=Paper=
{{Paper
|id=Vol-3768/paper7
|storemode=property
|title=A General Dialogue Framework for Logic-based Argumentation
|pdfUrl=https://ceur-ws.org/Vol-3768/paper7.pdf
|volume=Vol-3768
|authors=Loan Ho,Stefan Schlobach
|dblpUrl=https://dblp.org/rec/conf/comma/HoS24
}}
==A General Dialogue Framework for Logic-based Argumentation==
https://ceur-ws.org/Vol-3768/paper7.pdf
A General Dialogue Framework for Logic-based
Argumentation
Loan Ho1 , Stefan Schlobach1
1
Vrije Universiteit Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
Abstract
There is an extensive body of work in logic-based argumentation, which links logic and argumentation,
which is a potential solution to address inconsistencies or conflicting information in knowledge bases
(KBs) by offering dialogue games as proof procedures to determine and explain the acceptance of
propositions. Most existing work, though, focuses on specific logics (such as description logics, existential
rules, defeasible and propositional logics), has limitations of representational aspects, for selected
semantics and binary conflicts. In this paper, we generalise this work by introducing G-SAF, which
generalises the notions of arguments, dialogues and dialogue trees for more general logical reasoning
with inconsistencies, including the most common semantics and to facilitate reasoning with non-binary
conflicts using argumentation with collective attacks (SAFs).
Keywords
Argumentation, Inconsistency-tolerant reasoning, Explanation
1. Introduction
Argumentation is a potential solution for inconsistencies or conflicting information in knowledge
bases; it offers dialogue games as proof procedures to determine and explain the acceptance of
propositions. To benefit from argumentation in explaining and querying answers in (possibly
inconsistent) KBs, Logic-based argumentation frameworks ( LAFs) have emerged, linking logic
with abstract argumentation frameworks (AFs), providing argument game-based proofs.
Several different approaches to logical argumentation have been introduced in the literature.
Most previous LAFs frameworks link Dung’s AFs [1] to specific logics. For instance, these
works present an instantiation of AFs for classical logic [2], Description Logic [3] and for
existential rules (Datalog± ) [4, 5]. In [6, 7], LAFs with collective attacks are proposed to capture
non-binary conflicts in Datalog± , i.e., assuming that every conflict has more than two formulas.
In [8, 9], ASPIC/ASPIC+ is introduced for defeasible logic. Following [10], the logical formalism
in ASPIC+ is ill-defined, i.e., the contrariness relation is not general enough to consider n-ary
constraints. This issue is stated in [6] for Datalog± .
[11, 12] proposes sequence-based argumentation for (propositional) logic and provides non-
monotonic extensions for Gentzen-style proof systems in terms of argumentation-based. More-
over, the authors conclude [12] by wishing future work to include “the study of more expressive
formalisms, like those that are based on first-order logics”. In [13, 14] DeLP/DeLP with collective
ArgXAI-24: 2nd International Workshop on Argumentation for eXplainable AI
$ loanthuyho.cs@gmail.com (L. Ho); k.s.schlobach@vu.nl (S. Schlobach)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
41
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
Loan Ho et al. CEUR Workshop Proceedings 41–55
attacks are introduced for defeasible logic programming. However, [6] claims they cannot
instantiate DeLP for Datalog± , since it only considers ground rules. In [15, 16] assumption-
based argumentation (ABA)/ ABA with collective attacks for logic programming encodes a
single conflict/multi-conflicts for each assumption, while in [17], ABA is linked to Answer Set
Programming. In [15, 16], the focus is on flat ABA frameworks, i.e., it ignores the cases of the
inferred assumption being conflicting, which is allowed in the existential rules or other logics.
From the observation, earlier works focus on specific logics or have limitations of representa-
tional aspects. This paper, using abstract logic with no requirement on the language ℒ and CN
(a function from 2ℒ to 2ℒ ), provides a very flexible environment for logical argumentation. Like
our approach, the work of [18] proposed "non-flat" ABA frameworks with collective attacks.
While the work of [18] mainly focuses on representation considerations, our work further
studies proof procedures used in AFs with collective attacks.
To compute and explain answers w.r.t the semantics of logical reasoning, argumentation
provides dialogue models as proof procedures, but, there is no unifying approach for the most
common inconsistency-tolerant semantics. In the Datalog± context, the framework of [19]
focuses on ICR semantics (i.e., the query being entailed under the intersection of closure of
repairs).The dialogue model of [20, 4] considers queries entailed under IAR semantics (the
intersection of all repairs) or Brave semantics (some repairs 1 ), while AR semantics have not yet
been considered. This paper gives possible translations for the most common types of semantics
(as defined in Definition 2).
Previous proposed dialogue models are used in LAFs with binary attacks, which are not
generic enough to deal with collective attacks. In [21, 20, 4] the authors propose a dialogue
model as an abstract dialectic proof procedure and only involving arguments and binary attacks.
The models in [22, 23], applied to ABA, are limited to express the contrariness relation for
existential rules with non-binary conflicts. The models in [24, 25] related domain expert and a
system, but are only applied to a specific domain (agronomy).
Main contributions. The existing approaches are mostly restricted to: (1) specific logics
or limitations of representational aspects; (2) specific inconsistency-tolerant semantics; (3)
(dialectical) proof procedures used in AFs with binary attacks. This paper closes this gap: (1)
we propose a unifying framework, G-LAF, that allows a translation of KBs in a broad family of
logics into argumentation frameworks; (2) our unified approach includes inconsistency-tolerant
semantics considered in previous literature (e.g. IAR and Brave in the case of Datalog± ) as well
as new ones (AR in the case of Datalog± ); (3) we propose an extended version of the dialogue
model that applies to AFs/LAFs with collective attacks, generalizing the dialogue model used in
AFs/LAFs with binary attacks.
We only sketch the proofs for our main results in this paper, more details can be found in the
appendix.
2. Preliminaries
Most of our discussion applies to arbitrary logics (monotonic and non-monotonic) consisting of
a set ℒ (of sentences) and a function CN from 2ℒ to 2ℒ returning the logical consequences of a
1
The notion of repair corresponds to "maximal consistent subsets" in our work
42
Loan Ho et al. CEUR Workshop Proceedings 41–55
set of sentences (the consequence operator), that satisfies the axiom: Expansion 𝑋 ⊆ CN(𝑋),
Idempotence CN(CN(𝑋)) = CN(𝑋). Fix a logic (ℒ, CN) and a set of sentences 𝑋 ⊆ ℒ. We say
that:
• 𝑋 is consistent wrt (ℒ, CN) iff CN(𝑋) ̸= ℒ. It is inconsistent otherwise;
• A knowledge base (KB) is any subset 𝒦 of ℒ. A knowledge base may be inconsistent.
Reasoning in inconsistent KBs 𝒦 ⊆ ℒ amounts to:
1. Constructing maximal consistent subsets,
2. Applying classical entailment mechanism on a choice of the maximal consistent subsets.
Motivated by this idea, we give the following definition.
Definition 1. Let 𝒦 be a KB and 𝑋 ⊆ 𝒦 be a set of sentences. 𝑋 is a maximal (for set-inclusion)
consistent subset of 𝒦 iff 𝑋 is consistent, there is no 𝑋 ′ such that 𝑋 ⊂ 𝑋 ′ and 𝑋 ′ is consistent.
We denote the set of all maximal consistent subsets by MCS(𝒦).
Inconsistency-tolerant reasoning in KB has semantics to determine different types of answers.
Definition 2. Let 𝒦 be a KB and ⊢: 2MCS(𝒦) ↦→ MCS(𝒦) be an entailment mechanism. A sentence
𝑓 ∈ ℒ is entailed in
• all maximal consistent subsets, written 𝒦 ⊢⋃︀ MCS(𝒦) 𝑓 , iff for all Δ ∈ MCS(𝒦), 𝑓 ∈ CN(Δ);
• some maximal consistent subset, written 𝒦 ⊢MCS(𝒦) 𝑓 , iff for some Δ ∈ MCS(𝒦), 𝑓 ∈
CN(Δ);
• the
⋂︀ intersection of all maximal consistent subsets, written 𝒦 ⊢ MCS(𝒦) 𝑓 , iff for Ψ =
⋂︀
{Δ | Δ ∈ MCS(𝒦)}, 𝑓 ∈ CN(Ψ).
3. A General Framework of Explanatory Dialogues
In this section, we introduce: (1) A general logic-based argumentation with collective
attacks (G-SAF) translated from a KB. (2) General formal model of dialogue (for G-SAF)
considered as a dialectical proof procedure to determine and explain reasoning outcomes in KBs.
This dialogue model is inspired by previous works [26, 20, 21].
3.1. General Logic-based Argumentation
Arguments are built from a KB. They represent proofs for conclusions. Thus, an argument has
two parts: a support (also called a set of premisses) and a conclusion.
Definition 3 (G-SAF arguments). An argument induced from a KB 𝒦 is the pair 𝐴 = (Γ, 𝑓 )
such that Γ ⊆ 𝒦 and 𝑓 ∈ CN(Γ). For such argument, we denote by Sup(𝐴) = Γ the support set
of 𝐴 and by Con(𝐴) = 𝑓 the consequence of 𝐴.⋃︀We use Arg𝒦 to denote the set of arguments
constructed from 𝒦. Fix 𝒮 ⊆ Arg𝒦 , Cons(𝒮) = {Con(𝐴)} are the conclusions in 𝒮.
𝐴∈𝒮
43
Loan Ho et al. CEUR Workshop Proceedings 41–55
Some proposals for logic-based argumentation stipulate additionally that the argument’s
support is consistent and/or that none of its subsets entails the argument’s conclusion (see [27]).
However, such restrictions are not be substantial (although required for some specific logics).
To keep our framework as general as possible, we do not consider the extra restrictions for our
definition of arguments. We refer to [27, 11] for further justifications of this choice.
Different attack relations have been considered in the literature for logic-based argumentation.
However, some definitions of attack are not suitable to capture non-binary conflicts [8, 13, 15, 3,
17]. The attack definitions in [4, 28, 11] can capture non-binary conflicts, and these frameworks
can generate a large number (See [7, 6] for justifications in the case of Datalog± ). To overcome
this, we use the notion of collective attacks.
Definition 4 (Collective Attacks). Let 𝐴 = (Γ, 𝛼) be an argument and 𝒳 ⊆ Arg𝒦 be a set of
arguments.
• 𝒳 undercut-attacks 𝐴 iff there is Γ′ ⊆ Γ s.t 𝑋∈𝒳 {Con(𝑋)} ∪ Γ′ is inconsistent.
⋃︀
• 𝒳 rebuttal-attacks 𝐴 iff 𝑋∈𝒳 {Con(𝑋)} ∪ {𝛼} is inconsistent.
⋃︀
We can say that 𝒳 attacks 𝐴 for short. We use Att𝒦 ⊆ 2Arg𝒦 × Arg𝒦 to denote the set of attacks
induced from 𝒦.
For KBs with only binary conflicts (e.g. core DL-Lite dialects), we have that |𝒳 | = 1. We refer
this case to (Dung) AFs where they consider an attack between two arguments (i.e. a binary
attack).
Definition 5. Let 𝒦 be a KB, the corresponding general logic-based argumentation (G-SAF)
𝒜ℱ 𝒦 is the pair (Arg𝒦 , Att𝒦 ), where Arg𝒦 is the set of arguments induced from 𝒦 and Att𝒦 is
the set of attacks.
Its semantics are now defined as in the definition of argumentation semantics for SAFs [29].
Given a G-SAF 𝒜ℱ 𝒦 = (Arg𝒦 , Att𝒦 ) and 𝒮 ⊆ Arg𝒦 . 𝒮 attacks 𝒳 iff ∃𝐴 ∈ 𝒳 s.t. 𝒮 attacks
𝐴. 𝒮 defends 𝐴 if for each 𝒳 ⊆ Arg𝒦 s.t. 𝒳 attacks 𝐴, some 𝒮 ′ ⊆ 𝒮 attacks 𝒳 . An extension 𝒮
is called
• conflict-free if it does not attack itself.
• admissible (ad) if it is conflict-free and defends itself.
• complete (cm) if it is an admissible extension containing all arguments that it defends.
• preferred (pr) if it is an inclusion-maximal admissible extension.
• stable (st) if it is conflict-free and attacks every argument which is not in it.
• grounded (gr) if it is an inclusion-minimal complete extension.
Let Exts (𝒜ℱ 𝒦 ) denote the set of all extensions of 𝒜ℱ 𝒦 under the semantics s ∈
{ad, cm, st, pr, gr}. Let us define acceptability in our G-SAF:
Definition 6. Let 𝒜ℱ 𝒦 be the corresponding G-SAF of a KB 𝒦 and s ∈ {ad, st, pr}. A sentence
𝑓 ∈ ℒ is
44
Loan Ho et al. CEUR Workshop Proceedings 41–55
• credulously accepted under s iff for some ℰ ∈ Exts (𝒜ℱ 𝒦 ), 𝑓 ∈ Cons(ℰ).
• sceptically accepted under s iff for all ℰ ∈ Exts (𝒜ℱ 𝒦 ), 𝑓 ∈ Cons(ℰ).
• groundedly accepted under gr iff for some ℰ ∈ Extgr (𝒜ℱ 𝒦 ), 𝑓 ∈ Cons(ℰ).
Proposition 1 shows a relation between extensions of G-SAFs and maximal consistent subsets
of KBs.
Proposition 1. Let 𝒜ℱ 𝒦 be the corresponding G-SAF of a KB 𝒦. Then,
• the maximal consistent subset of 𝒦 coincides with the stable/ preferred extension of 𝒜ℱ 𝒦 ;
• the intersection of the maximal consistent subsets of 𝒦 coincides with the grounded extension
of 𝒜ℱ 𝒦 .
Proof 1. The idea of the proof is to show that every preferred extension is the set of arguments
generated from a MCS, that every such set of arguments is a stable extension, and that every stable
extension is a preferred extension.
The following is the main result of this section, which generalises related results of previous
works.
Theorem 1. Let 𝒜ℱ 𝒦 be the corresponding G-SAF of a KB 𝒦, 𝑓 ∈ ℒ a sentence and s ∈
{st, ad, pr}. Then,
• 𝒦 ⊢MCS(𝒦) 𝑓 if 𝑓 is credulously accepted under s.
• 𝒦 ⊢⋃︀ MCS(𝒦) 𝑓 if 𝑓 is sceptically accepted under s.
• 𝒦 ⊢⋂︀ MCS(𝒦) 𝑓 if 𝑓 is groundedly accepted under gr.
The proof of Theorem 1 follows Proposition 1.
To argue the quality of G-SAF, it can be shown that it satisfies the rationality postulates
introduced in [2, 30].
Definition 7. Let 𝒜ℱ 𝒦 be the corresponding G-SAF of a KB 𝒦. Wrt. s ∈ {ad, st, pr, gr},
𝒜ℱ 𝒦 is
1. closed under CN iff for all ℰ ∈ Exts (𝒜ℱ 𝒦 ), Cons(ℰ) = CN(Cons(ℰ));
2. consistent iff for all ℰ ∈ Exts (𝒜ℱ 𝒦 ), Cons(ℰ) is consistent.
Proposition 2. Wrt. to any semantics in {ad, st, pr, gr}, 𝒜ℱ 𝒦 satisfies consistency, closure.
To further illustrate the generality of our framework for monotonic and nonmonotonic logics,
let us consider the following example.
Example 1. Let 𝐴 be a set of propositional atoms. Any atoms 𝑎 ∈ 𝐴 is a well-formed formula
wrt. 𝐴. If 𝜑 and 𝛼 are well-formed formulas wrt. 𝐴 then ¬𝜑, 𝜑 ∧ 𝛼, 𝜑 ∨ 𝛼 are well-formulas wrt.
𝐴 (we also assume that the usual abbreviations →, ↔ are defined accordingly). Then ℒ𝑝 is the set
of well-formed formulas wrt. 𝐴. Let |= be the entailment relation, i.e., 𝜑 |= 𝛼 if all models of 𝜑 are
45
Loan Ho et al. CEUR Workshop Proceedings 41–55
models of 𝛼 in the propositional semantics. A consequence operator CN𝑝 : 2ℒ𝑝 → 2ℒ𝑝 is defined
for each 𝑋 ⊆ ℒ𝑝 by CN𝑝 (𝑋) = {𝛼 ∈ ℒ𝑝 | 𝑋 |= 𝛼}. The propositional logic can be defined as
(ℒ𝑝 , CN𝑝 ).
Consider the propositional atoms 𝐴1 = {𝑥, 𝑦} and the knowledge base 𝒦′ = {𝑥, 𝑦, 𝑥 → ¬𝑦} ⊆
ℒ𝑝 . The following set of arguments:
𝐶1 = ({𝑥}, 𝑥); 𝐶2 = ({𝑦}, 𝑦); 𝐶3 = ({𝑥 → ¬𝑦}, 𝑥 → ¬𝑦);
𝐶4 = ({𝑥, 𝑥 → ¬𝑦}, ¬𝑦); 𝐶5 = ({𝑦, 𝑥 → ¬𝑦}, ¬𝑥); 𝐶6 = ({𝑥, 𝑦}, 𝑥 ∧ 𝑦).
The attack relations:{(𝐶1 , 𝐶5 ), (𝐶5 , 𝐶1 ), (𝐶2 , 𝐶4 ), (𝐶4 , 𝐶2 ), (𝐶3 , 𝐶6 ), (𝐶6 , 𝐶3 )}.
The preferred extensions: {𝐶1 , 𝐶2 , 𝐶3}, {𝐶4 , 𝐶5 , 𝐶6 }.
Example 2. Let (ℒ𝑑 , CN𝑑 ) be a defeasible logic such as used in defeasible logic programming [13],
assumption-based argumentation (ABA) [15], ASPIC systems [8]. The language for defeasible
logic ℒ𝑑 includes a set of (strict and defeasible) rules and a set of literals. The rules is the form of
𝑥1 , . . . 𝑥𝑖 →𝑠 𝑥𝑖+1 (𝑥1 , . . . 𝑥𝑖 →𝑑 𝑥𝑖+1 ) where 𝑥1 , . . . 𝑥𝑖 , 𝑥𝑖+1 are literals and →𝑠 (denote strict
rules) and →𝑑 (denotes defeasible rules) are implication symbols.
The consequence operator CN𝑑 is a function from 2ℒ𝑑 to 2ℒ𝑑 such that for all 𝑋 ⊆ ℒ𝑑 , 𝑥 ∈
CN𝑑 (𝑋) iff there exists a sequence 𝑥1 , . . . , 𝑥𝑛 such that
1. 𝑥 is 𝑥𝑛 , and
2. for each 𝑥𝑖 ∈ {𝑥1 , . . . , 𝑥𝑛 },
• there is 𝑦1 , . . . , 𝑦𝑗 →𝑠 𝑥𝑖 ∈ 𝑋, or 𝑦1 , . . . , 𝑦𝑗 →𝑑 𝑥𝑖 ∈ 𝑋, such that {𝑦1 , . . . , 𝑦𝑗 } ⊆
{𝑥1 , . . . , 𝑥𝑖−1 },
• or 𝑥𝑖 ∈ 𝑋
Consider 𝒦′′ = {𝑎, ¬𝑏, 𝑎 →𝑠 ¬𝑐, ¬𝑏 ∧ ¬𝑐 →𝑑 𝑠, 𝑠 →𝑠 𝑡, 𝑎 ∧ 𝑡 →𝑑 𝑢}, the following is an
argument in defeasible logic 𝐵 = ({𝑎, ¬𝑏, 𝑎 →𝑠 ¬𝑐, ¬𝑏 ∧ ¬𝑐 →𝑑 𝑠}, 𝑠).
Example 3. Let (ℒ𝑑𝑎 , CN𝑑𝑎 ) be Datalog± [31].
Let Nt be a set of terms that contain variables, constants and function terms. An atom is of
the form 𝑃 (𝑡 ⃗), with 𝑃 a predicate name and ⃗𝑡 a vector of terms, which is ground if it contains
no variables. A database is a finite instance. A tuple-generating dependency (TGD) 𝜎 is a first-
order formula of the form ∀𝑥 ⃗ ∀𝑦 ⃗ , ⃗𝑦 ) → ∃𝑧
⃗ 𝜑(𝑥 ⃗ 𝜓(𝑥
⃗ , ⃗𝑧 ), where 𝜑(𝑥
⃗ , ⃗𝑦 ) and 𝜓(𝑥
⃗ , ⃗𝑧 ) are non-empty
conjunctions of atoms. A negative constraint (NC) 𝛿 is a rule of the form ∀𝑥
2 ⃗ 𝜑(𝑥 ⃗ ) → ⊥ where
𝜑(𝑥⃗ ) is a conjunction of atoms. The language ℒ𝑑𝑎 contain a set of facts, TGDs, and NCs.
Assume that |=ℒ𝑑𝑎 is an entailment of first-order formulas, i.e., Δ |=ℒ𝑑𝑎 𝛼 holds iff every model
of all elements in Δ is also a model of 𝛼. The consequence operator CN𝑑𝑎 is defined for 𝑋 ⊆ ℒ𝑑𝑎
by CN𝑑𝑎 (𝑋) = {𝛼 ∈ ℒ𝑑𝑎 | 𝑋 |=ℒ𝑑𝑎 𝛼} in the first-order semantics.
A knowledge base 𝒦 is a tuple (ℱ, ℛ, 𝒞) where a database ℱ, a set ℛ of TGDs and a set 𝒞 of
NCs.
Consider KB 𝒦 = {ℛ, 𝒞, ℱ} ⊆ ℒ, where:
ℛ ={𝑟1 : Le(𝑥) → Em(𝑥), 𝑟2 : Re(𝑥) → Em(𝑥),
2
We usually leave out the universal quantification, and refer to 𝜑(𝑥
⃗ , ⃗𝑦 ) and 𝜓(𝑥
⃗ , ⃗𝑧 ) as the body ad head of 𝜎.
46
Loan Ho et al. CEUR Workshop Proceedings 41–55
Table 1
Supports and conclusions of arguments
Argument Sup(𝐴𝑖 ) Con(𝐴𝑖 )
𝐴0 {t(v, KR)} t(v, KR)
𝐴1 {GC(KR)} GC(KR)
𝐴2 {GC(KR), t(v, KR)} FP(v)
𝐴3 {GC(KR), t(v, KR)} Re(v)
𝐴4 {t(v, KD)} t(v, KD)
𝐴5 {ta(v, KD)} ta(v, KD)
𝐴6 {UC(KD)} UC(KD)
𝐴7 {ta(v, KD), UC(KD)} TA(v)
𝐴9 {t(v, KR)} Le(v)
𝐴10 {t(v, KD)} Le(v)
𝐴11 {t(v, KR)} Em(v)
𝐴12 {t(v, KD)} Em(v)
𝐴13 {GC(KR), t(v, KR)} Em(v)
𝑟3 : FP(𝑥) → Re(𝑥), 𝑟5 : t(𝑥, 𝑦) → Le(𝑥),
𝑟4 : ta(𝑥, 𝑦) ∧ UC(𝑦) → TA(𝑥),
𝑟6 : t(𝑥, 𝑦) ∧ GC(𝑦) → FP(𝑥)}
𝒞 ={𝑐1 : TA(𝑥), Re(𝑥) → ⊥, 𝑐2 : Le(𝑥), TA(𝑥) → ⊥}
ℱ ={ta(v, KD), UC(KD), t(v, KR), GC(KR), t(v, KD)}
Table 1 shows the supports and conclusions of arguments induced from 𝒦.
The KB admits six MCSs (called repairs) ℬ1 , . . . , ℬ6 , e.g., ℬ1 = {ta(v, KD), UC(KD)}.
The corresponding G-SAF admits extensions: Extst/pr (𝒜ℱ) = {ℰ1 , . . . , ℰ6 }, where ℰ1 =
Args({ta(v, KD), UC(KD)}) 3 = {𝐴5 , 𝐴6 , 𝐴7 }, and ℰ2 , . . . , ℰ6 are obtained in an analogous
way. By Theorem 1, the extensions correspond to the repairs of the KBs, for example, ℰ1 corresponds
to ℬ1 .
Consider 𝑞1 = Re(v). We have that "Re(v) is a possible answer" since 𝑞1 is entailed in
ℬ2 , ℬ3 , ℬ5 . In other words, 𝑞1 is credulously accepted under st (pr) extensions.
3.2. Explanatory Dialogue Model for G-SAF
We have presented the translation of KBs into G-SAFs as an instantiation of SAFs. Subsequently,
we shift the problem of determining the entailment of a sentence 𝑓 to computing and explaining
the acceptance of arguments for 𝑓 . This section shows how to determine the acceptance of an
argument, in various argumentation semantics from an explanatory dialogue (or "dialogue" for
short).
Dialogues can be viewed as dialectical proof procedures of moving arguments as a 2-person
dialogue game. Informally, a dialectical proof procedure is formalised by a dialogue between
two players a proponent (P) and an opponent (O). The dialogue begins with P moving an initial
3
Fix ℱ ′ ⊆ ℱ , Args(ℱ ′ ) = {𝐴 ∈ Arg𝒦 | Sup(𝐴) ⊆ ℱ ′ }
47
Loan Ho et al. CEUR Workshop Proceedings 41–55
argument 𝐴 that it wants to put to the test. O and P take turns in moving arguments that attack
their counterpart’s last move. We adapt the notion of dialogue model4 from [21] for G-SAF:
• A topic language ℒ𝑡 , which is argument-based.
• A communication language ℒ𝑐 .
• An utterance is a pair 𝑢 = (PL, C), where PL is the player P (or O) and C ∈ ℒ𝑐 is the speech
act performed in the utterance. C is one of the following forms:
– claim(𝐴): The move advances an argument 𝐴 ∈ Arg𝒦 that supports a query 𝑞 of
a KB. We set arg(𝑢) = 𝐴.
– contrary(𝐴) (contrary(𝒮) resp.): The move advances an argument 𝐴 (a set of
arguments 𝒮 resp.) in Arg𝒦 that attack the previously advanced arguments. We set
arg(𝑢) = 𝐴 (arg(𝑢) = 𝒮 resp.).
– retract(𝐴, 𝑖): When O (P resp.) cannot advance any argument attacking (defend-
ing) the previously advanced arguments, it should retrace back and choose another
argument 𝐴 to start a new line of attack (defend resp.) at the 𝑖-th position. We set
arg(𝑢) = 𝐴.
For such utterance 𝑢, we set pl(𝑢) = PL. Let 𝒰 denote the set of utterances.
• A dialogue 𝐷G is a finite sequence 𝑢1 , . . . , 𝑢𝑘 of utterances such that the first utterance
𝑢1 is played by P, i.e. pl(𝑢1 ) = P. We denote by 𝒟< ∞ the set of all dialogues.
• A protocol Θ on 𝒰 specifying the legal moves at each stage of a dialogue. Formally, a
protocol on 𝒰 is a function Θ : 𝒟 ↦→ 2𝒰 such that 𝒟 ⊆ 𝒟< ∞ .
The elements of 𝒟 are called the legal finite dialogues. For 𝐷G ∈ 𝒟, the elements of Θ(𝐷G )
are called the moves allowed after 𝐷G . If 𝐷G is a legal dialogue and Θ(𝐷G ) = ∅, then 𝐷G
is said to be a terminated dialogue. Θ must satisfy the following condition: for all finite
dialogues 𝐷G and moves 𝑢, 𝐷G ∈ 𝒟 and 𝑢 ∈ Θ(𝐷G ) iff 𝐷G , 𝑢 ∈ 𝒟.
The dialogue represents a compact representation of a tree where nodes are arguments played
by both parties. To display all disclosed information from the dialogues, we introduce the notion
of dialogue trees in terms of a meta-level, which includes arguments and counter-arguments.
Definition 8. Given a dialogue 𝐷G = 𝑢1 , . . . , 𝑢𝑘 (for an argument 𝐴 ∈ Arg𝒦 ). The general
dialogue tree drawn from 𝐷G is a labelled tree 𝒯G = (𝒱, ℰ), where 𝒱 is the set of nodes and ℰ is
the set of hyperedges that are directed from nodes to their parent node. 𝒯G is defined as follows:
• The set of all nodes is defined as 𝒱 = {arg(𝑢𝑘 ) | 𝑢𝑘 ∈ 𝐷G } with arg(𝑢1 ) as the root node
of the tree s.t. arg(𝑢1 ) = 𝐴.
• ℰ is recursively defined as:
– ℰ1 = ∅ if the player P utters 𝑢1 ,
– ℰ𝑘−1 ∪ {(arg(𝑢𝑘−1 ), arg(𝑢𝑘 ))} if the player PL ∈ {P, O} utters 𝑢𝑘 .
4
Following [20, 21], this model uses Dung’s very abstract notion of argument, and they do not need utterances
to reflect particular procedures or forms of argument. Thus, the model has a rather small set of utterances.
48
Loan Ho et al. CEUR Workshop Proceedings 41–55
A branch in a dialogue tree may be finite or infinite. A finite branch represents a winning
debate (i.e., P wins) that ends with an argument by P against which O cannot attack. An infinite
branch represents a winning debate in which P counterattacks every attack of O, ad infinitum.
The requirement that "the proponent must counterattack every attack" does not guarantee that
the proponent does not attack itself. This further requirement is incorporated in the definition
of admissible dialogue tree: A dialogue tree is said to be admissible iff P wins and no argument
labels both a proponent and an opponent node.
Notice that, in admissible dialogue trees, it is not required that the opponent and proponent
nodes have no arguments in common. This is because the opponent can use the proponent’s
arguments against the proponent. If the opponent can attack the proponent using only the
proponent’s arguments, then the proponent loses. To win, the proponent must identify and
counter-attack each opponent’s attack with some culprit not part of their own defence.
To ensure credulous soundness, all possible O nodes must be considered. But if such a parent
node is already in the dialogue tree, then deploying it will not help O win the dialogues. For
this reason we call an admissible tree non-redundant [32] if there is no sequence of arguments
𝐴1 , . . . , 𝐴𝑚 with 𝐴𝑖+1 children of 𝐴𝑖 and 𝐴1 children of 𝐴𝑚 .
The following definition defines successful dialogues.
Definition 9. Let 𝒯G be the dialogue tree drawn from a dialogue 𝐷G for an argument 𝐴 ∈ Arg𝒦 .
𝐷G is called
• admissible-successful iff 𝒯G is admissible;
• preferred-successful iff it is admissible-successful;
• grounded-successful iff 𝒯G is admissible and finite;
• sceptical-successful iff 𝒯G is admissible and for no opponent node O in it there exists an
admissible dialogue tree for the argument labelling O.
We now determine the acceptance of an argument from its dialogues and dialogue trees.
Theorem 2. Let 𝐷G be a dialogue for an argument 𝐴 ∈ ArgG . Then 𝐴 is
• credulously accepted in some admissible extension of 𝒜ℱ G if 𝐷G is admissible-succesful;
• credulously accepted in some preferred extension of 𝒜ℱ G if 𝐷G is preferred-successful;
• groundedly accepted in 𝒜ℱ G if 𝐷G grounded-successful;
• sceptical accepted in all preferred extensions of 𝒜ℱ G if 𝐷G is sceptical-successful.
Proof 2. Theorem 2 generalizes known results about the abstract dispute trees of [32, 33] to the
dialogue trees of Definition 8.
The following is a direct corollary of Theorem 1, 2 and Definition 9, which shows how
to determine the entailment of a sentence wrt dialogues. The acceptance of a sentence 𝑓
corresponds to the acceptance of a set of arguments 𝒜 for 𝑓 .
Corollary 1. Let 𝒦 be a KB and 𝑓 be a sentence in ℒ. Let 𝒜ℱ 𝒦 be the corresponding G-SAF and
𝒜 ⊆ Arg𝒦 be the set of arguments for 𝑓 . Then, 𝑓 is
49
Loan Ho et al. CEUR Workshop Proceedings 41–55
P : claim(𝐴3 ) 𝐴3 : ⟨{t(v, KR), GC(KR)}, Re(v)⟩
O : contrary(𝐴7 )
𝐴7 : ⟨{t(v, KD), UC(KR)}, ta(v)⟩
P : contrary(𝐴9 )
P : contrary(𝐴10 )
𝐴9 : ⟨{t(v, KR)}, Le(v)⟩ 𝐴10 : ⟨{t(v, KD)}, Le(v)⟩
Figure 1: Left: a dialectical proof procedure for abstract level. Right: the corresponding dialogue tree; the line
indicates children conflict with their parents.
• credulously accepted, and 𝒦 ⊢MCS(𝒦) 𝑓 if there is a dialogue 𝐷G for some 𝐴 ∈ 𝒜 s.t. 𝐷G is
preferred-successful;
• groundedly accepted, and 𝒦 ⊢⋂︀ MCS(𝒦) 𝑓 if there is a dialogue 𝐷G for all 𝐴 ∈ 𝒜 s.t. 𝐷G is
grounded-successful;
• sceptically accepted, and 𝒦 ⊢⋃︀ MCS(𝒦) 𝑓 if
1. there is a dialogue 𝐷G for some 𝐴 ∈ 𝒜 such that 𝐷G is sceptical-successful;
2. or there is a dialogue 𝐷G for all 𝐴 ∈ 𝒜 s.t. 𝐷G is preferred-successful and 𝒜 is contained
in all preferred extensions of 𝒜ℱ 𝒦 .
Example 4 (Continue Example 3). Assume that the user wants to understand why "v is a
possible researcher". The deliver system considers a persuasion dialogue: P is persuading O to agree
that v is a researcher. The agreement can be reached through a dialogue 𝐷(𝐴3 ). Figure 1 shows
the dialogue tree 𝒯 drawn from the dialogue 𝐷(𝐴3 ).
4. Conclusions
In this paper, we generalized the translation of arbitrary logic into SAFs. We introduced an
argumentation dialogue framework and investigated the relation between its outcome and
inconsistency-tolerant semantics in KBs. Our dialogue framework functions as proof procedures
to compute and explain the acceptability of propositions. It can be applied to formalisms with
collective attacks, thereby we show that this framework can be a generalization of the dialogue
model used in AFs/LAFs with binary attacks. However, our framework has limitations: (1) Our
dialogue model is still defined abstractly, only considering arguments and attacks and ignoring
the internal structure of arguments; (2) Space complexity wrt the large input size, since the
procedure considers all arguments translated from KBs as input to compute attacks. In future,
we would address the limitations.
Acknowledgments
This work is partially supported by the Hybrid Intelligence programme (https://www.
hybrid-intelligence-centre.nl/), funded by a 10 year Zwaartekracht grant from the Dutch Min-
istry of Education, Culture and Science.
50
Loan Ho et al. CEUR Workshop Proceedings 41–55
References
[1] P. M. Dung, On the acceptability of arguments and its fundamental role in nonmonotonic
reasoning, logic programming and n-person games, Artif. Intell. (1995).
[2] M. D’Agostino, S. Modgil, Classical logic, argument and dialectic, Artif. Intell. 262 (2018)
15–51. URL: https://doi.org/10.1016/j.artint.2018.05.003. doi:10.1016/J.ARTINT.2018.
05.003.
[3] X. Zhang, Z. Lin, An argumentation framework for description logic ontology reasoning
and management, J. Intell. Inf. Syst. 40 (2013) 375–403.
[4] A. Arioua, M. Croitoru, S. Vesic, Logic-based argumentation with existential rules, Int. J.
Approx. Reason. 90 (2017) 76–106.
[5] L. Ho, S. Arch-int, E. Acar, S. Schlobach, N. Arch-int, An argumentative approach for
handling inconsistency in prioritized datalog± ontologies, AI Commun. (2022) 243–267.
[6] B. Yun, S. Vesic, M. Croitoru, Sets of attacking arguments for inconsistent datalog knowl-
edge bases, in: Computational Models of Argument - Proceedings of COMMA 2020, volume
326 of Frontiers in Artificial Intelligence and Applications, IOS Press, 2020, pp. 419–430.
[7] B. Yun, S. Vesic, M. Croitoru, Toward a More Efficient Generation of Structured Argumen-
tation Graphs, in: COMMA, 2018.
[8] H. Prakken, G. Vreeswijk, Logics for Defeasible Argumentation, Springer, 2002, pp. 219–
318.
[9] S. Modgil, H. Prakken, The ASPIC + framework for structured argumentation: a tutorial,
Argument Comput. 5 (2014) 31–62.
[10] L. Amgoud, Five weaknesses of ASPIC +, in: IPMU 2012 Proceedings, volume 299, Springer,
2012, pp. 122–131.
[11] O. Arieli, C. Straßer, Sequent-based logical argumentation, Argument Comput. 6 (2015)
73–99. URL: https://doi.org/10.1080/19462166.2014.1002536.
[12] O. Arieli, C. Straßer, Logical argumentation by dynamic proof systems, Theor. Comput.
Sci. 781 (2019) 63–91.
[13] A. J. García, G. R. Simari, Defeasible logic programming: Delp-servers, contextual queries,
and explanations for answers, Argument Comput. 5 (2014) 63–88.
[14] T. Alsinet, R. Béjar, L. Godo, A characterization of collective conflict for defeasible argu-
mentation, in: Computational Models of Argument: Proceedings of COMMA 2010, volume
216 of Frontiers in Artificial Intelligence and Applications, IOS Press, 2010, pp. 27–38.
[15] P. M. Dung, R. A. Kowalski, F. Toni, Assumption-Based Argumentation, Springer, 2009, pp.
199–218.
[16] M. König, A. Rapberger, M. Ulbricht, Just a matter of perspective, in: Computational
Models of Argument - Proceedings of COMMA 2022, volume 353 of Frontiers in Artificial
Intelligence and Applications, IOS Press, 2022, pp. 212–223.
[17] C. Schulz, F. Toni, Justifying answer sets using argumentation, Theory Pract. Log. Program.
16 (2016) 59–110.
[18] O. Arieli, J. Heyninck, Collective attacks in assumption-based argumentation, in: Proceed-
ings of the 39th ACM/SIGAPP Symposium on Applied Computing,SAC, ACM, 2024, pp.
746–753.
[19] A. Arioua, N. Tamani, M. Croitoru, Query answering explanation in inconsistent datalog±
51
Loan Ho et al. CEUR Workshop Proceedings 41–55
knowledge bases, in: In DEXA, volume 9261, Springer, 2015, pp. 203–219.
[20] A. Arioua, M. Croitoru, Dialectical characterization of consistent query explanation with
existential rules, in: Z. Markov, I. Russell (Eds.), Proceedings of FLAIRS, AAAI, 2016, pp.
621–625.
[21] H. Prakken, Coherence and flexibility in dialogue games for argumentation, J. Log. Comput.
15 (2005) 1009–1040.
[22] P. M. Thang, P. M. Dung, N. D. Hung, Towards argument-based foundation for sceptical
and credulous dialogue games, in: Proceedings of COMMA, volume 245, IOS Press, 2012,
pp. 398–409.
[23] X. Fan, F. Toni, A general framework for sound assumption-based argumentation dialogues,
Artif. Intell. 216 (2014) 20–54.
[24] A. Arioua, P. Buche, M. Croitoru, Explanatory dialogues with argumentative faculties over
inconsistent knowledge bases, Expert Systems with Applications 80 (2017) 244–262.
[25] A. Arioua, N. Tamani, M. Croitoru, P. Buche, Query failure explanation in inconsistent
knowledge bases using argumentation, in: Comma, 2014.
[26] P. E. Dunne, T. Bench-Capon, Two party immediate response disputes: Properties and
efficiency, Artificial Intelligence 149 (2003) 221–250.
[27] A. Hunter, Base logics in argumentation, in: Proceedings of COMMA, volume 216 of
Frontiers in Artificial Intelligence and Applications, IOS Press, 2010, pp. 275–286.
[28] F. Castagna, A dialectical characterisation of argument game proof theories for classical
logic argumentation, in: M. D’Agostino, F. A. D’Asaro, C. Larese (Eds.), Proceedings of
AIxIA, volume 3086, CEUR-WS.org, 2021.
[29] S. H. Nielsen, S. Parsons, A generalization of dung’s abstract framework for argumentation:
Arguing with sets of attacking arguments, in: Argumentation in Multi-Agent Systems,
2007, pp. 54–73.
[30] L. Amgoud, P. Besnard, Logical limits of abstract argumentation frameworks, J. Appl. Non
Class. Logics 23 (2013) 229–267.
[31] A. Calì, G. Gottlob, T. Lukasiewicz, A general datalog-based framework for tractable query
answering over ontologies, Journal of Web Semantics 14 (2012) 57–83.
[32] P. M. Thang, P. M. Dung, N. D. Hung, Towards a common framework for dialectical proof
procedures in abstract argumentation, J. Log. Comput. 19 (2009) 1071–1109.
[33] P. Dung, P. Mancarella, F. Toni, Computing ideal sceptical argumentation, Artificial
Intelligence 171 (2007) 642–674.
A. Proof for Proposition 1
Notation 1. Let 𝒦 be a KB, 𝒳 ⊆ 𝒦 be a set of formulas and 𝒮 ⊆ Arg𝒦 be a set of arguments.
Then,
• Args(𝒳 ) = {𝐴 ∈ Arg | Sup(𝐴) ⊆ 𝒳 } are the set of arguments generated by 𝒳 ,
• Base(𝒮) = Sup(𝐴) are the set of supports of arguments in 𝒮,
⋃︀
𝐴∈𝒮
• An argument 𝐵 is a subargument of argument 𝐴 iff Sup(𝐵) ⊆ Sup(𝐴). We denote the set
of subarguments of 𝐴 as Subs(𝐴).
52
Loan Ho et al. CEUR Workshop Proceedings 41–55
• 𝑋 is a minimal conflict of 𝒦 if 𝑋 ′ ⊊ 𝑋 implies 𝑋 ′ is consistent. We denote the set of
minimal conflicts of 𝒦 by Conflicts(𝒦).
We prove Proposition 1 through Lemma 1 and Lemma 2.
Lemma 1. Let 𝒦 be a KB, the corresponding G-LAF 𝒜ℱ 𝒦 . Then, the maximal consistent subset
of 𝒦 coincides with the stable/ preferred extension of 𝒜ℱ 𝒦 ;
Proof 3. To prove the lemma, we first prove it with the undercut-attacks. For the rebuttal-attacks,
the proof is similar to that of the undercut-attacks. We separate the proof of the lemma into two
parts:
1. We prove {Args(𝒜) | 𝒜 ∈ MCS(𝒦)} ⊆ Extst (𝒜ℱ 𝒦 )
2. We prove Extpr (𝒜ℱ 𝒦 ) ⊆ {Args(𝒜) | 𝒜 ∈ MCS(𝒦)}
1. We prove {Args(𝒜) | 𝒜 ∈ MCS(𝒦)} ⊆ Extst (𝒜ℱ 𝒦 ). Suppose that 𝒜 ∈ MCS(𝒦) (i.e., 𝒜 is
consistent) and ℰ = Args(𝒜), we show that ℰ is a stable extension of 𝒜ℱ 𝒦 , i.e., ℰ ∈ Extst (𝒜ℱ 𝒦 ).
Which means that, by definition, we prove that ℰ is conflict-free and attacks all arguments not
belonging to ℰ.
First, we prove that ℰ is conflict-free. Suppose for a contradiction that ℰ is not conflict-free.
There exists ℰ ′ ⊆ ℰ and an⋃︀argument 𝐴 ∈ ℰ such that ℰ ′ attacks 𝐴. By ⋃︀ Definition 4, there
exists 𝛽 ∈ Sup(𝐴) such that 𝐸∈ℰ ′ {Con(𝐸)} ∪ {𝛽} is inconsistent. Thus 𝐸∈ℰ ′ Sup(𝐸) ∪ {𝛽}
′
⋃︀
is inconsistent. Since ℰ ⊆ ℰ, it follows that 𝐸∈ℰ Sup(𝐸) ∪ {𝛽} is inconsistent. Thus 𝒜 is
inconsistent which is a contradiction to 𝒜 is consistent. Thus ℰ must be conflict-free.
Second, we prove that ℰ attacks all arguments not belonging to ℰ. Let 𝐶 ∈ Arg𝒦 ∖ ℰ and
𝛼 ∈ Sup(𝐶) ∖ 𝒜. Since⋃︀ 𝛼 ∈ / 𝒜 and 𝒜 is a maximal consistent subset of 𝒦, then
⋃︀ for every 𝒮 ⊆ ℰ =
Args(𝒜) such that 𝑆∈𝒮 Sup(𝑆) ∪ {𝛼} is inconsistent, there follows that 𝑆∈𝒮 {Con(𝑆)} ∪ {𝛼}
is inconsistent. By Definition 4, 𝒮 attacks 𝐶. Since ℰ = Args(𝒜), this proves that ℰ attacks all
arguments not belonging to ℰ. Since ℰ is conflict-free and attacks all arguments not belonging to
ℰ, it follows that ℰ is a stable extension.
2. We next prove Extpr (𝒜ℱ 𝒦 ) ⊆ {Args(𝒜) | 𝒜 ∈ MCS(𝒦)}. Suppose that ℰ ∈ Extpr (𝒜ℱ 𝒦 ),
we prove that there exists a maximal consistent subset 𝒜 such that ℰ = Args(𝒜). Which means
that, by definition, we prove that 𝒜 is maximal consistent.
First, suppose that 𝒜 = Base(ℰ) and we prove 𝒜 is consistent. Assume for a contradiction
that 𝒜 is inconsistent. Let {𝛽1 , . . . , 𝛽𝑚 } = 𝒜′ ⊆ 𝒜 be an inconsistent set and every proper
subset of it is consistent. Let 𝐶 ∈ ℰ be an argument such that 𝛽𝑚 ∈ Sup(𝐶) and 𝒮 be the set
′ ′
⋃︀ arguments generated from 𝒜 ∖{𝛽𝑚 } (i.e., 𝒮 = {𝑆
of ⋃︀ ∈ Arg | Sup(𝑆) ⊆ 𝒜 ∖{𝛽𝑚 }}) such that
𝑆∈𝒮 Sup(𝑆) ∪ {𝛽𝑚 } is inconsistent. It follows that 𝑆∈𝒮 {Con(𝑆)} ∪ {𝛽𝑚 } is inconsistent, then
𝒮 attacks 𝐶 (by Definition 4). Since Sup(𝐶) ⊆ 𝒜′ , 𝒮 attacks 𝐶. Since ℰ is conflict-free, we have
𝒮 ⊈ ℰ; and since ℰ is also an admissible set, there is a set of arguments 𝒟 ⊆ ℰ such that 𝒟 attacks
𝒮.
⋃︀ Then, there exist some arguments 𝑆𝑖 ∈ 𝒮 and 𝛽𝑖 ∈ Sup(𝑆𝑖 ), 𝑖 ∈ {1, . . . , 𝑚 − 1} such that
𝐷∈𝒟 {Con(𝐷)} ∪ {𝛽𝑖 } is inconsistent. Next, we will show 𝒟 attacks an argument in ℰ. Since
𝛽𝑖 ∈ Base(ℰ), there exists an argument 𝐹 ∈ ℰ s.t. 𝛽𝑖 ∈ Sup(𝐹 ), which implies 𝒟 attacks 𝐹 .
Clearly, 𝐶 is attacked by 𝒮 but cannot be defended, which is in contradiction with the fact that ℰ
is admissible. Hence 𝒜 is consistent.
53
Loan Ho et al. CEUR Workshop Proceedings 41–55
Second, we show that 𝒜 is maximal consistent, i.e., there is no maximal consistent subset
𝒜′ ∈ MCS(𝒦) such that 𝒜′ ⊆ 𝒜 and 𝒜′ is inconsistent. Assume the contrary. Since ℰ is a
preferred extension, it follows that Args(𝒜)∖Args(𝒜′ ) ̸= ∅, and let Ω ⊆ 𝒜∖𝒜′ and ℳ be the
set of arguments generated by Ω, i.e., ℳ = {𝑀 ∈ Arg | Sup(𝑀 ) ⊆ Ω}. Since Ω ⊆ 𝒜∖𝒜′ ,
it follows that ℳ ⊆ Args(𝒜)∖Args(𝒜′ ). By the first part of the proof, Args(𝒜′ ) is a stable
extension of 𝒜ℱ 𝒦 , there must be Φ ⊆ 𝒜′ such that the set of all arguments generated by Φ, that is
𝒳 = {𝑋 ∈ Args(𝒜′ ) | Sup(𝑋) ⊆ Φ}, attacks ℳ. Since ℰ is the preferred extension, there must
be a set of arguments ℋ ⊆ ℰ and ℋ attacks 𝒳 . Since ℋ ⊆ ℰ and 𝒜 = Base(ℰ), it follows that
Base(ℋ) ⊆ 𝒜. By assumption, i.e., 𝒜′ ⊆ 𝒜, it follows that 𝒜′ is inconsistent, which contradicts
the assumption. Hence 𝒜 is maximal consistent. We conclude that 𝒜 ∈ MCS(𝒦).
Since every stable extension is a preferred one [29], we can proceed as follows. From the first
item, we proved that every maximal consistent subset is a stable extension, thus the proposition
holds for preferred semantics. From the second item, we have that every preferred extension is a
maximal consistent subset, thus the proposition holds for stable semantics.
Lemma 2. Let 𝒦 be a KB, the corresponding G-LAF 𝒜ℱ 𝒦 . Then, the intersection of the maximal
consistent subsets of 𝒦 coincides with the grounded extension of 𝒜ℱ 𝒦 .
Proof 4. Let 𝒢 be the grounded extension of 𝒜ℱ 𝒦 and⋂︀𝒜 be the intersection of the maximal
consistent subsets of 𝒦. Let 𝒜′ ∈ MCS(𝒦) and 𝒜 = 𝒜′ ∈MCS(𝒦) 𝒜′ . By Lemma 1, 𝒜′ is a
stable extension of 𝒜ℱ 𝒦 . Since every stable extension is also a complete extension, and 𝒢 is the
minimal (w.r.t. set inclusion) complete extension of 𝒜ℱ 𝒦 , it follows that 𝒢 ⊆ Args(𝒜′ ). Thus
𝒢 ⊆ Args(𝒜).
In the other direction, let Δ ⊊ 𝒜. Then there is no 𝒞 ∈ Conflicts(𝒦) such that Δ ⊊ 𝒞 (If
Δ ⊊ 𝒞, 𝒞 ∖ Δ would be consistent and could be extended to a maximal consistent subset of 𝒦 that
would not contain Δ). Hence, there are no arguments that attack the argument 𝐷 ∈ Args(𝒜)
where Sup(𝐷) = Δ. Since 𝐷 has no attackers, then any extension trivially defends it, so 𝐷 must
be in 𝒢. It follows that Args(𝒜) ⊆ 𝒢.
B. Proof of Proposition 2
We prove each postulate:
1. Closure: Let us show that for every ℰ ∈ Exts (𝒜ℱ 𝒦 ), Cons(ℰ) = CN(Cons(ℰ)). From
Expansion axiom, it follows that Cons(ℰ) ⊆ CN(Cons(ℰ)).
In the other direction, we prove CN(Cons(ℰ)) ⊆ Cons(ℰ). Let 𝛼 ∈ CN(Cons(ℰ)). Since ℰ is
a preferred or stable extension, there is a set of sentences 𝒜 ⊆ 𝒦 such that ℰ = Args(𝒜) (by
Proposition 1). Thus ℰ = Args(Base(ℰ)). Since the supports of the arguments of ℰ include
the sentences in 𝒜, it follows that 𝛼 ∈ CN(𝒜). Hence, there is an argument 𝐴 ∈ ℰ such that
Con(𝐴) = 𝛼.
3. Consistency:
We prove that for ℰ ∈ Exts (𝒜ℱ 𝒦 ), Cons(ℰ) is consistent. First, we consider the case of
stable or preferred extensions. Let ℰ be a stable or preferred extension of 𝒜ℱ 𝒦 . By Proposition
1, there is a maximal consistent subset 𝒜 ∈ MCS(𝒦) such that ℰ = Args(𝒜). Since 𝒜 is a
54
Loan Ho et al. CEUR Workshop Proceedings 41–55
preferred maximal consistent subset, we see that CN(𝒜) is consistent. Since Cons(ℰ) = CN(𝒜)
and CN(𝒜) is consistent, it follows that Cons(ℰ) is consistent.
We consider the case of grounded semantics. We denote 𝒢 as the grounded extension of
𝒜ℱ 𝒦 . We prove that for every ℰ ∈ Extgr (𝒜ℱ 𝒦 ), Cons(ℰ) is consistent. Since the grounded
extension is a subset of the intersection of the preferred extensions, and since there is at least
one preferred extension ℰ1 , then 𝒢 ⊆ ℰ1 . From Proposition 1, there is a maximal consistent
subset 𝒜 ∈ MCS(𝒦) s.t. ℰ1 = Args(𝒜). By the first part of the proof, Cons(ℰ1 ) is consistent. It
follows that Cons(𝒢) is consistent.
C. Proof of Theorem 1
• 𝒦 ⊢MCS(𝒦) 𝑓 if 𝑓 is credulously accepted under s semantics.
• 𝒦 ⊢⋃︀ MCS(𝒦) 𝑓 if 𝑓 is sceptically accepted under s semantics.
• 𝒦 ⊢⋂︀ MCS(𝒦) 𝑓 if 𝑓 is groundedly accepted under gr semantics.
By Proposition 1, there exists a link between the set of maximal consistent subsets on 𝒦 and
the set of extensions on 𝒜ℱ 𝒦 . Obviously, for every sentence 𝑓 ∈ ℒ and maximal consistent
𝒜 ∈ MCS(𝒦), it is the case that 𝑓 ∈ CN(𝒜) iff 𝑓 ∈ Cons(Args(𝒜)). The theorem is now proven
as follows:
1. 𝒦 ⊢⋃︀ MCS(𝒦) 𝑓 iff for every 𝒜 ∈ MCS(𝒦), 𝑓 ∈ CN(𝒜) iff for every Args(𝒜) ∈ Ext(𝒜ℱ 𝒦 ),
𝑓 ∈ Cons(Args(𝒜)) iff 𝑓 is sceptically accepted.
2. 𝒦 ⊢MCS(𝒦) 𝑓 iff for some 𝒜 ∈ MCS(𝒦), 𝑓 ∈ CN(𝒜) iff for some Args(𝒜) ∈ Ext(𝒜ℱ),
𝑓 ∈ Cons(Args(𝒜)) iff 𝑓 is credulously accepted.
3. 𝒦 ⊢⋂︀ MCS(𝒦) 𝑓 iff for every 𝒜𝑖 ∈ MCS(𝒦), 𝑓 ∈ CN( 𝒜𝑖 ) iff for every Args(𝒜𝑖 ) ∈
⋂︀
Ext(𝒜ℱ 𝒦 ), 𝑓 ∈ Cons( Args(𝒜𝑖 )) iff 𝑓 is accepted under grounded semantics.
⋂︀
This ends the proof of Theorem 1.
D. Proof for Theorem 2
Proof 5. It is clear that 𝐴 is accepted in some admissible extension of 𝒜ℱ G if 𝐷G is admissible.
Let 𝒯G be the dialogue tree drawn from 𝐷G . The argument given in Definition 2 and Lemma 1
of [32] for binary attacks generalizes to collective attacks, implying that 𝐴 is accepted in some
admissible extension if 𝒯G is admissible which in turn holds iff P wins and no argument labels both
a proponent and an opponent node. By Definition 9, 𝐷G is admissible-successful iff 𝒯G is admissible.
Thus, the statement is proved.
𝐷G is preferred-successful if 𝐷G is admissible-successful. This result directly follows from the
results of [1] that states that an extension is preferred if it is admissible. Thus, 𝐴 is credulously
accepted in some preferred extension if 𝐷G is preferred-successful.
The other statement follows in a similar way as a straightforward generalization of Theorem
1 of [32] for the "grounded-successful" semantic; Definition 3.3 and Theorem 3.4 of [33] for the
"sceptical-successful" semantic.
55