Preferential approaches have been used to provide axiomatic foundations of non-monoto- nic and common sense reasoning [21, 43, 44, 39, 45, 40, 3, 36]. They have been extended to description logics (DLs), to deal with inheritance with exceptions in ontologies, by allowing for non-strict forms of inclusions, called typicality or defeasible inclusions, with different preferential semantics [31, 13, 32] and closure constructions [16, 15, 34, 6, 47, 18, 28].
In previous work , a concept-wise multipreference semantics for weighted knowledge bases has been proposed to account for preferences with respect to different concepts, by allowing, for a concept , a set of typicality inclusions of the form (meaning “the typical ’s are ’s” or “normally ’s are ’s”) with positive or negative weights. The concept-wise multipreference semantics has been first introduced as a semantics for ranked DL knowledge bases  and extended in  to weighted knowledge bases in the two-valued and fuzzy case, based on a different semantic closure construction, still in the spirit of Lehmann’s lexicographic closure  and Kern-Isberner’s c-interpretations [36, 37], but exploiting multiple preferences associated to concepts. A related semantics with multiple preferences has been proposed in the first-order logic setting by Delgrande and Rantsaudis , and an extension of DLs with defeasible role quantifiers and defeasible role inclusions has been developed by Britz and Varzinczak [14, 12], by associating multiple preferences to roles.
The concept-wise multipreference semantics has been proved to have some desired properties from the knowledge representation point of view in the two-valued case [24, 25]. In particular, it satisfies the KLM postulates of a preferential consequence relation. The properties of entailment in a fuzzy DL with typicality have not been studied so far. In this paper, a monotonic extension of a fuzzy with typicality is considered (called ) and the KLM properties of a preferential consequence relation are reformulated for this logic. Most of the postulates are satisfied, depending on the reformulation and on the chosen fuzzy combination functions.
The closure construction developed in  to define the models of a weighted defeasible knowledge base in the fuzzy case is reconsidered, by introducing a notion of faithful model of a weighted (fuzzy) knowledge base which is weaker than the notion of coherent fuzzy multipreference model in 
. This allows to capture the larger class of monotone non-decreasing activation functions in multilayer perceptrons. The properties of faithful multipreference entailment are discussed.
2 The description logic and fuzzy
Let be a set of concept names, a set of role names
and a set of individual names.
The set of concepts (or, simply, concepts) can be
- , and are concepts;
- if and are concepts, and , then are concepts.
A knowledge base (KB) is a pair , where is a TBox and is an ABox. The TBox is a set of concept inclusions (or subsumptions) , where are concepts. The ABox is a set of assertions of the form and where is a concept, an individual name in and a role name in .
An interpretation is defined as a pair where: is a domain—a set whose elements are denoted by —and is an extension function that maps each concept name to a set , each role name to a binary relation , and each individual name to an element . It is extended to complex concepts as follows:
, , ,
The notion of satisfiability of a KB in an interpretation and the notion of entailment are defined as follows:
Definition 1 (Satisfiability and entailment)
Given an interpretation :
- satisfies an inclusion if ;
- satisfies an assertion (resp., ) if (resp., ).
Given a KB , an interpretation satisfies (resp. ) if satisfies all inclusions in (resp. all assertions in ); is a model of if satisfies and .
A subsumption (resp., an assertion , ), is entailed by , written , if for all models of , satisfies .
Given a knowledge base , the subsumption problem is the problem of deciding whether an inclusion is entailed by .
Fuzzy description logics have been widely studied in the literature for representing vagueness in DLs [49, 48, 42, 8, 4], based on the idea that concepts and roles can be interpreted as fuzzy sets. As in Mathematical Fuzzy Logic  a formula has a degree of truth in an interpretation, rather than being either true or false, in a fuzzy DL axioms are associated with a degree of truth (usually in the interval ). In the following we shortly recall the semantics of a fuzzy extension of referring to the survey by Lukasiewicz and Straccia . We limit our consideration to a few features of a fuzzy DL and, in particular, we omit considering datatypes.
A fuzzy interpretation for is a pair where: is a non-empty domain and is fuzzy interpretation function that assigns to each concept name a function , to each role name a function , and to each individual name an element . A domain element belongs to the extension of to some degree in , i.e., is a fuzzy set.
The interpretation function is extended to complex concepts as follows:
, , ,
where and , , and are arbitrary but fixed t-norm, s-norm, implication function, and negation function, chosen among the combination functions of various fuzzy logics (we refer to  for details). For instance, in both Zadeh and Gödel logics , . In Zadeh logic and . In and Gödel logic if and otherwise; if and otherwise.
The interpretation function is also extended to non-fuzzy axioms (i.e., to strict inclusions and assertions of an knowledge base) as follows:
, , .
A fuzzy knowledge base is a pair where is a fuzzy TBox and a fuzzy ABox. A fuzzy TBox is a set of fuzzy concept inclusions of the form , where is an concept inclusion axiom, and . A fuzzy ABox is a set of fuzzy assertions of the form or , where is an concept, , , and . Following Bobillo and Straccia , we assume that fuzzy interpretations are witnessed, i.e., the sup and inf are attained at some point of the involved domain. The notions of satisfiability of a KB in a fuzzy interpretation and of entailment are defined in the natural way.
Definition 2 (Satisfiability and entailment for fuzzy KBs)
A fuzzy interpretation satisfies a fuzzy axiom (denoted ), as follows, for :
- satisfies a fuzzy inclusion axiom if ;
- satisfies a fuzzy assertion if ;
- satisfies a fuzzy assertion if .
Given a fuzzy KB , a fuzzy interpretation satisfies (resp. ) if satisfies all fuzzy inclusions in (resp. all fuzzy assertions in ). A fuzzy interpretation is a model of if satisfies and . A fuzzy axiom is entailed by a fuzzy knowledge base , written , if for all models of , satisfies .
3 Fuzzy with typicality:
In this section, we extend fuzzy with typicality concepts of the form , for a concept in fuzzy . The idea is similar to the extension of with typicality , but transposed to the fuzzy case. The extension allows for the definition of fuzzy typicality inclusions of the form , meaning that typical -elements are -elements with a degree grater than . A typicality inclusion , as in the two-valued case, stands for a KLM conditional implication [39, 40], but now it has an associated degree.
We call the extension of fuzzy with typicality. As in the two valued case, such as in , a preferential extension of with typicality , or in the propositional typicality logic, PTL  the typicality concept may be allowed to freely occur within inclusions and assertions, while the nesting of the typicality operator is not allowed.
We have to define the semantics of . Observe that, in a fuzzy interpretation , the degree of membership of the domain elements in a concept , induces a preference relation on , as follows:
Each has the properties of preference relations in KLM-style ranked interpretations , that is, is a modular and well-founded strict partial order. Let us recall that, is well-founded if there is no infinite descending chain , , of domain elements; is modular if, for all , implies ( or ). Well-foundedness holds for the induced preference defined by condition (1) under the assumption that fuzzy interpretations are witnessed  (see section 2) or that is finite. In the following, we will assume to be finite.
While each preference relation has the properties of a preference relation in KLM rational interpretations  (also called ranked interpretations), here there are multiple preferences and, therefore, fuzzy interpretations can be regarded as multipreferential interpretations, which have also been studied in the two-valued case [35, 24, 22, 28].
Each preference relation captures the relative typicality of domain elements wrt concept and may then be used to identify the typical -elements. We will regard typical -elements as the domain elements that are preferred with respect to relation among those such that .
Let be the crisp set containing all domain elements such that , that is, . One can provide a (two-valued) interpretation of typicality concepts in a fuzzy interpretation , by letting:
where and s.t. . When , we say that is a typical -element in .
Note that, if for some , is non-empty. This generalizes the property that, in the crisp case, implies .
Proposition 1 ( interpretation)
An interpretation is fuzzy interpretation, equipped with the valuation of typicality concepts given by condition (2) above.
The fuzzy interpretation implicitly defines a multipreference interpretation, where any concept is associated to a preference relation . This is different from the two-valued multipreference semantics in , where only a subset of distinguished concepts have an associated preference, and a notion of global preference is introduced to define the interpretation of the typicality concept , for an arbitrary . Here, we do not need to introduce a notion of global preference. The interpretation of any concept is defined compositionally from the interpretation of atomic concepts, and the preference relation associated to is defined from .
The notions of satisfiability in , model of an knowledge base, and entailment can be defined in a similar way as in fuzzy (see Section 2). In particular, given an knowledge base , an inclusion (with and ) is entailed from in (written ) if is satisfied in all models of . For instance, the fuzzy inclusion axiom is satisfied in a fuzzy interpretation if holds, which can be evaluated based on the combination functions of some specific fuzzy logic.
4 KLM properties of
In this section we aim at investigating the properties of typicality in and, in particular, verifying whether the KLM postulates of a preferential consequence relation [39, 40] are satisfied in . The satisfiability of KLM postulates of rational or preferential consequence relations [39, 40] has been studied for with defeasible inclusions and typicality inclusions in the two-valued case [13, 32]. The KLM postulates of a preferential consequence relation (i.e., reflexivity, left logical equivalence, right weakening, and, or, cautious monotonicity) can be reformulated for with typicality, by considering that a typicality inclusion stands for a conditional in KLM preferential logics, as follows:
(LLE) If and , then
(RW) If and , then
(AND) If and , then
(OR) If and , then
(CM) If and , then
For , is interpreted as equivalence of concepts and in the underlying description logic (i.e., in all interpretations ), while is interpreted as validity of the inclusion in (i.e., for all interpretations ).
How can these postulates be reformulated in the fuzzy case? First, we can interpret as the requirement that the fuzzy inclusion is valid in fuzzy (that is, is satisfied in all fuzzy interpretations), and as the requirement that the fuzzy inclusions and are valid in fuzzy . For the typicality inclusions, we have some options. We might interpret an inclusion as the fuzzy inclusion , or as the fuzzy inclusion .
The fuzzy inclusion axiom , is rather strong, as it requires that all typical -elements belong to with membership degree . On the other hand, is a weak condition, as it requires that all typical -elements belong to with some membership degree grater that . We will see that both options fail to satisfy one of the postulates. With the first option, the postulates can be reformulated as follows:
(LLE’) If and , then
(RW’) If and , then
(AND’) If and , then
(OR’) If and , then
(CM’) If and , then
We can prove that, in the well-known Zadeh logic and Gödel logic, the postulates above, with the exception of reflexivity , are satisfied in all interpretations.
In Zadeh logic and in Gödel logic any interpretation satisfies postulates and
Let be an interpretation in Zadeh logic, or in Gödel logic. Let us prove, as an example, that satisfies postulate (LLE’).
Assume that axioms , are valid in fuzzy and that is satisfied in . We prove that is satisfied in , that is .
From the validity of and , and in both Zadeh logic and in Gödel logic. Hence,
In Gödel logic, this implies that: for all , and , i.e., . Therefore, the preference relations and must be the same and . Hence, for all , and from , it follows that , that is, is satisfied in .
In Zadeh logic, (3) implies that, for all , and must hold, which implies that either or . It follows that, for all , either or . Hence, for all , it holds that , and from , it follows that is satisfied in .
For the other postulates that proof is similar and is omitted for lack of space. ∎
The meaning of the postulate (REFL’) is that the typical -elements must have a degree of membership in equal to , which may not be the case in an interpretation , when there is no domain element such that . Product and Lukasiewicz logics fail to satisfy postulates (REFL’) and (OR’), but they can be proven to satisfy all the other postulates.
The following corollary is a consequence of Proposition 2.
In Zadeh logic and in Gödel logic, entailment satisfies postulates and .
For instance, for , if and are entailed from a knowledge base in , then they are satisfied in all the models of . Hence, by Proposition 2, is as well satisfied in all the models of , i.e., it is entailed by . As reflexivity is not satisfied, the notion of entailment does not define a preferential consequence relation, under the proposed formulation of the postulates.
Let us consider the alternative formulation of the postulates in the fuzzy case, obtained by interpreting the typicality inclusion as the fuzzy inclusion axiom , that is:
(LLE”) If and , then
(RW”) If and , then
(AND”) If and , then
(OR”) If and , then
(CM”) If and , then
With this formulation of the postulates, it can be proven that in Zadeh logic and Gödel logic all postulates except for Cumulativity (CM”) are satisfied in all interpretations. Under this formulation, reflexivity (REFL”) is satisfied in all interpretations , as it requires that all typical -elements in have a degree of membership in higher than , which holds from the definition of .
In Zadeh logic and in Gödel logic, any interpretation satisfies postulates (REFL”), (LLE”), (RW”), (AND”) and (OR”).
Cumulativity is too strong in the formulation (CM”). From the hypothesis that is entailed from , we know that, in all models of , the typical -elements have some degree of membership in . However, the degree may be small and not enough to conclude that typical -elements are as well typical -elements (which is needed to conclude that is satisfied in , given that is satisfied in ). A weaker alternative formulation of Cumulativity can be obtained by strengthening the antecedent of (CM”) as follows:
(CM) If and , then
This postulate is satisfied by entailment in Zadeh logic and Gödel logic. We can then prove that:
In Zadeh logic and in Gödel logic, entailment satisfies postulates (REFL”), (LLE”), (RW”), (AND”), (OR”) and (CM).
As in the two valued case, the typicality operator introduced in is non-monotonic in the following sense: for a given knowledge base , from we cannot conclude that . Nevertheless, the logic is monotonic, that is, for two knowledge bases and , if , and then . is a fuzzy relative of the monotonic logic .
Although most of the postulates of a preferential consequence relation hold in , this typicality extension of fuzzy is rather weak, as it happens in the two-valued case with and, similarly, with the defeasible extension of in . A monotonic logic, in particular, does not allow to deal with irrelevance. From the fact that birds normally fly, one would like to be able to conclude that normally yellow birds fly (being the color irrelevant to flying). But this conclusions is to be retracted when coming to know that normally yellow birds do not fly. Retracting conclusions in face of new evidence is not possible in a monotonic logic.
In the two-valued case, this has led to the definition of non-monotonic defeasible DLs [16, 15, 34, 6, 17, 28], which exploit some closure construction (such as the rational closure  and the lexicographic closure ) or some notion of minimal entailment [5, 33]. In the next section we strengthen based on a closure construction similar to the one in , but exploiting a weaker notion of coherence, and we discuss its properties.
5 Strengthening : a closure construction
To overcome the weakness of rational closure (as well as of preferential entailment), Lehmann has introduced the lexicographic closure of a conditional knowledge base  which strengthens the rational closure by allowing further inferences. From he semantic point of view, in the propositional case, a preference relation is defined on the set of propositional interpretations, so that the interpretations satisfying conditionals with higher rank are preferred to the interpretations satisfying conditionals with lower rank and, in case of contradictory defaults with the same rank, interpretations satisfying more defaults with that rank are preferred. The ranks of conditionals used by the lexicographic closure construction are those computed by the rational closure construction  and capture specificity: the higher is the rank, the more specific is the default. In other cases, the ranks may be part of the knowledge base specification, such as for ranked knowledge bases in Brewka’s framework of basic preference descriptions , or might be learned from empirical data.
In this section, we consider weighted (fuzzy) knowledge bases, where typicality inclusions are associated to weights, and develop a (semantic) closure construction to strengthen entailment, which leads to a generalization of the notion of fuzzy coherent multipreference model in . The construction is related to the definition of Kern-Isberner’s c-interpretations [36, 37] which also include penalty points for falsified conditionals.
A weighted knowledge base , over a set of distinguished concepts, is a tuple , where is a set of fuzzy inclusion axiom, is a set of fuzzy assertions and is a set of weighted typicality inclusions , where each inclusion has a weight , a real number. As in , the typicality operator is assumed to occur only on the left hand side of a weighted typicality inclusion, and we call distinguished concepts those concepts occurring on the l.h.s. of some typicality inclusion . Arbitrary inclusions and assertions may belong to and .
Consider the weighted knowledge base , over the set of distinguished concepts , with empty ABox and with containing the inclusions:
The weighted TBox contains the following weighted defeasible inclusions:
contains the defeasible inclusions:
, - 70
and contains the defeasible inclusions:
The meaning is that a bird normally has wings, has feathers and flies, but having wings and feather (both with weight 50) for a bird is more plausible than flying (weight 20), although flying is regarded as being plausible. For a penguin, flying is not plausible (inclusion has a negative weight -50), while being a bird and being black are plausible properties of prototypical penguins and and have a positive weight (100 and 50, respectively). Similar considerations can be done for concept . Given Reddy who is red, has wings, has feather and flies (all with degree 1) and Opus who has wings and feather (with degree 1), is black with degree 0.8 and does not fly (), considering the weights of defeasible inclusions, we may expect Reddy to be more typical than Opus as a bird, but less typical as a Penguin.
We define the semantics of a weighted knowledge base trough a semantic closure construction, similar in spirit to Lehmann’s lexicographic closure , but more related to c-interpretations and, additionally, based on multiple preferences. The construction allows to select a subset of the interpretations, the interpretations whose induced preference relations , for the distinguished concepts , faithfully represent the defeasible part of the knowledge base .
Let be the set of weighted typicality inclusions associated to the distinguished concept , and let be a fuzzy interpretation. In the two-valued case, we would associate to each domain element and each distinguished concept , a weight of wrt in , by summing the weights of the defeasible inclusions satisfied by . However, as is a fuzzy interpretation, we not only distinguish between the typicality inclusions satisfied and those falsified by . For all inclusion , we need to consider the degree of membership of in . Furthermore, in comparing the weight of domain elements with respect to , we give higher preference to the domain elements belonging to with a degree grater than , with respect to those belonging to with degree .
For each domain element and distinguished concept , the weight of wrt in the interpretation is defined as follows:
where is added at the bottom of all real values.
The value of is when is not a -element (i.e., ). Otherwise, and the higher is the sum , the more typical is the element relative to concept . How much satisfies a typicality property depends on the value of , which weighted by in the sum. In the two-valued case, , and is the set of the weights of the typicality inclusions for satisfied by , if is a -element, and is , otherwise.
Let us consider again Example 1.
Let be an interpretation such that and
, i.e., Reddy flies, has wings and feather and is red (and ). Suppose further that and and , i.e., Opus does not fly, has wings and feather, and is black with degree 0.8. Considering the weights of the typicality inclusions for , and . This suggests that reddy is more typical as a bird than opus.
On the other hand, if we suppose and , we have: and . This suggests that reddy should be less typical as a penguin than opus.
We have seen in Section 3 that each fuzzy interpretation induces a preference relation for each concept and, in particular, it induces a preference for each distinguished concept . We further require that, if , then must be more typical than wrt , that is, the weight of wrt should be higher than the weight of wrt (and should satisfy more properties or more plausible properties of typical -elements with respect to ). This leads to the following definition of a fuzzy multipreference model of a weighted a knowledge base.
Definition 3 (Faithful (fuzzy) multipreference model of )
Let be a weighted knowledge base over . A faithful (fuzzy) multipreference model (fm-model) of is a fuzzy interpretation s.t.:
satisfies the fuzzy inclusions in and the fuzzy assertions in ;
for all , the preference is faithful to , that is:
Let us consider again Example 2.
Notice that, requiring that the converse of condition (5) also holds, gives the equivalence iff , a stronger condition which would make the notion of faithful multipreference model of above coincide with the notion of coherent fuzzy multipreference model of introduced in . Here, we have considered the weaker notion of faithfulness and a larger class of fuzzy multipreference models of a weighted knowledge base, compared to the class of coherent models. This allows a larger class of monotone non-decreasing activation functions in neural network models to be captured (for space limitation, for this result, we refer to , Sec. 7).
The notion of faithful multipreference entailment (fm-entailment) from a weighted knowledge base can be defined in the obvious way.
Definition 4 (fm-entailment)
A fuzzy axiom is fm-entailed from a fuzzy weighted knowledge base () if, for all fm-models of , satisfies .
From Proposition 3, the next corollary follows as a simple consequence.
In Zadeh logic and in Gödel logic, fm-entailment satisfies postulates (REFL”), (LLE”), (RW”), (AND”), (OR”) and (CM).
To conclude the paper let us infomally describe how fuzzy multipreference entailment deals with irrelevance and avoids inheritance blocking, properties which have been considered as desiderata for preferential logics of defeasible reasoning [50, 37].
For irrelevance, we have already considered an example: if typical birds fly, we would like to conclude that of typical yellow birds fly, as the property of being yellow is irrelevant with respect to flying. Observe, that in Example 2, we can conclude that Reddy is more typical than Opus as a bird (), as Opus does not fly, while Reddy flies. The relative typicality of Reddy and Opus wrt does not depend on the their color, and we would obtain the same relative preferences if reddy were yellow rather than red. A formal proof of the irrelevance property would require that the domain to be large enough to contain some typical bird which is yellow, requiring, as usual in the two valued case , a restriction to some canonical models.
The fuzzy multipreference entailment is not subject to it to the problem called by Pearl the “blockage of property inheritance” problem , and by Benferhat et al. the “drowning problem” . This problem affects rational closure and system Z , as well as rational closure refinements. Roughly speaking, the problem is that property inheritance from classes to subclasses is not guaranteed. If a subclass is exceptional with respect to a superclass for a given property, it does not inherit from that superclass any other property. For instance, referring to the typicality inclusions in Example 2, in the rational closure, typical penguins would not inherit the property of typical birds of having wings, being exceptional to birds concerning flying. On the contrary, in fuzzy multipreference models, considering again Example 2, the degree of membership of a domain element in concept , i.e., , is used to determine the weight of wrt (as the weight of typicality inclusion is positive. The higher is the value of , the higher the value of . Hence, provided the relevant properties of penguins (such as non-flying) remain unaltered, the more typical is as a bird, the more typical is as a Penguin. Notice that the weight of a domain element wrt is related to the interpretation of in by the faithfulness condition.
In this paper we have studied the properties of an extension of fuzzy with typicality, . We have considered some alternative reformulation of the KLM postulates of a preferential consequence relation for , showing that most of these postulates are satisfied, depending on the formulation considered and on the fuzzy logic combination functions. We have considered a (semantic) closure construction to strengthen , by defining a notion of faithful (fuzzy) multipreference model of a weighted knowledge base. Faithful models of a conditional (weighted) knowledge base are a more general class of models with respect to the coherent fuzzy multipreference models considered in  to provide a semantic interpretation of multilayer perceptrons. This allows to capture the larger class of monotone non-decreasing activation functions in multilayer perceptrons (a result which is not included in the paper due to space limitations and for which we refer to ). The paper studies the KLM properties of faithful multipreference entailment and discusses how the multipreference approach allows to deal with irrelevance and avoids inheritance blocking.
The strong relationship between the fuzzy extension of the multipreference semantics for weighted KBs and multilayer perceptrons 
, allows a deep neural network to be regarded as a conditional knowledge base, where neurons correspond to concepts and synaptic weights corresponds to the weights of conditionals. Conversely, a conditional KB (with typicality inclusionsand atomic) can be regarded as a deep network, and its weights can be learned from empirical data. This possibility of combining empirical knowledge and symbolic knowledge in the form of DL axioms motivates the study of the properties of this multipreference extension of a fuzzy DLs.
Undecidability results for fuzzy description logics with general inclusion axioms [2, 19, 9] have motivated restricting the logics to finitely valued semantics , and also motivate the investigation of decidable approximations of fm-entailment. Another issue is whether the multipreference semantics can provide a semantic interpretation of other neural network models, besides multilayer perceptrons and self-organising maps , whose multipreference semantics has been investigated in [29, 30].
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