key: cord-0044871-0bnlu31s
authors: Aguiló, Isabel; Massanet, Sebastia; Riera, Juan Vicente; Ruiz-Aguilera, Daniel
title: Modus Ponens Tollens for RU-Implications
date: 2020-05-15
journal: Information Processing and Management of Uncertainty in Knowledge-Based Systems
DOI: 10.1007/978-3-030-50143-3_61
sha: e8bdeb51f4e1fa3386448bb6c13977218f401936
doc_id: 44871
cord_uid: 0bnlu31s

In fuzzy rules based systems, fuzzy implication functions are usually considered to model fuzzy conditionals and to perform forward and backward inferences. These processes are guaranteed by the fulfilment of the Modus Ponens and Modus Tollens properties by the fuzzy implication function with respect to the considered conjunction and fuzzy negation. In this paper, we investigate which residual implications derived from uninorms satisfy both Modus Ponens and Modus Tollens properties with respect to the same t-norm and a fuzzy negation simultaneously. The most usual classes of uninorms are considered and many solutions are obtained which allow to model the fuzzy conditionals in a fuzzy rules based systems (and perform backward and forward inferences) with a unique residual implication derived from a uninorm.

Fuzzy implication functions have been extensively studied in the last decades (see [3, 4, 18] and references therein). There exist two main reasons to support the great effort made by the scientific community in this field. First, fuzzy implication functions have proved useful in many applications ranging from approximate reasoning to image processing, including fuzzy control, fuzzy relational equations, fuzzy DI-subsethood measures or computing with words, among other fields. The second reason is a direct consequence of their definition, which imposes only some monotonicities and corner conditions to ensure that they generalize the binary implication when restricted to {0, 1}. This fact opens a plethora of additional properties which, although they are studied from a theoretical point of view, are useful to obtain feasible and more adequate fuzzy implication functions in the applications.

Two of such additional properties are the (generalized) Modus Ponens and Modus Tollens. These properties are of paramount importance in approximate reasoning. Indeed, in any fuzzy rules based system, the fuzzy conditionals are usually modelled by fuzzy implication functions. However, in order to perform backward and forward inferences, the considered fuzzy implication functions must satisfy the aforementioned (generalized) Modus Ponens and Modus Tollens properties with respect to the conjunction and fuzzy negation considered in the system. These properties are usually carried out through the Compositional Rule of Inference (CRI) of Zadeh, based on the sup −T composition, where T is a tnorm (see for instance [5] or Chapter 7 in [3] ). Applying this approach, the (generalized) Modus Ponens and Modus Tollens are usually expressed by the following two functional inequalities:

T (x, I(x, y)) ≤ y, for all x, y ∈ [0, 1], T (N (y), I(x, y)) ≤ N (x), for all x, y ∈ [0, 1], where T is a t-norm, I a fuzzy implication function and N a fuzzy negation. These properties have been studied in the literature for the most usual families of fuzzy implication functions such as (S, N ), R, QL and D-implications derived from t-norms and t-conorms [2, 3, 16, 18, [23] [24] [25] or from uninorms [14, 15] . Even recently, a whole new line of research has been proposed in which the t-norm T is generalized to a more general conjunction such as a conjunctive uninorm [19] or an overlap function [8] , leading to the so-called U -Modus Ponens or O-Modus Ponens.

Although the functional inequalities of Modus Ponens and Modus Tollens have quite similar expressions, it is well-known that both properties are not equivalent. Thus, in [23, 24] , the simultaneous fulfillment of both properties was studied for the first time for some restricted classes of (S, N ), R, QL and Dimplications. It was proved that when the fuzzy negation N is a strong negation, both properties are equivalent if the fuzzy implication function satisfies the contrapositive symmetry with respect to N . The importance of the disposal of fuzzy implication functions satisfying both properties lies on the possibility of considering a unique implication to model fuzzy conditionals regardless of whether backward or forward inference processes have to be performed.

Following this line of research, in this paper, we analyze which residual implications derived from uninorms, or RU -implications for short, satisfy both the Modus Ponens and the Modus Tollens properties with respect to the same tnorm T and a fuzzy negation N (continuous, but not necessarily strong). It has to be said that the Modus Ponens property had been already studied for this family of uninorms in [15] and the Modus Tollens property was analyzed in [14] . However, while in [15] , the results were given in terms of the t-norm T U associated to the uninorm U , in [14] the results were presented for each class of uninorms separately. This fact makes it difficult to coordinate the results of both studies in order to find RU -implications satisfying both properties. This is the main goal of this paper in addition to find some cases for which the fulfillment of one property implies the fulfillment of the other one.

The paper is organized as follows. In the next section we recall some basic definitions and properties on fuzzy implication functions and uninorms. In Sect. 3, we introduce the so-called Modus Ponens Tollens property and we discuss its fulfillment when the fuzzy implication function satisfies the contrapositive symmetry with respect to a strong negation. After that, in Sect. 4, the Modus Ponens Tollens property is studied in depth for RU -implications depending on the class of the uninorm U considered. The paper ends with some conclusions and future work.

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 [11] ). We also suppose that some basic facts on uninorms are known (see for instance [9] ) as well as their most usual classes (see [13] for a complete survey), that is, uninorms in U min ( [9] ), representable uninorms ( [9] ), idempotent uninorms ( [6, 12, 22] ) and uninorms continuous in the open unit square ( [10] ).

We recall here only some facts on implications and uninorms in order to establish the necessary notation that we will use along the paper. If N is a fuzzy negation that is strictly decreasing and continuous, it will be called strict, and if it is involutive, N (N (x)) = x for all x ∈ [0, 1], then it will be called strong.

is called the natural negation of T or the negation induced by T . We will usually denote a uninorm with neutral element e and underlying t-norm T U and t-conorm S U by U ≡ T U , e, S U . For any uninorm it is satisfied that U (0, 1) ∈ {0, 1} and a uninorm U is called conjunctive if U (1, 0) = 0 and disjunctive when U (1, 0) = 1. On the other hand, let us recall the most usual classes of uninorms in the literature that will be used along the paper. We start with the class of uninorms in U min .

where T U is a t-norm, and S U is a t-conorm. We will denote a uninorm in U min with underlying t-norm T U , underlying t-conorm S U and neutral element e as U ≡ T U , e, S U min .

The class of idempotent uninorms, that satisfy U (x, x) = x for all x ∈ [0, 1], was characterized first in [6] for those uninorms with a lateral continuity and in [12] for the general case. An improvement of this last result was done in [22] as follows. 

Any idempotent uninorm U with neutral element e and associated function g will be denoted by U ≡ g, e ide and the class of idempotent uninorms will be denoted by U ide . Obviously, for any of these uninorms, the underlying t-norm is the minimum and the underlying t-conorm is the maximum.

U : [0, 1] 2 → [0, 1

Any representable uninorm U with neutral element e and additive generator h will be denoted by U ≡ h, e rep and the class of representable uninorms will be denoted by U rep . For any of these uninorms the underlying t-norm and tconorm are always strict. For all representable uninorm U , a strong negation can be defined from U as

Another studied class of uninorms is U cos , composed by all uninorms continuous in ]0, 1[ 2 . They were introduced and characterized in [10] as follows. representable uninorm R such that U can be represented as

, two continuous t-conorms S 1 and S 2 and a representable uninorm R such that U can be represented as

elsewhere.

Now we will recall residual implications from uninorms: RU -implications.

). Let U be a uninorm. The residual operation derived from U is the binary operation given by

U (x, 0) = 0 for all x < 1.

In this case I U is called an RU -implication.

This includes all conjunctive uninorms but also many disjunctive ones, for instance in the classes of representable uninorms (see [7] ) and idempotent uninorms (see [20] ).

Some properties of RU -implications have been studied involving the main classes of uninorms, those previously stated: uninorms in U min , idempotent uninorms and representable uninorms (for more details see [1, 3, 7, 17, 20, 21] ). The Modus Ponens and Modus Tollens properties have been studied for different types of implications, usuallly taking into account continuous t-norms T and continuous fuzzy negations N . If we consider RU -implications, (MP) and (MT) have been studied in depth in [15] and [14] , respectively. These properties are not equivalent in general, as it is stated in the following examples. First, we have a fuzzy implication function that satisfies (MP) but not (MT). 3 4 rep a representable uninorm with T U = T P , the product t-norm (with additive generator t U (x) = − ln(x) up to a multiplicative constant) and S U any strict t-conorm. Let us consider its residual implication I U which will be given later in Proposition 10. Let us also consider T = T P and the negation N (x) = 1−x 1+10x which belongs to the family of Sugeno negations with λ = 10. In this case, I U safisfies (MP) with respect to T by using Proposition 9 in [15] . However, I U does not satisfy (MT) with respect to T and N (just taking x = 0.7 and y = 0.5 in Eq. (MP)).

Next example provides an RU -implication that satisfies (MT) but not (MP). 1 2 rep be a representable uninorm with additive generator h(x) = ln x 1−x for all x ∈ [0, 1]. Let T be a t-norm whose expression is given by the ordinal sum T ≡ ( 0, 1 2 , T P , 1 2 , 1, T 1 ) with T 1 any continuous t-norm and let us consider the continuous fuzzy negation N given by

In this case, I U is given by

otherwise.

According to Proposition 5.3.20-(ii) in [14] , I U satisfies (MT) with respect to T and N . However, a simple computation shows that g : [0, 1] → [0, 1] given by

is not subadditive (for instance, take x = 0.3 and y = 0.2) and therefore, by using Proposition 10 in [19] , I U does not satisfy (MP) with respect to T .

Then, as we have seen, (MT) and (MP) are not equivalent in general, and the question about which fuzzy implication functions satisfy both properties with respect to the same t-norm T and fuzzy negation N is worthy to study.

property with respect to T and N whenever Eqs. (MP) and (MT) are satisfied simultaneously. Remark 1. Note that, when x ≤ y we have N (y) ≤ N (x) and then (MT) trivially holds in these cases. Similarly, (MP) is satisfied in these cases. Thus, both properties need to be checked only in points (x, y) ∈ [0, 1] 2 where y < x.

Anyway, a special case that can be considered is when I satisfies the contrapositive symmetry with respect to N . Contrapositive symmetry is a well known property, which is related to the Modus Ponens Tollens as it is stated in the following results. From the result above, in the case that a fuzzy implication function I satisfies (CP) with respect to N , only one of (MP) or (MT) needs to be checked in order to satisfy (MPT). Now, let us recall the result on contrapositive symmetry for residual implications derived from idempotent uninorms.

Consider U ≡ g, e ide an idempotent uninorm with g(0) = 1, I U its residual implication and N a strong negation. Then I U satisfies (CP) with respect to N if and only if g = N .

As a consequence of the previous result, we have infinite RU -implications that satisfy (MPT) for any t-norm T , by using Proposition 5.3.14 in [14] . Coming up next, we recall the case of (CP) for uninorms continuous in ]0, 1[ 2 . 

In this section we investigate the Modus Ponens Tollens property (MPT) for fuzzy implication functions derived from three well known classes of uninorms.

In this section we will deal with RU -implications derived from uninorms in U min , that is, uninorms U ≡ T U , e, S U min with neutral element e ∈]0, 1[. Recall that for this kind of uninorms, RU -implications have the following structure. [3] ). Let U ≡ T U , e, S U min a uninorm in U min and I U its residual implication. Then

For this family of RU -implications we have the following result. 

for some y ≤ e, then I U satisfies (MPT) with respect to T and N if and only if Property ( 1 ) is fulfilled and the following condition holds:

).

Property ( 1 ) and the following property holds: Example 5. Let us consider U ≡ T P , 1 2 , S U min with S U any t-conorm. Let T = T L and let us take

elsewhere, a continuous fuzzy negation with fixed point s = 1 3 . Note that in this case N T = N c (x) = 1 − x and N ( 1 4 ) < N T ( 1 4 ). Thereby, we are under the conditions of Case (iii) of the previous proposition, and then Properties ( 1 ) and ( 2 ) must be checked. A simple computation shows that g : [0, 1] → [ 1 2 , 1] given by g(x) = 1 − 1 2 e −x is subadditive and so Property ( 1 ) is fulfilled. With respect to Property ( 2 ), we obtain the inequality y 2x ≤ 1 − 2(x − y) which is valid for y < x < 1 2 . Consequently, I U satisfies (MPT) with respect to T L and N . Example 6. Let us consider U ≡ T L , 1 2 , S U min with S U any t-conorm. Let us consider T = T L and the same negation N used in the previous example. In this case, again it holds that N T = N c and it follows that 

In this section we will deal with RU -implications derived from idempotent uninorms, that is, uninorms U ≡ g, e ide with neutral element e ∈ [0, 1] and such that g(0) = 1. Recall that for this kind of uninorms, the corresponding RUimplications have the following structure.

Let U ≡ g, e ide be an idempotent uninorm with neutral element e ∈]0, 1[ and such that g(0) = 1. Then I U is given by

From results in [15] , for an idempotent uninorm U ≡ g, e ide with g(0) = 1, as T U = min, I U satisfies (MP) with respect to any t-norm. Therefore, we can write the following result. Thus, in the rest of the section, all the conditions in the results will be related to the fulfillment of (MT) (that was studied in [14] ), which will imply the fulfillment of (MPT). Now we will distinguish two cases depending on the value of g (1) . We will start with the case g(1) > 0. Proposition 8. Let U ≡ g, e ide with g(0) = 1 and g(1) > 0 and I U its residual implication. Let T be a t-norm and N a continuous fuzzy negation. If I U satisfies the (MPT) property with respect to T and N , then the following statements are true:

(i) T (N (y), y) = 0 for all y ≤ g (1) .

tive generator t : [0, 1] → [0, 1] and associated negation N T , which is given by N T (x) = t −1 (1 − t(x)), such that N (y) ≤ N T (y) for all y ≤ g (1) .

Although this result provides only necessary conditions on T , the following example gives infinite cases of residual implications I U from U an idempotent uninorm such that satisfy (MPT) with respect a t-norm T and a strong negation N . When N is strict, the following result provides an easier condition in order to verify the fulfillment of (MPT).

Let T be a t-norm, N a strict fuzzy negation, and U ≡ g, e ide be an idempotent uninorm with neutral element e ∈]0, 1[ with g(0) = 1, g(1) = 0 and I U its residual implication. Then I U satisfies (MPT) with respect to T and N if and only if g(x) ≤ N (x) for all x ≥ e. Example 9. Let us consider U ≡ g, 1 4 ide an idempotent uninorm where

otherwise, T = T P and N = N c . It is straightforward to prove that g(x) ≤ N (x) for all x ≥ 1 4 . Then, we are under the conditions of the previous result. Thus, I U satisfies (MPT) with respect to T and N .

In this section we will deal with RU -implications derived from representable uninorms, that is, from uninorms U ≡ h, e rep with neutral element e ∈]0, 1[. Let us recall in this case the expression of the residual implication derived from U . Proposition 10 (Theorem 5.4.10 in [3] ). Let U ≡ h, e rep be a representable uninorm with neutral element e ∈]0, 1[. Then I U is given by

For this kind of uninorms we will consider only continuous t-norms which are not an ordinal sum, namely, the minimum t-norm and continuous Archimedean t-norms. which has e = 1 2 as neutral element and additive generator h(x) = ln( x 1−x ). In this case, φ(x) = h(t −1 (x)) = ln 1−x x which is clearly subadditive. By applying Case (ii) of the previous proposition, we conclude that I U satisfies (MPT) with respect to T and N .

In this paper, we have studied the fulfillment of the so-called Modus Ponens Tollens property (MPT) by the family of RU -implications, i.e., we have analyzed which RU -implications satisfy at the same time the Modus Ponens and the Modus Tollens properties with respect to a t-norm T and a negation N . From this study, many solutions are available. On the one side, all RU -implications which satisfy the Modus Ponens property with respect to a t-norm T and the contrapositive symmetry with respect to a strong negation N are solutions of (MPT). On the other side, when N is not strong or the contrapositive symmetry is not satisfied, other solutions exist within RU -implications derived from uninorms in U min , representable uninorms and idempotent uninorms. For most of these families, necessary and sufficient conditions are presented and in some cases, it is shown that the fulfillment of the Modus Tollens property implies the fulfillment of the Modus Ponens property.

As future work, we want to complete the results presented in this paper by considering also continuous ordinal sum t-norms as T in some of the results presented in Sect. 4 and to deepen the study in the particular case of idempotent uninorms with g(0) = 1 and g(1) > 0.

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