Journal of Applied Mathematics

Volume 2012 (2012), Article ID 804623, 9 pages

doi:10.1155/2012/804623

Research Article

A Remark on Myhill-Nerode Theorem for Fuzzy Languages

Shou-feng Wang

School of Mathematics, Yunnan Normal University, Yunnan, Kunming 650500, China

Received 21 May 2012; Accepted 4 August 2012

Academic Editor: Abdel-Maksoud A. Soliman

Copyright © 2012 Shou-feng Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Regular fuzzy languages are characterized by some algebraic approaches. In particular, an extended version of Myhill-Nerode theorem for fuzzy languages is obtained.

1. Introduction

Fuzzy sets were introduced by Zadeh in [1] and since then have appeared in many fields of sciences. They have been studied within automata theory for the first time by Wee in [2]. More on recent development of algebraic theory of fuzzy automata and formal fuzzy languages can be found in the book Mordeson and Malik [3], the texts Malik et al. [4, 5], and Petković [6].

A fuzzy language is called regular if it can be recognized by a fuzzy automaton. In the texts Mordeson and Malik [3], Petković [6], Ignjatovic et al. [7], and Shen [8], regular fuzzy languages have been characterized by the principal congruences (principal right congruences, principal left congruences) determined by fuzzy languages, which are known as Myhill-Nerode theorem for fuzzy languages. Moreover, Petković [6] also considered the varieties of fuzzy languages and Ignjatovic and Ciric [9] considered regular operations of fuzzy languages.

Recently, Wang et al. [10] generalized the usual principal congruences (resp., principal right congruences, principal left congruences) to some kinds of generalized principal congruences (resp., generalized principal right congruences, generalized principal left congruences) determined by crisp languages by using prefix-suffix-free subsets (resp., prefix-free languages, suffix-free languages) and obtained some characterizations of regular crisp languages.

In this note, we will realize the idea of the text [10] for fuzzy languages. In other word, we characterize regular fuzzy languages by some kinds of generalized principal congruences (resp., generalized principal right congruences, generalized principal left congruences) determined by fuzzy languages. In particular, we obtain an extended version of Myhill-Nerode theorem for fuzzy languages.

2. Preliminaries

Throughout the paper, is a finite set which is called an alphabet and is the free monoid generated by , that is, the set of all words with letters from . The empty word is denoted by . The length of a word in is the number of letters appearing in and is denoted by . The complement of a subset of is the set . A subset of is cofinite if is finite. A nonempty subset of is called a suffix-free language over if no element in is a suffix of another element in . Prefix-free languages over can be defined dually. On the other hand, a nonempty subset of is called a prefix-closed language over if any prefix of an element in is also in .

As an analogue of prefix-free languages and suffix-free languages over , Wang et al. [10] introduced prefix-suffix-free subsets of . A subset of the set is called a prefix-suffix-free subset if for all words in , the following holds: if both and are in , then .

An equivalence on is called a right congruence if implies that for any . A left congruences can be defined dually. An equivalence is a congruence if it is a right congruence and also a left congruence.

A fuzzy subset of a set is a mapping . By and infimum and supremum in the unit segment will be denoted, respectively. Every element of can be considered as the following fuzzy subset of :

A fuzzy language over is a fuzzy subset of . A fuzzy language is regular if it is recognizable by a fuzzy automaton from the book [3]. For a fuzzy language over , the relations defined on by the following:

are called the principal right congruence (resp., principal left congruence, principal congruence) determined by , respectively.

Now, we state the well-known Myhill-Nerode theorem for fuzzy languages which gives some algebraic characterizations for regular fuzzy languages. Recall that the index of an equivalence on is the number of -classes of .

Theorem 2.1 (see [3, 6, 8], Myhill-Nerode theorem). For a fuzzy language over , the following statements are equivalent:

(1)is regular.(2) is of finite index.(3) is of finite index.(4) is of finite index.

In the sequel, we recall some operations of fuzzy languages. For two fuzzy languages and over , the union, intersection, product, and left and right quotients of and are defined, respectively, by the following:

Further, we also define left-right quotient of three fuzzy languages , and over by the following:

Observe that for any with the above notations.

On regular fuzzy languages, we have the following.

Lemma 2.2 (see [6]). Finite unions, intersections, products, and left-right quotients of regular fuzzy languages over are regular.

3. Main Result

In this section, we shall introduce some kinds of generalized principal (resp., right, left) congruences determined by fuzzy languages by using prefix-suffix-free subsets (resp., prefix-free languages, suffix-free languages) and give an extended version of Myhill-Nerode theorem for fuzzy languages.

Now, let be a prefix-free language, be a suffix-free language over , be a prefix-suffix-free subset of , and be a fuzzy language over , respectively. For a prefix-suffix-free subset , denote

Define the following relations on :

Then we have the following observations.

Proposition 3.1. The above are right congruences (resp., left congruences; congruence) on . Furthermore,

Proof. It is easy to check that (resp., ) is a right (resp., left) congruence, is a congruence, and

by their definitions. In the sequel, we show that is a right congruence and is a left congruence. Clearly, both and are equivalences. Now, let be two words in and . Then there exists a finite subset of such that for any in . Now, let be a word in and be the union of and . Then is in for any in . This implies that for any in whence since is finite. Thus, is a right congruence. Dually, is a left congruence.

Remark 3.2. Note that the above inclusions are all proper in general. For example, let and . Then . Define a fuzzy language over as follows:

where are in and . Then we have

Similarly, we can show that the remainder inclusions are all proper.

To obtain our main result, we need a series of lemmas. First, we recall the following alphabetic order “” on : For two words and in with different lengths, if , for two words with same length, the order is the lexicographic order. Observe that the alphabetic order is a well order on . We have the following result.

Lemma 3.3. Let be an infinite prefix-closed language over . Then there exists an infinite subset of , where .

Proof. Denote

Observe that is finite and is infinite, there exists and such that is infinite and . Denote

Then is infinite. Hence, there also exists and such that is infinite and . In general, for any positive integer , there exists and such that is infinite and . Let

Clearly, is infinite. We claim that . Let . Observe that

Therefore, there exists such that . And hence, . Since is prefix-closed, . This implies that .

Lemma 3.4. Let be a right congruence on and be the set of all -classes of . Then,

is prefix-closed.

Proof. Clearly, is in . Let be in and for some positive integer and in . Then, is not in . Suppose that is in . Then, . This implies that . On the other hand, since and is a right congruence, we have . Hence, is in and so . Thus, . This implies that whence is in .

Lemma 3.5. Let be a suffix-free language and be a fuzzy language over .

(1) is of finite index if and only if is of finite index.(2) is of finite index if and only if is of finite index.

Proof. (1)  Similar to the proof of Proposition  3.11 in [10].

(2) Observe that , the necessity holds. Conversely, if is of finite index and is of infinite index, then by Lemma 3.4, is infinite and prefix-closed. By Lemma 3.3, there exists an infinite subset

of , where . Since is of finite index, there exist two distinct elements such that . Therefore, there exists a finite subset of such that . Denote and take in satisfying . We assert that is in for any in . In fact, if for some in and in , then by the choice of , is a prefix of and so is a suffix of whence is in . A contradiction. Therefore, for any in , we have . This implies that .

Without loss of generality, we let with respect to the alphabetic order, and . Then, by the above discussions, and is in . Observe that is a subset of , in view of the definition of , . This implies that . A contradiction.

Lemma 3.6. Let be a finite suffix-free language over . Then is cofinite if and only if is maximal.

Proof. It follows from Lemma  3.14 in [10].

Lemma 3.7. Let be a finite prefix-suffix-free subset of and be a fuzzy language over . Then the following are equivalent:

(1) is of finite index.(2)The following fuzzy language over defined by

is regular.(3), where is regular and for any in .

Proof. (1) implies (2). Let be in , be in and . Then for any in , is in . Therefore,

whence . Thus,

Now, if is of finite index, then is regular for any in . Observe that

it follows that is regular from Lemma 2.2.

(2) implies (3). By (2), is regular. Let be the following fuzzy language over defined by

Then , as required.

(3) implies (1). If for some regular fuzzy language and a fuzzy language such that for any in , then is of finite index and . Observe that

is of finite index.

Remark 3.8. In general, for a given finite prefix-suffix-free subset of and a fuzzy language over , may be nonregular even if is of finite index. For example, let and . Define the following fuzzy language over as follows:

Clearly, for every in and so is trivially regular. By Lemma 3.7, is of finite index. However, for any pair in with different lengths, we have whence is not related to . Observe that is infinite, there are infinite -classes of and so is of infinite index. This implies that is nonregular by Theorem 2.1.

Now, we have our main theorem.

Theorem 3.9 (An extended version of Myhill-Nerode theorem). For a fuzzy language over , the following statements are equivalent:

(1) is regular.(2) is of finite index for some finite maximal suffix-free language over .(3) is of finite index for some finite maximal prefix-free language over .(4) is of finite index for some finite prefix-suffix-free subset of such that is cofinite.(5) is of finite index for some finite maximal prefix-free language over .(6) is of finite index for some finite maximal suffix-free language over .

Proof. (1) implies (2). Observe that is a maximal suffix-free language over and , the result follows from Theorem 2.1.

(2) implies (4). Observe that is a prefix-suffix-free subset of and , the result follows from Lemma 3.5 (1) and Lemma 3.6.

(4) implies (1). By Lemma 3.7 (3), there exists a regular fuzzy language and another fuzzy language such that and for any in . However, by (4), is finite, which implies that is also regular. In view of Lemma 2.2, is regular.

By symmetry, we can prove that the facts that (1) implies (3) and (3) implies (4). On the other hand, by Lemma 3.5 (2) and its dual, it follows that (3) is equivalent to (5) and (2) is equivalent to (6).

4. Conclusions

In this short note, we have obtained an extended version of Myhill-Nerode theorem for fuzzy languages (Theorem 3.9) which provides some algebraic characterizations of regular fuzzy languages. On the other hand, for a given prefix-suffix-free subset of , by Proposition 3.1 and Remark 3.8,

contains the class of regular fuzzy languages over as a proper subclass. In fact, Lemma 3.7 gives some characterizations of members in for a given finite prefix-suffix-free subset of . Thus the following questions could be considered as a future work. For a general prefix-suffix-free subset of , what can be said about ? For example, can we obtain some results parallel to Theorems  3.5 and 3.17 in [10]?

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