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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 825249, 16 pages
http://dx.doi.org/10.1155/2013/825249
Research Article

On Intuitionistic Fuzzy Context-Free Languages

1College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
2College of Sciences, Southwest Petroleum University, Chengdu 610500, China

Received 19 October 2012; Revised 15 December 2012; Accepted 15 December 2012

Academic Editor: Hak-Keung Lam

Copyright © 2013 Jianhua Jin et al. 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

Taking intuitionistic fuzzy sets as the structures of truth values, we propose the notions of intuitionistic fuzzy context-free grammars (IFCFGs, for short) and pushdown automata with final states (IFPDAs). Then we investigate algebraic characterization of intuitionistic fuzzy recognizable languages including decomposition form and representation theorem. By introducing the generalized subset construction method, we show that IFPDAs are equivalent to their simple form, called intuitionistic fuzzy simple pushdown automata (IF-SPDAs), and then prove that intuitionistic fuzzy recognizable step functions are the same as those accepted by IFPDAs. It follows that intuitionistic fuzzy pushdown automata with empty stack and IFPDAs are equivalent by classical automata theory. Additionally, we introduce the concepts of Chomsky normal form grammar (IFCNF) and Greibach normal form grammar (IFGNF) based on intuitionistic fuzzy sets. The results of our study indicate that intuitionistic fuzzy context-free languages generated by IFCFGs are equivalent to those generated by IFGNFs and IFCNFs, respectively, and they are also equivalent to intuitionistic fuzzy recognizable step functions. Then some operations on the family of intuitionistic fuzzy context-free languages are discussed. Finally, pumping lemma for intuitionistic fuzzy context-free languages is investigated.

1. Introduction

Intuitionistic fuzzy set (IFS) introduced by Atanassov [13], which emerges from the simultaneous consideration of the degrees of membership and nonmembership with a degree of hesitancy, has been found to be highly useful in dealing with problems with vagueness and uncertainty. The notion of vague set, proposed by Gau and Buehrer [4], is another generalization of fuzzy sets. However, Burillo and Bustince [5] showed that it is an equivalence of the IFS and studied intuitionistic fuzzy relations. Recently, IFS theory has supported a wealth of important applications in many fields such as fuzzy multiple attribute decision making, fuzzy pattern recognition, medical diagnosis, fuzzy control, and fuzzy optimization [610].

In classical theoretical computer science, it is well known that formal languages are very useful in the description of natural languages and programming languages. But they are not powerful in the processing of human languages. For this, Lee and Zadeh [11] introduced the notion of fuzzy languages and gave some characterizations, where fuzzy languages took values in the unit interval . Malik and Mordeson [1214] studied algebraic properties of fuzzy languages. They stated that fuzzy regular languages can be characterized by fuzzy finite automata, fuzzy regular expressions, and fuzzy regular grammars. Meanwhile, as one of the generators of fuzzy languages, fuzzy automata have been used to solve meaningful issues such as the model of computing with words [15], clinical monitoring [16], neural networks [17], and pattern recognition [18]. Also, fuzzy grammars, automata, and languages tend to the improvement of lexical analysis and simulating fuzzy discrete event dynamical systems and hybrid systems [14, 19].

As is well known, quantum logic was proved by Birkhoff and Von Neumann as a logic of quantum mechanics and is currently understood as a logic with truth values taken from an orthomodular lattice. To study quantum computation, Ying [20, 21] first proposed automata theory based on quantum logic where quantum automata are defined to be orthomodular lattice-valued generalization of classical automata. More systematic exposition of this theory appeared in [22, 23]. Moore and Crutchfield [24] defined quantum version of pushdown automata and regular and context-free grammars. He showed that the corresponding languages generated by quantum grammars and recognized by quantum automata have satisfactory properties in analogy to their classical counterparts. A basic framework of grammar theory on quantum logic was established by Cheng and Wang [25]. They proved that the set of lattice-valued quantum regular languages generated by lattice-valued quantum regular grammars coincides with that of lattice-valued quantum languages recognized by lattice-valued quantum automata. Then some algebraic properties of automata based on quantum logic were discussed by Qiu [26, 27]. To enhance the processing ability of fuzzy automata, the membership grades were extended to many general algebraic structures. For example, by combining the ideas in [2023] and the idea in Ying's another work on topology based on residuated lattice-valued logic [28], Qiu has primarily established automata theory based on complete residuated lattice-valued logic [2931]. And Li and Pedrycz [32] studied automata theory with membership values in lattice-ordered monoids. They showed that lattice-valued finite automata have more power to recognize fuzzy languages than that of classical fuzzy finite automata. Recently, Li [33] studied automata theory with membership values in lattices, introduced the technique of extended subset construction to prove the equivalence between lattice-valued finite automata and lattice-valued deterministic finite automata, and then presented a minimization algorithm of lattice-valued deterministic finite automata. On the basis of breadth-first and depth-first ways, Jin and Li [34] established a fundamental framework of fuzzy grammars based on lattices, which provided a necessary tool for the analysis of fuzzy automata.

Fuzzy context-free languages, more powerful than fuzzy regular languages, have also been studied and can be characterized by fuzzy pushdown automata with two distinct ways and fuzzy context-free grammars, respectively [14, 35]. As a continuation of the work in [2931], a fundamental framework of fuzzy pushdown automata theory based on complete residuated lattice-valued logic has been established in recent years by Xing et al. [36], and the work generalizes the previous fuzzy automata theory systematically studied by Mordeson and Malik to some extent. The pumping lemma for fuzzy context-free grammar theory in this setting was also investigated by Xing and Qiu [37].

Using the notions of IFSs and fuzzy finite automata, Jun [38, 39] presented the concept of intuitionistic fuzzy finite state machines as a generalization of fuzzy finite state machines, and Zhang and Li [40] discussed intuitionistic fuzzy recognizers, intuitionistic fuzzy finite automata, and intuitionistic fuzzy language. They showed that the languages recognized by intuitionistic fuzzy recognizers are regular, and the intuitionistic fuzzy languages recognized by the intuitionistic fuzzy finite automata and the intuitionistic fuzzy languages recognized by deterministic intuitionistic fuzzy finite automata are equivalent. Recently Chen et al. [41] utilized the intuitionistic fuzzy automata to deal with consumers' advertising involvement when considering the expression of an IFS characterized by a pair of membership degree and nonmembership degree is similar to human thinking logic with pros and cons. Due to pushdown automata being another kind of important computational models [15] and also motivated by the importance of grammars, languages and models theory [14], it stands to reason that we ought consider the notions of intuitionistic fuzzy pushdown automata, intuitionistic fuzzy context-free grammars, and fuzzy context-free languages because our discussion in this paper will provide a fundamental framework for studying intuitionistic fuzzy set theory on fuzzy pushdown automata and generators as well. How to characterize intuitionistic fuzzy context-free languages and its pumping lemma in this setting becomes open problems; however, there is no research on the algebraic characterization of intuitionistic fuzzy context-free languages. We will try to solve the problems in this paper. Additionally, some examples are given to illustrate the significance of the results. In particular, Example 35 presented in this paper will show that intuitionistic fuzzy pushdown automata have more power than fuzzy pushdown automata when comparing two distinct strings although the degrees of membership of these strings recognized by the underlying fuzzy pushdown automata are equal. Investigating intuitionistic fuzzy context-free languages will reduce the gap between the precision of formal languages and the imprecision of human languages.

The remaining parts of the paper are arranged as follows. Section 2 describes some basic concepts of IFSs. Section 3 gives the definitions of intuitionistic fuzzy pushdown automata with two distinct ways and their languages. It is investigated that, for any intuitionistic fuzzy pushdown automaton with final states (IFPDA, for short), there is a cover, which consists of a collection of classical pushdown automata, equivalent to the IFPDA. By introducing intuitionistic fuzzy recognizable step functions, it is shown that intuitionistic fuzzy pushdown automata with final states and empty stack are intuitionistic fuzzy recognizable step functions, respectively, and conversely any intuitionistic fuzzy recognizable step function can be recognized by an intuitionistic fuzzy pushdown automaton with final states or empty stack. It follows that intuitionistic fuzzy pushdown automata with final states and empty stack are equivalent. Section 4 studies intuitionistic fuzzy context-free grammars (IFCFGs) as a type of generator of intuitionistic fuzzy context-free languages (IFCFLs). The notions of intuitionistic fuzzy Chomsky normal form (IFCNF) and Greibach normal form (IFGNF) are proposed. The results of our study indicate that IFCFLs generated by IFCFGs are equivalent to those generated by IFGNFs and IFCNFs, respectively, and they are also equivalent to intuitionistic fuzzy recognizable step functions. The algebraic properties of IFCFLs are also discussed. Section 5 establishes pumping lemma for IFCFLs. Some examples are then given to illustrate the application of pumping lemma and the significance of IFCFLs. Finally, conclusions and directions for future work are presented in Section 6.

2. Basic Concepts

Definition 1 (see [40]). An intuitionistic fuzzy set in a nonempty set is an object having the form: where the functions and denote the degree of membership (i.e., ) and the degree of nonmembership ( ) of each element to the set , respectively, and the two quantities satisfy the following inequalities: For the sake of simplicity, we use the notation instead of . An intuitionistic fuzzy set will be abbreviated as an IFS.
Let be a family of IFSs in . Then the infimum and supremum operations of IFSs are defined as follows: where and denote supremum and infimum of real numbers in , respectively.
For two IFSs and , we say if and . In addition, if the IFS in satisfies the condition that, for any , , then reduces to a fuzzy set in . The difference between intuitionistic fuzzy sets and fuzzy sets is whether the sum of the degrees of membership and nonmembership of an element to a set equals one.
An IFR in is an intuitionistic fuzzy subset of ; that is, it is an expression given by where the mappings and satisfy An IFBR over is an IFS of . Let and be IFRs in and , respectively. Define the composition of IFRs, in , by for all .
Furthermore, if is an IFBR over , then its reflexive and transitive closure is , where , and , that is, for all .

Definition 2. Let be an IFS in . Then the image set of , denoted as , is given as where and .
For any , , the -cut set of is defined as And the support set of , denoted as supp , is defined by If supp is finite, then is called a finite IFS in .

3. Intuitionistic Fuzzy Pushdown Automata

It is well known that any language accepted by a pushdown automaton with final states can be accepted by a certain pushdown automaton with empty stack, and vice versa. As a natural generalization of pushdown automata, we give the notions of intuitionistic fuzzy pushdown automata with final states and empty stack, respectively, and then do research in the algebraic characterization of their intuitionistic fuzzy recognizable languages including decomposition form and representation theorem. Note that is the free monoid generated from the set with the operator of concatenation, where the empty string is identified with the identity of . And the length of the string is denoted by . .

Definition 3. An intuitionistic fuzzy pushdown automaton with final states (IFPDA, for short) is a seven tuple , where(i) is a finite nonempty set of states;(ii) is a finite nonempty set of input symbols;(iii) is a finite nonempty set of stack symbols;(iv) is a finite IFS in ;(v) is called the start stack symbol;(vi) and are intuitionistic fuzzy subsets in , which are called the intuitionistic fuzzy subsets of initial and final states, respectively.

Definition 4. An intuitionistic fuzzy pushdown automaton with empty stack (IFPD , for short) is a seven tuple , where and are the same as those in IFPDA , and represents an empty set.

Definition 5. Let be an IFPDA. Define an IFBR on , in the form of for any . Here, for any nonempty string , , , , and . is the reflexive and transitive closure of .
If no confusion, we denote and instead of and , respectively.

Definition 6. Let be an IFPDA. Then we call an intuitionistic fuzzy language accepted by with final states, where , , and are functions from to the unit interval , and = , , = , for any .

Definition 7. Let be an IFPD . Then we call an intuitionistic fuzzy language accepted by with empty stack, where , and are functions from to the unit interval , and = , , , = for any .

Lemma 8 (see [33]). Let be a lattice and a finite subset of . Then the -semilattice of generated by , written as , is finite, and the -semilattice of generated by , denoted as , is also finite, where , and .

Proposition 9. If can be accepted by some IFPDA , then is an IFS in , and the image set of is finite.

Proof. We have the following.
Claim 1 ( is an IFS in ).
It suffices to show that , for any , , .
Clearly, , = , , , , , , , , and , = , , , , , , , .
On the one hand, ; on the other hand, there exists a sequence , such that , , , , , ) , , . Hence , , , , , , , .
Therefore, , , , , , .
Claim 2 ( is finite).
In fact, let and . Then , and are finite sets by Lemma 8. Since is a finite IFS, for any , , , there exists a natural number such that , , , , , = , where , ; . By Lemma 8, is also finite. Since for any , . Hence is a finite subset of .
Similarly, it follows that is also a finite subset of .
Therefore, is finite.
If can be accepted by some IFPD , then, by Definition 7, for any , , , we have , = , , , , and , , , , , , .
In a similar manner, it is concluded that the following must be true.

Proposition 10. If can be accepted by some IFPD , then is an IFS in , and the image set of is finite.

Specially, the IFPDA will be abbreviated as , whenever and supp . Moreover, if and supp has only one element, then the IFPDA is a classical PDA.

For two IFPDAs and , we say that they are equivalent if they accept the same intuitionistic fuzzy language.

Proposition 11. Let be an IFS in a nonempty set . Then the following statements are equivalent:(i) can be accepted by an IFPDA ;(ii) can be accepted by a certain IFPDA , where .

Proof. (i) implies (ii). Construct an IFPDA as follows: , , where , . Define an IFS in by
Define an IFS in by
Define an IFS in by mappings , , , , , where , , , ; otherwise, and .
Then for any , , we have , , , , , , , , and , , , , , , , , , , , , , . Therefore, .
From the construction, it is clearly that can be denoted as .
(ii) implies (i). Suppose the IFS is accepted by the IFPDA . Then we construct an IFS in by
It follows that the IFPDA accepts .
Similarly, it is easily concluded that the following must be true.

Proposition 12. Let be an IFS in a nonempty set . Then the following statements are equivalent:(i) can be accepted by an IFPD ;(ii)There exists an IFPD recognizing , where .

There is especially a simple type of intuitionistic fuzzy pushdown automata, which is called intuitionistic fuzzy simple pushdown automata. The definition is given as follows.

Definition 13. An IFPDA is called an intuitionistic fuzzy simple pushdown automaton (IFSPDA) if the image set of is contained in the set .

Next any IFPDA is proven to be an equivalence of a certain IFSPDA by utilizing the generalized subset construction method. Noting that an IFS requires that the sum of the degrees of membership and nonmembership of an element to a set is no more than the natural number 1. So the proof technique is to some extent different from the technique of extended subset construction introduced by Li in [33], and it is not an easy task to conduct reasoning in the realm of the modified techniques.

Proposition 14. Let be an IFPDA. Then there exists an IFSPDA such that .

Proof. Let be an IFPDA. Then we construct an IFSPDA   as follows:(i) , where , , and ;(ii) ;(iii) is an IFS in , where the mappings are given as follows. For any , , and , , and whenever there exist and such that , , and . Otherwise, and ;(iv) is an IFS in . For any ,
Now, it is claimed that, for any , , and for any , , and whenever the following condition is satisfied.
(P1) There exist , and such that , , , and , , , . Otherwise, and .
It is proved by induction. In fact, if , then , and . Hence and .
Suppose the result still holds whenever , . If , and , then and .
Next, for any , , whenever (P1) is satisfied; that is, there exists a sequence of states , , such that , = , , = , , , where , .
Let , , . Then , = , , , , , , and , ,where , , , .
By assumption, , ,   , , and so , , , , .
Therefore, , , , , , , , .
, , = , , , , .
For any , , if (P1) is not satisfied, then it follows that , .
Hence, for any , , , we have , , , = , , , , , , , , , , ,    ,   , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .
Therefore, .

Clearly, an IFPDA is a generalization of a classical pushdown automaton (PDA). Next, it will be shown that any IFPDA can be characterized by a collection of pushdown automata. To describe the behavior of a pushdown automaton , we need to introduce the concept of instantaneous description. An instantaneous description is a three-tuple , which means that the automaton is in the state and has unexpended input and stack contents . An instantaneous description represents the configuration of a pushdown automaton at a given instant. To introduce the transition in a pushdown automaton in terms of instantaneous descriptions, we define as a binary relation on . We say if contains , where , , , , and . Furthermore, we define as the reflexive and transitive closure of . Then the language accepted by with final states is defined as

Definition 15. A collection of classical pushdown automata with final states is called a cover if the following conditions hold:(i) and imply ;(ii) .
For a cover , its recognized intuitionistic fuzzy language in is given by accepts , accepts , for all .

Theorem 16. Let be an IFS in . Then can be accepted by an IFPDA if and only if can be recognized by a cover .

Proof. If can be accepted by an IFPDA, then there exists an IFSPDA such that accepts by Proposition 14. Next we construct a cover where ; the mapping is given by , for all .
Clearly, the cover is well defined.
Next, we will show that can be recognized by the cover . In fact, we have if and only if ,for all with , for all . accepts , accepts .
Therefore, .
Conversely, suppose can be recognized by a cover Then we construct an IFSPDA , where