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Journal of Applied Mathematics
Volume 2012, Article ID 804623, 9 pages
http://dx.doi.org/10.1155/2012/804623
Research Article

A Remark on Myhill-Nerode Theorem for Fuzzy Languages

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, 𝐴 is a finite set which is called an alphabet and 𝐴 is the free monoid generated by 𝐴, that is, the set of all words with letters from 𝐴. The empty word is denoted by 1. The length of a word 𝑤 in 𝐴 is the number of letters appearing in 𝑤 and is denoted by |𝑤|. The complement of a subset 𝐿 of 𝐴 is the set 𝐿={𝑤𝐴𝑤𝐿}. A subset 𝐿 of 𝐴 is cofinite if 𝐿 is finite. A nonempty subset 𝑆 of 𝐴 is called a suffix-free language over 𝐴 if no element in 𝑆 is a suffix of another element in 𝑆. Prefix-free languages over 𝐴 can be defined dually. On the other hand, a nonempty subset 𝐿 of 𝐴 is called a prefix-closed language over 𝐴 if any prefix of an element in 𝐿 is also in 𝐿.

As an analogue of prefix-free languages and suffix-free languages over 𝐴, Wang et al. [10] introduced prefix-suffix-free subsets of 𝐴×𝐴. A subset Δ of the set 𝐴×𝐴 is called a prefix-suffix-free subset if for all words 𝑠,𝑡,𝑥,𝑦 in 𝐴, the following holds: if both (𝑠,𝑡) and (𝑠𝑥,𝑦𝑡) are in Δ, then 𝑥=𝑦=1.

An equivalence 𝜌 on 𝐴 is called a right congruence if 𝑥𝜌𝑦 implies that 𝑥𝑧𝜌𝑦𝑧 for any 𝑥,𝑦,𝑧𝐴. 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 𝛼 of a set 𝑋 is a mapping 𝛼𝑋[0,1]. By and infimum and supremum in the unit segment [0,1] will be denoted, respectively. Every element 𝑦 of 𝑋 can be considered as the following fuzzy subset of 𝑋: 𝑦(𝑥)=1for𝑥=𝑦,𝑦(𝑥)=0for𝑥𝑦.(2.1) A fuzzy language over 𝐴 is a fuzzy subset of 𝐴. A fuzzy language is regular if it is recognizable by a fuzzy automaton from the book [3]. For a fuzzy language 𝜆 over 𝐴, the relations defined on 𝐴 by the following: 𝑥𝑃𝜆(𝑟)𝑦if𝜆(𝑥𝑢)=𝜆(𝑦𝑢)forevery𝑢in𝐴,𝑥𝑃𝜆(𝑙)𝑦if𝜆(𝑢𝑥)=𝜆(𝑢𝑦)forevery𝑢in𝐴,𝑥𝑃𝜆𝑦if𝜆(𝑢𝑥𝑣)=𝜆(𝑢𝑦𝑣)forevery𝑢,𝑣𝐴,(2.2) are called the principal right congruence (resp., principal left congruence, principal congruence) determined by 𝜆, 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 𝜌 on 𝐴 is the number of 𝜌-classes of 𝐴.

Theorem 2.1 (see [3, 6, 8], Myhill-Nerode theorem). For a fuzzy language 𝜆 over 𝐴, the following statements are equivalent:(1)𝜆is regular.(2)𝑃𝜆 is of finite index.(3)𝑃𝜆(𝑟) is of finite index.(4)𝑃𝜆(𝑙) is of finite index.

In the sequel, we recall some operations of fuzzy languages. For two fuzzy languages 𝜆1 and 𝜆2 over 𝐴, the union, intersection, product, and left and right quotients of 𝜆1 and 𝜆2 are defined, respectively, by the following: 𝜆1𝜆2(𝑤)=𝜆1(𝑤)𝜆2𝜆(𝑤),1𝜆2(𝑤)=𝜆1(𝑤)𝜆2(𝜆𝑤),1𝜆2(𝑤)=𝑥𝑦=𝑤𝜆1(𝑥)𝜆2𝜆(𝑦)11𝜆2(𝑤)=𝑢𝐴𝜆2(𝑢𝑤)𝜆1,𝜆(𝑢)2𝜆11(𝑤)=𝑢𝐴𝜆2(𝑤𝑢)𝜆1.(𝑢)(2.3) Further, we also define left-right quotient of three fuzzy languages 𝜆1, 𝜆2 and 𝜆 over 𝐴 by the following: 𝜆11𝜆𝜆21𝜆(𝑤)=11𝜆𝜆21(𝑤).(2.4) Observe that (𝑠1𝜆𝑡1)(𝑤)=𝜆(𝑠𝑤𝑡) for any 𝑠,𝑡,𝑤𝐴 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 𝐴 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 𝑃 be a prefix-free language, 𝑆 be a suffix-free language over 𝐴, Δ be a prefix-suffix-free subset of 𝐴×𝐴, and 𝜆 be a fuzzy language over 𝐴, respectively. For a prefix-suffix-free subset Δ, denote ΩΔ=(𝑠𝑥,𝑦𝑡)(𝑠,𝑡)Δ,𝑥,𝑦𝐴,𝑁(Δ)=(𝑠,𝑡)Δ𝑠𝐴𝑡.(3.1) Define the following relations on 𝐴: 𝑥𝑃(𝑙)𝑃,𝜆𝑦if𝜆(𝑢𝑥)=𝜆(𝑢𝑦)forevery𝑢in𝑃𝐴,𝑥𝑃(𝑟)𝑆,𝜆𝑦if𝜆(𝑥𝑢)=𝜆(𝑦𝑢)forevery𝑢in𝐴𝑆,𝑥𝑃Δ,𝜆𝑦if𝜆(𝑢𝑥𝑣)=𝜆(𝑢𝑦𝑣)forevery(𝑢,𝑣)inΩΔ,𝑥𝑃(𝑟),𝑆,𝜆𝑦ifthereexistssomenitesubset𝐹of𝐴suchthat𝜆(𝑥𝑢)=𝜆(𝑦𝑢)forevery𝑢in𝐹𝐴𝑆,𝑥𝑃(𝑙),𝑃,𝜆𝑦ifthereexistssomenitesubset𝐹of𝐴suchthat𝜆(𝑢𝑥)=𝜆(𝑢𝑦)forevery𝑢in𝑃𝐴𝐹.(3.2) Then we have the following observations.

Proposition 3.1. The above 𝑃(𝑟)𝑆,𝜆,𝑃(𝑟),𝑆,𝜆(𝑟𝑒𝑠𝑝.,𝑃(𝑙)𝑃,𝜆,𝑃(𝑙),𝑃,𝜆;𝑃Δ,𝜆) are right congruences (resp., left congruences; congruence) on 𝐴. Furthermore, 𝑃𝜆(𝑟)𝑃(𝑟)𝑆,𝜆𝑃(𝑟),𝑆,𝜆,𝑃𝜆(𝑙)𝑃(𝑙)𝑃,𝜆𝑃(𝑙),𝑃,𝜆,𝑃𝜆𝑃Δ,𝜆.(3.3)

Proof. It is easy to check that 𝑃(𝑟)𝑆,𝜆 (resp., 𝑃(𝑙)𝑃,𝜆) is a right (resp., left) congruence, 𝑃Δ,𝜆 is a congruence, and 𝑃𝜆(𝑟)𝑃(𝑟)𝑆,𝜆𝑃(𝑟),𝑆,𝜆,𝑃𝜆(𝑙)𝑃(𝑙)𝑃,𝜆𝑃(𝑙),𝑃,𝜆,𝑃𝜆𝑃Δ,𝜆(3.4) by their definitions. In the sequel, we show that 𝑃(𝑟),𝑆,𝜆 is a right congruence and 𝑃(𝑙),𝑃,𝜆 is a left congruence. Clearly, both 𝑃(𝑟),𝑆,𝜆 and 𝑃(𝑙),𝑃,𝜆 are equivalences. Now, let 𝑥,𝑦 be two words in 𝐴 and 𝑥𝑃(𝑟),𝑆,𝜆𝑦. Then there exists a finite subset 𝐹 of 𝐴 such that 𝜆(𝑥𝑢)=𝜆(𝑦𝑢) for any 𝑢 in 𝐹𝐴𝑆. Now, let 𝑧 be a word in 𝐴 and 𝐹 be the union of {𝑤𝐴𝑧𝑤𝐹} and {1}. Then 𝑧𝑢 is in 𝐹𝐴𝑆 for any 𝑢 in 𝐹𝐴𝑆. This implies that 𝜆(𝑥𝑧𝑢)=𝜆(𝑦𝑧𝑢) for any 𝑢 in 𝐹𝐴𝑆 whence 𝑥𝑧𝑃(𝑟),𝑆,𝜆𝑦𝑧 since 𝐹 is finite. Thus, 𝑃(𝑟),𝑆,𝜆 is a right congruence. Dually, 𝑃(𝑙),𝑃,𝜆 is a left congruence.

Remark 3.2. Note that the above inclusions are all proper in general. For example, let 𝐴={𝑎},𝑆={𝑎2} and 𝐹={1,𝑎,𝑎2,𝑎3}. Then 𝐹𝐴𝑆=𝐴𝑎5. Define a fuzzy language over 𝐴 as follows: 𝑎𝜆(𝑤)=𝛼for𝑤2,𝑎3,𝜆(𝑤)=𝛽for𝑤𝐴𝑎2,𝑎3,(3.5) where 𝛼,𝛽 are in [0,1] and 𝛼𝛽. Then we have 𝑎3,𝑎4𝑃𝜆(𝑟),𝑎3,𝑎4𝑃(𝑟)S,𝜆,1,𝑎2𝑃(𝑟)𝑆,𝜆,1,𝑎2𝑃(𝑟),𝑆,𝜆.(3.6) 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 𝐴: For two words 𝑢 and 𝑣 in 𝐴 with different lengths, 𝑢<𝑣 if |𝑢|<|𝑣|, for two words with same length, the order is the lexicographic order. Observe that the alphabetic order is a well order on 𝐴. We have the following result.

Lemma 3.3. Let 𝐿 be an infinite prefix-closed language over 𝐴. Then there exists an infinite subset {1,𝑎1,𝑎1𝑎2,,𝑎1𝑎2,,𝑎𝑛,} of 𝐿, where 𝑎𝑖𝐴.

Proof. Denote Pre𝑓𝐴(𝐿)=𝑎𝐴𝑦𝐴𝑎𝑦𝐿.(3.7) Observe that 𝐴 is finite and 𝐿 is infinite, there exists 𝐿1𝐿 and 𝑎1𝐴 such that 𝐿1 is infinite and Pre𝑓𝐴(𝐿1)={𝑎1}. Denote 𝑎11𝐿1=𝑤𝐴𝑎1𝑤𝐿1.(3.8) Then 𝑎11𝐿1 is infinite. Hence, there also exists 𝐿2𝑎11𝐿1 and 𝑎2𝐴 such that 𝐿2 is infinite and Pre𝑓𝐴(𝐿2)={𝑎2}. In general, for any positive integer 𝑛, there exists 𝐿𝑛+1𝑎𝑛1𝐿𝑛 and 𝑎𝑛+1𝐴 such that 𝐿𝑛+1 is infinite and Pre𝑓𝐴(𝐿𝑛+1)={𝑎𝑛+1}. Let 𝐶=1,𝑎1,𝑎1𝑎2,𝑎1𝑎2𝑎3,,𝑎1𝑎2𝑎3𝑎𝑛,.(3.9) Clearly, 𝐶 is infinite. We claim that 𝐶𝐿. Let 𝑎1𝑎2𝑎3𝑎𝑛𝐶. Observe that 𝐿𝑛𝑎1𝑛1𝐿𝑛1𝑎1𝑛1𝑎1𝑛2𝐿𝑛2𝑎1𝑛1𝑎1𝑛2𝑎11𝐿1=𝑎1𝑎2𝑎𝑛11𝐿1,𝑎𝑛=Pre𝑓𝐴𝐿𝑛Pre𝑓𝐴𝑎1𝑎2𝑎𝑛11𝐿1.(3.10) Therefore, there exists 𝑦𝐴 such that 𝑎𝑛𝑦(𝑎1𝑎2𝑎𝑛1)1𝐿1. And hence, 𝑎1𝑎2𝑎𝑛1𝑎𝑛𝑦𝐿1𝐿. Since 𝐿 is prefix-closed, 𝑎1𝑎2𝑎𝑛1𝑎𝑛𝐿. This implies that 𝐶𝐿.

Lemma 3.4. Let 𝜌 be a right congruence on 𝐴 and {𝐿𝑖𝑖𝐼} be the set of all 𝜌-classes of 𝐴. Then, 𝐿𝜌=𝑠𝑖𝑠𝑖𝑖𝑠𝑡𝑒𝑙𝑒𝑎𝑠𝑡𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑖𝑛𝐿𝑖𝑤𝑖𝑡𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑡𝑜,𝑖𝐼(3.11) is prefix-closed.

Proof. Clearly, 1 is in 𝐿𝜌. Let 𝑠𝑗 be in 𝐿𝜌 and 𝑠𝑗=𝑎1𝑎2𝑎𝑡 for some positive integer 𝑡>1 and 𝑎1,𝑎2,,𝑎𝑡 in 𝐴. Then, 𝑎1𝑎2𝑎𝑡1 is not in 𝐿𝑗. Suppose that 𝑎1𝑎2𝑎𝑡1 is in 𝐿𝑘. Then, 𝑠𝑘𝑎1𝑎2𝑎𝑡1. This implies that 𝑠𝑘𝑎𝑡𝑎1𝑎2𝑎𝑡1𝑎𝑡=𝑠𝑗. On the other hand, since 𝑠𝑘𝜌𝑎1𝑎2𝑎𝑡1 and 𝜌 is a right congruence, we have 𝑠𝑘𝑎𝑡𝜌𝑠𝑗. Hence, 𝑠𝑘𝑎𝑡 is in 𝐿𝑗 and so 𝑠𝑘𝑎𝑡𝑠𝑗. Thus, 𝑠𝑘𝑎𝑡=𝑠𝑗=𝑎1𝑎2𝑎3𝑎𝑡1𝑎𝑡. This implies that 𝑠𝑘=𝑎1𝑎2𝑎3𝑎𝑡1 whence 𝑎1𝑎2𝑎3𝑎𝑡1 is in 𝐿𝜌.

Lemma 3.5. Let 𝑆 be a suffix-free language and 𝜆 be a fuzzy language over 𝐴.(1)𝑃{1}×𝑆,𝜆 is of finite index if and only if 𝑃(𝑟)𝑆,𝜆 is of finite index.(2)𝑃(𝑟)𝑆,𝜆 is of finite index if and only if 𝑃(𝑟),𝑆,𝜆 is of finite index.

Proof. (1)  Similar to the proof of Proposition  3.11 in [10].
(2) Observe that 𝑃(𝑟)𝑆,𝜆𝑃(𝑟),𝑆,𝜆, the necessity holds. Conversely, if 𝑃(𝑟),𝑆,𝜆 is of finite index and 𝑃(𝑟)𝑆,𝜆 is of infinite index, then by Lemma 3.4, 𝐿𝑃(𝑟)𝑆,𝜆 is infinite and prefix-closed. By Lemma 3.3, there exists an infinite subset 𝐶=1,𝑎1,𝑎1𝑎2,,𝑎1𝑎2𝑎𝑛,,(3.12) of 𝐿𝑃(𝑟)𝑆,𝜆, where 𝑎𝑖𝐴. Since 𝑃(𝑟),𝑆,𝜆 is of finite index, there exist two distinct elements 𝑥,𝑦𝐶 such that 𝑥𝑃(𝑟),𝑆,𝜆𝑦. Therefore, there exists a finite subset 𝐹 of 𝐴 such that 𝜆(𝑥𝑢)=𝜆(𝑦𝑢)forevery𝑢in𝐹𝐴𝑆. Denote 𝑇=max{|𝑓|𝑓𝐹} and take 𝑢 in 𝐴 satisfying |𝑢|>𝑇. We assert that 𝑢𝑣 is in 𝐹𝐴𝑆 for any 𝑣 in 𝐴𝑆. In fact, if 𝑢𝑣=𝑓𝑤 for some 𝑓 in 𝐹 and 𝑤 in 𝐴𝑆, then by the choice of 𝑢, 𝑓 is a prefix of 𝑢 and so 𝑣 is a suffix of 𝑤 whence 𝑤 is in 𝐴𝑆. A contradiction. Therefore, for any 𝑣 in 𝐴𝑆, we have 𝜆(𝑥𝑢𝑣)=𝜆(𝑦𝑢𝑣). This implies that 𝑥𝑢𝑃(𝑟)𝑆,𝜆𝑦𝑢.
Without loss of generality, we let 𝑥<𝑦 with respect to the alphabetic order, 𝑦=𝑎1𝑎2𝑎𝑡 and 𝑢=𝑎𝑡+1𝑎𝑡+𝑇+1. Then, by the above discussions, 𝑥𝑢𝑃(𝑟)𝑆,𝜆𝑦𝑢 and 𝑦𝑢 is in 𝐶. Observe that 𝐶 is a subset of 𝐿𝑃(𝑟)𝑆,𝜆, in view of the definition of 𝐿𝑃(𝑟)𝑆,𝜆, 𝑥𝑢𝑦𝑢. This implies that 𝑥𝑦. A contradiction.

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

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

Lemma 3.7. Let Δ be a finite prefix-suffix-free subset of 𝐴×𝐴 and 𝜆 be a fuzzy language over 𝐴. Then the following are equivalent:(1)𝑃Δ,𝜆 is of finite index.(2)The following fuzzy language 𝜆Δ over 𝐴 defined by 𝜆Δ(𝑤)=𝜆(𝑤)𝑓𝑜𝑟𝑤𝑁(Δ),𝜆Δ(𝑤)=0𝑓𝑜𝑟𝑤𝑁(Δ)(3.13) is regular.(3)𝜆=𝜆1𝜆2, where 𝜆1 is regular and 𝜆2(𝑤)=0 for any 𝑤 in 𝑁(Δ).

Proof. (1) implies (2). Let 𝑥,𝑦 be in 𝐴, (𝑠,𝑡) be in Δ and 𝑥𝑃Δ,𝜆𝑦. Then for any 𝑢,𝑣 in 𝐴, (𝑠𝑢,𝑣𝑡) is in ΩΔ. Therefore, 𝑠1𝜆𝑡1(𝑢𝑥𝑣)=𝜆(𝑠𝑢𝑥𝑣𝑡)=𝜆(𝑠𝑢𝑦𝑣𝑡)=𝑠1𝜆𝑡1(𝑢𝑦𝑣),(3.14) whence 𝑥𝑃𝑠1𝜆𝑡1𝑦. Thus, 𝑃Δ,𝜆(𝑠,𝑡)Δ𝑃𝑠1𝜆𝑡1.(3.15) Now, if 𝑃Δ,𝜆 is of finite index, then 𝑠1𝜆𝑡1 is regular for any (𝑠,𝑡) in Δ. Observe that 𝜆Δ=(𝑠,𝑡)Δ𝑠𝑠1𝜆𝑡1𝑡,(3.16) it follows that 𝜆Δ is regular from Lemma 2.2.
(2) implies (3). By (2), 𝜆Δ is regular. Let 𝜆2 be the following fuzzy language over 𝐴 defined by 𝜆2(𝑤)=0for𝑤𝑁(Δ),𝜆2(𝑤)=𝜆(𝑤)for𝑤𝑁(Δ).(3.17) Then 𝜆=𝜆Δ𝜆2, as required.
(3) implies (1). If 𝜆=𝜆1𝜆2 for some regular fuzzy language 𝜆1 and a fuzzy language 𝜆2 such that 𝜆2(𝑤)=0 for any 𝑤 in 𝑁(Δ), then 𝑃𝜆1 is of finite index and 𝑃Δ,𝜆2=𝐴×𝐴. Observe that 𝑃𝜆1𝑃Δ,𝜆1𝑃Δ,𝜆1𝜆2=𝑃Δ,𝜆,(3.18)𝑃Δ,𝜆 is of finite index.

Remark 3.8. In general, for a given finite prefix-suffix-free subset of 𝐴×𝐴 and a fuzzy language 𝜆 over 𝐴, 𝜆 may be nonregular even if 𝑃Δ,𝜆 is of finite index. For example, let 𝐴={𝑎,𝑏} and Δ={(𝑎,𝑏)}. Define the following fuzzy language 𝜆 over 𝐴 as follows: 1𝜆(𝑤)=0for𝑤𝑁(Δ),𝜆(𝑤)=|𝑤|+1for𝑤𝑁(Δ).(3.19) Clearly, 𝜆Δ(𝑤)=0 for every 𝑤 in 𝐴 and so 𝜆Δ is trivially regular. By Lemma 3.7, 𝑃Δ,𝜆 is of finite index. However, for any pair 𝑤1,𝑤2 in 𝑁(Δ) with different lengths, we have 𝜆(𝑤1)𝜆(𝑤2) whence 𝑤1 is not 𝑃𝜆 related to 𝑤2. Observe that 𝑁(Δ) is infinite, there are infinite 𝑃𝜆-classes of 𝐴 and so 𝑃𝜆 is of infinite index. This implies that 𝜆 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 𝜆 over 𝐴, the following statements are equivalent:(1)𝜆 is regular.(2)𝑃(𝑟)𝑆,𝜆 is of finite index for some finite maximal suffix-free language 𝑆 over 𝐴.(3)𝑃(𝑙)𝑃,𝜆 is of finite index for some finite maximal prefix-free language 𝑃 over 𝐴.(4)𝑃Δ,𝜆 is of finite index for some finite prefix-suffix-free subset Δ of 𝐴×𝐴 such that 𝑁(Δ) is cofinite.(5)𝑃(𝑙),𝑃,𝜆 is of finite index for some finite maximal prefix-free language 𝑃 over 𝐴.(6)𝑃(𝑟),𝑆,𝜆 is of finite index for some finite maximal suffix-free language 𝑆 over 𝐴.

Proof. (1) implies (2). Observe that {1} is a maximal suffix-free language over 𝐴 and 𝑃𝜆(𝑟)=𝑃(𝑟){1},𝜆, the result follows from Theorem 2.1.
(2) implies (4). Observe that {1}×𝑆 is a prefix-suffix-free subset of 𝐴×𝐴 and 𝑁({1}×𝑆)=𝐴𝑆, 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 𝜆1 and another fuzzy language 𝜆2 such that 𝜆=𝜆1𝜆2 and 𝜆2(𝑤)=0 for any 𝑤 in 𝑁(Δ). However, by (4), 𝑁(Δ) is finite, which implies that 𝜆2 is also regular. In view of Lemma 2.2, 𝜆 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 Δ of 𝐴×𝐴, by Proposition 3.1 and Remark 3.8, 𝔽Δ(𝐴)=𝜆𝜆isafuzzylanguageover𝐴suchthattheindexof𝑃Δ,𝜆isnite(4.1)

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

References

  1. L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965. View at Google Scholar
  2. W. G. Wee, On generalizations of adaptive algorithm and application of the fuzzy sets concept to pattern classification, [Ph.D. thesis], Purdue University, 1967.
  3. J. N. Mordeson and D. S. Malik, Fuzzy Automata and Languages, Chapman & Hall/CRC, 2002. View at Publisher · View at Google Scholar
  4. D. S. Malik, J. N. Mordeson, and M. K. Sen, “Products of fuzzy finite state machines,” Fuzzy Sets and Systems, vol. 92, no. 1, pp. 95–102, 1997. View at Publisher · View at Google Scholar
  5. D. S. Malik, J. N. Mordeson, and M. K. Sen, “Minimization of fuzzy finite automata,” Information Sciences, vol. 113, no. 3-4, pp. 323–330, 1999. View at Publisher · View at Google Scholar
  6. T. Petković, “Varieties of fuzzy languages,” in Proceedings of the 1st International Conference on Algebraic Informatics, pp. 197–205, Aristotle University of Thessaloniki, Thessaloniki, Greece, 2005.
  7. J. Ignjatovic, M. Ciric, S. Bogdanovic, and T. Petkovic, “Myhill-Nerode type theory for fuzzy languages and automata,” Fuzzy Sets and Systems, vol. 161, no. 9, pp. 1288–1324, 2010. View at Publisher · View at Google Scholar
  8. J. Shen, “Fuzzy language on free monoid,” Information Sciences, vol. 88, no. 1–4, pp. 149–168, 1996. View at Publisher · View at Google Scholar
  9. J. Ignjatovic and M. Ciric, “Formal power series and regular operations on fuzzy languages,” Information Sciences, vol. 180, no. 7, pp. 1104–1120, 2010. View at Publisher · View at Google Scholar
  10. S. F. Wang, Y. Q. Guo, and S. X. Xu, “𝒫 𝒮-regular languages,” International Journal of Computer Mathematics, vol. 88, no. 12, pp. 2464–2484, 2011. View at Publisher · View at Google Scholar