Bruno Yun
email: yun@lirmm.fr

Srdjan Vesic
email: vesic@cril.fr

Madalina Croitoru
email: croitoru@lirmm.fr

Ranking-Based Semantics for Sets of Attacking Arguments

Argumentation is a process of evaluating and comparing sets of arguments. Ranking-based semantics received a lot of attention recently. All of the semantics introduced so far are applicable to binary attack relations. In this paper, we study a more general case when sets of arguments can jointly attack an argument. We generalise existing postulates for rankingbased semantics to fit this framework, define a general variant of h-categoriser, prove that it converges for every argumentation framework and study the postulates it satisfies. We also study the link between binary and hypergraph version of hcategoriser.

Introduction

Argumentation is the process of reasoning in presence of inconsistent information by exchanging, evaluating or ranking interactive arguments. The most popular way of representing this argumentation process was proposed by Dung in his seminal paper [START_REF] Dung | On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic Programming and n-Person Games[END_REF] which sees arguments as nodes and attacks as edges in a directed graph. Then, one can apply the several argumentation semantics [START_REF] Baroni | An introduction to argumentation semantics[END_REF] to extract meaningful subsets of arguments called extensions. However, for many applications, this two level of acceptance (arguments belong or not to an extension) was not enough [START_REF] Leite | Social Abstract Argumentation[END_REF]. Against this background, a solution consisted in having many levels of acceptability by ranking arguments using the several ranking-based semantics defined in the literature [START_REF] Amgoud | Ranking-Based Semantics for Argumentation Frameworks[END_REF][START_REF] Cayrol | Graduality in Argumentation[END_REF][START_REF] Leite | Social Abstract Argumentation[END_REF][START_REF] Matt | A Game-Theoretic Measure of Argument Strength for Abstract Argumentation[END_REF] for the Dung's framework along with a set of desirable properties that they should satisfy [START_REF] Bonzon | A Comparative Study of Ranking-Based Semantics for Abstract Argumentation[END_REF][START_REF] Amgoud | Ranking-Based Semantics for Argumentation Frameworks[END_REF].

Classic Dung argumentation graphs only take into account a binary attack relation. Such binary attack relation implicitly assumes that the logical language used underneath the argument structure is closed under the "and" logical connective. To free oneself from such assumption, Nielsen and Parsons proposed a generalisation of the Dung framework with sets of attacking arguments [START_REF] Nielsen | A Generalization of Dung's Abstract Framework for Argumentation: Arguing with Sets of Attacking Arguments[END_REF]. In their framework arguments can jointly attack a single other argument. For instance, let us consider the deductive argumentation framework [START_REF] Besnard | A logic-based theory of deductive arguments[END_REF], also known as the (support, conclusion) representation, where support is a set of formulas and conclusion is a formula that follows from the support. Now, let us further consider the standard undercut attack relation where an argument attacks another based on the contradiction between the conclusion of the former and a formula in the support of the latter. However, if we are given a less expressive logical language without conjunction, we are unable to model attacks corresponding to ternary conflicts. Let x, y and z be three formulas of the logic that form a ternary conflict. In order to attack the argument ({z}, z), one would need to construct the argument ({x, y}, x ∧ y) which is impossible if the language does not include conjunction. To this end [START_REF] Verheij | Rules, Reasons, Arguments[END_REF][START_REF] Nielsen | A Generalization of Dung's Abstract Framework for Argumentation: Arguing with Sets of Attacking Arguments[END_REF] propose to have sets of attacking arguments (i.e. ({x}, x) and ({y}, y) jointly attack ({z}, z)). Furthermore, [START_REF] Nielsen | A Generalization of Dung's Abstract Framework for Argumentation: Arguing with Sets of Attacking Arguments[END_REF] extended the classical Dung semantics (stable, preferred, complete and grounded) to this new hypergraph framework. In view of this, we generalise ranking-based semantics for the hypergraph argumentation framework.

The structure of the paper is as follows. In Section 2, we recall the necessary definitions of a hypergraph argumentation framework and ranking-based semantics. In Section 3, we define some desirable properties that such rankingbased semantics should satisfy. In Section 4, we propose the first ranking-based semantics for hypergraphs argumentation frameworks following the insights from [START_REF] Besnard | A logic-based theory of deductive arguments[END_REF], we show that this new ranking-based semantics converges and that it is a proper generalisation of the categoriser semantics defined in [START_REF] Besnard | A logic-based theory of deductive arguments[END_REF] for classic Dung graphs.

Background

In the next definition, we extend Dung's abstract framework with sets of attacking arguments, i.e. sets of arguments can now jointly attack an argument. Definition 1 (Hypergraph argumentation framework). A hypergraph argumentation framework is a pair AS = (A, R) with A a finite and non-empty set of arguments and R ⊆ (2 A \ {∅}) × A. AS a. By abuse of notation, if S and S are two sets of arguments, we write that S σ AS S iff for every s ∈ S and every s ∈ S , we have s σ AS s . Let S be a set in (2 A \{∅}) and σ be a ranking-based semantics, the set min σ AS (S) is the set {s ∈ S | for every s ∈ S, s σ AS s}. Definition 3 (Path). Let AS = (A, R) ∈ AS and a ∈ A. We say that a sequence of attacks ((S 1 , t 1 ), . . . , (S n , t n )) is a path of size n from S 1 to a iff for every i ∈ {1, . . . , n}, (S i , t i ) ∈ R, t n = a and for every i ∈ {1, . . . , n -1}, it holds that t i ∈ S i+1 .

Notation. Let AS = (A, R) ∈ AS, S ∈ 2 A and a ∈ A, we say that S ∈ R - n (a)
iff there exists a path of size n from S to a. We say that S ∈ R - 1 (a) is a direct attacker of a and

S ∈ R - 2 (a) is a direct defender of a. Example 2 (Example 1 Cont'd).
The sequence (({d}, a), ({a, c}, e)) is a path of size 2 from {d} to e. Thus, {d} ∈ R - 2 (e) and {d} is a direct defender of e.

Properties

We first introduce the definition of an isomorphism between two hypergraph argumentation frameworks. Definition 4 (Isomorphism). An isomorphism between two hypergraph argumentation frameworks AS = (A, R) and

AS = (A , R ) is a bijective function γ : A → A such that for every S ∈ 2 A and a ∈ A, (S, a) ∈ R iff ({γ(s) | s ∈ S}, γ(a)) ∈ R .
With a slight abuse of notation, we will note AS = γ(AS).

In the rest of this section, we recall the properties for a ranking-based semantics σ that have been defined in the literature [START_REF] Amgoud | Ranking-Based Semantics for Argumentation Frameworks[END_REF][START_REF] Bonzon | A Comparative Study of Ranking-Based Semantics for Abstract Argumentation[END_REF] and we translate them for hypergraph argumentation frameworks. We now begin with the properties that can be translated straightforwardly.

The Abstraction property states that the name of the arguments should not be taken into account for the ranking. Property 1 (Abstraction). We say that σ satisfies Abstraction iff for any AS, AS ∈ AS and isomorphism γ such that AS = γ(AS), we have a σ AS b iff γ(a) σ AS γ(b) The Independence property states that two arguments with no paths connecting them should not influence each other. Property 2 (Independence). We say that σ satisfies Independence iff for any AS = (A, R), AS = (A , R ) in AS such that A ∩ A = ∅ and every a, b ∈ A we have a σ AS b iff a σ AS⊕AS b. The Void precedence property states that non-attacked arguments should be ranked higher than attacked arguments. Property 3 (Void precedence). We say that σ satisfies Void precedence iff for any AS

= (A, R) ∈ AS and a, b ∈ A such that R - 1 (a) = ∅ and R - 1 (b) = ∅ we have a σ AS b.
The Self-contradiction property states that selfcontradicting arguments should be ranked lower than non self-contradicting arguments. Property 4 (Self-contradiction). We say that σ satisfies Selfcontradiction iff for any AS = (A, R) ∈ AS and a, b ∈ A such that there exists S 1 ∈ R - 1 (a) with a ∈ S 1 and there exists no S 2 ∈ R - 1 (b) with b ∈ S 2 we have b σ AS a. The Cardinality precedence property states that if an argument a has more attackers than an argument b than it should be ranked lower than b. The intuition is that an argument attacked by more sets of arguments is seen as more conflicted (and thus weaker). Property 5 (Cardinality precedence). We say that σ satisfies Cardinality precedence iff for any AS

= (A, R) ∈ AS and a, b ∈ A such that |R - 1 (a)| > |R - 1 (b)| we have b σ AS a.
The Defense precedence property states that if a, b are two arguments with the same number of attackers and a is defended but b is not then a should be ranked higher than b. Property 6 (Defense precedence). We say that σ satisfies Defense precedence iff for any AS

= (A, R) ∈ AS and a, b ∈ A such that |R - 1 (a)| = |R - 1 (b)|, R - 2 (a) = ∅ and R - 2 (b) = ∅ we have b σ AS a.
The Total property states that two arguments should always be comparable. Property 7 (Total). We say that σ satisfies Total iff for any

AS = (A, R) ∈ AS and a, b ∈ A we have a σ AS b or b σ AS a.
The Non-attacked equivalence property states that two non-attacked arguments should be ranked equivalently. Property 8 (Non-attacked equivalence). We say that σ satisfies Non-attacked equivalence iff for any AS

= (A, R) ∈ AS and a, b ∈ A such that R - 1 (a) = R - 1 (b) = ∅ we have a σ
AS b Definition 5 (Simple and distributed defense). Let AS = (A, R) ∈ AS and a ∈ A. The defense of a is simple iff for every direct defender S of a, there is a unique S ∈ R - 1 (a) and s ∈ S such that (S, s ) ∈ R. The defense of a is distributed iff for every direct attacker S of a, there is at most one direct defender S of a such that (S , s) ∈ R where s ∈ S.

The Distributed-defense precedence property states that if argument a has a simple and distributed defense and argument b has a simple but not distributed defense than a should be higher ranked than b. Property 9 (Distributed-defense precedence). σ satisfies Distributed-defense precedence iff for any AS

= (A, R) ∈ AS, for any a, b ∈ A such that |R - 1 (a)| = |R - 1 (b)| and |R - 2 (a)| = |R - 2 (b)|
and that the defense of a is simple and distributed and that the defense of b is simple but not distributed we have a σ AS b. Definition 6 (Cycle). Let AS = (A, R) be a hypergraph argumentation framework, S ∈ (2 A \ {∅}) and s ∈ A. We say that a path from S to s is a cycle iff s ∈ S. An argumentation framework is called acyclic iff there is no cycle.

An attack (resp. defense) branch of an argument a is a hypergraph argumentation framework such that its fusion with the original argumentation framework should result in the addition of an even (resp. odd) path to a. Definition 7 (Added attack/defense branch). Let AS = (A, R) be a hypergraph argumentation framework and a ∈ A. P + (a) (resp. P -(a)) is a pair (A 0 , R 0 ) called a defense (resp. attack) branch added to a iff:

• A 0 = {a, x 1 , . . . , x k } and A ∩ A 0 = {a} • there exists n ∈ 2N (resp. n ∈ 2N + 1), S 1 ∈ (2 A0 \ {∅})
and an acyclic path ((S 1 , t 1 ), . . . , (S n , t n )) of size n such that t n = a.

• R 0 = {(S i , t i ) | i ∈ {1, . . . , n}} and ( n 1 S i ) ∪ {a} = A 0
The Strict addition of defense branch property states that adding a defense branch to an argument should increase its rank. Property 10 (Strict addition of defense branch). We say that σ satisfies Strict addition of defense branch iff for any AS , AS = (A, R) ∈ AS and any a ∈ A such that there exists an isomorphism γ with AS = γ(AS) we have γ(a) σ AS⊕AS ⊕P+(γ(a)) a. The Addition of defense branch property states that adding a defense branch to an attacked argument should increase its rank. Property 11 (Addition of defense branch). We say that σ satisfies Addition of defense branch iff for any AS , AS = (A, R) ∈ AS and any a ∈ A such that there exists an isomorphism γ with AS = γ(AS) and R - 1 (a) = ∅ we have γ(a) σ AS⊕AS ⊕P+(γ(a)) a. The Addition of attack branch property states that adding an attack branch to an argument should decrease its rank.

Property 12 (Addition of attack branch). We say that σ satisfies Addition of attack branch iff for any AS , AS = (A, R) ∈ AS and any a ∈ A such that there exists an isomorphism γ with AS = γ(AS) we have a σ AS⊕AS ⊕P-(γ(a)) γ(a). The Increase of attack branch property states that increasing the length of an attack branch of an argument a should increase the ranking of a.

Property 13 (Increase of attack branch). We say that σ satisfies Increase of attack branch iff for any AS , AS = (A, R) ∈ AS and any a ∈ A such that there exists an isomorphism γ with AS = γ(AS), for every S ∈ (2 A \ {∅}) such that for every s ∈ S, R - 1 (s) = ∅, there exists a path of size n from S to a with n ∈ 2N + 1 and there is no path of size m from S to a with m ∈ 2N then for every s ∈ S, γ(a) σ AS⊕AS ⊕P+(γ(s)) a. The Increase of defense branch property states that increasing the length of a defense branch of an argument a should decrease the ranking of a.

Property 14 (Increase of defense branch). We say that σ satisfies Increase of defense branch iff for any AS , AS = (A, R) ∈ AS and any a ∈ A such that there exists an isomorphism γ with AS = γ(AS), for every S ∈ (2 A \ {∅}) such that for every s ∈ S, R - 1 (s) = ∅, there exists a path of size n from S to a with n ∈ 2N and there is no path of size m from S to a with m ∈ 2N + 1 then for every s ∈ S, a σ AS⊕AS ⊕P+(γ(s)) γ(a). The Attack vs full defense property states that, in an acyclic hypergraph argumentation framework, an argument with only one direct attacker should be ranked lower than an argument without any attack branch.

Property 15 (Attack vs full defense). We say that σ satisfies Attack vs full defense iff for any AS

= (A, R) ∈ AS that is acyclic and a, b ∈ A such that |R - 1 (b)| = 1, R - 2 (b) = ∅, there exists no S ∈ (2 A \ {∅}) with for every s ∈ S, R - 1 (s) =
∅ and a path of size n from S to a with n ∈ 2N + 1 then a σ AS b. In the next properties, we make the assumption that the set of arguments should be compared w.r.t. their weakest arguments. We work under this hypothesis since, for an attack to exist, all arguments from S are necessary. Of course, another aggregation function could be used -most of the corresponding definitions could easily be changed to take this change into account.

The Quality precedence property states that if an argument a has an attacker S such that the weakest element of S is ranked higher than every weakest element of any attacker of b than a should be ranked lower than b.

Property 16 (Quality precedence). We say that σ satisfies Quality precedence iff for any AS = (A, R) ∈ AS and a, b ∈ A such that there exists S ∈ R - 1 (a) and for every S ∈ R - 1 (b) we have min σ AS (S) σ AS min σ AS (S ) then b σ AS a. Before introducing the next properties, we need to introduce a relation that compares sets of sets of arguments by inspiring ourselves from [START_REF] Amgoud | Ranking-Based Semantics for Argumentation Frameworks[END_REF].

Definition 8 (Group comparison). Let AS = (A, R) and G, G ⊆ (2 A \ {∅}). We say that G ≥ G iff there exists an injective function f : G → G such that for every g ∈ G , min σ AS (f (g )) σ AS min σ AS (g ). Notation. As usual, we use the notation

G > G iff |G | < |G| or there exists g ∈ G such that min σ AS (f (g )) σ
AS min σ AS (g ). The Counter-transitivity property states that if the attackers of a are better than the attackers of b w.r.t. the group comparison then b should be better or equal ranked than a.

Property 17 (Counter-transitivity). We say that σ satisfies Counter-transitivity iff for any AS

= (A, R) ∈ AS and a, b ∈ A such that R - 1 (a) ≥ R - 1 (b) then b σ AS a.
The Strict counter-transitivity property states that if the attackers of a are better than the attackers of b w.r.t. the group comparison then b should be ranked strictly better than a.

Property 18 (Strict counter-transitivity). We say that σ satisfies Strict counter-transitivity iff for any AS

= (A, R) ∈ AS and a, b ∈ A such that R - 1 (a) > R - 1 (b) then b σ AS a.

The nh-categoriser

As mentioned before, in each set of attacking arguments, all the components are necessary. Thus, an important intuition is that removing one argument from the set of attacking arguments would make the attack void. In the definition of the nh-categoriser, we thus consider the force of the set of attacking arguments to be the force of the weakest argument of the set. Of course, the approach that we use can be generalised with other aggregating methods.

Please note that we chose to generalise the h-categoriser because of several reasons: (1) the h-categoriser is a wellknown ranking-based semantics in the argumentation community, (2) it converges for every argumentation graph and (3) it satisfies a maximal (for set inclusion) number of postulates introduced by (Amgoud and Ben-Naim 2013).

Definition 9 (nh-categoriser). Let AS = (A, R) be a hypergraph argumentation framework. The nh-categoriser is the function C : A → (0, 1] defined as, for all a ∈ A : The ranking outputted by the nh-categoriser is expected as it only considers the minimum in each set of attacking arguments. As a result, although the argument c is only attacked by the set {b}, it is ranked lower than argument e which is attacked by both {d, c} and {a, c} because b is stronger than both c and a combined.

C(a) =    1 if R - 1 (a) = ∅ 1 1+ S∈R - 1 (
In the rest of this section, we consider the argumentation framework AS = (A, R) where A = {a 1 , . . . , a n }. We now answer the two following questions for the nhcategoriser: (1) "How many solutions exists?" and (2) "how to find them?"

We first transform the problem into a fixed point form. Let us consider v ∈ [0, 1] n such that:

v = F (v) = [f 1 (v), f 2 (v), . . . , f n (v)] T
(1) where the function F maps [0, 1] n to [0, 1] n , and for every i ∈ {0, . . . , n}, the function f i from [0, 1] n to [0, 1] is defined by the nh-categoriser function:

f i (v) =    1 if R - 1 (a i ) = ∅ 1 1+ S∈R - 1 (a i ) min a j ∈S fj (v) else (2)
The function F is continuous and non-increasing as for every two vectors u = (u 1 , . . . , u n ) and u

= (u 1 , . . . u n ) of [0, 1] n with u 1 ≤ u 1 , . . . , u n ≤ u n , F (u) ≥ F (u ) holds.
Proposition 1 (Solution existence). For any argumentation framework AS = (A, R), the nh-categoriser valuation defined in (2) has at least one solution in [0, 1] n .

Proof. The proof is similar to the one of [START_REF] Pu | Argument Ranking with Categoriser Function[END_REF] and relies on the equivalence result that function F has at least one fixed point. The proof uses Brouwer's fixed point theorem and the fact that [0, 1] n is homeomorphic to a closed ball and function that F is continuous on it.

Proposition 2 (Uniqueness of nh-categoriser valuation). Let AS = (A, R) be a hypergraph argumentation framework with A = {a 1 , . . . , a n } and R with sets of attacking arguments. Then, the scores of the nh-categoriser converge toward a unique solution v * ∈ [0, 1] n , which is the limit of the sequence of {v (k) } ∞ k=0 defined from an arbitrary selected v (0) in [0, 1] n and , for each k ≥ 1, generated by

v (k) = F (v (k-1) ) (3)
Proof. First, let us consider that u (0) = (0, . . . , 0), u (1) = F (u (0) ) = (1, . . . , 1) and u (k) = F (u (k-1) ) for each k ≥ 2.

We can easily check that

u (0) ≤ u (2) ≤ u (1)
(4) and that there exists 0 < ϕ < 1 such that ϕu (1) ≤ u (2) (5) This is true because every element of u ( 2) is strictly positive and ϕ can be initialised to the minimum element. Now, let us prove by induction that for all k ≥ 0, the following statement holds

u (0) ≤ u (2) ≤ • • • ≤ u (2k) ≤ • • • ≤ u (2k+1) ≤ • • • ≤ u (3) ≤ u (1) (6)
Base case: We showed in (4) that u (0) ≤ u (2) ≤ u (1) . Inductive step: Suppose that u 2k+1) .

(0) ≤ u (2) ≤ • • • ≤ u (2k) ≤ • • • ≤ u (2k+1) ≤ • • • ≤ u (3) ≤ u (1) is true. We need to prove that u (2k) ≤ u (2k+2) ≤ u (2k+3) ≤ u (
First, we show that u (2k+2) ≤ u (2k+1) . Since u (2k) ≤ u (2k+1) and that F is non increasing, we deduce that F (u (2k+1) ) ≤ F (u (2k) ) and that u (2k+2) ≤ u (2k+1) . Likewise, we show u (2k) ≤ u (2k+2) using the same reasoning.

Second, we show that u (2k+2) ≤ u (2k+3) ≤ u (2k+1) . Since we prove in the previous step that u (2k+2) ≤ u (2k+1) and u (2k) ≤ u (2k+2) , we deduce using the fact that F is non increasing that u (2k+2) ≤ u (2k+3) and u (2k+3) ≤ u (2k+1) respectively. This concludes the proof by induction.

From ( 5) and ( 6), we find that there exists ϕ such that ϕu (2k-1) ≤ u (2k) for each k ≥ 1. Now, let us denote

π k = sup{π such that πu (2k-1) ≤ u 2k) }. Then, 0 < ϕ ≤ π 1 ≤ • • • ≤ π k ≤ • • • ≤ 1. We now show that lim k→∞ π k = 1. We first show f i (πu) = 1 π+fi(u)(1-π) f i (u) for all i ∈ {1, 2, . . . , n}. f i (πu) = 1 1 + S∈R - 1 (ai) min aj ∈S f j (πu) = 1 1 + π S∈R - 1 (ai) min aj ∈S f j (u) = 1 + S∈R - 1 (ai) min aj ∈S f j (u) 1 + π S∈R - 1 (ai) min aj ∈S f j (u) × f i (u) = 1 1+π S∈R - 1 (a i ) min a j ∈S fj (u) 1+ S∈R - 1 (a i ) min a j ∈S fj (u) × f i (u) = 1 (1+ S∈R - 1 (a i ) min a j ∈S fj (u))π-π+1 1+ S∈R - 1 (a i ) min a j ∈S fj (u) × f i (u) = 1 π + 1-π 1+ S∈R - 1 (a i ) min a j ∈S fj (u) × f i (u) = 1 π + (1 -π)f i (u) × f i (u)
Then, there exists 0 < α < 1 and a continuous function

ψ(π) = 1 π+α(1-π) such that F (πu) ≤ ψ(π)F (u), ∀π ∈ [ϕ, 1[, u ∈ [ϕ, 1] n (7)
Then, we show that

u (2k+1) = F (u (2k) ) ≤ F (π k u (2k-1) ) ≤ ψ(π k )u (2k) ≤ ψ(π k )u (2k+2) (8)
By definition of π k , it holds that π k u (2k-1) ≤ u (2k) and since F is non decreasing, we deduce that F (u (2k) ) ≤ F (π k u (2k-1) ). Using (7), we conclude that 2k) . Using (6), we have that u (2k) ≤ u (2k+2) and π k u (2k) ≤ π k u (2k+2) . Now, we show that π k+1 ≥ 1 ψ(π k ) by contradiction. Suppose that this π k+1 < 1 ψ(π k ) . Using (8), we have that 2k+2) . Contradiction with the definition of π k+1 .

F (π k u (2k-1) ) ≤ π k F (u (2k-1) ) = π k u (
u (2k+1) ≤ ψ(π k )u (2k+2) and thus 1 ψ(π k ) u (2k+1) ≤ u (
We now show that

1 -π k+1 ≤ (1 -α)(1 -π k ) ≤ . . . ≤ (1 -α) k (1 -π 1 ) ≤ (1 -α) k (1 -ϕ) (9)
Using the previous result, we have that

π k+1 ≥ 1 ψ(π k ) = π k + α(1 -π k ) απ k -α -π k + 1 ≥ 1 -π k+1 (1 -α)(1 -π k ) ≥ 1 -π k+1
This is sufficient to prove (9) by noticing that ϕ ≤ π 1 . As 0 < α < 1, thus by (9) we have

lim k→∞ (1 -π k+1 ) = 0 ⇒ lim k→∞ π k = 1 (10)
There by (6), we get for any integer p ≥ 1

0 ≤ u (2k+2p) -u (2k) ≤ u (2k+1) -u (2k) ≤ (1 -π k )u (2k+1) ≤ (1 -π k )u (1) (11) Since [0, 1] n is normal, both {u (2k+1) } ∞ k=0 and {u (2k) } ∞
k=1 are convergence sequences. By ( 10) and (11), thus, there exists 2k+1) . Letting k → ∞ and combining with (12), it follows that F (u * ) = u * (it is a fixed point of F .

u * ∈ [0, 1] n such that lim k→∞ u (2k+1) = lim k→∞ u (2k) = u * (12) Hence, u (2k) ≤ u * ≤ u (2k+1) and u (2k) ≤ F (u * ) ≤ u (
We now show that the result holds for any arbitrary v (0) ∈ [0, 1] n . By induction, we have that for any k ≥ 1, u (2k) ≤ v (2k) ≤ u (2k-1) and u (2k) ≤ v (2k+1) ≤ u (2k+1)

v (k) → v * = u * as k → ∞. In particular, let v (0) = w * , where w * is any fixed point of F in [0, 1] n , then v (k) = w * for all k ≥ 1, and we get w * = u * . So F has a unique fixed point in [0, 1] n .
In Table 1, we show that nh-categoriser satisfies the same properties as h-categoriser. The proof is dropped because of lack of space.

Our goal is achieved as the nh-categoriser is a proper instantiation of the h-categoriser for hypergraph argumentation frameworks and satisfies the same properties as the regular h-categoriser. The set of properties of this paper can be used to select the most suitable ranking-based semantics for a specific task. There has been a lot of discussion about whether some properties are more desirable than others, for instance, the satisfaction of Abstraction is considered essential for most ranking-based semantics whereas other properties, such as Increase of attack branch, are considered optional.

In the next proposition, we show that there always exists a regular Dung's abstract argumentation framework such that the score on arguments with the h-categoriser are the same as the scores on arguments in the hypergraph argumentation framework with the nh-categoriser.

Definition 10 (Abstract argumentation framework). An abstract argumentation framework is a pair AS = (A , R ), where A is a finite and non-empty set of arguments and R ⊆ A × A is a set of binary attacks. Proof. This proof is an original work and is split in two parts: First, we exhibit how to build an abstract argumentation framework AS from an arbitrary hypergraph argumentation framework AS. Second, we show that aforementioned AS is such that Proposition 3 is satisfied.

• 1-Let AS = (A, R) be an arbitrary hypergraph argumentation framework. We build the abstract argumentation AS = (A , R ) such that:

-A = {a 1 , . . . , a m } ∪ {a m+1 , . . . , a n } -There is a bijection γ 1 from A to {a 1 , . . . , a m } and a bijection γ 2 from R to {a m+1 , . . . , a n }. -For every (S, t) ∈ R, we create the attack (γ 2 ((S, t)), γ 1 (t)) in R . -For every (S, t) ∈ R, we consider the set X S = {s ∈ S | for every s ∈ S, s nh AS s} and we chose x S ∈ X S . Then, for every S ∈ R - 1 (x S ), we create the attack (γ 2 ((S , x S )), γ 2 ((S, t))) in R .

• 2-We now show that the constructed argumentation framework AS is such that Proposition 3 is satisfied. Let X = {a m+1 , . . . , a n } and AS = (A , R ) be the restriction of AS to X , i.e. A = X and R = {(a, b) ∈ R |a ∈ X and b ∈ X }.

We now define the function f : X → R that returns for every element a in X , the minimum score of the attackers associated with the attack corresponding to a.

Namely, for every a ∈ X , f (a) = min a ∈W C(a ) where (W, t) = γ -1 2 (a) and γ -1 2 is the inverse function of γ 2 . We now prove that for every a ∈ A , it holds that f (a) = 2 (a). By construction, we know that there exists

x W ∈ W such that {b 1 , . . . , b p } = {γ 2 ((S , x W ))|S ∈ R - 1 (x W )}. Thus, f (a) = min a ∈W C(a ) f (a) = C(x W ) f (a) = 1 1 + S ∈R - 1 (x W ) min s ∈S C(s ) f (a) = 1 1 + (b,a)∈R f (b)
Since f satisfies the same formula as h-categoriser then f is identical to h-categoriser on AS [START_REF] Pu | Argument Ranking with Categoriser Function[END_REF]. Let us now consider the whole argumentation graph AS . Since there are no attacks from arguments in {a 1 , . . . , a m } to arguments in {a m+1 , . . . , a n }, the hcategoriser gives the same scores to {a m+1 , . . . , a n } in AS and in AS . Since for every argument a in {a 1 , . . . , a m }, the set of attackers of a in AS is {b ∈ {a

m+1 , . . . , a n } | γ -1 2 (b) = (S, γ -1 1 (a)) with S ∈ R - 1 (γ -1 1 (a)
)}, we have for every a ∈ {a 1 , . . . , a m }:

C (a) = 1 1 + b∈Att(a) C (b) C (a) = 1 1 + b∈Att(a) f (b) C (a) = 1 1 + b∈Att(a) min a ∈W C(a )
where

(W, t) = γ -1 2 (b) C (a) = 1 1 + S∈R - 1 (γ -1 1 (a)) min s∈S C(s) C (a) = C(γ -1 1 (a))
Proposition 3 is important as it indicates the computational relation between the nh-categoriser and the hcategoriser. Moreover, this proposition shows how one is able to move from an argumentation hypergraph to a regular argumentation directed graph without loss of generality. We would like to point out that for a hypergraph argumentation framework, there might be several abstract argumentation frameworks that satisfy Proposition 3.

Please note that although it seems intuitive that restricting the sets of attacking arguments to a binary attack from the weaker argument of the set would result in the same score with the h-categoriser than with the nh-categoriser, this idea does not work. For illustration purposes, we present one simple counter-example against this intuition. Consider the argumentation framework AS = (A, R) with such that A = {a, b, c, d} and R = {({d}, b), ({b, a}, c), ({d, b}, c)}. The score of the argument c in AS using the nh-categoriser is C(c) = 0.5. Now, if we use the aforementioned approach, we obtain the binary argumentation framework AS = (A, R ) such that R = {(b, c), (d, b)}. In AS , the score of c with the h-categoriser is C (c) = 0.66, which is not equal to the score of c given by the nh-categoriser. The approach described in the proof of Proposition 3 enables us to obtain a strength of 0.5, which is the strength of c using the nh-categoriser.

Discussion and Related Work

We introduced the notion of ranking-based semantics for argumentation frameworks with sets of attacking arguments (also called hypergraph argumentation frameworks). We motivated our work rooted in the need to account for extensions of classical Dung graphs encoding arguments over a logical language not necessarily closed under the "and" logical connective. In that context, our work aligns with existing works in the literature that extend ranking-based semantics to weighted argumentation framework [START_REF] Amgoud | Acceptability Semantics for Weighted Argumentation Frameworks[END_REF], support argumentation frameworks [START_REF] Amgoud | Evaluation of Arguments from Support Relations: Axioms and Semantics[END_REF] or bipolar argumentation frameworks [START_REF] Amgoud | On bipolarity in argumentation frameworks[END_REF]. Our contributions consist in (1) providing a set of 18 translated properties for ranking-based semantics that are directly applicable in the context of hypergraph argumentation framework, (2) introducing the nhcategoriser as the first ranking-based semantics for hypergraph argumentation framework, (3) giving the complete proof of the existence of a unique solution for the nhcategoriser and (4) bridging the link between ranking arguments in hypergraph argumentation frameworks and ranking arguments in the usual Dung's framework.

The focus of the argumentation community have recently been shifted to argumentation frameworks with sets of attacking arguments [START_REF] Flouris | A comprehensive study of argumentation frameworks with sets of attacking arguments[END_REF] and we believe that this paper will be of interest for the community.

Although the work of (Flouris and Bikakis 2019) completes the characterisation of such argumentation frameworks with the treatment of various interesting semantics not considered in the original publication [START_REF] Nielsen | A Generalization of Dung's Abstract Framework for Argumentation: Arguing with Sets of Attacking Arguments[END_REF] and functions allowing the transition between extensions and labellings along with their properties, it does not consider ranking-based semantics. To the extend of our knowledge, this paper is the only work that has been interested in the topic of applying ranking-based semantics to this setting.

Figure 1 :

 1 Figure 1: A hypergraph argumentation framework

  arguments obtained by the nb-categoriser is denoted by nh AS , where for every a, b ∈ A, a nh AS b iff C(a) ≥ C(b). Example 3 (Example 1 Cont'd). The nh-categoriser scores of arguments are C(a) ≈ 0.38, C(b) = 1, C(c) = 0.5, C(d) ≈ 0.65 and C(e) ≈ 0.53. We obtain the ranking: b nh AS d nh AS e nh AS c nh AS a.

  Notation. Let AS = (A , R ) be an abstract argumentation framework and a∈ A , Att(a) = {b ∈ A | (b, a) ∈ R }.Definition 11 (h-categoriser). Let AS = (A , R ) be an abstract argumentation framework. The h-categoriser is the function C : A → [0, 1] defined as, for all a ∈ A : Dung Equivalent Existence). Let AS = (A, R) be a hypergraph argumentation frameworks. Then, there exists at least one abstract argumentation framework AS = (A , R ), a subset {a 1 , . . . , a m } ⊆ A and a bijection γ : A → {a 1 , . . . , a m } such that for every a ∈ A, C(a) = C (γ(a)).

  f (b) . Let a be an arbitrary argument in A , {b 1 , . . . , b p } the set of attackers of a in AS and (W, t) = γ -1

Figure 2 :

 2 Figure 2: An abstract argumentation framework

Table 1 :

 1 Properties satisfied by the nh-categoriser

	. Then

Acknowledgments

We acknowledge the support of the DOCAMEX project.


Faustine Cantalloube

Olivier Absil

Thomas Bertram

Wolfgang Brandner

Christian Delacroix

Markus Feldt

Matthew A C Kenworthy

Martin Kulas

Julien Milli

Philip Neureuther

Orban De Xivry

High contrast imaging with ELT/METIS: The wind driven halo, from SPHERE to METIS

Keywords: AO4ELT-6 proceeding, ELT/METIS, Adaptive Optics, High-contrast imaging, Coronagraphy, Post-processing

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

High-contrast imaging (HCI) is one of the main challenges for future astronomical observation, especially to study exoplanet formation and evolution within their environment. In the context of the European Extremely Large Telescope (ELT) that is currently under construction at Cerro Armazones (Chile), all of the three first light instruments hold a high-contrast imaging mode. To achieve the contrast required for exoplanet detection, they are all equipped with specific coronagraphic technologies, working under adaptive optics (AO) correction, delivering images on which advanced post-processing techniques will be applied to further carve out the contrast.

The Mid-Infrared ELT Imager and Spectrograph, METIS 1, 2 * , is one of the three first light instruments of the ELT whose main science goal is to observe exoplanets and circumstellar disks. [START_REF] Quanz | Direct detection of exoplanets in the 3-10 µm range with e-elt/metis[END_REF] It is working in the thermal infrared (from L-band, at 3.7 µm to Q-band, at 19 µm) and equipped with a single conjugated adaptive optics (SCAO) system [START_REF] Bertram | Single conjugate adaptive optics for metis[END_REF][START_REF] Hippler | Single conjugate adaptive optics for the elt instrument metis[END_REF] analyzing the atmospheric turbulence with a pyramid wavefront sensor working in the Kband (2.2 µm), and with five types of coronagraphs [START_REF] Kenworthy | A review of high contrast imaging modes for metis[END_REF] designed to reach a L-band contrast of 3 × 10 -5 at 5 λ/D (requirement) to 1 × 10 -6 at 2 λ/D (goal). METIS has recently gone through its preliminary design review (PDR) in May 2019, and its first light at the ELT is due in 2026. Under good observing conditions, the highcontrast capabilities of the latest generation of HCI instruments are limited by non-common path aberrations (NCPA) resulting in the presence of bright speckles in the images hindering the presence of planetary companions. However, other limitations exist due to the structure of the telescope or due to turbulence conditions interacting with the whole instrument. [START_REF] Cantalloube | Peering through SPHERE Images: A Glance at Contrast Limitations[END_REF] Among those limitations, the wind driven halo (WDH) is an elongation of the point spread function (PSF) along the direction of the equivalent wind at the telescope pupil, that is revealed as a Further author information: (Send correspondence to F.C.) F.C.: E-mail: cantalloube@mpia.de, Telephone: 49 622 123 1234 butterfly-shaped pattern in the high-contrast images. This elongation occurs when the atmospheric turbulence is varying too fast to be fully corrected by the AO system. This effect has been clearly seen on the latest generation of HCI instruments such as VLT/SPHERE [START_REF] Beuzit | SPHERE: the exoplanet imager for the Very Large Telescope[END_REF][START_REF] Cantalloube | Origin of the asymmetry of the wind driven halo observed in highcontrast images[END_REF] or Gemini-S/GPI. [START_REF] Macintosh | The Gemini Planet Imager: from science to design to construction[END_REF][START_REF] Madurowicz | Asymmetries in adaptive optics point spread functions[END_REF] According to turbulence profiling data from Stereo-SCIDAR measurements, [START_REF] Osborn | Optical turbulence profiling with stereo-scidar for vlt and elt[END_REF] this situation happens up to 40% of the observing time at the Paranal observatory. The WDH is illustrated in Fig. 1 showing simulations with an ideal coronagraph [START_REF] Cavarroc | Fundamental limitations on Earth-like planet detection with extremely large telescopes[END_REF][START_REF] Sauvage | Analytical expression of long-exposure adaptive-optics-corrected coronagraphic image. first application to exoplanet detection[END_REF] for a SPHERElike system (left) and METIS-like system (right), in which only the contribution from AO residuals is taken into account. After introducing the origin of the WDH in the following section, we present simulated images for a SPHERElike instrument and for a METIS-like instrument to evaluate the impact of the WDH in the raw contrast reached in the delivered images when going from one to the other system.

THE WIND DRIVEN HALO

The wind driven halo (WDH) is due to the interaction between the turbulence decorrelation speed, parametrized by the turbulence coherence time, and the finite speed at which the AO loop is running, inducing delays in the correction, provoking the so-called AO servolag (or temporal bandwidth) error.

The adaptive optics delay

For an AO system running in closed loop with a classical integrator law, the AO servolag error is due to the delay between the measurement of the turbulent phase at the wavefront sensor (WFS) detector and the effective correction by the deformable mirror (DM). It is therefore dependent on the WFS integration time, the real time computer latency and the time the DM takes to be fully in shape. The AO loop frequency is the frequency at which a new turbulent phase is integrated at the WFS detector. The shape of the DM is therefore refreshed at the same frequency. For the VLT/SPHERE instrument, the routine AO loop frequency used for bright targets is 1380 Hz (set by the integration time needed at the WFS detector) and the AO-loop delay has been measured at 2.3 frames. For the ELT/METIS instrument, the AO loop frequency is limited to 1000 Hz (set by the maximum running frequency of the M5 DM) and, in the following, we assume a frame delay of 2.

If the atmospheric turbulence is varying faster than the AO loop delay, the corrected phase (or AO residual phase) will show strong low spatial frequency residuals.

Temporal variation of the atmospheric turbulence

The turbulence coherence time characterizes the temporal evolution of the atmospheric turbulence. It is defined [START_REF] Roddier | The effects of atmospheric turbulence in optical astronomy[END_REF] as τ 0 = 0.314 r 0 /v eq , where r 0 is the Fried parameters (defined at 500 nm and at zenith) and v eq is the equivalent wind velocity at the telescope pupil, that is to say the integrated wind speed weighted by the index structure constant (C 2 n ). [START_REF] Roddier | The effects of atmospheric turbulence in optical astronomy[END_REF] At the Paranal observatory, the median coherence time is 4.6 ms based on MASS-DIMM historical data from April 2016 to April 2019. At the Cerro Armazones, the median coherence time is 3.5 ms. [START_REF] Vernin | E-elt site characterization status[END_REF] When the turbulence coherence time is shorter than or close to the AO loop delay, the AO corrected phase shows strong low spatial frequency phase residuals along the equivalent wind direction. When propagated to the focal plane, those directional phase residuals provoke an elongation of the PSF, along the direction of the equivalent wind, at a contrast of up to 10 -4 . This scattered light, forming the WDH, is therefore revealed only when using a coronagraphic device that cancels most of the on-axis starlight (see Fig. 3).

When modeling the atmosphere as a finite number of turbulent layers located at different altitudes, under the Taylor hypothesis (the frozen flow hypothesis), the main temporal variation is due to the wind that translates each layer at a different speed and direction. Since high equivalent wind velocities are the main responsible for the WDH, the WDH appears most of the time in the presence of strong jet stream, a thin layer of a few kilometers width, located at about 12 km above sea level (near the tropopause) and whose speed can go from 20 up to 50 m/s, flowing west to east. We indeed observed on VLT/SPHERE 8 high-contrast images that the WDH is almost always aligned with the direction of the subtropical jet that crosses the Paranal observatory location and is tightly correlated to the presence of strong jet stream. Using Gemini-S/GPI 10 (located at Cerro Pachon, Chile) high-contrast images, a team correlated the WDH direction to the different wind layer direction and came to the same conclusions. [START_REF] Madurowicz | Characterization of lemniscate atmospheric aberrations in gemini planet imager data[END_REF] In addition, we observed in high-contrast images delivered by the latest generation of HCI instrument (equipped with both extreme AO, XAO, and advanced coronagraph designs) that the WDH is asymmetric, one wing of the butterfly-shaped pattern being wider and brighter than the other. We recently found out that this asymmetry is due to interference between delayed phase error and amplitude errors. [START_REF] Cantalloube | Origin of the asymmetry of the wind driven halo observed in highcontrast images[END_REF] The delayed phase error is caused by the AO servolag and the amplitude errors are caused by the scintillation effect that are phase errors translating into amplitude error due to Fresnel propagation through the atmosphere. Even though the scintillation effect is negligible for 8-m telescopes working in the near infrared (contrast of more than 10 -6 ), this is still enough to create this asymmetry that any type of coronagraph would reveal. This asymmetry increases with the scintillation, that is to say with smaller telescope diameter, larger wavelength and lower observing site (higher atmospheric layers). As this asymmetry is due to interference between the delayed phase error and the amplitude errors, it increases when the correlation between delayed phase error and amplitude errors increase, that is to say with slower equivalent wind velocity and/or smaller AO loop delay.

Power spectral density of the AO residuals

To illustrate the explanations from previous sections, Fig. 2 shows the phase (upper row) and the corresponding power spectral densities (lower row) for a VLT/SPHERE-like XAO system observing in H-band (centered at 1.6 µm). From left to right, Fig. 2 shows the case without AO correction (seeing limited), with an ideal AO correction (only limited by the fitting error due to the limited number of DM actuators), the case without AO servolag error (showing other AO errors such as chromaticity, anisoplanetism, aliasing and noise propagation), and the cases with servolag with or without its interference with amplitude errors.

Effect of the AO servolag error in high-contrast imaging

The contribution of the AO servolag error in the image is expected to be 10 4 fainter than the star. Consequently, without coronagraph, this extension of the PSF is completely hidden by the starlight. However, when using a coronagraph coupled with extreme AO, a raw contrast of 10 -4 can be reached in the AO-corrected area, thus revealing the wind driven halo very clearly in the image. Figure 3 shows the example of simulated (top row) and on-sky images (bottom row) of the SPHERE instrument that is equipped with an XAO and an apodized Lyot coronagraph [START_REF] Soummer | Apodized Pupil Lyot Coronagraphs for Arbitrary Apertures. III. Quasi-achromatic Solutions[END_REF][START_REF] Martinez | Design, analysis, and testing of a microdot apodizer for the apodized pupil lyot coronagraph[END_REF] (APLC): the non-coronagraphic image (left column) shows no hint of the PSF elongation whereas the coronagraphic image (right column) reveals clearly the wind driven halo. As shown in Fig. 3, the simulations of SPHERE images are consistent with the obtained on-sky images. In the following we therefore extrapolate from a SPHERE-like system to a METIS-like system to show how the raw contrast reached in the resulting images is affected by the WDH.

FROM VLT/SPHERE TO ELT/METIS

In this section, the effect of the wind driven halo observed in SPHERE images is predicted for a METIS-like system to verify how this error term will finally affect the raw contrast. To do so, we rely on analytical AO simulations [START_REF] Jolissaint | Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach[END_REF] and, in order to focus on the effect of the wind driven halo only, we only take into account the AO residuals in the image (i. e. we ignore NCPA or other exogenous error terms). From the residual phase screens produced, we simulate ideal coronagraphic images so that we are not biased by the type of coronagraph used as the coronagraphs designed for SPHERE and METIS are quite different and therefore show different inner working angle, response and limitations.

To produce the AO residual phase screens, we use two wind profiles showing a typical case with and without a strong jet stream. These wind profiles are real profiles measured at the Paranal observatory during operation on July 23 rd 2015, by the European Center for Medium-Range Weather Forecasts (ECMWF † ). The wind profiles consist of 19 layers distributed between 0.15 km to 26.5 km shown in Fig. 4. Using the altitude measured by the ECMWF, the corresponding Cn 2 profiles have been interpolated from the 35-layered median Cn 2 profile for the Cerro Armazones site provided by ESO (ESO-258292). To simulate the AO residuals for both instruments, the same profiles are used, assuming a target star of magnitude 4.5 in K-band and 4.7 in I-band (similar to the star 51 Eri), located at a zenith angle of 30 o , and with a seeing of 0.65" (equivalent to r 0 (30 o ) = 84 cm for METIS and r 0 (30 o ) = 21 cm for SPHERE). From these profiles, the turbulence coherence times are therefore 21 of about 11.4 ms in the case of absent jet stream and about 2.1 ms in the case of strong jet stream.

In the AO simulations, the frame delay is set to two so that the total delay of the AO loop is of 1.5 ms for SPHERE and 2 ms for METIS. We therefore expect to observe the WDH from a turbulence coherence time of about 2 ms and below. In practice, at Paranal observatory, we clearly observe the WDH on SPHERE images as soon as the coherence time is below 2 ms. n (left), wind speed (middle) and wind direction (right) used for the simulations. Two turbulence conditions are presented: with strong jet stream layers (red solid lines) and without jet stream layers (green solid lines). In both cases the same wind direction is used (orange solid line).

Table 1 gathers the fundamental differences between the VLT/SPHERE and ELT/METIS systems that have a significant impact on the AO-residuals. In the following we describe and simulate the effect of each of these differences on the wind driven halo. For both cases we considered a high-order DM with actuators distributed on a regular grid showing Gaussian influence functions. To simulate the high-contrast images from the AO residual phase screens, we considered the telescope pupil (including the central obstruction and spiders) and simulated images with an ideal coronagraph (without any undersized Lyot stop). In the ELT case, we ignored the segmentation of the primary mirror and considered a circular pupil in the inscribed circle of the actual ELT pupil. The simulated images are purely monochromatic at the observing wavelength. The resulting high-contrast images simulated for both systems under strong jet stream conditions are shown in Fig. 1. These two images are obtained from a stack of 200 instantaneous images generated from random AO residual phase screens. No photon noise is taken into account since the target star is bright enough and neither readout noise nor thermal background is taken into account.

Table 1. Main differences between the VLT/SPHERE and ELT/METIS (as of the current design for the ELT and for the METIS SCAO) systems that are taken into account for the simulations to highlight the effect of each parameters to the wind driven halo.

Parameters

VLT

Variation of the raw contrast in presence of WDH

In this section we derive the raw contrast as a function of the separation to the star, obtained from ideal coronagraphic images of infinite exposure simulated by accounting only for the AO residuals under the conditions of strong jet stream. Starting from the SPHERE-like instrument working in the H-band, we changed the parameters one-by-one to see the effect of each key parameter, until we reach the METIS-like configuration. Figure . 5 shows the mean azimuthal profile for each case.

1. When going from a telescope diameter of 8 m to 37 m, as expected we reach a higher angular resolution, but the small number of actuators with respect to the size of the pupil create a dark hole at worse contrast (red line);

2. When increasing the number of DM actuators (and sub-pupils of the WFS), we recover a better contrast at all separations and the cutoff frequency is almost located at the same physical separation (orange line);

3. When decreasing the AO loop frequency, the contrast decreases as expected since the AO servolag error, creating the WDH, is higher (yellow line);

4. When sensing at larger wavelength (from I to K-band), we suffer from less chromaticity hence the contrast increases at short separation (green line); 5. When going from a Shack-Hartmann wavefront sensor (SH-WFS) to a Pyramid wavefront sensor (Pyr-WFS), we observe an increase of the contrast in the dark hole since the Pyr-WFS is more sensitive, shows smaller noise propagation and is less sensitive to aliasing (blue line); 6. At last, when imaging at larger wavelength, from the H-band to the L-band, we observe a significant increase of the contrast in the dark hole since larger wavelengths are less affected by atmospheric turbulence (purple line). In that configuration, compared to SPHERE, for the same physical separation to the star, METIS gains slightly more than an order of magnitude in raw contrast. To highlight the effect of the AO servolag error in the raw contrast, Fig. 6 shows the raw contrast profiles for SPHERE and METIS in the direction of the WDH and in the direction perpendicular to the WDH (solid lines), to be compared to the raw contrast reached without AO servolag error (dash-dotted lines). At small separations, the gain for SPHERE is not obvious since the AO of SPHERE is dominated by chromaticity and noise propagation. However for METIS, the gain is non-negligible when no AO servolag error is present (almost one order of magnitude in the case without WDH, to two order of magnitude with WDH). This further confirms that the AO servolag error is dominating the raw contrast budget at small angular separation for METIS.

The requirement for METIS is to reach a post-processed contrast of 3.10 -5 at 5λ/D (103 mas) and the goal is to reach a post-process contrast of 1.10 -6 at 2λ/D (41 mas) in L-band. The AO servolag error is therefore not a limiting factor to reach the requirement, but it is a limiting factor for reaching the goals, unless we remove the WDH contribution, by either a better control scheme or by post-processing techniques. Overall, in terms of AO residuals, only the AO servolag error is an issue to reach the contrast goal for METIS, the other AO error terms are negligible to reach the goal.

Amount of WDH in the focal plane images

To account for the amount of WDH present in the images, we compare the power spectral densities (PSD) obtained with and without the AO servolag error, in the cases of METIS and SPHERE, for the two typical observing conditions with and without strong jet stream. Compared to the PSD, the WDH in the resulting focal plane image would be attenuated by the effect of the coronagraph (and is therefore coronagraph-dependent), but at separation larger than the coronagraph inner working angle, the PSD shows the absolute amount of light scattered into the WDH for a long exposure time.

Figure 7 shows the simulated PSD produced with weak (green upper box) and strong jet stream (red lower box) for both SPHERE-like (blue boxes) and METIS-like (purple boxes) systems. In the case of SPHERE, without the strong jet stream, 15.0% of the starlight within the corrected area is produced by the AO servolag error (about 2.1% of the starlight within the total field of view), and 69.7% with the strong jet stream (about 21.5% within the total field of view). In the case of METIS, without the strong jet stream, 51.3% of the starlight within the corrected area is produced by the AO servolag error (about 0.6% of the starlight within the total field of view), and 93.3% with the strong jet stream (about 8.0% of the starlight within the total field of view). Comparison of the raw contrast obtained with an ideal coronagraph as a function of the separation for a SPHERE-like instrument (blue lines) and a METIS-like instrument (purple lines). In the ideal cases without AO servolag error (dash-dotted lines), the profiles are the azimuthal average of the image. In presence of AO servolag error, the shaded area shows the raw contrast from the azimuthal minimum (perpendicular to the WDH) to the azimuthal maximum (along the wind direction). The goal and requirement of METIS are highlighted (resp. green and yellow lines), as well as the cutoff frequencies (vertical dotted lines) for SPHERE (843 mas in H-band) and METIS (773.5 mas in L-band).

The difference between the AO servolag contribution with respect to the total light and its contribution in the corrected area only is due to the fact that for METIS, the sub-pupils are 50 cm wide whereas they are 20 cm wide for SPHERE, making the DM fitting error a much higher contribution in the METIS case.

In the specific case without strong jet stream, the AO servolag signature is much more visible in the METIS case and is hidden by other dominant terms in the SPHERE case, as visible in Fig. 7.

In the case with strong jetstream, the AO servolag signature is indeed along the direction of the wind at the jet stream layer shown in Fig. 4. As expected, over the total amount of light, the WDH is less prominent in the METIS case compared to the SPHERE case, when taking into account all of the differences between both systems. The larger diameter of the telescope, the larger sensing wavelength and the use of a pyramid WFS do decrease the AO servolag error (and therefore the WDH), even though the AO loop speed is slightly slower for METIS than for SPHERE. In the corrected area however, the amount of WDH is higher for METIS than for SPHERE because the AO servolag error is dominant with respect to other AO errors in the case of METIS, which suffers less from chromaticity, anisoplanetism, noise propagation and aliasing. Note that these simulations do not take into account exogenous errors such as pupil fragmentation, [START_REF] Bonnet | Adaptive optics at the ESO ELT[END_REF] vibrations, calibration errors, non-common path aberrations or low wind effect, and are purely monochromatic.

Asymmetry of the WDH in the focal plane images

As previously, we computed the absolute amount of asymmetry directly using the simulated PSD that are shown in Fig. 7. We define the asymmetry factor as the difference between the light contained in the bright wing and the light contained in the faint wing, divided by the total amount of light in the two wings. [START_REF] Cantalloube | Origin of the asymmetry of the wind driven halo observed in highcontrast images[END_REF] An asymmetry factor of 100% is therefore a purely asymmetric WDH, showing only one wing, whereas an asymmetry factor of 0% is a purely symmetric WDH.

In the case of SPHERE, without the strong jet stream, other errors (such as anisoplanetism and chromaticity) are dominating over the servolag and therefore no asymmetry can be observed. With strong jet stream, the asymmetry is of 21.1%. In the case of METIS, as shown previously, even without strong jet stream, the AO servolag error dominates over other sources of error but the asymmetry is only of 1.4%. With strong jet stream, the asymmetry is of 21.3%, similar to that of SPHERE. Left column shows the PSDs including the AO servolag error, the middle column shows the PSDs without the AO servolag error and the right column the difference of the two, that is to say the AO servolag error only. In each box, the upper row is the case for a SPHERE-like system (blue boxes) and the lower row is the case for a METIS-like system (purple boxes).

As explained in Sect. 2, from SPHERE to METIS, we expect the asymmetry to decrease due to the larger telescope diameter but to increase due to the slower AO loop and larger imaging wavelength. All in all, the fraction of asymmetry is similar for a SPHERE-like system in H-band and a METIS-like system in L-band when enough WDH is visible in the images. However the Cerro Armazones that will host the ELT is higher than Cerro Paranal hosting the VLT. We therefore expect the ELT to undergo less scintillation and therefore show less asymmetry.

CONCLUSIONS

The wind driven halo (WDH) is a butterfly-shaped structure observed in high-contrast images provided by the latest generation of instruments dedicated to the detection and characterisation of exoplanets and circumstellar disks. This structure shows up at a contrast of about 10 -4 and is limiting the contrast performance of the instrument, especially for imaging extended sources, for which it is not possible to apply a spatial high-pass filter. The WDH is due to the limited temporal bandwidth of the AO correction and appears when the turbulence coherence time is lower than the absolute AO loop delay. From simulations of power spectral densities of AO residuals, using an analytical AO simulator and realistic turbulence conditions with strong jet stream, we computed the amount of starlight within the corrected area that belongs to the WDH: about 70% for the VLT/SPHERE instrument, consistent to what is observed on real on-sky data, and about 93% for the ELT/METIS instrument, since the AO servolag is the dominant AO error term.

The asymmetry of the WDH is due to the interference between the delayed phase error (introduced by the AO servolag error) and amplitude errors (introduced by scintillation). From simulations of the power spectral densities of AO residuals, we computed the amount of asymmetry of the WDH: about 20% for the VLT/SPHERE instrument, consistent to what is observed on real on-sky data, and also about 20% for the ELT/METIS instrument, since the parameters at play are balancing at the end.

Currently, the post-processed contrast performance of SPHERE is limited by the WDH that has an occurrence rate ranging from 30 to 40% of the observation time, according to the seasonal occurrence of strong jet stream above Paranal observatory. For ELT/METIS, the AO error budget is dominated by the AO servolag error and, under short turbulence coherence time, this will significantly affect the contrast performance in a preferential direction, even after applying post-processing techniques. This effect can be alleviated by specific AO control scheme, such as predictive control, or by post-processing techniques adapted to deal with the WDH. Furthermore, when assessing the performance of the instrument in terms of contrast curves, as it is usually done in the highcontrast community, one should provide not only the azimuthal average of the residuals, but also the minimum and maximum contrast reached as a function of the separation to the star to account for the WDH.

PERSPECTIVES

For high-contrast instruments currently in operation, there are various ways to upgrade the instrument in order to reduce the WDH in the final images. One way is to significantly decrease the AO-loop delay, which requires a faster wavefront sensor detector, hence a more sensitive wavefront sensor, along with a faster real-time computer and faster deformable mirror response. In that framework, SPHERE+, an upgrade of SPHERE, has been recently defined and proposed at the VLT-2030 conference (June 2019, Garching, Germany) and GPI 2.0 23 has been proposed and will be commissioned at the Gemini-North telescope in 2022. Another way is to make use of predictive control algorithms, based for instance on machine learning techniques. [START_REF] Males | Ground-based adaptive optics coronagraphic performance under closed-loop predictive control[END_REF][START_REF] Jensen-Clem | Demonstrating predictive wavefront control with the keck ii nearinfrared pyramid wavefront sensor[END_REF] This solutions is already routinely used at the Subaru/SCeXAO instrument. [START_REF] Sahoo | Subaru coronagraphic extreme-ao (scexao) wavefront control: current status and ongoing developments[END_REF] At last, dedicated post-processing techniques could be developed to tackle this specific signature.

As for the ELT/METIS instrument, the present study can be refined using more atmospheric profiles and more realistic AO simulations, in order to assess how much the WDH affects the contrast under various observing conditions. This study will have a direct impact on how to optimize the observing schedule to not be limited by this effect. On the SCAO design side, control algorithms to optimize the AO loop delay are currently being defined and predictive control solutions are being explored. [START_REF] Van Kooten | Performance of AO predictive control in the presence of non-stationary turbulence[END_REF] A similar study can be made for any other instrument foreseen to be installed on giant segmented mirror telescopes (GSMT), using the turbulence parameters of the chosen observing site.

A last aspect worth mentioning, is the effect of the WDH in PSF reconstruction outside of the HCI field. In the context of ELT instruments, we are currently on the way to reach such a precision in PSF reconstruction that the elongation provoked by the WDH, even without coronagraph device, might be present in the image and slightly bias high precision measurements such as astrometric measurements. [START_REF] Beltramo-Martin | PRIME: PSF Reconstruction and Identification for Multiplesource characterization Enhancement -application to Keck NIRC2 imager[END_REF]

Figure 1 .

 1 Figure 1. Illustration of the wind driven halo in high-contrast images. Simulations of SPHERE ideal coronagraphic images in H-band (left) and of METIS ideal coronagraphic images in L-band (right), both accounting only for the AO residuals under the same turbulence conditions. The images are cropped to 384 × 384 pixels and the color scale shows the contrast in magnitude (logarithmic scale).

  high wind and scin=lla=on

Figure 2 .

 2 Figure 2. Illustration of the AO servolag error for a SPHERE-like system, with five different scenarios (from left to right): Without AO correction (seeing limited), with a perfect AO correction only limited to the finite density of the DM (fitting error), AO residuals without the servolag error, AO residuals with servolag error and strong equivalent wind speed and AO residuals additionally taking into account the interference with amplitude errors. Top row (blue): Phases projected on the 8 m diameter telescope. Bottom row (red): Corresponding power spectral densities showing the spatial frequencies distribution of the AO residual terms.

Figure 3 .

 3 Figure 3. Illustration of the effect of the coronagraph revealing the wind driven halo in the in AO-corrected images. Top: Simulations of SPHERE images in broadband-H (centered around 1.6 µm) without (left) and with (right) its APLC coronagraph. Bottom: On-sky images from SPHERE in broadband-H without (left) and with (right) its APLC coronagraph.

Figure 4 .

 4 Figure 4. Profiles of C 2n (left), wind speed (middle) and wind direction (right) used for the simulations. Two turbulence conditions are presented: with strong jet stream layers (red solid lines) and without jet stream layers (green solid lines). In both cases the same wind direction is used (orange solid line).

Figure 5 .

 5 Figure 5. Raw contrast obtained from simulations of infinite exposure using an ideal coronagraph when adding one-by-one the main differences between SPHERE (thick black line) and METIS (thick purple line). All simulations account for the same atmospheric turbulence profiles shown at Fig. 4 (case of strong jet stream, in red).

Figure 6 .

 6 Figure 6. Comparison of the raw contrast obtained with an ideal coronagraph as a function of the separation for a SPHERE-like instrument (blue lines) and a METIS-like instrument (purple lines). In the ideal cases without AO servolag error (dash-dotted lines), the profiles are the azimuthal average of the image. In presence of AO servolag error, the shaded area shows the raw contrast from the azimuthal minimum (perpendicular to the WDH) to the azimuthal maximum (along the wind direction). The goal and requirement of METIS are highlighted (resp. green and yellow lines), as well as the cutoff frequencies (vertical dotted lines) for SPHERE (843 mas in H-band) and METIS (773.5 mas in L-band).

Figure 7 .

 7 Figure 7. Power spectral densities after the AO correction for the cases without jet stream (green upper box) and with strong jet stream (red lower box). Left column shows the PSDs including the AO servolag error, the middle column shows the PSDs without the AO servolag error and the right column the difference of the two, that is to say the AO servolag error only. In each box, the upper row is the case for a SPHERE-like system (blue boxes) and the lower row is the case for a METIS-like system (purple boxes).

† For more information, visit: https://www.ecmwf.int/