On the recognizability of Fuzzy languages I

Fuzzy Sets and Systems 157 (2006) 2394–2402www.elsevier.com/locate/fss

On the recognizability of Fuzzy languages I

Symeon Bozapalidisa,∗, Olympia Louscou-Bozapalidoub

aDepartment of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, GreecebSection of Mathematics and Informatics, Technical Institute of West Macedonia, Koila, Kozani, Greece

Received 16 June 2005; received in revised form 6 April 2006; accepted 6 April 2006Available online 19 May 2006

Abstract

Fuzzy language recognizability via finite monoids (called m-recognizability) is examined. Fuzzy languages computed by (max,min)-automata, (max, �L)-automata and (max, �D)-automata are m-recognizable (�L, �D are the Lucasiewicz intersectionand the drastic intersection, respectively). The syntactic monoid associated to each m-recognizable language can be effectivelyconstructed. Thus, the equality of two m-recognizable fuzzy languages is decidable. A pumping lemma is displayed.© 2006 Elsevier B.V. All rights reserved.

Keywords: Fuzzy automata; Language derivatives; Pumping lemma

1. Introduction

A classical result in mathematical language theory states that a language L ⊆ X∗ is recognizable if and only if it isthe inverse image of a subset of a finite monoid M through a morphism from X∗ to M. Our scope in the present paperis to examine the recognizability notion obtained by fuzzyfying the above characterization result.

Our main used structure is that of a monoid representation of a fuzzy language � : X∗ → [0, 1] which consists of amonoid M, a morphism h : X∗ → M and a fuzzy set � : M → [0, 1] so that � = � ◦ h.

To any fuzzy language � we associate its syntactic representation which is the most economical way torepresent �. Moreover, we show that this representation can be effectively constructed provided that � is m-recognizable(i.e., finitely representable).

The class of all m-recognizable fuzzy languages have nice closure properties, namely it is a convex set closed underany t-norm, t-conorm and negations operations as well as under inverse word morphisms.

It is worthwhile to point out that the equality problem for m-recognizable fuzzy languages is shown to be decidable.Automata weighted over [0, 1] whose behavior is computed with respect to a pair (∇, �) of a t-conorm ∇ and a t-norm� are considered (called (∇,�)-fuzzy automata). In the case that (∇, �) = (max, min), (max, �L), (max, �D) thebehaviors of the corresponding automata are m-recognizable fuzzy languages. Hence the equivalence problem for suchautomata is decidable. Finally, we state a pumping lemma in the present framework.

Fuzzy automata where transitions are weighted over lattice ordered monoids and the closure properties of thelanguages they recognize are investigated in [8].

∗ Corresponding author. Tel.: +30 2310997921.E-mail address: [email protected] (S. Bozapalidis).

0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2006.04.008

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S. Bozapalidis, O. Louscou-Bozapalidou / Fuzzy Sets and Systems 157 (2006) 2394 –2402 2395

In [7] infinite computations over (max, min) and (min, max)-automata are studied. An elegant presentation on thesubject of fuzzy context free languages can be found in [1–3].

2. Syntactic representations

The set X∗ of all words over an alphabet X, equipped with the word concatenation operation, becomes a monoidwhose unit element is the empty word ε. Subsets of X∗ are called languages over X.

Let � : X∗ → [0, 1] be a fuzzy language and define the relation ∼� on X∗ by

w ∼� w′ if �(�1w�2) = �(�1w′�2) for all �1, �2 ∈ X∗.

This relation is compatible with the operation of concatenation, that is

w ∼� w′ and u ∼� u′ imply wu ∼� w′u′.

Indeed, for all �1, �2 ∈ X∗ we have

�(�1(wu)�2) = �(�1w(u�2)) = �(�1w′(u�2)) = �((�1w

′)u�2)

= �((�1w′)u′�2) = �(�1(w

′u′)�2).

We call ∼� the syntactic congruence of � and the quotient monoid M� = X∗/ ∼� the syntactic monoid of �.Moreover, a fuzzy set �� : M� → [0, 1] is defined by setting ��([w]) = �(w), where [w] is the ∼�-class of the wordw ∈ X∗. Since [w] = [w′] implies �(w) = �(w′) we get that �� is well defined. It holds that � = �� ◦ h� whereh� : X∗ → M� is the monoid epimorphism sending every word w ∈ X∗ to its ∼�-class [w].

A representation of the fuzzy language � : X∗ → [0, 1] is a triple (M, h, �) where M is a monoid, h : X∗ → M isa morphism of monoids and � : M → [0, 1] is a fuzzy set so that � = � ◦ h.

According to the previous discussion every fuzzy language � : X∗ → [0, 1] admits a representation (M�, h�, ��).

Theorem 1. For any representation (M, h, �) (with h surjective) of the fuzzy language � : X∗ → [0, 1] there exists aunique epimorphism h̄ : M → M� rendering commutative the diagram

Proof. By the surjectivity of h, any element m ∈ M can be written as m = h(w), for some word w ∈ X∗. We seth̄(m) = h�(w). We show that

h(w) = h(w′) implies h�(w) = h�(w′).

2396 S. Bozapalidis, O. Louscou-Bozapalidou / Fuzzy Sets and Systems 157 (2006) 2394 –2402

Indeed, let �1, �2 ∈ X∗. Then

�(�1w�2) = (� ◦ h)(�1w�2) = �(h(�1w�2)) = �(h(�1)h(w)h(�2))

= �(h(�1)h(w′)h(�2)) = �(h(�1w′�2)) = (� ◦ h)(�1w

′�2)

= �(�1w′�2).

It comes that w ∼� w′ and so [w] = [w′] or h�(w) = h�(w′) as wanted. Thus h̄ is a well defined function. Moreoverfor any mi ∈ M , let wi ∈ X∗ so that mi = h(wi), i = 1, 2. Then h(w1w2) = h(w1)h(w2) = m1m2 and so

h̄(m1m2) = h�(w1w2) = h�(w1)h�(w2) = h̄(m1)h̄(m2),

i.e., h̄ is a morphism of monoids.The rest of the proof is left to the reader. �

Following Eilenberg ([4], vol. B) we say that the monoid M covers the monoid N (notation N < M) if there exist amonoid M̄ , a monomorphism � : M̄ → M and an epimorphism h : M̄ → N .

For the notions t-norm, t-conorm and negation used below see [6].

Theorem 2. Let �1, �2, � : X∗ → [0, 1] be fuzzy languages. Then

(i) For any operation � : [0, 1]2 → [0, 1] we have

M�1��2< M�1

× M�2

where �1��2 : X∗ → [0, 1] is defined by (�1��2)(w) = �1(w)��2(w), for all w ∈ X∗. In particular for anyt-norm � and t-conorm ∇ we have M�1��2

< M�1× M�2

, M�1∇�2< M�1

× M�2.

(ii) For any negation neg : [0, 1] → [0, 1] we have Mneg(�) = M�. In particular, M1−� = M�.(iii) If f : Y ∗ → X∗ is a morphism of monoids, then M�◦f < M�. Moreover, in the case that f is an epimorphism we

have that M�◦f �M�, that is the monoids M�◦f and M� are isomorphic.

Proof. (i) Consider the diagram

where h(w) = (h�1(w), h�2

(w)) and �(m1, m2) = ��1(m1)��2

(m2) for all w ∈ X∗, m1 ∈ M�1and m2 ∈ M�2

.Then

(� ◦ h)(w) = �(h(w)) = �(h�1(w), h�2

(w)) = ��1(h�1

(w))��2(h�2

(w))

= (��1◦ h�1

)(w)�(��2◦ h�2

)(w) = (�1��2)(w)

i.e., � ◦ h = �1��2. By Theorem 1, there results an epimorphism from the monoid

Im(h) = {(h�1(w), h�2

(w)) | w ∈ X∗}onto M�1��2

and item (i) is achieved.For (ii) we have

w ∼neg(�) w′ iff neg(�)(�1w�2) = neg(�)(�1w′�2) for all �1, �2 ∈ X∗,

iff neg(�(�1w�2)) = neg(�(�1w′�2)) for all �1, �2 ∈ X∗.


Since neg : [0, 1] → [0, 1] is an injective function, we get

�(�1w�2) = �(�1w′�2) for all �1, �2 ∈ X∗

and so w ∼� w′. In other words ∼neg(�)=∼� and so Mneg(�) = M�.Finally we shall establish the second statement of item (iii). For all w, w′, �1, �2 ∈ Y ∗ we have

w ∼�◦f w′ iff (� ◦ f )(�1w�2) = (� ◦ f )(�1w′�2),

iff �(f (�1w�2)) = �(f (�1w′�2)),

iff �(f (�1)f (w)f (�2)) = �(f (�1)f (w′)f (�2)) (f surjective),

iff �(�1f (w)�2) = �(�1f (w′)�2) for all �1, �2 ∈ X∗,iff f (w) ∼� f (w′).

In other words, the congruence ∼�◦f is the inverse image via h of the congruence ∼� and so the function sendingthe ∼�◦f -class of w ∈ Y ∗ to the ∼�-class of f (w) is an isomorphism of the monoid M�◦f onto the monoid M�. �

3. m-recognizability

A fuzzy language � : X∗ → [0, 1] is said to be m-recognizable if it has a finite representation (M, h, �). We denoteby m-Rec(X) the set of all m-recognizable fuzzy languages over X.

Remark. It should be noticed that any m-recognizable fuzzy language � : X∗ → [0, 1] has finite range.

Example 1. Let X = {x1, . . . , xn}. Any function � : X → [0, 1] can be uniquely extended by means of the minoperation into a function also denoted by �, from X∗ to [0, 1]

�(x1 · · · xk) = min(�(x1), . . . ,�(xk)).

The so obtained fuzzy language is m-recognizable. Indeed, for w ∈ X∗ we set

〈w〉�,xi= 1 if xi occurs in w and �(xi) > 0

= 0 else.

Then the set

M = {(〈w〉�,x1 , . . . , 〈w〉�,xn) | w ∈ X∗}with the boolean addition 1 + 1 = 1, becomes a monoid and the triple (M, h, �) with h : X∗ → M , h(w) =(〈w〉�,x1 , . . . , 〈w〉�,xn) and � : M → [0, 1], (〈w〉�,x1 , . . . , 〈w〉�,xn) = �(w) is a representation of �.

Taking into account Theorem 2, we get

Theorem 3. A fuzzy language � : X∗ → [0, 1] is m-recognizable if and only if its syntactic monoid M� is finite.

Combining the above result with Theorem 2 and the fact that if the finite monoid M covers the monoid N (i.e.,N < M) then N is again finite, we obtain

Proposition 1. (i) For any t-norm � and t-conorm ∇ we have �1, �2 ∈ m-Rec(X) implies �1��2, �1∇�2 ∈m-Rec(X).

(ii) For any negation function neg : [0, 1] → [0, 1] we have � ∈ m-Rec(X) implies neg(�) ∈ m-Rec(X). Inparticular

� ∈ m-Rec(X) implies 1 − � ∈ m-Rec(X).

(iii) For any monoid morphism f : X∗ → Y ∗ we have � ∈ m-Rec(Y ) implies � ◦ f ∈ m-Rec(X).


Proposition 2. The set m-Rec(X) is a convex set i.e., for all �, �′ ∈ [0, 1] � + �′ = 1 and �, �′ ∈ m-Rec(X) implies�� + �′�′ ∈ m-Rec(X).

Proof. We need the following auxiliary result:

Fact. For every �, �′ : X∗ → [0, 1] such that �(w) + �′(w)�1 for all w ∈ X∗, it holds that

�, �′ ∈ m-Rec(X) implies � + �′ ∈ m-Rec(X).

Indeed, a finite realization of � + �′ is (M� × M�′ , h, �), where

h : X∗ → M� × M�′ , h(w) = (h�(w), h�′(w))

and

� : M� × M�′ → [0, 1], �(m, m′) = ��(m) + ��′(m′).

Now, from �, �′ ∈ m-Rec(X) we get ��, �′�′ ∈ m-Rec(X) and since

(��)(w) + (�′�′)(w) = ��(w) + �′�′(w)�� + �′ = 1

the previous result can be applied: �� + �′�′ ∈ m-Rec(X). �

The cut languages of � : X∗ → [0, 1] at � ∈ [0, 1] are defined as follows:

[� > �] = {w|w ∈ X∗, �(w) > �}, [� = �] = {w | w ∈ X∗, �(w) = �}.It is worthwhile to point out that

Proposition 3. If � : X∗ → [0, 1] is m-recognizable, then all its cut languages are recognizable.

Proof. Since � has finite range and [� > �] = [� = �1] ∪ · · · ∪ [� = �k] where �1, . . . , �k are all the points of �(X∗)that are > �, it suffices to establish our statement only for the languages [� = �].

Recall that a language L ⊆ X∗ is recognizable if and only if the congruence

w ∼L w′ whenever �1w�2 ∈ L ⇔ �1w′�2 ∈ L for all �1, �2 ∈ X∗

has finite index (i.e., a finite number of classes cf. [4]). Assume now that w ∼� w′, that is �(�1w�2) = �(�1w′�2) for

all �1, �2 ∈ X∗. Then we have

�(�1w�2) = � ⇔ �(�1w′�2) = � for all �1, �2 ∈ X∗

or

�1w�2 ∈ [� = �] ⇔ �1w′�2 ∈ [� = �] for all �1, �2 ∈ X∗

or

w ∼[�=�] w′.

In other words ∼�⊆∼[�=�] and since ∼� has finite index, so does ∼[�=�], that is [� = �] is a recognizablelanguage. �

It should be noted that cut-recognizability does not imply m-recognizability. For instance, although the fuzzy language� : X∗ → [0, 1], �(w) = 1/2|w|, w ∈ X∗ has all its cut languages [� > �], [� = �] finite, hence recognizable, it isnot m-recognizable for the simple reason that it has non-finite range.

We denote by c-Rec(X) the class of all fuzzy languages � : X∗ → [0, 1] whose cut languages [� > �], [� = �](� ∈ [0, 1]) are recognizable.

Proposition 4. The class m-Rec(X) is a proper subset of the class c-Rec(X).


4. Automata weighted over [0,1]

In this section we show that there are remarkable m-recognizable fuzzy languages. For the notions of various fuzzyautomata see [5].

We need to recall some matter about fuzzy relations. Let � and ∇ be a t-norm and a t-conorm, respectively, andconsider a finite set Q. We denote by FRel(Q) the set of all fuzzy relations over Q i.e., of all functions R : Q × Q →[0, 1]. The composition of two such relations R, S : Q × Q → [0, 1] with respect to �, ∇ is given by the formula

(R ◦ S)(q, q ′) = ∇p∈Q

R(q, p)�S(p, q ′), q, q ′ ∈ Q.

The composition is associative whenever � is distributive over ∇ i.e., it holds x�(y∇z) = (x�y)∇(x�z) for allx, y, z ∈ [0, 1]. Under this assumption FRel(Q) with the above operation becomes a monoid whose unit element isthe diagonal DQ : Q × Q → [0, 1] with DQ(q, q ′) = 1 if q = q ′ and = 0 else.

An (∇, �)-automaton is a 5-tuple A = (Q, X, , I, T ) where Q is the finite set of states, X is the input alphabet, : X → FRel(Q) is the transition function and I, T : Q → [0, 1] are the initial and final fuzzy subsets.

The run mapping of A is the monoid morphism ∗ : X∗ → FRel(Q), defined by

∗(x1 · · · xn) = (x1) ◦ · · · ◦ (xn), x1, . . . , xn ∈ X, n > 0

∗(ε) = DQ

and the behavior of A is the fuzzy language |A| : X∗ → [0, 1]|A|(w) = ∇

p,q∈QI (p)�∗(w)(p, q)�T (q).

Proposition 5. Let A = (Q, X, , I, T ) be a (∇, �)-automaton and assume that the list below

∇q1,...,qn+1∈Q

(x1)(q1, q2)� · · · �(xn)(qn, qn+1), x1, . . . xn ∈ X, n�0

is finite. Then |A| : X∗ → [0, 1] is m-recognizable.

Proof. Our assumption ensures that

M = {(x1) ◦ · · · ◦ (xn) | x1, . . . , xn ∈ X, n�0}is a finite submonoid of FRel(Q) and thus (M, ∗, �) with

� : M → [0, 1], �(R) = ∇p,q∈Q

I (p)�R(p, q)�T (q), R ∈ M

is a finite representation of |A|. �

The Lucasiewicz intersection and the drastic intersection, are defined by

x�Ly = max(0, x + y − 1), x�Dy = x if y = 1

= y if x = 1

= 0 else.

Lemma 1. Let = {�1, . . . , �k} ⊆ [0, 1]. Then the following sets are finite. min = {min(�i1 , . . . , �in ) | i1, . . . , in ∈{1, . . . , k}, n�1}, max = {max(�i1 , . . . , �in ) | i1, . . . , in ∈ {1, . . . , k}, n�1}, �L

= {�i1�L · · · �L�in |i1, . . . , in ∈ {1, . . . , k}, n�1}, �D

= {�i1�D · · · �D�in | i1, . . . , in ∈ {1, . . . , k}, n�1}.

Proof. It is immediate that min and max are finite. In order to establish the finiteness of �L, let us assume (without

any loss) that �1 < 1, . . . , �r < 1, �r+1 = 1, . . . , �k = 1. We choose a positive integer s so that

�1, . . . , �r < 1 − 1

s


and set � = max(�1, . . . , �k). Then for every n > s and i1, . . . , in ∈ {1, . . . , r} we have

�i1�L · · · �L�in ��L · · · �L� = max(0, n� + 1 − n).

But n� + 1 − n < s� + 1 − s < 0 so that max(0, n� + 1 − n) = 0. Hence �i1�L · · · �L�in = 0. It turns out that

�L= {�i1�L · · · �L�im |m�s}.

Now, for each � ∈ [0, 1] we have ��D� = 1 if � = 1 and ��D� = 0 if � < 1. Thus

�D= {�i1�D · · · �D�im |m > r}. �

Theorem 4. The fuzzy languages computed by

(i) (max, min)-automata(ii) (max, �L)-automata

(iii) (max, �D)-automataare all m-recognizable.

Remark. It should be noted that the previous result for (max, min) is already known see [5].

Although the t-norm

[0, 1]2 −→ [0, 1], (x, y) �→ xy

is distributive over max and min, the associated fuzzy automata have non-m-recognizable behaviors. Take for instancethe automaton A = (Q = {q}, X = {�}, , I = {q}, T = {q}) with (�)(q, q) = 1

2 . Then |A|(�n) = 12n , n�0, which

is non-m-recognizable.

5. Some decidability results

We start with two lemmas.

Lemma 2. Let (M, h, �) be a finite representation of the fuzzy language � : X∗ → [0, 1] and set n = cardM .

(i) For any w ∈ X∗ there is a w̄ ∈ X∗ with |w̄|�n so that h(w) = h(w̄).(ii) Given �1, �2 ∈ X∗ we can determine �̄1, �̄2 ∈ X∗ with |�̄1|�n, |�̄2|�n so that |�1|�n, |�̄2|�n and �(�1w�2) =

�(�̄1w�̄2) for all w ∈ X∗.

Proof. (i) Assume that w = x1 · · · xk , k > n, xi ∈ X, 1� i�k. The elements h(x1), h(x1x2), . . . , h(x1x2 · · · xk) arenot pairwise distinct; since cardM = n, there are i, j ∈ {1, . . . , k}, i < j , so that h(x1 · · · xi) = h(x1 · · · xj ). But then

h(w) = h(x1 · · · xi · · · xj · · · xk)

= h(x1 · · · xi · · · xj )h(xj+1 · · · xk)

= h(x1 · · · xi)h(xj+1 · · · xk)

= h(x1 · · · xixj+1 · · · xk),

the result comes by repeatedly applying (if necessary) the above procedure.(ii) Choose �̄1, �̄2 as in item (i). Then for all w ∈ X∗ we have

�(�1w�2) = (� ◦ h)(�1w�2)

= �(h(�1w�2))

= �(h(�1)h(w)h(�2))

= �(h(�̄1)h(w)h(�̄2))

= �(�̄1w�̄2). �


Lemma 3. Let (M, h, �) be a finite representation of � : X∗ → [0, 1] and n = cardM .

(i) For every w ∈ X∗ we can find w̄ ∈ X∗ with |w̄|�n so that w ∼� w̄.(ii) Given w, w′ ∈ X∗, we can decide whether w ∼� w′ or not.

Proof. (i) Let w̄ as in item (i) of the previous lemma. Then for all �1, �2 ∈ X∗ we have

�(�1w�2) = �(h(�1)h(w)h(�2)) = �(h(�1)h(w̄)h(�2)) = �(�1w̄�2)

and so w ∼� w̄.(ii) We are going to show the following implication: if �(�1w�2) = �(�1w

′�2) for all �1, �2 ∈ X∗ with |�1|, |�2|�n

then

�(�1w�2) = �(�1w′�2) for all �1, �2 ∈ X∗.

Indeed, for any �1, �2 ∈ X∗, let �̄1, �̄2 as in item (ii), Lemma 10. Then

�(�1w�2) = �(�̄1w�̄2) = �(�̄1w′�̄2) = �(�1w

′�2).

Therefore, w ∼� w′ is decidable. �

Now, we are ready to show that

Theorem 5. Given an m-recognizable fuzzy language � : X∗ → [0, 1], its syntactic monoid M� can be effectivelyconstructed.

Proof. Let (M, h, �) be a finite representation of �, n = cardM and denote by [w] the ∼�-class of the word w ∈ X∗.Obviously

[w] = [w′] iff w ∼� w′

and thus, by Lemma 3(ii) the equality of two ∼�-classes is decidable. Therefore, we can effectively determine thedistinct elements [w1], . . . , [wk] of the finite list

[w]/w ∈ X∗, |w|�n.

Taking into account that

M� = {[w] | w ∈ X∗}as well as Lemma 3(i) and the previous discussion, we get

M� = {[w1], . . . , [wk]}.The multiplication of M� is given by

[wi][̇wj ] = [wp],where p is the unique integer between 1 and k such that

[wiwj ] = [wp].Hence the syntactic monoid M� is effectively constructed. �

Corollary 1. Given m-recognizable fuzzy languages �, �′ : X∗ → [0, 1] we can decide whether M� ≈ M�′ or not.

Proposition 6. Let (Mi, hi, �i ) be a finite representation of �i : X∗ → [0, 1], (i = 1, 2). If �1(u) = �2(u) for allu ∈ X∗, |u| < cardM1 + cardM2 then �1 = �2. Consequently, the equality of two m-recognizable fuzzy languagesis decidable.


Proof. By applying Lemma 2(i) to the monoid morphism h : X∗ → M1 ×M2, h(w) = (h1(w), h2(w)) we get that forall w ∈ X∗ there exists w̄ ∈ X∗ with |w̄| < cardM1 + cardM2 so that h(w) = h(w̄), i.e., such that hi(w) = hi(w̄)

(i = 1, 2). Hence for all w ∈ X∗ we obtain

�1(w) = �1(h1(w)) = �1(h1(w̄)) = �1(w̄) = �2(w̄) = �2(w). �

Corollary 2. The equivalence problem for

(i) (max, min)-automata(ii) (max, �L)-automata

(iii) (max, �D)-automatais decidable.

Using again the technique of Lemma 2, one can show that

Pumping Lemma. Given an m-recognizable fuzzy language � : X∗ → [0, 1], there is a natural number k > 0 suchthat for any w ∈ X∗, |w| > k, there is a factorization

w = w1w2w3, |w2| > 0

with the property

�(w1wn2w3) = �(w) for all n�0.

By this lemma, it is immediately seen that the fuzzy language � : X∗ → [0, 1] with

�(w) = 12 if w = �n�n, n�0

= 13 else

fails to be m-recognizable.

References

[1] P.R.J. Asveld, Algebraic aspects of families of fuzzy languages, Theoret. Comput. Sci. 293 (2003) 417–445.[2] P.R.J. Asveld, Fuzzy context-free languages, Part 1: generalized fuzzy context-free grammars, Theoret. Comput. Sci. 347 (2005) 167–190.[3] P.R.J. Asveld, Fuzzy context-free languages, Part 2: recognition and parsing algorithms, Theoret. Comput. Sci. 347 (1–2, 30) (2005) 191–213.[4] S. Eilenberg, Automata, Languages and Machines, vol. A, B, Academic Press, New York, 1974, 1977.[5] J. Mordeson, D. Malik, Fuzzy Automata and Languages: Theory and Applications, Chapman & Hall, CRC Press, London, Boca Raton, FL,

2002.[6] H.T. Nguyen, E.A. Walker, A First Course in Fuzzy Logic, CRC Press, Boca Raton, FL, 1977.[7] G. Rahonis, Infinite Fuzzy Computations Fuzzy Sets and Systems, vol. 153, 2005, pp. 275–288.[8] Yongming Li, Witold Pedrycz, Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids, Fuzzy

Sets and Systems 156 (2005) 68–92.

Documents

On the recognizability of Fuzzy languages I