Elsevier

Information Sciences

Volume 176, Issue 21, 3 November 2006, Pages 3232-3255
Information Sciences

Algebraic properties of LA-languages

https://doi.org/10.1016/j.ins.2005.10.003Get rights and content

Abstract

In this study, we introduce the concepts of L-valued regular substitution (LA-substitution), deterministic L-valued regular substitution (DLA-substitution), L-valued fuzzy homomorphism and its inverse images, homomorphism and its inverse images for a lattice-ordered monoid L. We also study the properties of LA-languages and DLA-languages under the above-mentioned algebraic operations. The algebraic characterization of the L-valued regular language is given.

Introduction

It is well known that the following approaches to represent a language (regular language, to be prise) L are equivalent [5]:

  • (1)

    L is recognized by deterministic finite automaton.

  • (2)

    L is recognized by nondeterministic finite automaton.

  • (3)

    L is described by regular expression.

  • (4)

    L is generated by regular grammar.

The same results hold for fuzzy regular languages with truth values in [0, 1] and with max-min composition [1], [2], [6], [7], [8], [14], [16], [18], [19], [22]. In [20], the algebraic definition of fuzzy regular language was given, however it did not demonstrate whether the family of fuzzy languages given by algebraic definition are equal to that of fuzzy languages induced by other methods similar to (1)–(4) above. On the other hand, for the generalized fuzzy languages with truth-values in a lattice-ordered monoid, it was shown in [11], [13] that (1) and (2) are not equivalent. That is to say, nondeterministic fuzzy automata with truth-values in a lattice-ordered monoid L which are called L-valued finite automata (LA, for short) are more powerful than deterministic fuzzy automata with truth-values in L which are called deterministic L-valued finite automata (DLA, for short). An important problem arises as to the algebraic characterization of LA and DLA. In this paper, the equivalence between the algebraic definition of the L-valued regular language and the former definition (corresponding to (3)) is shown. Since the equivalence between (2) and (3) has been shown in [11], [13] and the equivalence between (2) and (4) has been presented in [21], hence the algebraic definition of the L-valued regular language and the former three definitions (corresponding to (2), (3) and (4)) are equivalent. Then the problem in [20] has been solved. Based on the equivalent characterization of L-valued regular languages given above, we further consider some algebraic operations on the family of L-valued regular languages. It is well known that the family of regular languages is closed under regular substitution, homomorphism and its inverse images . Correspondingly, we introduce the notions of L-valued regular substitution (LA-substitution), deterministic L-valued regular substitution (DLA-substitution), L-valued fuzzy homomorphism and its inverse images, homomorphism and its inverse images. Then we consider the issue whether LA-languages are still LA-languages under these algebraic operations. Some results presented in [1] are also generalized.

The reader is also referred to [3], [5], [6], [15], [16], [23] as a useful reference material in the context of this study.

Section snippets

Preliminaries

We first introduce some basic concepts to be used within this paper.

Definition 2.1

[11], [13]

Given a lattice L, we use ∨ and ∧ to represent the supremum operation and infimum operation on L, respectively, with 0, 1 being the least and the largest element. Assume that there is a binary operation • (we call it multiplication) on L such that (L, •, e) is a monoid with identity e  L. We call L a lattice-ordered monoid (some modification of the notion of lattice-ordered monoid in [4]) if it satisfies the following two

Algebraic definition of LA-languages

Definition 3.1

For ant u  Σ, we define fu  F(Σ*) as followsfu(θ)=eifθ=u,0otherwise,for any θ  Σ* and I = fΛ, then fu is called a basic LA-language on Σ. Let E = {fu : u  Σ}∪{fΛ, ∅}.

Definition 3.2

Consider that a family F  F(Σ*) of L-valued languages satisfies the following conditions

  • (1)

    a  L, f  Fa f  F and f a  F (scalar operation), where af, fa is defined as following, respectively(af)(θ)=af(θ)and(fa)(θ)=f(θ)aforanyθΣ,

  • (2)

    f1, f2  Ff1  f2  F (union operation), where(f1f2)(θ)=f1(θ)f2(θ),θΣ,

  • (3)

    f1, f2  Ff1 f2  F (concatenation operation), where(f1f

The properties of LA-languages under some algebraic operations

Following Definition 3.1, Definition 3.3 and Theorem 3.1, we obtain the following corollary.

Corollary 4.1

[11], [13], [12]

The family of LA-languages is closed under the operations of scalar, union, concatenation and the Kleene closure. That is to say, if f and g are two LA-languages on Σ and a  L, then af, fa, f  g, fg and f* are also LA-languages.

Based on Definition 3.5, Definition 3.6, Definition 3.7, Theorem 2.3, Theorem 3.2 (see also Example 2.2), we have

Corollary 4.2

([11], [13], [12])

The family of DLA-languages is closed under the operations of

Conclusions

In this paper, we have introduced seven algebraic operations on the family of fuzzy languages: homomorphism, ELA-substitution, L-fuzzy homomorphism, DLA-substitution, LA-substitution, the inverse image of L-fuzzy homomorphism, the inverse image of homomorphism. Among these concepts, we know that homomorphism is not only a special instance of ELA-substitution but also a special instance of L-fuzzy homomorphism, ELA-substitution and L-fuzzy homomorphism are special instances of DLA-substitution,

Acknowledgements

The authors would like to thank the anonymous referees and Prof. Witold Pedrycz for their careful review of this paper and a number of valuable comments which improve the quality of this submission.

References (23)

  • J.E. Hopcroft et al.

    Introduction to Automata Theory, Languages and Computation

    (1979)
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    This work is supported by National Science Foundation of China (Grant No. 10571112, 60174016), “TRAPOYT” of China and 973 Program of China No. 2002CB312200.

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