=Paper=
{{Paper
|id=Vol-3197/paper12
|storemode=property
|title=From Weighted Conditionals with Typicality to a Gradual Argumentation Semantics and Back
|pdfUrl=https://ceur-ws.org/Vol-3197/paper12.pdf
|volume=Vol-3197
|authors=Laura Giordano
|dblpUrl=https://dblp.org/rec/conf/nmr/000122
}}
==From Weighted Conditionals with Typicality to a Gradual Argumentation Semantics and Back==
https://ceur-ws.org/Vol-3197/paper12.pdf
From Weighted Conditionals with Typicality
to a Gradual Argumentation Semantics and backβ
Laura Giordano1,*
1
DISIT - UniversitΓ 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(πΆ) β π· with positive or nega-
sion making and for explanation [61]. The argumentation tive weights, for distinguished concepts πΆ. 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 Lehmannβs lexico-
to deal with inheritance with exceptions in ontologies, graphic closure [53] and Kern-Isbernerβs 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(πΆ) β π· (meaning βthe typical πΆ-elements proven to have some desired properties from the knowl-
are π·-elements" or βnormally πΆβs are π·βs"), 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- βdrowning problem" [56, 9], and deals with βambiguity
tionals πΆ |βΌ π· [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 Kohonenβ 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 07β09, 2022, Haifa, Israel that most KLM postulates are satisfied, depending on
β
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
Β https://people.unipmn.it/laura.giordano/ (L. Giordano) Multilayer Perceptrons (MLPs), after training, in terms
Β© 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 βπ interpretation is defined as a pair πΌ = β¨Ξ, Β·πΌ β©
weighted KB which is associated to the neural network. where: Ξ is a domainβa set whose elements are denoted
The relationships between preferential and conditional by π₯, π¦, π§, . . . βand Β·πΌ is an extension function that maps
approaches to non-monotonic reasoning and argumen- each concept name πΆ β ππΆ to a set πΆ πΌ β Ξ, and each
tation semantics are strong. Let us just mention, the individual name π β ππΌ to an element ππΌ β Ξ. It is
work by Geffner and Pearl on Conditional Entailment, extended to complex concepts as follows:
whose proof theory is defined in terms of βargumentsβ
[34]. In this paper we aim at investigating the relation- β€πΌ = Ξ β₯πΌ = β
(Β¬πΆ)πΌ = ΞβπΆ πΌ
ships between the fuzzy multipreference semantics for (πΆ β π·) = πΆ β© π·πΌ
πΌ πΌ
(πΆ β π·)πΌ = πΆ πΌ βͺ π·πΌ
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 π-coherent fuzzy multipreference semantics. In βπ interpretation πΌ = β¨Ξ, Β·πΌ β©:
Section 5, we propose three new gradual semantics for - πΌ satisfies an inclusion πΆ β π· if πΆ πΌ β π·πΌ ;
a weighted argumentation graph (namely, a coherent, a - πΌ satisfies an assertion πΆ(π) if ππΌ β πΆ πΌ .
faithful and a π-coherent semantics) inspired by the fuzzy Given a knowledge base πΎ = (π―πΎ , ππΎ ), an interpreta-
preferential semantics of weighted conditionals and, in tion πΌ satisfies π―πΎ (resp. ππΎ ) if πΌ satisfies all inclusions
Section 6, we investigate the relationship of π-coherent in π―πΎ (resp. all assertions in ππΎ ); πΌ is a model of πΎ if
semantics with the family of gradual semantics studied by πΌ satisfies π―πΎ and ππΎ .
Amgoud and Doder. The relationships between weighted A subsumption πΉ = πΆ β π· (resp., an assertion
conditional knowledge bases and MLPs easily extend to πΆ(π)), is entailed by πΎ, written πΎ |= πΉ , if for all models
the proposed gradual semantics, which captures the sta- πΌ =β¨Ξ, Β·πΌ β© of πΎ, πΌ satisfies πΉ .
tionary behavior of MLPs. This is in agreement with
the previous results on the relationships between argu- Given a knowledge base πΎ, the subsumption problem is
mentation frameworks and neural networks by Garces, the problem of deciding whether an inclusion πΆ β π· is
Gabbay and Lamb [27] and by Potyca [57]. Section 7 sug- entailed by πΎ.
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 βπ and degree of truth, usually in the interval [0, 1]. In the follow-
fuzzy βπ ing we shortly recall the semantics of a fuzzy extension
of πβπ for the fragment βπ, 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 βπ, the boolean fragment datatypes, and restricting to constructs in βπ.
of πβπ [6], which does not allow for roles. Let ππΆ A fuzzy interpretation for βπ is a pair πΌ = β¨Ξ, Β·πΌ β©
be a set of concept names, and ππΌ a set of individual where: Ξ is a non-empty domain and Β·πΌ is fuzzy interpre-
names. βπ concepts (or, simply, concepts) can be defined tation function that assigns to each concept name π΄ β ππΆ
inductively as follows: a function π΄πΌ : Ξ β [0, 1], and to each individual name
π β ππΌ an element ππΌ β Ξ. A domain element π₯ β Ξ
β’ π΄ β ππΆ , β€ and β₯ are concepts;
belongs to concept π΄ with a membership degree π΄πΌ (ππΌ )
β’ if πΆ and π· are concepts, then πΆ βπ·, πΆ βπ·, Β¬πΆ
in [0, 1], i.e., π΄πΌ is a fuzzy set.
are concepts.
The interpretation function Β·πΌ is extended to complex
An βπ knowledge base πΎ is a pair (π―πΎ , ππΎ ), where π―πΎ concepts as follows:
is a TBox and ππΎ is an ABox. The TBox π―πΎ is a set β€πΌ (π₯) = 1, β₯πΌ (π₯) = 0,
of concept inclusions (or subsumptions) πΆ β π·, where (Β¬πΆ)πΌ (π₯) = βπΆ πΌ (π₯),
πΆ, π· are concepts. The ABox ππΎ is a set of assertions of (πΆ β π·)πΌ (π₯) = πΆ πΌ (π₯) β π·πΌ (π₯),
the form πΆ(π), where πΆ is a concept and π an individual (πΆ β π·)πΌ (π₯) = πΆ πΌ (π₯) β π·πΌ (π₯).
name in ππΌ .
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where π₯ β Ξ and β, β, β· and β are a t-norm, an s-norm, but now it has an associated degree. We call βπ F T the
an implication function, and a negation function, chosen extension of fuzzy βπ 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 π β π = [12] the nesting of the typicality operator is not allowed.
πππ{π, π}, π β π = πππ₯{π, π}, π β· π = πππ₯{1 β π, π} Observe that, in a fuzzy βπ interpretation πΌ = β¨Ξ, Β·πΌ β©,
and βπ = 1 β π. the degree of membership πΆ πΌ (π₯) of the domain elements
The interpretation function Β·πΌ is also extended to non- π₯ in a concept πΆ induces a preference relation <πΆ on Ξ,
fuzzy axioms (i.e., to strict inclusions and assertions of an as follows:
βπ knowledge base) as follows:
(πΆ β π·)πΌ = ππππ₯βΞ πΆ πΌ (π₯) β· π·πΌ (π₯), π₯ <πΆ π¦ iff πΆ πΌ (π₯) > πΆ πΌ (π¦) (1)
(πΆ(π))πΌ = πΆ πΌ (ππΌ ).
A fuzzy βπ knowledge base πΎ is a pair (π―π , ππ ) where Each <πΆ has the properties of preference relations in
π―π is a fuzzy TBox and ππ a fuzzy ABox. A fuzzy KLM-style ranked interpretations [52], that is, <πΆ is a
TBox is a set of fuzzy concept inclusions of the form modular and well-founded strict partial order, under the
πΆ β π· π π, where πΆ β π· is an βπ concept inclusion assumption that fuzzy interpretations are witnessed (see
axiom, π β {β₯, β€, >, <} and π β [0, 1]. A fuzzy ABox Section 2) or that Ξ is finite. Let us recall that, <πΆ is well-
ππ is a set of fuzzy assertions of the form πΆ(π)ππ, where founded if there is no infinite descending chain π₯1 <πΆ π₯0 ,
πΆ is an βπ concept, π β ππΌ , π β {β₯, β€, >, <} and π β π₯2 <πΆ π₯1 , π₯3 <πΆ π₯2 , . . . of domain elements; <πΆ is
[0, 1]. Following Bobillo and Straccia [10], we assume modular if, for all π₯, π¦, π§ β Ξ, π₯ <πΆ π¦ implies (π₯ <πΆ π§
that fuzzy interpretations are witnessed, i.e., the sup and or π§ <πΆ π¦).
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 DuprΓ© [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 πΌ satisfies a fuzzy βπ axiom πΈ (denoted Preference relation <πΆ captures the relative typical-
πΌ |= πΈ), as follows: ity of domain elements wrt concept πΆ and may then be
used to identify the typical πΆ-elements. We will regard
β’ πΌ satisfies a fuzzy βπ inclusion axiom πΆ β π· π π typical πΆ-elements as the domain elements π₯ that are
if (πΆ β π·)πΌ π π; preferred with respect to relation <πΆ among those such
β’ πΌ satisfies a fuzzy βπ assertion πΆ(π) π π if that πΆ πΌ (π₯) ΜΈ= 0. Let πΆ>0 πΌ
be the crisp set containing
πΆ πΌ (ππΌ )π π all domain elements π₯ such that πΆ πΌ (π₯) > 0, that is,
πΌ
πΆ>0 = {π₯ β Ξ | πΆ πΌ (π₯) > 0}. One can provide a
where for π β {β₯, β€, >, <} and π β [0, 1].
(two-valued) interpretation of typicality concepts T(πΆ)
Given a fuzzy βπ knowledge base πΎ = (π―π , ππ ), a
in a fuzzy interpretation πΌ, by letting:
fuzzy interpretation πΌ satisfies π―π (resp. ππ ) if πΌ satisfies
all fuzzy inclusions in π―π (resp. all fuzzy assertions in ππ ). {οΈ
1 if π₯ β πππ<πΆ (πΆ>0 πΌ
)
A fuzzy interpretation πΌ is a model of πΎ if πΌ satisfies π―π (T(πΆ))πΌ (π₯) = (2)
0 otherwise
and ππ . A fuzzy axiom πΈ is entailed by a fuzzy knowledge
base πΎ (i.e., πΎ |= πΈ) if for all models πΌ =β¨Ξ, Β·πΌ β© of πΎ, where πππ< (π) = {π’ : π’ β π and βπ§ β π s.t. π§ < π’}.
πΌ satisfies πΈ. When (T(πΆ))πΌ (π₯) = 1, we say that π₯ is a typical πΆ-
element in πΌ. Notice that, if πΆ πΌ (π₯) > 0 for some π₯ β Ξ,
πΌ
πππ<πΆ (πΆ>0 ) is non-empty.
3. Fuzzy βπ with typicality: βπ F T
Definition 3 (βπ F T interpretation). An βπ F T inter-
In this section, we describe an extension of fuzzy βπ
pretation πΌ = β¨Ξ, Β·πΌ β© is a fuzzy βπ interpretation, ex-
with typicality following [42, 36]. Typicality concepts
tended by interpreting typicality concepts as in (2).
of the form T(πΆ) are added, where πΆ is a concept in
fuzzy βπ. The idea is similar to the extension of πβπ The fuzzy interpretation πΌ = β¨Ξ, Β·πΌ β© implicitly defines
with typicality under the two-valued semantics [39] but a multipreference interpretation, where any concept πΆ is
transposed to the fuzzy case. The extension allows for associated to a preference relation <πΆ . This is different
the definition of fuzzy typicality inclusions of the form from the two-valued multipreference semantics in [41],
T(πΆ) β π· π π, meaning that typical πΆ-elements are where only the subset of distinguished concepts have an
π·-elements with a degree π such that πππ holds. In associated preference, and a notion of global preference <
the two-valued case, a typicality inclusion T(πΆ) β π· is introduced to define the interpretation of the typicality
stands for a KLM conditional implication πΆ |βΌ π· [51, 52], concept T(πΆ), for any arbitrary πΆ. Here, we do not need
129
to introduce a notion of global preference. The interpre- penguin, flying is not plausible (inclusion (π5 ) has nega-
tation of any βπ concept πΆ 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 (π4 )
erence relation <πΆ associated to πΆ is defined from πΆ πΌ . and (π6 ) have positive weights. Given an ABox in which
The notions of satisfiability in βπ F T, model of an βπ F T Reddy is red, has wings, has feather and flies (all with
knowledge base, and βπ F T entailment can be defined in degree 1) and Opus has wings and feather, does not fly
a similar way as in fuzzy βπ (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 βπ 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 πβπ 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 DuprΓ© [42] to strengthen tions <πΆπ , for the distinguished concepts πΆπ , coherently
πβπ 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 πΎ.
valued case, the construction is related to the definition Let π―πΆπ = {(ππβ , π€βπ )} be the set of weighted typicality
of Kern-Isbernerβs c-representations [47, 48], which in- inclusions ππβ = T(πΆπ ) β π·π,β associated to the distin-
clude penalty points for falsified conditionals. In the fuzzy guished concept πΆπ , and let πΌ = β¨Ξ, Β·πΌ β© be a fuzzy βπ 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 π₯ β Ξ and each distinguished
A weighted βπ F T knowledge base πΎ, over a set concept πΆπ , a weight ππ (π₯) of π₯ wrt πΆπ in πΌ, by summing
π = {πΆ1 , . . . , πΆπ } of distinguished βπ concepts, is a the weights of the defeasible inclusions for πΆπ satisfied
tuple β¨π―π , π―πΆ1 , . . . , π―πΆπ , ππ β©, where π―π is a set of fuzzy by π₯. However, as πΌ is a fuzzy interpretation, we also
βπ F T inclusion axiom, ππ is a set of fuzzy βπ F T as- need to consider, for all inclusions T(πΆπ ) β π·π,β β π―πΆπ ,
sertions and π―πΆπ = {(ππβ , π€βπ )} is a set of all weighted the degree of membership of π₯ in π·π,β . Furthermore, in
typicality inclusions ππβ = T(πΆπ ) β π·π,β for πΆπ , in- comparing the weight of domain elements with respect to
dexed by β, where each inclusion ππβ has weight π€βπ , a <πΆπ , we give higher preference to the domain elements
real number. As in [42], the typicality operator is assumed belonging to πΆπ (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 πΆπ (having membership
sion, and we call distinguished concepts those concepts degree 0).
πΆπ occurring in the l.h.s. of such inclusions. Arbitrary For each domain element π₯ β Ξ and distinguished
βπ F T inclusions and assertions may belong to π―π and concept πΆπ , the weight ππ (π₯) of π₯ wrt πΆπ in the βπ F T
ππ . Let us consider the following example of weighted interpretation πΌ = β¨Ξ, Β·πΌ β© is defined as follows:
βπ F T knowledge base adapted from [36]: {οΈ βοΈ π πΌ
β π€β π·π,β (π₯) if πΆππΌ (π₯) > 0
ππ (π₯) =
Example 1. Consider the weighted knowledge base πΎ = ββ otherwise
β¨π―π , π―π΅πππ , π―π ππππ’ππ , ππ β©, over the set of distinguished (3)
concepts π = {Bird , Penguin}, with the strict TBox
where ββ is added at the bottom of real values.
π―π containing the inclusion Black β Red β β₯ β₯ 1 ; the
The value of ππ (π₯) is ββ when π₯ is not a πΆ-element
weighted TBox π―π΅πππ containing the weighted defeasible
(i.e., πΆππΌ (π₯) = 0). Otherwise, πΆππΌ (π₯) > 0 and the higher
inclusions:
is the sum ππ (π₯), the more typical is the element π₯ rela-
(π1 ) T(Bird ) β Fly, +20
tive to the defeasible properties of πΆπ .
(π2 ) T(Bird ) β Has_Wings, +50
In [42] a notion of coherence is introduced, to force
(π3 ) T(Bird ) β Has_Feather , +50;
an agreement between the preference relations <πΆπ in-
π―π ππππ’ππ containing the weighted defeasible inclusions:
duced by a fuzzy interpretation πΌ, for each distinguished
(π4 ) T(Penguin) β Bird , +100
concept πΆπ , and the weights ππ (π₯) computed, for each
(π5 ) T(Penguin) β Fly, - 70
π₯ β Ξ, from the conditional knowledge base πΎ, given
(π6 ) T(Penguin) β Black , +50.
the interpretation πΌ. 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 βπ 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 πΎ = β¨π―π , π―πΆ1 , . . . , π―πΆπ , ππ β© be a weighted βπ F T as a conditional knowledge base, by mapping synaptic
knowledge base over π. A coherent (fuzzy) multi- connections to weighted conditionals, so that the input-
preference model (cfπ -model) of πΎ is a fuzzy βπ F T output model of the network can be regarded as a coherent-
interpretation πΌ = β¨Ξ, Β·πΌ β© s.t.: model of the associated conditional knowledge base [42].
β’ πΌ satisfies the fuzzy inclusions in π―π and the fuzzy
assertions in ππ ; 4. Yet another closure
β’ for all πΆπ β π, the preference <πΆπ is coherent to
π―πΆπ , that is, for all π₯, π¦ β Ξ,
construction: π-coherent
models
π₯ <πΆπ π¦ ββ ππ (π₯) > ππ (π¦) (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 πΌ wrt a KB, that we call π-coherence,
erence model (fm-model) of πΎ by replacing the coherence where π is a function from R to the interval [0, 1], i.e.,
condition (4) with the following faithfulness condition π : R β [0, 1]. We also establish it relationships with
(called weak coherence in [42], extended version): for all coherent and faithful models.
π₯, π¦ β Ξ,
Definition 5 (π-coherence). Let πΎ = β¨π―π , π―πΆ1 , . . . ,
π₯ <πΆπ π¦ β ππ (π₯) > ππ (π¦). π―πΆπ , ππ β© be a weighted βπ F T knowledge base, and π :
(5)
R β [0, 1]. A fuzzy βπ F T interpretation πΌ = β¨Ξ, Β·πΌ β© is
The weaker notion of faithfulness allows to define a larger π-coherent if, for all concepts πΆπ β π and π₯ β Ξ,
class of fuzzy multipreference models of a weighted βοΈ π πΌ
knowledge base, compared to the class of coherent mod- πΆππΌ (π₯) = π( π€β π·π,β (π₯)) (6)
els. This allows a larger class of monotone non-decreasing β
activation functions in neural network models to be cap-
π
tured, whose activation function is monotonically non- where π―πΆπ = {(T(πΆπ ) β π·π,β , π€β )} is the set of
decreasing (we refer to [42], extended version, Sec. 7). weighted conditionals for πΆ π .
Example 2. Referring to Example 1 above, To define π-coherent multipreference model of a knowl-
let us further assume that Bird I (reddy) = 1 , edge base πΎ, 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 π-coherence of an
Penguin I (opus) = 0 .8 . Clearly, πππππ¦ <π΅πππ πππ’π interpretation πΌ wrt the knowledge base πΎ.
and πππ’π <π ππππ’ππ πππππ¦. The interpretation πΌ to be Observe that, for all π₯ such that πΆπ (π₯) > 0, condition
faithful and coherent, as WBird (reddy) > WBird (opus) (6) above corresponds to condition πΆππΌ (π₯) = π(ππ (π₯)).
and WPenguin (opus) > WPenguin (reddy) While in coherent and faithful models the notion of weight
hold. On the contrary, if we had Penguin I ππ (π₯) considers, as a special case, the case πΆπ (π₯) = 0,
(reddy) = 0 .9 , the interpretation πΌ would not condition (6) imposes the same constraint to all domain
be faithful. For Penguin I (reddy) = 0 .8 , the elements π₯.
interpretation πΌ 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 π can be
Penguin I (opus) = Penguin I (reddy). described
βοΈπ by the following pair of equations: π’π =
π=1 π€ππ π₯π , and π¦π = π(π’π + ππ ), where π₯1 , . . . , π₯π
It has been shown [42] that the proposed semantics are the input signals and π€π1 , . . . , π€ππ are the weights of
allows the input-output behavior of a deep network (con- neuron π; ππ is the bias, π the activation function, and π¦π
sidered after training) to be captured by a fuzzy multi- is the output signal of neuron π. By adding a new synapse
preference interpretation built over a set of input stimuli, with input π₯0 = +1 βοΈ and synaptic weight π€π0 = ππ , one
through a simple construction which exploits the activity can write: π’π = π π=0 π€ππ π₯π , and π¦π = π(π’π ), where
level of neurons for the stimuli. Each unit β of π© can be π’π is called the induced local field of the neuron. The
associated to a concept name πΆβ and, for a given domain neuron can be represented as a directed graph, where the
Ξ of input stimuli, the activation value of unit β for a stim- input signals π₯1 , . . . , π₯π and the output signal π¦π of neu-
ulus π₯ is interpreted as the degree of membership of π₯ in ron π are nodes of the graph. An edge from π₯π to π¦π ,
concept πΆβ . The resulting preferential interpretation can labelled π€ππ , means that π₯π is an input signal of neuron
be used for verifying properties of the network by model π with synaptic weight π€ππ . A neural network can then
checking (e.g., T(Penguin) β Has_Wings β₯ 0.7, do be seen as βa directed graph consisting of nodes with
typical penguins have wings with degree β₯ 0.7?). interconnecting synaptic and activation links" [44].
131
Let us associate a concept name πΆπ to each unit π in a 5. Coherent, faithful and
deep neural network π© (possibly allowing for feedback),
π-coherent semantics for
and let us interpret, as in [42], a synaptic connection
between neuron β and neuron π with weight π€πβ as the weighted argumentation
conditional T(πΆπ ) β πΆπ with weight π€βπ = π€πβ . If
we assume that π is the activation function of all units There is much work in the literature concerning extension
in the network π© , then condition (6) characterizes the of Dungβs argumentation framework [29] with weights
stationary states of the network, where πΆππΌ (π₯) corresponds attached to arguments and/or to the attacks between argu-
to of neuron π for some input stimulus π₯ and ments. Many different proposals have been investigated
βοΈthe activation
π πΌ and compared in the literature. Let us just mention, for the
β π€β π·π,β (π₯) corresponds to the induced local field of
neuron π, where each π·π,β πΌ
(π₯) represents the input signal moment, the work by Cayrol and Lagasquie-Schiex [24],
π₯β , for input stimulus π₯. Janssen and Cock [46], Dunne et al. [30], Egilmez et al.
Of course, π-coherence could be easily extended to [31], Amgoud et al. [2], Amgoud and Doder [4], which
deal with different activation functions ππ , one for each also include extensive comparisons. In the following, we
concept πΆπ (i.e., for each unit π). The following proposi- propose some semantics for weighted argumentation with
tion establishes some relationships between π-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 πΎ. sections.
We consider a notion of weighted argumentation graph
Proposition 1. Let πΎ be a weighted conditional βπ F T as a triple πΊ = β¨π, β, πβ©, where π is a set of argu-
knowledge base and π : R β [0, 1]. (1) if π is a monoton- ments, β β π Γ π and π : β β R. This definition
ically non-decreasing function, a π-coherent fuzzy multi- of weighted argumentation graph corresponds to the defi-
preference model πΌ of πΎ is also a faithful-model of πΎ; (2) nition of weighted argument system in [30], but here we
if π is a monotonically increasing function, a π-coherent admit both positive and negative weights, while [30] only
fuzzy multipreference model πΌ of πΎ is also a coherent- allows for positive weights representing the strength of at-
model of πΎ. tacks. In our notion of weighted graph, a pair (π΄, π΅) β β
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-
β³π,Ξπ© of a deep neural network π© , built over the domain
tion semantics we consider in the following, as in the case
of input stimuli Ξ, is proven to be a coherent model of the of weighted conditionals, deals with both the positive and
knowledge base πΎ π© associated to π© , under the specified the negative weights in a uniform way. For the moment
conditions on the activation function π, and the assump- we do not include in πΊ a function determining the basic
tion that each stimulus in Ξ 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 πΊ = β¨π, β, πβ©,
well the analog of Proposition 2 in [42], extended version, we define a labelling of the graph πΊ as a function
stating that β³π,Ξ
π© is a faithful (or weakly-coerent) model
π : π β [0, 1] which assigns to each argument an ac-
of πΎ π© . ceptability degree, i.e., a value in the interval [0, 1]. Let
A notion of coherent/faithful/π-coherent multiprefer- R β (A) = {B | (B , A) β β}. When R β (A) = β
, argu-
ence entailment from a weighted βπ F T knowledge base ment π΄ has neither supports nor attacks.
πΎ can be defined in the obvious way (see [42, 36] for the For a weighted graph πΊ = β¨π, β, πβ© and a labelling
definitions of coherent and faithful (fuzzy) multiprefer- π, we introduce a weight πππΊ on π, as a partial function
ence entailment). The properties of faithful entailment πππΊ : π β R, assigning a positive or negative support to
have been studied in [36]. Faithful entailment is reason- the arguments π΄π β π such that R β (Ai ) ΜΈ= β
, as follows:
ably well-behaved: it deals with specificity and irrele- βοΈ
vance; it is not subject to inheritance blocking; it satisfies πππΊ (π΄π ) = π(π΄π , π΄π ) π(π΄π ) (7)
most KLM properties [51, 52], depending on their fuzzy (π΄π ,π΄π )ββ
reformulation and on the chosen combination functions.
As MLPs are usually represented as a weighted graphs When R β (Ai ) = β
, πππΊ (π΄π ) 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 πΊ as follows.
weighted knowledge bases considered above, to weighted
argumentation graphs. Definition 6. Given a weighted graph πΊ = β¨π, β, πβ©
and a labelling π:
132
β’ π is a coherent labelling of πΊ if, for all arguments An evaluation method for a graph πΊ = β¨π, π0 , β, πβ©
π΄, π΅ β π s.t. R β (A) ΜΈ= β
and R β (B ) ΜΈ= β
, is a triple π = β¨β, π, π β©, where1 :
π(π΄) < π(π΅) ββ πππΊ (π΄) < πππΊ (π΅); R Γ [0, 1] β R
β : βοΈ
π : +β π
π=0 R β R
β’ π is a faithfull labelling of πΊ if, for all arguments π : [0, 1] Γ π
ππππ(π) β [0, 1]
π΄, π΅ β π s.t. R β (A) ΜΈ= β
and R β (B ) ΜΈ= β
,
Function β is intended to calculate the strength of an
π(π΄) < π(π΅) β πππΊ (π΄) < πππΊ (π΅); attack/support by aggregating the weight on the edge be-
tween two arguments with the strength of the attacker/sup-
β’ for a function π : R β [0, 1], π is a π-coherent porter. Function π aggregates the strength of all attacks
labelling of πΊ if, for all arguments π΄ β π s.t. and supports to a given argument, and function π returns a
R β (A) ΜΈ= β
, π(π΄) = π(πππΊ (π΄)). 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 π is a function assigning
of arguments which do not have incoming edges in πΊ: π
to any graph πΊ = β¨π, π0 , β, πβ© a weighting π·πππΊ on π,
their labelling is arbitrary, provided the constraints on the π π
i.e., π·πππΊ : π β [0, 1], where π·πππΊ (π΄) represents the
labelings of all other arguments can be satisfied, depend-
strength of an argument π΄ (or its acceptability degree).
ing on the semantics considered.
A gradual semantics π is based on an evaluation
The definition of π-coherent labelling of πΊ is defined
method π iff, β πΊ = β¨π, π0 , β, πβ©, βπ΄ β π,
through a set of equations, as in Gabbayβs equational
approach to argumentation networks [32]. Here, we use π·πππΊ π π
(π΄) = π (π0 (π΄), π(β(π(π΅1 , π΄), π·πππΊ (π΅1 )), . . . ,
equations for defining the weights of arguments starting π
from the weights of attacks/supports. β(π(π΅π , π΄), π·πππΊ (π΅π )))
A π-coherent labelling of a weigthed graph πΊ can be
where B1 , . . . , Bn are all arguments attacking or support-
proven to be as well a coherent labelling or a faithful
ing π΄ (i.e., R β (A) = {B1 , . . . , Bn }).
labelling, under some conditions on the function π.
Let us consider the evaluation method π π =
Proposition 2. Given a weighted graph πΊ = β¨π, β, πβ©: β¨βππππ , ππ π’π , ππ β©, where the functions βππππ and ππ π’π
(1) A coherent labelling of πΊ is a faithful labelling of πΊ; are defined as in [4],βοΈi.e., βππππ (π₯, π¦) = π₯ Β· π¦ and
π
(2) if π is a monotonically non-decreasing function, a π- ππ π’π (π₯1 , . . . , π₯π ) = π=1 π₯π , but we let ππ π’π () to be
coherent labelling π of πΊ is a faithful labelling of πΊ; (3) undefined. We let ππ (π₯, π¦) = π₯ when π¦ is undefined, and
if π is a monotonically increasing function, a π-coherent ππ (π₯, π¦) = π(π¦) otherwise. The function ππ returns a
labelling π of πΊ is a coherent labelling of πΊ. 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 π΄ has neither
the property of a π-labelling that π(π΄) = π(πππΊ (π΄)), attacks nor supports (R β (A) = β
), ππ returns the basic
for all arguments π΄ with R β (A) ΜΈ= β
, as well as the prop- strength of π΄, π0 (π΄).
erties of π. The evaluation method π π = β¨βππππ , ππ π’π , ππ β© pro-
vides a characterization of the π-coherent labelling for an
argumentation graph, in the following sense.
6. π-coherent labellings and the
gradual semantics Proposition 3. Let πΊ = β¨π, β, πβ© be a weighted argu-
mentation graph. If, for some π0 : π β [0, 1], π is a
The notion of π-coherent labelling relates to the frame- gradual semantics of graph πΊβ² = β¨π, π0 , β, πβ© based
work of gradual semantics studied by Amgoud and Doder on the evaluation method π π = β¨βππππ , ππ π’π , ππ β©, then
π
[4] where, for the sake of simplicity, the weights of argu- π·πππΊ β² is a π-coherent labelling for πΊ.
ments and attacks are in the interval [0, 1]. Here, as we Vice-versa, if π is a π-coherent labelling for πΊ, then
have seen, positive and negative weights are admitted to there are a function π0 and a gradual semantics π based
represent the strength of attacks and supports. To define on the evaluation method π π = β¨βππππ , ππ π’π , ππ β©, such
an evaluation method for π-coherent labellings, we need that, for the graph πΊβ² = β¨π, π0 , β, πβ©, π·πππΊ π
β² β‘ π.
to consider a slightly extended definition of an evaluation
method for a graph πΊ in [4]. Following [4] we include a Amgoud and Doder [4] study a large family of determi-
function π0 : π β [0, 1] in the definition of a weighted native and well-behaved evaluation models for weighted
graph, where π0 assigns to each argument π΄ β π its 1 This definition is the same as in [4], but for the fact that in the
basic strength. Hence a graph πΊ becomes a quadruple domain/range of functions β and π interval [0, 1] is sometimes
πΊ = β¨π, π0 , β, πβ©. 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 πΊ with positive and tics, one can define a preferential structure starting from
negative weights, the evaluation method π π cannot be Ξ£ 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 π is monotonically increasing and continuous. In gen- like: "does normally argument π΄2 follows from argument
eral, there is not a unique semantics π based on π π , and π΄1 with a degree greater than 0.7?" This query can be
there is not a unique π-coherent labelling for a weighted formalized as a fuzzy inclusion T(π΄1 ) β π΄2 > 0.7.
graph πΊ, given a basic strength π0 . This is not surprising, In particular, let Ξ£ is a finite set of π-coherent labelings
considering that π-coherent labelings of a graph corre- π1 , π2 , . . . of a weighted graph πΊ = β¨π, β, πβ©, for some
spond to stationary states (or equilibrium states) in a deep function π. One can define a fuzzy multipreference in-
neural network [44]. terpretation over Ξ£ 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 ππ plays the role of a fit vector and each
argument. Synaptic positive and negative weights cor- argument π΄ in π can be interpreted as a concept name
respond to the strength of supports (when positive) and of the language. Let ππΆ = π and ππΌ = {π₯1 , π₯2 , . . .}.
attacks (when negative). In this view, π-coherent label- We assume that there is one individual name π₯π in the
ings, assigning to each argument a weight in the interval language for each labelling ππ β Ξ£, and define a fuzzy
πΊ
[0, 1], correspond to stationary states of the network, the multipreference interpretation πΌΞ£ = β¨Ξ£, Β·πΌ ) as follows:
solutions of a set of equations. This is in agreement with
previous results on the relationship between weighted β’ for all π₯π β ππΌ , π₯πΌπ = ππ ;
argumentation graphs and MLPs established by Garcez, β’ for all π΄ β ππΆ , π΄πΌ (π₯πΌπ ) = ππ (π΄).
Gabbay and Lamb [27] and, more recently, by Potyca [57]. πΊ
The fuzzy βπ interpretation πΌΞ£ induces a preference rela-
We refer to the conclusions for some comparisons.
tion <π΄π for each argument π΄π β π. For all ππ , ππ β Ξ£:
Unless the network is feedforward (and the correspond-
ing graph is acyclic), stationary states cannot be uniquely ππ <π΄π ππ iff π΄πΌπ (π₯πΌπ ) > π΄πΌπ (π₯πΌπ )
determined by an iterative process from an initial labelling iff ππ (π΄π ) > ππ (π΄π ).
π0 . On the other hand, a semantics π based on π π sat-
isfies some of the properties considered in [4], including Let πΎ πΊ 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- πΎ πΊ = {(T(π΄π ) β π΄π , π€π,π ) |
erly reformulated to deal with both positive and negative (π΄π , π΄π ) β β and π€((π΄π , π΄π )) = π€ππ }
weights (i.e., by replacing R β (x ) to π΄π‘π‘(π₯), for each It can be proven that:
argument π₯ in the formulation in [4]). However, a se-
mantics π based on π π cannot be expected to satisfy the Proposition 4. Let Ξ£ be a finite set of π-coherent la-
properties of neutrality, weakening, proportionality and belings of a weighted graph πΊ = β¨π, β, πβ©, for some
resilience. In fact, function ππ completely disregard the function π : R β [0, 1]. The following statements hold:
initial valuation π0 in graph πΊ = β¨π, π0 , β, πβ©, for those (i) If π is a monotonically increasing function and
arguments having incoming edges (even if their weight is π : R β (0, 1], then πΌ Ξ£ is a coherent (fuzzy)
0). So, for instance, it is not the same, for an argument to multipreference model of πΎ πΊ .
have a support with weight 0 or no support or attack at all: (ii) If π is a monotonically non-decreasing function,
neutrality does not hold. then πΌ Ξ£ is a faithful (fuzzy) multipreference model
A detailed analysis of the properties of this argumen- of πΎ πΊ .
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 Ξ£ of π-coherent labelings is
interpretations needed to guarantee the well-foundedness of the resulting
interpretation. In fact, in general, the set of all π-coherent
An interesting question is whether, given a set of possible labelings of πΊ might be infinite and, if Ξ£ is the set of all
labelings Ξ£ = {π1 , π2 , . . .} for a weighted argumentation π-coherent labelings of πΊ, there is no guarantee that the
graph πΊ, where each labelling ππ assigns to each argument
134
resulting fuzzy βπ F T interpretation is witnessed and the tation frameworks and neural networks, first investigated
preference relations <π΄π 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(πΆ) β π·, putation of arguments. The work by Potyca [57] con-
concepts πΆ and π· 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 βπ of πβπ considered in this paper, π· can be based on the logistic function to define an MLP-based
any boolean combination of concept names. The corre- semantics πMLP for a QBAF: for each argument π β π,
(π)
spondence between weighted conditionals T(π΄π ) β π΄π πMLP (π) = limπββ π π , when the limit exists, and is
in πΎ πΊ and weighted attacks/supports in the argumenta- (π)
undefined otherwise; where π π is a value in the interval
tion graph πΊ, suggests a possible generalization of the [0, 1], and π 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 π-coherent
concepts in the conditional semantics (e.g., by letting models fails to satisfy some of the properties in [2].
π(π΄1 β§ π΄2 ) = πππ{π(π΄1 ), π(π΄2 )}, 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 π-coherent labelings and the gradual semantics by
the work considering βsets of attacking (resp. supporting) Amgoud and Doder [4], by slightly extending their defini-
argumentsβ; 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 βthe force uation method π π and π-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 π-
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 π-labelling Ξ£ 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 Ξ£ by model checking over a preferen-
we introduce the notions of coherent/faithful/π-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, π-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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tics developed in the paper for weighted argumentation D. Nardi, and P.F. Patel-Schneider. The Descrip-
can 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.
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Undecidability results for fuzzy description logics with erties do we need for gradual argumentation? In
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Thanks to the anonymous referees for their helpful com- its of decidability in fuzzy description logics with
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in the context of the European Cooperation in Science & [14] S. Borgwardt and R. PeΓ±aloza. Undecidability
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