From Weighted Conditionals of Multilayer Perceptrons to a Gradual Argumentation Semantics

Abstract

A fuzzy multipreference semantics has been recently proposed for weighted conditional knowledge bases, and used to develop a logical semantics for Multilayer Perceptrons, by regarding a deep neural network (after training) as a weighted conditional knowledge base. Based on some different variants of this semantics, we propose some new gradual argumentation semantics, and relate them to the family of the gradual semantics. The relationships between weighted conditional knowledge bases and MLPs extend to the proposed gradual semantics to capture the stationary states of MPs, in agreement with previous results on the relationship between argumentation frameworks and neural networks.

1. Introduction

Argumentation is a reasoning approach which, in its different formulations and semantics, has been used in different contexts in the multi-agent setting, from social networks [ 1 ] to classification [ 2 ], and it is very relevant for decision making and for explanation [ 3 ]. The argumentation semantics are strongly related to other non-monotonic reasoning formalisms and semantics [ 4, 5 ].

Our starting point in this paper is a preferential semantics for commonsense reasoning which has been proposed for a description logic with typicality. Preferential description logics have been studied in the last fifteen years to deal with inheritance with exceptions in ontologies, based on the idea of extending the language of Description Logics (DLs), by allowing for non-strict forms of inclusions, called typicality or defeasible inclusions, of the form T() ⊑ (meaning “the typical -elements are -elements" or “normally ’s are ’s"), with different preferential semantics [ 6, 7 ] and closure constructions [ 8, 9, 10, 11, 12 ]. Such defeasible inclusions correspond to KLM conditionals |∼ [ 13, 14 ], and defeasible DLs inherit and extend some of the preferential semantics and the closure constructions developed within preferential and conditional approaches to commonsense reasoning [ 13, 15, 14, 16, 17 ].

In previous work [ 18 ], a concept-wise multipreference semantics for weighted conditional knowledge bases (KBs) has been proposed to account for preferences with respect to different concepts, by allowing a set of typicality inclusions of the form T() ⊑ with positive or negative weights, for distinguished concepts . The concept-wise multipreference semantics has been first introduced as a semantics for ranked DL knowledge bases [ 19 ] (where conditionals are associated a positive integer rank), and later extended to weighted conditional KBs (in the two-valued and in the fuzzy case), based on a different semantic closure construction, still in the spirit of Lehmann’s lexicographic closure [ 14 ] and Kern-Isberner’s c-representations [ 20, 21 ], but exploiting multiple preferences with respect to concepts.

The concept-wise multipreference semantics has been proven to have some desired properties from the knowledge representation point of view in the two-valued case [ 19, 22 ]: it satisfies the KLM properties of a preferential consequence relation [ 13, 14 ], it allows to deal with specificity and irrelevance and avoids inheritance blocking or the “drowning problem" [ 15, 17 ], and deals with “ambiguity preservation" [ 16 ]. The plausibility of the concept-wise multipreference semantics has also been supported [ 23, 24 ] by showing that it is able to provide a logical interpretation to Kohonen’ Self-Organising Maps [ 25 ], which are psychologically and biologically plausible neural network models. In the fuzzy case, the KLM properties of non-monotomic entailment have been studied in [ 26 ], showing that most KLM postulates are satisfied, depending on their reformulation and on the choice of fuzzy combination functions. It has been shown [ 18 ] that both in the two-valued and in the fuzzy case, the multi-preferential semantics allows to describe the input-output behavior of Multilayer Perceptrons (MLPs), after training, in terms of a preferential interpretation which, in the fuzzy case, can be proved to be a model (in a logical sense) of the weighted KB which is associated to the neural network.

The relationships between preferential and conditional approaches to non-monotonic reasoning and argumentation semantics are strong. Let us also mention, the work by Geffner and Pearl on Conditional Entailment, whose proof theory is defined in terms of “arguments” [ 16 ]. In this paper we aim at investigating the relationships between the fuzzy multipreference semantics for weighted conditionals and gradual argumentation semantics [ 27, 28, 29, 30, 31, 32, 33, 34 ]. To this purpose, in addition to the notions of coherent and faithful fuzzy multipreference semantics [ 18, 26 ], in Section 4, we introduce a notion of -coherent (fuzzy) multipreference semantics. In Section 5, we propose three new gradual semantics for a weighted argumentation graph (namely, a coherent, a faithful and a -coherent semantics) inspired by the fuzzy preferential semantics of weighted conditionals and, in Section 6, we investigate the relationship of -coherent semantics with the family of gradual semantics studied by Amgoud and Doder. The relationships between weighted conditional knowledge bases and MLPs easily extend to the proposed gradual semantics, which captures the stationary states of MLPs. This is in agreement with the previous results on the relationships between argumentation frameworks and neural networks by Garces, Gabbay and Lamb [35] and by Potyca [36]. Section 7 concludes the paper by suggesting a possible approach for defeasible reasoning building on a gradual semantics, as considered in an extended version of this paper [37]. 2. The description logic ℒ and fuzzy ℒ In this section we recall the syntax and semantics of the description logic ℒ [38] and of its fuzzy extension [39]. For sake of simplicity, we only focus on ℒ, the boolean fragment of ℒ, which does not allow for roles. Let be a set of concept names, and a set of individual names. The set of ℒ concepts (or, simply, concepts) can be defined inductively: - ∈ , ⊤ and ⊥ are concepts; - if and are concepts, and ∈ , then ⊓ , ⊔ , ¬ are concepts. An ℒ 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 (), where is a concept and an individual name in .

An ℒ 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 ⊆ ∆ , 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 ℒ interpretation = ⟨∆ , · ⟩: - satisfies an inclusion ⊑ if ⊆ ; - satisfies an assertion () if ∈ .

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 [40, 41, 39, 42, 43], based on the idea that concepts and roles can be interpreted as fuzzy sets. Formulas in Mathematical Fuzzy Logic [44] have a degree of truth in an interpretation rather than being true or false; similarly, axioms in a fuzzy DL have a degree of truth, usually in the interval [ 0, 1 ]. In the following we shortly recall the semantics of a fuzzy extension of ℒ for the fragment ℒ, referring to the survey by Lukasiewicz and Straccia [39]. We limit our consideration to a few features of a fuzzy DL, without considering roles, datatypes, and restricting to the language of ℒ.

A fuzzy interpretation for ℒ is a pair = ⟨∆ , · ⟩ where: ∆ is a non-empty domain and · is fuzzy interpretation function that assigns to each concept name ∈ a function : ∆ → [ 0, 1 ], and to each individual name ∈ an element ∈ ∆ . A domain element ∈ ∆ belongs to the extension of to some degree in [ 0, 1 ], i.e., is a fuzzy set.

The interpretation function · is extended to complex concepts as follows: ⊤ () = 1, ⊥ () = 0, (¬) () = ⊖ (), ( ⊓ ) () = () ⊗ (), ( ⊔ ) () = () ⊕ (). 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 [39] for details). For instance, in Zadeh logic ⊗ = {, }, ⊕ = {, }, ▷ = {1 − , } and ⊖ = 1 − .

The interpretation function · is also extended to non-fuzzy axioms (i.e., to strict inclusions and assertions of an ℒ knowledge base) as follows: ( ⊑ ) = ∈Δ () ▷ (), (()) = ( ).

A fuzzy ℒ 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 ℒ concept inclusion axiom, ∈ {≥ , ≤ , >, <} and ∈ [ 0, 1 ]. A fuzzy ABox is a set of fuzzy assertions of the form () , where is an ℒ concept, ∈ , ∈ {≥ , ≤ , >, <} and ∈ [ 0, 1 ]. Following Bobillo and Straccia [43], 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 ℒ axiom (denoted |= ), as follows, for ∈ {≥ , ≤ , >, <}: - satisfies a fuzzy ℒ inclusion axiom ⊑ if ( ⊑ ) ; - satisfies a fuzzy ℒ assertion () if ( ) ; Given a fuzzy ℒ 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 (i.e., |= ) if for all models =⟨∆ , · ⟩ of , satisfies . 3. Fuzzy ℒ with typicality: ℒ FT In this section, we describe an extension of fuzzy ℒ with typicality following [ 18, 26 ]. Typicality concepts of the form T() are added, where is a concept in fuzzy ℒ. The idea is similar to the extension of ℒ with typicality under the two-valued semantics [ 6 ] but transposed to the fuzzy case. The extension allows for the definition of fuzzy typicality inclusions of the form T() ⊑ , meaning that typical -elements are -elements with a degree greater than . A typicality inclusion T() ⊑ , as in the two-valued case, stands for a KLM conditional implication |∼ [ 13, 14 ], but now it has an associated degree. We call ℒFT the extension of fuzzy ℒ with typicality. As in the two-valued case, and in the propositional typicality logic, PTL, [45] the nesting of the typicality operator is not allowed.

Observe that, in a fuzzy ℒ interpretation = ⟨∆ , · ⟩, the degree of membership () of the domain elements in a concept , induces a preference relation < on ∆ , as follows: < iff () > () (1) Each < has the properties of preference relations in KLM-style ranked interpretations [ 14 ], 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 1 < 0, 2 < 1, 3 < 2, . . . 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 [43] (see Section 2) or that ∆ is finite. For simplicity, we will assume ∆ to be finite.

Each preference relation < has the properties of a preference relation in KLM rational interpretations [ 14 ] (also called ranked interpretations), but here there are multiple preferences and, therefore, fuzzy interpretations can be regarded as multipreferential interpretations, which have been also studied in the two-valued case [ 19, 46, 47 ]. 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 () ̸= 0. Let > 0 be the crisp set containing all domain elements such that () > 0, that is, > 0 = { ∈ ∆ | () > 0}. One can provide a (two-valued) interpretation of typicality concepts T() in a fuzzy interpretation , by letting: (T()) () = ︂{ 1 0 otherwise if ∈ < (> 0) (2) a typical -element in . where <() = { : ∈ and ∄ ∈ s.t. < }. When (T()) () = 1, we say that is

Note that, if () > 0 for some ∈ ∆ , < (> 0) is non-empty. pretation, extended by interpreting typicality concepts as in (2).

Definition 3 ( ℒFT interpretation). An ℒFT interpretation = ⟨∆ , · ⟩ is a fuzzy ℒ interentailment can be defined in a similar way as in fuzzy

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 [ 19 ], 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 T(), for an arbitrary . Here, we do not need to introduce a notion of global preference. The interpretation of any ℒ concept is defined compositionally from the interpretation of atomic concepts, and the preference relation < associated to is defined from . The notions of satisfiability in ℒFT, model of an ℒFT knowledge base, and ℒFT ℒ (see Section 2). 3.1. Strengthening ℒFT: a closure construction To overcome the weakness of preferential entailment, the rational closure [ 14 ] and the lexicographic closure of a conditional knowledge base [48] have been introduced, to allow for further inferences. In this section, we recall a closure construction introduced to strengthen ℒFT entailment for weighted conditional knowledge bases, where typicality inclusions are associated real-valued weights. In the two-valued case, the construction is related to the definition of Kern-Isberner’s c-representations [ 20, 21 ], which include penalty points for falsified conditionals. In the fuzzy case, the construction also relates to the fuzzy extension of rational closure by Casini and Straccia [49].

A weighted ℒFT knowledge base , over a set = {1, . . . , } of distinguished ℒ concepts, is a tuple ⟨ , 1 , . . . , , ⟩, where is a set of fuzzy ℒFT inclusion axiom, is a set of fuzzy ℒFT assertions and = {(ℎ, ℎ )} is a set of all weighted typicality inclusions ℎ = T() ⊑ ,ℎ for , indexed by ℎ, where each inclusion ℎ has weight ℎ , a real number. As in [ 18 ], the typicality operator is assumed to occur only on the left hand side of a may belong to and . Let us consider the following example from [ 26 ]. weighted typicality inclusion, and we call distinguished concepts those concepts occurring on the l.h.s. of some typicality inclusion T() ⊑ . Arbitrary ℒFT inclusions and assertions Example 1. Consider the weighted knowledge base = ⟨ , , , , ⟩, over the set of distinguished concepts = {Bird , Penguin, Canary }, with empty and containing, for instance, the inclusions:

Yellow ⊓ Black ⊑ ⊥ ≥ 1 Yellow ⊓ Red ⊑ ⊥ ≥ 1 Black ⊓ Red ⊑ ⊥ ≥ 1 The weighted TBox contains the following weighted defeasible inclusions: (1) T(Bird ) ⊑ Fly , +20 (2) T(Bird ) ⊑ Has_Wings, +50 (3) T(Bird ) ⊑ Has_Feather , +50; and contain, respectively, the following defeasible inclusions: (4) T(Penguin) ⊑ Bird , +100 (7) T(Canary ) ⊑ Bird , +100 (5) T(Penguin) ⊑ Fly , - 70 (8) T(Canary ) ⊑ Yellow , +30 (6) T(Penguin) ⊑ Black , +50; (9) T(Canary ) ⊑ Red , +20 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 (5) has negative weight -70), while being a bird and being black are plausible properties of prototypical penguins, and (4) and (6) have positive weights (100 and 50, respectively). Similar considerations can be done for concept Canary . Given an Abox in which Reddy is red, has wings, has feather and lfies (all with degree 1) and Opus has wings and feather (with degree 1), is black with degree 0.8 and does not fly ( Fly I (opus) = 0 ), considering the weights of defeasible inclusions, we may expect Reddy to be more typical than Opus as a bird, but less typical than Opus as a penguin.

The semantics of a weighted knowledge base is defined in [ 18 ] trough a semantic closure construction, similar in spirit to Lehmann’s lexicographic closure [48], but strictly related to c-representations and, additionally, based on multiple preferences. The construction allows a subset of the ℒFT interpretations to be selected, the interpretations whose induced preference relations < , for the distinguished concepts , faithfully represent the defeasible part of the knowledge base .

Let = {(ℎ, ℎ )} be the set of weighted typicality inclusions ℎ = T() ⊑ ,ℎ associated to the distinguished concept , and let = ⟨∆ , · ⟩ be a fuzzy ℒFT 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 do not only distinguish between the typicality inclusions satisfied or falsified by ; we also need to consider, for all inclusions T() ⊑ ,ℎ ∈ , the degree of membership of in ,ℎ. Furthermore, in comparing the weight of domain elements with respect to < , we give higher preference to the domain elements belonging to (with a degree greater than 0), with respect to those not belonging to (having membership degree 0).

For each domain element ∈ ∆ and distinguished concept , the weight () of wrt in the ℒFT interpretation = ⟨∆ , · ⟩ is defined as follows: () = ︂{ ∑︀ℎ ℎ ,ℎ() −∞ if () > 0 otherwise (3) where −∞

is added at the bottom of all real values.

The value of () is −∞ when is not a -element (i.e., () = 0). Otherwise, () > 0 and the higher is the sum (), the more typical is the element relative to the defeasible properties of . How much satisfies a typicality property T() ⊑ ,ℎ depends on the value of ,ℎ() ∈ [ 0, 1 ], which is weighted by ℎ in the sum. In the two-valued case, ,ℎ() ∈ {0, 1}, and () is the sum of the weights of the typicality inclusions for satisfied by , if is a -element, and is −∞ , otherwise.

Example 2. Let us consider again Example 1. Let be an ℒFT interpretation such that Fly I (reddy ) = (Has_Wings)I (reddy ) = (Has_Feather )I (reddy ) = 1 and Red I (red - dy ) = 1 , i.e., Reddy flies, has wings and feather and is red (and Black I (reddy ) = 0). Suppose further that Fly I (opus) = 0 and (Has_Wings)I (opus) = (Has_ Feather )I (opus) = 1 and Black I (opus) = 0 .8 , i.e., Opus does not fly, has wings and feather, and is black with degree 0.8. Considering the weights of typicality inclusions for Bird , WBird (reddy ) = 20 + 50 + 50 = 120 and WBird (opus) = 0 + 50 + 50 = 100. This suggests that reddy should be more typical as a bird than opus. On the other hand, if we suppose further that Bird I (reddy ) = 1 and Bird I (opus) = 0.8, then WPenguin (reddy ) = 100 − 70 + 0 = 30 and WPenguin (opus) = 0 .8 × 100 − 0 + 0 .8 × 50 = 120 , and Reddy should be less typical as a penguin than Opus.

In [ 18 ] a notion of coherence is introduced, to force an agreement between the preference relations < induced by a fuzzy interpretation , for each distinguished concept , and the weights () computed, for each ∈ ∆ , from the conditional knowledge base , given the interpretation . This leads to the following definition of a coherent fuzzy multipreference model of a weighted a ℒFT knowledge base.

Definition 4 (Coherent (fuzzy) multipreference model of [ 18 ]). Let = ⟨ , 1 , . . . , , ⟩ be a weighted ℒFT knowledge base over . A coherent (fuzzy) multipreference model (cf-model) of is a fuzzy ℒFT interpretation = ⟨∆ , · ⟩ s.t.: • satisfies the fuzzy inclusions in and the fuzzy assertions in ; • for all ∈ , the preference < is coherent to , that is, for all , ∈ ∆ , < ⇐⇒ () > () In a similar way, one can define a faithful (fuzzy) multipreference model (fm-model) of by replacing the coherence condition (4) with the following faithfulness condition (called weak coherence in [50]): for all , ∈ ∆ , < ⇒ () > (). (4) (5) The weaker notion of faithfulness allows to define 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, whose activation function is monotonically non-decreasing (we refer to [50], Sec. 7). Example 3. Referring to Example 2 above, where Bird I (reddy ) = 1 , Bird I (opus) = 0.8, let us further assume that PenguinI (reddy ) = 0 .2 and PenguinI (opus) = 0 .8 . Clearly, < and < . For the interpretation to be faithful, it is necessary that the conditions WBird (reddy ) > WBird (opus) and WPenguin (opus) > WPenguin (reddy ) hold; which is true. On the contrary, if it were PenguinI (reddy ) = 0 .9 , the interpretation would not be faithful. For PenguinI (reddy ) = 0 .8 , the interpretation would be faithful, but not coherent, as WPenguin (opus) > WPenguin (reddy ), but PenguinI (opus) = PenguinI (reddy ).

It has been shown [ 18, 50 ] that the proposed semantics allows the input-output behavior of a deep network (considered after training) to be captured by a fuzzy multipreference interpretation built over a set of input stimuli, through a simple construction which exploits the activity level of neurons for the stimuli. Each unit ℎ of can be associated to a concept name ℎ and, for a given domain ∆ of input stimuli, the activation value of unit ℎ for a stimulus is interpreted as the degree of membership of in concept ℎ. The resulting preferential interpretation can be used for verifying properties of the network by model checking.

For MLPs, the deep network itself can be regarded as a conditional knowledge base, by mapping synaptic connections to weighted conditionals, so that the input-output model of the network can be regarded as a coherent-model of the associated conditional knowledge base [ 18 ].

4. -coherent models

In this section we consider a new notion of coherence of a fuzzy interpretation wrt a KB, that we call -coherence, where is a function from R to the interval [ 0, 1 ], i.e., : R → [ 0, 1 ]. We also establish it relationships with coherent and faithful models.

Definition 5 ( -coherence). Let = ⟨ , 1 , . . . , , ⟩ be a weighted ℒFT knowledge base, and : R → [ 0, 1 ]. A fuzzy ℒFT interpretation = ⟨∆ , · ⟩ is -coherent if, for all concepts ∈ and ∈ ∆ , () = (∑︁ ℎ ,ℎ()) ℎ (6) where = {(T() ⊑ ,ℎ, ℎ )} is the set of weighted conditionals for . To define -coherent multipreference model of a knowledge base , we can replace the coherence condition (4) in Definition 4 with the notion of -coherence of an interpretation wrt the knowledge base .

Observe that, for all such that () > 0, condition (6) above corresponds to condition () = (()). While the notions of coherence and of weight () (of an element wrt a concept ) consider, as a special case, the case when () = 0, in condition (6) we impose the same constraint to all domain elements (including those with () = 0).

For Multilayer Perceptrons, let us associate a concept name to each unit in a deep network , and let us interpret, as in [ 18 ], a synaptic connection between neuron ℎ and neuron with weight ℎ as the conditional T() ⊑ with weight ℎ = ℎ. If we assume that is the activation function of all units in the network , then condition (6) characterizes the stationary states of MLPs, where () corresponds to the activation of neuron for some input stimulus and ∑︀ℎ ℎ ,ℎ() corresponds to the induced local field of neuron , which is obtained by summing the input signals to the neuron, 1, . . . , , weighted by the respective synaptic weights: ∑︀

ℎ=1 ℎℎ [51]. Here, each ,ℎ() corresponds to the input signal ℎ, for input stimulus . Of course, -coherence could be easily extended to deal with different activation functions , one for each concept (i.e., for each unit ).

Proposition 1. Let be a weighted conditional knowledge base and : R → [ 0, 1 ]. (1) if is a monotonically non-decreasing function, a -coherent fuzzy multipreference model of is also an fm-model of ; (2) if is a monotonically increasing function, a -coherent fuzzy multipreference model of is also an cf-model of .

All proofs can be found in the technical report [37]. Item 2 can be regarded as the analog of ,Δ of a deep neural Proposition 1 in [ 18, 50 ], where the fuzzy multi-preferential interpretation ℳ network , built over the domain of input stimuli ∆ , is proven to be a coherent model of the knowledge base associated to , under the specified conditions on the activation function , and the assumption that each stimulus in ∆ corresponds to a stationary state in the neural ,Δ network. Item 1 in Proposition 1 is as well the analog of Proposition 2 in [50] stating that ℳ is a faithful (or weakly-coerent) model of .

A notion of coherent/faithful/ -coherent multipreference entailment from a weighted ℒFT knowledge base can be defined in the obvious way (see [ 18, 26 ] for the definitions of coherent and faithful (fuzzy) multipreference entailment). The properties of faithful entailment have been studied in [ 26 ]. Faithful entailment is reasonably well-behaved: it deals with specificity and irrelevance; it is not subject to inheritance blocking; it satisfies most KLM properties [ 13, 14 ], depending on their fuzzy reformulation and on the chosen combination functions.

As MLPs are usually represented as a weighted graphs [51], whose nodes are units and whose edges are the synaptic connections between units with their weight, it is very tempting to extend the different semantics of weighted knowledge bases considered above, to weighted argumentation graphs.

5. Coherent, faithful and -coherent semantics for weighted argumentation graphs

There is much work in the literature concerning extension of Dung’s argumentation framework [ 4 ] with weights attached to arguments and/or to the attacks between arguments. Many different proposals have been investigated and compared in the literature. Let us just mention [ 27, 28, 29, 30, 31, 33 ] for the moment, which also include extensive comparisons. In the following, we will propose some semantics for weighted argumentation with the purpose of establishing some links with the semantics of conditional knowledge bases considered in the previous section.

In the following, we will consider a notion of weighted argumentation graph as a triple = ⟨, ℛ, ⟩, where is a set of arguments, ℛ ⊆ × and : ℛ → R. This definition of weighted argumentation graph corresponds to the definition of weighted argument system in [ 29 ], but here we admit both positive and negative weights, while [ 29 ] only allows for positive weights representing the strength of attacks. In our notion of weighted graph, a pair (, ) ∈ ℛ can be regarded as a support relation when the weight is positive and an attack relation when the weight is negative, and it leads to bipolar argumentation [52]. The argumentation semantics we will consider in the following, as in the case of weighted conditionals, deals with both the positive and the negative weights in a uniform way. For the moment we do not include in a function determining the basic strength of arguments [ 31 ].

Given a weighted argumentation graph = ⟨, ℛ, ⟩, we define a labelling of the graph as a function : → [ 0, 1 ] which assigns to each argument and acceptability degree, i.e., a value in the interval [ 0, 1 ]. Let R− (A) = {B | (B , A) ∈ ℛ}. When R− (A) = ∅, argument has neither supports nor attacks.

For a weighted graph = ⟨, ℛ, ⟩ and a labelling , we introduce a weight on , as a partial function : → R, assigning a positive or negative support to the arguments ∈ such that R− (Ai ) ̸= ∅, as follows: () =

∑︁ (,)∈ℛ ( , ) ( ) (7) When R− (Ai ) = ∅, () is let undefined.

We can now exploit this notion of weight of an argument to define different argumentation semantics for a graph as follows.

Definition 6. Given a weighted graph = ⟨, ℛ, ⟩ and a labelling : • is a coherent labelling of if, for all arguments , ∈ s.t. R− (A) ̸= ∅ and R− (B ) ̸= ∅,

() < () ⇐⇒ • is a faithfull labelling of if, for all arguments , ∈ s.t. R− (A) ̸= ∅ and R− (B ) ̸= ∅,

() < (); () < () ⇒ • for a function : R → [ 0, 1 ], is a -coherent labelling of if, for all arguments ∈ s.t. R− (A) ̸= ∅, () = ( ()).

() < (); These definitions do not put any constraint on the labelling of arguments which do not have incoming edges in : their labelling is arbitrary, provided the constraints on the labelings of all other arguments can be satisfied, depending on the semantics considered.

The definition of -coherent labelling of is defined through a set of equations, as in Gabbay’s equational approach to argumentation networks [53]. Here, we use equations for defining the weights of arguments starting from the weights of attacks/supports.

A -coherent labelling of a weigthed graph can be proven to be as well a coherent labelling or a faithful labelling, under some conditions on the function .

Proposition 2. Given a weighted graph = ⟨, ℛ, ⟩: (1) A coherent labelling of is a faithful labelling of ; (2) if is a monotonically non-decreasing function, a -coherent labelling of is a faithful labelling of ; (3) if is a monotonically increasing function, a -coherent labelling of is a coherent labelling of .

The proof is similar to the one of Proposition 1, and can be found in [37]. It exploits the property of a -labelling that () = ( ()), for all arguments with R− (A) ̸= ∅, as well as the properties of .

6. -coherent labellings and the gradual semantics

The notion of -coherent labelling relates to the framework of gradual semantics studied by Amgoud and Doder [ 33 ] where, for the sake of simplicity, the weights of arguments and attacks are in the interval [ 0, 1 ]. Here, as we have seen, positive and negative weights are admitted to represent the strength of attacks and supports. To define an evaluation method for -coherent labellings, we need to consider a slightly extended definition of an evaluation method for a graph in [ 33 ]. Following [ 33 ] we include a function 0 : → [ 0, 1 ] in the definition of a weighted graph, where 0 assigns to each argument ∈ its basic strength. Hence a graph becomes a quadruple = ⟨, 0, ℛ, ⟩.

An evaluation method for a graph = ⟨, 0, ℛ, ⟩ is a triple = ⟨ℎ, , ⟩, where1: ℎ : R × [ 0, 1 ] → R : ⋃︀+=∞0 R → R : [ 0, 1 ] × () → [ 0, 1 ] Function ℎ is intended to calculate the strength of an attack/support by aggregating the weight on the edge between two arguments with the strength of the attacker/supporter. Function aggregates the strength of all attacks and supports to a given argument, and function returns a value for an argument, given the strength of the argument and aggregated weight of its attacks and supports.

As in [ 33 ], a gradual semantics is a function assigning to any graph = ⟨, 0, ℛ, ⟩ a weighting on , i.e., : → [ 0, 1 ], where () represents the strength of an argument (or its acceptability degree).

A gradual semantics is based on an evaluation method iff, ∀ = ⟨, 0, ℛ, ⟩, ∀ ∈ , () = ( 0(), (ℎ( (1, ), (1)), . . . , ℎ( (, ), ())) (8) where B1 , . . . , Bn are all arguments attacking or supporting (i.e., R− (A) = {B1 , . . . , Bn }).

Let us consider the evaluation method = ⟨ℎ, , ⟩, where the functions ℎ and are defined as in [ 33 ], i.e., ℎ(, ) = · and (1, . . . , ) = ∑︀ =1 , but we let () to be undefined . We let (, ) = when is undefined, and (, ) = () otherwise. The function returns a value which is independent from the first argument, when the second argument is not undefined (i.e., there is some support/attack for the argument). When has neither attacks nor supports (R− (A) = ∅), returns the basic strength of , 0().

The evaluation method = ⟨ℎ, , ⟩ provides a characterization of the -coherent labelling for an argumentation graph, in the following sense.

Proposition 3. Let = ⟨, ℛ, ⟩ be a weighted argumentation graph. If, for some 0 : → [ 0, 1 ], is a gradual semantics of graph ′ = ⟨, 0, ℛ, ⟩ based on the evaluation method = ⟨ℎ, , ⟩, then ′ is a -coherent labelling for .

Vice-versa, if is a -coherent labelling for , then there are a function 0 and a gradual semantics based on the evaluation method = ⟨ℎ, , ⟩, such that, for the graph ′ = ⟨, 0, ℛ, ⟩,

′ ≡ .

1This definition is the same as in [ 33 ], but for the fact that in the domain/range of functions ℎ and interval [ 0, 1 ] is sometimes replaced by R.

The proof can be found in [37].

Amgoud and Doder [ 33 ] study a large family of determinative and well-behaved evaluation models for weighted graphs in which attacks have positive weights in the interval [ 0, 1 ]. For weighted graph with positive and negative weights, the evaluation method cannot be guaranteed to be determinative, even under the conditions that is monotonically increasing and continuous. In general, there is not a unique semantics based on , and there is not a unique -coherent labelling for a weighted graph , given a basic strength 0. This is not surprising, considering that -coherent labelings of a graph correspond to stationary states (or equilibrium states) [51] in a deep neural network.

A deep neural network can than be seen as a weighted argumentation graph, with positive and negative weights, where each unit in the network is associated to an argument, and the activation value of the unit can be regarded as the weight (in the interval [ 0, 1 ]) of the corresponding argument. Synaptic positive and negative weights correspond to the strength of supports (when positive) and attacks (when negative). In this view, -coherent labelings, assigning to each argument a weight in the interval [ 0, 1 ], correspond to stationary states of the network, the solutions of a set of equations. This is in agreement with previous results on the relationship between weighted argumentation graphs and MLPs established by Garcez, Gabbay and Lamb [35] and, more recently, by Potyca [36]. We refer to [37] for comparisons.

Unless the network is feedforward (and the corresponding graph is acyclic), stationary states cannot be uniquely determined by an iterative process from the values of input units (that is, from an initial labelling 0). On the other hand, a semantics based on satisfies some of the properties considered in [ 33 ], including anonymity, independence, directionality, equivalence and maximality, provided the last two properties are properly reformulated to deal with both positive and negative weights (i.e., by replacing R− (x ) to (), for each argument in the formulation in [ 33 ]). However, a semantics based on cannot be expected to satisfy the properties of neutrality, weakening, proportionality and resilience. In fact, function completely disregard the initial valuation 0 in graph = ⟨, 0, ℛ, ⟩, for those arguments having some incoming edge (even if their weight is 0). So, for instance, it is not the same, for an argument to have a support with weight 0 or no support or attack at all: neutrality does not hold.

7. Conclusions

In this paper, drawing inspiration from a fuzzy preferential semantics for weighted conditionals, which has been introduced for modeling the behavior of Multilayer Perceptrons [ 18 ], we develop some semantics for weighted argumentation graphs, where positive and negative weights can be associated to pairs of arguments. In particular, we introduce the notions of coherent/faithful/ coherent labellings, and establish some relationships among them. While in [ 18 ] a deep neural network is mapped to a weighted conditional knowledge base, a deep neural network can as well be seen as a weighted argumentation graph, with positive and negative weights, under the proposed semantics. In this view, -coherent labellings correspond to stationary states in the network (where each unit in the network is associated to an argument and the activation value of the unit can be regarded as the weight of the corresponding argument). This is in agreement with previous work on the relationship between argumentation frameworks and neural networks first investigated by Garcez, et al. [35] and recently by Potyca [36]. See in [37] for comparisons.

The proposed approach suggests interesting directions for future work. On the one hand, the generality of the fuzzy conditional logic, where in T() ⊑ , and are boolean concepts, suggests a simple approach to deal with attacks/supports by boolean combination of arguments, based on the fuzzy semantics of weighted conditionals [37]. On the other hand, it has been shown in [37] that, under suitable conditions on , a multipreference model can be constructed over a (finite) set of -labelling Σ . This allows (fuzzy) conditional formulas over arguments to be validated by model checking over a preferential model. For instance, the property: "does normally argument 2 follows from argument 1 with a degree greater than 0.7?" can be formalized by the fuzzy inclusion T(1) ⊑ 2 > 0.7. Whether this approach can be extended to the other gradual semantics, and under which conditions on the evaluation method, is subject of future work.

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