On fuzzy multiset regular languages

The purpose of present work is to study some algebraic aspect of fuzzy multiset regular languages. In between, we show the equivalence of multiset regular language and fuzzy multiset regular language. Finally, we introduce the concept of pumping lemma for fuzzy multiset regular languages, which we use to establish a necessary and sufficient condition for a fuzzy multiset language to be non-constant.


Introduction
In the late 1960s after the introduction of fuzzy set theory by Zadeh [36], fuzzy automata and fuzzy languages are introduced and studied as a means for bridging the gap between the precision of formal languages and natural languages (cf., [2,8,9,13,14,18,19,24]).Such studies were initiated by Wee and Fu [34] and Santos [27,28].In the last few years many researchers greatly contributed the considerable literature to this field, such as Malik and Mordeson [19,20,22] have established a fundamental framework of algebraic fuzzy automata theory.Thereafter, there were number of authors having contributed to this field (cf., [15,24,26,29,30,35]).In application point of view, fuzzy regular language have been shown to be useful in numerous engineering applications such as learning systems, computing with word, fuzzy discrete event systems, pattern recognition, database theory, lexical analysis in programming language compilation, user-interface translations, circulation design and text editing (cf., [1,7,16,17,25]).In recent years their application have been further extended to include parallel processing, image generation and compression, type theory for object-oriented languages, DNA computing, etc.In 1989, Blizard [4] introduced the concept of multiset, which is a collection of elements in which elements may occur more than once, is a generalization of a set.In recent years, multiset processing has appeared frequently in various areas of mathematics, computer science, biology and biochemistry (cf., [3,5,21,23,31]).The researchers have also used the concept of multisets in automata theory and introduced the concepts of multiset finite automata and multiset languages (cf., [6,10,11,12]), which play an important role in various areas of theoretical computer science, as in membrane computing.
On the way of extending the concepts of automata theroy and formal languages to multiset finite automata and multiset languages, Wang, Yin and Gu [33] introduced the concept of fuzzy multiset finite automata and fuzzy multiset language as the extension of multiset finite automata and multiset languages.In [33], it is shown that if a fuzzy multiset language is generated by a fuzzy multiset regular grammar, it can be accepted by a fuzzy multiset finite automaton, and vice-versa.Recently, Tiwari, Gautam and Dubey [32] shown that a deterministic fuzzy multiset finite automaton is equally powerful as a fuzzy multiset finite automaton in the sense of acceptance of a fuzzy multiset language.Also, in [32], the minimal realizations (minimal deterministic fuzzy multiset automata) for a given fuzzy muliset language is studied.But, there is a silence on equivalence of multiset regular languages and fuzzy multiset regular languages.In this paper, we initiate such study for fuzzy multiset regular languages.This paper is organized as follows.In Section 2, we recall some basic concepts of multisets, multiset finite automata, and multiset languages.In Section 3, we study the fuzzy multiset automata and fuzzy multiset languages.Specifically, we demonstrate the equivalence of multiset regular language and fuzzy multiset regular language.Finally in Section 4, we introduce the concept of pumping lemma for fuzzy multiset regular languages, which is used to establish a necessary and sufficient condition for a given fuzzy multiset language to be non-constant.

Preliminaries
In this section, we collect some notions and notations associated with multisets and multiset finite automata which are required in the subsequent sections.The concepts related to multisets, we use in this paper are fairly standard and can be found in the literature (cf., [4,6,32,33]).We begin with the following.Definition 2.1.If Σ is a finite alphabet, then α : Σ → N is a multiset over Σ, where N denotes the set of natural numbers including 0. The α norm of Σ is defined by |α| = ∑ a∈Σ α(a).
We shall denote by Σ ⊕ , the set of all multiset over Σ.The multiset 0 Σ ∈ Σ ⊕ is defined by 0 Σ (a) = 0, ∀a ∈ Σ.For b ∈ Σ, we shall denote by < b >, a singleton multiset, and is defined by For two multisets α, β ∈ Σ ⊕ , the operations inclusion ⊆, addition ⊕ and difference ⊖ are defined as follows : Obviously, Σ ⊕ is a commutative monoid with identity element 0 Σ with respect to ⊕.
International Scientific Publications and Consulting Services Now, we recall the following concepts of multiset finite automata and multiset languages from [6].
Definition 2.2.A multiset finite automaton (MFA) is a 5-tuple M = (Q, Σ, δ , q 0 , F), where (i) Q and Σ are nonempty finite sets called the state-set and input-set, respectively; A configuration of a MFA M is a pair (p, α), where p and α denote current state and current multiset, respectively.The transition in a multiset finite automaton are described with the help of configurations.The transition from configuration (p, α) leads to configuration (q, β ) if there exists a multiset γ ∈ Σ ⊕ with γ ⊆ α, q ∈ δ (p, γ) and β = α ⊖ γ, and is denoted by (p, α) → (q, β ).We shall denote by → * , the reflexive and transitive closure of this operation.
3 Fuzzy multiset regular languages In this section, we recall the concepts and properties of fuzzy multiset finite automaton and fuzzy multiset regular language from [32,33].Thereafter, we introduce λ -cut of a fuzzy multiset language and show that λ -cut of a given fuzzy multiset regular language is a multiset regular language.Definition 3.1.[32] A fuzzy multiset finite automaton (FMFA) is a 5-tuple M = (Q, Σ, δ , σ , τ), where (i) Q and Σ are nonempty finite sets called the state-set and input-set, respectively; is a map called the fuzzy set of initial states; and (iv) τ : Q → [0, 1] is a map called the fuzzy set of final states.
A configuration of FMFA M is a pair (p, α), where p and α denote current state and current multiset, respectively.The transition in a FMFA are described with the help of configurations.The transition from configuration (p, α) leads to configuration (q, β ) with membership value k ∈ for some n ≥ 0, there exists (n + 1) states q 0 , ..., q n and (n + 1) multisets α 0 , ..., α n such that q 0 = p, q n = q, α 0 = α, α n = β and Definition 3.2.[32] (i) For a given set Σ, a fuzzy multiset language is a map f : (iii) A fuzzy multiset language which is accepted by a FMFA is called fuzzy multiset regular language.Now, we have the following.
Theorem 3.1.Given a multiset regular language L ⊆ Σ ⊕ , the characteristic function 1 L of L is a fuzzy multiset regular language.
Following is an example for above result.

It can be easily verify that f M
International Scientific Publications and Consulting Services Theorem 3.2.Given L ⊆ Σ ⊕ .Let characteristic function 1 L of L be a fuzzy multiset regular language.Then L is a multiset regular language on Σ.
Proof.Given fuzzy multiset language 1 L is regular, i.e., there exists a Now, α ∈ L if and only if there exists q ∈ Q such that σ (q) = 1 and µ M ((q, α) L q be a multiset language accepted by M q , whereby L q ⊆ L. Then ∪ q∈Q L q ⊆ L. Also, let α ∈ L. Then Hence L is regular as each L q is regular.International Scientific Publications and Consulting Services Theorem 3.3.Given a fuzzy multiset regular language f : Σ ⊕ → [0, 1] and λ ∈ [0, 1], λ ∧ f is a fuzzy multiset regular language.
Proof.Since f is fuzzy multiset regular language on Σ ⊕ , there exists For a fuzzy multiset language f : Σ ⊕ → [0, 1] and λ ∈ (0, 1], λ − cut of fuzzy multiset language is given by This lead us to the following result for λ − cut of fuzzy multiset language.Theorem 3.4.Let f : Σ ⊕ → [0, 1] be a fuzzy multiset regular language on Σ ⊕ .Then for all λ ∈ Im( f ), f λ is a multiset regular language on Σ ⊕ .Proof.Let f : Σ ⊕ → [0, 1] be a fuzzy multiset regular language.Then there exists a Thus, for all α ∈ f λ , there exists q α ∈ Q such that Again, let L M qα be the multiset language of M q α and β ∈ L M qα .Then δ α (q α , β ) ∈ τ λ and τ(δ α (q α , β )) ≥ λ , i.e., Conversly, let β ∈ f λ , i.e., there exists q β ∈ Q such that Then, 4 Pumping lemma for fuzzy multiset regular languages In this section, we introduce the concept of pumping lemma and use it to establish a necessary and sufficient condition for fuzzy multiset languages to be non-constant.

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This leads us to establish a necessary and sufficient condition for fuzzy multiset language to be non-constant.Case (ii) : If f M (γ) < f M (α).In this case, if |α| < n, then α and γ are the required words.Now, if |α| ≥ n.Then proceeding as in case (i), we can show that there exist words α 1 and α 2 , such that |α 1 | < n and f M (α 2 ) < f M (α 1 ).
The converse is immediate.

Conclusion
This paper is towards the study of fuzzy multiset regular languages.We tried to associate a multiset regular languages to a given fuzzy multiset regular languages and vice-versa.Meanwhile, we introduced λ -cut of a fuzzy multiset language and show that λ -cut of a fuzzy multiset language is also a multiset regular language.Finally, we introduced the concept of pumping lemma for fuzzy multiset regular languages, which we use to establish a necessary and sufficient condition for fuzzy multiset languages to be non-constant.Theory established in this paper may provide some new direction in the theory of fuzzy multiset finite automata and fuzzy multiset languages with hope that members of the scientific community will find useful applications for these theories in near future.

Example 2 . 1 .
Let Σ = {a, b, c, d} be a set of symbols and we have number of objects but they are not distinguishable except their label of a, b, c or d.For example, we have three balls with the label a and two balls with the label b, four with the label c, but no ball with the label d.Moreover, we are not allowed to put additional labels to distinguish three a ′ s.Therefore a natural representation of the situation form multiset α = (3/a, 2/b, 4/c, 0/d) can be denoted by denote the reflexive and transitive closure of

Example 3 . 1 .
Let Q = {m, n, r, s,t}, Σ = {a, b} and transition diagram is given in Figure1.International Scientific Publications and Consulting Services