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| Paper | |
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 edit
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| description | scientific paper published in CEUR-WS Volume 3197 |
| id | Vol-3197/paper12 |
| wikidataid | Q117341442→Q117341442 |
| title | From Weighted Conditionals with Typicality to a Gradual Argumentation Semantics and Back |
| pdfUrl | https://ceur-ws.org/Vol-3197/paper12.pdf |
| dblpUrl | https://dblp.org/rec/conf/nmr/000122 |
| volume | Vol-3197→Vol-3197 |
| session | → |
Freitext
From Weighted Conditionals with Typicality to a Gradual Argumentation Semantics and Back
From Weighted Conditionals with Typicality
to a Gradual Argumentation Semantics and back⋆
Laura Giordano1,*
1
DISIT - Università del Piemonte Orientale, Alessandria, Italy
Abstract
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.
Keywords
Defeasible Reasoning, Gradual Argumentation, Fuzzy Description Logics
1. Introduction Geffner and Pearl [34], Benferhat et al.[9].
In previous work [42], a concept-wise multiprefer-
Argumentation 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(𝐶) ⊑ 𝐷 with positive or nega-
sion making and for explanation [61]. The argumentation tive weights, for distinguished concepts 𝐶. The concept-
semantics 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 clo-
scription logics have been studied in the last fifteen years sure construction, still in the spirit of Lehmann’s lexico-
to 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(𝐶) ⊑ 𝐷 (meaning “the typical 𝐶-elements proven to have some desired properties from the knowl-
are 𝐷-elements" or “normally 𝐶’s are 𝐷’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 con-
structions, by Casini and Straccia [20, 21] and other re- sequence relation [51, 52], it allows to deal with specificity
searchers [40, 11, 23]. Such defeasible inclusions cor- and irrelevance and avoids inheritance blocking or the
respond to Kraus, Lehmann and Magidor (KLM) condi- “drowning problem" [56, 9], and deals with “ambiguity
tionals 𝐶 |∼ 𝐷 [51, 52], and defeasible DLs inherit and preservation" [34]. The plausibility of the concept-wise
extend some of the preferential semantics and closure multipreference semantics has also been supported [38]
constructions developed within preferential and condi- by showing that it is able to provide a logical interpreta-
tional approaches to commonsense reasoning by Kraus, tion to Kohonen’ Self-Organising Maps [49], which are
Lehmann and Magidor [51], Pearl [56], Lehmann [52], psychologically and biologically plausible neural network
models. In the fuzzy case, the KLM properties of non-
NMR 2022: 20th International Workshop on Non-Monotonic Reason- monotomic entailment have been studied in [36], showing
ing, August 07–09, 2022, Haifa, Israel that most KLM postulates are satisfied, depending on
⋆
You can use this document as the template for preparing your pub- their reformulation and on the choice of fuzzy combina-
lication. We recommend using the latest version of the ceurart
tion functions. It has been shown [42] that both in the
style.
*
Corresponding author. two-valued and in the fuzzy case, the multi-preferential
$ laura.giordano@uniupo.it (L. Giordano) semantics allows to describe the input-output behavior of
https://people.unipmn.it/laura.giordano/ (L. Giordano) Multilayer Perceptrons (MLPs), after training, in terms
© 2022 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0
International (CC BY 4.0). of a preferential interpretation which, in the fuzzy case,
CEUR Workshop Proceedings (CEUR-WS.org)
1613-0073
CEURWorkshopProceedingshttp://ceur-ws.orgISSN
127
�can be proven to be a model (in a logical sense) of the An ℒ𝒞 interpretation is defined as a pair 𝐼 = ⟨Δ, ·𝐼 ⟩
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 𝑥, 𝑦, 𝑧, . . . —and ·𝐼 is an extension function that maps
approaches to non-monotonic reasoning and argumen- each concept name 𝐶 ∈ 𝑁𝐶 to a set 𝐶 𝐼 ⊆ Δ, and each
tation semantics are strong. Let us just mention, the individual name 𝑎 ∈ 𝑁𝐼 to an element 𝑎𝐼 ∈ Δ. It is
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- ⊤𝐼 = Δ ⊥𝐼 = ∅ (¬𝐶)𝐼 = Δ∖𝐶 𝐼
ships between the fuzzy multipreference semantics for (𝐶 ⊓ 𝐷) = 𝐶 ∩ 𝐷𝐼
𝐼 𝐼
(𝐶 ⊔ 𝐷)𝐼 = 𝐶 𝐼 ∪ 𝐷𝐼
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 𝜙-coherent fuzzy multipreference semantics. In ℒ𝒞 interpretation 𝐼 = ⟨Δ, ·𝐼 ⟩:
Section 5, we propose three new gradual semantics for - 𝐼 satisfies an inclusion 𝐶 ⊑ 𝐷 if 𝐶 𝐼 ⊆ 𝐷𝐼 ;
a weighted argumentation graph (namely, a coherent, a - 𝐼 satisfies an assertion 𝐶(𝑎) if 𝑎𝐼 ∈ 𝐶 𝐼 .
faithful and a 𝜙-coherent semantics) inspired by the fuzzy Given a knowledge base 𝐾 = (𝒯𝐾 , 𝒜𝐾 ), an interpreta-
preferential semantics of weighted conditionals and, in tion 𝐼 satisfies 𝒯𝐾 (resp. 𝒜𝐾 ) if 𝐼 satisfies all inclusions
Section 6, we investigate the relationship of 𝜙-coherent in 𝒯𝐾 (resp. all assertions in 𝒜𝐾 ); 𝐼 is a model of 𝐾 if
semantics with the family of gradual semantics studied by 𝐼 satisfies 𝒯𝐾 and 𝒜𝐾 .
Amgoud and Doder. The relationships between weighted A subsumption 𝐹 = 𝐶 ⊑ 𝐷 (resp., an assertion
conditional knowledge bases and MLPs easily extend to 𝐶(𝑎)), is entailed by 𝐾, written 𝐾 |= 𝐹 , if for all models
the proposed gradual semantics, which captures the sta- 𝐼 =⟨Δ, ·𝐼 ⟩ of 𝐾, 𝐼 satisfies 𝐹 .
tionary behavior of MLPs. This is in agreement with
the previous results on the relationships between argu- Given a knowledge base 𝐾, the subsumption problem is
mentation frameworks and neural networks by Garces, the problem of deciding whether an inclusion 𝐶 ⊑ 𝐷 is
Gabbay and Lamb [27] and by Potyca [57]. Section 7 sug- entailed by 𝐾.
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 Strac-
semantics. cia [59], Stoilos [58], Lukasiewicz and Straccia [55],
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 be-
ing true or false; similarly, axioms in a fuzzy DL have a
2. The description logic ℒ𝒞 and degree of truth, usually in the interval [0, 1]. In the follow-
fuzzy ℒ𝒞 ing we shortly recall the semantics of a fuzzy extension
of 𝒜ℒ𝒞 for the fragment ℒ𝒞, referring to the survey by
In this section we recall the syntax and semantics of a Lukasiewicz and Straccia [55]. We limit our considera-
description logic and of its fuzzy extension [55]. For sake tion to a few features of a fuzzy DL, without considering
of simplicity, we only focus on ℒ𝒞, the boolean fragment datatypes, and restricting to constructs in ℒ𝒞.
of 𝒜ℒ𝒞 [6], which does not allow for roles. Let 𝑁𝐶 A fuzzy interpretation for ℒ𝒞 is a pair 𝐼 = ⟨Δ, ·𝐼 ⟩
be a set of concept names, and 𝑁𝐼 a set of individual where: Δ is a non-empty domain and ·𝐼 is fuzzy interpre-
names. ℒ𝒞 concepts (or, simply, concepts) can be defined tation function that assigns to each concept name 𝐴 ∈ 𝑁𝐶
inductively as follows: a function 𝐴𝐼 : Δ → [0, 1], and to each individual name
𝑎 ∈ 𝑁𝐼 an element 𝑎𝐼 ∈ Δ. A domain element 𝑥 ∈ Δ
• 𝐴 ∈ 𝑁𝐶 , ⊤ and ⊥ are concepts;
belongs to concept 𝐴 with a membership degree 𝐴𝐼 (𝑎𝐼 )
• if 𝐶 and 𝐷 are concepts, then 𝐶 ⊓𝐷, 𝐶 ⊔𝐷, ¬𝐶
in [0, 1], i.e., 𝐴𝐼 is a fuzzy set.
are concepts.
The interpretation function ·𝐼 is extended to complex
An ℒ𝒞 knowledge base 𝐾 is a pair (𝒯𝐾 , 𝒜𝐾 ), where 𝒯𝐾 concepts as follows:
is a TBox and 𝒜𝐾 is an ABox. The TBox 𝒯𝐾 is a set ⊤𝐼 (𝑥) = 1, ⊥𝐼 (𝑥) = 0,
of concept inclusions (or subsumptions) 𝐶 ⊑ 𝐷, where (¬𝐶)𝐼 (𝑥) = ⊖𝐶 𝐼 (𝑥),
𝐶, 𝐷 are concepts. The ABox 𝒜𝐾 is a set of assertions of (𝐶 ⊓ 𝐷)𝐼 (𝑥) = 𝐶 𝐼 (𝑥) ⊗ 𝐷𝐼 (𝑥),
the form 𝐶(𝑎), where 𝐶 is a concept and 𝑎 an individual (𝐶 ⊔ 𝐷)𝐼 (𝑥) = 𝐶 𝐼 (𝑥) ⊕ 𝐷𝐼 (𝑥).
name in 𝑁𝐼 .
128
�where 𝑥 ∈ Δ and ⊗, ⊕, ▷ and ⊖ are a t-norm, an s-norm, but now it has an associated degree. We call ℒ𝒞 F T the
an implication function, and a negation function, chosen extension of fuzzy ℒ𝒞 with typicality. As in the two-
among the combination functions of fuzzy logics (we refer valued case, and in the propositional typicality logic, PTL,
to [55] for details). For instance, in Zadeh logic 𝑎 ⊗ 𝑏 = [12] the nesting of the typicality operator is not allowed.
𝑚𝑖𝑛{𝑎, 𝑏}, 𝑎 ⊕ 𝑏 = 𝑚𝑎𝑥{𝑎, 𝑏}, 𝑎 ▷ 𝑏 = 𝑚𝑎𝑥{1 − 𝑎, 𝑏} Observe that, in a fuzzy ℒ𝒞 interpretation 𝐼 = ⟨Δ, ·𝐼 ⟩,
and ⊖𝑎 = 1 − 𝑎. the degree of membership 𝐶 𝐼 (𝑥) of the domain elements
The interpretation function ·𝐼 is also extended to non- 𝑥 in a concept 𝐶 induces a preference relation <𝐶 on Δ,
fuzzy axioms (i.e., to strict inclusions and assertions of an as follows:
ℒ𝒞 knowledge base) as follows:
(𝐶 ⊑ 𝐷)𝐼 = 𝑖𝑛𝑓𝑥∈Δ 𝐶 𝐼 (𝑥) ▷ 𝐷𝐼 (𝑥), 𝑥 <𝐶 𝑦 iff 𝐶 𝐼 (𝑥) > 𝐶 𝐼 (𝑦) (1)
(𝐶(𝑎))𝐼 = 𝐶 𝐼 (𝑎𝐼 ).
A fuzzy ℒ𝒞 knowledge base 𝐾 is a pair (𝒯𝑓 , 𝒜𝑓 ) where Each <𝐶 has the properties of preference relations in
𝒯𝑓 is a fuzzy TBox and 𝒜𝑓 a fuzzy ABox. A fuzzy KLM-style ranked interpretations [52], that is, <𝐶 is a
TBox is a set of fuzzy concept inclusions of the form modular and well-founded strict partial order, under the
𝐶 ⊑ 𝐷 𝜃 𝑛, where 𝐶 ⊑ 𝐷 is an ℒ𝒞 concept inclusion assumption that fuzzy interpretations are witnessed (see
axiom, 𝜃 ∈ {≥, ≤, >, <} and 𝑛 ∈ [0, 1]. A fuzzy ABox Section 2) or that Δ is finite. Let us recall that, <𝐶 is well-
𝒜𝑓 is a set of fuzzy assertions of the form 𝐶(𝑎)𝜃𝑛, where founded if there is no infinite descending chain 𝑥1 <𝐶 𝑥0 ,
𝐶 is an ℒ𝒞 concept, 𝑎 ∈ 𝑁𝐼 , 𝜃 ∈ {≥, ≤, >, <} and 𝑛 ∈ 𝑥2 <𝐶 𝑥1 , 𝑥3 <𝐶 𝑥2 , . . . of domain elements; <𝐶 is
[0, 1]. Following Bobillo and Straccia [10], we assume modular if, for all 𝑥, 𝑦, 𝑧 ∈ Δ, 𝑥 <𝐶 𝑦 implies (𝑥 <𝐶 𝑧
that fuzzy interpretations are witnessed, i.e., the sup and or 𝑧 <𝐶 𝑦).
inf are attained at some point of the involved domain. The As there are multiple preferences, fuzzy interpretations
notions of satisfiability of a KB in a fuzzy interpretation can be regarded as multipreferential interpretations, which
and of entailment are defined in the natural way. have been also studied in the two-valued case by Giordano
and Theseider Dupré [41], by Delgrande and Rantsoudis
Definition 2 (Satisfiability and entailment). A fuzzy in- [28], by Giordano and Gliozzi [37], by Casini et al. [19].
terpretation 𝐼 satisfies a fuzzy ℒ𝒞 axiom 𝐸 (denoted Preference relation <𝐶 captures the relative typical-
𝐼 |= 𝐸), as follows: ity of domain elements wrt concept 𝐶 and may then be
used to identify the typical 𝐶-elements. We will regard
• 𝐼 satisfies a fuzzy ℒ𝒞 inclusion axiom 𝐶 ⊑ 𝐷 𝜃 𝑛 typical 𝐶-elements as the domain elements 𝑥 that are
if (𝐶 ⊑ 𝐷)𝐼 𝜃 𝑛; preferred with respect to relation <𝐶 among those such
• 𝐼 satisfies a fuzzy ℒ𝒞 assertion 𝐶(𝑎) 𝜃 𝑛 if that 𝐶 𝐼 (𝑥) ̸= 0. Let 𝐶>0 𝐼
be the crisp set containing
𝐶 𝐼 (𝑎𝐼 )𝜃 𝑛 all domain elements 𝑥 such that 𝐶 𝐼 (𝑥) > 0, that is,
𝐼
𝐶>0 = {𝑥 ∈ Δ | 𝐶 𝐼 (𝑥) > 0}. One can provide a
where for 𝜃 ∈ {≥, ≤, >, <} and 𝑛 ∈ [0, 1].
(two-valued) interpretation of typicality concepts T(𝐶)
Given a fuzzy ℒ𝒞 knowledge base 𝐾 = (𝒯𝑓 , 𝒜𝑓 ), a
in a fuzzy interpretation 𝐼, by letting:
fuzzy interpretation 𝐼 satisfies 𝒯𝑓 (resp. 𝒜𝑓 ) if 𝐼 satisfies
all fuzzy inclusions in 𝒯𝑓 (resp. all fuzzy assertions in 𝒜𝑓 ). {︂
1 if 𝑥 ∈ 𝑚𝑖𝑛<𝐶 (𝐶>0 𝐼
)
A fuzzy interpretation 𝐼 is a model of 𝐾 if 𝐼 satisfies 𝒯𝑓 (T(𝐶))𝐼 (𝑥) = (2)
0 otherwise
and 𝒜𝑓 . A fuzzy axiom 𝐸 is entailed by a fuzzy knowledge
base 𝐾 (i.e., 𝐾 |= 𝐸) if for all models 𝐼 =⟨Δ, ·𝐼 ⟩ of 𝐾, where 𝑚𝑖𝑛< (𝑆) = {𝑢 : 𝑢 ∈ 𝑆 and ∄𝑧 ∈ 𝑆 s.t. 𝑧 < 𝑢}.
𝐼 satisfies 𝐸. When (T(𝐶))𝐼 (𝑥) = 1, we say that 𝑥 is a typical 𝐶-
element in 𝐼. Notice that, if 𝐶 𝐼 (𝑥) > 0 for some 𝑥 ∈ Δ,
𝐼
𝑚𝑖𝑛<𝐶 (𝐶>0 ) is non-empty.
3. Fuzzy ℒ𝒞 with typicality: ℒ𝒞 F T
Definition 3 (ℒ𝒞 F T interpretation). An ℒ𝒞 F T inter-
In this section, we describe an extension of fuzzy ℒ𝒞
pretation 𝐼 = ⟨Δ, ·𝐼 ⟩ is a fuzzy ℒ𝒞 interpretation, ex-
with typicality following [42, 36]. Typicality concepts
tended by interpreting typicality concepts as in (2).
of the form T(𝐶) are added, where 𝐶 is a concept in
fuzzy ℒ𝒞. The idea is similar to the extension of 𝒜ℒ𝒞 The fuzzy interpretation 𝐼 = ⟨Δ, ·𝐼 ⟩ implicitly defines
with typicality under the two-valued semantics [39] but a multipreference interpretation, where any concept 𝐶 is
transposed to the fuzzy case. The extension allows for associated to a preference relation <𝐶 . This is different
the definition of fuzzy typicality inclusions of the form from the two-valued multipreference semantics in [41],
T(𝐶) ⊑ 𝐷 𝜃 𝑛, meaning that typical 𝐶-elements are where only the subset of distinguished concepts have an
𝐷-elements with a degree 𝑚 such that 𝑚𝜃𝑛 holds. In associated preference, and a notion of global preference <
the two-valued case, a typicality inclusion T(𝐶) ⊑ 𝐷 is introduced to define the interpretation of the typicality
stands for a KLM conditional implication 𝐶 |∼ 𝐷 [51, 52], concept T(𝐶), for any arbitrary 𝐶. Here, we do not need
129
�to introduce a notion of global preference. The interpre- penguin, flying is not plausible (inclusion (𝑑5 ) has nega-
tation of any ℒ𝒞 concept 𝐶 is defined compositionally tive weight -70), while being a bird and being black are
from the interpretation of atomic concepts, and the pref- plausible properties of prototypical penguins, and (𝑑4 )
erence relation <𝐶 associated to 𝐶 is defined from 𝐶 𝐼 . and (𝑑6 ) have positive weights. Given an ABox in which
The notions of satisfiability in ℒ𝒞 F T, model of an ℒ𝒞 F T Reddy is red, has wings, has feather and flies (all with
knowledge base, and ℒ𝒞 F T entailment can be defined in degree 1) and Opus has wings and feather, does not fly
a similar way as in fuzzy ℒ𝒞 (see Section 2). (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
3.1. Strengthening ℒ𝒞 F T: some closure
a penguin.
constructions
To overcome the weakness of preferential entailment, the The semantics of a weighted knowledge base is defined
rational closure [52] and the lexicographic closure of a in [42] trough a semantic closure construction, which al-
conditional knowledge base [53] have been introduced. In lows a subset of the 𝒜ℒ𝒞 F T interpretations to be selected,
this section, we recall a closure construction introduced namely, the interpretations whose induced preference rela-
by Giordano and Theseider Dupré [42] to strengthen tions <𝐶𝑖 , for the distinguished concepts 𝐶𝑖 , coherently
𝒜ℒ𝒞 F T entailment for weighted conditional knowledge or faithfully represent the defeasible part of the knowledge
bases, and then we consider some variants. In the two- base 𝐾.
valued case, the construction is related to the definition Let 𝒯𝐶𝑖 = {(𝑑𝑖ℎ , 𝑤ℎ𝑖 )} be the set of weighted typicality
of Kern-Isberner’s c-representations [47, 48], which in- inclusions 𝑑𝑖ℎ = T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ associated to the distin-
clude penalty points for falsified conditionals. In the fuzzy guished concept 𝐶𝑖 , and let 𝐼 = ⟨Δ, ·𝐼 ⟩ be a fuzzy ℒ𝒞 F T
case, the construction also relates to the fuzzy extension interpretation. In the two-valued case, we would associate
of rational closure by Casini and Straccia [22]. to each domain element 𝑥 ∈ Δ and each distinguished
A weighted ℒ𝒞 F T knowledge base 𝐾, over a set concept 𝐶𝑖 , a weight 𝑊𝑖 (𝑥) of 𝑥 wrt 𝐶𝑖 in 𝐼, by summing
𝒞 = {𝐶1 , . . . , 𝐶𝑘 } of distinguished ℒ𝒞 concepts, is a the weights of the defeasible inclusions for 𝐶𝑖 satisfied
tuple ⟨𝒯𝑓 , 𝒯𝐶1 , . . . , 𝒯𝐶𝑘 , 𝒜𝑓 ⟩, where 𝒯𝑓 is a set of fuzzy by 𝑥. However, as 𝐼 is a fuzzy interpretation, we also
ℒ𝒞 F T inclusion axiom, 𝒜𝑓 is a set of fuzzy ℒ𝒞 F T as- need to consider, for all inclusions T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ ∈ 𝒯𝐶𝑖 ,
sertions and 𝒯𝐶𝑖 = {(𝑑𝑖ℎ , 𝑤ℎ𝑖 )} is a set of all weighted the degree of membership of 𝑥 in 𝐷𝑖,ℎ . Furthermore, in
typicality inclusions 𝑑𝑖ℎ = T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ for 𝐶𝑖 , in- comparing the weight of domain elements with respect to
dexed by ℎ, where each inclusion 𝑑𝑖ℎ has weight 𝑤ℎ𝑖 , a <𝐶𝑖 , we give higher preference to the domain elements
real number. As in [42], the typicality operator is assumed belonging to 𝐶𝑖 (with a degree greater than 0), with re-
to occur only on the l.h.s. of a weighted typicality inclu- spect to those not belonging to 𝐶𝑖 (having membership
sion, and we call distinguished concepts those concepts degree 0).
𝐶𝑖 occurring in the l.h.s. of such inclusions. Arbitrary For each domain element 𝑥 ∈ Δ and distinguished
ℒ𝒞 F T inclusions and assertions may belong to 𝒯𝑓 and concept 𝐶𝑖 , the weight 𝑊𝑖 (𝑥) of 𝑥 wrt 𝐶𝑖 in the ℒ𝒞 F T
𝒜𝑓 . Let us consider the following example of weighted interpretation 𝐼 = ⟨Δ, ·𝐼 ⟩ is defined as follows:
ℒ𝒞 F T knowledge base adapted from [36]: {︂ ∑︀ 𝑖 𝐼
ℎ 𝑤ℎ 𝐷𝑖,ℎ (𝑥) if 𝐶𝑖𝐼 (𝑥) > 0
𝑊𝑖 (𝑥) =
Example 1. Consider the weighted knowledge base 𝐾 = −∞ otherwise
⟨𝒯𝑓 , 𝒯𝐵𝑖𝑟𝑑 , 𝒯𝑃 𝑒𝑛𝑔𝑢𝑖𝑛 , 𝒜𝑓 ⟩, over the set of distinguished (3)
concepts 𝒞 = {Bird , Penguin}, with the strict TBox
where −∞ is added at the bottom of real values.
𝒯𝑓 containing the inclusion Black ⊓ Red ⊑ ⊥ ≥ 1 ; the
The value of 𝑊𝑖 (𝑥) is −∞ when 𝑥 is not a 𝐶-element
weighted TBox 𝒯𝐵𝑖𝑟𝑑 containing the weighted defeasible
(i.e., 𝐶𝑖𝐼 (𝑥) = 0). Otherwise, 𝐶𝑖𝐼 (𝑥) > 0 and the higher
inclusions:
is the sum 𝑊𝑖 (𝑥), the more typical is the element 𝑥 rela-
(𝑑1 ) T(Bird ) ⊑ Fly, +20
tive to the defeasible properties of 𝐶𝑖 .
(𝑑2 ) T(Bird ) ⊑ Has_Wings, +50
In [42] a notion of coherence is introduced, to force
(𝑑3 ) T(Bird ) ⊑ Has_Feather , +50;
an agreement between the preference relations <𝐶𝑖 in-
𝒯𝑃 𝑒𝑛𝑔𝑢𝑖𝑛 containing the weighted defeasible inclusions:
duced by a fuzzy interpretation 𝐼, for each distinguished
(𝑑4 ) T(Penguin) ⊑ Bird , +100
concept 𝐶𝑖 , and the weights 𝑊𝑖 (𝑥) computed, for each
(𝑑5 ) T(Penguin) ⊑ Fly, - 70
𝑥 ∈ Δ, from the conditional knowledge base 𝐾, given
(𝑑6 ) T(Penguin) ⊑ Black , +50.
the interpretation 𝐼. This leads to the following definition
The meaning is that a bird normally has wings, has feath-
of a coherent fuzzy multipreference model of a weighted
ers and flies, but having wings and feather (both with
a ℒ𝒞 F T knowledge base.
weight 50) for a bird is more plausible than flying (weight
20), although flying is regarded as being plausible. For a
130
�Definition 4 (Coherent (fuzzy) multipreference model). For MLPs, the deep network itself can be regarded
Let 𝐾 = ⟨𝒯𝑓 , 𝒯𝐶1 , . . . , 𝒯𝐶𝑘 , 𝒜𝑓 ⟩ be a weighted ℒ𝒞 F T as a conditional knowledge base, by mapping synaptic
knowledge base over 𝒞. A coherent (fuzzy) multi- connections to weighted conditionals, so that the input-
preference model (cf𝑚 -model) of 𝐾 is a fuzzy ℒ𝒞 F T output model of the network can be regarded as a coherent-
interpretation 𝐼 = ⟨Δ, ·𝐼 ⟩ s.t.: model of the associated conditional knowledge base [42].
• 𝐼 satisfies the fuzzy inclusions in 𝒯𝑓 and the fuzzy
assertions in 𝒜𝑓 ; 4. Yet another closure
• for all 𝐶𝑖 ∈ 𝒞, the preference <𝐶𝑖 is coherent to
𝒯𝐶𝑖 , that is, for all 𝑥, 𝑦 ∈ Δ,
construction: 𝜙-coherent
models
𝑥 <𝐶𝑖 𝑦 ⇐⇒ 𝑊𝑖 (𝑥) > 𝑊𝑖 (𝑦) (4)
In this section we consider a new notion of coherence of a
In a similar way, one can define a faithful (fuzzy) multipref- fuzzy interpretation 𝐼 wrt a KB, that we call 𝜙-coherence,
erence model (fm-model) of 𝐾 by replacing the coherence where 𝜙 is a function from R to the interval [0, 1], i.e.,
condition (4) with the following faithfulness condition 𝜙 : R → [0, 1]. We also establish it relationships with
(called weak coherence in [42], extended version): for all coherent and faithful models.
𝑥, 𝑦 ∈ Δ,
Definition 5 (𝜙-coherence). Let 𝐾 = ⟨𝒯𝑓 , 𝒯𝐶1 , . . . ,
𝑥 <𝐶𝑖 𝑦 ⇒ 𝑊𝑖 (𝑥) > 𝑊𝑖 (𝑦). 𝒯𝐶𝑘 , 𝒜𝑓 ⟩ be a weighted ℒ𝒞 F T knowledge base, and 𝜙 :
(5)
R → [0, 1]. A fuzzy ℒ𝒞 F T interpretation 𝐼 = ⟨Δ, ·𝐼 ⟩ is
The weaker notion of faithfulness allows to define a larger 𝜙-coherent if, for all concepts 𝐶𝑖 ∈ 𝒞 and 𝑥 ∈ Δ,
class of fuzzy multipreference models of a weighted ∑︁ 𝑖 𝐼
knowledge base, compared to the class of coherent mod- 𝐶𝑖𝐼 (𝑥) = 𝜙( 𝑤ℎ 𝐷𝑖,ℎ (𝑥)) (6)
els. This allows a larger class of monotone non-decreasing ℎ
activation functions in neural network models to be cap-
𝑖
tured, whose activation function is monotonically non- where 𝒯𝐶𝑖 = {(T(𝐶𝑖 ) ⊑ 𝐷𝑖,ℎ , 𝑤ℎ )} is the set of
decreasing (we refer to [42], extended version, Sec. 7). weighted conditionals for 𝐶 𝑖 .
Example 2. Referring to Example 1 above, To define 𝜙-coherent multipreference model of a knowl-
let us further assume that Bird I (reddy) = 1 , edge base 𝐾, we can replace the coherence condition
Bird I (opus) = 0.8, that Penguin I (reddy) = 0 .2 and (4) in Definition 4 with the notion of 𝜙-coherence of an
Penguin I (opus) = 0 .8 . Clearly, 𝑟𝑒𝑑𝑑𝑦 <𝐵𝑖𝑟𝑑 𝑜𝑝𝑢𝑠 interpretation 𝐼 wrt the knowledge base 𝐾.
and 𝑜𝑝𝑢𝑠 <𝑃 𝑒𝑛𝑔𝑢𝑖𝑛 𝑟𝑒𝑑𝑑𝑦. The interpretation 𝐼 to be Observe that, for all 𝑥 such that 𝐶𝑖 (𝑥) > 0, condition
faithful and coherent, as WBird (reddy) > WBird (opus) (6) above corresponds to condition 𝐶𝑖𝐼 (𝑥) = 𝜙(𝑊𝑖 (𝑥)).
and WPenguin (opus) > WPenguin (reddy) While in coherent and faithful models the notion of weight
hold. On the contrary, if we had Penguin I 𝑊𝑖 (𝑥) considers, as a special case, the case 𝐶𝑖 (𝑥) = 0,
(reddy) = 0 .9 , the interpretation 𝐼 would not condition (6) imposes the same constraint to all domain
be faithful. For Penguin I (reddy) = 0 .8 , the elements 𝑥.
interpretation 𝐼 would be faithful, but not coher- To see the relation between this semantics and Mul-
ent, as WPenguin (opus) > WPenguin (reddy), but tilayer Perceptrons, consider that a neuron 𝑘 can be
Penguin I (opus) = Penguin I (reddy). described
∑︀𝑛 by the following pair of equations: 𝑢𝑘 =
𝑗=1 𝑤𝑘𝑗 𝑥𝑗 , and 𝑦𝑘 = 𝜙(𝑢𝑘 + 𝑏𝑘 ), where 𝑥1 , . . . , 𝑥𝑛
It has been shown [42] that the proposed semantics are the input signals and 𝑤𝑘1 , . . . , 𝑤𝑘𝑛 are the weights of
allows the input-output behavior of a deep network (con- neuron 𝑘; 𝑏𝑘 is the bias, 𝜙 the activation function, and 𝑦𝑘
sidered after training) to be captured by a fuzzy multi- is the output signal of neuron 𝑘. By adding a new synapse
preference interpretation built over a set of input stimuli, with input 𝑥0 = +1 ∑︀ and synaptic weight 𝑤𝑘0 = 𝑏𝑘 , one
through a simple construction which exploits the activity can write: 𝑢𝑘 = 𝑛 𝑗=0 𝑤𝑘𝑗 𝑥𝑗 , and 𝑦𝑘 = 𝜙(𝑢𝑘 ), where
level of neurons for the stimuli. Each unit ℎ of 𝒩 can be 𝑢𝑘 is called the induced local field of the neuron. The
associated to a concept name 𝐶ℎ 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 𝑥1 , . . . , 𝑥𝑛 and the output signal 𝑦𝑘 of neu-
ulus 𝑥 is interpreted as the degree of membership of 𝑥 in ron 𝑘 are nodes of the graph. An edge from 𝑥𝑗 to 𝑦𝑘 ,
concept 𝐶ℎ . The resulting preferential interpretation can labelled 𝑤𝑘𝑗 , means that 𝑥𝑗 is an input signal of neuron
be used for verifying properties of the network by model 𝑘 with synaptic weight 𝑤𝑘𝑗 . 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].
131
� Let us associate a concept name 𝐶𝑖 to each unit 𝑖 in a 5. Coherent, faithful and
deep neural network 𝒩 (possibly allowing for feedback),
𝜙-coherent semantics for
and let us interpret, as in [42], a synaptic connection
between neuron ℎ and neuron 𝑖 with weight 𝑤𝑖ℎ as the weighted argumentation
conditional T(𝐶𝑖 ) ⊑ 𝐶𝑗 with weight 𝑤ℎ𝑖 = 𝑤𝑖ℎ . If
we assume that 𝜙 is the activation function of all units There is much work in the literature concerning extension
in the network 𝒩 , then condition (6) characterizes the of Dung’s argumentation framework [29] with weights
stationary states of the network, where 𝐶𝑖𝐼 (𝑥) corresponds attached to arguments and/or to the attacks between argu-
to of neuron 𝑖 for some input stimulus 𝑥 and ments. Many different proposals have been investigated
∑︀the activation
𝑖 𝐼 and compared in the literature. Let us just mention, for the
ℎ 𝑤ℎ 𝐷𝑖,ℎ (𝑥) corresponds to the induced local field of
neuron 𝑖, where each 𝐷𝑖,ℎ 𝐼
(𝑥) represents the input signal moment, the work by Cayrol and Lagasquie-Schiex [24],
𝑥ℎ , for input stimulus 𝑥. Janssen and Cock [46], Dunne et al. [30], Egilmez et al.
Of course, 𝜙-coherence could be easily extended to [31], Amgoud et al. [2], Amgoud and Doder [4], which
deal with different activation functions 𝜙𝑖 , one for each also include extensive comparisons. In the following, we
concept 𝐶𝑖 (i.e., for each unit 𝑖). The following proposi- propose some semantics for weighted argumentation with
tion establishes some relationships between 𝜙-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 𝐾. sections.
We consider a notion of weighted argumentation graph
Proposition 1. Let 𝐾 be a weighted conditional ℒ𝒞 F T as a triple 𝐺 = ⟨𝒜, ℛ, 𝜋⟩, where 𝒜 is a set of argu-
knowledge base and 𝜙 : R → [0, 1]. (1) if 𝜙 is a monoton- ments, ℛ ⊆ 𝒜 × 𝒜 and 𝜋 : ℛ → R. This definition
ically non-decreasing function, a 𝜙-coherent fuzzy multi- of weighted argumentation graph corresponds to the defi-
preference model 𝐼 of 𝐾 is also a faithful-model of 𝐾; (2) nition of weighted argument system in [30], but here we
if 𝜙 is a monotonically increasing function, a 𝜙-coherent admit both positive and negative weights, while [30] only
fuzzy multipreference model 𝐼 of 𝐾 is also a coherent- allows for positive weights representing the strength of at-
model of 𝐾. tacks. In our notion of weighted graph, a pair (𝐴, 𝐵) ∈ 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-
ℳ𝑓,Δ𝒩 of a deep neural network 𝒩 , built over the domain
tion semantics we consider in the following, as in the case
of input stimuli Δ, is proven to be a coherent model of the of weighted conditionals, deals with both the positive and
knowledge base 𝐾 𝒩 associated to 𝒩 , under the specified the negative weights in a uniform way. For the moment
conditions on the activation function 𝜙, and the assump- we do not include in 𝐺 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 𝐺 = ⟨𝒜, ℛ, 𝜋⟩,
well the analog of Proposition 2 in [42], extended version, we define a labelling of the graph 𝐺 as a function
stating that ℳ𝑓,Δ
𝒩 is a faithful (or weakly-coerent) model
𝜎 : 𝒜 → [0, 1] which assigns to each argument an ac-
of 𝐾 𝒩 . ceptability degree, i.e., a value in the interval [0, 1]. Let
A notion of coherent/faithful/𝜙-coherent multiprefer- R − (A) = {B | (B , A) ∈ ℛ}. When R − (A) = ∅, argu-
ence entailment from a weighted ℒ𝒞 F T knowledge base ment 𝐴 has neither supports nor attacks.
𝐾 can be defined in the obvious way (see [42, 36] for the For a weighted graph 𝐺 = ⟨𝒜, ℛ, 𝜋⟩ and a labelling
definitions of coherent and faithful (fuzzy) multiprefer- 𝜎, we introduce a weight 𝑊𝜎𝐺 on 𝒜, as a partial function
ence entailment). The properties of faithful entailment 𝑊𝜎𝐺 : 𝒜 → R, assigning a positive or negative support to
have been studied in [36]. Faithful entailment is reason- the arguments 𝐴𝑖 ∈ 𝒜 such that R − (Ai ) ̸= ∅, as follows:
ably well-behaved: it deals with specificity and irrele- ∑︁
vance; it is not subject to inheritance blocking; it satisfies 𝑊𝜎𝐺 (𝐴𝑖 ) = 𝜋(𝐴𝑗 , 𝐴𝑖 ) 𝜎(𝐴𝑗 ) (7)
most KLM properties [51, 52], depending on their fuzzy (𝐴𝑗 ,𝐴𝑖 )∈ℛ
reformulation and on the chosen combination functions.
As MLPs are usually represented as a weighted graphs When R − (Ai ) = ∅, 𝑊𝜎𝐺 (𝐴𝑖 ) is let undefined.
[44], whose nodes are units and whose edges are the We can now exploit this notion of weight of an argu-
synaptic 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 𝐺 as follows.
weighted knowledge bases considered above, to weighted
argumentation graphs. Definition 6. Given a weighted graph 𝐺 = ⟨𝒜, ℛ, 𝜋⟩
and a labelling 𝜎:
132
� • 𝜎 is a coherent labelling of 𝐺 if, for all arguments An evaluation method for a graph 𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩
𝐴, 𝐵 ∈ 𝒜 s.t. R − (A) ̸= ∅ and R − (B ) ̸= ∅, is a triple 𝑀 = ⟨ℎ, 𝑔, 𝑓 ⟩, where1 :
𝜎(𝐴) < 𝜎(𝐵) ⇐⇒ 𝑊𝜎𝐺 (𝐴) < 𝑊𝜎𝐺 (𝐵); R × [0, 1] → R
ℎ : ⋃︀
𝑔 : +∞ 𝑛
𝑛=0 R → R
• 𝜎 is a faithfull labelling of 𝐺 if, for all arguments 𝑓 : [0, 1] × 𝑅𝑎𝑛𝑔𝑒(𝑔) → [0, 1]
𝐴, 𝐵 ∈ 𝒜 s.t. R − (A) ̸= ∅ and R − (B ) ̸= ∅,
Function ℎ is intended to calculate the strength of an
𝜎(𝐴) < 𝜎(𝐵) ⇒ 𝑊𝜎𝐺 (𝐴) < 𝑊𝜎𝐺 (𝐵); attack/support by aggregating the weight on the edge be-
tween two arguments with the strength of the attacker/sup-
• for a function 𝜙 : R → [0, 1], 𝜎 is a 𝜙-coherent porter. Function 𝑔 aggregates the strength of all attacks
labelling of 𝐺 if, for all arguments 𝐴 ∈ 𝒜 s.t. and supports to a given argument, and function 𝑓 returns a
R − (A) ̸= ∅, 𝜎(𝐴) = 𝜙(𝑊𝜎𝐺 (𝐴)). value for an argument, given the strength of the argument
and aggregated weight of its attacks and supports.
These definitions do not put any constraint on the labelling
As in [4], a gradual semantics 𝑆 is a function assigning
of arguments which do not have incoming edges in 𝐺: 𝑆
to any graph 𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩ a weighting 𝐷𝑒𝑔𝐺 on 𝒜,
their labelling is arbitrary, provided the constraints on the 𝑆 𝑆
i.e., 𝐷𝑒𝑔𝐺 : 𝒜 → [0, 1], where 𝐷𝑒𝑔𝐺 (𝐴) represents the
labelings of all other arguments can be satisfied, depend-
strength of an argument 𝐴 (or its acceptability degree).
ing on the semantics considered.
A gradual semantics 𝑆 is based on an evaluation
The definition of 𝜙-coherent labelling of 𝐺 is defined
method 𝑀 iff, ∀ 𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩, ∀𝐴 ∈ 𝒜,
through a set of equations, as in Gabbay’s equational
approach to argumentation networks [32]. Here, we use 𝐷𝑒𝑔𝐺 𝑆 𝑆
(𝐴) = 𝑓 (𝜎0 (𝐴), 𝑔(ℎ(𝜋(𝐵1 , 𝐴), 𝐷𝑒𝑔𝐺 (𝐵1 )), . . . ,
equations for defining the weights of arguments starting 𝑆
from the weights of attacks/supports. ℎ(𝜋(𝐵𝑛 , 𝐴), 𝐷𝑒𝑔𝐺 (𝐵𝑛 )))
A 𝜙-coherent labelling of a weigthed graph 𝐺 can be
where B1 , . . . , Bn are all arguments attacking or support-
proven to be as well a coherent labelling or a faithful
ing 𝐴 (i.e., R − (A) = {B1 , . . . , Bn }).
labelling, under some conditions on the function 𝜙.
Let us consider the evaluation method 𝑀 𝜙 =
Proposition 2. Given a weighted graph 𝐺 = ⟨𝒜, ℛ, 𝜋⟩: ⟨ℎ𝑝𝑟𝑜𝑑 , 𝑔𝑠𝑢𝑚 , 𝑓𝜙 ⟩, where the functions ℎ𝑝𝑟𝑜𝑑 and 𝑔𝑠𝑢𝑚
(1) A coherent labelling of 𝐺 is a faithful labelling of 𝐺; are defined as in [4],∑︀i.e., ℎ𝑝𝑟𝑜𝑑 (𝑥, 𝑦) = 𝑥 · 𝑦 and
𝑛
(2) if 𝜙 is a monotonically non-decreasing function, a 𝜙- 𝑔𝑠𝑢𝑚 (𝑥1 , . . . , 𝑥𝑛 ) = 𝑖=1 𝑥𝑖 , but we let 𝑔𝑠𝑢𝑚 () to be
coherent labelling 𝜎 of 𝐺 is a faithful labelling of 𝐺; (3) undefined. We let 𝑓𝜙 (𝑥, 𝑦) = 𝑥 when 𝑦 is undefined, and
if 𝜙 is a monotonically increasing function, a 𝜙-coherent 𝑓𝜙 (𝑥, 𝑦) = 𝜙(𝑦) otherwise. The function 𝑓𝜙 returns a
labelling 𝜎 of 𝐺 is a coherent labelling of 𝐺. value which is independent from the first argument, when
the second argument is not undefined (i.e., there is some
The proof is similar to the one of Proposition 1. It exploits support/attack for the argument). When 𝐴 has neither
the property of a 𝜙-labelling that 𝜎(𝐴) = 𝜙(𝑊𝜎𝐺 (𝐴)), attacks nor supports (R − (A) = ∅), 𝑓𝜙 returns the basic
for all arguments 𝐴 with R − (A) ̸= ∅, as well as the prop- strength of 𝐴, 𝜎0 (𝐴).
erties of 𝜙. The evaluation method 𝑀 𝜙 = ⟨ℎ𝑝𝑟𝑜𝑑 , 𝑔𝑠𝑢𝑚 , 𝑓𝜙 ⟩ pro-
vides a characterization of the 𝜙-coherent labelling for an
argumentation graph, in the following sense.
6. 𝜙-coherent labellings and the
gradual semantics Proposition 3. Let 𝐺 = ⟨𝒜, ℛ, 𝜋⟩ be a weighted argu-
mentation graph. If, for some 𝜎0 : 𝒜 → [0, 1], 𝑆 is a
The notion of 𝜙-coherent labelling relates to the frame- gradual semantics of graph 𝐺′ = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩ based
work of gradual semantics studied by Amgoud and Doder on the evaluation method 𝑀 𝜙 = ⟨ℎ𝑝𝑟𝑜𝑑 , 𝑔𝑠𝑢𝑚 , 𝑓𝜙 ⟩, then
𝑆
[4] where, for the sake of simplicity, the weights of argu- 𝐷𝑒𝑔𝐺 ′ is a 𝜙-coherent labelling for 𝐺.
ments and attacks are in the interval [0, 1]. Here, as we Vice-versa, if 𝜎 is a 𝜙-coherent labelling for 𝐺, then
have seen, positive and negative weights are admitted to there are a function 𝜎0 and a gradual semantics 𝑆 based
represent the strength of attacks and supports. To define on the evaluation method 𝑀 𝜙 = ⟨ℎ𝑝𝑟𝑜𝑑 , 𝑔𝑠𝑢𝑚 , 𝑓𝜙 ⟩, such
an evaluation method for 𝜙-coherent labellings, we need that, for the graph 𝐺′ = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩, 𝐷𝑒𝑔𝐺 𝑆
′ ≡ 𝜎.
to consider a slightly extended definition of an evaluation
method for a graph 𝐺 in [4]. Following [4] we include a Amgoud and Doder [4] study a large family of determi-
function 𝜎0 : 𝒜 → [0, 1] in the definition of a weighted native and well-behaved evaluation models for weighted
graph, where 𝜎0 assigns to each argument 𝐴 ∈ 𝒜 its 1 This definition is the same as in [4], but for the fact that in the
basic strength. Hence a graph 𝐺 becomes a quadruple domain/range of functions ℎ and 𝑔 interval [0, 1] is sometimes
𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩. replaced by R.
133
�graphs in which attacks have positive weights in the in- a value in the interval [0, 1] with respect to a given seman-
terval [0, 1]. For weighted graph 𝐺 with positive and tics, one can define a preferential structure starting from
negative weights, the evaluation method 𝑀 𝜙 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 𝜙 is monotonically increasing and continuous. In gen- like: "does normally argument 𝐴2 follows from argument
eral, there is not a unique semantics 𝑆 based on 𝑀 𝜙 , and 𝐴1 with a degree greater than 0.7?" This query can be
there is not a unique 𝜙-coherent labelling for a weighted formalized as a fuzzy inclusion T(𝐴1 ) ⊑ 𝐴2 > 0.7.
graph 𝐺, given a basic strength 𝜎0 . This is not surprising, In particular, let Σ is a finite set of 𝜙-coherent labelings
considering that 𝜙-coherent labelings of a graph corre- 𝜎1 , 𝜎2 , . . . of a weighted graph 𝐺 = ⟨𝒜, ℛ, 𝜋⟩, for some
spond to stationary states (or equilibrium states) in a deep function 𝜙. One can define a fuzzy multipreference in-
neural 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] describ-
ment, 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 𝜎𝑖 plays the role of a fit vector and each
argument. Synaptic positive and negative weights cor- argument 𝐴 in 𝒜 can be interpreted as a concept name
respond to the strength of supports (when positive) and of the language. Let 𝑁𝐶 = 𝒜 and 𝑁𝐼 = {𝑥1 , 𝑥2 , . . .}.
attacks (when negative). In this view, 𝜙-coherent label- We assume that there is one individual name 𝑥𝑗 in the
ings, assigning to each argument a weight in the interval language for each labelling 𝜎𝑗 ∈ Σ, and define a fuzzy
𝐺
[0, 1], correspond to stationary states of the network, the multipreference interpretation 𝐼Σ = ⟨Σ, ·𝐼 ) as follows:
solutions of a set of equations. This is in agreement with
previous results on the relationship between weighted • for all 𝑥𝑗 ∈ 𝑁𝐼 , 𝑥𝐼𝑗 = 𝜎𝑗 ;
argumentation graphs and MLPs established by Garcez, • for all 𝐴 ∈ 𝑁𝐶 , 𝐴𝐼 (𝑥𝐼𝑗 ) = 𝜎𝑗 (𝐴).
Gabbay and Lamb [27] and, more recently, by Potyca [57]. 𝐺
The fuzzy ℒ𝒞 interpretation 𝐼Σ induces a preference rela-
We refer to the conclusions for some comparisons.
tion <𝐴𝑖 for each argument 𝐴𝑖 ∈ 𝒜. For all 𝜎𝑗 , 𝜎𝑘 ∈ Σ:
Unless the network is feedforward (and the correspond-
ing graph is acyclic), stationary states cannot be uniquely 𝜎𝑗 <𝐴𝑖 𝜎𝑘 iff 𝐴𝐼𝑖 (𝑥𝐼𝑗 ) > 𝐴𝐼𝑖 (𝑥𝐼𝑘 )
determined by an iterative process from an initial labelling iff 𝜎𝑗 (𝐴𝑖 ) > 𝜎𝑗 (𝐴𝑖 ).
𝜎0 . On the other hand, a semantics 𝑆 based on 𝑀 𝜙 sat-
isfies some of the properties considered in [4], including Let 𝐾 𝐺 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- 𝐾 𝐺 = {(T(𝐴𝑖 ) ⊑ 𝐴𝑗 , 𝑤𝑗,𝑖 ) |
erly reformulated to deal with both positive and negative (𝐴𝑗 , 𝐴𝑖 ) ∈ ℛ and 𝑤((𝐴𝑗 , 𝐴𝑖 )) = 𝑤𝑗𝑖 }
weights (i.e., by replacing R − (x ) to 𝐴𝑡𝑡(𝑥), for each It can be proven that:
argument 𝑥 in the formulation in [4]). However, a se-
mantics 𝑆 based on 𝑀 𝜙 cannot be expected to satisfy the Proposition 4. Let Σ be a finite set of 𝜙-coherent la-
properties of neutrality, weakening, proportionality and belings of a weighted graph 𝐺 = ⟨𝒜, ℛ, 𝜋⟩, for some
resilience. In fact, function 𝑓𝜙 completely disregard the function 𝜙 : R → [0, 1]. The following statements hold:
initial valuation 𝜎0 in graph 𝐺 = ⟨𝒜, 𝜎0 , ℛ, 𝜋⟩, for those (i) If 𝜙 is a monotonically increasing function and
arguments having incoming edges (even if their weight is 𝜙 : R → (0, 1], then 𝐼 Σ is a coherent (fuzzy)
0). So, for instance, it is not the same, for an argument to multipreference model of 𝐾 𝐺 .
have a support with weight 0 or no support or attack at all: (ii) If 𝜙 is a monotonically non-decreasing function,
neutrality does not hold. then 𝐼 Σ is a faithful (fuzzy) multipreference model
A detailed analysis of the properties of this argumen- of 𝐾 𝐺 .
tation semantics is left for an extended version of this
work. The proof of item (i) is similar to the proof of Proposition
1 in [42] (extended version with proofs). The proof of
item (ii) is similar to the proof of Proposition 2 therein.
7. Back to conditional The restriction to a finite set Σ of 𝜙-coherent labelings is
interpretations needed to guarantee the well-foundedness of the resulting
interpretation. In fact, in general, the set of all 𝜙-coherent
An interesting question is whether, given a set of possible labelings of 𝐺 might be infinite and, if Σ is the set of all
labelings Σ = {𝜎1 , 𝜎2 , . . .} for a weighted argumentation 𝜙-coherent labelings of 𝐺, there is no guarantee that the
graph 𝐺, where each labelling 𝜎𝑖 assigns to each argument
134
�resulting fuzzy ℒ𝒞 F T interpretation is witnessed and the tation frameworks and neural networks, first investigated
preference relations <𝐴𝑖 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 learn-
erential 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 com-
tions 3.1 and 4, in a typicality inclusion T(𝐶) ⊑ 𝐷, putation of arguments. The work by Potyca [57] con-
concepts 𝐶 and 𝐷 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 ℒ𝒞 of 𝒜ℒ𝒞 considered in this paper, 𝐷 can be based on the logistic function to define an MLP-based
any boolean combination of concept names. The corre- semantics 𝜎MLP for a QBAF: for each argument 𝑎 ∈ 𝒜,
(𝑘)
spondence between weighted conditionals T(𝐴𝑖 ) ⊑ 𝐴𝑗 𝜎MLP (𝑎) = lim𝑘→∞ 𝑠𝑎 , when the limit exists, and is
in 𝐾 𝐺 and weighted attacks/supports in the argumenta- (𝑘)
undefined otherwise; where 𝑠𝑎 is a value in the interval
tion graph 𝐺, suggests a possible generalization of the [0, 1], and 𝑘 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 𝜙-coherent
concepts in the conditional semantics (e.g., by letting models fails to satisfy some of the properties in [2].
𝜎(𝐴1 ∧ 𝐴2 ) = 𝑚𝑖𝑛{𝜎(𝐴1 ), 𝜎(𝐴2 )}, using the mini- In this work we have investigated the relationships be-
mum t-norm as in Zadeh fuzzy logic). This also relates to tween 𝜙-coherent labelings and the gradual semantics by
the work considering “sets of attacking (resp. supporting) Amgoud and Doder [4], by slightly extending their defini-
arguments”; 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 eval-
been studied by Yun and Vesic, by considering “the force uation method 𝑀 𝜙 and 𝜙-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 rela-
would 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
fuzzy extension of rational closure by Casini and Straccia
8. Conclusions [22].
In this paper, drawing inspiration from a fuzzy preferen- The paper discusses an approach for defeasible reason-
tial semantics for weighted conditionals, which has been ing over a weighted argumentation graph building on 𝜙-
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 𝜙-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 preferen-
we introduce the notions of coherent/faithful/𝜙-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, 𝜙-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
135
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