Interval-valued intuitionistic fuzzy implications – Construction, properties and representability
Introduction
Since [33], [54], [63], [64], such and many other extensions of Fuzzy Logic (FL) have been studied (see also [30], [44]) by providing logical interpretation of information obtained by fuzzy connectives and by reducing the uncertainty and the imprecision from the membership grades.
This work focuses on the integration of two mathematically equivalent but semantically different approaches: (i) Interval-Valued Fuzzy Logic (IVFL), dealing with representations in terms of fuzzy connectives on U = [0, 1] and (ii) Intuitionistic Fuzzy Logic (IFL), here based on Atanassov’s intuitionistic fuzzy sets introduced in [2], [6], which were defined as a pair of membership functions by relaxing the complementary operation stated over fuzzy sets in FL.
In the former, we obtained the narrowest interval containing the range of such fuzzy connectives together with the correctness of interval computations based on such interval-valued fuzzy connectives, as pointed out by Hickey et al. in [34]. Moreover, the measure of vagueness and uncertainty is reflected in the diameter of an interval membership degree. On the other hand, the latter considers not only to what degree an element belongs to a particular set defined by an Atanassov’s intuitionistic fuzzy connective but also to what degree this element does not belong to the corresponding set. Thus, the hesitation related to the dual construction is modelled by an inequality, i.e., the difference between an interval membership degree and its interval non-membership degree is less or equal to the degenerate interval [1, 1].
As proposed by Atanassov and Gargov [7], in 1989, interval-valued Atanassov’s intuitionistic fuzzy sets (IVIFSs) extend the Atanassov’s intuitionistic fuzzy sets (IFSs) by considering a membership function and a non-membership function whose values, which are not necessarily complementary, can be given as intervals rather than real numbers. Based on the Atanassov’s IVIFSs, we consider the Interval-valued Atanassov’s Intuitionistic Fuzzy Logic (IVIFL) integrating concepts from IVFL and IFL as representability of aggregation and dual Atanassov’s intuitionistic fuzzy connectives.
In such logical context, following previous works, see [13], [52], this paper considers the canonical representation [55] and idempotent aggregation class, in order to map Atanassov’s intuitionistic fuzzy connective class into interval-valued Atanassov’s intuitionistic fuzzy connective class. Thus, as the main contribution, the paper introduces the class of interval-valued Atanassov’s intuitionistic fuzzy implications (IVI-implications) obtained by representable idempotent aggregation functions, interval-valued implications (IV-implications) and their dual constructions, the interval-valued fuzzy coimplications (IV-coimplications). By applying projection functions and considering the sets of diagonal elements and degenerate intervals the main properties of fuzzy implications are preserved by IVI-implications. Additionally, the properties from the Atanassov’s intuitionistic approach related to Atanassov’s intuitionistic index are also investigated.
Our motivation is to contribute with the study of IVIFIs as foundations to inference systems in the context of IVIFL [5]. In [3], [5], [49], different operators are defined over the interval valued Atanassov’s intuitionistic fuzzy sets and their basic algebraic properties are studied. Such research area has received much attention from theoretical [4], [38], [39], [47] and applied approach in the fields of multi-criteria decision [20], [35], [61], [67], group decision [48], [60], [65], computational game theory [38] and aggregation functions [47], [46], [62], modelling of fuzzy-valued information systems [36], [62], [65], [66].
In order to consider different models to the membership degree and to the non-membership degree in systems based on IVFL, theoretical and applied research are frequently based on aggregation functions, which play an important role in statistics, regression analysis, pattern recognition, decision sciences and image processing, mainly reported in [58]. So, several aggregation function classes have been studied in detail in the fuzzy literature [19].
In [16], an Atanassov’s intuitionistic fuzzy implication (IFI) operator recovers J. Fodor’s definition [31] of a fuzzy implication operator when the sets are fuzzy, generalizing the Atanassov’s operator [4]. Additionally, by [21], [22], [23], the definition of fuzzy connectives are undertaken by such theories. In [27], a new implication class obtained by binary aggregation operators based on t-norms on the unit interval is introduced. In [29], aggregation functions integrating concepts of interval-valued fuzzy and Atanassov’s intuitionistic fuzzy set theory were discussed. In [19], generalized Atanassov’s operators are studied, including the OWA aggregation operators. Recently, in [15], the media operator is defined by aggregation functions whose values depend on central inputs.
Following such approaches, we make use of interval-valued idempotent aggregation functions in order to define representable (co)implications and representable Atanassov’s intuitionistic fuzzy implications, extending the work proposed in [16]. Such extension may contribute to interpret the truth and non-truth of conditional rules in inference systems based on IVFL and IVIFL, respectively.
The paper presents an N-dual structure of an representable implication generated from a set M of aggregation functions and mutual dual fuzzy implications I and IN, based on the isomorphism between and by mapping the notions presented in [16, Definition 3] to the canonical representation [14]. We discuss under which conditions the representable implications preserve the main properties of interval implications, showing the applicability of N-dual interval approach for representable implications.
Another contribution provides two equivalent characterizations of an IVI-implication . On one hand, -representable Atanassov’s intuitionistic implication which is generated by a set of representable idempotent IV-aggregation functions and a pair of an IV-implication and its corresponding dual IV-coimplication. It also discusses some properties of -representable IVI implications. On the other hand, IVI-implications generated by -representable IV-implications. Thus, the conditions under which pairs of degenerate intervals are preserved by -representable IVI-implications are also studied.
The paper is organized as follows. Section 2 contains preliminary definitions, discussing the notions of interval representations of real functions and presenting definitions and several results concerned with fuzzy negations, duality relationship and aggregation operators. The basic concepts of fuzzy (co)implications are studied in Section 3, including main properties and some examples.
Section 4 reports definitions of Atanassov’s intuitionistic fuzzy connectives, focusing on the study of Atanassov’s intuitionistic fuzzy implications generated by aggregation functions and a pair of an implication and its corresponding dual coimplication, obtained by a strong fuzzy negation, as conceived in [16]. Interval-valued (co)implications are studied in Section 5, where firstly, some results related to the interval extensions of fuzzy (co)implications complement previous works, see e.g. [12], [13], [14], extending their main properties from FL approach.
Thus, in Section 6, this paper presents a study of IV-(co)implications generated by IV-aggregation functions and by mutual dual fuzzy (co)implications, proving that several analogous properties of fuzzy (co)implications are also held for such representable (co)implications.
In Section 7, interval-valued Atanassov’s intuitionistic fuzzy implications generated by IV-aggregation functions and a pair of an IV-implication and corresponding IV-coimplication are discussed. In addition, such construction can also be obtained by representable implications.
We conclude by pointing out the main results of this paper, the ongoing work and some final remarks.
Section snippets
Interval representations
Consider the real unit interval and let be the set of subintervals of U, that is, . The interval set has two projections , defined by l(X) = l([a, b]) = a and r(X) = r([a, b]) = b and denoted by X and , respectively.
In the following, we report the component-wise Kulisch–Miranker partial order (also called product order), defined by:
In addition, we also consider as the binary relation given by:which was based in the
Fuzzy (co)implications
Coimplication functions are conceived as N-dual structures of implication functions extending the classical (co)implications [9], [18], [31], [42]: Definition 3.1 The binary operator (J)I: [0, 1]2 → [0, 1] is a fuzzy (co)implication if it satisfies the boundary conditions:56, Section 3.1
Since Definition 3.1 imposes a very weak condition for a binary function to be a fuzzy (co)implication, several other extra properties are considered in the literature, in
Atanassov’s intuitionistic fuzzy (co)implications
A formal treatment of Atanassov’s intuitionistic fuzzy sets are studied in [6], [22], [23], [26], [31]. In this section, firstly, some basic concepts are introduced in order to study Atanassov’s intuitionistic fuzzy implications generated by aggregation functions, following the work of Bustince at. al. in [16].
Let be the set of all Atanassov’s intuitionistic fuzzy membership degrees. For all , it holds that:
- (i)
and ; and
- (ii)
Interval-valued (co)implications
Since real numbers may be identified with degenerate intervals in the context of interval mathematics, the boundary conditions which must be satisfied by the classical fuzzy implications can be naturally extended to interval fuzzy degrees, whenever degenerate intervals are considered, see e.g. [32]. The IV-(co)implications are considered, grounded by the canonical representation. Definition 5.1 The binary function is called an IV-(co)implication iff it satisfies the boundary conditions given by
Representable IV-(co)implications generated from aggregations and pairs of dual operators
An expression for IV-(co)implication generated from aggregation functions and mutual dual fuzzy (co)implications is studied in this section, including the presentation of constraints assuring that such interval functions are well defined and may extend the canonical representation in the sense of [14]. Such study extends the previous results presented in [50]. Definition 6.1 Let I be an implication, J be a coimplication, N be a fuzzy negation and be a set of binary idempotent aggregation functions. Then,
Interval-valued Atanassov’s intuitionistic fuzzy implications
Based on the work introduced in [6] and later in [22], this section briefly studies the interval-valued Atanassov’s intuitionistic fuzzy sets, their main concepts and properties. Thus, IVI-implications generated by -representable IV-implications are defined. In the following, IVI-implications generated by aggregation functions and pairs of mutual dual IV-implications are discussed, showing the conditions under which such constructions are equivalent.
Let
Conclusion and final remarks
Interval-valued Atanassov’s intuitionistic fuzzy sets have been studied as a natural generalization of Atanassov’s intuitionistic fuzzy sets modelling the uncertainty due to the lack of information and the hesitation in the definition of a membership and non-membership function. Dealing with representable fuzzy implication generated from aggregation and dual interval operators, the paper introduces a representable Atanassov’s intuitionistic fuzzy implication extending the definition introduced
Acknowledgments
This work is supported by CNPq (under the Process Numbers 480832/2011-0 and 307681/2012-2) and FAPERGS (under the Process Number 11/1520-1 of Edital PqG 02/2011).
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