=Paper=
{{Paper
|id=Vol-3086/paper8
|storemode=property
|title=From Weighted Conditionals of Multilayer Perceptrons to a Gradual Argumentation Semantics
|pdfUrl=https://ceur-ws.org/Vol-3086/paper8.pdf
|volume=Vol-3086
|authors=Laura Giordano
|dblpUrl=https://dblp.org/rec/conf/aiia/000121
}}
==From Weighted Conditionals of Multilayer Perceptrons to a Gradual Argumentation Semantics==
<pdf width="1500px">https://ceur-ws.org/Vol-3086/paper8.pdf</pdf>
<pre>
From Weighted Conditionals of Multilayer
Perceptrons to a Gradual Argumentation
Semantics
Laura Giordano1
1
    DISIT - Università del Piemonte Orientale, Italy


                                         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(𝐶) ⊑ 𝐷 (meaning “the typical
𝐶-elements are 𝐷-elements" or “normally 𝐶’s are 𝐷’s"), with different preferential semantics
[6, 7] and closure constructions [8, 9, 10, 11, 12]. Such defeasible inclusions correspond to KLM
conditionals 𝐶 |∼ 𝐷 [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(𝐶) ⊑ 𝐷 with positive or
negative weights, for distinguished concepts 𝐶. The concept-wise multipreference semantics has
been first introduced as a semantics for ranked DL knowledge bases [19] (where conditionals

5th Workshop on Advances In Argumentation In Artificial Intelligence (AIxIA 2021)
" laura.giordano@uniupo.it (L. Giordano)
                                       © 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
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                  ISSN 1613-0073       CEUR Workshop Proceedings (CEUR-WS.org)
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 seman-
tics 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 𝜙-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 𝜙-coherent semantics) inspired by the fuzzy preferential semantics of
weighted conditionals and, in Section 6, we investigate the relationship of 𝜙-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 ℒ𝒞 and fuzzy ℒ𝒞
In this section we recall the syntax and semantics of the description logic 𝒜ℒ𝒞 [38] and of its
fuzzy extension [39]. For sake of simplicity, we only focus on ℒ𝒞, the boolean fragment of 𝒜ℒ𝒞,
which does not allow for roles. Let 𝑁𝐶 be a set of concept names, and 𝑁𝐼 a set of individual
names. The set of ℒ𝒞 concepts (or, simply, concepts) can be defined inductively:
- 𝐴 ∈ 𝑁𝐶 , ⊤ and ⊥ are concepts;
- if 𝐶 and 𝐷 are concepts, and 𝑟 ∈ 𝑁𝑅 , then 𝐶 ⊓ 𝐷, 𝐶 ⊔ 𝐷, ¬𝐶 are concepts.
An ℒ𝒞 knowledge base (KB) 𝐾 is a pair (𝒯𝐾 , 𝒜𝐾 ), where 𝒯𝐾 is a TBox and 𝒜𝐾 is an ABox.
The TBox 𝒯𝐾 is a set of concept inclusions (or subsumptions) 𝐶 ⊑ 𝐷, where 𝐶, 𝐷 are concepts.
The ABox 𝒜𝐾 is a set of assertions of the form 𝐶(𝑎), where 𝐶 is a concept and 𝑎 an individual
name in 𝑁𝐼 .
    An ℒ𝒞 interpretation is defined as a pair 𝐼 = ⟨∆, ·𝐼 ⟩ where: ∆ is a domain—a set whose
elements are denoted by 𝑥, 𝑦, 𝑧, . . . —and ·𝐼 is an extension function that maps each concept
name 𝐶 ∈ 𝑁𝐶 to a set 𝐶 𝐼 ⊆ ∆, and each individual name 𝑎 ∈ 𝑁𝐼 to an element 𝑎𝐼 ∈ ∆. It is
extended to complex concepts as follows:

               ⊤𝐼 = ∆                    ⊥𝐼 = ∅                  (¬𝐶)𝐼 = ∆∖𝐶 𝐼
               (𝐶 ⊓ 𝐷)𝐼 = 𝐶 𝐼 ∩ 𝐷𝐼                          (𝐶 ⊔ 𝐷)𝐼 = 𝐶 𝐼 ∪ 𝐷𝐼

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 ℒ𝒞 interpretation 𝐼 = ⟨∆, ·𝐼 ⟩:
   - 𝐼 satisfies an inclusion 𝐶 ⊑ 𝐷 if 𝐶 𝐼 ⊆ 𝐷𝐼 ;
   - 𝐼 satisfies an assertion 𝐶(𝑎) if 𝑎𝐼 ∈ 𝐶 𝐼 .
Given a KB 𝐾 = (𝒯𝐾 , 𝒜𝐾 ), an interpretation 𝐼 satisfies 𝒯𝐾 (resp. 𝒜𝐾 ) if 𝐼 satisfies all
inclusions in 𝒯𝐾 (resp. all assertions in 𝒜𝐾 ); 𝐼 is a model of 𝐾 if 𝐼 satisfies 𝒯𝐾 and 𝒜𝐾 .
   A subsumption 𝐹 = 𝐶 ⊑ 𝐷 (resp., an assertion 𝐶(𝑎)), is entailed by 𝐾, written 𝐾 |= 𝐹 , if
for all models 𝐼 =⟨∆, ·𝐼 ⟩ of 𝐾, 𝐼 satisfies 𝐹 .

Given a knowledge base 𝐾, the subsumption problem is the problem of deciding whether an
inclusion 𝐶 ⊑ 𝐷 is entailed by 𝐾.
   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
𝒜ℒ𝒞 for the fragment ℒ𝒞, 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 ℒ𝒞.
   A fuzzy interpretation for ℒ𝒞 is a pair 𝐼 = ⟨∆, ·𝐼 ⟩ where: ∆ is a non-empty domain and ·𝐼 is
fuzzy interpretation function that assigns to each concept name 𝐴 ∈ 𝑁𝐶 a function 𝐴𝐼 : ∆ →
[0, 1], and to each individual name 𝑎 ∈ 𝑁𝐼 an element 𝑎𝐼 ∈ ∆. A domain element 𝑥 ∈ ∆ belongs
to the extension of 𝐴 to some degree in [0, 1], i.e., 𝐴𝐼 is a fuzzy set.
   The interpretation function ·𝐼 is extended to complex concepts as follows:
      ⊤𝐼 (𝑥) = 1,                ⊥𝐼 (𝑥) = 0,               (¬𝐶)𝐼 (𝑥) = ⊖𝐶 𝐼 (𝑥),
                𝐼         𝐼
      (𝐶 ⊓ 𝐷) (𝑥) = 𝐶 (𝑥) ⊗ 𝐷 (𝑥),  𝐼                 (𝐶 ⊔ 𝐷)𝐼 (𝑥) = 𝐶 𝐼 (𝑥) ⊕ 𝐷𝐼 (𝑥).
where 𝑥 ∈ ∆ 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 𝑎 ⊗ 𝑏 = 𝑚𝑖𝑛{𝑎, 𝑏}, 𝑎 ⊕ 𝑏 = 𝑚𝑎𝑥{𝑎, 𝑏},
𝑎 ▷ 𝑏 = 𝑚𝑎𝑥{1 − 𝑎, 𝑏} and ⊖𝑎 = 1 − 𝑎.
   The interpretation function ·𝐼 is also extended to non-fuzzy axioms (i.e., to strict inclusions
and assertions of an ℒ𝒞 knowledge base) as follows:
(𝐶 ⊑ 𝐷)𝐼 = 𝑖𝑛𝑓𝑥∈Δ 𝐶 𝐼 (𝑥) ▷ 𝐷𝐼 (𝑥),                (𝐶(𝑎))𝐼 = 𝐶 𝐼 (𝑎𝐼 ).
   A fuzzy ℒ𝒞 knowledge base 𝐾 is a pair (𝒯𝑓 , 𝒜𝑓 ) where 𝒯𝑓 is a fuzzy TBox and 𝒜𝑓 a fuzzy
ABox. A fuzzy TBox is a set of fuzzy concept inclusions of the form 𝐶 ⊑ 𝐷 𝜃 𝑛, where 𝐶 ⊑ 𝐷
is an ℒ𝒞 concept inclusion axiom, 𝜃 ∈ {≥, ≤, >, <} and 𝑛 ∈ [0, 1]. A fuzzy ABox 𝒜𝑓 is a set
of fuzzy assertions of the form 𝐶(𝑎)𝜃𝑛, where 𝐶 is an ℒ𝒞 concept, 𝑎 ∈ 𝑁𝐼 , 𝜃 ∈ {≥, ≤, >, <}
and 𝑛 ∈ [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 𝐼 satisfies a
fuzzy ℒ𝒞 axiom 𝐸 (denoted 𝐼 |= 𝐸), as follows, for 𝜃 ∈ {≥, ≤, >, <}:
   - 𝐼 satisfies a fuzzy ℒ𝒞 inclusion axiom 𝐶 ⊑ 𝐷 𝜃 𝑛 if (𝐶 ⊑ 𝐷)𝐼 𝜃 𝑛;
   - 𝐼 satisfies a fuzzy ℒ𝒞 assertion 𝐶(𝑎) 𝜃 𝑛 if 𝐶 𝐼 (𝑎𝐼 )𝜃 𝑛;
Given a fuzzy ℒ𝒞 KB 𝐾 = (𝒯𝑓 , 𝒜𝑓 ), a fuzzy interpretation 𝐼 satisfies 𝒯𝑓 (resp. 𝒜𝑓 ) if 𝐼 satisfies
all fuzzy inclusions in 𝒯𝑓 (resp. all fuzzy assertions in 𝒜𝑓 ). A fuzzy interpretation 𝐼 is a model
of 𝐾 if 𝐼 satisfies 𝒯𝑓 and 𝒜𝑓 . A fuzzy axiom 𝐸 is entailed by a fuzzy knowledge base 𝐾 (i.e.,
𝐾 |= 𝐸) if for all models 𝐼 =⟨∆, ·𝐼 ⟩ of 𝐾, 𝐼 satisfies 𝐸.


3. Fuzzy ℒ𝒞 with typicality: ℒ𝒞 F T
In this section, we describe an extension of fuzzy ℒ𝒞 with typicality following [18, 26]. Typicality
concepts of the form T(𝐶) are added, where 𝐶 is a concept in fuzzy ℒ𝒞. The idea is similar
to the extension of 𝒜ℒ𝒞 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(𝐶) ⊑ 𝐷 𝜃 𝑛, meaning that typical 𝐶-elements are 𝐷-elements with a degree greater than
𝑛. A typicality inclusion T(𝐶) ⊑ 𝐷, as in the two-valued case, stands for a KLM conditional
implication 𝐶 |∼ 𝐷 [13, 14], but now it has an associated degree. We call ℒ𝒞 F T the extension
of fuzzy ℒ𝒞 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 ℒ𝒞 interpretation 𝐼 = ⟨∆, ·𝐼 ⟩, the degree of membership 𝐶 𝐼 (𝑥) of the
domain elements 𝑥 in a concept 𝐶, induces a preference relation <𝐶 on ∆, as follows:
                                   𝑥 <𝐶 𝑦 iff 𝐶 𝐼 (𝑥) > 𝐶 𝐼 (𝑦)                                 (1)
Each <𝐶 has the properties of preference relations in KLM-style ranked interpretations [14], that
is, <𝐶 is a modular and well-founded strict partial order. Let us recall that, <𝐶 is well-founded if
there is no infinite descending chain 𝑥1 <𝐶 𝑥0 , 𝑥2 <𝐶 𝑥1 , 𝑥3 <𝐶 𝑥2 , . . . of domain elements;
<𝐶 is modular if, for all 𝑥, 𝑦, 𝑧 ∈ ∆, 𝑥 <𝐶 𝑦 implies (𝑥 <𝐶 𝑧 or 𝑧 <𝐶 𝑦). Well-foundedness
holds for the induced preference <𝐶 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 <𝐶 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 <𝐶 captures the
relative typicality of domain elements wrt concept 𝐶 and may then be used to identify the typical
𝐶-elements. We will regard typical 𝐶-elements as the domain elements 𝑥 that are preferred with
respect to relation <𝐶 among those such that 𝐶 𝐼 (𝑥) ̸= 0. Let 𝐶>0
                                                                𝐼 be the crisp set containing all

domain elements 𝑥 such that 𝐶 (𝑥) > 0, that is, 𝐶>0 = {𝑥 ∈ ∆ | 𝐶 𝐼 (𝑥) > 0}. One can provide
                                𝐼                  𝐼

a (two-valued) interpretation of typicality concepts T(𝐶) in a fuzzy interpretation 𝐼, by letting:
                                                                   𝐼 )
                                         {︂
                                             1    if 𝑥 ∈ 𝑚𝑖𝑛<𝐶 (𝐶>0
                          (T(𝐶))𝐼 (𝑥) =                                                        (2)
                                             0    otherwise
where 𝑚𝑖𝑛< (𝑆) = {𝑢 : 𝑢 ∈ 𝑆 and ∄𝑧 ∈ 𝑆 s.t. 𝑧 < 𝑢}. When (T(𝐶))𝐼 (𝑥) = 1, we say that 𝑥 is
a typical 𝐶-element in 𝐼.
   Note that, if 𝐶 𝐼 (𝑥) > 0 for some 𝑥 ∈ ∆, 𝑚𝑖𝑛<𝐶 (𝐶>0
                                                     𝐼 ) is non-empty.


Definition 3 (ℒ𝒞 F T interpretation). An ℒ𝒞 F T interpretation 𝐼 = ⟨∆, ·𝐼 ⟩ is a fuzzy ℒ𝒞 inter-
pretation, extended by interpreting typicality concepts as in (2).
   The fuzzy interpretation 𝐼 = ⟨∆, ·𝐼 ⟩ implicitly defines a multipreference interpretation, where
any concept 𝐶 is associated to a preference relation <𝐶 . This is different from the two-valued
multipreference semantics in [19], where only a subset of distinguished concepts have an associ-
ated preference, and a notion of global preference < is introduced to define the interpretation
of the typicality concept T(𝐶), for an arbitrary 𝐶. Here, we do not need to introduce a notion
of global preference. The interpretation of any ℒ𝒞 concept 𝐶 is defined compositionally from
the interpretation of atomic concepts, and the preference relation <𝐶 associated to 𝐶 is defined
from 𝐶 𝐼 . The notions of satisfiability in ℒ𝒞 F T, model of an ℒ𝒞 F T knowledge base, and ℒ𝒞 F T
entailment can be defined in a similar way as in fuzzy ℒ𝒞 (see Section 2).

3.1. Strengthening ℒ𝒞 F T: a closure construction
To overcome the weakness of preferential entailment, the rational closure [14] and the lexico-
graphic 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 𝒜ℒ𝒞 F T
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 ℒ𝒞 F T knowledge base 𝐾, over a set 𝒞 = {𝐶1 , . . . , 𝐶𝑘 } of distinguished ℒ𝒞
concepts, is a tuple ⟨𝒯𝑓 , 𝒯𝐶1 , . . . , 𝒯𝐶𝑘 , 𝒜𝑓 ⟩, where 𝒯𝑓 is a set of fuzzy ℒ𝒞 F T inclusion axiom,
𝒜𝑓 is a set of fuzzy ℒ𝒞 F T assertions and 𝒯𝐶𝑖 = {(𝑑𝑖ℎ , 𝑤ℎ𝑖 )} is a set of all weighted typicality
inclusions 𝑑𝑖ℎ = T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ for 𝐶𝑖 , indexed by ℎ, where each inclusion 𝑑𝑖ℎ has weight 𝑤ℎ𝑖 , a
real number. As in [18], 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 𝐶𝑖 occurring on
the l.h.s. of some typicality inclusion T(𝐶𝑖 ) ⊑ 𝐷. Arbitrary ℒ𝒞 F T inclusions and assertions
may belong to 𝒯𝑓 and 𝒜𝑓 . Let us consider the following example from [26].
Example 1. Consider the weighted knowledge base 𝐾 = ⟨𝒯𝑓 , 𝒯𝐵𝑖𝑟𝑑 , 𝒯𝑃 𝑒𝑛𝑔𝑢𝑖𝑛 , 𝒯𝐶𝑎𝑛𝑎𝑟𝑦 , 𝒜𝑓 ⟩,
over the set of distinguished concepts 𝒞 = {Bird , Penguin, Canary}, with empty 𝒜𝑓 and 𝒯𝑓
containing, for instance, the inclusions:
  Yellow ⊓ Black ⊑ ⊥ ≥ 1             Yellow ⊓ Red ⊑ ⊥ ≥ 1             Black ⊓ Red ⊑ ⊥ ≥ 1
The weighted TBox 𝒯𝐵𝑖𝑟𝑑 contains the following weighted defeasible inclusions:
   (𝑑1 ) T(Bird ) ⊑ Fly, +20              (𝑑2 ) T(Bird ) ⊑ Has_Wings, +50
   (𝑑3 ) T(Bird ) ⊑ Has_Feather , +50;
𝒯𝑃 𝑒𝑛𝑔𝑢𝑖𝑛 and 𝒯𝐶𝑎𝑛𝑎𝑟𝑦 contain, respectively, the following defeasible inclusions:
   (𝑑4 ) T(Penguin) ⊑ Bird , +100               (𝑑7 ) T(Canary) ⊑ Bird , +100
   (𝑑5 ) T(Penguin) ⊑ Fly, - 70                (𝑑8 ) T(Canary) ⊑ Yellow , +30
   (𝑑6 ) T(Penguin) ⊑ Black , +50;              (𝑑9 ) 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 (𝑑5 ) has negative
weight -70), while being a bird and being black are plausible properties of prototypical penguins,
and (𝑑4 ) and (𝑑6 ) 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
flies (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 𝒜ℒ𝒞 F T interpretations to be selected, the interpretations whose induced preference
relations <𝐶𝑖 , for the distinguished concepts 𝐶𝑖 , faithfully represent the defeasible part of the
knowledge base 𝐾.
   Let 𝒯𝐶𝑖 = {(𝑑𝑖ℎ , 𝑤ℎ𝑖 )} be the set of weighted typicality inclusions 𝑑𝑖ℎ = T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ
associated to the distinguished concept 𝐶𝑖 , and let 𝐼 = ⟨∆, ·𝐼 ⟩ be a fuzzy ℒ𝒞 F T interpretation.
In the two-valued case, we would associate to each domain element 𝑥 ∈ ∆ and each distinguished
concept 𝐶𝑖 , a weight 𝑊𝑖 (𝑥) of 𝑥 wrt 𝐶𝑖 in 𝐼, by summing the weights of the defeasible inclusions
satisfied by 𝑥. However, as 𝐼 is a fuzzy interpretation, we do not only distinguish between
the typicality inclusions satisfied or falsified by 𝑥; we also need to consider, for all inclusions
T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ ∈ 𝒯𝐶𝑖 , the degree of membership of 𝑥 in 𝐷𝑖,ℎ . Furthermore, in comparing the
weight of domain elements with respect to <𝐶𝑖 , we give higher preference to the domain elements
belonging to 𝐶𝑖 (with a degree greater than 0), with respect to those not belonging to 𝐶𝑖 (having
membership degree 0).
   For each domain element 𝑥 ∈ ∆ and distinguished concept 𝐶𝑖 , the weight 𝑊𝑖 (𝑥) of 𝑥 wrt 𝐶𝑖
in the ℒ𝒞 F T interpretation 𝐼 = ⟨∆, ·𝐼 ⟩ is defined as follows:
                                   {︂ ∑︀ 𝑖 𝐼
                                         ℎ 𝑤ℎ 𝐷𝑖,ℎ (𝑥)       if 𝐶𝑖𝐼 (𝑥) > 0
                         𝑊𝑖 (𝑥) =                                                               (3)
                                      −∞                     otherwise

where −∞ is added at the bottom of all real values.
   The value of 𝑊𝑖 (𝑥) is −∞ when 𝑥 is not a 𝐶-element (i.e., 𝐶𝑖𝐼 (𝑥) = 0). Otherwise, 𝐶𝑖𝐼 (𝑥) > 0
and the higher is the sum 𝑊𝑖 (𝑥), the more typical is the element 𝑥 relative to the defeasible
properties of 𝐶𝑖 . How much 𝑥 satisfies a typicality property T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ depends on the value
     𝐼 (𝑥) ∈ [0, 1], which is weighted by 𝑤 𝑖 in the sum. In the two-valued case, 𝐷 𝐼 (𝑥) ∈ {0, 1},
of 𝐷𝑖,ℎ                                     ℎ                                      𝑖,ℎ
and 𝑊𝑖 (𝑥) is the sum of the weights of the typicality inclusions for 𝐶 satisfied by 𝑥, if 𝑥 is a
𝐶-element, and is −∞, otherwise.

Example 2. Let us consider again Example 1. Let 𝐼 be an ℒ𝒞 F T 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 fur-
ther 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 <𝐶𝑖 induced by a fuzzy interpretation 𝐼, for each distinguished concept 𝐶𝑖 , and the
weights 𝑊𝑖 (𝑥) computed, for each 𝑥 ∈ ∆, from the conditional knowledge base 𝐾, given the
interpretation 𝐼. This leads to the following definition of a coherent fuzzy multipreference model
of a weighted a ℒ𝒞 F T knowledge base.

Definition 4 (Coherent (fuzzy) multipreference model of 𝐾 [18]). Let 𝐾 = ⟨𝒯𝑓 , 𝒯𝐶1 , . . . ,
𝒯𝐶𝑘 , 𝒜𝑓 ⟩ be a weighted ℒ𝒞 F T knowledge base over 𝒞. A coherent (fuzzy) multipreference
model (cf𝑚 -model) of 𝐾 is a fuzzy ℒ𝒞 F T interpretation 𝐼 = ⟨∆, ·𝐼 ⟩ s.t.:
    • 𝐼 satisfies the fuzzy inclusions in 𝒯𝑓 and the fuzzy assertions in 𝒜𝑓 ;
    • for all 𝐶𝑖 ∈ 𝒞, the preference <𝐶𝑖 is coherent to 𝒯𝐶𝑖 , that is, for all 𝑥, 𝑦 ∈ ∆,

                                    𝑥 <𝐶𝑖 𝑦 ⇐⇒ 𝑊𝑖 (𝑥) > 𝑊𝑖 (𝑦)                                 (4)

In a similar way, one can define a faithful (fuzzy) multipreference model (fm-model) of 𝐾 by
replacing the coherence condition (4) with the following faithfulness condition (called weak
coherence in [50]): for all 𝑥, 𝑦 ∈ ∆,

                                  𝑥 <𝐶𝑖 𝑦 ⇒ 𝑊𝑖 (𝑥) > 𝑊𝑖 (𝑦).                                   (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 Penguin I (reddy) = 0 .2 and Penguin I (opus) = 0 .8 . Clearly, 𝑟𝑒𝑑𝑑𝑦 <𝐵𝑖𝑟𝑑
𝑜𝑝𝑢𝑠 and 𝑜𝑝𝑢𝑠 <𝑃 𝑒𝑛𝑔𝑢𝑖𝑛 𝑟𝑒𝑑𝑑𝑦. For the interpretation 𝐼 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 Penguin I (reddy) = 0 .9 , the interpretation 𝐼 would
not be faithful. For Penguin I (reddy) = 0 .8 , the interpretation 𝐼 would be faithful, but not
coherent, as WPenguin (opus) > WPenguin (reddy), but Penguin I (opus) = Penguin I (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 𝒩 can be associated to a concept name 𝐶ℎ and, for a
given domain ∆ of input stimuli, the activation value of unit ℎ for a stimulus 𝑥 is interpreted as
the degree of membership of 𝑥 in concept 𝐶ℎ . 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. 𝜙-coherent models
In this section we consider a new notion of coherence of a fuzzy interpretation 𝐼 wrt a KB, that
we call 𝜙-coherence, where 𝜙 is a function from R to the interval [0, 1], i.e., 𝜙 : R → [0, 1]. We
also establish it relationships with coherent and faithful models.

Definition 5 (𝜙-coherence). Let 𝐾 = ⟨𝒯𝑓 , 𝒯𝐶1 , . . . , 𝒯𝐶𝑘 , 𝒜𝑓 ⟩ be a weighted ℒ𝒞 F T knowledge
base, and 𝜙 : R → [0, 1]. A fuzzy ℒ𝒞 F T interpretation 𝐼 = ⟨∆, ·𝐼 ⟩ is 𝜙-coherent if, for all
concepts 𝐶𝑖 ∈ 𝒞 and 𝑥 ∈ ∆,
                                             ∑︁
                                𝐶𝑖𝐼 (𝑥) = 𝜙(     𝑤ℎ𝑖 𝐷𝑖,ℎ𝐼
                                                             (𝑥))                             (6)
                                                ℎ

where 𝒯𝐶𝑖 = {(T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ , 𝑤ℎ𝑖 )} is the set of weighted conditionals for 𝐶𝑖 .

To define 𝜙-coherent multipreference model of a knowledge base 𝐾, we can replace the coherence
condition (4) in Definition 4 with the notion of 𝜙-coherence of an interpretation 𝐼 wrt the
knowledge base 𝐾.
   Observe that, for all 𝑥 such that 𝐶𝑖 (𝑥) > 0, condition (6) above corresponds to condition
  𝐼
𝐶𝑖 (𝑥) = 𝜙(𝑊𝑖 (𝑥)). While the notions of coherence and of weight 𝑊𝑖 (𝑥) (of an element 𝑥 wrt a
concept 𝐶𝑖 ) consider, as a special case, the case when 𝐶𝑖 (𝑥) = 0, in condition (6) we impose the
same constraint to all domain elements 𝑥 (including those with 𝐶𝑖 (𝑥) = 0).
   For Multilayer Perceptrons, let us associate a concept name 𝐶𝑖 to each unit 𝑖 in a deep network
𝒩 , and let us interpret, as in [18], a synaptic connection between neuron ℎ and neuron 𝑖 with
weight 𝑤𝑖ℎ as the conditional T(𝐶𝑖 ) ⊑ 𝐶𝑗 with weight 𝑤ℎ𝑖 = 𝑤𝑖ℎ . If we assume that 𝜙 is the
activation function of all units in the network 𝒩 , then condition (6) characterizes the stationary
states of MLPs, where 𝐶𝑖𝐼 (𝑥) corresponds to the activation of neuron 𝑖 for some input stimulus
𝑥 and ℎ 𝑤ℎ𝑖 𝐷𝑖,ℎ 𝐼 (𝑥) corresponds to the induced local field of neuron 𝑖, which is obtained by
      ∑︀
summing
∑︀𝑛       the input signals to the neuron, 𝑥1 , . . . , 𝑥𝑛 , weighted by the respective synaptic weights:
                                    𝐼
   ℎ=1 𝑖ℎ 𝑥ℎ [51]. Here, each 𝐷𝑖,ℎ (𝑥) corresponds to the input signal 𝑥ℎ , for input stimulus 𝑥.
       𝑤
Of course, 𝜙-coherence could be easily extended to deal with different activation functions 𝜙𝑖 ,
one for each concept 𝐶𝑖 (i.e., for each unit 𝑖).

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

All proofs can be found in the technical report [37]. Item 2 can be regarded as the analog of
Proposition 1 in [18, 50], where the fuzzy multi-preferential interpretation ℳ𝑓,Δ
                                                                              𝒩 of a deep neural
network 𝒩 , built over the domain of input stimuli ∆, is proven to be a coherent model of the
knowledge base 𝐾 𝒩 associated to 𝒩 , under the specified conditions on the activation function
𝜙, and the assumption that each stimulus in ∆ corresponds to a stationary state in the neural
network. Item 1 in Proposition 1 is as well the analog of Proposition 2 in [50] stating that ℳ𝑓,Δ
                                                                                              𝒩
is a faithful (or weakly-coerent) model of 𝐾 𝒩 .
   A notion of coherent/faithful/𝜙-coherent multipreference entailment from a weighted ℒ𝒞 F T
knowledge base 𝐾 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 𝜙-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
𝐺 = ⟨𝒜, ℛ, 𝜋⟩, where 𝒜 is a set of arguments, ℛ ⊆ 𝒜 × 𝒜 and 𝜋 : 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 (𝐴, 𝐵) ∈ 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 𝐺 a
function determining the basic strength of arguments [31].
   Given a weighted argumentation graph 𝐺 = ⟨𝒜, ℛ, 𝜋⟩, we define a labelling of the graph 𝐺
as a function 𝜎 : 𝒜 → [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 𝐴
has neither supports nor attacks.
   For a weighted graph 𝐺 = ⟨𝒜, ℛ, 𝜋⟩ and a labelling 𝜎, we introduce a weight 𝑊𝜎𝐺 on 𝒜, as a
partial function 𝑊𝜎𝐺 : 𝒜 → R, assigning a positive or negative support to the arguments 𝐴𝑖 ∈ 𝒜
such that R − (Ai ) ̸= ∅, as follows:
                                             ∑︁
                               𝑊𝜎𝐺 (𝐴𝑖 ) =          𝜋(𝐴𝑗 , 𝐴𝑖 ) 𝜎(𝐴𝑗 )                      (7)
                                          (𝐴𝑗 ,𝐴𝑖 )∈ℛ

When R − (Ai ) = ∅, 𝑊𝜎𝐺 (𝐴𝑖 ) is let undefined.
  We can now exploit this notion of weight of an argument to define different argumentation
semantics for a graph 𝐺 as follows.
Definition 6. Given a weighted graph 𝐺 = ⟨𝒜, ℛ, 𝜋⟩ and a labelling 𝜎:
    • 𝜎 is a coherent labelling of 𝐺 if, for all arguments 𝐴, 𝐵 ∈ 𝒜 s.t. R − (A) ̸= ∅ and
      R − (B ) ̸= ∅,
                              𝜎(𝐴) < 𝜎(𝐵) ⇐⇒ 𝑊𝜎𝐺 (𝐴) < 𝑊𝜎𝐺 (𝐵);
    • 𝜎 is a faithfull labelling of 𝐺 if, for all arguments 𝐴, 𝐵 ∈ 𝒜 s.t. R − (A) ̸= ∅ and
      R − (B ) ̸= ∅,
                               𝜎(𝐴) < 𝜎(𝐵) ⇒ 𝑊𝜎𝐺 (𝐴) < 𝑊𝜎𝐺 (𝐵);
    • for a function 𝜙 : R → [0, 1], 𝜎 is a 𝜙-coherent labelling of 𝐺 if, for all arguments 𝐴 ∈ 𝒜
      s.t. R − (A) ̸= ∅, 𝜎(𝐴) = 𝜙(𝑊𝜎𝐺 (𝐴)).
These definitions do not put any constraint on the labelling of arguments which do not have
incoming edges in 𝐺: 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 𝜙-coherent labelling of 𝐺 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 𝜙-coherent labelling of a weigthed graph 𝐺 can be proven to be as well a coherent labelling
or a faithful labelling, under some conditions on the function 𝜙.
Proposition 2. Given a weighted graph 𝐺 = ⟨𝒜, ℛ, 𝜋⟩: (1) A coherent labelling of 𝐺 is a faithful
labelling of 𝐺; (2) if 𝜙 is a monotonically non-decreasing function, a 𝜙-coherent labelling 𝜎 of 𝐺
is a faithful labelling of 𝐺; (3) if 𝜙 is a monotonically increasing function, a 𝜙-coherent labelling
𝜎 of 𝐺 is a coherent labelling of 𝐺.
The proof is similar to the one of Proposition 1, and can be found in [37]. It exploits the property
of a 𝜙-labelling that 𝜎(𝐴) = 𝜙(𝑊𝜎𝐺 (𝐴)), for all arguments 𝐴 with R − (A) ̸= ∅, as well as the
properties of 𝜙.
6. 𝜙-coherent labellings and the gradual semantics
The notion of 𝜙-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 𝜙-coherent
labellings, we need to consider a slightly extended definition of an evaluation method for a graph
𝐺 in [33]. Following [33] we include a function 𝜎0 : 𝒜 → [0, 1] in the definition of a weighted
graph, where 𝜎0 assigns to each argument 𝐴 ∈ 𝒜 its basic strength. Hence a graph 𝐺 becomes a
quadruple 𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩.
   An evaluation method for a graph 𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩ is a triple 𝑀 = ⟨ℎ, 𝑔, 𝑓 ⟩, where1 :
            R × [0, 1] → R
        ℎ : ⋃︀
        𝑔 : +∞       𝑛
              𝑛=0 R → R
        𝑓 : [0, 1] × 𝑅𝑎𝑛𝑔𝑒(𝑔) → [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 𝑔
aggregates the strength of all attacks and supports to a given argument, and function 𝑓 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 𝑆 is a function assigning to any graph 𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩ a
weighting 𝐷𝑒𝑔𝐺  𝑆 on 𝒜, i.e., 𝐷𝑒𝑔 𝑆 : 𝒜 → [0, 1], where 𝐷𝑒𝑔 𝑆 (𝐴) represents the strength of an
                                   𝐺                           𝐺
argument 𝐴 (or its acceptability degree).
  A gradual semantics 𝑆 is based on an evaluation method 𝑀 iff, ∀ 𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩, ∀𝐴 ∈ 𝒜,
           𝑆                                    𝑆                                 𝑆
        𝐷𝑒𝑔𝐺 (𝐴) = 𝑓 (𝜎0 (𝐴), 𝑔(ℎ(𝜋(𝐵1 , 𝐴), 𝐷𝑒𝑔𝐺 (𝐵1 )), . . . , ℎ(𝜋(𝐵𝑛 , 𝐴), 𝐷𝑒𝑔𝐺 (𝐵𝑛 )))                            (8)
where B1 , . . . , Bn are all arguments attacking or supporting 𝐴 (i.e., R − (A) = {B1 , . . . , Bn }).
   Let us consider the evaluation method 𝑀 𝜙 = ⟨ℎ𝑝𝑟𝑜𝑑 , 𝑔𝑠𝑢𝑚 , 𝑓𝜙 ⟩, where the functions          ℎ𝑝𝑟𝑜𝑑
and 𝑔𝑠𝑢𝑚 are defined as in [33], i.e., ℎ𝑝𝑟𝑜𝑑 (𝑥, 𝑦) = 𝑥 · 𝑦 and 𝑔𝑠𝑢𝑚 (𝑥1 , . . . , 𝑥𝑛 ) = 𝑛𝑖=1 𝑥𝑖 , but
                                                                                         ∑︀
we let 𝑔𝑠𝑢𝑚 () to be undefined. We let 𝑓𝜙 (𝑥, 𝑦) = 𝑥 when 𝑦 is undefined, and 𝑓𝜙 (𝑥, 𝑦) = 𝜙(𝑦)
otherwise. The function 𝑓𝜙 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
𝐴 has neither attacks nor supports (R − (A) = ∅), 𝑓𝜙 returns the basic strength of 𝐴, 𝜎0 (𝐴).
   The evaluation method 𝑀 𝜙 = ⟨ℎ𝑝𝑟𝑜𝑑 , 𝑔𝑠𝑢𝑚 , 𝑓𝜙 ⟩ provides a characterization of the 𝜙-coherent
labelling for an argumentation graph, in the following sense.

Proposition 3. Let 𝐺 = ⟨𝒜, ℛ, 𝜋⟩ be a weighted argumentation graph. If, for some 𝜎0 : 𝒜 →
[0, 1], 𝑆 is a gradual semantics of graph 𝐺′ = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩ based on the evaluation method
𝑀 𝜙 = ⟨ℎ𝑝𝑟𝑜𝑑 , 𝑔𝑠𝑢𝑚 , 𝑓𝜙 ⟩, then 𝐷𝑒𝑔𝐺
                                    𝑆 is a 𝜙-coherent labelling for 𝐺.
                                      ′

   Vice-versa, if 𝜎 is a 𝜙-coherent labelling for 𝐺, then there are a function 𝜎0 and a gradual
semantics 𝑆 based on the evaluation method 𝑀 𝜙 = ⟨ℎ𝑝𝑟𝑜𝑑 , 𝑔𝑠𝑢𝑚 , 𝑓𝜙 ⟩, such that, for the graph
𝐺′ = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩, 𝐷𝑒𝑔𝐺  𝑆 ≡ 𝜎.
                             ′

    1
     This definition is the same as in [33], but for the fact that in the domain/range of functions ℎ and 𝑔 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 𝐺 with positive and negative weights, the evaluation method 𝑀 𝜙 cannot be
guaranteed to be determinative, even under the conditions that 𝜙 is monotonically increasing and
continuous. In general, there is not a unique semantics 𝑆 based on 𝑀 𝜙 , and there is not a unique
𝜙-coherent labelling for a weighted graph 𝐺, given a basic strength 𝜎0 . This is not surprising,
considering that 𝜙-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, 𝜙-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 𝜎0 ). On the other hand, a semantics 𝑆 based on 𝑀 𝜙 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 𝐴𝑡𝑡(𝑥), for each argument 𝑥 in the formulation
in [33]). However, a semantics 𝑆 based on 𝑀 𝜙 cannot be expected to satisfy the properties of
neutrality, weakening, proportionality and resilience. In fact, function 𝑓𝜙 completely disregard
the initial valuation 𝜎0 in graph 𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩, 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/𝜙-
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, 𝜙-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(𝐶) ⊑ 𝐷, 𝐶 and 𝐷 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 𝜙, a multipreference model can be constructed over
a (finite) set of 𝜙-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 𝐴2 follows from argument 𝐴1 with a degree greater than 0.7?" can be formalized by the
fuzzy inclusion T(𝐴1 ) ⊑ 𝐴2 > 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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