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 nonmonotonic extensions for Gentzen-style proof systems in terms of argumentation-based. Moreover, 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) 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] assumptionbased 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 representational aspects. This paper, using abstract logic with no requirement on the language ℒ and CN (a function from 2 ℒ to 2 ℒ ), 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 repairs1 ), 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.

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 2 ℒ to 2 ℒ returning the logical consequences of a 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 ⊢: 2 MCS(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).

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].

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. 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.

Definition 3 (G-SAF arguments). An argument induced from a KB

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 ⊆ Arg 𝒦 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 Att 𝒦 ⊆ 2 Arg 𝒦 × Arg 𝒦 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 (Arg 𝒦 , Att 𝒦 ), where Arg 𝒦 is the set of arguments induced from 𝒦 and Att 𝒦 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 = (Arg 𝒦 , Att 𝒦 ) and 𝒮 ⊆ Arg 𝒦 . 𝒮 attacks 𝒳 iff ∃𝐴 ∈ 𝒳 s.t. 𝒮 attacks 𝐴. 𝒮 defends 𝐴 if for each 𝒳 ⊆ Arg 𝒦 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 Ext s (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 • credulously accepted under s iff for some ℰ ∈ Ext s (𝒜ℱ 𝒦 ), 𝑓 ∈ Cons(ℰ ).

• sceptically accepted under s iff for all ℰ ∈ Ext s (𝒜ℱ 𝒦 ), 𝑓 ∈ Cons(ℰ ).

• groundedly accepted under gr iff for some ℰ ∈ Ext gr (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 [ Example 2. Let (ℒ 𝑑 , CN 𝑑 ) 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 ℒ 𝑑 includes a set of (strict and defeasible) rules and a set of literals.

𝑟 6 : t(𝑥, 𝑦) ∧ GC(𝑦) → FP(𝑥)} 𝒞 ={𝑐 1 : TA(𝑥), Re(𝑥) → ⊥, 𝑐 2 : 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) ℬ 1 , . . . , ℬ 6 , e.g., ℬ 1 = {ta(v, KD), UC(KD)}. The corresponding G-SAF admits extensions: Ext st/pr (𝒜ℱ) = {ℰ 1 , . . . , ℰ 6 }, where ℰ 1 = Args({ta(v, KD), UC(KD)})3 = {𝐴 5 , 𝐴 6 , 𝐴 7 }, and ℰ 2 , . . . , ℰ 6 are obtained in an analogous way. By Theorem 1, the extensions correspond to the repairs of the KBs, for example, ℰ 1 corresponds to ℬ 1 .

Consider 𝑞 1 = Re(v). We have that "Re(v) is a possible answer" since 𝑞 1 is entailed in ℬ 2 , ℬ 3 , ℬ 5 . In other words, 𝑞 1 is credulously accepted under st (pr) extensions.

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 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 ℒ 𝑡 , which is argument-based.

• A communication language ℒ 𝑐 .

• An utterance is a pair 𝑢 = (PL, C), where PL is the player P (or O) and C ∈ ℒ 𝑐 is the speech act performed in the utterance. C is one of the following forms:

claim(𝐴): The move advances an argument 𝐴 ∈ Arg 𝒦 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 Arg 𝒦 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 (defending) 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 𝐷 G is a finite sequence 𝑢 1 , . . . , 𝑢 𝑘 of utterances such that the first utterance 𝑢 1 is played by P, i.e. pl(𝑢 1 ) = 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 ↦ → 2 𝒰 such that 𝒟 ⊆ 𝒟 < ∞ .

The elements of 𝒟 are called the legal finite dialogues. For 𝐷 G ∈ 𝒟, the elements of Θ(𝐷 G ) are called the moves allowed after 𝐷 G . If 𝐷 G is a legal dialogue and Θ(𝐷 G ) = ∅, then 𝐷 G is said to be a terminated dialogue. Θ must satisfy the following condition: for all finite dialogues 𝐷 G and moves 𝑢, 𝐷 G ∈ 𝒟 and 𝑢 ∈ Θ(𝐷 G ) iff 𝐷 G , 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 𝐷 G = 𝑢 1 , . . . , 𝑢 𝑘 (for an argument 𝐴 ∈ Arg 𝒦 ). The general dialogue tree drawn from 𝐷 G is a labelled tree 𝒯 G = (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. 𝒯 G is defined as follows:

• The set of all nodes is defined as 𝒱 = {arg(𝑢 𝑘 ) | 𝑢 𝑘 ∈ 𝐷 G } with arg(𝑢 1 ) as the root node of the tree s.t. arg(𝑢 1 ) = 𝐴. • ℰ is recursively defined as:

-ℰ 1 = ∅ if the player P utters 𝑢 1 , -ℰ 𝑘−1 ∪ {(arg(𝑢 𝑘−1 ), arg(𝑢 𝑘 ))} if the player PL ∈ {P, O} utters 𝑢 𝑘 .

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] We now determine the acceptance of an argument from its dialogues and dialogue trees.

Theorem 2. Let 𝐷 G be a dialogue for an argument 𝐴 ∈ Arg G . Then 𝐴 is

• credulously accepted in some admissible extension of 𝒜ℱ G if 𝐷 G is admissible-succesful; • credulously accepted in some preferred extension of 𝒜ℱ G if 𝐷 G is preferred-successful; • groundedly accepted in 𝒜ℱ G if 𝐷 G grounded-successful; • sceptical accepted in all preferred extensions of 𝒜ℱ G if 𝐷 G 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 . 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(𝐴 3 ). Figure 1 shows the dialogue tree 𝒯 drawn from the dialogue 𝐷(𝐴 3 ).

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.