Uninorm based residual implications satisfying the Modus Ponens property with respect to a uninorm

doi:10.1016/j.fss.2018.09.014

Fuzzy Sets and Systems

Volume 359, 15 March 2019, Pages 22-41

https://doi.org/10.1016/j.fss.2018.09.014 Get rights and content

Abstract

In any fuzzy rules based system the inference management is usually carried out by the so-called fuzzy implication functions. In this framework, the Modus Ponens property becomes essential to make forward inferences and it is well known that this inference rule is guaranteed when the conjunction and the implication function used in the process satisfy the corresponding functional inequality. Such inequality has been extensively studied for many kinds of implication functions when the conjunction is modelled by a t-norm. However, the use of conjunctive uninorms to model conjunctions is an increasingly widespread option in fuzzy systems and for this reason the study of the Modus Ponens with respect to a conjunctive uninorm U instead of a t-norm T, which we call here U-Modus Ponens or U-conditionality, becomes very important. In this paper this new property is deeply analyzed and it is shown that usual implications derived from t-norms and t-conorms do not satisfy it, but many solutions appear among those implications derived from uninorms. In particular, the U-Modus Ponens for the case of residual implications derived from uninorms, or RU-implications, is investigated in detail when the uninorm U lies in any of the four most usual classes of uninorms.

Introduction

The deductive reasoning of human thinking in forward inferences is usually carried out through the inference rule of the Modus Ponens, that looks like $P & (P \to Q) \equiv Q$ where P and Q are classical propositions. In the framework of fuzzy logic and trying to model this type of reasoning, the Zadeh's compositional rule of inference is usually considered to manage forward inferences. In this process, the Modus Ponens rule derives into the inequality: $T (x, I (x, y)) \leq y for all x, y \in [0, 1],$ where T is an operation modelling the fuzzy conjunction and I is an operation modelling the fuzzy conditional. The above functional inequality is known as the Modus Ponens property and also as the property of T-conditionality. Thus, due to its essential role in the fuzzy inference process, the inequality given in (1) has been extensively studied by many authors (see for instance, [2], [3], [23], [25], [35], [36], [37], [38]). It is worth to mention that the Modus Ponens property is also important in other applications such as Fuzzy Mathematical Morphology and Image Processing (see [14], [15]).

Usually, the fuzzy conjunction is modelled by a (continuous or at least left-continuous) t-norm T and the fuzzy conditional is modelled by a fuzzy implication function I. Thus, the main studies are related to implications derived from t-norms and t-conorms. Specifically, the cases of residual implications and $(S, N)$ -implications were investigated in detail in [2], [35], [36], and QL and D-implications in [37]. For all these cases see also Section 7.4 in [3] or also the chapter book [29] and the references therein.

However, it is becoming more and more common the use of uninorms instead of t-norms in fuzzy systems. In fact, although uninorms were firstly introduced in the framework of aggregation functions (see [13], [39]), they have been also used as logical operators due to the fact that they are always conjunctive or disjunctive. Considering their structure, uninorms can be viewed as generalizations of both t-norms and t-conorms and, in this framework, they have proved to be useful in many fields like fuzzy expert systems ([9]), neural networks ([5]), fuzzy mathematical morphology and image processing ([14], [15]), and also in fuzzy logic in general (see [30] and the references therein). In particular, uninorms have been used also to generate new classes of fuzzy implication functions, generalizing those derived from t-norms and t-conorms, that is, RU-implications, $(U, N)$ -implications (see for instance [1], [4], [8], [31], [32], [33]) and also QL and D-implications derived from uninorms (see [22]). Recently, T-conditionality was already studied for two of these kinds of implication functions: for RU-implications and $(U, N)$ -implications (see [21]).

Note that in this line, conjunctive uninorms are commonly considered as fuzzy conjunctions and consequently, the use of a conjunctive uninorm U instead a t-norm T in the Modus Ponens property appears in a natural way and becomes interesting for the mentioned applications. Taking into account these considerations, the idea of this paper is to investigate the generalization of the Modus Ponens obtained by substituting the t-norm T by a conjunctive uninorm U. We refer to this new property as the U-Modus Ponens property or as U-conditionality by obvious reasons. The study of U-conditionality is carried out for a uninorm U and a fuzzy implication function I in general. However, initial results prove that the considered uninorm needs to be conjunctive and the implication function can not be any of the usual implications derived from t-norms and t-conorms. This fact leads us to focus our investigation in the case of RU-implications. We extend our study to RU-implications derived from the four most usual classes of uninorms, that is, from uninorms in $U_{\min}$ , from idempotent uninorms, from representable uninorms, and from uninorms continuous in the open unit square. We will see that in all these cases we obtain many new solutions. Note that some initial ideas on this study were recently proposed without proofs by the same authors in [26], [27]. Thus, the current paper can be considered as an extended version of these two congress communications, that have been enlarged by including the proofs of all the results, the correction of some little mistakes, and the inclusion of new examples.

The paper is organized as follows. After this introduction, Section 2 is devoted to some preliminaries in order to make the paper as self-contained as possible. Section 3 deals with the Modus Ponens with respect to a uninorm U, including some general results for any kind of implication functions as well as some particular ones for the case of RU-implications. Then, the following sections give a detailed study of the U-conditionality for RU-implications derived from the four most usual classes of uninorms, that is, from uninorms in $U_{\min}$ (Section 4), from idempotent uninorms (Section 5), from representable uninorms (Section 6), and from uninorms continuous in the open unit square (Section 7). Finally, the paper ends with Section 8 devoted to some conclusions and future work.

Section snippets

Preliminaries

We will suppose the reader to be familiar with the theory of t-norms, t-conorms and fuzzy negations (all necessary results and notations can be found in [17]). We also suppose that some basic facts on uninorms are known (see for instance [13]) as well as their most usual classes, that is, uninorms in $U_{\min}$ and $U_{\max}$ ([13]) and idempotent uninorms ([7], [19], [34]). For more information on the studied classes of uninorms, see the recent survey in [20]. We recall here only some facts on

Modus Ponens with respect to a uninorm

As we have already commented, the main goal of the paper is to study the Modus Ponens in fuzzy systems whenever the conjunction is modelled by a conjunctive uninorm. That is, we want to generalize the Modus Ponens property by substituting the t-norm T by a conjunctive uninorm U. So, let us introduce first the definition in general.

Definition 6

Let I be a fuzzy implication function and U a uninorm. It is said that I satisfies the Modus Ponens property with respect to U, or that I satisfies the U-Modus Ponens

Case of RU-implications obtained from a uninorm in $U_{\min}$

In this section we want to deal with residual implications derived from uninorms in $U_{\min}$ . Thus, we will take a uninorm $U_{0} \in U_{\min}$ , $I_{U_{0}}$ its residual implication and we will study when $I_{U_{0}}$ is a U-conditional with respect to a conjunctive uninorm U. Let us recall first how are this kind of implications, that can be found for instance in [8] (see also [32] for the version recalled in the next proposition).

Proposition 5

([32]) Let $U_{0} \equiv {〈 T_{0}, e_{0}, S_{0} 〉}_{\min}$ be a uninorm in $U_{\min}$ with neutral element $e_{0} \in] 0, 1 [$ . Then its residual

Case of RU-implications obtained from a uninorm in $U_{ide}$

Let us deal in this section with residual implications derived from idempotent uninorms and let us recall first the structure of this kind of implications. First, recall that the residual operator derived from a uninorm $U_{0}$ is an implication function if and only if $U_{0} (x, 0) = 0$ for all $x < 1$ (see [8], or Proposition 1 in the preliminaries). Consequently, in our case when $U_{0}$ is idempotent, say $U_{0} \equiv {〈 g_{0}, e_{0} 〉}_{ide}$ with $0 < e_{0} < 1$ , then it is necessary to have $g_{0} (0) = 1$ and so this condition will be always assumed

Case of RU-implications obtained from a uninorm in $U_{rep}$

In this section we want to deal with residual implications derived from representable uninorms. Let us recall first the description of this kind of implications that can be found in [8] (see also [3]). Suppose that $U_{0}$ is a conjunctive representable uninorm with neutral element $e_{0} \in] 0, 1 [$ and additive generator h. Denote by $U_{0}^{⁎}$ the disjunctive representable uninorm with the same additive generator h. Then both, the residuated RU-implication derived from $U_{0}$ and the one derived from $U_{0}^{⁎}$ , coincide

Case of RU-implications obtained from a uninorm in $U_{\cos}$

Let us deal in this section with residual implications derived from uninorms continuous in the open unit square $] 0, 1 [^{2}$ . However, since there are two different classes of these uninorms, we will divide our study into two subsections, one devoted to uninorms in $U_{\cos, \min}$ and the other to uninorms in $U_{\cos, \max}$ .

Conclusions and future work

Forward inferences schemes in approximate reasoning are based on the Modus Ponens property, also called T-conditionality given by Equation (1). In this paper we have extended such property to the so-called U-Modus Ponens or U-conditionality, given by Equation (5), just by substituting the t-norm T by a conjunctive uninorm U. Fixed a uninorm U, we have studied in this paper which fuzzy implication functions satisfy U-conditionality leading to the fact that the most usual ones derived from

Acknowledgements

This paper has been partially supported by the Spanish Grant TIN2016-75404-P, AEI/FEDER, UE.

References (39)

I. Aguiló et al.
A characterization of residual implications derived from left-continuous uninorms
Inf. Sci.
(2010)
C. Alsina et al.
When $(S, N)$ -implications are $(T, T_{1})$ -conditional functions?
Fuzzy Sets Syst.
(2003)
M. Baczyński et al.
(U,N)-implications and their characterizations
Fuzzy Sets Syst.
(2009)
E. Czogala et al.
Associative monotonic operations in fuzzy set theory
Fuzzy Sets Syst.
(1984)
B. De Baets
Idempotent uninorms
Eur. J. Oper. Res.
(1999)
B. De Baets et al.
Van Melle's combining function in MYCIN is a representable uninorm: an alternative proof
Fuzzy Sets Syst.
(1999)
P. Drygaś
On properties of uninorms with underlying t-norm and t-conorm given as ordinal sums
Fuzzy Sets Syst.
(2010)
P. Drygaś et al.
A characterization of a class of uninorms with continuous underlying operators
Fuzzy Sets Syst.
(2016)
S.K. Hu et al.
The structure of continuous uninorms
Fuzzy Sets Syst.
(2001)
J. Martín et al.
On locally internal monotonic operations
Fuzzy Sets Syst.
(2003)

M. Mas et al.

RU and $(U, N)$ -implications satisfying Modus Ponens

Int. J. Approx. Reason.

(2016)

M. Mas et al.

Two types of implications derived from uninorms

Fuzzy Sets Syst.

(2007)

M. Mas et al.

Modus Ponens and Modus Tollens in discrete implications

Int. J. Approx. Reason.

(2008)

M. Mas et al.

A characterization of $(U, N), R U, Q L$ and D-implications derived from uninorms satisfying the law of importation

Fuzzy Sets Syst.

(2010)

S. Massanet et al.

On a new class of fuzzy implications: h-implications and generalizations

Inf. Sci.

(2011)

D. Ruiz-Aguilera et al.

Distributivity of residual implications over conjunctive and disjunctive uninorms

Fuzzy Sets Syst.

(2007)

D. Ruiz-Aguilera et al.

S- and R-implications from uninorms continuous in $] 0, 1 [^{2}$ and their distributivity over uninorms

Fuzzy Sets Syst.

(2009)

R.R. Yager et al.

Uninorm aggregation operators

Fuzzy Sets Syst.

(1996)

M. Baczyński et al.

Fuzzy Implications

(2008)

Cited by (13)

MP and MT properties of fuzzy inference with aggregation function
2024, Engineering Applications of Artificial Intelligence
As the two basic fuzzy inference models, fuzzy modus ponens (FMP) and fuzzy modus tollens (FMT) have the important application in artificial intelligence. In order to solve FMP and FMT problems, Zadeh proposed a compositional rule of inference (CRI) method. This paper aims mainly to investigate the validity of $A$ -compositional rule of inference (ACRI) method, as a generalized CRI method based on aggregation functions, from a logical view and an interpolative view, respectively. Specifically, the modus ponens (MP) and modus tollens (MT) properties of ACRI method are discussed in detail. It is shown that the aggregation functions to implement FMP and FMT problems provide more generality than the t-norms, uninorms and overlap functions as well-known the laws of $T$ -conditionality, $U$ -conditionality and $O$ -conditionality, respectively. Moreover, two examples are also given to illustrate our theoretical results. Especially, Example 6.2 shows that the output $B^{'}$ in FMP (FMT) problem is close to $B$ ( $D^{C}$ ) with our proposed inference method when the fuzzy input and the antecedent of fuzzy rule are near (the fuzzy input near with the negation of the succedent in fuzzy rule).
Investigations of T-power based implications satisfying some functional equations related to reasoning schemes
2023, International Journal of Approximate Reasoning
In fuzzy logic, the fuzzy reasoning rules generalized hypothetical syllogism (GHS), generalized modus tollens (GMT) and generalized reduction to absurdity (GRA) play an indispensable role in selecting appropriate fuzzy implications in specific tasks. In order to enhance the set of fuzzy implications satisfying these three properties to a greater extent, in this work, we investigate in detail under what conditions the T-power based implications will satisfy (GHS), (GMT) and (GRA). Then the necessary and sufficient conditions are given. The results obtained can not only provide a theoretical basis for selecting fuzzy implication satisfying these properties, but also help us further insight the behaviour of this new kind of fuzzy implication.
On interval fuzzy implications derived from interval additive generators of interval t-norms
2023, International Journal of Approximate Reasoning
Yager's f-generated implication is one important class of fuzzy implication, which is based on additive generators of continuous Archimedean t-norms. Considering the related interval extension, this paper presents an analysis of interval fuzzy implications derived from interval additive generators of interval t-norms, called $F$ -generated interval fuzzy implications. The relationship between Yager's f-generated implications and $F$ -generated interval fuzzy implications is discussed. Then We study the basic properties of $F$ -generated interval fuzzy implications and discuss some general forms of classical logic tautologies (i.e. law of importation, contrapositive symmetry, and distributivity over interval t-norms or interval t-conorms). Meanwhile, the relationships between $F$ -generated interval fuzzy implications and several families of interval fuzzy implications are obtained.
On functional equations and inequalities related to some reasoning schemes that involve fuzzy implications
2022, Fuzzy Sets and Systems
In this paper, we investigate some functional equations and inequalities which are associated with recognized reasoning schemes like hypothetical syllogism, modus ponens, modus tollens and reduction to absurdity. First, we recall facts and theorems that might be found in the literature. Next, we show some properties and relationships that take place between these equations and inequalities. Moreover, we present some of their solutions among well-known families of fuzzy implications – R-implications, $(S, N)$ -implications and Yager's implications.
Modus Ponens property of T-power based implications
2022, Fuzzy Sets and Systems
Among the great bunch of additional properties which fuzzy implications may satisfy, the so-called invariance with respect to T-powers highlights due to its applications in approximate reasoning. This property ensures the invariance of the truth value of the fuzzy implication when both the antecedent and the consequent are modified using the same quantifier modeled by using powers of t-norms. In order to find the fuzzy implications that satisfy this property, a new class of fuzzy implications named T-power based implications has been proposed. In this paper we study when this kind of fuzzy implication satisfies the Modus Ponens property with respect to a continuous t-norm $T^{♯}$ . Then a characterization theorem on this topic in terms of residual implication generated from a summand of $T^{♯}$ is presented.
Constructions of uninorms on bounded lattices by means of t-subnorms and t-subconorms
2021, Fuzzy Sets and Systems
Citation Excerpt :
Uninorms on the unit interval, introduced by Yager and Rybalov [20], are one of the most important aggregation operators generalizing the notions of t-norms and t-conorms. In recent years, there has been growing interest in the study of uninorms [10,15,17]. Uninorms have been found to be a useful tool in many different fields, such as expert systems, neural networks, and fuzzy logics [8,11,18,19].
In the literature on uninorms on lattices, the proof of associativity of a uninorm is usually split into all possible cases, which takes up a lot of space. In this article, we use commutativity to reduce the complexity in the proof of associativity. Moreover, some methods for constructing uninorms on an arbitrary bounded lattice are introduced. These methods generalize some construction methods in the literature.

View all citing articles on Scopus

View full text

Uninorm based residual implications satisfying the Modus Ponens property with respect to a uninorm

Abstract

Introduction

Section snippets

Preliminaries

Modus Ponens with respect to a uninorm

Case of RU-implications obtained from a uninorm in Umin

Case of RU-implications obtained from a uninorm in Uide

Case of RU-implications obtained from a uninorm in Urep

Case of RU-implications obtained from a uninorm in Ucos

Conclusions and future work

Acknowledgements

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Eur. J. Oper. Res.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Int. J. Approx. Reason.

Fuzzy Sets Syst.

Int. J. Approx. Reason.

Fuzzy Sets Syst.

Inf. Sci.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Implications

Case of RU-implications obtained from a uninorm in $U_{\min}$

Case of RU-implications obtained from a uninorm in $U_{ide}$

Case of RU-implications obtained from a uninorm in $U_{rep}$

Case of RU-implications obtained from a uninorm in $U_{\cos}$