Andreas Herzig
email: andreas.herzig@irit.fr

Antonio Yuste Ginel
email: antonioyusteginel@gmail.com

Multi-Agent Abstract Argumentation Frameworks With Incomplete Knowledge of Attacks

We introduce a multi-agent, dynamic extension of abstract argumentation frameworks (AFs), strongly inspired by epistemic logic, where agents have only partial information about the conflicts between arguments. These frameworks can be used to model a variety of situations. For instance, those in which agents have bounded logical resources and therefore fail to spot some of the actual attacks, or those where some arguments are not explicitly and fully stated (enthymematic argumentation). Moreover, we include second-order knowledge and common knowledge of the attack relation in our structures (where the latter accounts for the state of the debate), so as to reason about different kinds of persuasion and about strategic features. This version of multi-agent AFs, as well as their updates with public announcements of attacks (more concretely, the effects of these updates on the acceptability of an argument) can be described using S5-PAL, a well-known dynamic-epistemic logic. We also discuss how to extend our proposal to capture arbitrary higher-order attitudes and uncertainty.

Introduction

Argumentation has been shown to be a powerful perspective to deal with some problems within the field of artificial intelligence [Bench-Capon and Dunne, 2007], such as reasoning on the basis of conflicting information, autonomous agents interaction, and, more recently, bridging machine and human reasoning [START_REF] Modgil | [END_REF]. A prominent tool in this context are abstract argumentation frameworks (AFs), introduced by [Dung, 1995]. AFs are just a new way to look at directed graphs, where nodes represent arguments and arrows represent some kind of conflict relation among them, usually called the attack relation. AFs enables formal reasoning about a fundamental question in argumentation theory, namely, given a set of (possibly) conflicting arguments: which of them are to be accepted by a rational agent? Context. Although they have shown to be extremely fruitful, AFs are notably abstract in their original formulation and they therefore face many limitations when modelling argumentative scenarios. They in particular do not account for multiple agents and for dynamics, both of which are essential to real-life argumentation. Many proposals attempted to overcome these two limitations. As to the former, there is an already well-established tradition of applying argumentation techniques to multi-agent systems [START_REF] Maudet | [END_REF]Carrera and Iglesias, 2015]. Simultaneously, dynamics has received increasing attention during the last decade [START_REF] Doutre | Constraints and changes: a survey of abstract argumentation dynamics[END_REF]. Recent studies on strategic argumentation [START_REF] Thimm | Strategic argumentation in multi-agent systems[END_REF] encompass both aspects. Here, the main research question is how agents (should) select their next movement during a dialogue in order to accomplish their goal. A recurrent idea in this area is the use of opponent models [START_REF] Rienstra | Opponent models with uncertainty for strategic argumentation[END_REF][START_REF] Thimm | Strategic argumentation in multi-agent systems[END_REF], i.e., letting agents have beliefs about other agents' views of the underlying AF, as a means for computing their strategy. Contribution. This paper introduces a novel multi-agent and dynamic extension of AFs that is strongly inspired by epistemic logic. There are a number of features that distinguish our approach from previous ones. First, the usual focus when modelling partial information is considering awareness of arguments, rather than attacks, the customary object of knowledge [START_REF] Schwarzentruber | Building an epistemic logic for argumentation[END_REF][START_REF] Thimm | Strategic argumentation in multi-agent systems[END_REF][START_REF] Rahwan | Mechanism design for abstract argumentation[END_REF][START_REF] Rienstra | Opponent models with uncertainty for strategic argumentation[END_REF]. We follow the less explored perspective, considered for instance in [Dyrkolbotn and Pedersen, 2016], that models an agent's view as a (possibly incomplete) set of attacks. This option fits well in debates where some arguments are not fully stated (enthymemes), or where agents have bounded reasoning resources, and hence fail to spot some of the conflicts among arguments. Second, we include simple opponent-models: second-order views of the attack relation, i.e., what an agent knows that another agent knows, in order to make our structures appealing for the study of strategical issues. Third, we also take into account the notion of public view, accounting for the state of a debate.1

Background

We recall AFs and their stable extensions, as well as their encoding in propositional logic. Our focus on the stable semantics is by no means restrictive and we may replace it by any other semantics that can be captured in propositional logic [START_REF] Besnard | [END_REF]. We also briefly recall epistemic logic and public announcement logic.

Argumentation Frameworks

A Dung argumentation framework (AF) is a graph R = A, R where A is a finite set (the set of arguments) and R ⊆ A × A is a relation on A (the attack relation).

A stable extension of an AF R = A, R is a set of arguments E ⊆ A such that (1) there are no a and b in E such that (a, b) ∈ R (E is conflict-free), and (2) for every argument b ∈ A \ E there exists a ∈ E such that (a, b) ∈ R (any argument outside the extension is attacked by the extension). An argument a is sceptically accepted (resp. credulously accepted) if it belongs to every (resp. at least one) stable extension.

AFs in Propositional Logic

AFs can be encoded in propositional logic, as first proposed in [START_REF] Besnard | Checking the acceptability of a set of arguments[END_REF] and extensively discussed in [Gabbay, 2011;[START_REF] Besnard | [END_REF]. We suppose that the propositional variables contain attack variables r a,b capturing that (a, b) ∈ R. Given a set of arguments A,

Att A = {r a,b : (a, b) ∈ A × A} is the set of all attack variables. Then the theory of R = A, R is the Boolean formula θ(R) = (a,b)∈R r a,b ∧ (a,b)∈(A×A)\R ¬r a,b . The theory of R correctly captures the attack relation: (a, b) ∈ R if and only if θ(R) |= r a,b . Moreover, it is com- plete w.r.t. attack variables: for every r a,b ∈ Att A , either θ(R) |= r a,b or θ(R) |= ¬r a,b .
Extensions can also be characterised in propositional logic if we add furthermore acceptance variables in a , to be read "a is accepted" or "a is in". For a fixed set of arguments A, the set of propositional variables of our language is therefore Prp A = Att A ∪ {in a : a ∈ A}. Valuations (noted v, v , etc.) are subsets of Prp A , and satisfaction of Boolean formulas ϕ built from Prp A is defined in the standard way and noted v |= ϕ as usual. As as shown in [START_REF] Besnard | Checking the acceptability of a set of arguments[END_REF], the stable semantics of an AF R = A, R can then be characterised by the Boolean formula

Stable = a∈A in a ↔ b∈A (r b,a → ¬in b ) .
The models of that formula correspond to the extensions under the stable semantics: for every valuation

v ⊆ Prp A , v |= θ(R) ∧ Stable if and only if the set E v = {a ∈ A : in a ∈ v} is a stable extension of R.

Public Announcement Logic

The language of public announcement logic (PAL) extends that of propositional logic by epistemic and public announcement operators. We consider the S5 version of PAL with common knowledge, in brief, S5-PAL [Ditmarsch et al., 2007, Chapter 4]. Its language is defined by the following grammar:

ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | K i ϕ | C Agt ϕ | [ϕ!]ϕ
where p ranges over a countable set of propositional variables Prp and i ranges over a finite set of agents Agt. In order to avoid tedious exceptions in what follows, we assume that |Agt| ≥ 2. The formula K i ϕ reads "i knows that ϕ" and C Agt ϕ reads "it is common knowledge among all agents that ϕ". The latter enables reasoning about the common ground in a conversation [START_REF] Clark | [END_REF]]. The program ϕ! is the public announcement of ϕ, where we suppose that announcements are truthful: ϕ can be announced only if ϕ is true. The formula [ψ!]ϕ reads "if ψ can be announced then ϕ is true afterwards". The dual ψ! ϕ abbreviates ¬[ψ!]¬ϕ and therefore reads "ψ can be announced and ϕ is true afterwards". The "everybody knows" operator E Agt ϕ abbreviates i∈Agt K i ϕ. The dual of K i ϕ is noted Ki ϕ and defined as

¬K i ¬ϕ. An S5-model is a triple M = W, {∼ i } i∈Agt , V such that W is a non-empty set, every ∼ i ⊆ W × W is an equivalence relation, and V : W -→ ℘(Prp) is a valuation. A pointed S5- model is a couple M, w such that M = W, {∼ i } i∈Agt , V
is an S5-model and w ∈ W . The truth conditions for the modal operators are:

M, w |= K i ϕ if M, v |= ϕ for every v s.th. (w, v) ∈ ∼ i M, w |= C Agt ϕ if M, v |= ϕ for every v s.th. (w, v) ∈ i∈Agt ∼ i * M, w |= [ψ!]ϕ if M, w |= ψ implies M ψ! , w |= ϕ
where i∈Agt ∼ i * is the reflexive and transitive closure of the union of the equivalence relations ∼ i and where

M ψ! = W ψ! , {∼ ψ! i } i∈Agt , V ψ! is the update of M by ψ!, defined by: W ψ! = {v ∈ W : M, v |= ψ} ; ∼ ψ! i = ∼ i ∩ (W ψ! × W ψ! ); and V ψ! = V | W ψ! (the restriction of V to W ψ! ).
Axiomatically, every K i obeys the axioms

K i , K i ϕ → ϕ, K i (ϕ ∧ ψ) ↔ (K i ϕ ∧ K i ψ), K i ϕ → K i K i ϕ, and ¬K i ϕ → K i ¬K i ϕ.
The common knowledge operator C Agt is also an S5 operator. Moreover, common knowledge implies individual knowledge C Agt ϕ → E Agt ϕ and satisfies the induction axiom C Agt (E Agt ϕ∨E Agt ¬ϕ) → (C Agt ϕ∨C Agt ¬ϕ) [START_REF] Herzig | On the axiomatisation of common knowledge[END_REF].

In the sequel we are going to use a fragment of the language of PAL where announcements are restricted to epistemic formulas of the form K i ϕ.2 Such public announcements capture, at least partly, public announcements made by one of the agents in Agt. In contrast, in standard PAL it is considered that announcements are made by an agent who is not in Agt and is therefore 'outside the system' [Herzig, 2017].

Multi-Agent AFs

Given an AF R = A, R , the subset of the set of all attacks an agent knows is the agent's view of R. For example, suppose agent 1 knows that b and b both attack a and that c (resp. c ) attacks b (resp. b ), and suppose agent 2 only knows Agents argue by informing each other about attacks. We suppose that they do so in a truthful and sincere manner: when 1 announces that b attacks a then b indeed attacks a, and moreover 1 knows that. In Figure 1, both 1 and 2 can announce that c attacks b because they know this; and only 1 can announce that b attacks a while 2 cannot. These inform acts are public announcements: when i has put forward a piece of information then it becomes part of the common ground of the dialogue. In epistemic logic terms: the piece of information becomes common knowledge [START_REF] Clark | [END_REF]].

c 1,2 -→ b 1 -→ a 1 ←-b 1 ←-c
In the above example we have only considered first-order knowledge: knowledge about the world, namely about the attack relation. We may also consider second-order knowledge: knowledge about other agents' knowledge. Suppose e.g. that 1 knows that 2 knows that c attacks b. Then 1 knows that 2 is able to announce that c attacks b. Therefore 1 knows that her announcement that b attacks a can be countered by 2 and will not contribute to get a publicly rejected.

We now define the agents' first-order views, their secondorder views, and the common ground of the conversation.

First-Order Views

Given a background AF R = A, R and an agent

i ∈ Agt, i's view of R is an AF R i = A, R i that is a subgraph of R, i.e., such that R i ⊆ R. (C1)
The agent-annotated AF of Figure 1

Second-Order Views

Second-order views account for what agents know about other agent's knowledge. In strategic argumentation terms, they are one-depth opponent models without uncertainty [START_REF] Rienstra | Opponent models with uncertainty for strategic argumentation[END_REF].

(R i ) j = A, (R i ) j is agent j's pos- itive view of i's positive view. 3 When (a, b) is in j's view of i's view then j knows that i knows that a attacks b.
Just as for first-order knowledge, we require that j's knowledge about i's knowledge and ignorance is correct, i.e., that (R i ) j ⊆ R i . Moreover, what j knows about i's knowledge must also be known by j herself, i.e., we should also have (R i ) j ⊆ R j . Put together, we require that

(R i ) j ⊆ R i ∩ R j .
(C2)

3 So we abstract away from, e.g., the attacks of which j knows that i knows they are false.

We furthermore suppose that the agents are introspective, i.e., the inclusion also holds in the other sense when j equals i:

(R i ) i = R i . (C3)
We could go beyond second-order views and define arbitrary higher-order views of AFs. We do not do so here because second-order views suffice to account for strategic dialogues of the above kind where an agent knows she can win a debate thanks to her knowledge of other agents' knowledge. It would also pose some technical difficulties because higherorder views would a priori make the representation of multiagent AFs infinite. We comment more on this in Section 6.1.

Public Views

A public view of R is yet another AF R pub = A, R pub . We require R pub to be a subgraph of all individual first-order and second-order views:

R pub ⊆ i,j∈Agt (R i ) j . (C4) Then R pub ⊆ i∈Agt R i ∩ i,j∈Agt (R i ) j
follows with condition (C2). Let us compare our public views to the notion of state of a debate in [Dyrkolbotn and Pedersen, 2016]. The latter is a pair

A, R d where i∈Agt R i ⊆ R d ⊆ i∈Agt R i .
Hence, the fact that a pair (a, b) is known by everyone is a sufficient condition to be part of the state of a debate, while the same fact is a necessary condition for (a, b) to be part of the public view.

Multi-Agent AFs

Putting all these ingredients together we define multi-agent AFs as quintuplets of the form

M = A, R, {R i } i∈Agt , {(R i ) j } i,j∈Agt , R pub
such that the above conditions (C1), (C2), (C3), (C4) hold. Every A, R i is i's view of A, R , every A, (R i ) j is j's view of i's view of A, R , and A, R pub is a public view of A, R .

Interpretation. As mentioned, our choice to model partial knowledge of an AF by partial knowledge of the attack relation is rather non-standard, the more usual choice being to model incomplete information through partial awareness of arguments [START_REF] Schwarzentruber | Building an epistemic logic for argumentation[END_REF][START_REF] Thimm | Strategic argumentation in multi-agent systems[END_REF][START_REF] Rahwan | Mechanism design for abstract argumentation[END_REF][START_REF] Rienstra | Opponent models with uncertainty for strategic argumentation[END_REF]. Both alternatives seem to be orthogonal, and our choice is mainly for the sake of exploration. Nevertheless, we believe it is natural in the context of enthymematic arguments where the hypotheses together with background knowledge entail the conclusion: the agents' unawareness of some of the background hypotheses (enthymeme) explains their incomplete knowledge of the attack relation. Besides, even when arguments are explicitly and fully stated by participants in a debate, non-omniscient agents can fail to see part of the attack relation due to their limited logical skills. One may for example consider resource-bounded agents that are good (sound) reasoners equipped with an incomplete set of inference rules, as proposed in [Konolige, 1984].

Extensions of a Multi-Agent AF

What is accepted in the current state of the debate is determined by the stable extensions of the public view: a public stable extension of the multi-agent AF M = A, R, {R i } i∈Agt , {(R i ) j } i,j∈Agt , R pub is a set of arguments E pub that is a stable extension of A, R pub . Then, an argument is publicly-sceptically (resp. publicly-credulously) accepted iff it is sceptically (resp. credulously) accepted w.r.t. A, R pub . This definition can be connected with the distinction between public persuasion and private persuasion that is made in the introduction of [Dupin de [START_REF] De Saint-Cyr | Argumentation update in YALLA (yet another logic language for argumentation)[END_REF]. In this sense, public persuasion amounts to public acceptance (or rejection) while private persuasion is related to second-order views.

Updating a Multi-Agent AF

The agents' incomplete knowledge of the attack relation evolves when one of them announces an attack known by her. That attack then becomes part of what is public (and consequently also part of the agents' first-and second-order views).

An announcement is a triple consisting of an agent and two arguments, noted i:(a, b) and read "i announces that a attacks b".

Given a multi-agent AF M = A, R, {R j } j∈Agt , {(R j ) k } j,k∈Agt , R pub and an announcement i:(a, b), the update of M by i:(a, b) incorporates the attack (a, b) into all views, resulting in a new multi-agent AF M +i:(a, b). The update function has to be partial because agents can only make truthful and sincere announcements. It is defined on M if (a, b) ∈ R i and is undefined otherwise. When it is defined then the result of the update is

M +i:(a, b) = A, R, {R j ∪{(a, b)}} j∈Agt , {(R j ) k ∪{(a, b)}} j,k∈Agt , R pub ∪{(a, b)} .
It can be easily checked that if M is a multi-agent AF satisfying the above conditions (C1), (C2), (C3), (C4) then M +i:(a, b) is either undefined or is again a multi-agent AF satisfying (C1), (C2), (C3), (C4).

Characterising Multi-Agent AFs and their Updates in Epistemic Logic

Just as an AF can be captured in propositional logic, a multiagent AF can be captured in epistemic logic: the formula

K i r a,b expresses that (a, b) ∈ R i ; likewise, ¬K i r a,b ex- presses that (a, b) / ∈ R i .
We call formulas of the form K i r a,b first-order attack variables. Formulas of the form either K i r a,b or ¬K i r a,b are first-order attack literals. Similarly, formulas of the form K j K i r a,b are second-order attack variables. Note that we should not encode (a, b) / ∈ R i as K i ¬r a,b because in her current view, i may not know that a attacks b, but may learn this later on.

Characterisation of Multi-Agent AFs

Given an AF R = A, R , agent i's theory of R is the conjunction of first-order attack variables

θ i (R) = (a,b)∈R K i r a,b ∧ (a,b)∈(A×A)\R ¬K i r a,b .
The agents' theories are complete w.r.t. first-order attack variables: either θ i (R) |= K i r a,b or θ i (R) |= ¬K i r a,b . Moreover, θ i (R) is an epistemic characterisation of the attack relation: a attacks b if and only if θ i (R) |= K i r a,b . The other way round, completeness of θ i (R) implies that a does not attack b if and only if θ i (R) |= ¬K i r a,b . The formulas θ i (.) are useful to capture i's view of the AF in epistemic logic: we will typically apply θ i (.) to R i and write θ i ( A, R i ).

Second-order views account for what agents know about other agents' knowledge and can nicely be captured in epistemic logic by nesting epistemic operators. For example,

K j K i r a,b expresses that (a, b) ∈ (R i ) j , and K j r a,b ∧ ¬K j K i r a,b expresses that (a, b) ∈ R j and (a, b) / ∈ (R i ) j
. Given an AF R = A, R , agent j's theory about agent i is the following conjunction of second-order attack variables:

θ i;j (R) = (a,b)∈R K j K i r a,b ∧ (a,b)∈(A×A)\R ¬K j K i r a,b .
The theory θ i;j (R) is complete w.r.t. second-order attack variables: either

θ i;j (R) |= K j K i r a,b or θ i;j (R) |= ¬K j K i r a,b . It characterises the attack relation: (a, b) ∈ R if and only if θ i;j (R) |= K j K i r a,b
. We will use secondorder knowledge expressed in θ i;j (.) to reason about j's view of i's view, typically writing θ i;j ( A, (R i ) j ).

Finally, the public theory of R is captured by means of the common knowledge operator:

θ pub (R) = (a,b)∈R C Agt r a,b ∧ (a,b)∈(A×A)\R ¬C Agt r a,b
. Again, the public theory is complete w.r.t. common knowledge of the attack relation and also characterises the attack relation. Again, we typically write θ pub ( A, R pub ).

Let us put all this together: let M = A, R, {R i } i∈Agt , {(R i ) j } i,j∈Agt , R pub be a multi-agent AF. Then the epistemic theory of M is the conjunction of the above theories:

θ(M) = θ( A, R ) ∧   i∈Agt θ i ( A, R i )   ∧   i,j∈Agt θ i;j ( A, (R i ) j )   ∧ θ pub ( A, R pub ). Proposition 1. For every multi-agent AF M = A, R, {R i } i∈Agt , {(R i ) j } i,j∈Agt , R pub we have: • (a, b) ∈ R iff θ(M) → r a,b is S5 valid; • (a, b) / ∈ R iff θ(M) → ¬r a,b is S5 valid; • (a, b) ∈ R i iff θ(M) → K i r a,b is S5 valid; • (a, b) / ∈ R i iff θ(M) → ¬K i r a,b is S5 valid; • (a, b) ∈ (R i ) j iff θ(M) → K j K i r a,b is S5 valid; • (a, b) / ∈ (R i ) j iff θ(M) → ¬K j K i r a,b is S5 valid; • (a, b) ∈ R pub iff θ(M) → C Agt r a,b is S5 valid; • (a, b) / ∈ R pub iff θ(M) → ¬C Agt r a,b is S5 valid.

Characterisation of Extensions

We characterise the public stable extensions of a multi-agent AF in epistemic logic as follows:

Stable pub = a∈A in a ↔ b∈A (C Agt r b,a → ¬in b ) .
Our definition requires some explanations: why didn't we insert the common knowledge operator into the definition of stable extensions of Section 2 in a different way? For example, we could have defined "a is in" as "every b that is in is also known to not attack a". However, this is too strong a requirement in debates where the agents learn the attack relation step by step: then no argument a can ever be accepted because the agent may learn later on in the debate about some attack of a that she currently does not know. Proposition 2. Let M be a multi-agent AF. Then:

• E pub is a public stable extension of M iff θ(M)∧Stable pub ∧ a∈E pub in a ∧ a∈A\E pub ¬in a is satisfiable in S5. • If R pub = R, then θ(M) → (Stable pub ↔ Stable) is S5 valid. • a ∈ A is publicly-sceptically accepted (resp. publicly- credulously accepted) iff (θ(M) ∧ Stable pub ) → in a is S5-valid (resp. θ(M)∧Stable pub ∧in a is S5-satisfiable).

Characterisation of Updates

Let us show how updates can be characterised in S5-PAL. Proposition 3. Let M be a multi-agent AF. Then (a, b)

∈ R i iff θ(M) → K i r a,b ! is S5-PAL valid.
Proposition 4. Let M be a multi-agent AF such that (a, b) ∈ R i . Let M +i:(a, b) be its update by i:(a, b). For every (c, d) ∈ A × A:

• (c, d) ∈ R j ∪{(a, b)} iff θ(M) → [K i r a,b !]K j r c,d is S5-PAL valid; • (c, d) ∈ (R j ) k ∪{(a, b)} iff θ(M) → [K i r a,b !]K k K j r c,d is S5-PAL valid; • (c, d) ∈ R pub ∪{(a, b)} iff θ(M) → [K i r a,b !]C Agt r c,d is S5-PAL valid.

An Example

Let us take up the agent-annotated AF of Figure 1, and suppose the following informal reading for the involved arguments, representing a debate on public health policies to stop the spreading of COVID-19. a: "everybody should wear, at least, a surgical mask." b: "there are not enough surgical masks." c: "home-made masks can replace surgical masks." b : "surgical masks don't prevent people from getting infected." c : "surgical masks prevent people from infecting others."

Let M be the multi-agent AF where the first-order knowledge is the one depicted in Figure 1 and where moreover 1 has some second-order knowledge about 2's knowledge: we suppose that 1 knows that 2 knows that c attacks b. Suppose that the public view is initially empty and that 1's goal is that a is publicly-sceptically rejected (meaning that a must be attacked by a publicly-sceptically accepted argument).

Observe that this goal is dishonest, since according to 1's view a should be accepted. Agent 1 can start the debate in two different ways: either by announcing that b attacks a, or by announcing that b attacks a. We will formally show below that 1 knows that the first announcement will not contribute to the achievement of her goal. Therefore 1 should better announce that b attacks a: even if 1 is not sure that she will attain her goal, this is at least possible for her. In terms of the planning literature, the announcement 1:(b , a) is therefore a strong plan to achieve 1's goal.

Alternatively In dynamic epistemic logic terms, we have that

θ(M) → K 1 r b,a ! K 2 r c,b ! C Agt r b,a ∧ C Agt r c,b ∧ (x,y) / ∈{(b,a),(c,b)} ¬C Agt r x,y
is S5-PAL valid. Moreover, 1 knows this thanks to her second-order knowledge:

θ(M) → K 1 K 1 r b,a ! K 2 r c,b ! C Agt r b,a ∧ C Agt r c,b ∧ (x,y) / ∈{(b,a),(c,b)} ¬C Agt r x,y
is also S5-PAL valid. Therefore

θ(M) → K 1 K 1 r b,a ! K 2 r c,b ! Stable pub → (in a ∧ ¬in b )
is S5-PAL valid.

Discussion

Let us now discuss how multi-agent AFs can be extended in two different directions, so as to include, respectively, thirdand higher-order epistemic information and uncertainty.

Higher-Order Views

We have only considered the agents' first-and second-order views of an AF. As we have mentioned in Section 3.2, one may wish to go beyond this and use the full power of epistemic logic where epistemic operators in formulas can be arbitrarily nested. Let us sketch here how this can be achieved by using S5-models directly in order to represent multi-agent AFs with arbitrary higher-order views. Define the characteristic valuation of an AF R = A, R as v R = {r a,b : (a, b) ∈ R}. A pointed S5-model for R is a pointed S5-model M, w such that (i) M, w |= θ(R) and (ii) for every u ∈ W and every a ∈ A either in a ∈ V (u) or in a / ∈ V (u) (that is, the valuation of in a -variables is uniform throughout W ). We can rewrite condition (i) more semantically as V (w)∩Att A = v R . Now, all the information contained in a multi-agent AF can be extracted from M, w . More in detail, agent i's view of R can be defined as

R i = {(a, b) ∈ A × A : M, w |= K i r a,b }.
Again, this can be expressed without making any reference to the epistemic language as: R i = ( u∈∼i[w] V (u)) ∩ Att A , where ∼ i [w] = {u ∈ W : w ∼ i u}. Analogous formulations can be made for (R i ) j and R pub . Condition (C1) is guaranteed by reflexivity of ∼ i . Second-order views are then defined as C2) is guaranteed by reflexivity of ∼ j and ∼ i . Condition (C3) is guaranteed by reflexivity and transitivity of ∼ i . The public view is defined as C4) is guaranteed by the definition of common knowledge. Agents' n-order views of R could be defined in general as

(R i ) j = {(a, b) ∈ A × A : M, w |= K j K i r a,b }. Condition (
R pub = {(a, b) ∈ A × A : M, w |= C Agt r a,b }. Condition (
(((R i0 ) ... ) in ) = {(a, b) ∈A × A : M, w |= K in K in-1 . . . K i0 r a,b }
where ((R i0 ) ... ) in ) represents what agent i n knows that agent i n-1 knows... that agent i 0 knows about R. The updated multi-agent AF M +i:(a, b) would be then the one induced by M Kir a,b ! . Analogous results to those of Proposition 2 can be formulated and proved in this setting.

Uncertainty

AFs assume perfect knowledge about the attack relation and the existence of arguments. Recently, some works have relaxed this assumption, incomplete argumentation frameworks being one of the most popular approaches [Baumeister et al., 2018]. Our multi-agent AFs are strongly linked to a well-known subclass of incomplete AFs, namely, attackincomplete AFs, previously studied under the denomination partial AFs [Cayrol et al., 2007]. An (attack-)incomplete AF (IAF) is a tuple A, R, R ? such that R, R ? ⊆ A × A and R ∩ R ? = ∅. Intuitively, R captures the view of a single agent while R ? captures the set of attacks that the agent sees as uncertain (she does not know whether they hold). The fundamental notion when reasoning about IAFs is that of a completion: an AF A, R * such that R ⊆ R * ⊆ R ∪ R ? . Semantic notions (extensions and argument acceptability) are then combined in different ways with that of completion (see [Baumeister et al., 2018;[START_REF] Fazzinga | [END_REF]). There are some obvious differences between our approach and IAFs. First, IAFs lack a multi-agent component. Second, IAFs fit better a doxastic interpretation (rather than an epistemic one), since there is no direct way to express that an attack actually holds, and thus to distinguish between true and false beliefs. Third, our multi-agent AFs lack any mechanism for modelling uncertainty.

Combining both approaches seems a promising line of research. By mixing multi-agency with uncertainty we obtain, besides the already discussed attack relations, four new different kinds: R i ? the set of attacks that i is uncertain about;

(R i ? ) j the attacks that j knows that i is uncertain about; (R i ) j ? the attacks that j does not know whether i knows; and (R i ? ) j ? the attacks that j does not know whether i knows whether. Again, epistemic logic can help us to throw some light on these notions, by providing a list of intuitive constraints that they must satisfy. Let us illustrate this idea. The fact that (a, b) ∈ R i ? is naturally encoded in epistemic logic as Ki r a,b ∧ Ki ¬r a,b . Hence (a, b) ∈ R i ? ∪ R i is encoded as ( Ki r a,b ∧ Ki ¬r a,b ) ∨ K i r a,b which is S5-equivalent to Ki r a,b . Following a similar reasoning, we can deduce that (a, b) ∈ (R j ) i ? ∪ (R j ) i is encoded as Ki K j r a,b . Since Ki K j r a,b → Ki r a,b is S5-valid, it seems reasonable to require that (R j ) i ? ∪ (R j ) i ⊆ R i ∪ R i ? .

7 Related Work and Future Goals

Interactions among formal argumentation and epistemic logic are receiving increasing attention. These interactions can be classified into two groups: (i) the study of the notion of justified or argument-based belief [START_REF] Shi | [END_REF][START_REF] Grossi | Justified beliefs by justified arguments[END_REF]] and (ii) epistemic reasoning about AFs [START_REF] Schwarzentruber | Building an epistemic logic for argumentation[END_REF][START_REF] Sakama | [END_REF][START_REF] Proietti | Dynamic epistemic logics for abstract argumentation[END_REF], in which the current work is framed. The main differences with [START_REF] Sakama | [END_REF] is that we use a simpler-and thus less expressive-logical encoding and that we do take into account several aspectssuch as multi-agent scenarios and an explicit modelling of the dynamics of argumentation-that were left aside in [START_REF] Sakama | [END_REF]. As for [START_REF] Schwarzentruber | Building an epistemic logic for argumentation[END_REF][START_REF] Proietti | Dynamic epistemic logics for abstract argumentation[END_REF], we change the object of knowledge (from awareness of arguments to attacks), and we show that, at least to some extent, there is no need for incorporating the full Kripke semantics of epistemic logic in order to capture some of the strategic features of argumentation.

We have studied the relation between multi-agent AFs and multiagent S5, a standard logic of knowledge. The question of whether the current approach can be extended to belief in a relatively simple manner remains open. Such an extension requires the integration of a mechanism for belief revision in order to model argumentative/doxastic dynamics. There are other paths for future research. Besides developing in detail the ideas sketched in Section 6, we intend to define dialogues based on our multi-agent AFs and apply our current results to several reasoning tasks as, for instance, computing the next movement within a dialogue.

Figure 1 :

 1 Figure 1: An agent-annotated AF where the attacks (b, a), (b , a) and (c , b ) are known by 1 but not by 2

  depicts 1's and 2's views. We have R 1 = {(c, b), (b, a), (c , b ), (b , a)} and R 2 = {(c, b)}.

  , M can be updated by first 1:(b, a) and then 2:(c, b), resulting in M +1:(b, a) +2:(c, b), for which the public view is (R pub ∪{(b, a)})∪{(c, b)} = {(b, a), (c, b)}.

Results will be stated throughout this paper without proof due to space limitations, but they can be found in Antonio Yuste Ginel's forthcoming PhD dissertation.Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence

We could have introduced the fragment only, but preferred to introduce the full PAL-language in order to follow its standard presentation.Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence

Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence

Acknowledgments

Andreas Herzig is partially supported by the EU ICT-48 2020 project TAILOR (No. 952215). The research activity of Antonio Yuste-Ginel is supported by the predoctoral grant MECDFPU 2016/04113.


Andreas Herzig
email: herzig@irit.fr

Antonio Yuste-Ginel
email: antonioyusteginel@gmail.com

On the Epistemic Logic of Incomplete Argumentation Frameworks

We study the relation between two existing formalisms: incomplete argumentation frameworks (IAFs) and epistemic logic of visibility (ELV). We show that the set of completions of a given IAF naturally corresponds to a specific equivalence class of possible worlds within the model of visibility. This connection is further strengthened in two directions. First, we show how to reduce argument acceptance problems of IAFs to ELV model-checking problems. Second, we highlight the epistemic assumptions that underlie IAFs by providing a minimal epistemic logic for IAFs.

Introduction

Argumentation is a well-established research area within the field of Artificial Intelligence. It has been proved useful for a variety of purposes (Bench-Capon and Dunne 2007; [START_REF] Atkinson | Towards artificial argumentation[END_REF], including (but not limited to): providing an operational semantics for diverse non-monotonic logics, establishing a bridge between human and machine reasoning, and improving artificial agents' interaction. At the abstract level, that is, without analysing the structure and nature of arguments, Dung's argumentation frameworks (AFs) are by far the most studied formal construct [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. Conceptually, AFs respond to one fundamental question of (general) argumentation theory: given a set of possibly conflicting arguments, which subsets of arguments are to be accepted by a rational agent?

AFs are, however, too abstract for some purposes. Consequently, they have been extended in many different directions, including the handling of preferences [START_REF] Amgoud | A new approach for preference-based argumentation frameworks[END_REF], and more refined forms of interaction between arguments [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation frameworks[END_REF]. One of the main assumptions that AFs come equipped with is perfect knowledge about the structure of the modelled discussion (about the structure of the argumentation framework). The formal argumentation community has recently made efforts to relax this assumption. This has been done using quantitative methods, mainly probabilistic [START_REF] Li | Probabilistic argumentation frameworks[END_REF], and qualitative ones. Among the second type, incomplete argumentation frameworks (IAFs) are prominent [START_REF] Coste-Marquis | On the merging of Dung's argumentation systems[END_REF]Baumeister, Neugebauer, and Rothe 2018;Baumeister et al. 2018b;[START_REF] Fazzinga | Revisiting the notion of extension over incomplete abstract argumentation frameworks[END_REF]). An IAF partitions both the set of arguments A and the set of attacks R of an AF into two sets, distinguishing arguments (attacks) known with certainty from those perceived as uncertain by the agent.

In this paper, we pursue a simple research question: what is the epistemic logic underlying IAFs? Our answer, presented in Section 3, consists in establishing a strong connection between IAFs and the logic of visibility (ELV) [START_REF] Herzig | Possible worlds semantics based on observation and communication[END_REF]. We further exploit this connection in two different senses. First, we show how it can be used to reduce acceptance reasoning tasks of IAFs to ELV model-checking problems (Section 4). Second, we provide a minimal epistemic logic for IAFs that makes the epistemic assumptions underlying them explicit (Section 5). For space reasons, results are stated without proof; they can be found in Antonio Yuste-Ginel's forthcoming PhD dissertation.

Preliminaries

Abstract Argumentation Frameworks

An abstract argumentation framework (AF) is a directed graph, i.e., a pair (A, R) where A is a finite set whose elements stand for arguments and R ⊆ A×A represents some kind of conflict-based relation among arguments, typically, an attack relation. AFs were first studied by [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], where the author proposed to capture the informal notion of argument acceptability through different argumentation semantics. For the sake of brevity, we limit ourselves to stable semantics, but our analysis applies to all semantics that can be captured in propositional logic (see [START_REF] Besnard | Logical theories and abstract argumentation: A survey of existing works[END_REF]). A stable extension of an AF (A, R) is a set of arguments E ⊆ A s.th.: (i) there are no x, y ∈ E s.th. (x, y) ∈ R ('E is conflict-free'); and (ii) for every x ∈ A \ E, there is a y ∈ E such that (y, x) ∈ R ('E attacks every argument outside itself').

Incomplete AFs

An IAF is a tuple IAF = (A, A ? , R, R ? ) s.th. A ∩ A ? = ∅; R, R ? ⊆ (A ∪ A ? ) × (A ∪ A ? ); and R ∩ R ? = ∅. (A, R) is the definite part of IAF while (A ? , R ? ) is the uncertain part of IAF. We assume that A ∪ A ? is finite. A completion of IAF is any pair (A * , R * ) s.th.:

• A ⊆ A * ⊆ A ∪ A ? ; and • R |A * ⊆ R * ⊆ (R ∪ R ? ) |A * where we note R |Y = R ∩ (Y ×Y ) the restriction of a rela- tion R ⊆ X×X to Y ⊆ X.
All the classic reasoning problems regarding AFs were generalized to IAFs and their complexity was characterised (Baumeister et al. 2018b;[START_REF] Baumeister | Acceptance in incomplete argumentation frameworks[END_REF]. We here focus on argument acceptance problems. Classic acceptance problems for standard AFs can be generalized to IAFs as follows. The set of all stable extensions of (A, R) is noted st(A, R).

stable-Possible-Sceptical-Acceptance (st-PSA) Given: An incomplete AF IAF = (A, A ? , R, R ? ) and an argument a ∈ A. Question: Does there exist a completion (A * , R * ) of IAF s.th. for all E ∈ st(A * , R * ), a ∈ E? stable-Necessary-Credulous-Acceptance (st-NCA) Given: An incomplete AF IAF = (A, A ? , R, R ? ) and an argument a ∈ A. Question: Is it true that for all completions (A * , R * ) of IAF there is an E ∈ st(A * , R * ) s.th. a ∈ E?

The other two variants of the problem, stable-Possible-Credulous-Acceptance (st-PCA) and stable-Necessary-Sceptical-Acceptance (st-NSA), are obtained by changing the quantifiers of the definitions above in the obvious way; see [START_REF] Baumeister | Acceptance in incomplete argumentation frameworks[END_REF] for details.

Epistemic Logic of Visibility

Epistemic logic of visibility or observability was born as a lightweight alternative to standard epistemic logic [START_REF] Van Der Hoek | Knowledge and control[END_REF][START_REF] Herzig | Possible worlds semantics based on observation and communication[END_REF][START_REF] Cooper | A lightweight epistemic logic and its application to planning[END_REF]. We consider its singleagent version and adapt it to our purposes. Let Σ be a finite set of arguments, called the signature. We define the set of atoms for signature Σ in two steps:

Prop Σ = {aw x | x ∈ Σ} ∪ {r x,y | (x, y) ∈ Σ×Σ} ∪ {in x | x ∈ Σ}; ATM Σ = Prop Σ ∪ {Sp | p ∈ Prop Σ }.
Intuitively, aw x means that the agent is aware of argument x; r x,y means that x attacks y; in x means that argument x is in the extension; and Sp means that the agent sees the value of p (equivalently, she knows whether p). We use the term awareness in a rather loose sense here. Among different possible interpretations, it can be seen as the arguments the agent's opponent is aware of. This explains why an agent can be aware of an argument without knowing whether she is aware of it. Moreover, note that Σ does not need to coincide with A ∪ A ? : there can be arguments whose awareness the agent knows not to hold. The epistemic language for signature Σ is generated by the following BNF:

ϕ ::= α | ¬ϕ | (ϕ ∧ ϕ) | Kϕ (α ∈ ATM Σ )
where Kϕ reads "the agent knows that ϕ". 1 We also consider the dual K = ¬K¬, read "the agent considers possible 1 As shown in [START_REF] Herzig | Possible worlds semantics based on observation and communication[END_REF], in ELV the operator K can be recursively eliminated from any visibility formula. However, we prefer to include it for presentational purposes. that...".

The single-agent visibility model for ATM Σ is the pair

M vis = (2 ATM Σ , ∼), where the accessibility relation ∼ ⊆ 2 ATM Σ × 2 ATM Σ is such that v ∼ v iff for every p ∈ Prop Σ : 1. if Sp ∈ v then (p ∈ v iff p ∈ v ); 2. Sp ∈ v iff Sp ∈ v .
Formulas are interpreted at states of M vis , i.e., at propositional valuations. We only state the clause for the epistemic operator:

M vis , v |= Kϕ iff v ∼ v implies M vis , v |= ϕ.
Note that the second constraint in the definition of the visibility model guarantees that K is a fully introspective operator (Kϕ → KKϕ and ¬Kϕ → K¬Kϕ are both valid in the visibility model). Moreover, Kp ↔ (Sp ∧ p) is valid too.

Incomplete AFs and Visibility Models

Relations between IAFs and possible worlds semantics were spotted in the literature (Baumeister et al. 2018a;Baumeister, Neugebauer, and Rothe 2018;[START_REF] Baumeister | Acceptance in incomplete argumentation frameworks[END_REF], and just recently explored in more depth by reducing IAFs to general epistemic logic [START_REF] Proietti | Dynamic epistemic logics for abstract argumentation[END_REF]. Nevertheless, a basic research question about the relation between these two formalisms remains still open: what is the epistemic logic underlying IAFs? We provide an answer by connecting IAFs to the single-agent logic of visibility. Formally, we show a one-to-one correspondence between completions of a given IAF and a specific equivalence class of the visibility model. For the sake of simplicity, we dispense with acceptance variables in this section, that is, we assume that Prop Σ = {aw x | x ∈ Σ} ∪ {r x,y | (x, y) ∈ Σ×Σ}.

Let IAF = (A, A ? , R, R ? ) be an IAF with A∪A ? ⊆ Σ. We associate to IAF its single-agent propositional valuation

v IAF = {Saw x | x ∈ Σ \ A ? } ∪ {aw x | x ∈ A} ∪ {Sr x,y | (x, y) ∈ (Σ×Σ) \ R ? } ∪ {r x,y | (x, y) ∈ R}.
The other way round, for every propositional valuation v ⊆ ATM Σ we define its induced AF (A v , R v ) by:

A v = {x ∈ Σ | aw x ∈ v}; R v = {(x, y) ∈ A v ×A v | r x,y ∈ v}.

From IAFs to the Visibility Model

Using the notions we have just defined, we can provide the first clear connection between IAFs and the visibility model. Proposition 1. Let IAF = (A, A ? , R, R ? ) be an IAF s.th.

A ∪ A ? ⊆ Σ, let v IAF be its propositional valuation, and let M vis = (2 ATM Σ , ∼) be the single-agent visibility model for the set of atoms ATM Σ , then:

• For each completion (A * , R * ) of IAF there is a u ∈ ∼ [v IAF ] s.th. (A * , R * ) = (A u , R u ). • For each u ∈ ∼ [v IAF ] there is a completion (A * , R * ) of IAF s.th. (A * , R * ) = (A u , R u ).
Hence completions can be understood as possible worlds of the visibility model (i.e., as propositional valuations over ATM Σ ) that are indistinguishable for the formalized agent.

From the Visibility Model to IAFs

The visibility model for a signature Σ contains all IAFs that can be built from Σ. Formally, we say that (A, A ? , R, R ? ) is built over Σ whenever A ∪ A ? ⊆ Σ.

Proposition 2. Let Σ be a signature, let M vis = (2 ATM Σ , ∼) be the single-agent visibility model for ATM Σ . For every IAF = (A, A ? , R, R ? ) built over Σ there is a u ∈ 2 ATM Σ s.th. u = v IAF .

In the other direction things get more subtle. Given a valuation v ⊆ ATM Σ , we define the IAF represented by v as

IAF (v) = (A(v), A ? (v), R(v), R ? (v))
, where:

A(v) = {x ∈ Σ | v |= Kaw x }; A ? (v) = {x ∈ Σ | v |= ¬Saw x }; R(v) = {(x, y) ∈ Σ×Σ | v |= Kr x,y ∧ Kaw x ∧ Kaw y }; R ? (v) = {(x, y) ∈ Σ×Σ | v |= ¬Sr x,y ∧ Kaw x ∧ Kaw y }.
Informally, IAF (v) is the IAF where certain arguments are those that the agent knows that she is aware of; uncertain arguments are those whose awareness the agent is not sure about; and analogously for attacks (conditionally on the awareness of the involved arguments). It is easy to check that, for each valuation v, (i) IAF (v) is indeed an IAF; and (ii) the set of all completions of IAF (v) equals the set of AFs

(A u , R u ) such that u ∈ ∼ [v].
Moreover, given IAF, it holds that

IAF (v IAF ) = IAF.
In words, we can go from IAFs to states of the visibility model and back without loosing any information. However, this is not true if we start from the set of all valuations: it does not hold that for any u, v IAF (u) = u. The reason is that there are some valuations containing "defective information": there can be attacks that are true in all indistinguishable valuations, and hence known, even if the agent knows that she is not aware of one of the involved arguments. These pieces of defective information are cleaned out by applying the function IAF . Alternatively we could impose that valuations should satisfy the property Kr x,y → ( Kaw x ∧ Kaw y ).

Checking Acceptance via Model-Checking

As an application of the correspondence between IAFs and the logic of visibility, we can reduce acceptance problems regarding the former to model-checking problems of the latter. This can be done modulo some propositional logic encoding of standard argumentation semantics. We illustrate this idea for the case of stable semantics. Let us first recall how stable extensions of (awareness-relativised) AFs can be captured in propositional logic. We make use of the shorthand r aw x,y = r x,y ∧ aw x ∧ aw y and define the formula

Stable = x∈Σ (in x → aw x ) ∧ aw x → (in x ↔ ¬ y∈Σ (in y ∧ r aw y,x ) .
As shown in (Doutre, Maffre, and McBurney 2017, Proposition 1), we have that

v |= Stable iff E v = {x ∈ Σ | in x ∈ v} is a stable extension of (A v , R v ).
Using this encoding and the correspondence between completions and accessible states in the visibility model of Proposition 1 we obtain: Proposition 3. Let IAF = (A, A ? , R, R ? ) be an IAF built over Σ, let v IAF be its propositional valuation, let a ∈ A and let M vis = (2 ATM Σ , ∼) be the single-agent visibility model for the set of atoms ATM Σ , then:

• The answer to the st-NSA (stable-Necessary-Sceptical-Acceptance) problem with input IAF and a ∈ A is yes iff

M vis , v IAF |= K(Stable → in a ). • The answer to the st-PCA (stable-Possible-Credulous- Acceptance) problem with input IAF and a ∈ A is yes iff M vis , v IAF |= K(Stable ∧ in a ).
Curiously enough, capturing st-PSA and st-NCA in the logic of visibility requires augmenting the language, jumping from the single-agent version to the two-agent version. The underlying reason for this is that while the definitions of st-PCA and st-NSA use a ∃∃ (resp. ∀∀) pattern for quantifying over completions and extensions, the definitions of the other two reasoning tasks alternate quantifiers. Our solution is to modularize the formalized agent: the agent consists of a part for reasoning about completions (named 1 or the epistemic part) and another one for reasoning about extensions (named 2 or the argumentative part).

First of all, we generalize the set of logical atoms so as to include a finite, non-empty set of agents Agt:

ATM Σ Agt = Prop Σ ∪ {S i p | i ∈ Agt, p ∈ Prop Σ }
, where Prop Σ is the set of propositional variables as defined in Section 2.3.

Note that we do not need to parametrize awareness variables for each agent. As mentioned, we are going to modularize the agent regarding his epistemic and argumentative reasoning, but not her awareness. In the remaining of this section, we shall restrict our attention to Agt = {1, 2}. The multi-agent epistemic language extends the epistemic language used before by including a different epistemic operator K i for each agent in Agt. Nevertheless, since agent 2 represents the "argumentative part" of the formalize agent, we use 2 instead of K 2 , for conceptual clarity. We also use ♦ 2 as a shorthand for ¬ 2 ¬.

The multi-agent visibility model is the pair

M vis = (2 ATM Σ Agt , {∼ i } i∈Agt ) where each ∼ i ⊆ 2 ATM Σ Agt × 2 ATM Σ Agt is such that v ∼ i v iff for every p ∈ Prop Σ , 1. if S i p ∈ v then (p ∈ v iff p ∈ v ); 2. S j p ∈ v iff S j p ∈ v for every j ∈ Agt.
We adapt the notion of propositional valuation associated to an IAF to the two-agent language as follows. Let IAF = (A, A ? , R, R ? ) be an IAF s.th. A ∪ A ? ⊆ Σ. Define

v IAF = {S 1 aw x | x ∈ Σ \ A ? } ∪ {S 1 r x,y | (x, y) ∈ (Σ×Σ) \ R ? } ∪ {S 2 aw x | x ∈ Σ} ∪ {S 2 r x,y | (x, y) ∈ Σ×Σ} ∪ {aw x | x ∈ A} ∪ {r x,y | (x, y) ∈ R}.
This extends the associated valuation of the single-agent visibility language so as to include visibility atoms indexed with agent 2 for all awareness and attack variables. Conceptually, 2 is a part of the agent that always reasons under the assumption that the underlying AF is completed. Proposition 4. Let IAF = (A, A ? , R, R ? ) be an IAF built over Σ, let v IAF be its propositional valuation, let M vis be the visibility model for ATM Σ {1,2} , and let a ∈ A, then: • The answer to the st-NCA (stable-Necessary-Credulous-Acceptance) problem with input IAF and a ∈ A is yes iff

M vis , v IAF |= K 1 ♦ 2 (Stable ∧ in a )
. • The answer to the st-PCA (stable-Possible-Sceptical-Acceptance) problem with input IAF and a ∈ A is yes iff M vis , v IAF |= K1 2 (Stable → in a ).

A Minimal Epistemic Logic for IAFs

As it is well-known, modal operators interpreted over equivalence relations (as the one we have just used in the logic of visibility) validate S5 axioms, which informally correspond to a factive, fully introspective kind of knowledge; see e.g. (Fagin et al. 2004, Chapter 2). However, one may want to use epistemic logic to reason about AFs without embracing such strong principles. Interestingly, the connection we have established in Section 3 can be used to unravel a minimal epistemic logic for IAFs. By doing so, we clarify the main epistemic assumptions underlying IAFs. We start defining the logic semantically. Let Σ be an argument signature, Prop Σ its associated set of propositions, and Agt a finite, non-empty set of agents. A Kripke model is a tuple (W, R, V ) where W = ∅ is a set of possible worlds, R : Agt → ℘(W ×W ) assigns to each agent an epistemic accessibility relation R i , and V : Prop Σ → ℘(W ) is a valuation function. Since the current discussion is orthogonal to acceptance problems, we dispense again with acceptance variables and assume Prop Σ = {aw x | x ∈ Σ} ∪ {r x,y | (x, y) ∈ Σ×Σ}. Given a Kripke model (W, R, V ) and a world u ∈ W , we define the AF associated to u as the tuple (A u , R u ), where

A u = {x ∈ Σ | u ∈ V (aw x )} and R u = {(x, y) ∈ Σ×Σ | u ∈ V (r x,y )} ∩ (A u ×A u ).
Given the epistemic multi-agent language, the concepts of truth (M, w |= ϕ), validity in a model (M |= ϕ), and validity (|= ϕ) are the usual ones in epistemic modal logic, see e.g. (Fagin et al. 2004, Chapter 2). Moreover, an IAF-friendly Kripke model is a Kripke model where for every w ∈ W , every x, y ∈ Σ, and every i ∈ Agt:

(AWAR) if R i [w]∩V (r x,y ) = ∅, then R i [w]∩V (aw x ) = ∅ and R i [w] ∩ V (aw y ) = ∅; and (
COMP) there is an incomplete AF, IAF, built over Σ s.th.:

the set of completions of IAF is equal to

{(A v , R v ) | v ∈ R i [w]}.
Informally, an IAF-friendly Kripke model is just a Kripke model where (i) if an agent considers possible that x attacks y then she considers possible being aware of both x and y; and (ii) the set of i-successors of each world represents the set of completions of an incomplete AF. Now, consider the following two schemas:

(awar) Ki r x,y → ( Ki aw x ∧ Ki aw y );

(comp) ( Ki l 1 ∧ • • • ∧ Ki l n ) → Ki (l 1 ∧ • • • ∧ l n ),
where {l 1 , ..., l n } is a consistent set of Prop Σ -literals: elements of Prop Σ or their negations (hence the set cannot contain both p and ¬p). Note that (comp) is true in every world of the single-agent visibility model and, moreover, it is the contra-positive of one of the axioms of the logic of visibility of [START_REF] Herzig | Possible worlds semantics based on observation and communication[END_REF].

A sound and complete axiomatization with respect to IAF-friendly Kripke models can be obtained by simply adding all instances of (awar) and (comp) as well as axiom D: K i ϕ → Ki ϕ to the minimal modal proof system K.2 A key for obtaining this result is the following claim: Proposition 5. A Kripke model M = (W, R, V ) for Prop Σ is an IAF-friendly model iff R i is serial for every i ∈ Agt and all instances of (awar) and (comp) are valid in M .

Discussion. The fact that the logic of IAF-friendly Kripke models is KD + (comp) + (awar) reveals some interesting things. First, the minimal epistemic attitude underlying IAFs is a form of belief for which consistency is required (axiom D). Second, IAFs assume that all combinations of uncertain elements are considered as doxastically/epistemically possible by the agent (axiom (comp)). Equivalently, the belief/knowledge operator distributes over disjunctions of consistent literals.

Conclusion

To the best of our knowledge, this is the first paper studying the relations between IAFs and ELV, and it can be understood as an improvement of the reduction of IAFs to general epistemic logic provided in [START_REF] Proietti | Dynamic epistemic logics for abstract argumentation[END_REF]. We have briefly sketched two applications of this connection that seem conceptually valuable, as they dig into an epistemic interpretation of IAFs. As to future work, our findings can be used to provide a multi-agent generalization of IAFs. More precisely, we are now able to systematically investigate different multi-agent generalizations of IAFs, corresponding to different epistemic attitudes. A further perspective is to investigate the dynamics of IAFs in dynamic extensions of epistemic logic and ELV (cf. (Herzig and Yuste Ginel 2021)), taking inspiration from what has been done for Dung AFs in [START_REF] Doutre | A dynamic logic framework for abstract argumentation: adding and removing arguments[END_REF][START_REF] Doutre | Abstract argumentation in dynamic logic: Representation, reasoning and change[END_REF].

Proceedings of the 18th International Conference on Principles of Knowledge Representation and ReasoningMain Track

We recall that axiom D is the modal schema axiomatizing seriality of each relation Ri and conceptually corresponding to consistency of each agents' belief. It is needed to axiomatise the logic of IAF-friendly Kripke models because, by definition, every IAF has a non-empty set of completions. Proceedings of the 18th International Conference on Principles of Knowledge Representation and Reasoning Main Track

Acknowledgements

Andreas Herzig is partially supported by the EU ICT-48 2020 project TAILOR (No. 952215). Antonio Yuste-Ginel gratefully acknowledges funding received from the PhD grant No. MECDFPU 2016/04113.