A fuzzy multipreference semantics has been recently proposed for weighted conditional knowledge bases with typicality, 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 different variants of this semantics, we propose some new gradual argumentation semantics, and relate them to the family of the gradual semantics. The paper also suggests an approach for defeasible reasoning over a weighted argumentation graph, building on the proposed semantics.
In previous work [42], a concept-wise multipreferArgumentation is a reasoning approach which, in its differ- ence semantics for weighted conditional knowledge bases ent formulations and semantics, has been used in different (KBs) has been proposed to account for preferences with contexts in the multi-agent setting, from social networks respect to different concepts, by allowing a set of typicality [54] to classification [ 5], and it is very relevant for deci- inclusions of the form T() ⊑ with positive or negasion making and for explanation [61]. The argumentation tive weights, for distinguished concepts . The conceptsemantics are strongly related to other non-monotonic wise multipreference semantics has been first introduced reasoning formalisms and semantics [29, 1]. as a semantics for ranked DL knowledge bases [41], where
Our starting point in this paper is a preferential seman- conditionals are given a positive integer rank, and later
tics for commonsense reasoning which has been proposed extended to weighted conditional KBs, in the two-valued
for a description logic with typicality. Preferential de- and in the fuzzy case, based on a different semantic
closcription logics have been studied in the last fifteen years sure construction, still in the spirit of Lehmann’s
lexicoto deal with inheritance with exceptions in ontologies, graphic closure [53] and Kern-Isberner’s c-representations
based on the idea of extending the language of Descrip- [47, 48], but exploiting multiple preferences with respect
tion Logics (DLs), by allowing for non-strict forms of to concepts.
inclusions, called typicality or defeasible inclusions, of The concept-wise multipreference semantics has been
the form T() ⊑ (meaning “the typical -elements proven to have some desired properties from the
knowlare -elements" or “normally ’s are ’s"), with dif- edge representation point of view in the two-valued case
ferent preferential semantics [39, 18] and closure con- [41]: it satisfies the KLM properties of a preferential
constructions, by Casini and Straccia [
ECURWorkshpPcedingt:/u-w.ISN16307 CEUR Workshop Proceedings (CEUR-WS.org) can be proven to be a model (in a logical sense) of the An ℒ interpretation is defined as a pair = ⟨Δ, · ⟩ weighted KB which is associated to the neural network. where: Δ is a domain—a set whose elements are denoted
The relationships between preferential and conditional by , , , . . . —and · is an extension function that maps tation semantics are strong. Let us just mention, the ienadcihvicdounaclenpatmnaeme ∈ ∈to taonaelseemtent ⊆ ∈ Δ. It is approaches to non-monotonic reasoning and argumen- Δ, and each work by Geffner and Pearl on Conditional Entailment, extended to complex concepts as follows: whose proof theory is defined in terms of “arguments” [34]. In this paper we aim at investigating the relation- ⊤ = Δ ⊥ = ∅ (¬) = Δ∖ ships between the fuzzy multipreference semantics for ( ⊓ ) = ∩ ( ⊔ ) = ∪ weighted conditionals and gradual argumentation seman- The notion of satisfiability of a KB in an interpretation tics [24, 46, 30, 31, 2, 7, 4, 60]. To this purpose, in addi- and the notion of entailment are defined as follows: tion to the notions of coherent [42] and faithful [36] fuzzy multipreference semantics, in Section 4, we introduce a Definition 1 (Satisfiability and entailment). Given an notion of -coherent fuzzy multipreference semantics. In ℒ interpretation = ⟨Δ, · ⟩: Section 5, we propose three new gradual semantics for - satisfies an inclusion ⊑ if ⊆ ; a weighted argumentation graph (namely, a coherent, a - satisfies an assertion () if ∈ . faithful and a -coherent semantics) inspired by the fuzzy Given a knowledge base = ( , ), an interpretapreferential semantics of weighted conditionals and, in tion satisfies (resp. ) if satisfies all inclusions Section 6, we investigate the relationship of -coherent in (resp. all assertions in ); is a model of if semantics with the family of gradual semantics studied by satisfies and .
Amgoud and Doder. The relationships between weighted A subsumption = ⊑ (resp., an assertion
conditional knowledge bases and MLPs easily extend to ()), is entailed by , written |= , if for all models
the proposed gradual semantics, which captures the sta- =⟨Δ, · ⟩ of , satisfies .
tionary behavior of MLPs. This is in agreement with
the previous results on the relationships between argu- Given a knowledge base , the subsumption problem is
mentation frameworks and neural networks by Garces, the problem of deciding whether an inclusion ⊑ is
Gabbay and Lamb [27] and by Potyca [57]. Section 7 sug- entailed by .
gests a possible approach for defeasible reasoning over Fuzzy description logics have been widely studied in
an argumentation graph, building on the proposed gradual the literature for representing vagueness in DLs by
Stracsemantics. cia [59], Stoilos [58], Lukasiewicz and Straccia [
A preliminary version of this work has been pre- Borgwardt et al. [13], Bobillo and Straccia [10], based
sented in [35]. For the proofs of the results we refer on the idea that concepts and roles can be interpreted as
to https://arxiv.org/abs/2110.03643v2. fuzzy sets. Formulas in Mathematical Fuzzy Logic [26]
have a degree of truth in an interpretation rather than
being true or false; similarly, axioms in a fuzzy DL have a
2. The description logic ℒ and degree of truth, usually in the interval [0, 1]. In the
followfuzzy ℒ ing we shortly recall the semantics of a fuzzy extension
of ℒ for the fragment ℒ, referring to the survey by
In this section we recall the syntax and semantics of a Lukasiewicz and Straccia [
are concepts. The interpretation function · is extended to complex concepts as follows: ⊤ () = 1, ⊥ () = 0, (¬) () = ⊖ (), ( ⊓ ) () = () ⊗ (), ( ⊔ ) () = () ⊕ ().
An ℒ knowledge base 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 . where ∈ Δ and ⊗ , ⊕ , ▷ and ⊖ are a t-norm, an s-norm, but now it has an associated degree. We call ℒ and ⊖ = 1
− .
an implication function, and a negation function, chosen
among the combination functions of fuzzy logics (we refer
to [
Observe that, in a fuzzy ℒ interpretation = ⟨Δ, · ⟩, the degree of membership () of the domain elements fuzzy axioms (i.e., to strict inclusions and assertions of an as follows:
The interpretation function · is also extended to non- in a concept induces a preference relation < on Δ,
ℒ knowledge base) as follows: (()) = ( ).
( ⊑ ) = ∈Δ () ▷ (),
A fuzzy ℒ knowledge base is a pair ( , ) where is a fuzzy TBox and a fuzzy ABox. TBox is a set of fuzzy concept inclusions of the form A fuzzy ⊑ , 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 [10], 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. terpretation satisfies a fuzzy |= ), as follows: ℒ axiom (denoted if ( ⊑ ) ; • satisfies a fuzzy
( ) • satisfies a fuzzy ℒ inclusion axiom ⊑ ℒ assertion () if satisfies . where for ∈ {≥ , ≤ , >, <} and ∈ [0, 1].
Given a fuzzy ℒ knowledge base = ( , ), 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 =⟨Δ, · ⟩ of ,
ℒ with typicality: ℒ FT In this section, we describe an extension of fuzzy ℒ with typicality following [42, 36]. 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 [39] 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 such that holds. In the two-valued case, a typicality inclusion T() ⊑ stands for a KLM conditional implication |∼ [51, 52], < iff () > () (1) or < ).
modular and well-founded strict partial order, under the assumption that fuzzy interpretations are witnessed (see
< is wellfounded if there is no infinite descending chain 1 < 0, 2 < 1, 3 < 2, . . . of domain elements; < is modular if, for all , , ∈ Δ, < implies ( <
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 = { ∈
Δ | () > 0}. One can provide a (two-valued) interpretation of typicality concepts T() in a fuzzy interpretation , by letting: (T()) () = ︂{ 1 0 if ∈ < (> 0)
(2) < (> 0) is non-empty. where <() = { : ∈ and ∄ ∈ s.t. < }.
When (T()) () = 1, we say that is a typical element in . Notice that, if () > 0 for some ∈ Δ, Definition 3 ( ℒ
FT interpretation). An ℒFT intertended by interpreting typicality concepts as in (2). pretation = ⟨Δ, · ⟩ is a fuzzy ℒ interpretation, ex
The fuzzy interpretation = ⟨Δ, · ⟩ implicitly defines a multipreference interpretation, where any concept is associated to a preference relation < . This is different from the two-valued multipreference semantics in [41], where only the 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 any arbitrary . Here, we do not need to introduce a notion of global preference. The interpre- penguin, flying is not plausible (inclusion (5) has negatation 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 ℒ a similar way as in fuzzy ℒ (see Section 2). knowledge base, and ℒFT entailment can be defined in
FT, model of an ℒFT 3.1. Strengthening ℒ constructions
tive weight -70), while being a bird and being black are plausible properties of prototypical penguins, and (4) and (6) have positive weights. Given an ABox in which Reddy is red, has wings, has feather and flies (all with degree 1) and Opus has wings and feather, does not fly (with degree 1), and is black with degree 0.8, 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. of Kern-Isberner’s c-representations [47, 48], which in- inclusions ℎ = T() ⊑ ,ℎ associated to the distinℒ
rational closure [52] and the lexicographic closure of a conditional knowledge base [53] have been introduced. In this section, we recall a closure construction introduced by Giordano and Theseider Dupré [42] to strengthen
FT entailment for weighted conditional knowledge bases, and then we consider some variants. In the twovalued case, the construction is related to the definition typicality inclusions ℎ = clude penalty points for falsified conditionals. In the fuzzy case, the construction also relates to the fuzzy extension of rational closure by Casini and Straccia [22]. ℒ
A weighted ℒFT knowledge base , over a set = {1, . . . , } of distinguished ℒ concepts, is a tuple ⟨ , 1 , . . . , , ⟩, where is a set of fuzzy
FT inclusion axiom, is a set of fuzzy ℒ sertions and = {(ℎ, ℎ )} is a set of all weighted FT asT() ⊑ ,ℎ for , indexed by ℎ, where each inclusion ℎ has weight ℎ , a real number. As in [42], the typicality operator is assumed to occur only on the l.h.s. of a weighted typicality inclusion, and we call distinguished concepts those concepts occurring in the l.h.s. of such inclusions. Arbitrary ℒ
FT inclusions and assertions may belong to and ℒ F.TLektnuoswcolendsgideebratsheeafdoalplotewdinfrgoemxa[3m6p]l:e of weighted inclusions: Example 1. Consider the weighted knowledge base = ⟨ , , , ⟩, over the set of distinguished concepts =
{Bird , Penguin}, with the strict TBox containing the inclusion Black ⊓ Red ⊑ ⊥ ≥ weighted TBox containing the weighted defeasible 1 ; the (1) T(Bird ) ⊑ Fly , +20 (2) T(Bird ) ⊑ Has_Wings , +50 (3) T(Bird ) ⊑ Has_Feather , +50; (4) T(Penguin) ⊑ Bird , +100 (5) T(Penguin) ⊑ Fly , - 70 (6) T(Penguin) ⊑ Black , +50. containing the weighted 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
in [42] trough a semantic closure construction, which allows a subset of the ℒ namely, the interpretations whose induced preference relations < , for the distinguished concepts , coherently or faithfully represent the defeasible part of the knowledge
FT interpretations to be selected, base .
Let = {(ℎ, ℎ )} be the set of weighted typicality guished concept , and let = ⟨Δ, · ⟩ be a fuzzy ℒ interpretation. In the two-valued case, we would associate FT to each domain element ∈ Δ and each distinguished concept , a weight () of wrt in , by summing the weights of the defeasible inclusions for satisfied by . However, as is a fuzzy interpretation, 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). interpretation = ⟨Δ, · ⟩ is defined as follows:
For each domain element ∈ Δ and distinguished concept , the weight () of wrt in the ℒFT () = ︂{ ∑︀ℎ ℎ ,ℎ() −∞ if () > 0
Let = ⟨ , 1 , . . . , , ⟩ be a weighted ℒFT knowledge base over . A coherent (fuzzy) multipreference model (cf-model) of is a fuzzy ℒFT interpretation = ⟨Δ, · ⟩ s.t.: • satisfies the fuzzy inclusions in and the fuzzy assertions in ; • for all ∈ , the preference < is coherent to , that is, for all , ∈ Δ, < ⇐⇒ () > ()
For MLPs, the deep network itself can be regarded as a conditional knowledge base, by mapping synaptic connections to weighted conditionals, so that the inputoutput model of the network can be regarded as a coherentmodel of the associated conditional knowledge base [42].
(4) (5)
In a similar way, one can define a faithful (fuzzy) multipref- fuzzy interpretation wrt a KB, that we call -coherence, erence model (fm-model) of by replacing the coherence where is a function from R to the interval [0, 1], i.e., condition (4) with the following faithfulness condition : R → [0, 1]. We also establish it relationships with (called weak coherence in [42], extended version): for all coherent and faithful models. , ∈ Δ,
< ⇒ () > ().
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 nondecreasing (we refer to [42], extended version, Sec. 7).
Definition 5 ( -coherence). Let = ⟨ , 1 , . . . , , ⟩ be a weighted ℒFT knowledge base, and : R → [0, 1]. A fuzzy ℒFT interpretation = ⟨Δ, · ⟩ is -coherent if, for all concepts ∈ and ∈ Δ, () = (∑︁ ℎ ,ℎ()) ℎ
(6) where = {(T() ⊑ ,ℎ, ℎ )} is the set of weighted conditionals for .
Example 2. Referring to Example 1 above, To define -coherent multipreference model of a knowllet us further assume that Bird I (reddy ) = 1 , edge base , we can replace the coherence condition Bird I (opus ) = 0.8, that PenguinI (reddy ) = 0 .2 and (4) in Definition 4 with the notion of -coherence of an PenguinI (opus ) = 0 .8 . Clearly, < interpretation wrt the knowledge base . and < . The interpretation to be Observe that, for all such that () > 0, condition faithful and coherent, as WBird (reddy ) > WBird (opus ) (6) above corresponds to condition () = (()). and WPenguin (opus ) > WPenguin (reddy ) While in coherent and faithful models the notion of weight hold. On the contrary, if we had PenguinI () considers, as a special case, the case () = 0, (reddy ) = 0 .9 , the interpretation would not condition (6) imposes the same constraint to all domain be faithful. For PenguinI (reddy ) = 0 .8 , the elements . interpretation would be faithful, but not coher- To see the relation between this semantics and Mulent, as WPenguin (opus ) > WPenguin (reddy ), but tilayer Perceptrons, consider that a neuron can be PenguinI (opus ) = PenguinI (reddy ). described by the following pair of equations: = ∑︀=1 , and = ( + ), where 1, . . . ,
It has been shown [42] that the proposed semantics are the input signals and 1, . . . , are the weights of allows the input-output behavior of a deep network (con- neuron ; is the bias, the activation function, and sidered after training) to be captured by a fuzzy multi- is the output signal of neuron . By adding a new synapse preference interpretation built over a set of input stimuli, with input 0 = +1 and synaptic weight 0 = , one through a simple construction which exploits the activity can write: = ∑︀=0 , and = (), where level of neurons for the stimuli. Each unit ℎ of can be is called the induced local field of the neuron. The associated to a concept name ℎ and, for a given domain neuron can be represented as a directed graph, where the Δ of input stimuli, the activation value of unit ℎ for a stim- input signals 1, . . . , and the output signal of neuulus is interpreted as the degree of membership of in ron are nodes of the graph. An edge from to , concept ℎ. The resulting preferential interpretation can labelled , means that is an input signal of neuron be used for verifying properties of the network by model with synaptic weight . A neural network can then checking (e.g., T(Penguin) ⊑ Has_Wings ≥ 0.7, do be seen as “a directed graph consisting of nodes with typical penguins have wings with degree ≥ 0.7?). interconnecting synaptic and activation links" [44].
Let us associate a concept name to each unit in a deep neural network (possibly allowing for feedback), and let us interpret, as in [42], 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 There is much work in the literature concerning extension in the network , then condition (6) characterizes the of Dung’s argumentation framework [29] with weights stationary states of the network, where () corresponds attached to arguments and/or to the attacks between arguto the activation of neuron for some input stimulus and ments. Many different proposals have been investigated ∑︀ℎ ℎ ,ℎ() corresponds to the induced local field of and compared in the literature. Let us just mention, for the neuron , where each ,ℎ() represents the input signal moment, the work by Cayrol and Lagasquie-Schiex [24], ℎ, for input stimulus . Janssen and Cock [46], Dunne et al. [30], Egilmez et al.
Of course, -coherence could be easily extended to [31], Amgoud et al. [2], Amgoud and Doder [4], which deal with different activation functions , one for each also include extensive comparisons. In the following, we concept (i.e., for each unit ). The following proposi- propose some semantics for weighted argumentation with tion establishes some relationships between -coherent, the purpose of establishing some links with the semantics faithful and coherent fuzzy multipreference models of a of conditional knowledge bases considered in the previous weighted conditional knowledge base . sections.
We consider a notion of weighted argumentation graph Proposition 1. Let be a weighted conditional ℒFT as a triple = ⟨, ℛ, ⟩, where is a set of arguknowledge base and : R → [0, 1]. (1) if is a monoton- ments, ℛ ⊆ × and : ℛ → R. This definition ically non-decreasing function, a -coherent fuzzy multi- of weighted argumentation graph corresponds to the defipreference model of is also a faithful-model of ; (2) nition of weighted argument system in [30], but here we if is a monotonically increasing function, a -coherent admit both positive and negative weights, while [30] only fuzzy multipreference model of is also a coherent- allows for positive weights representing the strength of atmodel of . tacks. In our notion of weighted graph, a pair (, ) ∈ ℛ can be regarded as a support relation when the weight is Item 2 can be regarded as the analog of Proposition 1 positive and an attack relation when the weight is negative, in [42], where the fuzzy multi-preferential interpretation and it leads to bipolar argumentation [3]. The argumenta,Δ of a deep neural network , built over the domain tion semantics we consider in the following, as in the case oℳf input stimuli Δ, is proven to be a coherent model of the of weighted conditionals, deals with both the positive and knowledge base associated to , under the speciefid the negative weights in a uniform way. For the moment conditions on the activation function , and the assump- we do not include in a function determining the basic tion that each stimulus in Δ corresponds to a stationary strength of arguments [2]. state in the neural network. Item 1 in Proposition 1 is as Given a weighted argumentation graph = ⟨, ℛ, ⟩, well the analog of Proposition 2 in [42], extended version, we define a labelling of the graph as a function ,Δ is a faithful (or weakly-coerent) model : → [0, 1] which assigns to each argument an acstating that ℳ of . ceptability degree, i.e., a value in the interval [0, 1]. Let
A notion of coherent/faithful/ -coherent multiprefer- R− (A) = {B | (B , A) ∈ ℛ}. When R− (A) = ∅, arguence entailment from a weighted ℒFT knowledge base ment has neither supports nor attacks. can be defined in the obvious way (see [ 42, 36] for the For a weighted graph = ⟨, ℛ, ⟩ and a labelling definitions of coherent and faithful (fuzzy) multiprefer- , we introduce a weight on , as a partial function ence entailment). The properties of faithful entailment : → R, assigning a positive or negative support to have been studied in [36]. Faithful entailment is reason- the arguments ∈ such that R− (Ai ) ̸= ∅, as follows: ably well-behaved: it deals with specificity and irrelevance; it is not subject to inheritance blocking; it satisfies () = ∑︁ ( , ) ( ) (7) most KLM properties [51, 52], depending on their fuzzy ( ,)∈ℛ reformulation and on the chosen combination functions.
As MLPs are usually represented as a weighted graphs When R− (Ai ) = ∅, () is let undefined. [44], whose nodes are units and whose edges are the We can now exploit this notion of weight of an argusynaptic connections between units with their weight, ment to define different argumentation semantics for a it is very tempting to extend the different semantics of graph as follows. weighted knowledge bases considered above, to weighted argumentation graphs.
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 ) ̸= ∅,
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] () < () ⇐⇒
() < (); • is a faithfull labelling of if, for all arguments , ∈ s.t. R− (A) ̸= ∅ and R− (B ) ̸= ∅,
() < () ⇒ () < (); attack/support by aggregating the weight on the edge between two arguments with the strength of the attacker/sup• for a function : R → [0, 1], is a -coherent porter. Function aggregates the strength of all attacks labelling of if, for all arguments ∈ s.t. and supports to a given argument, and function returns a R− (A) ̸= ∅, () = ( ()). value for an argument, given the strength of the argument and aggregated weight of its attacks and supports.
As in [4], 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( (, ), ())) 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 [ 4], 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.
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 denfied through a set of equations, as in Gabbay’s equational approach to argumentation networks [32]. 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 property of a -labelling that () = ( ()), for all arguments with R− (A) ̸= ∅, as well as the properties of .
Proposition 3. Let = ⟨, ℛ, ⟩ be a weighted argumentation graph. If, for some 0 : → [0, 1], is a The notion of -coherent labelling relates to the frame- gradual semantics of graph ′ = ⟨, 0, ℛ, ⟩ based work of gradual semantics studied by Amgoud and Doder on the evaluation method = ⟨ℎ, , ⟩, then [4] where, for the sake of simplicity, the weights of argu- ′ is a -coherent labelling for . ments and attacks are in the interval [0, 1]. Here, as we Vice-versa, if is a -coherent labelling for , then have seen, positive and negative weights are admitted to there are a function 0 and a gradual semantics based represent the strength of attacks and supports. To define on the evaluation method = ⟨ℎ, , ⟩, such an evaluation method for -coherent labellings, we need that, for the graph ′ = ⟨, 0, ℛ, ⟩, ′ ≡ . to consider a slightly extended definition of an evaluation method for a graph in [4]. Following [4] we include a Amgoud and Doder [4] study a large family of determifunction 0 : → [0, 1] in the definition of a weighted native and well-behaved evaluation models for weighted graph, where 0 assigns to each argument ∈ its 1This definition is the same as in [ 4], but for the fact that in the basic strength. Hence a graph becomes a quadruple domain/range of functions ℎ and interval [0, 1] is sometimes = ⟨, 0, ℛ, ⟩. replaced by R. graphs in which attacks have positive weights in the in- a value in the interval [0, 1] with respect to a given semanterval [0, 1]. For weighted graph with positive and tics, one can define a preferential structure starting from negative weights, the evaluation method cannot be Σ to evaluate conditional properties of the argumentation guaranteed to be determinative, even under the conditions graph. This would allow, for instance, to verify properties that is monotonically increasing and continuous. In gen- like: "does normally argument 2 follows from argument eral, there is not a unique semantics based on , and 1 with a degree greater than 0.7?" This query can be there is not a unique -coherent labelling for a weighted formalized as a fuzzy inclusion T(1) ⊑ 2 > 0.7. graph , given a basic strength 0. This is not surprising, In particular, let Σ is a finite set of -coherent labelings considering that -coherent labelings of a graph corre- 1, 2, . . . of a weighted graph = ⟨, ℛ, ⟩, for some spond to stationary states (or equilibrium states) in a deep function . One can define a fuzzy multipreference inneural network [44]. terpretation over Σ by adopting the construction used in
A deep neural network can indeed be seen as a weighted [42] to build a fuzzy multipreference interpretation over argumentation graph, with positive and negative weights, the set of input stimuli of a neural network, where each where each unit in the network is associated to an argu- input stimulus was associated to a fit vector [ 50] describment, and the activation value of the unit can be regarded ing the activity levels of all units for that input. Here, as the weight (in the interval [0, 1]) of the corresponding each labelling plays the role of a fit vector and each argument. Synaptic positive and negative weights cor- argument in can be interpreted as a concept name respond to the strength of supports (when positive) and of the language. Let = and = {1, 2, . . .}. attacks (when negative). In this view, -coherent label- We assume that there is one individual name in the ings, assigning to each argument a weight in the interval language for each labelling ∈ Σ, and define a fuzzy [0, 1], correspond to stationary states of the network, the multipreference interpretation Σ = ⟨Σ, · ) as follows: solutions of a set of equations. This is in agreement with previous results on the relationship between weighted • for all ∈ , = ; argumentation graphs and MLPs established by Garcez, • for all ∈ , ( ) = (). Gabbay and Lamb [27] and, more recently, by Potyca [57]. The fuzzy ℒ interpretation Σ induces a preference relaWe refer to the conclusions for some comparisons. tion < for each argument ∈ . For all , ∈ Σ:
Unless the network is feedforward (and the corresponding graph is acyclic), stationary states cannot be uniquely < iff ( ) > () determined by an iterative process from an initial labelling iff () > (). 0. On the other hand, a semantics based on satisfies some of the properties considered in [ 4], including Let be the conditional knowledge base extracted from anonymity, independence, directionality, equivalence and the weighted argumentation graph, as follows: maximality, provided the last two properties are prop- = {(T() ⊑ , ,) | erly reformulated to deal with both positive and negative ( , ) ∈ ℛ and (( , )) = } weights (i.e., by replacing R− (x ) to (), for each It can be proven that: argument in the formulation in [4]). However, a semantics based on cannot be expected to satisfy the Proposition 4. Let Σ be a finite set of -coherent laproperties of neutrality, weakening, proportionality and belings of a weighted graph = ⟨, ℛ, ⟩, for some resilience. In fact, function completely disregard the function : R → [0, 1]. The following statements hold: initial valuation 0 in graph = ⟨, 0, ℛ, ⟩, for those (i) If is a monotonically increasing function and arguments having incoming edges (even if their weight is : R → (0, 1], then Σ is a coherent (fuzzy) 0). So, for instance, it is not the same, for an argument to multipreference model of . have a support with weight 0 or no support or attack at all: (ii) If is a monotonically non-decreasing function, neutrality does not hold. then Σ is a faithful (fuzzy) multipreference model
A detailed analysis of the properties of this argumen- of . tation semantics is left for an extended version of this work.
labelings Σ = { 1, 2, . . .} for a weighted argumentation graph , where each labelling assigns to each argument
1 in [42] (extended version with proofs). The proof of item (ii) is similar to the proof of Proposition 2 therein.
The restriction to a finite set Σ of -coherent labelings is needed to guarantee the well-foundedness of the resulting interpretation. In fact, in general, the set of all -coherent labelings of might be infinite and, if Σ is the set of all -coherent labelings of , there is no guarantee that the resulting fuzzy ℒFT interpretation is witnessed and the tation frameworks and neural networks, first investigated preference relations < is well-founded. by Garcez, et al. [27] and recently by Potyca [57].
While this allows (fuzzy) conditional formulas over The work by Garcez, et al. [27] combines value-based arguments to be validated by model checking over a pref- argumentation frameworks [8] and neural-symbolic learnerential model, whether this approach can be extended to ing systems by providing a translation from argumentation the other gradual semantics, and under which conditions networks to neural networks with 3 layers (input, output on the evaluation method, is subject of future work. layer and one hidden layer). This enables the accrual of
Observe also that, in the conditional semantics in Sec- arguments through learning as well as the parallel comtions 3.1 and 4, in a typicality inclusion T() ⊑ , putation of arguments. The work by Potyca [57] conconcepts and are not required to be concept names, siders a quantitative bipolar argumentation frameworks but they can be complex concepts. In particular, in the (QBAFs) similar to [7] and exploits an influence function fragment ℒ of ℒ considered in this paper, can be based on the logistic function to define an MLP-based any boolean combination of concept names. The corre- semantics MLP for a QBAF: for each argument ∈ , spondence between weighted conditionals T() ⊑ MLP () = lim→∞ (), when the limit exists, and is in and weighted attacks/supports in the argumenta- undefined otherwise; where () is a value in the interval tion graph , suggests a possible generalization of the [0, 1], and represents the iteration. The paper studies structure of the weighted argumentation graph by allow- convergence conditions both in the discrete and in the ing attacks/supports by a boolean combination of argu- continuous case, as well as the semantic properties of ments. The labelling of arguments in the set [0, 1] can MLP-based semantics, and proves that all properties for indeed be extended to boolean combinations of arguments the QBAF semantics proposed in [2] are satisfied. As we using the fuzzy combination functions, as for boolean have seen in Section 6, our semantics based on -coherent concepts in the conditional semantics (e.g., by letting models fails to satisfy some of the properties in [2]. (1 ∧ 2) = { (1), (2)}, using the mini- In this work we have investigated the relationships bemum t-norm as in Zadeh fuzzy logic). This also relates to tween -coherent labelings and the gradual semantics by the work considering “sets of attacking (resp. supporting) Amgoud and Doder [4], by slightly extending their definiarguments”; i.e., several argument together attacking (or tions to deal with positive and negative weights to capture supporting) an argument. Indeed, for gradual semantics, the strength of supports and of attacks. A correspondence the sets of attacking arguments framework (SETAF) has between the gradual semantics based on a specific evalbeen studied by Yun and Vesic, by considering “the force uation method and -coherent labelings has been of the set of attacking (resp. supporting) arguments to be established. Differently from the Fuzzy Argumentation the force of the weakest argument in the set" [60]. This Frameworks by Jenssen et al. [46], where an attack relawould correspond to interpret the set of arguments as a tion is a fuzzy binary relation over the set of arguments, conjunction, using minimum t-norm. here we have considered real-valued weights associated to pairs of arguments. Our semantics also relates to the 8. Conclusions fuzzy extension of rational closure by Casini and Straccia [22].
In this paper, drawing inspiration from a fuzzy preferen- The paper discusses an approach for defeasible reasontial semantics for weighted conditionals, which has been ing over a weighted argumentation graph building on introduced for modeling the behavior of Multilayer Per- coherent labelings. This allows a multipreference model ceptrons [42], we develop some semantics for weighted ar- to be constructed over a (finite) set of -labelling Σ and gumentation graphs, where positive and negative weights allows (fuzzy) conditional formulas over arguments to can be associated to pairs of arguments. In particular, be validated over Σ by model checking over a preferenwe introduce the notions of coherent/faithful/ -coherent tial model. Whether this approach can be extended to labelings of a graph, and establish some relationships the other gradual semantics, and under which conditions among them. While in [42] a deep neural network is on the evaluation method, requires further investigation mapped to a weighted conditional knowledge base, a deep for future work. The paper also suggests an approach to neural network can as well be seen as a weighted argumen- deal with attack/supports by a boolean combination of tation graph, with positive and negative weights, under the arguments, by exploiting the fuzzy semantics of weighted proposed semantics. In this view, -coherent labelings conditionals. correspond to stationary states in the network (where each The correspondence between Abstract Dialectical unit in the network is associated to an argument and the Frameworks [17] and Nonmonotonic Conditional Logics activation value of the unit can be regarded as the weight has been studied by Heyninck, Kern-Isberner and Thimm of the corresponding argument). This is in agreement [45], with respect to the two-valued models, the stable, with previous work on the relationship between argumen- the preferred semantics and the grounded semantics of ADFs. Whether the coherent/faithfull/ -coherent seman- [6] F. Baader, D. Calvanese, D.L. McGuinness, tics developed in the paper for weighted argumentation D. Nardi, and P.F. Patel-Schneider. The Descripcan be reformulated for a (weighted) Abstract Dialecti- tion Logic Handbook - Theory, Implementation, and cal Frameworks, and which are the relationships with the Applications. Cambridge, 2007. work in [45], also requires investigation for future work. [7] P. Baroni, A. Rago, and F. Toni. How many prop
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