K-operators: An approach to the generation of interval-valued fuzzy implications from fuzzy implications and vice versa

Abstract

This paper introduces the interval-valued fuzzy implications generated from fuzzy implications and from K-operators showing that this construction generalizes the canonical representation of fuzzy implications. Such generalization allows to recover the usual fuzzy implication properties when dealing with interval-valued fuzzy implications. In addition, the condition under which the converse construction, i.e. the generation of fuzzy implications by action of K-operators and interval-valued fuzzy implications, is discussed. Moreover, we analysed their conjugate construction obtained by action of interval automorphisms. Through commutative diagrams, the associativity related to compositions of conjugate, dual and K-operators performed over fuzzy implications is shown, also preserving their main properties.

Introduction

Based on the Atanassov’s operator [1], also called K-operator, distinct ways to define transformations from interval-valued fuzzy sets (IvFSs) to fuzzy sets are introduced. By associating an index α in the unitary interval with the amplitude of an interval X, such operator is able to interpret the interval membership degree of an element in an Atanassov’s intuitionistic fuzzy set (IFS), see [2], [3].

According to [4, Def. 2 and Lemma 1], the collections of IFSs and IvFSs, respectively denoted by $\mathcal{C}(\text{IFSs})$ and $\mathcal{C}(\text{IvFSs})$, are closed related, since an IvFS is equivalent to an intuitionistic fuzzy set, and both are equivalent to L-fuzzy sets in the sense of Goguen [21]. Thus, an analogous methodology based on K-operators can provide similar interpretation of an interval-valued Atanassov’s intuitionistic fuzzy set (IvIFS).

Following the work of Bustince et al. [11], in which to construct interval-valued fuzzy connectives, such as t-norms, t-conorms and fuzzy negations, it makes use of K-operators, this work considers the construction of interval-valued fuzzy implications (IvFIs). We investigate the class of interval-valued fuzzy implications which can be obtained in the same methodology, making use of K-operators and fuzzy implications, in such way that dual and conjugate operators are preserved by such construction. In addition, the converse construction is also considered, meaning that fuzzy implications can be expressed by interval-valued implications and K-operators, preserving the main properties of fuzzy implications.

Preliminary results of this study were published in [32], stating that IvFIs can be obtained from K_α-operators acting on a pair (I_a, I_b) of fuzzy implications. The resulting expression, named (I_a, I_b, K_α, K_β)-implication, may be intuitively interpreted in terms of interval amplitudes of such fuzzy connectives. It also supports distinct K-operator definitions and contributes to extend main properties of fuzzy connectives to their interval approach. Such construction generalizes the canonical representation of fuzzy implications. In addition, we also analysed their interval conjugate construction, defined by the action of interval automorphisms preserving their main properties.

Towards to the converse construction, the new results of our work are concentrated in making use of K-operators and IvFIs to generate fuzzy implications. In particular, such converse construction is described, by taking into account K-operators acting on IvFIs, which preserves degenerate intervals. The conditions under which the main properties of FIs are preserved in terms of Interval-valued Fuzzy Logic (IvFL) are described in this paper.

In addition, based on the concept of interval amplitudes, the dual expression of an K-operator enables the study of dual constructions of IvFIs, in the class of interval-valued fuzzy coimplications (IvFCs). A coimplication named (I_aN, I_bN, K_α, K_β)-coimplication is introduced from K_α-operators acting on a pair (I_aN, I_bN) of N-dual implications, as the N-dual construction of an (I_a, I_b, K_α, K_β)-implication.

In addition, through the commutative diagram presented, it makes it easy to understand how the associative property related to compositions of conjugate, dual and K-operators performed over fuzzy implications, preserves and guarantees that their main properties are verified.

Intervals have been used to model the uncertainty of a specialist’s information related to truth values in the fuzzy propositional calculus in IvFL.

As it is well known, the study in the strict sense of IvFIs and similarity IvFCs are two important issues as theoretical foundation, underlying the basis of if_then rules in intelligent systems based on IvFL. Such research area has been widely applied to pattern recognition, cluster analysis, image processing and decision making, see [24], [13], [35].

IvFIs could be used to extend the use of fuzzy implications in, for instance, neuro-fuzzy systems, in fuzzy systems control and making-decision, as one can observe in [31], [37]. See also [38], in which multi-criteria fuzzy systems use gradual rules and fuzzy arithmetic to treat pieces of information that may involve imprecision/vagueness by making use of residuated implications.

In [30], the authors examine some properties of interval-valued fuzzy relations in the context of Atanassov’s operators and decomposable operations over IvFSs. See also additional references with recent applications of Atanassov’s operator in IvFL and IvFI [41], [40], [12], [25].

Recently, many works with significant results have considered the concept of amplitudes based on K-operators to deal with fuzzy connectives on interval lattices. In [17], the Atanassov’s operator was considered together with representable Atanassov’s Intuitionistic De Morgan triples in order to generate new De Morgan triples, providing an extension of the notions of De Morgan triples and automorphisms on unitary interval for IFSs.

In [8], a class of fuzzy multisets with a fixed number of memberships is studied, also making use of a K-operator. An interesting study of generalized Atanassov’s operators is presented in [29].

In [15], the Atanassov’s operator acting on intervals, produces results that recover ordered weighted averaging (OWA) aggregation operators from K-operators. In addition, this work introduces the generalized Atanassov’s operators that, in particular, includes OWA operators.

This paper extends the work presented in [11] obtaining interval-valued fuzzy connectives (t-norms, t-conorms and negations) starting from a K-operators. Properties as the law of contradiction and the law of excluded middle, including idempotency, absorption, and distributiveness are discussed. It is shown that some of such connectives preserve, in certain conditions, the amplitude (or Atanassov’s intuitionistic index) of the intervals on which they act. This paper follows the same construction proposed by Bustince et al. in [11], focus on IvFIs.

The structure of the paper is as follows. In Section 2, a brief review of interval representations of fuzzy connective is presented, focusing on interval-valued fuzzy negation and concerned with the duality principle. Main concepts of K-operators and interesting results about representable interval-valued t-norms and t-conorms obtained by using K-operators, based on [11], conclude this revision.

Starting with definitions of fuzzy implications and their interval version, in Section 3 interval extended properties of fuzzy implications are considered in order to show two main contributions of this paper:

• generation of IvFI from K-operators and a pair of FIs, by introducing the ${\mathbb{I}}_{(I_a\text{,}{I}_b)}$-implication; in addition, it is also discussed under which conditions such expression of fuzzy implications preserves degenerate intervals, duality principle and main properties of fuzzy implications;

• generation of FIs from K-operators and an IvFI $\mathbb{I}$, by introducing the $(\mathbb{I}\text{,}K)$-implication. In this context, a study related to main properties of IvFI which are preserved by a ${\mathcal{I}}_{(\mathbb{I}\text{,}K)}$-operator was also developed.

Thus, the above study underlies an interval approach for automorphisms, enabling generation of new classes of IvFIs which can be expressed by K-operators and FIs, which are discussed in Section 6. The action of an automorphism on a K-operator generating another K-operator is also presented. As another main contribution, the representation of the conjugate of an interval-valued ${\mathbb{I}}_{(I_a\text{,}{I}_b)}$-implication, is defined as a composition between the conjugate of a K-operator and the pair of IFs $I_a^{\rho }$ and $I_b^{\rho }$, i.e., the corresponding conjugate-implications of I_a and I_b. In the reverse construction, we also show that automorphisms preserve the $(\mathbb{I}\text{,}K)$-operator, which means, the conjugate of an $(\mathbb{I}\text{,}K)$-implication is also an $(\mathbb{I}\text{,}K)$-implication.

And finally, the last section provides some conclusions and further work.