general fuzzy automaton and let λ is a fuzzy normal subgroup of Q and (Q, ) be a group. Then λ is a fuzzy normal kernel of F if and only if λ(p r 1 ) δ ((q k, µ t (q k)), x, p) δ ((q, µ t (q)), x, r) λ(k) for all p, q, k, r Q, x Σ.
6 62 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Proof. Let λ be a fuzzy normal kernel of F. We prove the theorem by induction on x = n. Let n = 0. Then x = Λ. If p = q k, r = q, then since λ is a fuzzy normal subgroup, we have δ ((q k, µ t (q k)), x, p) δ ((q, µ t (q)), x, r) λ(k) = λ(k) λ(q k q 1 ) = λ(p r 1 ) If p q k or r q, then δ ((q k, µ t (q k)), x, p) δ ((q, µ t (q)), x, r) λ(k) = 0 λ(p r 1 ) Thus the result holds for n = 0. Suppose that the result holds for all x Σ such that x = n 1. Let x = ya, y Σ, y = n 1, n > 0. Then we have δ ((q k, µ t (q k)), x, p) δ ((q, µ t (q)), x, r) λ(k) = δ ((q k, µ t (q k)), ya, p) δ ((q, µ t (q)), ya, r) λ(k) = ( u Q δ ((q k, µ t (q k)), y, u) δ((u, µ t (u)), a, p)) ( v Q δ ((q, µ t (q)), y, v) δ((v, µ t (v)), a, r)) λ(k) = u Q δ v Q ((q k, µ t (q k)), y, u) δ((u, µ t (u)), a, p) δ ((q, µ t (q)), y, v) δ((v, µ t (v)), a, r) λ(k) u Q v Q λ(u v 1 ) δ((u, µ t (u)), a, p) δ((v, µ t (v)), a, r) u Q v Q λ(v 1 u) δ((v v 1 u, µ t (v v 1 u)), a, p) δ((v, µ t (v)), a, r) = u Q v Q λ(k ) δ((v k, µ t (v k )), a, p) δ((v, µ t (v)), a, r) λ(p r 1 ) The converse is trivial. Definition 2.4. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and let λ be a fuzzy subset of Q and (Q, ) be a group. λ is called fuzzy kernel of F if the following conditions hold: i)λ is a fuzzy subgroup of Q, ii)λ(p) δ((q, µ t (q)), x, p) λ(q) for all p, q Q, x Σ. Theorem 2.5. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a max-min general fuzzy automaton and let λ is a fuzzy subgroup of Q and (Q, ) be a group. Then λ is a fuzzy kernel of F if and only if λ(p) δ ((q, µ t (q)), x, p) λ(q) for all p, q Q, x Σ. Proof. The proof is similar to that of Theorem 2.3.
7 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Definition 2.6. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and (Q, ) be a group. F is called adjustable if δ((p q, µ t (p q)), x, p) δ((p, µ t (p)), x, k) for all p, q, r, k Q, x Σ. Theorem 2.7. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable general fuzzy automaton, (Q, ) be a group and let λ be a fuzzy normal kernel of F and ν be a fuzzy kernel of F. Then λ ν is a fuzzy kernel of F. Proof. Since λ is a fuzzy normal subgroup of Q and ν is a fuzzy subgroup of Q, it follows that λ ν is a fuzzy subgroup of Q and λ ν = ν λ. Since p = (p r 1 ) r, then we have (λ ν)(p) λ(p r 1 ) ν(r) ( δ((a b, µ t (a b)), x, p) δ((a, µ t (a)), x, r) λ(b)) ( δ((a, µ t (a)), x, r) ν(a)) for all a, b, p Q, x Σ. Since F is adjustable, so we have δ((a b, µ t (a b)), x, p) δ((a, µ t (a)), x, r) Then we have ( δ((a b, µ t (a b)), x, p) δ((a, µ t (a)), x, r) λ(b)) ( δ((a, µ t (a)), x, r) ν(a)) Thus for all p, q Q, x Σ, = δ((a b, µ t (a b)), x, p) λ(b) ν(a) (λ ν)(p) { δ((a b, µ t (a b)), x, p) λ(b) ν(a) a, b Q, a b = q} = δ((q, µ t (q)), x, p) ( {λ(b) ν(a) a, b Q, a b = q}) = δ((q, µ t (q)), x, p) (ν λ)(q) = δ((q, µ t (q)), x, p) (λ ν)(q) Hence λ ν is a fuzzy kernel of F. Theorem 2.8. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable general fuzzy automaton and (Q, ) be a group. If λ and ν are fuzzy normal kernels of F, then λ ν is a fuzzy normal kernel of F.
8 64 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Proof. Since λ and ν are fuzzy normal subgroups of Q, it follows that λ ν is a fuzzy normal subgroup of Q and λ ν = ν λ. Since p r 1 = (p q 1 ) (q r 1 ), then we have (λ ν)(p r 1 ) λ(p q 1 ) ν(q r 1 ) ( δ((a b c, µ t (a b c)), x, p) δ((a b, µ t (a b)), x, q) λ(c)) ( δ((a b, µ t (a b)), x, q) δ((a, µ t (a)), x, r) ν(b)) for all a, b, c, p, r Q, x Σ. Since F is adjustable, so we have δ((a b c, µ t (a b c)), x, p) δ((a b, µ t (a b)), x, q) Then we have δ((a b c, µ t (a b c)), x, p) δ((a b, µ t (a b)), x, q) λ(c) δ((a b, µ t (a b)), x, q) δ((a, µ t (a)), x, r) ν(b) = δ((a b c, µ t (a b c)), x, p) δ((a, µ t (a)), x, r) λ(c) ν(b) Thus for all p, q, r, k Q, x Σ, we have (λ ν)(p r 1 ) { δ((a b c, µ t (a b c)), x, p) δ((a, µ t (a)), x, r) λ(c) ν(b) b, c Q, b c = k} = δ((a k, µ t (a k)), x, p) δ((a, µ t (a)), x, r) ( {λ(c) ν(b) b, c Q, b c = k}) = δ((a k, µ t (a k)), x, p) δ((a, µ t (a)), x, r) (ν λ)(k) = δ((a k, µ t (a k)), x, p) δ((a, µ t (a)), x, r) (λ ν)(k) Hence λ ν is a fuzzy normal kernel of F. Theorem 2.9. Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton, (Q, ) be a group and let λ be a fuzzy normal kernel of F with Supp(λ) = Q. Then there exists a general fuzzy automaton say H and there exists a homomorphism from F to H.
9 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Proof. Now λ is a fuzzy normal subgroup of Q. Define λ p : Q [0, 1] by λ p (q) = λ(p q 1 ) for all q Q. Let E = {λ p p Q}. Then E is a group with respect to the binary operation λ p λ q = λ p q for all λ p, λ q E. Define Q/λ : E [0, 1] by (Q/λ)(λ p ) = λ(p) for all λ p E. Thus Q/λ is a fuzzy subgroup of E. Let G = Supp(Q/λ). Define σ : (G [0, 1]) Σ G [0, 1] by σ((λ p, µ t (λ p )), x, λ q ) = { δ((a, µ t (a)), x, b) a, b Q, λ a = λ p, λ b = λ q } for all λ p, λ q G. Define f : Q G by f(q) = λ q for all q Q and let g : Σ Σ be idendity map. Then we have f(p q) = λ p q = λ p λ q = f(p) f(q) for all p, q Q. Thus (f, g) is a homomorphism from F to H = (G, Σ, R, Z, ω, σ, F 1, F 2 ). Definition Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and let λ be a fuzzy kernel of F and (Q, ) be a group. A fuzzy set ν of Q is called a fuzzy normal kernel of λ if the following conditions hold: i) ν λ and ν is a fuzzy normal subgroup of λ, ii) ν(p r 1 ) δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) ν(k) for all p, k, r Q, q Supp(λ), x Σ. Definition Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be a general fuzzy automaton and (Q, ) be a group. F is called multiplicative if having the following properties: i) there exists the element x 0 of Σ such that δ((e, µ t (e)), x 0, e) > 0 ii) δ((q, µ t (q)), x, p r) = δ((q, µ t (q)), x 0, p) δ((e, µ t (e)), x, r), iii) δ((p 1 p 1 2, µt (p 1 p 1 2 )), x 0, q 1 q 1 2 ) = δ((p 1, µ t (p 1 )), x 0, q 1 ) δ((p 2, µ t (p 2 )), x 0, q 2 ) for all p, q, r, p 1, p 2, q 1, q 2 Q, x Σ. Theorem Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable multiplicative general fuzzy automaton and (Q, ) be an abelian group. Let λ be a fuzzy kernel of F, ν be a fuzzy normal kernel of λ and Supp(λ) Φ. If ν is a fuzzy normal subgroup of Q, then ν is a fuzzy normal kernel of F. Proof. Since F is multiplicative, for all p, q, r, k Q, x Σ, we have δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) = δ((q k, µ t (q k)), x, p e) δ((q, µ t (q)), x, r e) = ( δ((q k, µ t (q k)), x 0, p) δ((e, µ t (e)), x, e)) ( δ((q, µ t (q)), x 0, r) δ((e, µ t (e)), x, e)) = ( δ((q k, µ t (q k)), x 0, p e) δ((e, µ t (e)), x, e)) ( δ((q, µ t (q)), x 0, r) δ((e, µ t (e)), x, e))
10 66 Alg. Struc. Appl. Vol. 4 No. 2 (2017) = δ((q, µ t (q)), x 0, p) δ((k 1, µ t (k 1 )), x 0, e) δ((e, µ t (e)), x, e) δ((q, µ t (q)), x 0, r) Now for any q Q, b Supp(λ), q = b (q 1 b) 1. Thus we have δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) = δ((b (q 1 b) 1 k, µ t (b (q 1 b) 1 k)), x, p e) δ((e q, µ t (e q)), x, r e) = δ((b, µ t (b)), x 0, p) δ((q 1 b, µ t (q 1 b)), x 0, e) δ((k 1, µ t (k 1 )), x 0, e) δ((e, µ t (e)), x, e) δ((q 1, µ t (q 1 )), x 0, r) = δ((b, µ t (b)), x 0, p) δ((k 1, µ t (k 1 )), x 0, e) δ((e, µ t (e)), x, e) ( δ((q 1 b, µ t (q 1 b)), x 0, e) δ((q 1, µ t (q 1 )), x 0, r)) = δ((b, µ t (b)), x 0, p) δ((k 1, µ t (k 1 )), x 0, e) δ((e, µ t (e)), x, e) δ((b, µ t (b)), x 0, r) Since ν is a fuzzy normal kernel of λ, for all p, q, r Q, b Supp(λ), x Σ, we have ν(p r 1 ) δ((b k, µ t (b k)), x, p) δ((b, µ t (b)), x, r) ν(k) = δ((b k, µ t (b k)), x, p e) δ((b, µ t (b)), x, r e) ν(k) = δ((b, µ t (b)), x 0, p) δ((k 1, µ t (k 1 )), x 0, e) δ((b, µ t (b)), x 0, r) δ((e, µ t (e)), x, e) ν(k) = δ((q k, µ t (q k)), x, p) δ((q, µ t (q)), x, r) ν(k) Hence ν is a fuzzy normal kernel of F. Theorem Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable multiplicative general fuzzy automaton, (Q, ) be a group, λ be a fuzzy subset of Q and Supp(λ) Φ. Then the following statements are equivalent: i) λ is a fuzzy normal kernel of F ii) λ is a fuzzy normal subgroup of Q and λ(q) δ((p, µ t (p)), x 0, q) λ(p), for all p, q Q. Proof. (i ii) Since λ is a fuzzy normal kernel of F, we have: λ(q) = λ(q e 1 ) δ((e p, µ t (e p)), x 0, q) δ((e, µ t (e)), x 0, e) λ(p) = δ((p, µ t (p)), x 0, q) λ(p) (Since F is adjustable, for all p, q Q, we have δ((p, µ t (p)), x 0, q) = δ((e p, µ t (e p)), x 0, q) δ((e, µ t (e)), x 0, e) )
11 Alg. Struc. Appl. Vol. 4 No. 2 (2017) (ii i) Since F is multiplicative, for all p, q, r, k Q, x Σ, we have δ((q k, µ t (q k)), x, p r) δ((q, µ t (q)), x, p) λ(k) = δ((q k, µ t (q k)), x, p r e) δ((q, µ t (q)), x, p e) λ(k) = ( δ((q k, µ t (q k)), x 0, p r) δ((e, µ t (e)), x, e)) ( δ((q, µ t (q)), x 0, p) δ((e, µ t (e)), x, e)) λ(k) = δ((q, µ t (q)), x 0, p) δ((k 1, µ t (k 1 )), x 0, r 1 ) δ((e, µ t (e)), x, e)) λ(k) = δ((q, µ t (q)), x 0, p) δ((e k 1, µ t (e k 1 )), x 0, e r 1 ) δ((e, µ t (e)), x, e)) λ(k) = δ((q, µ t (q)), x 0, p) ( δ((e, µ t (e)), x 0, e) δ((k, µ t (k)), x 0, r)) δ((e, µ t (e)), x, e)) λ(k) δ((k, µ t (k)), x 0, r) λ(k) λ(r) λ(p r p 1 ) Now let p 1, p 2, q, k Q, x Σ. Since Q is a group, then there exists a unique element r Q such that p 1 = p 2 r. Thus λ(p 1 p 1 2 ) = λ(p 2 r p 1 2 ) δ((q k, µ t (q k)), x, p 2 r) δ((q, µ t (q)), x, p 2 ) λ(k) Hence λ is a fuzzy normal kernel of F. = δ((q k, µ t (q k)), x, p 1 ) δ((q, µ t (q)), x, p 2 ) λ(k) Theorem Let F = (Q, Σ, R, Z, ω, δ, F 1, F 2 ) be an adjustable multiplicative general fuzzy automaton and (Q, ) be a group. Then there exists a semigroup homomorphism f : Q F P (Q) and there exists a function g : Σ F P (Q) such that δ((q, µ t (q)), x, p) = f(q)(p) g(x)(e), for all p, q Q, x Σ.
12 68 Alg. Struc. Appl. Vol. 4 No. 2 (2017) Proof. Define f : Q F P (Q) by f(q) = µ q for all q Q, where µ q : Q [0, 1] is defined by µ q (p) = δ((q, µ t (q)), x 0, p) for all p Q. Since F is multiplicative, for all p 1, p 2, q 1, q 2 Q, we have µ q1 q 2 (p 1 p 2 ) = δ((q 1 q 2, µ t (q 1 q 2 )), x 0, p 1 p 2 ) = δ((q 1, µ t (q 1 )), x 0, p 1 ) δ((q 1 2, µt (q 1 2 )), x 0, p 1 2 ) = δ((q 1, µ t (q 1 )), x 0, p 1 ) δ((q 2, µ t (q 2 )), x 0, p 2 ) Also we have = µ q1 (p 1 ) µ q2 (p 2 ) (µ q1 µ q2 )(p) = {µ q1 (q) µ q2 (r) q, r Q, p = q r} = µ q1 q 2 (p) for all p Q. Hence f(q 1 q 2 ) = f(q 1 ) f(q 2 ) For x Σ, define the fuzzy subset λ x of Q by λ x (q) = δ((e, µ t (e)), x, q) for all q Q. Define g : Σ F P (Q) by g(x) = λ x for all x Σ. Since F is multiplicative, for all p, q Q, x Σ, we have f(q)(p) g(x)(e) = µ q (p) λ x (e) = δ((q, µ t (q)), x 0, p) δ((e, µ t (e)), x, e) = δ((q, µ t (q)), x, p) 3. Conclusion In this paper, we defined the concepts of fuzzy normal kernel of a general fuzzy automaton, fuzzy kernel of a general fuzzy automaton, adjustable, multiplicative. Then we obtained the relationships between them. 4. Acknowledgments The author wish to sincerely thank the referees for several useful comments.
13 Alg. Struc. Appl. Vol. 4 No. 2 (2017) References [1] M. Doostfatemeh and S. C. Kremer, New directions in fuzzy automata, International Journal of Approximate Reasoning, 38 (2005), [2] M. Horry and M. M. Zahedi, Fuzzy subautomata of an invertible general fuzzy automaton, Annals of fuzzy sets, fuzzy logic and fuzzy systems, 2(2)(2013), [3] J. Jin, Q. Li, Y. Li, Algebric properties of L-fuzzy finite automata, Information Sciences, 234 (2013), [4] Y. Li, W. Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in latticeordered monoids, Fuzzy Sets and Systems, 156 (2005), [5] J. N. Mordeson and D. S. Malik, Fuzzy automata and languages, theory and applications, Chapman and Hall/CRC, London/Boca Raton, FL, [6] D. S. Malik, J. N. Mordeson and M. K. Sen, On subsystems of fuzzy finite state machines, Fuzzy Sets and Systems, 68 (1994), [7] M. Mizumoto, J. Tanaka and K. Tanaka, Some consideration on fuzzy automata, J. Compute. Systems Sci. 3 (1969), [8] W. Omlin, K. K. Giles and K. K. Thornber, Equivalence in knowledge representation: automata, rnns, and dynamic fuzzy systems, Proc. IEEE, 87(9)(1999), [9] W. Omlin, K. K. Thornber and K. K. Giles, Fuzzy finite-state automata can be deterministically encoded into recurrent neural networks, IEEE Trans. Fuzzy Syst. 5(1)(1998), [10] E. S. Santos, Realization of fuzzy languages by probabilistic, max-prod and maximin automata, Inform. Sci. 8 (1975), [11] W. G. Wee, On generalization of adaptive algorithm and application of the fuzzy sets concept to pattern classif ication, Ph.D. dissertation Purdue University, IN, [12] M. M. Zahedi, M. Horry and Kh. Abolpor, Bifuzzy (General) topology on max-min general fuzzy automata, Advanced in Fuzzy Mathematics, 3(1)(2008), Mohammad Horry Department of mathematics, Shahid Chamran university, Kerman, Iran. mhori@tvu.ac.ir
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