MINIMAL INTUITIONISTIC GENERAL L-FUZZY AUTOMATA

Transcript

1 italian journal of pure applied mathematics n ( ) 155 MINIMAL INTUITIONISTIC GENERAL L-UZZY AUTOMATA M. Shamsizadeh M.M. Zahedi Department of Mathematics Kerman Graduate University of Advanced Technology Kerman Iran s: zahedi Abstract. In this paper we present an intuitionistic general L-fuzzy automaton (IGLA) based on lattice valued intuitionistic fuzzy sets [2]. In this note, we define (α, β)- language, (α, β)-complete, (α, β)-accessible, (α, β)-reduced for an IGLA over a bounded complete lattice L, where α, β L α L N(β). In particular, we prove a theorem which is generalization of Myhill-Nerode theorem in ordinary deterministic automata. In other words for any recognizable (α, β)-language over a bounded complete lattice L, there exist minimal (α, β)-complete deterministic IGLA, which preserve (α, β)-language, where α, β L α L N(β). Also, we show that for any given (α, β)-language L, the minimal (α, β)-complete deterministic IGLA recognizing L is isomorphic with threshold (α, β) to any (α, β)-complete, (α, β)-accessible, deterministic, (α, β)-reduced IGLA recognizing L. Moreover, we give some examples to clarify these notions. inally, by using these notions, we give some theorems obtain some results. Keywords: Max-min intuitionistic general L-fuzzy automata; (α, β)-language; (α, β)- reduced; Minimal intuitionistic general L-fuzzy automata; (α, β)-isomorphic. 1. Introduction The theory of fuzzy sets was introduced by L.A. Zadeh in 1965 [40]. W.G. Wee [35] introduced the idea of fuzzy automata. E.T. Lee L.A. Zadeh [21] in 1969 gave the concept of fuzzy finite state automata. Thereafter, there were a considerable number of authors, such as Mordeson Malik [22], [23], Topencharov Peeva [33] others having contributed to this field. uzzy finite automata have many important applications such as in learning system, pattern recognition, neural networks data base theory [14], [15], [22], [25], [26], [29], [36]. Atanassove [1] has extended the notion of fuzzy sets to the intuitionistic fuzzy sets (IS) by

2 156 m. shamsizadeh, m.m. zahedi adding non-membership value, which may express more accurate flexible information as compared with fuzzy sets. Intuitionistic fuzzy set theory has many applications in several subject, see [5], [6], [9], [11], [16], [18], [19], [34], [7]. Using the notion of intuitionistic fuzzy sets, W.L. Jun [17] introduced the notion of intuitionistic fuzzy finite state machines as a generalization of fuzzy finite state machines. Based on the papers [17], [18], Zhang Li [41] discussed intuitionistic fuzzy recognizers intuitionistic fuzzy finite automata. K. Atanassov S. Stoeva generalized the concept of IS to intuitionistic L-fuzzy sets [2] where L is an appropriate lattice. A. Tepavcevic T. Gerstenkorn gave a new definition of lattice valued intuitionistic fuzzy sets in [32]. Thus, on the basis of lattice valued intuitionistic fuzzy sets, Yang et al. [38] introduced the concepts of latticevalued intuitionistic fuzzy finite state machines. In 2004, M. Doostfatemeh S.C. Kremer [10] extended the notion of fuzzy automata gave the notion of general fuzzy automata. In 2014, M. Shamsizadeh M.M. Zahedi [31] gave the notion of max-min intuitionistic general fuzzy automata. We will use bounded complete lattices as the structures of truth values. Note that usually the real unit interval [0, 1] is used in the literature (the reader not familiar with lattices may, without any harm, substitute [0, 1] for bounded complete lattices throughout the paper). Our paper deals with intuitionistic general fuzzy automata over a bounded complete lattice L, where endowed with a t-norm T, a t-conorm S, the least element 0 the greatest element 1, denoted by L = (L, L, T, S, 0, 1). State minimization is a fundamental problem in automata theory. There are many papers on the minimization problem of fuzzy finite automata. or example, minimization of mealy type of fuzzy finite automata in discussed in [4], minimization of fuzzy finite automata with crisp final states without outputs in studied in [3], minimizing the deterministic finite automaton with fuzzy (final) states in [24] minimization of fuzzy machines becomes the subject of [28], [27], [30], [33]. Myhill-Nerode s theorem has been extended to fuzzy regular language also an algorithm is given for minimizing the deterministic finite automaton with fuzzy (final) states in [20], [24]. It is important to find the minimal intuitionistic general L-fuzzy automata that recognizes the same language as a given language. In this note, for a given complete lattice L = (L, L, T, S, 0, 1), we define an (α, β)- language, where α, β L, α < L N(β). urthermore, we show that for any maxmin IGLA, there exist (α, β)-complete, (α, β)-accessible deterministic max-min IGLA recognizing L( ), where α, β L, α < L N(β). Also, we prove a theorem which is generalization of Myhill-Nerode theorem in ordinary deterministic automata. In other words, we have shown that for any (α, β)-language, there exist minimal (α, β)-complete deterministic intuitionistic general L-fuzzy automata (IGLA). Also, we define an (α, β)-reduced IGLA. We show that for any given (α, β)-language L, the minimal (α, β)-complete deterministic IGLA recognizing L is isomorphic with threshold (α, β) to any (α, β)-complete, (α, β)- accessible, deterministic, (α, β)-reduced IGLA recognizing L. Moreover we give some new notions results as mentioned in the abstract some examples to clarify these new notions.

3 minimal intuitionistic general L-fuzzy automata Preliminaries In this section we give some definitions that we need in the sequel. Assume that E is an universal set. A fuzzy set A on E is characterized by the same symbol A as a function A : E [0, 1] where A(u) [0, 1] is the membership degree of the element u E [40]. Definition 2.1 [1] Let A be a given subset on E. (IS) A + on E is an object of the following form An intuitionistic fuzzy set A + = { x, µ A (x), ν A (x) x E}, where the functions µ A : E [0, 1] ν A : E [0, 1] define the value of membership the value of non-membership of the element x E to the set A, respectively, for every x E, 0 µ A (x) + ν A (x) 1. Obviously, every ordinary fuzzy set {(x, µ A (x)) x E} has an intuitionistic form { x, µ A (x), 1 µ A (x) x E}. If π A (x) = 1 µ A (x) ν A (x), then π A (x) is the value of non-determinacy (uncertainty) of the membership of element x E to the set A. In the case of ordinary fuzzy sets, where ν A (x) = 1 µ A (x), we have π A (x) = 0 for every x E. In the rest of this paper, we denote the set A + by A. Let L = (L, L, 0, 1) be a bounded (complete) lattice. By an L-fuzzy set A on E we mean a function A : E L [12]. Definition 2.2 [37] Let L = (L, L, 0, 1) be a bounded lattice. A binary operation T : L L L is a lattice t-norm (Lt-norm) if it satisfies the following conditions: 1. T (1, x) = x, 2. T (x, y) = T (y, x), 3. T (x, T (y, z)) = T (T (x, y), z), 4. if w L x y L z, then T (w, y) L T (x, z). Definition 2.3 [37] Let L = (L, L, 0, 1) be a bounded lattice. A binary operation S : L L L is a lattice t-conorm (Lt-conorm) if it satisfies the following conditions: 1. S(0, x) = x, 2. S(x, y) = S(y, x), 3. S(x, S(y, z)) = S(S(x, y), z), 4. if w L x y L z, then S(w, y) L S(x, z).

4 158 m. shamsizadeh, m.m. zahedi In the rest of this paper, we denote D(x 1, D n 1 (x 2,..., x n+1 )) by D n (x 1, x 2,..., x n+1 ) where D is an Lt-norm or an Lt-conorm on lattice L, D 0 (x) = x D 1 (x 1, x 2 ) = D(x 1, x 2 ), for any n 1. We recall from [8] that L : {(x, y) [0, 1] 2 0 x+y 1} is a complete lattice with the order defined by (x 1, x 2 ) (y 1, y 2 ) if only if x 1 y 1 y 2 x 2. Definition 2.4 [2] Let X be a nonempty set L be a complete lattice with an involutive order reversing unary operation N : L L. An intuitionistic L-fuzzy set is an object of the form A = {(x, µ(x), ν(x)) x E}, where µ ν are functions µ : E L, ν : E L, such that for all x X, µ(x) N(ν(x)). We use the abbreviation ILS for intuitionistic L-fuzzy set. Definition 2.5 [13] Let L X consider the relation L on X, where x L y if only if, for all z X, xz L yz L. rom now on, we let L = (L, L, T, S, 0, 1) be a complete lattice, where endowed with an Lt-norm T, an Lt-conorm S, the least element element 0 the greatest element 1, also with an involutive order reversing unary operation N : L L. Note 2.6 Let A, B L. In this note we assume that A < L B if only if A L B A B. We also assume that A L B if only if B L A. 3. Intuitionistic general L-fuzzy automata Definition 1.1 An intuitionistic general L-fuzzy automaton (IGLA) is a tentuple machine denoted by = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ), where Q is a set of states, X is a finite set of input symbols, X = {a 1, a 2,..., a m }, R is the ILS of start states, R = {(q, µ t 0 (q)) q R}, where R is a finite subset of Q, Z is a finite set of output symbols, Z = {b 1, b 2,..., b l }, δ : (Q L L) X Q L L is the augmented transition function, ω : (Q L L) Z L L is the output function, 1 = (1 T, 1 S ), where 1 T : L L L is an Lt-norm it is called the membership assignment function. 1 T (µ, δ 1 ) as is seen, is motivated by two parameters µ δ 1, where µ is the membership value of a predecessor δ 1 is the membership value of a transition. Moreover, 1 S : L L L is an Ltconorm, where is the dual of 1 T respect to the involutive negation it is called non-membership assignment function. 1 S (ν, δ 2 ) as is seen, is motivated by two

5 minimal intuitionistic general L-fuzzy automata 159 parameters ν δ 2, where ν is the non-membership value of a predecessor δ 2 is the non-membership value of a transition. In this definition, the process that takes place upon the transition from the state q i to q j on an input a k is given by: (1.1) (1.2) µ t+1 (q j ) = δ 1 ((q i, µ t (q i ), ν t (q i )), a k, q j ) = T 1 (µ t (q i ), δ 1 (q i, a k, q j )), ν t+1 (q j ) = δ 2 ((q i, µ t (q i ), ν t (q i )), a k, q j ) = S 1 (ν t (q i ), δ 2 (q i, a k, q j )), thus (1.3) δ((q i, µ t (q i ), ν t (q i )), a k, q j ) = ( δ 1 ((q i, µ t (q i ), ν t (q i )), a k, q j ), δ 2 ((q i, µ t (q i ), ν t (q i )), a k, q j )), where, δ(q i, a k, q j ) = (δ 1 (q i, a k, q j ), δ 2 (q i, a k, q j )), which δ is an ILS. It means that the membership value of the state q j at time t + 1 is computed by the function 1 T using both the membership value of q i at time t the membership value of the transition. Also, the non-membership value of the state q j at time t + 1 is computed by function 1 S using both the non-membership value of q i at time t the non-membership value of the transition. Considering (1.1), (1.2) Definitions 2.2, , δ is an ILS. 2 = (2 T, 2 S ), where 2 T : L L L is an Lt-norm it is called the membership assignment output function. 2 T (µ, ω 1 ) as is seen, is motivated by two parameters µ ω 1, where µ is the membership value of present state ω 1 is the membership value of an output function. Moreover, 2 S : L L L is an Lt-conorm where it is the dual of 2 T respect to the involutive negation is called non-membership assignment output function. 2 S (ν, ω 2 ) as is seen, is motivated by two parameters ν ω 2, where ν is the non-membership value of present state ω 2 is the non-membership value of an output function. Then, we have (1.4) (1.5) ω 1 ((q i, µ t (q i ), ν t (q i )), b k ) = T 2 (µ t (q i ), ω 1 (q i, b k )), ω 2 ((q i, µ t (q i ), ν t (q i )), b k ) = S 2 (ν t (q i ), ω 2 (q i, b k )), thus, (1.6) ω((q i, µ t (q i ), ν t (q i )), b k ) = ( ω 1 ((q i, µ t (q i ), ν t (q i )), b k ), ω 2 ((q i, µ t (q i ), ν t (q i )), b k )), where, ω(q i, b k ) = (ω 1 (q i, b k ), ω 2 (q i, b k )), which ω is an ILS. It means that the output membership value of the state q i at time t is computed by the function 2 T using both the membership value of q i at time t the membership value of the output function, also output non-membership value of the state q i at time t is computed by function 2 S using both the non-membership value of q i at time t the non-membership value of the output function.

6 160 m. shamsizadeh, m.m. zahedi Considering (1.4), (1.5) Definitions 2.2, , ω is an ILS. 3 = (3 T S, 3 ST ), where 3 ST : L L is an Lt-norm it is called the multi-non-membership function. The multi- non-membership resolution function resolves the multi-non-membership active states assigns a single nonmembership value to them. Moreover, 3 T S : L L is an Lt-conorm, where it is the dual of 3 ST respect to the involutive negation it is called the multimembership function. The multi-membership resolution function resolves the multi-membership active states assigns a single membership value to them. 4 = (4 T S, 4 ST ), where 4 ST : L L, is an Lt-norm it is called the multi-non-membership output function. The multi-non-membership resolution output function resolves the output multi-non-membership active state assigns a single output non-membership value to it. Moreover, 4 T S : L L, is an Lt-conorm, where it is the dual of 4 ST respect to the involutive negation is multi-membership output function. The multi-membership resolution output function resolves the output multi-membership active state assigns a single output membership value to it. Let Q act (t i ) be the set of all active states at time t i for all i 0. We have Q act (t 0 )= R Q act (t i )={(q, µ t i (q), ν t i (q)) (q, µ t i 1 (q ), ν t i 1 (q )) Q act (t i 1 ), a X, δ(q, a, q), µ t i (q) > L 0} for all positive integer i. Since Q act (t i ) is an ILS, to say that a state q belongs to Q act (t i ), we write q Domain(Q act (t i )) for simplicity of notation we denote it by q Q act (t i ). The combination of the operations of functions 1 T 3 T S (1 S 3 ST ) on a multi-membership (multi- non-membership) state q j will lead to the multimembership (multi-non-membership) resolution algorithm. Also, the set of all transition of IGLA is denoted by. Algorithm: Multi-membership resolution (multi-non-membership resolution) for transition function. If there are several simultaneous transitions to the active state q j at time t+1, then the following algorithm will assign a unified membership value (non-membership value) to that 1. Each transition membership value (transition non-membership value) δ 1 (q i, a k, q j ) (δ 2 (q i, a k, q j )) together with µ t (q i ) (ν t (q i )), will be processed by the membership (non-membership) assignment function 1 T (1 S ) will produce a new membership value (non-membership value) as follows: δ 1 ((q i, µ t (q i ), ν t (q i )), a k, q j ) = T 1 (µ t (q i ), δ 1 (q i, a k, q j )), ( δ2 ((q i, µ t (q i ), ν t (q i )), a k, q j ) = S 1 (ν t (q i ), δ 2 (q i, a k, q j )) ). 2. These new membership (non-membership) values are not necessarily equal. Hence, they will be processed by 3 T S (3 ST ), which is called the multimembership (multi-non-membership) resolution function. By some manipulation on the product results by 1 T (1 S ), we obtain just one element, say

7 minimal intuitionistic general L-fuzzy automata 161 the instantaneous membership value (non-membership value) of the active state q j µ t+1 (q j ) = (3 T S ) n 1 (x 1, x 2,..., x n ), (ν t+1 (q j ) = (3 ST ) n 1 (x 1, x 2,..., x n )), where n is the number of simultaneous transitions to the active state q j at time t + 1 x i = T 1 (µ t (q i ), δ 1 (q i, a k, q j ))(x i = S 1 (ν t (q i ), δ 2 (q i, a k, q j ))), 1 i n, δ 1 (q i, a k, q j )(δ 2 (q i, a k, q j )) is the membership (non-membership) value of transition from q i to q j upon input a k, µ t (q i ) (ν t (q i )) is the membership (non-membership) value of q i at time t, Algorithm: Multi-membership resolution (Multi-non-membership resolution) for output function If there are several simultaneous output to the active state q i at time t, the following algorithm will assign a unified membership value (non-membership value) to that 1. Each output membership value (non-membership value) ω 1 (q i, b k ) (ω 2 (q i, b k )) together with µ t (q i ) (ν t (q i )), will be processed by the membership (non-membership) assignment function 2 T (2 S ) will produce a new output membership value (non-membership value) as follows: ω 1 ((q i, µ t (q i ), ν t (q i )), b k ) = T 2 (µ t (q i ), ω 1 (q i, b k )), ( ω2 ((q i, µ t (q i ), ν t (q i )), b k ) = S 2 (ν t (q i ), ω 2 (q i, b k )) ). 2. These new output membership (non-membership) values are not necessarily equal. Hence, they will be processed by 4 T S (4 ST ), which is called the output multi-membership (multi-non-membership) resolution function. By some manipulation on the product results by 2 T (2 S ), we obtain just one element, say the instantaneous output membership value (non-membership value) of the active state q i. where ω1(q t i ) = (4 T S ) n 1 (x 1, x 2,..., x n ), (ω2(q t i ) = (4 ST ) n 1 (x 1, x 2,..., x n )), n is the number of simultaneous outputs to the active state q i at time t, x j = T 2 (µ t (q i ), ω 1 (q i, b j ))(x j = S 2 (ν t (q i ), ω 2 (q i, b j ))), 1 j n, for some a k X q i is an active state at time t, ω 1 (q i, b j )(ω 2 (q i, b j )) is the membership (non-membership) value of output from q i on b k,

8 162 m. shamsizadeh, m.m. zahedi µ t (q i ) (ν t (q i )) is the membership (non-membership) value of q i at time t, ω t 1(q i ) (ω t 2(q i )) is the output membership (non-membership) value of q i at time t. Remark 1.2 We let for all q Q such that q / R, µ t 0 (q) = 0 ν t 0 (q) = 1 for all q R, µ t 0 (q) > L 0. Note 1.3 In this paper, we assume that the max-min IGLA has a finite number of states. Definition 1.4 Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) be an IGLA. We define the max-min intuitionistic general L-fuzzy automaton = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) such that δ : Q act X Q L L, where Q act = {Q act (t 0 ), Q act (t 1 ), Q act (t 2 ),...} for every i 0, { (1.7) δ 1 ((q, µ t i (q), ν t i 1 if p = q, (q)), Λ, p) = 0 otherwise, (1.8) δ 2 ((q, µ t i (q), ν t i (q)), Λ, p) = { 0 if p = q, 1 otherwise. Also for every i 0, δ 1((q, µ t i (q), ν t i (q)), u i+1, p) = δ 1 ((q, µ t i (q), ν t i (q)), u i+1, p) δ 2((q, µ t i (q), ν t i (q)), u i+1, p) = δ 2 ((q, µ t i (q), ν t i (q)), u i+1, p) recursively, δ 1((q, µ t 0 (q)), u 1 u 2...u n, p) = (1.9) { δ 1 ((q, µ t 0 (q)), u 1, p 1 ) δ 1 ((p 1, µ t 1 (p 1 ), ν t 1 (p 1 )), u 2, p 2 )... δ 1 ((p n 1, µ t n 1 (p n 1 ), ν t n 1 (p n 1 )), u n, p) p 1 Q act (t 1 ),..., p n 1 Q act (t n 1 )}, δ 2((q, µ t 0 (q)), u 1 u 2...u n, p) = (1.10) { δ 2 ((q, µ t 0 (q)), u 1, p 1 ) δ 2 ((p 1, µ t 1 (p 1 ), ν t 1 (p 1 )), u 2, p 2 )... δ 2 ((p n 1, µ t n 1 (p n 1 ), ν t n 1 (p n 1 )), u n, p) p 1 Q act (t 1 ),..., p n 1 Q act (t n 1 )}, in which u i X for all 1 i n assume that u i+1 is the entered input at time t i, for all 0 i n Minimal intuitionistic general L-fuzzy automata Definition 4.1 Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) be a max-min IGLA. Suppose that α, β L α L N(β). Then we say that

9 minimal intuitionistic general L-fuzzy automata is (α, β)-complete, if for any q Q, a X there exists p Q such that δ 1 (q, a, p) > L α δ 2 (q, a, p) < L β, 2. q Q is (α, β)-accessible if there exist p R, x X such that δ 1((p, µ t 0 (p)), x, q) > L α, δ 2((p, µ t 0 (p)), x, q) < L β, 3. is (α, β)-accessible if for any q Q, q is an (α, β)-accessible. Definition 4.2 Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) be a max-min IGLA. Then we say that is deterministic if there exists a unique p 0 R such that µ t 0 (p 0 ) > L 0, for any q Q, a X there exists at most one p Q such that δ 2 (q, a, p) < L 1. Example 4.3 Consider the complete lattice L = (L, L, T, S, 0, 1) as in igure 1, where L = {0, a, b, c, d, 1} N(0) = 1, N(1) = 0, N(a) = b, N(b) = a, N(c) = d, N(d) = c. igure 1: The complete lattice L of Example 4.3 Let the max-min IGLA = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) as in igure 2, where Q = {q 0, q 1, q 2, q 3 }, X = {u, v}, R = {(q 0, 1, 0)}, Z = {o} δ : Q X Q : L L is defined as follows: δ(q 0, u, q 1 ) = (a, 0) δ(q 1, v, q 3 ) = (a, b) δ(q 2, u, q 1 ) = (1, 0) δ(q 2, v, q 0 ) = (c, 0) δ(q, x, q ) = (0, 1) for all other (q, x, q ) Q X Q ω : Q Z : L L is defined by: ω(q 1, o) = (0, 1) ω(q, e) = (1, 0) for all other (q, e) Q Z. q 1 is (0, α)-accessible, (d, β)-accessible, where α {a, b, c, d, 1} β {a, b, c, d}, q 3 is (0, 1)- accessible, (0, c)- accessible, (d, c)- accessible but q 3 is not (d, 1)- accessible, since d N(1) = 0. is not (α, β)-accessible, not (α, β)- complete, for any (α, β) L in which α L N(β). Clearly, it is deterministic.

10 164 m. shamsizadeh, m.m. zahedi igure 2: The IGLA of Example 4.3 Theorem 4.4 Let = (Q, X, {p}, Z, δ, ω, 1, 2, 3, 4 ) be a deterministic maxmin IGLA. If δ 1((p, µ t 0 (p)), x, q) δ 1((p, µ t 0 (p)), x, q ) > L α, δ 2((p, µ t 0 (p)), x, r) δ 1((p, µ t 0 (p)), x, r ) < L β, then q = q = r = r, where q, q, r, r Q, x X, α, β L α L N(β). Proof. irst, let x = Λ. Then we have δ 1 ((p, µ t 0 (p)), Λ, q) δ 1 ((p, µ t 0 (p)), Λ, q ) > L α, δ 2 ((p, µ t 0 (p)), Λ, r) δ 2 ((p, µ t 0 (p)), Λ, r ) < L β, which implies that p = q = q = r = r. Thus the theorem holds for x = Λ. Now, we continue the proof for any x X x / Λ by induction on x. Suppose that x = 1. Then δ 1 ((p, µ t 0 (p)), x, q) = T 1 (µ t 0 (p), δ 1 (p, x, q)) > L α δ 1 ((p, µ t 0 (p)), x, q ) = T 1 (µ t 0 (p), δ 1 (p, x, q )) > L α δ 2 ((p, µ t 0 (p)), x, r) = S 1 (ν t 0 (p), δ 1 (p, x, r)) < L β These imply that δ 2 ((p, µ t 0 (p)), x, r ) = S 1 (ν t 0 (p), δ 1 (p, x, r )) < L β. δ 1 (p, x, q) δ 1 (p, x, q ) > L α δ 2 (p, x, r) δ 2 (p, x, r ) < L β. Since δ 1 (p, x, q) δ 1 (p, x, q ) > L α, then δ 2 (p, x, q) δ 2 (p, x, q ) < L N(α) L 1. Then by Definition 4.2, q = q = r = r. Now, suppose the claim holds for all y X such that y = n 1 n > 1. Let x = ya, where y X, a X y = n 1. Then δ 1((p, µ t 0 (p)), x, q) = { δ 1((p, µ t 0 (p)), y, p ) δ 1 ((p, µ t 0+n 1 (p ), ν t 0+n 1 (p )), a, q) p Q} > L α,

11 minimal intuitionistic general L-fuzzy automata 165 δ 1((p, µ t 0 (p)), x, q ) = { δ 1((p, µ t 0 (p)), y, p ) So, there exist d, d Q such that δ 1 ((p, µ t 0+n 1 (p ), ν t 0+n 1 (p )), a, q ) p Q} > L α. { δ 1((p,µ t 0 (p)), y, p ) δ 1 ((p, µ t 0+n 1 (p ), ν t 0+n 1 (p )), a, q) p Q} = δ 1((p, µ t 0 (p)), y, d) δ 1 ((d, µ t 0+n 1 (d), ν t 0+n 1 (d)), a, q) > L α, { δ 1((p,µ t 0 (p)), y, p ) δ 1 ((p, µ t 0+n 1 (p ), ν t 0+n 1 (p )), a, q ) p Q} = δ 1((p, µ t 0 (p)), y, d ) δ 1 ((d, µ t 0+n 1 (d ), ν t 0+n 1 (d )), a, q ) > L α. Also there exist s, s Q such that δ 2((p, µ t 0 (p)), x, r) = { δ 2((p, µ t 0 (p)), y, p ) δ 2 ((p, µ t 0+n 1 (p ), ν t 0+n 1 (p )), a, r) p Q} = δ 2((p, µ t 0 (p)), y, s) δ 2 ((s, µ t 0+n 1 (s), ν t 0+n 1 (s)), a, r) < L β, δ 2((p, µ t 0 (p)), x, r ) = { δ 2((p, µ t 0 (p)), y, p ) δ 2 ((p, µ t 0+n 1 (p ), ν t 0+n 1 (p )), a, r ) p Q} = δ 2((p, µ t 0 (p)), y, s ) δ 2 ((s, µ t 0+n 1 (s ), ν t 0+n 1 (s )), a, r ) < L β. Therefore, by the induction hypothesis, s = s = d = d. Hence δ 1 ((d, µ t 0+n 1 (d), ν t 0+n 1 (d)), a, q) δ 1 ((d, µ t 0+n 1 (d), ν t 0+n 1 (d)), a, q ) > L α, δ 2 ((s, µ t 0+n 1 (s), ν t 0+n 1 (s)), a, r) δ 2 ((s, µ t 0+n 1 (s), ν t 0+n 1 (s)), a, r ) < L β, imply that δ 1 (d, a, q) δ 1 (d, a, q ) > L α δ 2 (s, a, r) δ 2 (s, a, r ) < L β. Then q = q = r = r. Hence the claim is hold. Corollary 4.5 Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) be a max-min IGLA. Notice that if is an (α, β)-complete deterministic max-min IGLA, then for any p Q, a X there exists exactly one state q Q such that δ 1 (p, a, q) > L α δ 2 (p, a, q) < L β, where α, β L α L N(β). Example 4.6 Consider the complete lattice L = (L, L, T, S, 0, 1) as in Example 4.3, where L = {0, a, b, c, d, 1}. Let the max-min IGLA = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) as in igure 3, where Q = {q 0, q 1, q 2, q 3 }, X = {u}, R = {(q 0, a, b)}, Z = {o} δ : Q X Q : L L is defined as follows:

12 166 m. shamsizadeh, m.m. zahedi igure 3: The IGLA of Example 4.6 δ(q 0, u, q 1 ) = (c, d) δ(q 1, u, q 2 ) = (1, 0) δ(q 2, u, q 1 ) = (c, 0) δ(q 3, u, q 2 ) = (1, 0) δ(q, x, q ) = (0, 1) for all other (q, x, q ) Q X Q ω : Q Z L L is defined by: ω(q 1, o) = (1, 0) ω(q, e) = (0, 1) for all other (q, e) Q Z. Then is (a, b)-complete deterministic. Definition 4.7 Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) be a max-min IGLA. Then the (α, β)-language recognized by is a subset of X defined by: (4.1) L α,β ( ) = {x X δ 1((p, µ t 0 (p)), x, q) ω 1 ((q, µ t 0+ x + x (q)), b) > L α, δ 2((p, µ t 0 (p)), x, q) ω 2 ((q, µ t 0+ x + x (q)), b ) < L β, for some p R, q Q, b, b Z}. Definition 4.8 Let X be a nonempty finite set. Then subset L of X is called recognizable (α, β)- language, if there exists a max-min IGLA such that L = L α,β ( ), where α, β L α L N(β). Theorem 4.9 Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) be a max-min IGLA. Then there exists an (α, β)- complete IGLA c such that L α,β ( ) = L α,β ( c ), where α, β L α L N(β). Proof. Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) does not be an (α, β)-complete max-min IGLA. Let Q c = Q {t}, where t is an element such that t / Q. Consider γ, η L, where γ > L α, η < L β γ L N(η). We now give an algorithm in which the output is an (α, β)-complete max-min IGLA for a given (α, β)-incomplete max-min IGLA as input.

13 minimal intuitionistic general L-fuzzy automata 167 Algorithm: (to make (α, β)-complete) Step 1 input incomplete = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ), where Q = {q 1, q 2,..., q n }, X = {a 1, a 2,..., a m }, Step 2 i = 1, Step 3 j = 1, if i n go to the next step, else go to Step 7, Step 4 if j m, then T = {q δ 1 (q i, a j, q) > L α} else i = i + 1 go to Step 3, Step 5 if T =, then δ c 1(q i, a j, t) = γ, δ c 2(q i, a j, t) = η, j = j + 1, go to Step 4, else go to Step 6, Step 6 for q T if δ 2 (q i, a j, q) < L β, then j = j + 1 go to Step 4, else T = T {q} go to Step 5, Step 7 δ1(p, c a, q) = δ 1 (p, a, q), δ2(p, c a, q) = δ 2 (p, a, q), for all p, q Q, a X, δ1(t, c a, t) = γ, δ2(t, c a, t) = η for all a X, { Step 8 Q c = Q {t}, ω1(p, c ω 1 (p, b) if p t, b) = 0 if p = t, { ω2(p, c ω 2 (p, b) if p t, b) = 1 if p = t, Step 9 output c = (Q c, X, R, Z, δ c, ω c, 1, 2, 3, 4 ). It is easy to see that the max-min IGLA c = (Q c, X, R, Z, δ c, ω c, 1, 2, 3, 4 ) is (α, β)- complete. It is clear that L α,β ( ) L α,β ( c ). Let x = u 1 u 2...u k+1 L α,β ( c ). Then there exist p R, q Q c, b, b Z such that δ c 1 ((p, µ t 0 (p)), x, q) ω c 1((q, µ t 0+ x + x (q)), b) > L α, δ c 2 ((p, µ t 0 (p)), x, q) ω c 2((q, µ t 0+ x + x (q)), b ) < L β. These imply that q Q there exist p 1, p 2,..., p k, p 1, p 2,..., p k Q such that (4.2) δ c 1((p, µ t 0 (p)), u 1, p 1 )... δ c 1((p k, µ t 0+ x (p k ), ν t 0+ x (p k )), u k, q) > L α. (4.3) δ c 2((p, µ t 0 (p)), u 1, p 1)... δ c 2((p k, µ t 0+ x (p k), ν t 0+ x (p k)), u k, q) < L β. Definition 2.2 (4.2) imply that µ t 0 (p) > L α, δ c 1(p, u 1, p 1 ) > L α, δ c 1(p 1, u 2, p 2 ) > L α,..., δ c 1(p k, u k, q) > L α. Now, suppose that p j, 1 j k, be the first state that δ c 1(p j, u j, p j+1 ) > L α δ 1 (p j, u j, p j+1 ) was undefined. Then p j+1 = t. Therefore p j+1 = p j+2 =... = q = t, which is a contradiction. Hence δ 1((p, µ t 0 (p)), x, q) ω 1 ((q, µ t 0+ x + x (q)), b) > L α.

14 168 m. shamsizadeh, m.m. zahedi In a similar manner we obtain δ 2((p, µ t 0 (p)), x, q) ω 2 ((q, µ t0+ x (q), ν t0+ x (q)), b ) < L β. Then the claim is hold. Example 4.10 Consider the complete lattice L = (L, L, T, S, 0, 1) defined in Example 4.3, max-min IGLA = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) as in igure 4, igure 4: The IGLA of Example 4.10 where Q = {q 0, q 1, q 2, q 3 }, X = {a 0 = u, a 1 = v}, R = {(q 0, 1, 0)}, Z = {o} δ : Q X Q L L is defined as follows: δ(q 0, u, q 1 ) = (a, 0) δ(q 1, v, q 3 ) = (a, d) δ(q 2, u, q 1 ) = (1, 0) δ(q 2, v, q 0 ) = (c, d) δ(q, x, q ) = (0, 1) for all other (q, x, q ) Q X Q ω : Q Z L L is defined by: ω(q 1, o) = (1, 0) ω(q, e) = (0, 1) for all other (q, e) Q Z. Now, considering the complete algorithm in the proof of Theorem 4.9, α = a, β = d, we have Stage 1 Let i = 0, j = 0. Then T = δ c 1(q 0, u, t) = c, δ c 2(q 0, u, t) = 0. Stage 2 Let i = 0, j = 1. Then T = δ c 1(q 0, v, t) = c, δ c 2(q 0, v, t) = 0. Stage 3 Let i = 1, j = 0. Then T = δ c 1(q 1, u, t) = c, δ c 2(q 1, u, t) = 0. Stage 4 Let i = 1, j = 1. Then T = δ c 1(q 1, v, t) = c, δ c 2(q 1, u, t) = 0. Stage 5 Let i = 2, j = 0. Then T = {q 1 } we have δ 2 (q 2, u, q 1 ) = 0 < L d. Stage 6 Let i = 2, j = 1. Then T = {q 0 }. Since δ 2 (q 2, v, q 0 ) = d, thus T = T {q 0 } =. Then δ c 1(q 2, v, t) = c, δ c 2(q 2, v, t) = 0. Stage 7 Let i = 3, j = 0. Then T = δ c 1(q 3, u, t) = c, δ c 2(q 3, u, t) = 0. Stage 8 Let i = 3, j = 1. Then T = δ c 1(q 3, v, t) = c, δ c 2(q 3, v, t) = 0.

15 minimal intuitionistic general L-fuzzy automata 169 Therefore, we have an (a, d)-complete c = (Q c, X, R, Z, δ c, ω c, 1, 2, 3, 4 ) as in igure 5. igure 5: The (a, d)-complete c of Example 4.10 Theorem 4.11 Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) be a max-min IGLA R. Then there exists a deterministic max-min IGLA d such that L α,β ( ) = L α,β ( d ), where α, β L α L N(β). Proof. Let (4.4) I x = {q Q q R such that δ 1((p, µ t 0 (p)), x, q ) > L α δ 2((p, µ t 0 (p)), x, q ) < L β}, for all x X. Then I Λ = {q Q q R}. Let Q d = {I x x X }. Define δ d : Q d X Q d L L, where { γ 1 if I x = I ya, δ d1 (I y, a, I x ) = 0 otherwise, { η 1 if I x = I ya, δ d2 (I y, a, I x ) = 1 otherwise, ω d : Q d Z d L L, where (4.5) ω d1 (I x, e) = (4.6) ω d2 (I x, e) = { γ 2 if x L α,β ( ), 0 otherwise, { η 2 if x L α,β ( ), 1 otherwise,

16 170 m. shamsizadeh, m.m. zahedi where γ 1, η 1, γ 2, η 2 L, γ 1 γ 2 > L α, η 1 η 2 < L β, γ 1 L N(η 1 ) γ 2 L N(η 2 ). Consider µ t 0 (I Λ ) = {µ t 0 (q) q I Λ }, ν t 0 (I Λ ) = {ν t 0 (q) q I Λ } Z d = {e}. Now, suppose that d = (Q d, X, I Λ, Z d, δ d, ω d, 1, 2, 3, 4 ). It is clear that δ d is well defined. Now, we show that ω d is well defined. Let I x = I y e Z d. If x L α,β ( ), then there exist q R, p Q, b, b Z such that δ 1((q, µ t 0 (q)), x, p) ω 1 ((p, µ t 0+ x + x (p)), b) > L α, δ 2((q, µ t 0 (q)), x, p) ω 2 ((p, µ t 0+ x + x (p)), b ) < L β. Thus p I x = I y. Therefore δ 1((q, µ t 0 (q)), y, p) > L α δ 2((q, µ t 0 (q)), y, p) < L β for some q R. Also ω 1 (p, b) > L α ω 2 (p, b) < L β. Hence y L α,β ( ). In a similar way, if y L α,β ( ), then x L α,β ( ). Hence ω d (I x, e) = ω d (I y, e). Since there exists q R such that µ t 0 (q) > L 0, then µ t 0 (I Λ ) > L 0. It is easy to see that the max-min IGLA d is deterministic. We show that Lα,β ( ) = L α,β ( sd ). Let x = u 1u 2...u k+1 L α,β ( ). Then there exist q R, p Q, b, b Z such that δ 1((q, µ t 0 (q)), x, p) ω 1 ((p, µ t 0+ x + x (p)), b) > L α, δ 2((q, µ t 0 (q)), x, p) ω 2 ((p, µ t 0+ x + x (p)), b ) < L β. Since δ 1((q, µ t 0 (q)), u 1...u k+1, p) > L α δ 2((q, µ t 0 (q)), u 1...u k+1, p) < L β, then µ t 0 (q) > L α, ν t 0 (q) < L β. Thus µ t 0 (I Λ ) > L α ν t 0 (I Λ ) < L β. Also δ d1((i Λ, µ t 0 (I Λ ), ν t 0 (I Λ )), u 1, I u1 ) = 1 T (µ t 0 (I Λ ), δ d1 (I Λ, u 1, I u1 )) L α, thus µ t 1 (I u1 ) L α. Also we have δ d1 ((I u1, µ t 1 (I u1 ), ν t 1 (I u1 )), u 2, I u1 u 2 ) = 1 T (µ t 1 (I u1 ), δ d1 (I u1, u 2, I u1 u 2 )) L α. So if we continue this process, then by some manipulation we get that δ d1((i Λ, µ t 0 (I Λ ), ν t 0 (I Λ )), x, I x ) L α. Also, we have δ d2((i Λ, µ t 0 (I Λ ), ν t 0 (I Λ )), u 1, I u1 ) = 1 S (ν t 0 (I Λ ), δ d2 (I Λ, u 1, I u1 )) L β, then ν t 1 (I u1 ) L β. Therefore δ d2 ((I u1, µ t 1 (I u1 ), ν t 1 (I u1 )), u 2, I u1 u 2 ) L β.

17 minimal intuitionistic general L-fuzzy automata 171 By continuing this process we obtain that δ d2((i Λ, µ t 0 (I Λ ), ν t 0 (I Λ )), x, I x ) L β. Also we have ω d1 (I x, e) = γ 2 ω d2 (I x, e) = η 2. Hence x L α,β ( d ). suppose that x L α,β ( d ). Then Now, δ d1((i Λ, µ t 0 (I Λ ), ν t 0 (I Λ )), x, I x ) ω d1 ((I x, µ t 0+ x (I x ), ν t 0+ x (I x )), e) > L α, δ d2((i Λ, µ t 0 (I Λ ), ν t 0 (I Λ )), x, I x ) ω d2 ((I x, µ t 0+ x (I x ), ν t 0+ x (I x )), e) < L β. Since ω d1 (I x, e) > L α ω d2 (I x, e) < L β, then by (4.5) (4.6) x L α,β ( ). Theorem 4.12 Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) be a max-min IGLA R. Then there exists an (α, β)-accessible max-min IGLA a such that L α,β ( ) = L α,β ( a ), where α, β L α L N(β). Proof. By Theorem 4.11, without loss of generality we assume that is deterministic. Let S={q Q q be an (α, β)-accessible state}, R a ={(q, µ t 0 (q)) q S R}, Z a = Z, δ a = δ S X S, ω a = ω S Z, i.e., δ a is the restriction of δ to S X S ω a is the restriction of ω to S Z. Then the max-min IGLA a is (α, β)-accessible. It is clear that L α,β ( a ) L α,β ( ). Now, let x = u 1 u 2...u k+1 L α,β ( ). Then there exist q R, p Q, b Z such that δ 1((q, µ t 0 (q)), x, p) ω 1 ((p, µ t 0+ x + x (p)), b) > L α, δ 2((q, µ t 0 (q)), x, p) ω 2 ((p, µ t 0+ x + x (p)), b ) < L β. Therefore, there exist p 1, p 2,..., p k, p 1, p 2,..., p k Q such that δ 1((q, µ t 0 (q)), u 1, p 1 )... δ 1((p k, µ t k (p k ), ν t k (p k )), u k+1, p) > L α, δ 2((q, µ t 0 (q)), u 1, p 1)... δ 2((p k, µ t k (p k), ν t k (p k)), u k+1, p) < L β. These imply that q R a, since is a deterministic, then p 1 = p 1, p 2 = p 2,..., p k = p k. Thus p 1, p 2,..., p k, p S. Hence δ a1((q, µ t 0 (q)), x, p) ω a1 ((p, µ t0+ x (p), ν t0+ x (p)), b) > L α, δ a2((q, µ t 0 (q)), x, p) ω a2 ((p, µ t0+ x (p), ν t0+ x (p)), b ) < L β. Then x L α,β ( a ). Hence L α,β ( ) L α,β ( a ). Thus the claim is hold.

18 172 m. shamsizadeh, m.m. zahedi Example 4.13 Consider the complete lattice L = (L, L, T, S, 0, 1) defined in Example 4.3, max-min IGLA = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) as in igure 6, igure 6: The IGLA of Example 4.13 where Q = {q 0, q 1, q 2, q 3, q 4 }, X = {u, v}, R = {(q 0, 1, 0)}, Z = {o}. It is clear that L a,b ( ) = {u, v} uv{u, v} {u, v} vu{u, v}. Considering the proof of Theorems 4.9, 4.11, 4.12, we obtain an (a, b)-complete, deterministic (a, b)-accessible max-min IGLA cda as in igure 7, such that La,b ( cda ) = La,b ( ). igure 7: The (a, b)-complete, deterministic (a, b)-accessible cda 4.13 of Example

19 minimal intuitionistic general L-fuzzy automata 173 Definition 4.14 Let = (Q, X, {q 0 }, Z, δ, ω, 1, 2, 3, 4 ) be an (α, β)-accessible, (α, β)-complete, deterministic max-min IGLA, where α, β L α L N(β). We define a relation on Q by q 1 ρ α,β q 2, if only if the following two sets are equal (i) {w X δ 1((q 1, µ t i (q 1 ), ν t i (q 1 )), w, q) ω 1 ((q, µ ti+ w (q), ν ti+ w (q)), b) > L α, δ 2((q 1, µ t i (q 1 ), ν t i (q 1 )), w, q) ω 2 ((q, µ ti+ w (q), ν ti+ w (q)), b ) < L β, for some b, b Z, q Q} (ii) {w X δ 1((q 2, µ t j (q 2 ), ν t j (q 2 )), w, q) ω 1 ((q, µ tj+ w (q), ν tj+ w (q)), b) > L α, δ 2((q 2, µ t j (q 2 ), ν t j (q 2 )), w, q) ω 2 ((q, µ tj+ w (q), ν tj+ w (q)), b ) < L β, for some b, b Z, q Q}, where q 1 Q act (t i ) q 2 Q act (t j ). It is clear that ρ α,β is an equivalence relation. Definition 4.15 We say that the (α, β)-accessible, (α, β)-complete, deterministic max-min IGLA, where α, β L α L N(β), is (α, β)-reduced if q 1 ρ α,β q 2 implies that q 1 = q 2, for any q 1, q 2 Q. Let = (Q, X, {q 0 }, Z, δ, ω, 1, 2, 3, 4 ) be an (α, β)-accessible, (α, β)- complete, deterministic max-min IGLA, where α, β L α L N(β). Now, suppose that ρ α,β be the equivalence relation defined in Definition Consider Q/ρ α,β = {qρ α,β q Q} R/ρ α,β = q 0 ρ α,β, µ t 0 (q 0 ρ α,β ) = µ t 0 (q 0 ρ α,β ) = ν t 0 (q 0 ). We define δ ρ : Q/ρ α,β X Q/ρ α,β L L by: γ 1 ifδ 1 (q 1, a, q 2) > L α & δ 2 (q 1, a, q 2) < L β, δ ρ1 (q 1 ρ α,β, a, q 2 ρ α,β ) = where q 2ρ α,β q 2 0 otherwise, η 1 if δ 1 (q 1, a, q 2) > L α & δ 2 (q 1, a, q 2) < L β, δ ρ2 (q 1 ρ α,β, a, q 2 ρ α,β ) = where q 2ρ α,β q 2 1 otherwise, where γ 1, η 1 L, γ 1 > L α, η 1 < L β γ 1 L N(η 1 ). Define ω ρ : Q/ρ α,β Z L L by: { γ2 ifω (4.7) ω ρ1 (q 1 ρ α,β 1 (q 1, b ) > L α & δ 2 (q 1, b ) < L β,, b) = 0 otherwise, (4.8) ω ρ2 (q 1 ρ α,β, b) = { η2 ifω 1 (q 1, b ) > L α & δ 2 (q 1, b ) < L β, 1 otherwise, for some b, b Z γ 2, η 2 L, γ 2 > L α, η 2 < L β γ 2 L N(η 2 ). Theorem 4.16 Suppose that α, β L, α L N(β). Then the following properties hold:

20 174 m. shamsizadeh, m.m. zahedi 1. δ ρ ω ρ are well defined, 2. /ρ α,β = (Q/ρ α,β, X, q 0 ρ α,β, Z, δ ρ, ω ρ, 1, 2, 3, 4 ) is an (α, β)-reduced maxmin IGLA, 3. L α,β ( /ρ α,β ) = L α,β ( ). Proof. 1. Let q 1 ρ α,β, q 2 ρ α,β, p 1 ρ α,β, p 2 ρ α,β Q/ρ α,β, q 1 ρ α,β = p 1 ρ α,β q 2 ρ α,β = p 2 ρ α,β. Then q 1 ρ α,β p 1 q 2 ρ α,β p 2. Therefore A = {w X δ 1((q 1, µ t i (q 1 ), ν t i (q 1 )), w, q) ω 1 ((q, µ t i+ w (q), ν t i+ w (q)), b) > L α, δ 2((q 1, µ t i (q 1 ), ν t i (q 1 )), w, q) ω 2 ((q, µ t i+ w (q), ν t i+ w (q)), b ) < L β, for some b, b Z, q Q} = {w X δ 1((p 1, µ t j (p 1 ), ν t j (p 1 )), w, q) ω 1 ((q, µ t j+ w (q), ν t j+ w (q)), b) > L α, δ 2((p 1, µ t j (p 1 ), ν t j (p 1 )), w, q) ω 2 ((q, µ t j+ w (q), ν t j+ w (q)), b ) < L β, for some b, b Z, q Q} = B. Let δ ρ1 (q 1 ρ α,β, a, q 2 ρ α,β ) = γ δ ρ2 (q 1 ρ α,β, a, q 2 ρ α,β ) = η, where γ, η L γ L N(η). Then there exists q 2 Q such that q 2ρ α,β q 2 δ 1 (q 1, a, q 2) > L α δ 2 (q 1, a, q 2) < L β. We have w A if only w B, for all w X. This holds in particular for all w ax. Then {aw ax δ 1 ((q 1, µ t i (q 1 ), ν t i (q 1 )), a, q 2) δ 1((q 2, µ t i+1 (q 2), ν t i+1 (q 2)), w, q) ω 1 ((q, µ t i+ w (q), ν t i+ w (q)), b) > L α, δ 2 ((q 1, µ t i (q 1 ), ν t i (q 1 )), a, q 2) δ 2((q 2, µ t i+1 (q 2), ν t i+1 (q 2)), w, q) for some b, b Z, q, q 2 Q} = ω 2 ((q, µ t i+ w (q), ν t i+ w (q)), b ) < L β, {aw ax δ 1((p 1, µ t j (p 1 ), ν t j (p 1 )), a, p ) δ 1((p, µ t j+1 (p ), ν t j+1 (p )), w, q) These imply that ω 1 ((q, µ t j+ w (q), ν t j+ w (q)), b) > L α, δ 2 ((p 1, µ t j (p 1 ), ν t j (p 1 )), a, p ) δ 2((p, µ t j+1 (p ), ν t j+1 (p )), w, q) for some b, b Z, q, p Q}. ω 2 ((q, µ t j+ w (q), ν t j+ w (q)), b ) < L β, {w X δ 1((q 2, µ t i+1 (q 2), ν t i+1 (q 2)), w, q) ω 1 ((q, µ t i+ w (q), ν t i+ w (q)), b) > L α, δ 2((q 2, µ t i+1 (q 2), ν t i+1 (q 2)), w, q) ω 2 ((q, µ t i+ w (q), ν t i+ w (q)), b ) < L β, for some b, b Z, q Q} = {w X δ 1((p, µ t j+1 (p ), ν t j+1 (p )), w, q) ω 1 ((q, µ t j+ w (q), ν t j+ w (q)), b) > L α, δ 2((p, µ t j+1 (p ), ν t j+1 (p )), w, q) ω 2 ((q, µ t j+ w (q), ν t j+ w (q)), b ) < L β, for some b, b Z, q Q}.

21 minimal intuitionistic general L-fuzzy automata 175 Thus p ρ α,β q 2ρ α,β q 2 ρ α,β p 2 i.e., p ρ α,β p 2. Also, is (α, β)-complete deterministic, therefore δ 1 (p 1, a, p ) > L α δ 2 (p 1, a, p ) < L β. Hence So δ ρ is well defined. δ ρ1 (p 1 ρ α,β, a, p 2 ρ α,β ) = γ δ ρ2 (p 1 ρ α,β, a, p 2 ρ α,β ) = η. Now, let q 1 ρ α,β = p 1 ρ α,β. If ω ρ1 (q 1 ρ α,β, b) = γ ω ρ2 (q 1 ρ α,β, b) = η, where γ, η L γ L N(η ), then ω 1 (q 1, b ) > L α ω 2 (q 1, b ) < L β, for some b Z ρ, b, b Z. Since is (α, β)-accessible, then there exists a positive integer j such that µ t j (q 1 ) > L α ν t j (q 1 ) < L β. These imply that Λ A. Therefore Λ B. Then ω 1 (q 2, b ) > L α ω 2 (q 2, b ) < L β, for some b, b Z. Hence Then ω ρ is well defined. ω ρ1 (q 2 ρ α,β, b) = γ ω ρ2 (q 2 ρ α,β, b) = η. 2. Let (q 1 ρ α,β )ρ α,β (q 2 ρ α,β ). Now, we have to show that q 1 ρ α,β = q 2 ρ α,β. It suffices to show that q 1 ρ α,β q 2. Let for w X δ 1((q 1, µ t i (q 1 ), ν t i (q 1 )), w, p 1 ) ω 1 ((p 1, µ t i+ w (p 1 ), ν t i+ w (p 1 )), b) > L α, δ 2((q 1, µ t i (q 1 ), ν t i (q 1 )), w, p 1 ) ω 2 ((p 1, µ t i+ w (p 1 ), ν t i+ w (p 1 )), b ) < L β, for some b, b Z. Then δ ρ1((q 1 ρ α,β, µ t i (q 1 ρ α,β ), ν t i (q 1 ρ α,β )), w, p 1 ρ α,β ) δ ρ2((q 1 ρ α,β, µ t i (q 1 ρ α,β ), ν t i (q 1 ρ α,β )), w, p 1 ρ α,β ) for some b, b Z. Therefore δ ρ1((q 2 ρ α,β, µ t j (q 2 ρ α,β ), ν t j (q 2 ρ α,β )), w, p 1ρ α,β ) δ ρ2((q 2 ρ α,β, µ t j (q 2 ρ α,β ), ν t j (q 2 ρ α,β )), w, p 1ρ α,β ) ω ρ1 ((p 1 ρ α,β, µ t i+ w (p 1 ρ α,β ), ν t i+ w (p 1 ρ α,β )), b) > L α, ω ρ2 ((p 1 ρ α,β, µ t i+ w (p 1 ρ α,β ), ν t i+ w (p 1 ρ α,β )), b ) < L β, ω ρ1 ((p 1ρ α,β, µ t j+ w (p 1ρ α,β ), ν t j+ w (p 1ρ α,β )), b) > L α, ω ρ2 ((p 1ρ α,β, µ t j+ w (p 1ρ α,β ), ν t j+ w (p 1ρ α,β )), b ) < L β, p 1ρ α,β Q/ρ α,β for some b, b Z. Since is (α, β)-accessible, then there exists a positive integer j such that µ t j (q 2 ) > L α ν t j (q 2 ) < L β. Thus δ 1((q 2, µ t j (q 2 ), ν t j (q 2 )), w, p) ω 1 ((p, µ t j + w (p), ν t j + w (p)), b) > L α, δ 2((q 2, µ t j (q 2 ), ν t j (q 2 )), w, p) ω 2 ((p, µ t j + w (p), ν t j + w (p)), b ) < L β,

22 176 m. shamsizadeh, m.m. zahedi where pρ α,β p 1 for some b, b Z. The converse follows in a similar manner. Hence {w X δ 1((q 1, µ t i (q 1 ), ν t i (q 1 )), w, q) ω 1 ((q, µ t i+ w (q), ν t i+ w (q)), b) > L α, δ 2((q 1, µ t i (q 1 ), ν t i (q 1 )), w, q) ω 2 ((q, µ t i+ w (q), ν t i+ w (q)), b ) < L β, for some b, b Z, q Q} = {w X δ 1((q 2, µ t j (q 2 ), ν t j (q 2 )), w, q) ω 1 ((q, µ t j+ w (q), ν t j+ w (q)), b) > L α, δ 2((q 2, µ t j (q 2 ), ν t j (q 2 )), w, q) ω 2 ((q, µ t j+ w (q), ν t j+ w (q)), b ) < L β, for some b, b Z, q Q}. Hence the claim is hold. 3. Considering the proof of 2, it is trivial. Example 4.17 Consider the complete lattice L = (L, L, T, S, 0, 1) defined in Example 4.3, the max-min IGLA cda = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) as in igure 7. By the Definition 4.14, I uv ρ a,b I uvu ρ a,b I vuv ρ a,b I vu. Then the max-min IGLA /ρ a,b has the state diagram as in igure 8. igure 8: The reduced /ρ a,b of Example 4.17 Definition 4.18 Let 1 = (Q 1, X, R 1, Z 1, δ, ω, 1, 2, 3, 4 ) 2 = (Q 2, X, R 2, Z 2, δ, ω, 1, 2, 3, 4 ) be two max-min IGLA. A homomorphism from 1 onto 2 with threshold (α, β), where α, β L α L N(β), is a function ξ from Q 1 onto Q 2 such that for every q, q Q 1, u X b 1, b 2 Z the following conditions hold:

23 minimal intuitionistic general L-fuzzy automata (µ t 0 Q1 (q ) > L α & ν t 0 Q 1 (q ) < L β) if only if (µ t 0 Q2 (ξ(q )) > L α & ν t 0 Q 2 (ξ(q )) < L β), 2. (δ 1 (q, u, q ) > L α & δ 2 (q, u, q ) < L β) if only if (δ 1(ξ(q ), u, ξ(q )) > L α & δ 2(ξ(q ), u, ξ(q )) < L β), 3. (ω 1 (q, b 1 ) > L α & ω 2 (q, b 2 ) < L β) implies (ω 1(ξ(q ), b) > L α & ω 2 (ξ(q ), b 2 ) < L β) for some b, b Z. We say that ξ is an isomorphism with threshold (α, β) if only if ξ is a homomorphism with threshold (α, β) that is one-one (ω 1 (q, b 1 ) > L α ω 2 (q, b 2 ) < L β) if only if (ω 1(ξ(q ), b) > L α ω 2(ξ(q ), b 2 ) < L β), for some b, b Z. Definition 4.19 Let L be an (α, β)-language, where α, β L α L N(β). A relation R L on X is defined by: for any two strings x y in X, xr L y if, for all z X, either xz, yz L or xz, yz / L. Definition 4.20 Let = (Q, X, {q 0 }, Z, δ, ω, 1, 2, 3, 4 ) be a deterministic max-min IGLA. Then for any α, β L, where α L N(β), define a relation R α,β on X. or strings x y in X, xr α,β y if only if there exists q Q such that δ 1((q 0, µ t 0 (q 0 )), x, q) > L α δ 2 ((q 0, µ t 0 (q 0 )), x, q) < L β, iff (q 0 )), y, q) > L α δ 2 ((q 0, µ t 0 (q 0 )), y, q) < L β. Now, we show that R α,β is an equivalence relation. It is clear that xr α,β x if xr α,β y, then yrα,β x. Now, let xr α,β y yrα,β z. Suppose that there exists q Q such that δ 1((q 0, µ t 0 (q 0 )), x, q) > L α δ 2((q 0, µ t 0 (q 0 )), x, q) < L β. Then (q 0 )), y, q) > L α δ 2((q 0, µ t 0 (q 0 )), y, q) < L β,, since yr α,β z, then δ 1((q 0, µ t 0 (q 0 )), z, q) > L α δ 2((q 0, µ t 0 (q 0 )), z, q) < L β. Similarity we can obtain the converse, i.e., (q 0 )), z, q) > L α δ 2((q 0, µ t 0 (q 0 )), z, q) < L β, implies that (q 0 )), x, q) > L α δ 2((q 0, µ t 0 (q 0 )), x, q) < L β. Then xr α,β z. Hence the claim is hold.

24 178 m. shamsizadeh, m.m. zahedi Note 4.21 By Definition 4.20, for any (α, β)-complete deterministic maxmin IGLA, the number of classes of equivalence relation R α,β is less than or equal to the number of states of. Theorem 4.22 Let α, β L, where α L N(β). Suppose that L be a recognizable (α, β)-language where L = L α,β ( ), for some (α, β)-complete deterministic max-min IGLA = (Q, X, {q 0 }, Z, δ, ω, 1, 2, 3, 4 ). Then for a given equivalence class [w] R α,β such that [w] R α,β finite union of equivalence classes of R α,β of R α,β there is an equivalence class [w] RL of R L of the relation R L is a [w] RL. Each equivalence class [w] RL. Proof. Let [w] R α,β be an equivalence class of R α,β. Now, suppose that x [w] R. α,β Since is an (α, β)-complete, then there exists q Q such that (q 0 )), x, q) > L α δ 2((q 0, µ t 0 (q 0 )), x, q) < L β. Therefore (q 0 )), w, q) > L α δ 2((q 0, µ t 0 (q 0 )), w, q) < L β. By the (α, β)-complete property of, for any y X, there exists q Q such that (q 0 )), xy, q ) (q 0 )), wy, q ) > L α, δ 2((q 0, µ t 0 (q 0 )), xy, q ) δ 2((q 0, µ t 0 (q 0 )), wy, q ) < L β. If there exist b, b Z such that ω 1 (q, b) > L α ω 2 (q, b) < L β, then xy, wy L otherwise xy, wy / L. Then xr L w. Thus x [w] RL [w] R α,β [w] RL. Suppose that [w] RL is an equivalence class of R L. It is obvious that if w / L, then for any x [w] RL, x / L if w L, then for any x [w] RL, x L. If w / L, then for any q Q, b, b Z we have or (q 0 )), w, q) ω 1 ((q, µ t 0+ w + w (q)), b) L α, δ 2((q 0, µ t 0 (q 0 )), w, q) ω 2 ((q, µ t 0+ w + w (q)), b ) L β. If (q 0 )), w, q) > L α δ 2((q 0, µ t 0 (q 0 )), w, q) < L β, then The set (4.9) ω 1 (q, b) L α or ω 2 (q, b) L β. S = {q Q (q 0 )), x, q) > L α, δ 2((q 0, µ t 0 (q 0 )), x, q) < L β, x [ω] RL }, is finite. Then we have ω 1 (q, b) L α or ω 2 (q, b) L β for all q S b Z, or for all q S there exist b, b Z such that ω 1 (q, b) > L α ω 2 (q, b ) < L β. Therefore the equivalence class [w] RL of R L is a finite union of the equivalence classes [w] R α,β of R α,β, where q S.

25 minimal intuitionistic general L-fuzzy automata 179 Theorem 4.23 Let L be a recognizable (α, β)-language. Then there is an (α, β)- complete deterministic IGLA m such that L α,β ( m) = L m is a minimal automaton, where α, β L α L N(β). Proof. Let = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) be an (α, β)-complete deterministic max-min IGLA such that L = L α,β ( ). By Theorem 4.22, we have that the number of equivalence classes of R L is finite. Let Q m be the set of equivalence classes of R L i.e., Q m = {[w] [w] is an equivalence class of R L }. Let R m = {([Λ], γ 1, γ 2 )}, where γ 1, γ 2 L, γ 1 > L α, γ 2 < L β γ 1 L N(γ 2 ). Define δ m : Q m X Q m L L by { γ1 if [za] = [x], (4.10) δ m1 ([z], a, [x]) = 0 otherwise, (4.11) δ m2 ([z], a, [x]) = { γ2 if [za] = [x], 1 otherwise, Also define ω m : Q m {b} L L by { γ1 if w L, (4.12) ω m1 ([w], b) = 0 otherwise, (4.13) ω m2 ([w], b) = { γ2 if w L, 1 otherwise. It is clear that m = (Q m, X, R m, Z = {b}, δ m, ω m, 1, 2, 3, 4 ) is an (α, β)- complete deterministic IGLA. Clearly, δ m ω m are well defined. Let w L. Then, we have δ m1(([λ], µ t 0 ([Λ]), ν t 0 ([Λ])), w, [w]) > L α δ m2(([λ], µ t 0 ([Λ]), ν t 0 ([Λ])), w, [w]) < L β, ω m1 ([w], b) = γ 1 > L α, ω m2 ([w], b) = γ 2 < L β. These imply that w L α,β ( m). Now, suppose that w L α,β ( m). Then there exist [z] Q m, b Z such that (4.14) δ m1(([λ], µ t 0 ([Λ]), ν t 0 ([Λ])), w, [z]) ω m1 (([z], µ t 0+ w ([z]), ν t 0+ w ([z])), b)> L α, (4.15) δ m2(([λ], µ t 0 ([Λ]), ν t 0 ([Λ])), w, [z]) ω m2 (([z], µ t 0+ w ([z]), ν t 0+ w ([z])), b)< L β.

26 180 m. shamsizadeh, m.m. zahedi Considering (4.10), (4.11), (4.14) (4.15) we have [z] = [w]. Then ω m1 ([w], b) > L α ω m2 ([w], b) < L β. Then w L. Hence L α ( m) = L. By Theorem 4.22, we have that the number of equivalence classes of R L is less than or equal to that of equivalence classes of R α,β relation R α,β holds., by Note 4.21, the number of classes of equivalence is less than or equal to the number of states of. Hence the claim Example 4.24 Let the complete lattice L = (L, L, T, S, 0, 1) defined in Example 4.3, the max-min IGLA = (Q, X, R, Z, δ, ω, 1, 2, 3, 4 ) as in igure 6. Considering Definition 4.19, we find [u], [v], [uv] = [vu], [u 2 ] = [u], [v 2 ] = [v], [uv 2 ] = [uv] [uvu] = [uv]. Then we have m as in igure 9. It is igure 9: The minimal m of Example 4.24 obvious that /ρ a,b as in igure 8, m as in igure 9, have the same number of states. Theorem 4.25 or every recognizable (α, β) language L, the minimal max-min IGLA defined in proof of Theorem 4.23, is (α, β)-reduced, where α, β L α L N(β). Proof. Let be an (α, β)-accessible, (α, β)-complete, deterministic max-min IGLA. Suppose that q 1 = [u 1 ], q 2 = [u 2 ] be equivalence classes of R L. Now, let q 1 ρ α,β q 2. Then A={w X δ 1(([u 1 ], µ t i ([u 1 ]), ν t i ([u 1 ])), w, q) ω 1 ((q, µ t i+ w (q), ν t i+ w (q)), b)> L α, δ 2(([u 1 ], µ t i ([u 1 ]), ν t i ([u 1 ])), w, q) ω 2 ((q, µ t i+ w (q), ν t i+ w (q)), b ) < L β, for some b, b Z, q is an equivalence classes of R L } ={w X δ 1(([u 2 ], µ t j ([u 2 ]), ν t j ([u 2 ])), w, q) ω 1 ((q, µ t j+ w (q), ν t j+ w (q)), b)> L α, δ 2(([u 2 ], µ t j ([u 2 ]), ν t j ([u 2 ])), w, q) ω 2 ((q, µ t j+ w (q), ν t j+ w (q)), b ) < L β,

27 minimal intuitionistic general L-fuzzy automata 181 for some b, b Z, q is an equivalence classes of R L } = B. Therefore w A w B, for all w X, this implies that u 1 w L u 2 w L, for all w X. Then [u 1 ] = [u 2 ]. Hence is an (α, β)-reduced. Theorem 4.26 Let L be a recognizable (α, β)-language, where α, β L α L N(β). Let m be the max-min IGLA defined in the proof of Theorem 4.23, be an (α, β)-complete, (α, β)-accessible, deterministic (α, β)-reduced max-min IGLA. Then m are isomorphism with threshold (α, β). Proof. Let m = (Q,, X, {[Λ]}, {b}, δ m, ω m, 1, 2, 3, 4 ) = (Q, X, {q 0 }, Z, δ, ω, 1, 2, 3, 4 ). Define ξ : Q Q m by ξ(q) = [u], where (q 0 )), u, q) > L α δ 2((q 0, µ t 0 (q 0 )), u, q) < L β. Since is (α, β)-accessible, then µ t 0 Q (q 0 ) > L α ν t 0 Q (q 0) < L β. Also, µ t 0 Qm ([Λ]) > L α ν t 0 Q m ([Λ]) < L β. Let q 1, q 2 Q q 1 = q 2. Then q 1 ρ α,β q 2. Therefore {w X δ 1((q 1, µ t i (q 1 ), ν t i (q 1 )), w, q) ω 1 ((q, µ t i+ w (q), ν t i+ w (q)), b) > L α, δ 2((q 1, µ t i (q 1 ), ν t i (q 1 )), w, q) ω 2 ((q, µ t i+ w (q), ν t i+ w (q)), b ) < L β, for some b, b Z, q Q} = {w X δ 1((q 2, µ t j (q 2 ), ν t j (q 2 )), w, q) ω 1 ((q, µ t j+ w (q), ν t j+ w (q)), b) > L α, δ 2((q 2, µ t j (q 2 ), ν t j (q 2 )), w, q) ω 2 ((q, µ t j+ w (q), ν t j+ w (q)), b ) < L β, for some b, b Z, q Q}. Since µ t i (q 1 ) > L α, ν t i (q 1 ) < L β µ t j (q 2 ) > L α, ν t j (q 2 ) < L β, then there exist u, v X, where u = i, v = j, (q 0 )), u, q 1 ) > L α δ 2((q 0, µ t 0 (q 0 )), u, q 1 ) < L β, (q 0 )), v, q 2 ) > L α δ 2((q 0, µ t 0 (q 0 )), v, q 2 ) < L β. Thus uz L if only if vz L for all z X. Then [u] = [v], i.e., ξ(q 1 ) = ξ(q 2 ). Hence ξ is well defined. Let [u] Q m. By the (α, β)-complete property of, there exists q Q such that (q 0 )), u, q) > L α δ 2((q 0, µ t 0 (q 0 )), u, q) < L β. Then ξ(q) = [u]. Therefore ξ is surjective. Now, let δ 1 (q, a, q ) > L α δ 2 (q, a, q ) < L β. Since is (α, β)- accessible, then there exists u X such that (q 0 )), u, q) > L α δ 2((q 0, µ t 0 (q 0 )), u, q) < L β.