On closure properties of -valued linear languages
Introduction
With the passing of time, some generalizations for formal languages have been proposed. They introduce new features, such as: Randomness (probabilistic languages [1], [2], [3]) and uncertainty and imprecision (fuzzy languages [4], [5], [6], [7], [8], [9]).
Fuzzy languages is a well-developed theory. Indeed, Lee and Zadeh [5] introduced the 4 types of fuzzy grammars and their respective languages. These classes are fuzzy recursively enumerable [7], [10], fuzzy context-sensitive [5], fuzzy context-free [5], [8], [9] and fuzzy regular languages [10], [11], [12], [13], [14]. Moreover, in [12] the class of fuzzy linear languages was studied, therefore all class of languages in extended Chomsky hierarchy (Fig. 1) has a fuzzy version.
In [15], [16] Daowen Qiu introduces the automata theory based on complete residuated lattice-valued logic and in [17], he presents a version of Pumping lemma for such automata. The notion of equivalence between automata is presented in [18], whereas reduction and minimization of Mealy machines in [19]. Further works provide for this setting notions of: Context-free grammar [20], pushdown automata [21], [22] and Turing machines [23]. Thus, regular (), context-free (), context-sensitive () and recursively enumerable classes () of languages are extended to the complete residuated lattice-valued context.
Other works propose further algebraic structures in contrast to residuated lattices in order to generalize the theory of formal languages. For example, the study of Malik, Mordeson, and Sen uses Gödel structures [24], in [25], [26] Li and Pedrycz adopted the structure of lattice-ordered monoid and in a distributive lattice, and Ignjatović et al. using arbitrary partially ordered set [27], Bozapalidis and Louscou-Bozapalidou [28], [29] uses finite monoids.
From the point of view of languages the theory established by Qiu in [15], [16], can be called theory of languages based on complete lattice-valued logic, or simply theory of languages -valued. In this paper we extend the notion of linear languages and linear grammars to -valued grammars and also propose some operations on the resulting languages, and based on [7], [9] we present some closure properties.
Section snippets
Preliminaries
This section provides the required concepts to read this paper.
-valued languages and operations
A -valued language on Σ is an -valued set on , i.e. it is a set of the form: , the class of all -valued languages on an alphabet Σ is denoted by . Some usual operations on such languages are defined bellow:
Definition 18 Let be -valued languages on Σ. Then, -valued union: . -valued intersection: . -valued concatenation: such that: the -valued reverse
-valued linear grammars
Fuzzy extensions for linear grammars were studied in [12], [41]. In what follows we propose -valued extensions. At the end of this session we provide a characterization for -valued linear languages using -valued automata. In the sequel consider the set, , of all words produced by a linear grammar: .
Definition 21 Let be a complete residuated lattice. A structure is an -valued linear grammar (-LG), if are those introduced in Definition 15 and is an
Some closure properties of -valued linear languages
In this section, inspired by works [7], [9], [12], we will investigate some closure properties for the class of -valued linear languages. Here we adopt the same viewpoint of [7], i.e., we are interested to realize the -valued operations on the class -LinL. We start with the operation of union.
Theorem 6 The class -LinL is closed under the -valued union.
Proof Suppose that and are -valued linear languages, thus there exist and such that and , without
Final remarks
In his original papers Qiu [15], [16] established the fundamental framework based on machines for the theory of -valued languages, after that many further works were proposed [17], [18], [20], [23]. In general models of machines (or automata) are used to describe the -valued languages, in this paper we extended the class of linear languages using an extension -valued of linear grammars to describe languages.
Based [5], [8] we have introduced some operations on the class of -valued languages
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This study was funded by CNPq - National Council for Scientific and Technological Development (Brazil) within the project 312053/2018-5 and 307781/2016-0, and also by CAPES-Print 88887.363001/2019-00
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