Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids

Fuzzy Sets and Systems 156 (2005) 68 – 92 www.elsevier.com/locate/fss

Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids夡 Yongming Lia, b,∗ , Witold Pedryczc a Institute of Fuzzy Systems, College of Mathematics and Information Science, Shaanxi Normal University,

Xi’an 710062, PR China b Department of Automatical Control, Northwestern Polytechnical University, Xi’an 710072, PR China c Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada T6G2V4

Received 5 December 2003; received in revised form 3 February 2005; accepted 7 April 2005 Available online 27 April 2005 Communicated by Mordeson

Abstract We study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice-ordered monoid L. For these L-fuzzy finite automata (L-FFA, for short), we provide necessary and sufficient conditions for the extendability of the state-transition function. It is shown that nondeterministic L-FFA (NL-FFA, for short) are more powerful than deterministic L-FFA (DL-FFA, for short). Then, we give necessary and sufficient conditions for the simulation of an NL-FFA by an equivalent DL-FFA. Next, we turn to the closure properties of languages defined by L-FFAs: we establish closure under the regular operations and provide conditions for closure under intersection and reversal, in particular we show that the family of fuzzy languages accepted by DL-FFAs is not closed under Kleene closure operation, and the family of fuzzy languages accepted by NL-FFAs is not closed under complement operation. Furthermore, we introduce the notions of L-fuzzy regular expressions and give the Kleene theorem for NL-FFAs. The description of DL-FFAs by L-fuzzy regular expressions is also given. Finally, we investigate the level structures of L-FFAs. Our results provide some insight as to what extend properties of L-FFAs and their languages depend on the algebraic properties of L. © 2005 Elsevier B.V. All rights reserved. Keywords: Fuzzy finite automaton; Lattice-order monoid; Fuzzy language; Closure property; Fuzzy regular expression; Level structure 夡

This work is supported by National Science Foundation of China (Grant No. 60174016, 10226023), “TRAPOYT” of China and National 973 Foundation Research Program (Grant No. 2002CB312200). ∗ Corresponding author. Department of Mathematics, Shaanxi Normal University, Xi’an 710062, PR China. Fax: +86 29 530 7025. E-mail addresses: [email protected] (Y. Li), [email protected] (W. Pedrycz). 0165-0114/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2005.04.004

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1. Introduction The concept of fuzzy automata was introduced by Santos [31–33] and Wee [39,40] in the late 1960s. Since finite automata constitute a mathematical model of computation, fuzzy finite automata may be considered as an extended model which includes notions like “vagueness” and “imprecision”, i.e., notions frequently encountered in the study of natural languages. So investigating fuzzy finite-state automata [24] might reduce the gap between formal languages as studied in classical automata theory [9] on the one hand and natural languages on the other hand. Research on fuzzy languages accepted by fuzzy finitestate devices originated in the early 1970s by Zadeh and Lee [43,13], and Thomason and Marionos [36]. Algebraic properties of fuzzy languages have been studied, among others, by Shen [35], and by Mordeson and Malik[21,22,24]. For early overviews of the theory of fuzzy finite-state machines we refer to [10,38]; a much more recent one is [24]. Also, fuzzy finite automata have many important applications such as in learning systems, the model of computing with words, pattern recognition and data base theory [24,26,20,42]. Recently, Qiu and Asveld proposed to study fuzzy automata based on residuated logic in [28,29,2,25], where the proposed method provided a tool to study fuzzy automata in the frame of many-valued logic. Finite-state automata are the mathematical models to recognize formal languages in the theory of classical computation, and the former proposed fuzzy automata with membership values in unit interval [0, 1] with max–min composition, we call them classical fuzzy automata in this paper, were just the simple generalization of classical automata in this respect. Classical fuzzy automata are mostly drawn from fuzzifying preliminarily the classical case, and therefore have poor level structure, i.e., they could recognize only regular languages from the point of view of level structure. To overcome this kind of problem, we study automata theory with membership values in more general structures, such as lattice-order monoids [5], and set the formal models of computing with words on the L-fuzzy finite automata (L-FFAs) with membership values in a lattice-ordered monoid L. We give some comments in this respect as follows. It is well-known that in classical automata theory, the following four approaches to represent a language (regular language) L are equivalent: (i) (ii) (iii) (iv)

L is recognized by deterministic finite-state automaton. L is recognized by nondeterministic finite-state automaton. L is described by regular expression. L is generated by regular grammar.

The same results hold for fuzzy regular languages with membership values in [0,1] and with max– min composition. For L-fuzzy languages, we shall show that the similar results are not valid for some lattice-ordered monoids which the fuzzy sets take values in. The nondeterministic L-FFAs or NL-FFAs are more powerful than deterministic L-FFAs or DL-FFAs in recognizing fuzzy languages. We also give some sufficient and necessary conditions to guarantee that these two notions are equivalent. On the other hand, as shown by Mockoˇ ˇ r and Belohl ˇ avek ´ in [4,25], from the point of view of level structure, classical fuzzy finite automata can only recognize regular languages, that is to say, classical fuzzy finite automata are equivalent to classical finite automata in the sense of recognizing crisp languages from the point of view of level structure. The same problem appears in the fuzzy automata based on residuated logic in which fuzzy automata take membership values in a complete residuated lattice. But we shall show that L-FFA can recognize more languages than regular languages from the point of view of level structure, and thus have more power to recognize fuzzy languages than classical fuzzy finite state automata. Considering

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the regular operations on fuzzy languages, it will be shown that (a) the family of the fuzzy languages of DL-FFAs is closed under union, intersection, concatenation, scalar product and complement, but not Kleene closure, (b) the family of the fuzzy languages accepted by NL-FFAs is closed under union, intersection, concatenation, scalar product and Kleene closure, but not complement. This phenomenon also demonstrates some special properties of fuzzy finite-state automata. Another problem concerns a description of DL-FFA or NL-FFA by regular expressions. In the paper, we introduce the concept of L-fuzzy regular expressions. This concept provides not only a necessary tool for the analysis and synthesis of fuzzy automata, but also forms a vehicle for a recursive generation of the family of fuzzy languages accepted by fuzzy automata from certain simple fuzzy languages. Since the families of fuzzy languages accepted by various models of fuzzy automata are, in general, nondenumerable, the concept of regular fuzzy expressions could provide the necessary insights into the study of the structure of such families. In fact, Santos studied regular fuzzy expressions in [34]. Since the nondeterministic fuzzy finite automata are not equivalent to deterministic fuzzy finite automata, as shown in this paper. The proof of Santos in [34] completed within the framework of deterministic fuzzy finite automata does not hold there. More specifically, the proof of Santos only applies to the cases where the nondeterministic fuzzy automata are equivalent to deterministic fuzzy automata. Here, we demonstrate that the nondeterministic fuzzy automata can be represented by regular fuzzy expression in a more generalized frame—L-FFAs. Kleene theorem holds for NL-FFAs and L-fuzzy regular expressions, but not for DL-FFAs. Having this mind, we also give another more direct description of the regular expressions for DL-FFAs. The results of this paper show that L-FFAs exhibit some essential difference with fuzzy automata if the different kinds of lattice-ordered monoids in which L-FFAs take membership values are chosen, and they can recognize more extensive classes of formal languages and fuzzy languages. This also forms the essential feature of L-FFAs. We refer to [5,9,11] for formal language theory, and [1,10,12,24,38] for fuzzy finite automata in the following sections. The rest of the paper is arranged as follows. In Section 2, we first review and study some facts about fuzzy extension principle. In Section 3, we introduce the notion of L-fuzzy finite automata and the languages recognized by L-fuzzy finite automata. For these L-fuzzy finite automata, we provide necessary and sufficient conditions for the extendability of the state-transition function. It is shown that NL-FFA are more powerful than DL-FFA. Then we give necessary and sufficient conditions for the simulation of an NL-FFA by an equivalent DL-FFA. Next we turn to the closure properties of languages defined by L-FFAs in Section 4: we establish closure under the regular operations and provide conditions for closure under intersection and reversal, in particular we show that the family of fuzzy languages accepted by DL-FFAs is not closed under Kleene closure operation, and the family of fuzzy languages accepted by NL-FFAs is not closed under complement operation. In Section 5, we introduce the notions of L-fuzzy regular expressions and give the Kleene theorem for NL-FFAs. The description of DL-FFAs by L-fuzzy regular expressions is also given. In Section 6, we investigate the level structures of L-FFAs. Finally, some conclusions are concerned in Section 7. 2. Fuzzy extension of a function We first give some basic concepts to be used within this paper. Given a lattice L, we use ∨, ∧ to represent the supremum operation and infimum operation on L, respectively. We require L to have the least and the largest elements in this paper: those will be denoted

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as 0, 1, respectively. Assume that there is a binary operation Q (we call it multiplication) on L such that (L, Q, e) is a monoid with identity e ∈ L. We call L an po-monoid (some modifications of the notion of partially ordered monoid in [5]) if it satisfies the following two conditions for arbitrary a, b, x ∈ L: (C-1) a Q0 = 0Qa = 0, (C-2) a
b ⇒ a Qx
bQx and x Qa
x Qb. If L is an po-monoid and it satisfies the distributive laws, i.e., for arbitrary a, b, c ∈ L: (C-3) a Q(b ∨ c) = (a Qb) ∨ (a Qc), and (b ∨ c)Qa = (bQa) ∨ (cQa). then we call L a lattice-ordered monoid or l-monoid [5]. Moreover, if L is a complete lattice, and it satisfies the
following
infinite distributive
laws for
arbitrary a ∈ L and {bt }t∈T of L for an index set T: (C-4) a Q( t bt ) = t (a Qbt ), and ( t bt )Qa = t (bt Qa), then we call L a quantale [30,15,18]. If the distributive laws in (C-4) hold only for countable set {bt }, then L is called a countable l-monoid. For an l-monoid, we only concern with the multiplication Q and finite supremum operation ∨, in what follows, an l-monoid is denoted as (L, Q, ∨). And if we deal with the subalgebra L1 of an l-monoid (L, Q, ∨), it means that L1 is a nonempty subset of L and L1 is closed under the multiplication and finite supremum of L. We do not concern with the infimum operation in L1 . It is instructive to come up with several examples of l-monoids. Example 2.1. (1) Let (L, ∧, ∨) be a distributive lattice, and let Q = ∧, then L is an l-monoid, and the identity of multiplication is 1. (2) Let (L, Q, ∨) be an l-monoid, the identity is e. We use L(n) to denote all n × n matrices with values in L. The multiplication, denoted as ◦, is defined as matrix multiplication or as in Zadeh’s terminology, sup − Q composition; and ∨ is the pointwise-∨. That
is, for two n × n matrices, A = (aij ) and B = (bij ), with values in L, let A◦B = C = (cij ), then cij = nk=1 (aik Qbkj ), and let A∨B = D = (dij ), then dij = aij ∨bij . Then (L(n), ◦, ∨) is also an l-monoid, the identity is the diagonal-matrix E = diag(e, e, . . . , e) with e as the diagonal element. In general, the multiplication on L(n) is not commutative, even if the multiplication on L is commutative. (3) Let Q be any uninorm [19,41] on [0, 1]; if 0Q1 = 0, then ([0, 1], Q, ∨) is a commutative l-monoid. In particular, if Q is a t-norm on [0, 1], then ([0, 1], Q, ∨) is a commutative l-monoid with identity e = 1. (4) For any set X, let Rel(X) denote the set of all binary relations on X. If we denote the relation composition as ◦, then (Rel(X), ◦, ) is a quantale. For more examples of quantales, we refer to [30,15–18]. (5) Let L = [0, ∞) ∪ {∞} such that a
∞ for any a ∈ [0, ∞), L is also denoted as [0, ∞], then ∨ is the maximum operation. Let Q be the multiplication of real numbers, that is,   a · b, a Qb = 0,  ∞,

a, b = ∞, a = 0 or b = 0, a  = 0, b = ∞ or a = ∞, b  = 0.

Then (L, Q, ∨) is a commutative quantale, with identity 1, and the largest element equals ∞. (6) Complete residuated lattices [28,2,25] are special kinds of quantales, where its multiplication is commutative and the identity is the same as the largest element.

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Assume that L is an po-monoid, and let F (X) = {A : X → L|A is an L-fuzzy subset of X} denote the powerset of L-fuzzy subsets of X, where an L-fuzzy subset of X is just a function A : X → L with codomain L. Proposition 2.1. Let L be a lattice, and (L, Q, e) be a monoid with identity e ∈ L. If e = 1, then condition (C-2) implies condition (C-1), and a Qb
a ∧ b for any a, b ∈ L. Proof. For any a, b ∈ L, observe that a Qb
a Q1 = a, a Qb
1Qb = b, thus a Qb
a ∧ b.



We study the fuzzy extension of a function in the following way. Given two sets X, Y and a function f : X → Y , for any quantale L, according to the Zadeh’s extension principle, there is an extension of f to the power set of L-fuzzy subsets F (X) to F (Y ), which is also denoted by the same letter f : F (X) → F (Y ), and for each A ∈ F (X), y ∈ Y , it is defined as follows:  f (A)(y) = {A(x)|x ∈ f −1 (y)}.  From the representation theorem of L-fuzzy subsets, since A =
∈L
A
, where A
= {x ∈ X|A(x) 
} is the
-cut of A, and
A
is defined as follows for arbitrary x ∈ X, 
, x ∈ A

A
(x) = , 0, x ∈ / A
this extension can also defined as 
f (A
), f (A) =

(1)


∈L

where f (A
) represents the image set of A
under f, that is, f (A
) = {f (x)|x ∈ A
}. Consider the last observation of Zadeh’s extension by Eq. (1), we further extend a fuzzy function from a crisp set to the power set of L-fuzzy subsets as g : X → F (Y ). First, we extend g to the power set of subsets of X, denoted as the same letter g : 2X → F (Y ), as follows for arbitrary X1 ⊆ X and y ∈ Y ,  g(x)(y), g(X1 )(y) = x∈X1

where 2X denotes the power set of X. Furthermore, we extend it to the power set of L-fuzzy subsets of X, also denoted as the same letter g : F (X) → F (Y ), and for each A ∈ F (X), it is defined as follows:

  g(A) = g
A
=
g(A
), (2)
∈L


∈ L

where
g(A
) ∈ F (Y ) is defined reasonable as follows: 
g(A
)(y) =
Qg(x)(y). x∈A


We give an explicit formula for this extension in the following theorem.

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Theorem 2.1. For any A ∈ F (X) and y ∈ Y , we can define formula (2) in the following form:  g(A)(y) = {A(x)Qg(x)(y)|x ∈ X}. Proof. Since the multiplication is distributive over infinite joins


in L, we have g(A)(y) = [ x∈A
g(x)(y)] = {
Qg(x)(y)|A(x) 
, x ∈ X,
∈ L} = {A(x)Qg(x)(y)|x ∈ X}. 

(3)



∈L
Q

In particular, if X is a finite set, then the function g in formula (3) is also well defined for any pomonoid L. Therefore, for a given fuzzy function g : X → F (Y ), we shall use the form (3) to represent the extension function for the case of po-monoid in the following sections. 3. L-fuzzy finite automata and their languages We start with the definition of L-fuzzy finite automata for a specify po-monoid (L, Q, e). Definition 3.1. An L-fuzzy finite automaton or L-FFA is a five tuple, A = (Q, , , 0 , 1 ),

where Q,  are finite nonempty sets,  : Q ×  → F (Q) is a mapping, and 0 , 1 : Q → L are two L-fuzzy sets of Q. The elements of Q are called states, and the elements of  denote (input) symbols, respectively.  is called a fuzzy transition function and 0 , 1 are called fuzzy initial state and fuzzy final state, respectively. Let ∗ denote the set of all words of finite length over  and let  denote the empty word. Then ∗ is the free monoid generated by  with concatenation as binary operation. For  ∈ ∗ , let || denote the length of . First, we extend  on Q × ∗ . In order to do that, we first extend  on F (Q) × F () using the method introduced in the previous section, where we consider the elements of F () as fuzzy (input) symbols or words [42]. For any A ∈ F (Q) and B ∈ F (), we can define a mapping  from F (Q) × F () to F (Q × ) by the product operation (A, B)(q, u) = A(q)QB(u). If we use the extension function in the form (3), then we can extended  onto F (Q) × F (), denoted  : F (Q) × F () → F (Q), which is defined as in the following form,   (A, B)(q) = [(A, B)(p, u)Q(p, u)(q)] = [A(p)QB(u)Q(p, u)(q)]. (4) p∈Q,u∈

p∈Q,u∈

Now the whole fuzzy symbol in F (Q) × F () is taken into account, and not only a single symbol in Q ×  as that of  : Q ×  → F (Q). For convenience, we also write (q, u, p) as (q, u)(p), that is, (q, u, p) = (q, u)(p). We then can naturally induce the following extension of  on F ()∗ , denoted ∗ : F (Q) × F ()∗ → F (Q), (i) ∀A ∈ F (Q), ∗ (A, ) = A, (ii) ∀ ∈ F ()∗ , C ∈ F (), ∗ (A, C) = (∗ (A, ), C),

(5)

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where F ()∗ denotes the set of words (fuzzy languages) strings over F (). Therefore, the transition function ∗ can be used to cope with the natural languages with fuzzy meaning as the set of words string over F () [42]. Using expression (5), we can extend  on ∗ , denoted in sequel as ∗ : Q × ∗ → F (Q), as ∗ (q, u) = (∗ (q, ), u),

where q ∈ Q,  ∈ ∗ , u ∈ , and ∗ (q, ) ∈ F (Q). The explicit expression of ∗ comes in the form (i) ∀p ∈ Q, if p = q, then ∗ (q, , p) = e, otherwise ∗ (q, , p) = 0;  [∗ (q, , r)Q(r, u, p)]. (ii) ∀ ∈ ∗ , u ∈ , ∗ (q, u, p) =

(6)

r∈Q

As far as extension ∗ is concerned, for any  ∈ ∗ , if  = 1 2 , then it should satisfy the following:  ∗ (q, 1 2 , p) = [∗ (q, 1 , r)Q∗ (r, 2 , p)]. (7) r∈Q

To ensure Eq. (7) holds, we need L to be an l-monoid. Theorem 3.1. The following conditions are equivalent: (i) (L, Q, ∨) is an l-monoid, that is to say, the multiplication is distributive to finite joins, a Q(b ∨ c) = (a Qb) ∨ (a Qc), (b ∨ c)Qa = (bQa) ∨ (cQa). (ii) For any L-FFA, A = (Q, , , 0 , 1 ), and for any p, q ∈ Q, 1 , 2 ∈ ∗ ,  ∗ (q, 1 2 , p) = [∗ (q, 1 , r)Q∗ (r, 2 , q)]. (8) r∈Q

(iii) For any L-FFA, A = (Q, , , 0 , 1 ), and for any p, q ∈ Q,  = u1 u2 · · · uk ∈ ∗ ,  ∗ (q, , p) = [(q, u1 , q1 )Q · · · Q(qk−1 , uk , p)].

(9)

q1 ,q2 ,...,qk−1 ∈Q

Proof. (i) ⇒ (ii): We prove Eq. (8) by induction on the length of 2 . If |2 | = 0, then 2 = , and ∗ (q, 1 2 , p) = ∗ (q, 1 , p) = ∗ (q, 1 , p) = ∗ (q, 1 , p)Qe = ∗ (q, 1 , p)Q∗ (p, , p) =

∗ ∗ ∗ ∗ r∈Q ( (q, 1 , r)Q (r, , p)) = r∈Q ( (q, 1 , r)Q (r, 2 , p)). Eq. (8) holds for 2 with length 0. Assume that Eq. (8) holds for any 2 with length k  0. Let 2 ∈ ∗ with length k + 1, then we can write 2 as 2 u, where 2 ∈ ∗ with length k and u ∈ .
Then from the definition of ∗ and  ∗   by assumption, we have, ∗ (q, 1 2 , p) = ∗ (q, 1 2 u, p) = r  ∈Q ( (q, 1 2 , r )Q(r , u, p)) =



(∗ (q, 1 , r)Q∗ (r, 2 , r  ))Q(r  , u, p)) = r  ∈Q r∈Q (∗ (q, 1 , r)Q∗ (r, 2 , r  )Qk (r  , r  ∈Q (( r∈Q


  ∗ (q,  , r)Q( ∗   ∗ ∗ (  u, p)) =  ∈Q ( (r, 2 , r )Q(r , u, p))) = 1 r∈Q r r∈Q ( (q, 1 , r)Q (r, 2 , p))
= r∈Q (∗ (q, 1 , r)Q∗ (r, 2 , q)). By induction, Eq. (8) holds for any  ∈ ∗ . (ii)⇒(i): Construct an L-FFA, A = (Q, , , 0 , 1 ), as follows, Q = {q0 , q1 , q2 , q3 , q4 },  = {u1 , u2 , u3 }, (q0 , u1 , q1 ) = b, (q0 , u1 , q2 ) = c, (q1 , u2 , q3 ) = e, (q2 , u2 , q3 ) = e, (q3 , u3 , q4 ) = a, and (q, u, p) = 0 in other case. Let 0 = {q0 }, 1 = {q4 }.

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∗ Let 1 = u1 , 2 = u2 u3 , then ∗ (q0 , 1 2 , q4 ) = ∗ (q0 , u1 u2 u3 , q4 ) = z∈Q
[ (q∗0 , u1 u2 , z)Q ∗ ∗ ∗ ∗  (z, u3 , q4 )] =  (q0 , u1 u2 , q3 )Q (q3 , u3 , q4 ). Note that  (q0 , u1 u2 , q3 ) = z∈Q [ (q0 , u1 , z)Q ∗ (q ,   , q ) = (b ∨ c)Qa. ∗ (z, u2 , q3 )] = (bQe) ∨ (cQ
e) = b ∨ c, ∗ (q3 , u3 , q4 ) = a, thus 
0 1 2 4 On the other hand, D = z∈Q [∗ (q0 , 1 , z)Q(z, 2 , q4 )] = z∈Q [∗ (q0 , u1 , z)Q(z, u2 u3 , q4 )] = [∗ (q0 , u1 , q1 )Q(q1 , u2 u3 , q4 )] ∨ [∗ (q0 , u1 , q2 )Q(q2 , u2 u3 , q4 )] = [bQ(q1 , u2 u3 , q4 )] ∨ [cQ (q2 , u2 u3 , q4 )], and (q1 , u2 u3 , q4 ) = (q1 , u2 , q3 )Q(q3 , u3 , q4 ) = eQa = a, (q2 , u2 u3 , q4 ) = (q2 , u2 , q3 )Q(q3 , u3 , q4 ) = eQa = a, thus D = (bQa) ∨ (cQa). Since ∗ (q0 , 1 2 , q4 ) = D, we conclude that (b ∨ c)Qa = (bQa) ∨ (cQa). If we take A1 = (Q, , 1 , 0 , 1 ) as, where Q, , 0 , 1 are the same as those in A, 1 (q0 , u1 , q1 ) = a, 1 (q1 , u2 , q2 ) = 1 (q1 , u2 , q3 ) = e, 1 (q2 , u3 , q4 ) = b, 1 (q3
, u3 , q4 ) = c, and 1 (q, u, p) = 0 in other case. Let 1 = u1 , 2 = u2 u3 . Since ∗1 (q0 , 1 2 , q4 ) = z∈Q [∗1 (q0 , 1 , z)Q∗1 (z, 2 , q4 )], we conclude that a Q(b ∨ c) = (a Qb) ∨ (a Qc). This shows that the distributive laws of multiplication over ∨ is satisfied. Similarly, we can prove the implications of (i)⇒(iii) and (iii)⇒(i).  Without any explicit specification, in what follows, we will be referring to L as a lattice-ordered monoid. In this case the extension ∗ always satisfies Eqs. (8) and (9). Definition 3.2. Suppose that A = (Q, , , 0 , 1 ) is an L-FFA. Then the L-valued language fA ∈ F (∗ ) accepted by A or recognized by A is defined as follows:  [0 (q)Q1 (∗ (q, ))], fA () = q∈Q

where  ∈ ∗ and 1 has been extended onto F (Q). By formula (3), the explicit expression of fA reads as  [0 (q)Q∗ (q, , p)Q1 (q)]. fA () = p,q∈Q

We also denote fA () as 0 ◦  ◦ 1 , that is,  [0 (q)Q∗ (q, , p)Q1 (p)]. fA () = 0 ◦  ◦ 1 =

(10)

p,q∈Q

F (∗ )

is called an L-language on , an L-language which is accepted by an L-FFA The element f ∈ is called an L-FFA-language. For two L-FFAs A1 and A2 , we say that they are equivalent if they accept the same L-FFA-language, that is, fA1 = fA2 . From the definition of  and its extension form (5), we can also define the language fAw ∈ F (F ()∗ ) based on words accepted by A as, for any ∈ F ()∗ ,  fAw ( ) = [0 (A)Q1 (∗ (A, ))]. (11) A∈F (Q)

Using this formula, we can provide a formal model of computing with words based on the notion of L-FFA as done in [42].

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As to classical automata theory, we have the notions of deterministic finite automata and nondeterministic finite automata. The notion of an L-FFA is a generalization of the notion of nondeterministic automaton: instead of sets of initial and final states we have fuzzy sets of initial and final states; instead of a (bivalent) transition relation we have fuzzy transition relation. So we call L-FFA as in Definition 3.1 nondeterministic L-FFA or NL-FFA. The L-FFA is nondeterministic in nature: there may be nonzero truth degrees that the automaton can go to more than one state (given a state and input symbol). In the following we are going to present a simple deterministic counterpart of the notion of an NL-FFA. Definition 3.3. A deterministic L-fuzzy finite automaton (DL-FFA, for short) is a five tuple, A = (Q, , , 0 , 1 ), such that  : Q ×  → Q, 0 , 1 : Q → L. The extension of  onto ∗ is realized in the same way as we encounter in the classical case. In this case, the extension of  on F (Q) × F () → F (Q) is just the Zadeh’s
extension. Then the L-language fA ∈ F (∗ ) accepted by a DL-FFA is defined as, ∀ ∈ ∗ , fA () = q∈Q [0 (q)Q1 (∗ (q, ))]. Note that our definition differs from the usual definition of a deterministic automaton only in that the initial and the final states are fuzzy sets. This, however, makes it possible to accept words to certain truth degrees, and thus to recognize L-language. There are other forms to define the notion of deterministic fuzzy automata, for example, in [23]. We do not discuss them in this paper. Moreover, in the definition of a DL-FFA, we can require that fuzzy initial state or fuzzy final state are crisp as stated in the following theorem. Theorem 3.2. For an L-language f on , the following three conditions are equivalent: (i) f can be accepted by a D L-FFA. (ii) There exists a D L-FFA A1 = (Q1 , , , y0 , 1 ) with crisp initial state y0 such that f = fA1 . (iii) There exists a D L-FFA A2 = (Q2 , , , 0 , F ) with crisp final states F ⊆ Q2 such that f = fA2 . Proof. (i)⇒(ii) Since f can be accepted by a DL-FFA, there is a DL-FFA A = (Q, , , 0 , 1 ) such that f = fA . Let L1 = R(0 ), where R(0 ) denotes the image set of function 0 , and let L2 =
Q { T |T is a finite subset of L1 }, then L2 is finite. Let Q1 = L2 , the set of all mappings from Q to L2 . Since both Q and L2 are
finite sets, Q1 is also a finite set. The transition : Q1 ×  → Q1 is defined as, (y, u)(q) = (q  ,u)=q y(q  ), y0 = 0 ∈ Q1 , 1 : Q1 → L is defined as 1 (y) =
q∈Q [y(q)Q1 (q)]. Then A1 = (Q1 , , , y0 , 1 ) is a DL-FFA with crisp initial state y0 . First, from the definition of the extension of
fuzzy transition function as in Eq. (6), it is readily
verified that the extension of

is defined as ∗ (y, ) = ∗ (q  ,)=q y(q  ). Then fB () = 1 ( (0 , )) = q∈Q [ (0 , )(q)Q1 (q)] =


 ∗ q∈Q (q  ,)=q [0 (q )Q1 (q)] = q∈Q [0 (q)Q1 ( (q, ))] = fA () = f (). (ii)⇒(iii) The proof is similar to that of “(i)⇒(ii)”. Let L1 = R(1 ), the image set of function
Q1 1 , and L2 = { T |T is a finite subset of L1 }, then L2 is finite. Let
Q2 = L2 , then Q2 is a finite set. The transition : Q2 ×  → Q2 is defined as, (z, u)(y) = (y  ,u)=y z(y ), 0 : Q2 → L is defined as 0 (z) = z(y0 ), and F = {1 } ⊆ Q2 . Then A2 = (Q2 , , , 0 , F ) is a DL-FFA with crisp finite states F ⊆ Q1 . We show fA2 = f and thus complete the proof. First, from the definition of the extension of fuzzy transition function as in Eq. (6), it is easily verified that the extension of is defined

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77




as ∗ (z, ) = ∗ (y  ,)=y z(y  ). Then fA2 () = { (z)|z ∈ Q2 and ∗ (z, ) = 1 } = {z(y0 )|z ∈ Q2
  and ∗ (z, ) = 1 }. Noting ∗ (z,

∗ (y  , ) = y} = 1 (y) for
) = 1 means that∗ {z(y )|y ∈ Q1 and any y ∈ Q1 . Hence fA2 () = {z(y0 )|z ∈ Q2 and (z, ) = 1 } = 1 ( ∗ (y0 , )) = fA1 () = f (). (iii)⇒(i) Obvious.  But, unlike the nonfuzzy case, we shall show that NL-FFA and DL-FFA are not equally powerful in the following propositions. Lemma 3.1. Suppose that A is a D L-FFA, then the range of images set of fA is a finite subset of L, that is, the set R(fA ) = {fA ()| ∈ ∗ } is finite. Proof. By Theorem 3.2, we can assume that A = (Q, , , q0 , 1 ) is a DL-FFA with crisp initial state. Note that fA () = 1 (∗ (q0 , )), and ∗ (q0 , ) ∈ Q. Then {fA ()| ∈ ∗ } ⊆ {1 (q)|q ∈ Q}. Since Q is a finite set, {1 (q)|q ∈ Q} is also a finite subset of L. Therefore, R(f ) = {fA ()| ∈ ∗ } is a finite subset of L.  Theorem 3.3. If an L-language can be accepted by a D L-FFA, then it can also be accepted by an N L-FFA. The converse does not hold. Proof. Suppose that an L-language f is accepted by a DL-FFA, A = (Q, , , q0 , 1 ), that is f = fA . Construct an NL-FFA B = (Q, , , {e/q0 }, 1 ), where = eQ, that is,  e, (q, u) = p,

(q, u, p) = 0, (q, u) = p and {e/q0 } denotes a fuzzy point which takes value e at point {q0 }, and 0 in other case. Then it could be easily verified that f = fB . Therefore, f can be accepted by an NL-FFA. The following Example 3.1 shows us an L-language which can be accepted by an NL-FFA but no DL-FFA. Example 3.1. Let L = [0, 1], Q is the usual product operation of numbers. Let A = (Q, , , 0 , 1 ) be defined as, Q = {q1 , q2 },  = {u}, 0 = 0.8/q1 + 0.2/q2 , 1 = 0.1/q1 + 0.7/q2 , (q1 , u) = 0.5/q2 , (q2 , u) = 0.5/q1 , where for Q1 ⊆ Q, q∈Q1 a/q denotes an L-fuzzy set of Q taking values a at the point q ∈ Q1 and 0 at the remainder elements of Q. Then for any nonnegative integer n, we have (qi , u2n , qi ) = 1/4n , (qi , u2n+1 , qi ) = 0, (i = 1, 2), (qi , u2n , qj ) = 0, (qi , u2n+1 , qj ) = 1/22n+1 , (i  = j ).

Therefore, n

fA (u ) = 0 ◦ un ◦ 1 =



0.14/4m , 0.28/4m ,

n = 2m, n = 2m + 1.

This implies that fA has infinite values, that is, R(fA ) is an infinite subset of L. By Lemma 3.1, fA cannot be accepted by a DL-FFA.

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Lemma 3.2. Suppose that (L, Q, ∨) is an l-monoid, L is a finite subset of L, then the subalgebra of    (L, Q, ∨) generated by L is a finite subset iff the subsetL = {li1 Q · · · Qlik |lij ∈ L , 1
k
n, n  1} is finite. 

Proof. Since the distributive laws of multiplication over ∨, L is defined as follows: n

   L = mi |mi ∈ L , i = 1, . . . , n, n  0 . i=1

From [14], we know that the free join-semilattice generated by a finite set is also finite. This implies   that L is finite iff L is finite.  Theorem 3.4. For any N L-FFA, A, there is an equivalent D L-FFA iff the l-monoid L satisfies the   following conditions: for any finite subset L of L, the subalgebra of (L, Q, ∨) generated by L is finite. Proof. “ If” part. Suppose that L satisfies the conditions in the theorem. Let A = (Q, , , 0 , 1 ) be any NL-FFA, and let L2 = L1 , the subalgebra of L generated by L1 , where L1 = {(q, u, p)|q, p ∈ Q, u ∈ } ∪ {0 (q)|q ∈ Q}.

Since L1 is a finite subset of L, L2 is also finite. Constructing a DL-FFA, B
= (Z, , , z0 , 1 ) as, Z = L2 Q = {g|Q → L2 | g is a function}, then Z is a finite set; (z, u)(q) = p∈Q [z(p)Q(p, u, q)],
z0 = 0 , 1 (z) = z ◦ 1 = q∈Q [z(q)Q1 (q)]. We prove fA = fB .
First, ∀ ∈ ∗ , ∀z ∈ Z, we prove that ∗ (z, ) = z ◦  , that is ∗ (z, )(q) = p∈Q [z(p)Q∗ (p, , q)], by induction on the length of . Let n = ||, if n = 0, then ∗ (z, ) = z = z◦ ∗ is obvious. Let n−1 = |1 | and ∗ (z, 1 ) = z ◦ 1 holds. Let  = 1 u, where u ∈ , then ∗ (z, ) = ∗ (z, 1 u) = ( ∗ (z, 1 ), u) =

∗ (z, 1 ) ◦ u = z ◦ 1 ◦ u = z ◦ (1 ◦ u ) = z ◦ 1 u = z ◦  . Therefore, fB () = 1 ( ∗ (z0 , )) = 1 (z0 ◦  ) = z0 ◦  ◦ 1 = 0 ◦  ◦ 1 = fA (). “Only if” part. Suppose that there is a finite subset L1 of L such that the subalgebra of L generated by L1 is infinite. We construct an NL-FFA, A, such that the L-language accepted by A, fA , has infinite values in L. Thus there is no DL-FFA to accept fA . Let L1 = {l1 , . . . , lm }, and Q = {q1 , q2 },  = {u1 , . . . , um }, (q1 , ui , q2 ) = li = (q2 , ui , q1 ), (q1 , ui , q1 ) = 0 = (q2 , ui , q2 ), i = 1, . . . , m, 0 = 1 = e, that is, ∀q ∈ Q, 0 (q) = e = 1 (q). Then, for  = ui1 · · · uik ∈ ∗ , i = 1, 2, we have  l Q · · · Qlik , k is an odd integer and d = 2, or k is an even integer and d = 1 . ∗ (qi , , qd ) = i1 0 otherwise By Lemma 3.2, the set {∗ (qi , , qj )|i, j = 1, 2,  ∈ ∗ } is an infinite subset of L. Notice that, for  = ui1 · · · uik ∈ ∗ , we have  ∗ (q, , p) = li1 Q · · · Qlik . fA () = 0 ◦  ◦ 1 = q,p∈Q

Therefore, {fA ()| ∈ ∗ } is an infinite subset of L.



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Example 3.2. Let L = [0, 1], ∨ is the usual maximum operation. If Q is defined as a Qb = a ∧ b or a Qb = 0∨(a +b −1), then the correspondence lattice-ordered monoids satisfy the conditions in Theorem 3.4. If Q is the usual product operation, then the corresponding lattice-ordered monoid does not satisfy the condition in Theorem 3.4. Corollary 3.1. If (L, ∨, ∧) is a distributive lattice, then for each N L-FFA, there exists an equivalent D L-FFA, and for each D L-FFA, there exists an equivalent N L-FFA. In particular, any (nondeterministic) fuzzy finite automaton is equivalent to deterministic fuzzy finite automaton [4]. Proof. If (L, ∨, ∧) is a distributive lattice, then for any finite subset L of L, the subalgebra of (L, ∧, ∨) generated by L is also finite. In fact, the free distributive lattice generated by a finite set is finite, see Refs. [3,7,14]. From Theorem 3.4, we conclude the corollary. 

4. Regular operations on the family of languages of L-FFAs Recall we call an L-language accepted by an L-FFA the L-FFA language. Then a family of L-FFAlanguages on  forms a subset of F (∗ ). Now, we study some regular operations on this family in this section. Definition 4.1. Let f, f1 , f2 ∈ F (∗ ) be L-languages. (1) The union of f1 and f2 , denoted f1 ∪ f2 , is defined as, (f1 ∪ f2 )() = f1 () ∨ f2 () for any  ∈ ∗ . (2) The intersection of f1 and f2 , denoted f1 ∩ f2 , is defined as, (f1 ∩ f2 )() = f1 () ∧ f2 () for any  ∈ ∗ .
(3) The concatenation of f1 and f2 , denoted f1 f2 , is defined as, (f1 f2 )() = 1 2 = [f1 ()Qf2 ()] for any  ∈ ∗ . Then the concatenation operation satisfies the associative laws. (4) The Kleene closure of f, denoted f ∗ , is defined as, f ∗ = I ∪ f ∪ ff ∪ · · · ∪ f n ∪ · · ·, where I denotes the language of empty word, that is to say,  I () =

e, 0,

 = ,  = 

and f n is inductively defined as, f 0 = I , and f n = f n−1 f for n  1. (5) The reversal of f, denoted f −1 , is defined as f −1 () = f (−1 ), ∀ ∈ ∗ , where for  = u1 u2 · · · uk , −1 = uk · · · u2 u1 . (6) For a ∈ L, the scalar product of f and a, denoted af and fa, is defined as, (af )() = a Qf () and (f a)() = f ()Qa for any  ∈ U ∗ . Remark 4.1. From the definition of Kleene closure of f, it is necessary to require that L is closed under countable supremum. To emphasize this fact, we provide the following example.

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Example 4.1. Suppose that L1 = [0, ∞], as given in Example 2.1(5). Considering the l-monoid L1 (2), as given in Example 2.1(2). Let          ∞  1 1 1 0 ∞ ∞ 1 0 0 0 n , , , L2 = . 1 0 2 ∞ ∞ 0 1 0 0 ∞ n=1 n Furthermore, let L denote the subalgebra of L1 (2) generated by L2 , that is, L = L2 . Then for any positive integer n and m,       1 1 1 0 1 0 m ∈ L. ∈ < L and 1 0 2n 0 2n ∞ m Note that  1 0 so



0 ∞ 1 0

 ∈ / L,

  0  n = 1, 2, . . . 2n 

has not the least upper bound in L. If we let f : ∗ → L be defined as   1 0 , ∀ ∈ ∗ , f () = 0 2 then f n () =



1 0

0 2n



for any integer n. Since n

I () ∨ f () ∨ · · · ∨ f () ∨ · · · =

  1 0

  0  n = 0, 1, 2, . . . 2n 

has no the least upper bound in L, the Kleene closure of f is not well defined in this case. However, if the identity is the same as the largest element, then the Kleene closure is well defined as shown in the following proposition. Proposition 4.1. If e = 1 in an l-monoid L, then for any L-language f ∈ F (∗ ), the Kleene closure of f is well defined. Proof. From Proposition 2.1, we have a Qb
a ∧ b. Thus, if  = u1 u2 · · · uk , then it is easily verified that f n
f ∨ f 2 ∨ · · · ∨ f k+1 whenever n > k. Therefore, ∀ ∈
∗ , if || = k, then f ∗ () = I () ∨ k+1 k f () ∨ · · · ∨ f (). Note that for any positive integer k, f () = 1 ···k = [f (1 )Q · · · Qf (k )] is well defined, therefore f ∗ is also well defined.  Due to the above facts, we assume that L is a countable l-monoid if e  = 1. As to the Kleene closure of f, we note the following properties.

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81

Proposition 4.2. For an L-language f ∈ F (∗ ), the following conditions are equivalent: (i) f = f ∗ ; (ii) f ()  e and ∀1 , 2 ∈ ∗ , f (1 2 )  f (1 )Qf (2 ). Proof. (i)⇒(ii)
Since I
f ∗ = f , f ()  e. Furthermore, ∀1 , 2 ∈ ∗ , we have f (1 2 ) = f ∗ (1 2 )  (ff )(1 2 ) =   =1 2 [f ( )Qf ( )]  f (1 )Qf (2 ).
∗ , we have (ff )() = (ii)⇒(i) Since f ( )  e, I
f . Furthermore, ∀  ∈   1 2 = [f (1 )Qf (2 )[

n ∗ 1 2 = f (1 2 ) = f (), this shows that ff
f . Then ∀n  1, f
f , and thus f
f . The reverse ∗ ∗ inequality f  f is obvious, hence f = f .  Corollary 4.1. If f : ∗ → L is a monoid homomorphism, then f ∗ = f . Theorem 4.1. (i) The family of N L-FFA languages is closed under the operations of union, concatenation and scalar product. (ii) For a countable l-monoid L or an l-monoid with the largest element as the identity, the family of N L-FFA languages is closed under the Kleene closure. Proof. (i) Suppose that A1 = (Q1 , , 1 , 10 , 11 ) and A2 = (Q2 , , 2 , 20 , 21 ) are two NL-FFAs, the L-languages they accepted are fA1 and fA2 , respectively. Since we may rename the states of an NL-FFA at will without changing the language accepted we may assume that the two sets Q1 and Q2 are disjoint. (1) We first consider the union operation of NL-FFA-languages. Construct an NL-FFA, A = (Q, , , 0 , 1 ) as, Q = Q1 ∪ Q2 , 0 : Q → L and 1 : Q → L are defined, respectively, as 0 (q) =

10 (q), 20 (q),

q ∈ Q1 , q ∈ Q2 ;

1 (q) =

11 (q),

q ∈ Q1 ,

21 (q),

q ∈ Q2 .

 : Q ×  → F (Q) is defined as follows:

  1 (q, u, p), (q, u, p) = 2 (q, u, p),  0

q, p ∈ Q1 , q, p ∈ Q2 , otherwise.

Then the extension of  is,  ∗  1 (q, , p), ∗  (q, , p) = ∗2 (q, , p),  0

q, p ∈ Q1 , q, p ∈ Q2 , otherwise.



[0 (q)Q∗ (q, , p)Q1 (p)] = Therefore, ∀ ∈ ∗ , fA () = 0 ◦  ◦ 1 = q,p∈Q q,p∈Q1
[10 (q)Q∗1 (q, , p)Q11 (p)] ∨ q,p∈Q2 [20 (q)Q∗2 (q, , p)Q21 (p)] = fA1 () ∨ fA2 (). Hence, fA is the union of fA1 and fA2 .

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(2) Considering the concatenation operation. Construct an NL-FFA, A = (Q, , , 0 , 1 ) as, Q = Q1 ∪ Q2 , 0 : Q → L and 1 : Q → L are defined, respectively, as  1  0, q ∈ Q1 , 0 (q), q ∈ Q1 , 0 (q) =  (q) = 0, q ∈ Q2 ; 1 21 (q), q ∈ Q2 .  : Q ×  → F (Q) is defined as follows:

  (q, u, p), q, p ∈ Q1 ,    1 2 (q, u, p), q, p ∈ Q2 , (q, u, p) = q ∈ Q1 , p ∈ Q2 ,  a,   0, q ∈ Q2 , p ∈ Q 1 ,

where a = q1 ∈Q1 [1 (q, u, q1 )Q11 (q1 )Q20 (p)] ∨ q2 ∈Q2 [11 (q)Q20 (q2 )Q1 (q2 , u, p)]. Then the extension of  is ( = ),  e, q = p, ∗  (q, , p) = 0, q = p;  ∗  (q, , p), q, p ∈ Q1 ,    1∗ 2 (q, , p), q, p ∈ Q2 , ∗  (q, , p) = b, q ∈ Q1 , p ∈ Q2 ,    0, q ∈ Q2 , p ∈ Q 1 ,

where b = 1 2 = q1 ∈Q1 ,q2 ∈Q2 [1 (q, 1 , q1 )Q11 (q1 )Q20 (q2 )Q1 (q2 , 2 , p)].


Therefore, ∀ ∈ ∗ , fA () = 0 ◦  ◦ 1 = q,p∈Q [0 (q)Q∗ (q, , p)Q1 (p)] = 1 2 = [ q,p∈Q1

[10 (q)Q∗1 (q, 1 , p)Q11 (p)] ∨ q,p∈Q2 [20 (q)Q∗2 (q, 2 , p)Q21 (p)]] = 1 2 = [fA1 (1 )QfA2 (2 )], that is, fA = fA1 fA2 . (3) Considering the scalar product of a ∈ L and f = fA , where A = (Q, , , 0 , 1 ) is an NLFFAA. Let a A = (Q, , , a 0 , 1 ), and Aa = (Q, , , 0 , 1 a), where a 0 : Q → L is defined as a 0 (q) = a Q0 (q) and 1 a : Q → L is defined as 1 a(q) = 1 (q)Qa for any q ∈ X. Then a A and Aa are two NL-FFAs and it is readily verified that fa A = afA = af and fAa = fA a = f a. Hence LA is closed under the operation of the scalar product. (ii) Considering the Kleene closure operation. (a)We first assume that L is a countable l-monoid. Suppose that A = (Q, , , 0 , 1 ) is an NL-FFA. Construct an NL-FFA, B = (Z, , , 0 , 1 ) as, Z = Q ∪ {S}, 0 : Z → L and 1 : Z → L are defined as follows:   0 (z), z ∈ Q, 1 (z), z ∈ Q, 0 (z) =  (z) = e, z = S; 1 e ∨ c, z = S,
where c = d ∨ d 2 ∨ · · · ∨ d n ∨ · · ·, and d = q∈Q [0 (q)Q1 (q)], d 2 = d Qd, and for n  1, d n+1 = d n Qd.

: Z ×  → F (Z)is defined as  z1 , z2 ∈ Q, a(z1 , u, z2 ), 
[  (q) Q a(q, u, z )], z

(z1 , u, z2 ) = 0 2 1 = S, z2 ∈ Q,  q∈Q 0, z2 = S,

Y. Li, W. Pedrycz / Fuzzy Sets and Systems 156 (2005) 68 – 92

where a : Q × + × Q → L is defined as,    a(z1 , , z2 ) = ∗

83

[∗ (z1 , 1 , q1 )Q1 (q1 )Q0 (q2 )

k  0 1 ···k = q1 ,...,q2(k−1) ∈Q

Q · · · Q (q2(k−1) , k , z2 )],

where + = ∗ − {}. Then the extension of is defined as follows (  = ):  e, z1 = z2 ,

∗ (z1 , , z2 ) = 0, z1 = z2 ;  z1 , z2 ∈ Q, a(z1 , , z2 ), 
[  (q) Q a(q,  , z )], z1 = S, z2 ∈ Q,

∗ (z1 , , z2 ) = 0 2  q∈Q 0, z2 = S. Therefore, ∀ ∈ + ,



fB () = 0 ◦  ◦ 1 = [ z1 ,z2 ∈Z 0 (z1 )Q ∗ (z1 , , z2 )Q1 (z2 )] = k  0 1 ···k = q,q1 ,q2 ,...,qk−1 ,p∈Q [∗ (q, 1 , q1 )Q1 (q1 )Q0 (q2 ) · · · Q∗ (q2(k−1) , k , p)] = fA∗ ().
Note that fB () = 0 ◦ 1 = z∈Z [0 (z)Q1 (z)] = e ∨ d ∨ d 2 ∨ · · · ∨ d n ∨ · · · = fA∗ (). It follows that fB = fA∗ . (b) In this case, we assume that L is an l-monoid with e = 1. Let A = (Q, , , 0 , 1 ) be an NL-FFA. Construct an NL-FFA, B = (Z, , , 0 , 1 ) as, Z and are the same as in the case (a), and   0 (z), z ∈ Q, 1 (z), z ∈ Q, 0 (z) =  (z) = 1, z = S; 1 1, z = S.

Then, for u ∈ , a(z1 , u, z2 ) = (z1 , u, z2 ) ∨ q∈Q [1 (z1 )Q0 (q)Q(q, u, z2 )] ∨ q∈Q [(z1 , u, q)Q
1 (q)Q0 (z2 )] ∨ q,p∈Q [1 (z1 )Q0 (q)Q(q, u, p)Q1 (p)Q0 (z2 )]. Note that e = 1, the construction in the case (a) is well defined, the remainder of the proof is similar to that in the case (a).  Suppose that A1 = (Q1 , , 1 , 10 , 11 ), A2 = (Q2 , , 2 , 20 , 21 ) are two NL-FFAs, the languages they accepted are fA1 , fA2 , respectively. We construct an NL-FFA, A = (Q, , , 0 , 1 ) as, Q = Q1 × Q2 ; ((q1 , q2 ), u, (y1 , y2 )) = 1 (q1 , u, y1 ) ∧ 2 (q2 , u, y2 );0 (q1 , q2 ) = 10 (q1 ) ∧ 20 (q2 ), 1 (q1 , q2 ) = 11 (q1 ) ∧ 21 (q2 ). The language accepted by A is assumed as fA . A is called the tensor product of ∗ A1 and A2 . Then the extension of  is defined as, ∗ (x,
, x) = e,
and for y = x,  (x, , y) = 0; + ∗ while for any u = u1 u2 · · · uk ∈  ,  (x, , y) = i=1,...,k−1 qi1 ∈Q1 ,qi2 ∈Q2 [(1 (q10 , u1 , q11 ) ∧ 2 (q20 , u1 , q22 ))Q · · · Q(1 (q1,k−1 ,
uk , p1 ) ∧ 2 (q2,k−1 , uk , p2 ))], where x = (q10 , q20 ), y = (p1 , p2 ). Thus, fA () = 0 ◦  ◦ 1 = q1 ,p1 ∈Q1 ; q2 ,p2 ∈Q2 [10 (q1 )Q20 (q2 )Q∗ ((q1 , q2 ), , (p1 , p2 ))Q11 (p1 )Q 21 (p2 )]. In general, the above fA is not the intersection of fA1 and fA2 . We arrive at the following characterization. Theorem 4.2. Given any two N L-FFAs, A1 and A2 , let A denote the tensor product of A1 and A2 . Then fA = fA1 ∩ fA2 iff the operation Q is just the infimum operation ∧, that is, (L, ∧, ∨) is a distributive lattice.

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Proof. “If” part: Since ∧ satisfies commutative laws, fA () = 0 ◦  ◦ 1 = q1 ,p1 ∈Q1 ,q2 ,p2 ∈Q2 [10 (q1 )∧
20 (q2 ) ∧ ∗1 (q1 , , p1 ) ∧ ∗2 (q2 , , p2 ) ∧ 11 (p1 ) ∧ 21 (p2 )] = q1 ,p1 ∈Q1 [10 (q1 ) ∧ ∗1 (q1 , , p1 ) ∧ 11 (p1 )] ∧
2 ∗ 2 q2 ,p2 ∈Q2 [0 (q2 ) ∧ 2 (q2 , , p2 ) ∧ 1 (p2 )] = fA1 () ∧ fA2 (). Conversely, if for any two NL-FFAs, A1 , A2 , and their tensor product A, we always have fA = fA1 ∩ fA2 . We want to prove that Q is equal to the infimum operation ∧. We first prove that e = 1, where e is the identity of L. Take A1 = ({x, y}, {u}, 1 , 10 , 11 ), A2 = ({z, t}, {u}, 2 , 20 , 21 ), where 1 (x, u, x) = e, 1 (x, u, y) = 1, and 0 in other cases. 10 = 11 = e; 2 (z, u, t) = 1, 2 (t, u, t) = e, and 0 in other points, 20 = 21 = e. Then fA1 (uu) = 1, fA2 (uu) = 1. Note the tensor product of A1 and A2 is defined as, A = ({(x, z), (x, t), (y, z), (y, t)}, {u}, , 0 , 1 ), where ((x, z), u, (x, t)) = e, ((x, t), u, (x, t)) = e, ((x, t), u, (y, t)) = e, ((x, z), u, (y, t)) = 1, 0 in other cases and 0 = 1 = e. Then fA (uu) = e. Since fA = fA1 ∩ fA2 , fA (uu) = fA1 (uu) ∧ fA2 (uu). Therefore, e = 1. Next, we prove that the multiplication on L satisfies the idempotent laws, that is, ∀a ∈ L, a Qa = a. Take B1 = ({x, y}, {u}, 1 , 10 , 11 ), B2 = ({z, t}, {u}, 2 , 20 , 21 ), where 10 , 11 , 20 , 21 are the same as those in A1 and A2 , and 1 (x, u, x) = a, 1 (x, u, y) = 1, and 0 in other points; 2 (z, u, t) = 1, 2 (t, u, t) = a, and 0 in other cases. Then fB1 (uu) = a, fB2 (uu) = a. The tensor product of B1 and B2 is defined as, B = ({(x, z), (x, t), (y, z), (y, t)}, {u}, , 0 , 1 ), where ((x, z), u, (x, t)) = a, ((x, t), u, (x, t)) = a, ((x, t), u, (y, t)) = a, ((x, z), u, (y, t)) = 1, and 0 in other points; 0 = 1 = 1. Then fB (uu) = a Qa. Since fB = fB1 ∩ fB2 , fB (uu) = fB1 (uu) ∧ fB2 (uu). Thus a Qa = a. Then we have shown that e = 1 and ∀a ∈ L, a Qa = a. Therefore, ∀a, b ∈ L, a ∧ b = (a ∧ b)Q(a ∧ b)
a Qb. Form Proposition 2.1, we have a ∧ b  a Qb. Hence, a Qb = a ∧ b.  Suppose that A = (Q, , , 0 , 1 ) is an NL-FFA. Construct an NL-FFA B = (Q, , , 1 , 0 ), ∗ ∗ −1 where : Q ×  → F (Q)
is defined as ∗(q, u,−1p) = (p, u, q). Then (q, , p) =  (p,  , q) and fB () = 1 ◦  ◦ 0 = p,q∈Q [1 (p)Q (p,  , q)Q0 (q)]. In general, the above fB is not fA−1 , but we have the following characterization. Theorem 4.3. For any N L-FFA, A, and the above N L-FFA B constructing from A, fB = fA−1 iff Q satisfies the commutative laws. In particular, if (L, ∧, ∨) is a distributive lattice, then the family of N L-FFA-languages is closed under reversal operation.
Proof. “If” part. Since the multiplication is commutative, we have fB () = 1 ◦  ◦ 0 = q,p∈Q [1 (p)Q


∗ (p, , q)Q0 (q)] = q,p∈Q [0 (q)Q∗ (q, −1 , p)Q1 (p)] = fA (−1 ). Thus, fB = fA−1 . “Only if” part. If the multiplication does not satisfy the commutative laws, that is, there are a, b ∈ L such that ab  = ba. We construct an NL-FFA, A, such that fB  = fA−1 in the following. Take Q,  as two nonempty sets, and  = e, 0 = b, 1 = a. For any u ∈ , we have fA−1 (u) = fA (u) = 0 ◦ u ◦ 1 = ba, fB (u) = 1 ◦ u ◦ 0 = ab. Since ab = ba, we have fB (u)  = fA−1 (u). Therefore, fB = fA−1 .  Remark 4.2. If we denote the set of DL-FFA languages on , the L-languages accepted by DL-FFAs, by D L −F F A, then the family D L −F FA is closed under union, concatenation, scalar product, intersection, and reversal operations. The proofs are similar to those of N L−F FA, except in the proof of concatenation,

Y. Li, W. Pedrycz / Fuzzy Sets and Systems 156 (2005) 68 – 92

85

we should take A1 with crisp final states and A2 with crisp initial state, where N L − F FA denotes the family of NL-FFA languages. However, the following example illustrates that D L − F FA is not closed under Kleene closure. Example 4.2. Let L = ([0, 1], Q, ∨),  = {0, 1}, and an L-language f : ∗ → L is defined as, 1 ,  = 0m , m > 0,  2 f () = 13 ,  = 1m , m > 0,   0 otherwise. Then f ∗ () = 1, and for any m > 0, f ∗ (0m ) = 21 , f ∗ (1m ) = 13 , f ∗ ((01)m ) = f ∗ ((10)m ) = 61m , and thus R(f ∗ ) is infinite. f ∗ is not a DL-FFA language by Lemma 3.1. But f is a DL-FFA language which 1/3 can be accepted by a DL-FFA A = (Q, , , q0 , 1 ), where Q = {q0 , q1 , q2 , q3 }, 1 = 1/2 q1 + q2 , and  is defined by (q0 , 0) = q1 , (q0 , 1) = q2 , (q1 , 0) = q1 , (q1 , 1) = q3 , (q2 , 0) = q3 , (q2 , 1) = q3 , (q3 , 0) = (q3 , 1) = q3 . Therefore, DL-FFA languages are not closed under Kleene closure. Remark 4.3. For an l-monoid L, if there exists a negation c on L, that is, there is a mapping c : L → L satisfying a
b ⇒ c(a)  c(b) and cc(a) = a for any a, b ∈ L, then for any L-language f on , we can define the complement of f, denoted f¯, as f¯() = c(f ()). Then it is obvious that D L − F FA is closed under complement. But on the contrary, N L − F FA is not generally closed under complement, we can use Example 4.2 to illustrate this fact. We take linear negation on unit interval, that is, for any a ∈ [0, 1], c(a) = 1 − a. Take f ∗ as the NL-FFA in Example 4.2 on  = {0, 1}, let g denote the complement of f ∗ . We show that g is not an N L − F FA in the following. Take  = 01, then g(m ) = 1 − f ∗ (m ) = 1 − 61m . Obviously, if m > n, then g(m ) > g(n ). We show that if g is an N L − F FA, then there exits m > n such that g(m )
g(n ), and thus a contradiction appears. Assume that g is an N L − F FA, then there is an NL-FFA A = (Q, , , 0 , 1 ) such that fA = g. Take any integer m > |Q| + 1, where |Q| denotes the counting number of the set Q, then g(m ) = fA (m ) =
∗ m q,p∈Q 0 (q) (q,  , p)1 (q). Since Q is finite and [0, 1] is in linear order, there exist q, p ∈ Q such that g(m ) = 0 (q)∗ (q, m , p)1 (p). From the definition of the extension ∗ , there exist q2 , . . . , qm ∈ Q such that ∗ (q, m , p) = (q, , q2 ) · · · (qm , , p). Noting m > |Q|+1, there exist 2
i < j
m such that qi = qj . Then ∗ (q, m , p)
(q, , q1 ) · · · (qi−1 , , qi )(qj , , qj +1 ) · · · (qm , , p)
∗ (q, n , p), where n = m−j +1 < m. For these two integers m and n, we have m > n and g(m ) = 0 (q)∗ (q, m , p) 1 (p)
0 (q)∗ (q, n , p)1 (p)
fA (n ) = g(n ). 5. Fuzzy regular expressions with membership values in lattice-ordered monoids and their languages Definition 5.1. Let  be a finite nonempty set. The family LR of L-fuzzy regular expressions or regular L-expressions over  is defined inductively as follows: (i) ∅ ∈ LR; (ii)  ∈ LR;

86

(iii) (iv) (v) (vi) (vii) (viii)

Y. Li, W. Pedrycz / Fuzzy Sets and Systems 156 (2005) 68 – 92

u ∈ LR, for all u ∈ ; (a
) ∈ LR and (
a) ∈ LR, for all a ∈ L and
∈ LR; (
1 +
2 ) ∈ LR, for all
1 ,
2 ∈ LR; (
1 ◦
2 ) ∈ LR, for all
1 ,
2 ∈ LR; (
∗ ) ∈ LR, for all
∈ LR; There are no other regular L-expressions other than those given in steps (i)–(vii).

In writing regular L-expressions we can omit many parentheses if we assume that ∗ has higher precedence than concatenation, scalar or +, and that concatenation has higher precedence than scalar or +, and that scalar has higher precedence than +. For example, (((0.5a)((0.2b)∗ )) + (0.8a)) may be written 0.5a(0.2b)∗ + 0.8a. We may also abbreviate the expression rr ∗ by r + and rrr ∗ by r ++ and so on. Definition 5.2. For every
∈ LR, ||
|| is the L-language over  defined recursively as follows: (1) ||
|| = f
, for all
∈  ∪ {∅, }, where f
: ∗ → L is just the characteristic function of
, that is, f
() = e if  =
and 0 otherwise. (2) ||a
|| = a||
|| and ||
a|| = ||
||a, for all a ∈ L and
∈ LR. (3) ||
1 +
2 || = ||
1 || ∨ ||
2 ||, for all
1 ,
2 ∈ LR. (4) ||
1 ◦
2 || = ||
1 || ||
2 ||, for all
1 ,
2 ∈ LR. (5) ||
∗ || = ||
||∗ for all
∈ LR. In the following, we call ||
|| for the element
∈ LR the regular L-languages over . Let ||LR|| = {||
|||
∈ LR}. Theorem 5.1. If
∈ LR, then ||
|| is an N L-FFA-language. Proof. Follows from the Theorem 4.1 and the fact that for each
∈  ∪ {∅, }, f
is an NL-FFA-language over .  Theorem 5.2. If f is an N L-FFA-language over , then f = ||
|| for some
∈ LR. Proof. Let f = fA for an NL-FFA A = (Q, , ,
0 , 1 ). For q, p ∈ Q and S ⊆ Q, define f (q, p, S) : ∗ → L as, for  = u1 u2 · · · ul , f (q, p, S)() = {(q, u1 , q2 )Q(q2 , u2 , q3
)Q · · · Q(ql , ul , p)|qi ∈ S for i = 2, . . . , l}. Obviously, f (q, p, Q)() = ∗ (q, , p) and thus fA = q,p∈Q (0 (q)f (q, p, Q) 1 (p)). We first prove that f (q, p, S) = ||(q, p, S)|| for any subset S of Q, with regular expression (q, p, S). Then the regular expression for f, denoted (f ), is (f ) = +q,p∈Q ((0 (q)(q, p, Q))1 (p)).

The proof is by induction on the number of |S|, and involves the counting of the elements of the set S. Inductive base: |S| = 0, that is, S = ∅, then   (q, u, p), f (q, p, ∅)() = e,  0

 = u ∈ ,  =  and q = p,

otherwise.

Y. Li, W. Pedrycz / Fuzzy Sets and Systems 156 (2005) 68 – 92

87

If
we define q,p = e if q = p and q,p = 0 if q  = p, then it is easily verified that f (q, p, ∅) = u∈ ((q, u, p)fu )∨ q,p f = ||+u∈ ((q, u, p)u)+ q,p ||. That is, (q, p, ∅) = +u∈ ((q, u, p)u)+ q,p . Inductive step: Assume that for all S ⊆ Q with at most m elements and for all q, p ∈ S, the fuzzy set f (q, p, S) has regular expression (q, p, S), i.e., f (q, p, S) = ||(q, p, S)||. Let Y be a subset of Q possessing exactly m + 1 elements. We set Yz = Y − {z}. Now define the fuzzy set g : ∗ → L as follows:

  ∗ f (q, p, Yz ) ∨ (f (q, z, Yz )(f (z, z, Yz )) f (z, p, Yz )) . g= z∈Y

z∈Y

By the inductive hypothesis, the fuzzy set g is regular. In order to complete the proof of the theorem, we need to show that g = f (q, p, Y ). By the definition of g, it is
not hard to see that g
f (q, p, Y ). Conversely, suppose  = u1 u2 · · · ul ∈ ∗ , since f (q, p, Y )() =  {(q, u1 , q2 )Q(q2 , u2 , q3 )Q · · · Q(ql ,
ul , p)|qi ∈ Y for i = 2, . . . , l} =
{(q, u1 , q2 )Q(q2 , u2 , q3 )Q · · ·
Q(ql , ul , p)|{q2 , . . . ,
ql }
Y } ∨ {(q, u1 , q2 )Q(q2 , u2 , q3 )Q · · · Q (ql , ul , p)|{q2 , . . . , ql } = Y } =
z∈Y f (q, p, Yz )() ∨ {(q, u1 , q2 )Q(q2 , u2 , q3 )Q · · · Q(ql , ul , p)| {q , . . . , q } = Y }. Obviously, 2 l z∈Y f (q, p, Yz )
g. To finish the
proof, we further show that
{(q, u1 , q2 )Q(q2 , u2 , q3 )Q · · · Q(ql , ul , p)|{q2 , . . . , ql } = Y }
z∈Y (f (q, z, Yz )(f (z, z, Yz ))∗ f (z, p, Yz ))() in the following. Since {q2 , . . . , ql } = Y , there are two cases for the sequence q2 , q3 , . . . , ql : (1) qi = qj for any i = j , and (2) there exists i = j such that qi = qj . If qi  = qj for any i  = j , then (q, u1 , q2 )
f (q, q2 , Yq2 )(u1 ) and (q2 , u2 , q3 )Q · · · Q(ql , ul , p)
f (q2 , p, Yq2 )(u2 · · · ul ) and thus (q, u1 , q2 )Q · · · Q(ql , ul , p)
f (q, q2 , Yq2 )f (q2 , p, Yq2 )() = f (q, q2 , Yq2 )f (q2 , p, Yq2 )()

∗ z∈Y (f (q, z, Yz )(f (z, z, Yz )) f (z, p, Yz ))(). If qi = qj for some i  = j , let z = qi = qj ∈ Y . Let i = min{k|qk = z} and j = max{k|qk = z}, then for any k < i or k > j , qk  = z and qi = qj = z. In this case, (q, u1 , q2 )Q · · · Q(ql , ul , p) = ((q, u1 , q2 )Q · · · Q(qi−1 , ui−1 , qi ))Q((qi , ui , qi+1 )Q · · · Q (qj −1 , uj −1 , qj ))Q((qj , uj , qj +1 )Q · · · Q(ql , ul , p)) = a QbQc, where a = (q, u1 , q2 )Q · · · Q(qi−1 , ui−1 , qi ), b = (qi , ui , qi+1 Q · · · Q(qj −1 , uj −1 , qj ), c = (qj , uj , qj +1 )Q · · · Q(ql , ul , p). Then a
f (q, z, Yz ) (u1 · · · ui−1 ), b
(f (z, z, Yz ))∗ (ui · · · uj −1 ) and c
f (z, p, Yz ) (uj · · · ul ), which shows ∗ that a QbQc
f (q, z, Yz ) (f
(z, z, Yz )) f (z, p, Yz )(). Therefore, f (q, p, Y )
g. Hence f (q, p, Y ) = g. Thus fA coincides with q,p∈Q (0 (q)f (q, p, Q)1 (p)). We conclude that fA is named by a regular L-expression. Hence fA is regular.  Corollary 5.1 (Kleene Theorem). For any L-language f over a finite set , f can be accepted by an L-FFA iff f can be described by a regular L-expression. Corollary 5.2. ||LR|| = N L–FFA, and D L–FFA is a proper subset of ||LR||. In fact, in the proof of Theorem 5.2, we also give a procedure to deduce the regular expression of an L-FFA language. We give some examples to illustrate this fact. Example 5.1. (i) Regular expression for Example 3.1. (1) (q1 , q1 , ∅) = ; (q1 , q2 , ∅) = 0.5u; (q2 , q1 , ∅) = 0.5u; (q2 , q2 , ∅) = .

88

Y. Li, W. Pedrycz / Fuzzy Sets and Systems 156 (2005) 68 – 92

(2) (q1 , q1 , {q1 }) = ; (q1 , q1 , {q2 }) =  + 0.25uu; (q1 , q2 , {q1 }) = 0.5u; (q1 , q1 , {q2 }) = 0.5u; (q2 , q1 , {q1 }) = 0.5u; (q2 , q1 , {q1 }) = 0.5u; (q2 , q2 , {q1 }) =  + 0.25uu; (q2 , q2 , {q2 }) = ; (3) (q1 , q1 , Q) =  + ( + 0.25uu)++ + 0.25uu( + 0.25uu)∗ ; (q1 , q2 , Q) = 0.5u( + 0.25uu)∗ ; (q2 , q1 , Q) = 0.5u + 0.5u( + 0.25uu)+ ; (q2 , q2 , Q) =  + 0.25uu( + 0.25uu)∗ . (4) (f ) = 0.08(q1 , q1 , Q) + 0.56(q1 , q2 , Q) + 0.02(q1 , q1 , Q) + 0.14(q2 , q2 , Q) = 0.14 + 0.28u( + 0.25uu)∗ + 0.035uu( + 0.25uu)∗ + 0.08( + 0.25uu)∗ . (ii) Regular expression for Example 4.2.   1 + 1 + 1 + 1 + ∗ ∗ 0 + 1 (f ) = 0 + 1 , (f ) = . 2 3 2 3 Since the family of DL-FFA languages is not closed under Kleene closure, Kleene theorem does not hold for DL-FFA languages and L-fuzzy regular expressions. But, we still can give the L-fuzzy regular expressions for DL-FFA languages in the following forms. Let R denote the classical regular expressions on , that is, R = {0, 1}R, then S = {u1
1 + · · · + uk
k |u1 , . . . , uk ∈ L,
1 , . . . ,
k ∈ R} is called the DL-FFA’s regular expressions. For a DL-FFA’s regular expression u1
1 + · · · + uk
k , its corresponding L-language over  is defined by ||u1
1 + · · · +
uk
k ||() = {ui | ∈ ||
i ||}, for any  ∈ ∗ . Then from Theorem 6.3 in the next section, we have Theorem 5.3. ||S || = D L–FFA. 6. The relationship between L-FFA-languages and regular languages Suppose that f is an L-language on . For  ∈ L − {0}, the -cut of f, denoted f , is, f = { ∈ ∗ | f () }. Denoted the family of NL-FFA-languages on  as N L − F FA, then N L − F FA ⊆ F (∗ ). Let N L − F FA = {f |f ∈ N L − F FA}, and the family of all regular languages [9] on  is denoted as R.  Here we study the relationship between R and N L − F FA . Let us use the notation N L − F F A0 = ∈L N L − F FA .

Proposition 6.1. For an L-language f ∈ F (∗ ), if f is an N L-FFA-language, then supp(f ) = { ∈ ∗ |f ()  = 0} is a regular language. Proof. There is an NL-FFA A = (Q, , , q0 , ) such that fA = f . Then, we construct a nondeterministic finite automaton As = (Q, , s , q0 , s ) as, (q, u) = {p ∈ Q|(q, u, p) > 0} and s = {q ∈ Q|(q) > 0}. Now it is easy to show that the language accepted by As is just supp(f ).  Theorem 6.1. ∀ ∈ L − {0}, R ⊆ N L − F FA ⊆ N L − F FA0 . Proof. Suppose L ∈ R, then there is a deterministic finite automaton A = (Q, , , q0 , F ) such that the language accepted by A is L. Constructing an NL-FFA, B = (Q, , , 0 , 1 ), as,   , q ∈ F, e, q = q0 , 0 (q) =  (q) = 0, q = q0 ; 1 0, q ∈ / F;

Y. Li, W. Pedrycz / Fuzzy Sets and Systems 156 (2005) 68 – 92



(q, u, p) =

e, 0,

89

p ∈ (q, u), p∈ / (q, u).

Then, we have  fB () =

,

0,

 ∈ L, ∈ / L.

Therefore, (fB ) = L, and then R ⊆ N L − F FA ⊆ N L − F FA0 .  Theorem 6.2. If the l-monoid L satisfies the conditions of Theorem 3.4, then ∀f ∈ N L–FFA, ∀ ∈ L–{0}, f ∈ R. That is, N L–F FA ⊆ R, and thus N L–F FA0 = R. Proof. If L satisfies the conditions of Theorem 3.4, then f can be accepted by a DL-FFA, A = (Q, , , q0 , 1 ). That is to say, ∀ ∈ ∗ , f () = 1 (∗ (q0 , )). Let F = {q ∈ Q | 1 (q) }. Considering the crisp finite automaton A = (Q, , , q0 , F ), if we denote L(A) the language accepted by A, then  ∈ L(A) ⇔ ∗ (q0 , ) ∈ F ⇔ 1 (∗ (q0 , ))  ⇔ f () . That is, L(A) = f . Therefore, f ∈ R.  Remark 6.1. Due to the results of Theorem 6.2, the languages recognized by classical fuzzy automata (in this case, L = [0, 1]) are the same as regular languages in the level structure. This phenomenon reflects the limitation of the power of classical fuzzy finite automata to recognize fuzzy languages. This fact is also valid for L-FFA in which L is a quantale with the largest element as the identity, in particular a complete residuated lattice. We give a level characterization of the languages that can be accepted by a D L-FFA. Theorem 6.3. An L-language f ∈ F (∗ ) can be accepted by a D L-FFA iff R(f ) is a finite subset of L and ∀ ∈ R(f ), f is a regular language on . We need the following lemma to prove Theorem 6.3. Lemma 6.1. If an N
L-FFA, A = (Q, , , q0 , ) satisfies the conditions,  : Q ×  → 2Q ,  : Q → L, q0 ∈ Q, fA () = {(q) | q ∈ ∗ (q0 , )}, then there is an D L-FFA, B = (Y, , , y0 , ) such that fA = fB . Proof. Take Y = 2Q
, and : Y ×  → Y is defined as, (Z, u) = ∪q∈Z (q,∗u), y0 = {q0∗}, : Y → L is taken as, (Z) = q∈Z (q). Then B = (Y, , , y0 , ) is a DL-FFA, and (y0 , ) =  (q0 , ). Thus

fB () = ( ∗ (y0 , )) = q∈ ∗ (y0 ,) (q) = q∈∗ (q0 ,) (q) = fA (), that is, fB = fA .  Proof of Theorem 6.3. “Only if” part is from Theorem 6.2 and Lemma 3.1. “If” part. For any  ∈ R(f ), since f is a regular language on , there is a finite automata A = (Q , ,  , q , F ) such that the language accepted by A is f . Assume that if 1  = 2 , then Q1 ∩Q2 = ∅. Constructing an NL-FFA, A = (Q, , , q0 , ) as, Q = ∪∈R(F ) (Q − q } ∪ {q0 }, q0 ∈ / ∪∈R(F ) Q ,

90

Y. Li, W. Pedrycz / Fuzzy Sets and Systems 156 (2005) 68 – 92

 : Q → L is taken as,

 (q) =

,


{ ∈ R(f )|q ∈ F },

And  : Q ×  → 2Q is taken as  { (q, u)}, (q, u) = { (q , u)| ∈ R(f )},

q ∈ F , q = q0 .

q ∈ Q , q = q0 .


Then ∗ (q0 , ) = ∪∈R(f ) ∗ (s , ). If  = , then fA ()
= (∗ (q0 , )) = (q0 ) = { ∈ R(f
)|q ∈ F } = f (); and if  = , then fA () = (∗ (q0 , )) = {(q) | q ∈ ∗ (s , ),  ∈ R(f )} = { ∈ R(f ) |  ∈ f } = f (). Hence, fA = f . From Lemma 6.1, f can be accepted by a DL-FFA.  Theorem 6.4. If e = 1, and there are a, b ∈ L, such that a > e, b < e and ab = ba = e, then R is a proper subset of N L − F FAe , and thus the set R is also the proper subset of N L − F FA0 . Proof. Take A = (Q, , , 0 , 1 ) as, Q = {q1 , q2 },  = {u1 , u2 },  is defined as, (q1 , u1 , q1 ) = a, (q1 , u2 , q2 ) = (q2 , u2 , q2 ) = b, and 0 in other points, 0 = qe1 , 1 = qe2 . Then we have,  fA () =

a m Qbn , 0

n  = um 1 u2 , m  0, n  1,

otherwise.

n m n m n Therefore, (fA )e = { ∈ ∗ | fA ()  e} = {um 1 u2 | a Qb  e} = {u1 u2 | m  n  1}, while the later is not a regular language, which shows that R is a proper subset of N L − F FAe and N L − F FA0 . 

In fact, if we take L = [0, ∞] (as in Example 2.1(5)), then ∀ ∈ L − {∞}, the set R is a proper subset of N L − F F A . This phenomenon shows that the family of NL-FFA languages is bigger than the family of regular languages from the point of view of level structures. But the following theorem shows that N L − F F A is still the proper subset of F (∗ ). Theorem 6.5. (i) If f ∈ N L−F FA and R(f ) ⊆ {0, e}, then f is a regular language, that is, { | f () = e} is a crisp regular language. (ii) N L − F FA is the proper subset of F (∗ ). Proof. (i) Since f ∈ N L − F FA and the images of f, R(f ) ⊆ {0, e}, there is an NL-FFA, A = (Q, , , 0 , 1 ), such that f = fA and we can make A satisfy the following conditions: 0 = qe0 , 1 = e ∗ ∗ q1 , where q0 , q1 ∈ Q. Then  ∈  , fA () =  (q0 , , q1 ). Constructing a finite automaton, B = (Q, , , {q0 }, {q1 }), where p ∈ (q, u) ⇔ (q, u, p) = e, then the language accepted by B is { | f () = e}. Therefore, { | f () = e} is a regular language. (ii) Since there are nonregular languages on , assumed one of them is L, then f ∈ F (∗ ) is defined as, if  ∈ L, then f () = e, and f () = 0 in other cases. From (i), f ∈ / N L − F FA. Hence N L − F FA ∗ is a proper subset of F ( ). 

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7. Conclusion In this paper, we have studied the properties of finite automata whose membership values are in an l-monoid, and discussed languages induced by such automata. Some interesting results are obtained, such as, the extension ability of state transition function is equivalent to the distributive laws of l-monoid in which the finite automata take values. We have demonstrated that the nondeterministic L-FFA or NL-FFA are not equivalent to the deterministic L-FFA or DL-FFA in the sense of maintaining tha same ability of recognizing fuzzy languages. Some sufficient and necessary conditions for these two notions to be equivalent are also provided. It is not surprising that the family of NL-FFA-languages is closed under regular operations such as join, concatenation and Kleene closure, but to retain closure under meet and reversal operation, some auxiliary conditions are needed. We have introduced fuzzy regular expressions and their languages. We proved the equivalence (Kleene theorem) between fuzzy regular expressions and fuzzy finite automata in the sense of recognizing of fuzzy languages. Some fuzzy regular expressions for deterministic fuzzy finite automata are also discussed. Up to the level structures, NL-FFAs have the power to recognize more kinds of fuzzy languages than the classical fuzzy finite automata, the later have the equivalent power with crisp finite automata. Undoubtedly, much more work remains to be completed along this line. As we discussed in the introduction, we do not even know whether we can describe the NL-FFA languages or DL-FFA languages by regular grammars. Some other related researches such as minimization of L-FFA [27,37], analysis of special (algebraic) properties of L-FFA languages viewed from the stand of level structure, and the formal model of Computing With Words based on L-FFA, are of particular relevance. Acknowledgements The authors would like to thank the anonymous referees for their careful reading of this paper and for a number of valuable comments which improved the quality of this paper. References [1] P.R.J. Asveld, A bibliography on fuzzy automata, grammars and languages, Bull. European Assoc. Theoret. Comput. Sci. (58) (1996) 187–196 [ISSN 0252-9742]. [2] P.R.J. Asveld, Algebraic aspects of families of fuzzy languages, Theoret. Comput. Sci. 293 (2003) 417–445. [3] R. Balbes, P. Dwinger, Distributive Lattices, University of Mossouri Press, Columbia, Missouri, 1974. [4] R. Bˇelohlávek, Determinism and fuzzy automata, Inform. Sci. 142 (2002) 205–209. [5] G. Birkhoff, Lattice Theory (1940), (3rd) Ed., American Mathematical Society, Providence, RI, USA, 1973. [6] S. Eilenberg, Automata, Languages, and Machines, vol. A, B, Academic Press, New York, 1974. [7] G. Grätzer, General Lattice Theory, Academic Press, New York, San Franciso, 1978. [8] U. Höhle, On the fundamentals of fuzzy set theory, J. Math. Anal. Appl. 201 (1996) 786–826. [9] J.E. Hopcroft, J.D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley, New York, 1979. [10] A. Kandel, S.C. Lee, Fuzzy Switching and Automata: Theory and Applications, Arnold, London, 1980. [11] W. Kuich, A. Salomaa, Semiring, Automata, Language, Springer, Berlin, Heidelberg, New York, 1986. [12] H.V. Kumbhojkar, S.R. Chaudhari, Fuzzy recognizers and recognizable sets, Fuzzy Sets and Systems 131 (2002) 381–392. [13] E.T. Lee, L.A. Zadeh, Note on fuzzy languages, Inform. Sci. 1 (1969) 421–434.

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