A study on the relationship between relaxed metrics and indistinguishability operators

In 1982, Trillas introduced the notion of indistinguishability operator with the main aim of fuzzifying the crisp notion of equivalence relation. In the study of such a class of operators, an outstanding property must be stressed. Concretely, there exists a duality relationship between indistinguishability operators and metrics. The aforesaid relationship was deeply studied by several authors that introduced a few techniques to generate metrics from indistinguishability operators and vice versa. In the last years, a new generalization of the metric notion has been introduced in the literature with the purpose of developing mathematical tools for quantitative models in computer science and artificial intelligence. The aforesaid generalized metrics are known as relaxed metrics. The main purpose of the present paper is to explore the possibility of making explicit a duality relationship between indistinguishability operators and relaxed metrics in such a way that the aforementioned classical techniques to generate both concepts, one from the other, can be extended to the new framework.


Introduction
Throughout this paper, we will assume that the reader is familiar with the basics of triangular norms (see Klement et al. 2000 for a deeper treatment of the topic). Trillas (1982) introduced the notion of T -indistinguishability operator with the aim of fuzzifying the classical (crisp) notion of equivalence relation. Let us recall that, according to Trillas (1982) (see also Klement et al. 2000;Recasens 2010), given a t-norm T : [0, 1]×[0, 1] → [0, 1], a T -indistinguishability operator on a nonempty set X is a fuzzy relation E : X × X → [0, 1] satisfying for all x, y, z ∈ X the following conditions Communicated   A T -indistinguishability operator E is said to separate points provided that E(x, y) = 1 ⇒ x = y for all x, y ∈ X .
In the literature, the relationship between metrics and T -indistinguishability operators has been studied in depth for several authors (De Baets and Mesiar 2002;Gottwald 1992;Höhle 1993;Klement et al. 2000;Ovchinnikov 1984;Recasens 2010;Valverde 1985). Let us recall a few facts about metric spaces in order to explicitly state the aforesaid relationship. Following Bukatin et al. (2014), a pseudometric on a nonempty set X is a function d : X × X → [0, ∞] such that, for all x, y, z ∈ X , the following properties hold: (y, z).
A pseudo-metric d on X is called pseudo-ultrametric if it satisfies, in addition, for all x, y, z ∈ X the following inequality: (iv) d(x, z) ≤ max{d(x, y), d(y, z)}.
Of course, a pseudo-metric (pseudo-ultrametric) d on X is called a metric (ultrametric) provided that it satisfies, in addition, the following axiom for all x, y ∈ X : Regarding the relationship between (pseudo-)metrics and indistinguishability operators, the next results make it explicit. The first one introduces a technique that allows to construct (pseudo-)metrics from indistinguishability operators.

2) For any T -indistinguishability operator E on X , the function d E is a pseudo-metric on X . (3) For any T -indistinguishability operator E on X that separates points, the function d E is a metric on X.
The second one introduces a method to revert the technique given by Theorem 1, i.e., to construct now indistinguishability operators from (pseudo-)metrics.
for all x, y ∈ X . If d is a pseudo-metric on X , then E d is a T * -indistinguishability operator. Moreover, the T *indistinguishability operator E d separates points if and only if d is a metric on X .
In the last years, a few generalizations of the metric notion have been introduced in the literature with the purpose of developing suitable mathematical tools for quantitative models in computer science and artificial intelligence. Concretely, the notion of dislocated metric, dislocated ultrametric, weak partial (pseudo-)metric and partial (pseudo-)metric has been studied and applied to logic programming in Seda (2010, 2011), domain theory in Heckmann (1999), Romaguera and Valero (2009), Romaguera and Valero (2012), denotational semantics in Matthews (1994), Shahza and Valero (2013) and asymptotic complexity of programs in Alghamdi et al. (2013), respectively. Each of the preceding generalized metric notions can be retrieved as a particular case of a new notion, called relaxed metric, which has been introduced recently in Bukatin et al. (2014). Let us recall, according to Bukatin et al. (2014), the notion of relaxed metric.
Definition 1 A relaxed pseudo-metric on a nonempty set X is a function d : X × X → [0, ∞] which satisfies for all x, y, z the following: We will say that a relaxed pseudo-metric d on a nonempty set satisfies the small self-distances (SSD for short) property in the spirit of Heckmann (1999) whenever d(x, x) ≤ d(x, y) for all x, y ∈ X . Moreover, a relaxed pseudo-metric d is a relaxed metric provided that it satisfies the following separation property for all x, y ∈ X : Furthermore, a relaxed (pseudo-)metric d on X will be called a relaxed (pseudo-)ultrametric if satisfies in addition, for all x, y, z, the following inequality: The following example gives an instance of a relaxed pseudo-metric which is not a pseudo-metric.
A straightforward computation shows that d X is a relaxed pseudo-metric on X which satisfies the SSD property. However, d X is not a pseudo-metric, since d X (x, x) = 1 2 . Observe that defining d X (1, 1) = 1 and d X (x, y) = d X (x, y) otherwise, we obtain an instance of relaxed pseudo-metric which is not a pseudo-metric and, in addition, it does not satisfy the SSD property.
Recently, it has been discussed that the notion of indistinguishability operator and relaxed metric is closely related. Indeed, in Bukatin et al. (2014) and Demirci (2011) it has been stated that the logical counterpart for relaxed metrics is, in some sense, a generalized indistinguishability operator. On account of Bukatin et al. (2014) (see also Höhle 1992), the notion of generalized indistinguishability operator related to relaxed metrics can be formulated as follows: Definition 2 Let X be a nonempty set and let T : [0, 1] × [0, 1] → [0, 1] be a t-norm. A relaxed T -indistinguishability operator E on X is a fuzzy relation E : X × X → [0, 1] satisfying the following properties for any x, y, z ∈ X : Moreover, a relaxed T -indistinguishability operator E satisfies the small self-indistinguishability (SSI for short) property provided that for all x, y ∈ X . Furthermore, a relaxed T -indistinguishability operator E is said to separate points provided that E(x, y) = E(x, x) = E(y, y) ⇒ x = y for all x, y ∈ X .
Notice that the notion of T -indistinguishability operator is retrieved as a particular case of relaxed T -indistinguishability operator whenever the relaxed T -indistinguishability operator satisfies also the reflexivity. In fact, a relaxed indistinguishability operator is an indistinguishability operator if and only if it is reflexive. The same occurs when we consider T -indistinguishability operators that separate points. Furthermore, it must be stressed that M-valued equalities are exactly relaxed T -indistinguishability operators with the SSI property when the t-norm T is left-continuous and the underlaying GL-monoid M is exactly ([0, 1], ≤, T ) (see, for instance, Höhle 1998Höhle , 1992.
The next examples give instances of relaxed indistinguishability operators which are not indistinguishability operators.
Example 2 Fix k ∈]0, 1[. Consider the fuzzy binary relation E k : R + → [0, 1] defined by E(x, y) = k for x, y ∈ R + . It is obvious that E k is a relaxed T Minindistinguishability operator which is not a T Min -indistinguishability operator because E(x, x) = k = 1 for each x ∈ R + . Notice that E k satisfies the SSI property, but it does not separate points.
Example 3 Let be a nonempty alphabet. Denote by ∞ the set of all finite and infinite sequences over .
for all u, v ∈ ∞ , where l(v, w) denotes the longest common prefix between v and w. Of course, we have adopted the convention that 2 −∞ = 0. Then, it is not hard to check that E is a relaxed T Min -indistinguishability operator which is not a T Min -indistinguishability operator. Notice that E (u, u) = 1 − 1 2 l(u) for all u ∈ ∞ and that E (u, u) = 1 ⇔ u ∈ ∞ . It is clear that E satisfies the SSI property and separates points. Clearly, E is not a T Min -indistinguishability operator because E (u, u) < 1 for each x ∈ F .
The following example shows that there are relaxed indistinguishability operators that do not satisfy the SSI property.
Example 4 Let X be the set considered in Example 1. Define the fuzzy binary relation E X : A straightforward computation yields that E X is a relaxed T P -indistinguishability operator which does not separate points, where T P denotes the product t-norm. Moreover, E X does not satisfy the SSI property, since E X (1, 3) = 1 2 and E X (1, 1) = 1 4 .
Motivated, on the one hand, by the exposed facts and, on the other hand, by the utility of generalized metrics in computer science and artificial intelligence, the target of this paper is to study deeply the relationship between both concepts, relaxed indistinguishability operators and relaxed metrics, and try to extend the methods given in Theorems 1 and 2 to this new context.

From relaxed indistinguishability operators to relaxed metrics
In this section, we focus our work on the possibility of extending Theorem 1 to the relaxed framework. To this end, we will structure our study in two subsections. The first one, Sect. 2.1, will be devoted to make clear the relationship between relaxed metrics and relaxed T Min -indistinguishability operators, where T Min stands for the minimum t-norm. The second one, Sect. 2.2, will be devoted to specify the correspondence between relaxed T -indistinguishability operators and relaxed metrics whenever one considers t-norms T with additive generator.

Relaxed T Min indistinguishability
According to Zadeh (1971) (see also Recasens 2010), the relationship between T Min -indistinguishability operators and metrics is given by the next result.
Proposition 1 Let X be a nonempty set and let E : X × X → [0, 1] be a fuzzy relation. Then, the following assertions are equivalent: (1) E is a T Min -indistinguishability operator. ( Moreover, E separates points if and only if d E is a ultrametric on X .
Next, we show that the preceding result can be easily extended to our new context. Proposition 2 Let X be a nonempty set and let E be a fuzzy relation on X . Then, the following assertions are equivalent:

Moreover, E separates points if and only if d E is a relaxed ultrametric on X .
Proof for all x, y ∈ X . Next, fix x, y, z ∈ X . Then, we have that the next inequality Moreover, the preceding inequality is equivalent to the next one: At the same time, the above inequality is equivalent to the following one: Furthermore, the last inequality is equivalent to Therefore, E is a relaxed T Min -indistinguishability operator if and only if d E is a relaxed pseudo-ultrametric on X .
Finally, it is clear that for all x, y ∈ X . It follows that E separates points if and only if d E is a relaxed ultrametric on X .

Corollary 1 Let X be a nonempty set and let E be a T Minindistinguishability operator on X . Then, E fulfills the SSI property if and only if d E fulfills the SSD property.
Proof Since E is a T Min -indistinguishability operator, we have that for all x, y ∈ X . So taking x = y in the preceding inequality, we obtain that for all x, z ∈ X . Thus, every T Min -indistinguishability operator satisfies the SSI property. Clearly, for all x, y ∈ X . Therefore, E fulfills the SSI property if and only if d E fulfills the SSD property.
It must be pointed out that relaxed T Min -indistinguishability operators match up with -valued equalities in the sense of Fourman and Scott (1979) when the underlaying Heyting algebra is exactly ([0, 1], ≤).
The next example illustrates Theorem 2.
Example 5 Consider the relaxed T Min -indistinguishability operator E introduced in Example 3. Proposition 2 guarantees that the function d E given by for all u, v ∈ ∞ is a relaxed pseudo-ultrametric. Since E satisfies the SSI property and separates points, we get by Corollary 1 that the relaxed pseudo-ultrametric d E is, in fact, a relaxed ultrametric which fulfills the SSD property.
Observe that the preceding facts agree with those pointed out in Bukatin et al. (2014) (see also Matthews 1994).

T-norms with additive generator and relaxed T-indistinguishabilities
In this section, we study the duality relationship that exists between relaxed metrics and relaxed indistinguishability operators when the t-norm under consideration admits an additive generator. Notice that the study developed in Sect. 2.1 considers relaxed T Min -indistinguishability operators and that the t-norm T Min does not admit additive generator. In particular, we wonder whether Theorem 1 can be stated in our more general framework. The next result provides an affirmative answer to the posed question. Although few parts of the proof run following the same arguments to those given in the proof of Theorem 1, we have included all of them for the sake of completeness.
for all x, y ∈ X . If T is a t-norm, then the following assertions are equivalent: (1) T * ≤ T .
for all x, y ∈ X . To this end, note that if E is a relaxed T -indistinguishability operator, then E is also a relaxed T *indistinguishability operator. Thus, we have that for all x, y, z ∈ X .
Since f T * is an additive generator of the t-norm T * , we have that for all u, v ∈ [0, 1]. It follows that for all x, y, z ∈ X . Whence, we have that for all x, y, z ∈ X .
Next, we distinguish two possible cases: Hence, we obtain that (E(y, z)). Therefore,

Case 2 f T * (E(x, y))+ f T * (E(y, z)) /
∈ Ran( f T * ). Then, the fact that f T * is an additive generator of the t-norm T * , and thus that Ran( f T * ) is relatively closed, implies that f T * (E(x, y) Therefore, we conclude that the function d f T * E is a relaxed pseudo-metric on X .
(2) ⇒ (3) Since E is a T -indistinguishability operator, we have that the function d E is a relaxed pseudo-metric on X . Moreover, on the one hand, the fact that E separates points guarantees that E(x, y) = E(x, x) = E(y, y) ⇔ x = y.
On the other hand, the fact that f T * is an additive generator of the t-norm T * implies that f T * is strictly decreasing and, hence, injective. So we obtain that T (a, b) whenever a = 1 or b = 1. So we can assume that a, b ∈ [0, 1[. Now, fix three different elements x, y, z ∈ X and define the fuzzy binary relation E on X as follows: (E(z, y)).

Taking into account that
for all u, v ∈ [0, 1] and that f E(z, y)).
as we claim.
It is worth pointing out that Theorems 1 and 3 disclose a surprising connection (equivalence) between indistinguishability operators and the relaxed ones.
In Bezdek and Harris (1978), Gottwald (1992), Valverde (1985) (see also Mesiar 1997, 2002), the subsequent characterization was given in order to establish the relationship between indistinguishability operators and (pseudo-)metrics. Concretely, the aforesaid characterization states the following. E(x, y) for all x, y ∈ X . If T is a t-norm, then the following assertions are equivalent:

Theorem 4 Let X be a nonempty set and let E be a fuzzy binary relation on X . Let d E be the function defined by d
(1) T L ≤ T , where T L denotes the Lukasiewicz t-norm. (

2) For any T -indistinguishability operator, the function d f T L E is a pseudo-metric on X . (3) For any T -indistinguishability operator that separates points, the function d f T L E is a metric on X.
Taking in Theorem 3, T * as the Lukasiewicz t-norm T L and the function f T * as the function f T L : [0, 1] → [0, ∞] given by f T L (x) = 1 − x for all x ∈ [0, 1] we obtain as a particular case the following results, one of them, Corollary 2, providing an extension of Theorem 4.

Corollary 2 Let X be a nonempty set and let E
is a t-norm, then the following assertions are equivalent: (1) T L ≤ T . (

2) For any relaxed T -indistinguishability operator, the function d E is a relaxed pseudo-metric on X . (3) For any relaxed T -indistinguishability operator that separates points, the function d E is a relaxed metric on X .
When we consider in Theorem 3 the t-norm T as the minimum t-norm T M and the function f T * as an additive generator of any t-norm T * , we retrieve as a particular case the following result.

Corollary 3 Let X be a nonempty set and let E be a relaxed T Min -indistinguishability operator on X . Then, the function d f T * E is a relaxed pseudo-metric on X for any additive generator f T * of a t-norm T * .
Of course, the preceding results agree with Theorem 1 because every relaxed pseudo-ultrametric is a relaxed pseudo-metric.
If we consider in Theorem 3 the t-norm T * as the Drastic product T D and the function f T * the additive generator of T D given by f T D (x) = 2 − x if x ∈ [0, 1[ and f (1) = 0, then we get as a consequence the following result.

Corollary 4 Let X be a nonempty set and let T be a t-norm. If E is a relaxed T -indistinguishability operator on X , then the function d f T D E is a relaxed pseudo-metric on X .
Clearly, if we consider in Corollaries 3 and 4 indistinguishability operators that separate points, then the obtained relaxed pseudo-metrics become relaxed metrics.
Clearly, Theorem 3 provides a technique to generate relaxed pseudo-metrics from relaxed indistinguishability operators. Observe that in spite of the aforementioned equivalence between Theorems 1 and 3, the new technique gives instances of relaxed pseudo-metric which are not pseudometrics. The following examples illustrate the exposed facts. A straightforward computation yields that E X is a relaxed T P -indistinguishability operator which does not separate points. Moreover, assertion (2) in Theorem 3 [or assertion (2) in Corollary 2] gives that the function d y) for all x, y ∈ X is a relaxed pseudo-metric on X . Of course, since E X does not separate points Theorem 3 (or Corollary 2) provides that d E X is a relaxed pseudo-metric that is not a relaxed metric. Moreover, it is clear that E X is not reflexive and, thus, that d is not a pseudo-metric on X .
It must be stressed that there are relaxed T -indistinguishability operators that do not satisfy the SSI property such as Example 4 shows. Taking into account the aforementioned fact, we obtain, from Theorem 3, the result below.
for all x, y ∈ X and f T * is strictly increasing, we have that for all x, y ∈ X . Therefore, E fulfills the SSI property if and only if d f T * E fulfills the SSD property. In light of Corollary 5, we can elucidate that the relaxed pseudo-metric d f T L E X given in Example 7 does not satisfy the SSD property, since the relaxed T L -indistinguishability operator E X does not satisfy the SSI property. However, Corollary 5 guarantees that the relaxed pseudo-metric, introduced in Example 6, d f T P E Min induced by the T P -indistinguishability operator E X fulfills the SSD property, since E Min fulfills the SSI property.

From relaxed metrics to relaxed indistinguishability operators
This section is devoted to explore the possibility of developing a technique that allows to induce relaxed indistinguishability operators from relaxed pseudo-metrics. In particular, we inquire if Theorem 2 can be stated in our more general framework. The next result answers the framed question. Although the proof of assertion (1) in the aforesaid result runs following the same arguments to those given in the proof of Theorem 2, we have included it for the sake of completeness.
Theorem 5 Let X be a nonempty set and let T be a continuous Archimedean t-norm with additive generator f T :

If d is a relaxed pseudo-metric on X and E d is the binary fuzzy relation defined by
for all x, y ∈ X , then the following assertions hold: (1) E d is a relaxed T -indistinguishability operator.
(2) If E d is a relaxed T -indistinguishability operator that separates points for every relaxed pseudo-metric d on X , then T is strict. (

3) E d is a relaxed T -indistinguishability operator that separates points provided that d is a relaxed metric on X if and only if T is strict. (4) d is a relaxed metric on X provided that E d is a relaxed T -indistinguishability operator that separates points. (5) If d is a relaxed metric such that d(x, y) ≤ f T (0)
for all x, y ∈ X , then E d is a relaxed T -indistinguishability operator that separates points. (6) E d is a relaxed T -indistinguishability operator which satisfies the SSI property provided that the relaxed pseudo-metric d on X satisfies the SSD property.
Next, we show that Thus, Since the T -indistinguishability operator E d separates points, we deduce that x = y and, thus, that d is a relaxed metric on X .
(5) Since d(x, y) ≤ f T (0) for all x, y ∈ X , we have that for all x, y ∈ X . Consider x, y ∈ X such that Since f −1 T is strictly decreasing on Ran( f T ), we have that The fact that d is a relaxed metric yields that x = y. Therefore, the relaxed T -indistinguishability operator E d separates points.
(6) Assume that d is a relaxed pseudo-metric on X that fulfills the SSD property. Then, d(x, x) for all x, y ∈ X .
The following examples show that, in general, the relaxed T -indistinguishability operator E d provided by Theorem 5 does not separate points and does not enjoy the SSI property.
Example 8 Consider the Lukasiewicz t-norm T L , which is continuous and not a strict Archimedean. Let f T L be the additive generator of T L given by f T L (x) = 1 − x for all x ∈ [0, 1]. Then, the pseudo-inverse f Consider the relaxed metric d f T L + introduced in the proof of assertion (2) in Theorem 5. Then, assertion (1) in Theorem 5 gives that E d + is a relaxed T L -indistinguishability operator. Nevertheless, E d + does not separate points. Indeed, Example 9 Consider the product t-norm T P , which is continuous and Archimedean. Moreover, the pseudo-inverse f for all x, y ∈ X . It follows that Whence, we conclude that the SSD property is not satisfied by the relaxed pseudo-metric.
It must be stressed that the boundedness condition in Corollary 6 is always satisfied whenever the t-norm under consideration is strict.
We end the paper showing that the boundedness of the relaxed pseudo-metric cannot be deleted in Theorem 6.

Conclusions and future work
In the last years, many generalized metrics have been introduced in the literature with the purpose of developing mathematical tools for quantitative models in computer science and artificial intelligence. All the aforementioned metrics are particular cases of the notion of relaxed pseudo-metric. In this paper, we have studied a duality relationship between relaxed pseudo-metrics and a new class of indistinguishability operators that we have called relaxed indistinguishability operators, in such a way that the celebrated techniques to generate classical pseudo-metrics from indistinguishability operators, and vice versa, can be retrieved as a particular case. A few differences between the classical framework and the new one have been exposed.
Among the aforesaid generalized metrics, it is worth mentioning the so-called partial pseudo-metrics which satisfy the SSD property and a modified triangle inequality. Concretely, if p is a partial pseudo-metric on X , then p(x, z) ≤ p(x, y) + p(y, z) − p(y, y) for all x, y, z ∈ X . Besides, in the literature, as we have mentioned in Sect. 1, a generalized indistinguishability operator, known as M-valued equality (in the sense of Höhle 1998Höhle , 1992, can be found. Specifically, if E is an M-valued equality on X , then the following transitivity is hold: T (E(x, y), E(y, y) → T E(y, z)) ≤ E(x, z), for all x, y, z ∈ X and where → T denotes the T -residuum. In Bukatin et al. (2014) and Demirci (2011), partial pseudometrics have been proposed as the logical counterpart for M-valued equalities. Motivated, on the one hand, by the preceding exposed fact and, on the other hand, by the fact that the triangle inequality (1) is a refinement of that one fulfilled by a relaxed pseudo-metric, it seems natural to try to explore in depth, as a future research, whether (1) and (2) are really dual and, thus, whether the techniques exposed in the present paper can be adapted in such a way that partial pseudo-metrics can be generated from M-valued equalities and vice versa.