Algebraic properties of LA-languages☆
Introduction
It is well known that the following approaches to represent a language (regular language, to be prise) are equivalent [5]:
- (1)
is recognized by deterministic finite automaton.
- (2)
is recognized by nondeterministic finite automaton.
- (3)
is described by regular expression.
- (4)
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 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[11], [13]
Algebraic definition of LA-languages
Definition 3.1 For ant u ∈ Σ, we define fu ∈ F(Σ*) as followsfor 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 ∀a ∈ L, f ∈ F⇒a f ∈ F and f a ∈ F (scalar operation), where af, fa is defined as following, respectively ∀f1, f2 ∈ F⇒f1 ∪ f2 ∈ F (union operation), where ∀f1, f2 ∈ F⇒f1 f2 ∈ F (concatenation operation), where
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 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.[11], [13], [12]
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 The family of DLA-languages is closed under the operations of([11], [13], [12])
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.
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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.