Guillaume Aucher
email: guillaume.aucher@irisa.fr

Expedition in the Update Universe

Dynamic epistemic logic (DEL) is a logic dealing with knowledge and belief change based on the concepts of event model and product update. The product update accounts for the way we update our knowledge and beliefs about situations when events occur. However, DEL does not account for the way we update our knowledge and beliefs about events when other events occur. Indeed, events are assumed to occur instantaneously in DEL and this idealization precludes to study this kind of update. We provide a logical analysis of updates without this assumption. It leads us to identify a graph structure for events based on their relative dependence of occurence and to introduce a generic product update. The DEL product update is a specific instance of this generic product update.

Introduction

It is commonly believed that only our knowledge and beliefs about situations can be updated, whereas our knowledge and beliefs about events cannot. This common belief implies that what we represent has always a manichean nature: on the one hand we have situations and on the other hand we have events, and the occurrence of events update our knowledge and beliefs about situations. The most prominent logical formalisms of knowledge representation and reasoning are all based on this approach [START_REF] Reiter | Knowledge in Action: Logical Foundations for Specifying and Implementing Dynamical Systems[END_REF][START_REF] Van Benthem | Logical Dynamics of Information and Interaction[END_REF].

As we shall see, this manichean distinction is not fine enough to account for the dynamics of knowledge and beliefs. In fact, our knowledge and beliefs about events can also be updated and this can be demonstrated by the following scenario. Assume that there are two barrels of wine: barrel 1 and barrel 2. Barrel 1 is being filled with wine but Ann and Bob do not know which of these barrels is being filled. Clearly, this filling of barrel 1 with wine is an event, perceived identically by Ann and Bob. Now, assume that the wine waiter privately announces to Bob that it is actually barrel 1 which is being filled. Again, clearly, this announcement is another event, perceived differently by Ann and Bob. Then, as a result of this second event, Bob knows that barrel 1 is being filled but Ann still does not know which barrel is being filled. So, Bob's knowledge and beliefs of the first event (the filling with wine) has been updated by his perception of the second event (the announcement). This scenario cannot be directly represented in DEL because only situations, and not events, can be updated by events. This stems from the assumption that events are implicitly assumed to be instantaneous in DEL, thus leading to a new situation, and our perception of an event can be updated only if this event lasts long enough, obviously. Hence, this idealization precludes the study of important logical dynamics like the one of the barrel example. However, this assumption can be perfectly removed from the DEL framework. Once we remove it, we realize that the fact that events and not only situations can be updated by other events is only the 'tip of the iceberg' and many other logical dynamics start to appear. In particular, we realize that events have an internal and rich structure based on their relative dependence of occurrence. Moreover, this structure constrains and determines the updates which are possible and a generic kind of product updates can then be identified. A contribution of this article is to provide a formal account of these logical dynamics by eliciting a series of principles. These principles will guide us for defining our formal framework.

Organization of the article. In Section 2, we briefly recall DEL. In Section 3, we analyze the structure of events by means of various examples and we elicit a number of intuitive principles about events. We use them in Section 4 for motivating our formal definitions of event structures and generic product updates. We end Section 4 by providing an example of scenario which cannot (or hardly) be modeled in DEL. We conclude and discuss our approach in Section 5.

Dynamic Epistemic Logic

Dynamic epistemic logic (DEL) is a relatively recent non-classical logic introduced by [START_REF] Baltag | The logic of public announcements and common knowledge and private suspicions[END_REF]. It extends ordinary modal epistemic logic [START_REF] Hintikka | Knowledge and Belief, An Introduction to the Logic of the Two Notions[END_REF] by the inclusion of event models to describe actions/events, and a product update operator that defines how epistemic models are updated as the consequence of executing actions described through event models. For more details about DEL, see [START_REF] Baltag | Logic for epistemic programs[END_REF][START_REF] Van Benthem | Logical Dynamics of Information and Interaction[END_REF].

Epistemic Models

In the rest of this article, A := {1, . . . , N } is a finite set of indices called agents and P 0 is a set of propositional letters called atomic facts which describe static situations.

Definition 1 (Language L (P)). Let P be a set of propositional letters. We define the language L (P) inductively by the following grammar in BNF:

L (P) : ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) |
j ϕ where p ranges over P and j over A. When P = P 0 , L (P 0 ) is called the epistemic language. We will use the following abbreviations: ϕ ∨ ψ := ¬(¬ϕ ∧ ¬ψ) and ϕ → ψ := ¬ϕ ∨ ψ. To save parenthesis, we use the following ranking of binding strength: j , ¬, ∧, ∨, → ( i.e., j binds stronger than ¬, which binds stronger than ∧, etc.). For example, j ¬p ∧ q → r ∨ s means (( j (¬p)) ∧ q) → (r ∨ s).

If E = {ϕ 1 , . . . , ϕ n }, we write E := ϕ 1 ∧ . . . ∧ ϕ n and E := ϕ 1 ∨ . . . ∨ ϕ n .
A (pointed) epistemic model (M, w) represents how the actual world represented by w is perceived by the agents. Atomic facts are used to state properties of this actual world. Intuitively, wR j v means that in world w agent j considers that world v might be the actual world w.

Definition 2 (P-model and epistemic model). Let P be a set of propositional letters. A P-model is a tuple M = (W, R, V ) where:

-W is a non-empty set; -R : A → 2 W ×W assigns an accessibility relation to each agent; -V : P → 2 W is a valuation which assigns a subset of W to each atomic event of P.

If w, v ∈ W , we write wR j v for (w, v) ∈ R(j), and R j (w) denotes {v ∈ W | wR j v}. We write w ∈ M for w ∈ W and (M, w) is called a pointed Pmodel. When P = P 0 , M is called an epistemic model and (M, w) is called a pointed epistemic model (w often represents the actual world). The class of pointed P-models is denoted C(P).

As one can easily notice, a P-model is an 'ordinary' Kripke model. Then, the epistemic language can be used to describe and state properties of epistemic models.

Definition 3 (Epistemic logic). Let P be a set of propositional letters. We define the satisfaction relation ⊆ C(P) × L (P) inductively as follows. In the truth conditions below, (M, w) ∈ C(P) is any pointed P-model and ϕ, ψ ∈ L (P).

M, w p iff w ∈ V (p) M, w ¬ψ iff it is not the case that M, w ψ M, w ϕ ∧ ψ iff M, w ϕ and M, w ψ M, w j ϕ iff for all v ∈ R j (w), we have M, v ϕ
We write M ϕ when M, w ϕ for all w ∈ M. If S ⊆ L (P), we write M, w S (M S) when for all ϕ ∈ S, M, w ϕ (resp. M ϕ). The triple L (P) , C(P), is a logic called the epistemic logic based on P.

The formula j ϕ reads as "agent j believes ϕ". Its truth conditions are defined in such a way that agent j believes ϕ holds in a possible world when ϕ holds in all the worlds agent j considers possible in this possible world.

Example 1 ('Barrel' example). Assume that there are two agents Ann and Bob and that there are two barrels of wine: barrel 1 and barrel 2. So, we have A := {A, B} with A standing for Ann and B standing for Bob, and P 0 := {p 0 , q 0 } with p 0 standing for 'barrel 1 is full' and q 0 for 'barrel 2 is full'. The situation is such that barrel 1 is not full and barrel 2 is full, but Ann and Bob do not know which one is full. This situation is depicted in the pointed epistemic model (M 0 , w 0 ) = (W, R, V, w 0 ) of Fig. 1 (left). We have W = {w 0 , v 0 } and the circled world w represents the actual world. Possible worlds are labeled by the propositional letters of P 0 that are true at these worlds. The accessibility relations are represented by arrows indexed by A or B: an arrow indexed by A (or B) from a world u to a world u means that (u, u ) ∈ R A (resp. (u, u ) ∈ R B ). So, we have M 0 , w 0 A (p 0 ∨ q 0 ) ∧ B (p 0 ∨ q 0 ): 'Ann and Bob both know that one of the two barrels is full'. The situation where both barrels are full and both Ann and Bob know it is represented in the pointed epistemic model (N 0 , u 0 ) of Fig. 1 (right).

Event Models

A pointed event model (E, e) represents how the actual event represented by e is perceived by the agents. Intuitively, eR α j f means that while the possible event represented by e is occurring, agent j considers possible that the possible event represented by f is actually occurring.

Definition 4 (Event model

). An event model is a tuple E = (W α , R α , Pre, Post) where:

-W α is a finite and non-empty set of possible events; -R α : A → 2 W α ×W α assigns an accessibility relation to each agent; -Pre : W α → L (P 0 ) is a precondition function which assigns to each possible event a formula of L (P 0 ); -Post : W α → (P 0 → L (P 0 )) is a postcondition function which assigns to each possible event a function from P 0 to L (P 0 ).

If e, f ∈ W α , we write eR α j f for (e, f ) ∈ R α (j), and R α j (e) denotes {f ∈ W α | eR α j f }. We write e ∈ E for e ∈ W α , and (E, e) is called a pointed event model (e often represents the actual event).

Our definition of event models corresponds to the definition of [START_REF] Hans Van Ditmarsch | Semantic results for ontic and epistemic change[END_REF]. It embeds the definition of [START_REF] Van Benthem | Logics of communication and change[END_REF] based on the notion of substitutions.

Example 2 ('barrel' example). Assume that barrel 1 is being filled with wine. Ann and Bob do not know whether it is barrel 1 or barrel 2 which is being filled. This event and its perception by the agents Ann and Bob is represented in Fig. 2 (left). We use the same notations for the possible events and the accessibility relations as in Fig. 1. The preconditions are such that Pre(e) = ¬p 0 (and Pre(f ) = ¬q 0 ): barrel 1 (resp. barrel 2) is not full when barrel 1 (resp. barrel 2) is being filled with wine. The postconditions are such that Post(e)(p 0 ) = and Post(e)(q 0 ) = q 0 (and Post(f )(p 0 ) = p 0 and Post(f )(q 0 ) = ): when the filling of barrel 1 (resp. barrel 2) terminates, barrel 1 (resp. barrel 2) is full, the other barrel remaining in the same state.

Product Update

The DEL product update of [START_REF] Baltag | The logic of public announcements and common knowledge and private suspicions[END_REF] is defined as follows. This update yields a new epistemic model M⊗E representing how the new situation which was previously represented by M is perceived by the agents after the occurrence of the event represented by E.

Definition 5 (Product update). Let M = (W, R, V ) be an epistemic model and let E = (W α , R α 1 , . . . , R α N , Pre, Post
) be an event model. We define the epistemic model M ⊗ E = (W ⊗ , R ⊗ , V ⊗ ) as follows (with the proviso that W ⊗ is not empty): for all p ∈ P 0 and all j ∈ A,

-W ⊗ := (w, e) ∈ W × W α | M, w Pre(e) ; -(v, f ) ∈ R ⊗ j (w, e) iff v ∈ R j (w) and f ∈ R α j (e); -(w, e) ∈ V ⊗ (p) iff M, w
Post(e)(p).

If (M, w) and (E, e) are pointed epistemic and event models. If M, w Pre(e), we define the pointed epistemic model (M, w) ⊗ (E, e) = (M ⊗ E, (w, e)).

Example 3. The product update of (M 0 , w 0 ) by (E, e) results in the epistemic model represented on the extreme right of Fig. 3. This epistemic model is in fact bisimilar to the epistemic model (N 0 , u 0 ) of Fig. 1. In this epistemic model, we have that N 0 , u 0 (p 0 ∧ q 0 ) ∧ A (p 0 ∧ q 0 ) ∧ B (p 0 ∧ q 0 ): 'both barrels are full and Ann and Bob both know it'.

DEL Definition 6 ([4]

). We define the language L del inductively by the following grammar in BNF:

L del : ϕ ::= p | ¬ϕ | (ϕ ∧ ϕ) | j ϕ | [E, e]ϕ
where p ranges over P 0 , j over A and (E, e) over C α . Intuitively, [E, e]ϕ reads as 'ϕ will hold after the occurence of the event represented by (E, e)' and E, e ϕ reads as 'the event represented by (E, e) is executable in the current situation and ϕ will hold after its execution'.

Definition 7 (Dynamic epistemic logic). We define the satisfaction relation ⊆ C(P 0 )×L del inductively as follows. In the truth conditions below, (M, w) ∈ C(P) is any pointed P-model and ϕ ∈ L (P).

M, w [E, e]ϕ iff if M, w Pre(e) then (M, w) ⊗ (E, e) ϕ
The other truth conditions for the Boolean and modal cases are identical to those of Definition 3. The triple L del , C (P 0 ) , is a logic called dynamic epistemic logic (DEL).

Proposition 1 ([4]

). DEL is as expressive as the epistemic logic based on P 0 .

Analyzing the Structure of Events

In this section, we discuss and analyze two examples from which we elicit a series of principles about events. These principles, whose some of them are obvious, are introduced to motivate the formal definitions of Section 4. They will allow us to show that the dependence graph of Definition 8 can be any directed graph, and not necessarily a tree or a bipartite graph for example.

In philosophy, the exact definition of an event is a moot topic [START_REF] Wilson | Action[END_REF] and we do not intend to take any stance in this debate. Here, we are not so much interested in the nature of events but rather in their logical and internal structure. Our examples will always be chosen so that events are indisputable events.

The 'Barrel' Example

Assume at time t 1 that there are two barrels of wine: barrel 1 and barrel 2. Barrel 2 is full and barrel 1 is being filled with wine, but Ann and Bob do not know which of these barrels is being filled. However, they know that one of them is full (and therefore cannot be filled) but they do not know which one. Clearly, this filling of barrel 1 with wine is an event, perceived identically by Ann and Bob. We consider two Scenarios from which we are going to elicit a series of principles.

Scenario 1. Assume that during barrel 1 is filled the wine waiter privately announces to Bob that it is actually barrel 1 which is being filled. Again, clearly, this announcement is another event, perceived differently by Ann and Bob. Then, as a result of this second event, Bob knows that barrel 1 is being filled but Ann still does not know which barrel is being filled. So, Bob's knowledge and beliefs of the first event (the filling with wine) has been updated by his perception of the second event (the announcement).

It is important to modify the event model while it is being executed, as opposed to modifying the Kripke model that is obtained by applying the event model because if we sever the relation between (w, e 1 ) and (w, e 2 ) after the update, this means that the event modeled by e 1 has already ended and therefore the update is on the resulting situation and not on the perception of the event.

What we want to model is an update about the perception of the event itself while this event is occuring, not after. This example leads us to state the following principle:

Principle 1: Our knowledge about ongoing events can be updated by the perception of other events.

Moreover, this announcement depends on the fact that barrel 1 is being filled and not on the precondition of this event, i.e. the fact that barrel 1 is not full. In this scenario, the nature of the event is "barrel 1 is being filled with wine", its precondition is "barrel 1 is not full" and its postcondition is "barrel 1 is full". This example entails that the very nature of events has to be taken into account when knowledge of events is updated by the perception of other events. This is captured by the following principle:

Principle 2: The occurrence of events sometimes depends on the nature of other events and not on their preconditions. Scenario 2. Assume at time t 2 that the wine waiter publicly announces that barrel 1 is not full. As a result of this announcement, at time t 3 , Ann and Bob both know that barrel 2 is full. From this new piece of information, they can infer at time t 5 that it is barrel 1 which is being filled and not barrel 2 (since the latter is full). Hence, from this example, we can state the following principle:

Principle 3: Our knowledge about a situation or an event can update our knowledge of another event while this other event is occuring.

Moreover, Ann and Bob may not make immediately the inference that it is barrel 1 which is being filled, but only as an afterthought at time t 4 . Hence, Principle 4: After learning a new piece of information, we do not always update immediately our knowledge to take it into account.

Note that Principle 4 is very much related to well-known problems in epistemic logic dealing with bounded rationality and logical omniscience (for more details on these problems, see [7, Chap. 9], [8, p. 157-168] or [START_REF] Halpern | Dealing with logical omniscience: Expressiveness and pragmatics[END_REF]). In fact, for some time, Ann and Bob may entertain the inconsistent possibility that barrel 2 is full and that at the same time it is being filled. So, Principle 5: We may consider possible at the same time that some event is occurring and that its precondition does not hold.

Formalizing the 'barrel' example. We can formalize the example by introducing the following sets of propositional letters:

-P 0 := {p 0 , q 0 }, where p 0 stands for "Barrel 1 is full" and q 0 for "Barrel 2 is full". -P 1 := {p 1 }, where p 1 stands for "The wine waiter truthfully announces that barrel 1 is not full". -P 2 := {p 2 , q 2 }, where p 2 stands for "Barrel 1 is being filled" and q 2 for "Barrel 2 is being filled". -P 3 := {p 3 }, where p 3 stands for "The wine waiter truthfully announces that barrel 1 is being filled".

Then, we can represent the dependence between these sets of propositional letters by the graph (P, S) of Fig. 4 (where P := {P 0 , P 1 , P 2 , P 3 } and S ⊆ P × P is defined in Fig. 4). An edge (P i , P j ) ∈ S means that the events described by P i depend on the events/situations described by P j . More precisely, an edge is set from P i to P j when the preconditions for the occurrence of any atomic event of P i depends on the truth value of formula(s) of L (P j ) or that the occurrence of the atomic events of P i will affect in some way or another the occurrence of the atomic events of P j or their knowledge and beliefs (to be more concrete, see all subsequent examples, and in particular the 'traffic lights' example).

Note that we have an arrow from P 0 to P 2 . This arrow is motivated by the example that we used to introduce Principle 3: our knowledge about a situation can also update our perception/knowledge about an ongoing event. For each edge (P , P) ∈ S, we can define (P , P)-preconditions functions Pre P ,P : P → 2 L(P) and (P , P)-postconditions functions Post P ,P : P → P-> 2 L(P) . The reading of Pre P ,P (p) := ϕ is 'the precondition of the atomic event p is ϕ'; the reading of Post P ,P (q)(p) := ϕ is 'p holds after the occurrence of the atomic event q if, and only if, ϕ held before this occurrence'.

-Pre P1,P0 (p 1 ) := ¬p 0 : the wine waiter can truthfully announce that barrel 1 is not full only if it is indeed not full. Post P1,P0 (p 1 )(p) := p for all p ∈ P 0 : the announcement of the wine waiter does not change the actual state of the world. -Pre P2,P0 (p 2 ) := ¬p 0 and Pre P2,P0 (q 2 ) := ¬q 0 : barrel 1 and barrel 2 can be filled only if they are not full. Post P2,P0 (p 2 )(p 0 ) := and Post P2,P0 (p 2 )(q 0 ) := q 0 : after the filling of barrel 1, it is full, the status of barrel 2 remains unchanged. Post P2,P0 (q 2 )(p 0 ) := p 0 and Post P2,P0 (q 2 )(q 0 ) := : after the filling of barrel 2, it is full, the status of barrel 1 remains unchanged.

-Pre P3,P2 (p 3 ) := p 2 : the wine waiter can truthfully announce that barrel 1 is being filled only if it is indeed being filled. Post P3,P2 (p 3 )(p) := p for all p ∈ P 2 : the announcement of the wine waiter does not change the actual state of the world. -Pre P3,P0 (p 3 ) := ¬p 0 : the wine waiter can truthfully announce that barrel 1 is being filled only if it is not full (so that it can indeed be filled).

Post P3,P0 (p 3 )(p) := p for all p ∈ P 0 : the announcement of the wine waiter does not change the actual state of the world. -Pre P0,P2 (p 0 ) := ¬p 2 and Pre P0,P2 (q 0 ) := ¬q 2 : if one of the barrels is full, it is not possible that it is being filled. Post P0,P2 (p )(p) := p for all p ∈ P 2 and p ∈ P 0 .

Note that there is no arrows towards P 1 nor P 3 . This is because these announcements are instantaneous, and therefore it is not possible that the agents' beliefs about them change while they are occurring, unlike the filling event of P 2 . This said, we could add arrows from, say, P 0 to P 1 if we considered that the agents can assess the truthfulness of the announcement, which can be a lie, before applying the product update on the event model, and may then revise it beforehand, based on their beliefs about the barrels.

The 'Traffic Lights' Example

We consider a naïve representation of a traffic lights system on a road. This example would be classically modeled by means of timed-automata [START_REF] Baier | Principles of model checking[END_REF], but we follow here the modeling approach of DEL to investigate what this example implies in term of representational requirements (formalized by our principles) for a DEL style modeling approach based on event models and product updates.

Assume that there are n traffic lights on a road. Each traffic light can be either 'green', 'yellow' or 'red' and only one of them at the same time. The color changes and goes from green via yellow to red and then back to green. Between any two of these states, a timer counts the time that elapses and eventually changes the traffic light from one state to the next after a certain amount of time. Then, each time the state of a light changes (from 'green' to 'yellow', from 'yellow' to 'red', or from 'red' to 'green'), the corresponding timer starts (timer 'yellow', timer 'red' or timer 'green'). Multiple timers run at the same time and they can be arbitrarily many. So, Principle 6: Arbitrary many events can occur at the same time and in parallel.

Moreover, we assume that there is a synchronization between the different traffic lights: when the 'red' timer of light k starts, the 'green' timer of traffic light k + 1 ends and the traffic light k + 1 goes to state 'yellow' (and then the yellow timer of traffic light k + 1 starts). This synchronization is set in order to ease the flow of cars on the road so that cars do not stop at each traffic light.

When a pedestrian comes at traffic light k and presses the 'crossing button', the 'green' timer changes its timer mode and goes to another mode in order to shorten the amount of time that the pedestrian will have to wait. As a result, and in order to synchronize the other traffic lights on the road, the timer mode of traffic light k + 1 also changes mode if its timer is currently in its usual 'green' mode, so as to keep the synchronization between the different traffic lights. (To be really precise, this change of timer mode of traffic light k should also affect the timer mode of traffic light k -1 in order to keep the system fully synchronized.) Hence, Principle 7: There can be an arbitrarily long chain of events, each event depending on the occurrence of the previous one.

Moreover, if the pedestrian presses the 'crossing button' of traffic light k when it is green, the new green timer mode will affect not only the green timer mode of traffic light k + 1 but also the color of traffic light k (when the new green timer of traffic light k ends). Therefore, Principle 8: The occurrence of an event can have effects on multiple situations or types of events.

Formalizing the 'Traffic lights' example. We can formalize the example by introducing the following sets of propositional letters: for all k ∈ {1, . . . , n}, -P k := gTimer k , yTimer k , rTimer k , gTimer k , where gTimer k (resp. yTimer k , rTimer k , gTimer k ) stands for "the green (resp. yellow, red, modified green) timer of traffic light k is running".

-P n+k := {Press k },
where Press k stands for "a pedestrian is pressing the crossing button of traffic light k while it is green".

-P 0 := {green k , yellow k , red k | k ∈ {1, . . . , n}},
where green k (resp. yellow k , red k ) stands for "traffic light k is green (resp. yellow, red)".

Then, we can represent the dependence between these sets of propositional letters by the graph (P, S) of Fig. 5 (where P := {P i | i ∈ {0, . . . , 2n}} and S ⊆ P × P is defined in Fig. 5). An edge (P , P) ∈ S means that the events described by P depend on the events/situations described by P. We spell out the precondition and postcondition functions. We only do it for the edges of the form (P k , P 0 ) and (P 0 , P k ), where k ∈ {1, . . . , n}. We define (P , P)-preconditions functions Pre P ,P : P → 2 L(P) and (P , P)-postconditions functions Post P ,P : P → P-> 2 L(P) as follows: for all k ∈ {1, . . . , n}, they determine the factual nature of situations in epistemic models. Also, we 'externalize' the precondition and postcondition functions that were fused with the event model in DEL. So, on the one hand, we have event types represented by nodes in the dependence graph and on the other hand we have the different pre/postconditions between these events. These relative pre/postconditions determine in general the different edges of the dependence graph: an edge is set from a node P to a node P when the preconditions for the occurrence of any atomic event of P depends on the truth value of formulas of L (P) or when the occurrence of the atomic events of P will affect in some way or another the occurrence of the atomic events of P. This leads us to the following definition.

Definition 8 (Dependence graph, event structure). A dependence graph is an irreflexive directed graph (P, S) such that P is a family of disjoint sets of propositional letters. These sets are called event types and their elements atomic events or facts. Let (P, S) be a dependence graph. If (P , P) ∈ S, a (P , P)-precondition function is a mapping Pre P ,P : P → 2 L(P) . We denote by PRE P ,P the set of all (P , P)-precondition functions; a (P , P)-postcondition function is a mapping Post P ,P : P → P → 2 L(P) .

We denote by POST P ,P the set of all (P , P)-postcondition functions.

An event structure (P, S, PRE, POST) is a dependence graph (P, S) together with two sets of precondition and postcondition functions PRE := {Pre P ,P ∈ PRE P ,P | (P , P) ∈ S} and POST := {Post P ,P ∈ POST P ,P | (P , P) ∈ S}.

A dependence graph is a directed graph without specific constraint except its irreflexivity. It seems natural to wonder whether it is in fact a specific kind of graph: a tree, a chain, a clique,. . . The other principles can help us answering this question. Indeed, we learn from Principle 1 that there can be more than three event types. In fact, Principle 7 even indicates us that the number of nodes in the dependence graph can be arbitrary. Moreover, from Principle 3, we infer that there can be cycles in the dependence graph and it turns out that our two examples illustrate this phenomenon. Hence, the dependence graph is in general not a tree. Finally, we learn from Principle 8 that there can be multiple outgoing edges from a given node of the dependence graph. Therefore, it is not a chain either in general. So, from this analysis, we conclude that we cannot impose any particular constraint on the definition of this dependence graph and we state that it can be any kind of directed graph.

A Generic Product Update

In this section, (P, S, PRE, POST) is an event structure and (P , P) ∈ S. Each edge of an event structure induces a product update. To define it, we first need to recover the pre/postconditions for each world of a P -model from the (P , P)pre/postcondition functions associated to an event structure.

Definition 9 ((P , P)-precondition and postcondition functions of a Pmodel). Let M := (W , R , V ) be a P -model.

-The (P , P)-precondition function of M , Pre M P ,P : W → 2 L(P) , is such that for all w ∈ W , ) , is such that for all w ∈ W , all p ∈ P,

Post M P ,P (w )(p) := p ∈P {Post P ,P (p )(p) | w ∈ V (p )} (1)
Note that the range of our precondition and postcondition functions are sets of formulas, and not single formulas like for event models (see Definition 4). This generalization of the DEL framework is meaningful. Indeed, there is no particular reason that the occurrence of an event depends on a property definable in modal logic by a single formula. The precondition of an event is implicitly determined by the class of pointed epistemic models in which this event can occur. 1 This class of epistemic models is often infinite and there is no reason that it should be definable by a single formula. In general, and especially in an infinite setting, it is quite possible that an event occurs in a class of epistemic models which is only definable by an infinite set of formulas [6, Sect. 2.6-3.3].

Definition 10 (Generic product update). Let M = (W, R, V ) be a P-model and let M = (W , R , V ) be a P -model. The (P , P)-product update of M and M is the P-model M M = (W , R , V ) defined as follows (with the proviso that W is not empty): for all p ∈ P 0 and all j ∈ A,

-W := (w, w ) ∈ W × W | M, w Pre M P ,P (w ) ; -(v, v ) ∈ R j (w, w ) iff v ∈ R j (w) and v ∈ R j (w ); -(w, w ) ∈ V (p) iff M, w
ϕ for some ϕ ∈ Post M P ,P (w )(p).

The following example is a concrete example that the standard event update cannot account for: if everything was put on the same level, we could not account for the update of events by other events and then subsequently the update of events by the situation. This example will be discussed once again in the next section.

Example 4 ('Barrel' example). In Fig. 6 we represent the generic product update that occurred in Scenario 1 of Section 3.1, whereby Ann and Bob's perception of the ongoing event was updated by their perception of another event (namely the fully private announcement to Bob that barrel 1 is being filled). In Fig. 7, we represent the Scenario 2 of Section 3.1. At each line, we represent the situations and the events that occur at the corresponding time stamp as they are defined in Scenario 2. To simplify notations, edges are represented without arrows, so the reader must assume that all arrows are bidirectional and that there are reflexive arrows indexed by all agents at each node. When there are no arrows, this means that edges are bidirectional and reflexive. We start at time t 1 with two models, one showing knowledge and beliefs concerning q 0 , p 0 , the other one concerning p 2 and q 2 (which actually represent events). Then an announcement is added (represented by w 1 : p 1 ) such that in the system state at time t 2 we have three models. Then two of the models "amalgamate" by a product update leading to the next system state at time t 3 with two models. At time t 4 , the situation updates in a backward fashion the perception of the event, leading to the final situation at time t 5 .

Discussion

One may still argue that DEL can already handle our examples. Because we deal with lasting events, what seems to be needed are propositions stating the status of events in the epistemic models, such as "(the filling of barrel 1) has ended", "(the filling of barrel 2) is still happening" together with constraints such as "(the filling of barrel 2) is still happening"→"barrel 2 is not full". Then, the standard DEL setting can also handle the updates of knowledge of events by the perception of other events. In our approach, the uncertainty of the occurrences of the events and the uncertainty of basic facts are captured in separate models initially and the updates are on each model separately. However, they can be put in the same model if the propositions about the events can be expressed in the language. In a sense, our whole framework and the event structure that we have elicited could simply be 'flattened' by adding some sort of predicates about events that would be formalized by specific propositional letters. Even if that would work out from a formal point of view, this ad hoc solution is very far to be satisfactory from a conceptual and modeling point of view. Indeed, the intuitive insights that we have elicited by means of our principles would then be disguised under the form of (meta-)predicates and constraints between propositional letters. These predicates in disguise and constraints would actually encode our dependence graphs and event structures.

One may then argue that our examples could be dealt with by existing extensions of DEL such as temporal DEL with past. In particular, Scenario 2 of the 'barrel' example could be reformulated in terms of uncertainty about which actual event history the agents are in. This kind of modeling is however subject to problems which are inherent to any state-based models such as all dynamic and process logics [START_REF] Harel | Dynamic Logic[END_REF][START_REF] Pratt | Process logic[END_REF]. It is hardly possible to express in these logics that "barrel 1 is being filled" and to model Scenario 1. One could express it in an ad hoc way by adding the propositional letter "barrel 1 is being filled" in the language, but we would need to also update its truth value when the filling ends and we would need another specific update to formalize this ending of the filling event. Likewise, Scenario 1 would be possible in temporal DEL with past only if we had that propositional letter in our model and language. In fact, we would need again to somehow encode our event structure. Moreover, the kind of modular reasoning with bounded rationality which occurs in Scenario 2 at time t 4 would not be really captured with this type of state-based and history-based logic.

One may then argue that it is not clear exactly how specific scenarios are supposed to be modeled with dependence graphs and event structures. The answer is that this problem is not inherent to our approach but applies to any modeling approach of epistemic scenarios and in particular already with epistemic and event models. There is no procedure or algorithm for constructing neither epistemic or event models nor dependence graphs or event structure. So it is a general problem of epistemic modeling. We have striven to give some guidelines that would help modelers to build their models and dependency graphs, but the general problem of how to model epistemic situations is still more at the stage of an art than a science for the moment.

We have demonstrated that the current modeling approach of DEL is not adequate enough to account for certain information dynamics. This defect should not be ignored and dismissed, even if the examples that we have chosen to illustrate it were, intentionally, extreme, borderline and different from the usual examples encountered in DEL.

Extending [START_REF] Aucher | BMS revisited[END_REF], we identified various principles that events fulfill, by means of examples. They led us to motivate the formal definitions of dependence graph and event structure. These should be the main ingredients for a genuine logical framework. Yet, before defining this general framework, the preliminary logical analysis presented so far was necessary to be carried out in order to identify the key features that needed to be formalized and included as well as highlight the weaknesses of the current application of the DEL modeling approach, based on event models and product updates.

Fig. 1 :

 1 Fig. 1: Pointed epistemic models (M0, w0) and (N0, u0)

Fig. 2 :

 2 Fig. 2: Pointed event models (E, e) and (F, g)

Fig. 3 :

 3 Fig. 3: Product update of (M0, w0) by (E, e)

Fig. 4 :

 4 Fig. 4: Dependence graph for the 'barrel' example

  Pre M P ,P (w ) := p ∈P {Pre P ,P (p ) | w ∈ V (p )} -The (P , P)-postcondition function of M , Post M P ,P : W → P → 2 L(P

Fig. 6 :

 6 Fig. 6: Barrel example: Scenario 1

Fig. 7 :

 7 Fig. 7: Barrel example: Scenario 2

[START_REF] Billot | Multiagent belief revision[END_REF] defined independently from the DEL community a variant of the DEL framework where preconditions are replaced by classes of pointed epistemic models.

Acknowledgments. I thank Johan van Benthem and two anonymous referees for helpful comments. I thank Sabine Frittella for a discussion which enabled to find out a defect.

The Update Universe

The principles that we have elicited in Section 3 lead us to define what we call an event structure and a generic product update. An event structure captures the dependence relation between different types of events, based on their relative pre/postconditions, while the generic product update deals with the dynamics of knowledge and beliefs within the frame of a given event structure.

Event Structure

Because of Principle 2, the very nature of events plays a role in the dynamics of knowledge and beliefs. Our idea is to define formally an 'event' model completely identically to the way we define an epistemic model. The propositional letters for 'event' models will determine the factual nature of events, just as


Martin Pajot

Victor Duffort

Edouard Capoen

Anne-Sophie Mamede

Rose-Noëlle Vannier

Influence of the Strontium Content on the Performance of La 1-x Sr x MnO 3 /Bi 1.5 Er 0.5 O 3 Composite Electrodes for Low Temperature Solid Oxide Fuel Cells

Keywords: Low temperature solid oxide fuel cells, Stabilized bismuth oxides, Lanthanum strontium manganese oxides

Introduction

Solid Oxide Fuel Cells (SOFCs) are key components in the transition towards new energy sources with reduced carbon emissions. Due to their ability to convert the chemical energy stored in hydrogen into electrical and thermal energy with a 90 % efficiency [START_REF] Ellamla | Current Status of Fuel Cell Based Combined Heat and Power Systems for Residential Sector[END_REF] without the need for noble metal catalyst, SOFCs are a corner stone in the development of hydrogen as an energy vector. However, the current technology based on yttrium stabilized zirconia (YSZ) electrolytes requires high (> 700 °C) operating temperatures, [START_REF] Gao | A Perspective on Low-Temperature Solid Oxide Fuel Cells[END_REF] resulting in costly engineering, long startup time and reduced life span. The development of affordable, flexible and sustainable SOFCs based on more conductive electrolyte materials would greatly facilitate the integration of renewable energies into the power grid and allow applications in the field of transportation.

Decreasing the operating temperature can be achieved through two approaches;

optimization (electrolyte thickness, electrode microstructure) of the high temperature cells or the design of cells using more efficient materials. The mediocre ionic conductivity of YSZ typically limits the operating temperature around 700 °C, even with very thin electrolytes. On the other hand, it was shown that stabilized ceria and stabilized bismuth oxides, two very good families of oxide ion conductor plagued with stability issues under fuel conditions, could be combined to produce electrolytes with satisfactory performances at temperature as low as 350 °C. [START_REF] Wachsman | Lowering the Temperature of Solid Oxide Fuel Cells[END_REF] This research spurred interest in the search for new electrodes able to perform at reduced temperature such as the perovskite Ba 0.5 Sr 0.5 Co 0.8 Fe 0.2 O 3- 4 or the pyrochlore Bi 2 Ru 2 O 7 . [START_REF] Ahn | High-Performance Bilayered Electrolyte Intermediate Temperature Solid Oxide Fuel Cells[END_REF] La 1-x Sr x MnO 3 (LSM) is one of the most studied high temperature SOFC cathode material. [START_REF] Skinner | Recent Advances in Perovskite-Type Materials for Solid Oxide Fuel Cell Cathodes[END_REF][START_REF] Jiang | Development of Lanthanum Strontium Manganite Perovskite Cathode Materials of Solid Oxide Fuel Cells: A Review[END_REF][START_REF] Tsipis | Electrode Materials and Reaction Mechanisms in Solid Oxide Fuel Cells: A Brief Review[END_REF] Due to its purely electronic conductivity it needs to be used in composite with oxide ion conductors, typically YSZ, to provide some level of ionic conduction. The reaction of oxygen reduction then takes place at the triple phase boundary (TPB), i.e. at the junction of the electronic conductor (LSM), the ionic conductor (YSZ) and gaseous oxygen. After many years of research on these composite cathodes the exact mechanism of the oxygen reduction reaction (ORR) is still debated, but it is generally accepted that the main limiting steps are the migration of oxygen species on the surface of LSM and their transfer into the stabilized zirconia. [START_REF] Heuveln | Electrode Properties of Sr-Doped LaMnO3 on Yttria-Stabilized Zirconia II. Electrode Kinetics[END_REF][START_REF] Kim | Characterization of LSM-YSZ Composite Electrode by Ac Impedance Spectroscopy[END_REF] In the case of LSM/YSZ composite electrodes, it was shown that increasing the strontium content in La 1-x Sr x MnO 3 to about x ≈ 0.5 was both beneficial for the electronic conductivity and for the polarization resistance. [START_REF] Kim | Characterization of LSM-YSZ Composite Electrode by Ac Impedance Spectroscopy[END_REF][START_REF] Lee | Electrochemical Characteristics of La1-xSrxMnO3 for Solid Oxide Fuel Cell[END_REF][START_REF] Mizusaki | Electronic Conductivity, Seebeck Coefficient, Defect and Electronic Structure of Nonstoichiometric La1-xSrxMnO3[END_REF] However compositions in the range x = 0.15 -0.2 are typically selected because of mechanical compatibility with YSZ. [START_REF] Sakaki | Ln1-XSrxMnO3(Ln=Pr, Nd, Sm and Gd) as the Cathode Material for Solid Oxide Fuel Cells[END_REF][START_REF] Hammouche | Crystallographic, Thermal and Electrochemical Properties of the System La1-xSrxMnO3 for High Temperature Solid Electrolyte Fuel Cells[END_REF] Moreover, a small A-site deficiency is usually introduced to prevent the formation of La 2 Zr 2 O 7 and SrZrO 3 . [START_REF] Yokokawa | Thermodynamic Representation of Nonstoichiometric Lanthanum Manganite[END_REF][START_REF] Jiang | An Investigation of Shelf-Life of Strontium Doped LaMnO3 Materials[END_REF] The benefits of combining LSM with highly conductive and ORR active stabilized bismuth oxide phases rather than the typical YSZ have been identified early on. [START_REF] Jiang | Nanoscale Bismuth Oxide Impregnated (La,Sr)MnO3 Cathodes for Intermediate-Temperature Solid Oxide Fuel Cells[END_REF][START_REF] Li | La0.74Bi0.10Sr0.16)MnO3-δ-(Bi2O3)0.7(Er2O3)0.3 Composite Cathodes for Intermediate Temperature Solid Oxide Fuel Cells[END_REF][START_REF] Jiang | La0.85Sr0.15MnO3 -δ Infiltrated Y0.5Bi1.5O3 Cathodes for Intermediate-Temperature Solid Oxide Fuel Cells[END_REF][START_REF] Jiang | Bismuth Oxide-Coated (La,Sr)MnO3 Cathodes for Intermediate Temperature Solid Oxide Fuel Cells with Yttria-Stabilized Zirconia Electrolytes[END_REF][START_REF] Li | La0.84Sr0.16MnO3-δ Cathodes Impregnated with Bi1.4Er0.6O3 for Intermediate-Temperature Solid Oxide Fuel Cells[END_REF][START_REF] Jiang | Electrochemical Characteristics of Solid Oxide Fuel Cell Cathodes Prepared by Infiltrating (La,Sr)MnO3 Nanoparticles into Yttria-Stabilized Bismuth Oxide Backbones[END_REF] As expected, the overall ORR mechanism is similar for ESB and YSZ composite electrodes since both phases are pure ionic conductors. These electrodes became interesting candidates for low temperature applications when it was found that using erbium stabilized bismuth oxide (ESB) at both the cathode and electrolyte led to a 60 % decrease in the cathodic area specific resistance (ASR) over the same electrode deposited on stabilized ceria. [START_REF] Lee | Interfacial Modification of La0.80Sr0.20MnO3-δ-Er0.4Bi0.6O3 Cathodes for High Performance Lower Temperature Solid Oxide Fuel Cells[END_REF] Full cell test of these electrodes on ESB/GDC bilayer electrolyte demonstrated an impressive 1 W cm -2 at 650 °C. [START_REF] Lee | Interfacial Modification of La0.80Sr0.20MnO3-δ-Er0.4Bi0.6O3 Cathodes for High Performance Lower Temperature Solid Oxide Fuel Cells[END_REF] Several groups subsequently studied the stability and optimized the performances of this composite cathode by modifying the ESB/LSM ratio or the microstructure of the electrode by infiltration and nanoscaling. [START_REF] Ai | Highly Active and Stable Er0.4Bi1.6O3 Decorated La0.76Sr0.19MnO3+δ Nanostructured Oxygen Electrodes for Reversible Solid Oxide Cells[END_REF][START_REF] Huang | Electrochemical Performance and Stability of LSM-ESB Cathode[END_REF][START_REF] Kim | Er0.4Bi1.6O3-δ -La0.8Sr0.2MnO3-δ Nano-Composite as a Low-Temperature Firing Cathode of Solid Oxide Fuel Cell[END_REF][START_REF] Painter | Durability of (La0.8Sr0.2)0.95MnO3-δ-(Er0.2Bi0.8)2O3 Composite Cathodes for Low Temperature SOFCs[END_REF][START_REF] Park | Development of Nanostructured La0.8Sr0.2MnO3-δ-Er0.4Bi1.6O3 Cathodes via an Infiltration Process with Different Polymeric Agents for Intermediate Temperature Solid Oxide Fuel Cells[END_REF][START_REF] Park | Enhancing Performance of La0.8Sr0.2MnO3-δ-Infiltrated Er0.4Bi1.6O3 Cathodes via Controlling Wettability and Catalyst Loading of the Precursor Solution for IT-SOFCs[END_REF] In all these studies the strontium content of the La 1-x Sr x MnO 3 phases was kept close to x ≈ 0.2, i.e. the optimized composition for mechanical compatibility with YSZ in high temperature SOFCs.

In this paper, differently from previous work on LSM/ESB composite electrodes, we explore the whole solid solution of La 1-x Sr x MnO 3 , rather than focusing on x ≈ 0.2, in order to evaluate if the optimum strontium concentration for low temperature LSM/ESB electrodes is different from the optimized composition of high temperature LSM/YSZ electrodes. We synthesized La 1-x Sr x MnO 3 with x = 0.15, 0.3, 0.4, 0.5, 0.6, 0.8 by solid state reaction and characterized the structure, the oxygen stoichiometry and the conductivity as function of temperature under air atmosphere. We chose Bi 1.5 Er 0.5 O 3 rather than the more commonly used Bi 1.6 Er 0.4 O 3 , because despite a small decrease in ionic conductivity induced by the higher erbium content, this phase offers a much higher stability under 600 °C. [START_REF] Iwahara | Formation High Oxide Ion Conductive Phases in the Sintered Oxides of the System Bi2O3-Ln2O3 (Ln = La-Yb)[END_REF][START_REF] Kruidhof | Bismuth Oxide Based Ceramics with Improved Electrical and Mechanical Properties: Part II. Structural and Mechanical Properties[END_REF] The LSM/ESB electrode composition and thickness were optimized by minimizing the area specific resistance (ASR) on LSM/ESB|ESB|LSM/ESB symmetric cells with x = 0.15. These optimized conditions were then used to systematically study the influence of the strontium content on the ASR of the composite electrode. Furthermore, we use low energy ion scattering (LEIS) to study the surface composition of La 1-x Sr x MnO 3 with x = 0.15 and 0.4,

showing that differently from literature data reported at high temperature, SrO segregation is not an issue below 800 °C.

Experimental

Material preparation.

Bi 1.5 Er 0.5 O 3 (ESB) was synthesized following a previously described protocol. [START_REF] Kruidhof | Bismuth Oxide Based Ceramics with Improved Electrical and Mechanical Properties: Part I. Preparation and Characterisation[END_REF] with intermediate grindings.

The particle size was reduced by ball milling in acetone media for 2 h at 1000 rpm.

The milled powders were then used directly for the ink preparation (see below) or isostatically pressed (200 MPa) into pellets. ESB was sintered at 850 °C for 12 h to be used as electrolytes and the various LSM compositions were sintered at 1400 °C for resistivity measurements.

Ink elaboration and deposition.

Electrodes deposition was carried out following a standard protocol. [START_REF] Rolle | Ca3Co4O9+δ, a Growing Potential SOFC Cathode Material: Impact of the Layer Composition and Thickness on the Electrochemical Properties[END_REF] To prepare the inks, the LSM and ESB powders were ball milled with 2 wt.% of dispersant (T01 Cerlase) in ethanol for 5 h at 500 rpm. After evaporation of the ethanol, the powder was stirred with a binder (Mixtrio, EC3, Ferro Couleurs, France, 65 wt.%) and homogenized using a threecylinder roll mill. In a second step, symmetric cells made of a 10 mm diameter and 1.8 mm thickness ESB dense electrolyte were prepared by deposition of the ink on each face of the ESB pellet using a DEK 248 screen-printer. To allow subsequent layer deposition, the sample was dried at 80 °C between each deposition. Finally, the as prepared symmetric cells were annealed at 500 °C for 1 h to remove the organics and successively at 800 °C for 2 h with a 5

°C min -1 heating rate between each plateau.

Characterization techniques.

Powder X-ray diffraction (PXRD) patterns were recorded on Bruker D8 diffractometers equipped with Lynx-Eye Position Sensitive detectors, in the - implementation of the Bragg-Brentano geometry using Cu-K  radiation and a nickel K  filter.

High temperature patterns were recorded using an Anton Paar HTK1200 chamber under a flow of dry air. Structural information was extracted using Rietveld analysis within the FullProf software suite. [START_REF] Rodríguez-Carvajal | Recent Advances in Magnetic Structure Determination by Neutron Powder Diffraction[END_REF] Thermogravimetric analyses (TGA) were performed in platinum crucibles using a

Setaram Setsys thermobalance equipped with a differential thermal analysis (DTA) sample rod under argon or air atmospheres. Buoyancy and convection effects were accounted for using a proper blank in order to resolve the small mass losses due to oxygen stoichiometry evolution in LSM materials.

Electrochemical impedance spectra (EIS) were collected using a Solartron 1260 frequency response analyzer at open circuit voltage with a 50 mV sinusoidal excitation in the 10 6 -0.1 Hz frequency range on symmetric cells. Gold grids were used as current collectors.

Area specific resistances (ASR) were obtained by fitting the spectra with the Zplot (Scribner Associates) software and normalized to the total surface of the electrodes. Resistivity measurements were performed on dense LSM pellets using linear 4-probe technique and a Biologic SP150 galvanostat. Both types of samples were heated in tubular furnaces using homemade measurement rigs. Thermal stability was checked by taking two measurements at 15 min intervals after a fixed (1 h) stabilization time.

The electrodes microstructure was studied using a Hitachi S3400N scanning electron microscope (SEM) in backscattered electron (BSE) mode in order to obtain a good contrast between the LSM (dark) and ESB (clear) phases.

Low Energy Ion Scattering (LEIS) measurements were performed in a Qtac100 instrument (IONTOF GmbH). The samples, high density pellets with freshly ground surface annealed in air at 800 °C for 72 h, were cleaned with an oxygen plasma at room temperature and analysed using a 5 keV Ne + primary ion beam directed perpendicularly to the target surface. The surface, sub surface and bulk spectra were obtained at different ionic dose:

3.6x10 14 , 9.5x10 15 and 6.0x10 [START_REF] Jiang | An Investigation of Shelf-Life of Strontium Doped LaMnO3 Materials[END_REF] ion cm -2 , respectively.

Results and discussion

Raw materials characterization.

The purity of the different compounds was checked by PXRD. The pattern of the coprecipitated Bi 1.5 Er 0.5 O 3 (Fig. S1-a) is perfectly explained by the expected fluorite-type structure (Fm 3 m) with a = 5.4852(1) Å in good agreement with previously reported values. [START_REF] Kruidhof | Bismuth Oxide Based Ceramics with Improved Electrical and Mechanical Properties: Part I. Preparation and Characterisation[END_REF] The ionic conductivity of the prepared ESB was verified on dense pellet (Fig. S1-b).

The structure of all the LSM compositions studied here are based on the simple cubic perovskite structure (Pm 3 m). The 12-coordinated A-site is occupied by statistically distributed Sr 2+ /La 3+ cations and manganese atoms are located in the B-site. Small discrepancies are observed in the reported structural parameters throughout the literature due to the influence of the synthesis and/or annealing conditions of LSM. In our case, samples with a strontium content of x = 0.4 and lower crystalize with a hexagonal distortion (R 3 c) while samples with a strontium content of x = 0.5 and higher are orthorhombic (Pnma). This change of symmetry is clearly evidenced by the splitting of the reflection located at 2 = 47 °(Fig. 1-a) for x ≥ 0.5. In order to facilitate the comparison between the two symmetries, the cell parameters are reported in the pseudo-cubic cell corresponding to the aristotype (Fig. 1b).

The Goldschmidt factors of the La 1-x Sr x MnO 3 series evolve from 0.95 for x = 0 to 1.04 for x = 1, hence for La-rich compositions the A-site cations are smaller than the 12coordinated site in an ideal perovskite and the evolution of the cell parameters are dominated by the decrease of the manganese radius as it is oxidized from Mn 3+ to Mn 4+ through the substitution of La 3+ by Sr 2+ . Conversely, the A-site cations of Sr-rich compositions are too big to fit in the 12-coordinated site of the cubic perovskite structure and SrMnO 3 is expected to crystalize in a hexagonal perovskite structural type. This explains the growth of the impurity peaks visible on the XRD pattern of La 0.4 Sr 0.6 MnO 3 and La 0.2 Sr 0.8 MnO 3 . Using Rietveld refinement we were able to estimate that these peaks account for about 2 and 6 wt.% of SrMnO 3 with a 4H-hexagonal perovskite structure (P6 3 /mmc). This impurity is usually associated to high temperature synthesis of Sr-rich LSM oxides. However due to the low levels of impurity we kept these samples rather than using a different synthetic route that could potentially result in a very different microstructure. The main reason explaining the discrepancies observed throughout the LSM literature is probably an insufficient characterization of the oxygen overstoichiometry of these oxides.

Due to oxidation of trivalent manganese ions, these oxides can readily pick up small quantities of oxygen depending on the composition or thermal treatment history. Hence a more accurate way to write the LSM formula is La 1-x Sr x MnO 3+ . However, the presence of interstitial oxygen was not seen by neutron diffraction showing that the oxidation mechanism did not proceed through oxo anions insertion but resulted in cationic vacancies. [START_REF] Mitchell | Structural Phase Diagram of La1-xSrxMnO3+ δ: Relationship to Magnetic and Transport Properties[END_REF] Thus, from a structural point of view, the accurate formula should be (La 1-x Sr x Mn) 1-y O 3 with all oxygen sites fully occupied and a cationic vacancy concentration y=1-3/(3+ δ ¿)¿. As will be shown later, this mechanism has important implications on the surface composition of LSM, however for clarity reasons, the initial writing La 1-x Sr x MnO 3+ was used. A more detailed discussion on the oxidation mechanism by cationic vacancy formation, including the mathematical treatment necessary to the evaluation of the manganese average oxidation number by TGA, is given in the supplementary data.

Based on previous work showing that stoichiometric (i.e.  = 0) samples were obtained by annealing La 1-x Sr x MnO 3+ compositions with x ≤ 0.5 at 900 °C under argon atmosphere, [START_REF] Andersen | Determination of Stoichiometry in Lanthanum Strontium Manganates(III)(IV) by Wet Chemical Methods[END_REF][START_REF] Poulsen | Defect Chemistry Modelling of Oxygen-Stoichiometry, Vacancy Concentrations, and Conductivity of (La1-xSrx)YMnO3±δ[END_REF] we evaluated the oxygen overstoichiometry of our samples by thermal analysis under argon (Fig. 2). While the three hexagonal oxides exhibit well defined step-like mass losses, orthorhombic samples with x = 0.5, 0.6 show almost no irreversible mass variation. This clearly relates the hexagonal symmetry to the oxygen overstoichiometry.

Moreover we verified that argon annealed samples were all orthorhombic as previously decribed. [START_REF] Andersen | Determination of Stoichiometry in Lanthanum Strontium Manganates(III)(IV) by Wet Chemical Methods[END_REF] In the case of La 0.4 Sr 0.6 MnO 3 and La 0.2 Sr 0.8 MnO 3 , the higher oxidation state of manganese offers less resistance to the reduction and the small mass loss for temperature higher than 700 °C reflects the formation of understoichiometric oxides ( < 0). Interestingly while the process is reversible for x = 0.6 a small irreversibility is observed for x = 0.8. Since electrical conductivity of LSM materials is influenced by the oxygen stoichiometry, it was measured for each sample under air to simulate the conditions of the ASR evaluation (Fig. 3). In a first step, we checked that the oxygen stoichiometry of the samples does not evolve when heated under air in the temperature range considered for electrochemical performance evaluation, i.e. RT -800 °C. This was confirmed by the absence of irreversible mass variations for all the compositions under air by thermogravimetry as reported in Figure S2. The high temperature electronic conductivity of these materials is quite complex due to the interplay of carrier concentration, magnetic interaction and structure. As shown in Figure 3, the conductivity increases progressively with the strontium content up to x = 0.6. Despite the temperature dependence of log(T) not being exactly linear we observe a decrease of the activation energy with increasing strontium content, from 0.17 eV for x = 0.15 to 0.05 eV for x = 0.4. This behavior is typical of heavily doped semiconductors, where the increase in charge carrier usually triggers a change from semiconductor to a metallic behavior. [START_REF] Imada | Metal-Insulator Transitions[END_REF] As expected 39,40 metallic behavior is observed for x > 0.175 as evidenced by the change of slope below 100 °C for x = 0.3 and 0.4. In our case, however, the metallic behavior does not seem to survive to the disappearance of the ferromagnetic order reported around 90 °C for these two compositions. Further increase in Sr-content tends to decrease the Curie point of LSM which explains why the transition is not seen for x ≥ 0.5. [START_REF] Urushibara | Insulator-Metal Transition and Giant Magnetoresistance in La1-xSrMnO3[END_REF][START_REF] Martin | Magnetism and Structural Distortion in the La0.7Sr0.3MnO3 Metallic Ferromagnet[END_REF] The relevant parameters for all compositions are summarized in Table 1.

Electrode performance

The chemical stability of the ESB/LSM interface was investigated by mixing Careful inspection of the electrochemical impedance spectrum of La 0.85 Sr 0.15 MnO 3 /ESB symmetric cells at 678 °C reveals the presence of at least three components (Fig. 5-a, upper pane). Each phenomenon was modeled using a constant phase element in parallel with a resistive element in order to extract their temperature dependence (Fig. 5-b). In agreement with previously published results on similar compositions, [START_REF] Lee | Interfacial Modification of La0.80Sr0.20MnO3-δ-Er0.4Bi0.6O3 Cathodes for High Performance Lower Temperature Solid Oxide Fuel Cells[END_REF][START_REF] Park | Enhancing Performance of La0.8Sr0.2MnO3-δ-Infiltrated Er0.4Bi1.6O3 Cathodes via Controlling Wettability and Catalyst Loading of the Precursor Solution for IT-SOFCs[END_REF] we obtained two thermally activated phenomena with different activation energy and frequency range. One temperature independent phenomena was also observed at low frequencies which is the typical signature of gaseous O 2 diffusion limitation. Since this contribution is strongly related to the instrumentation, due to the large gold current collectors and static atmosphere used in this study, it was not accounted in the actual ASR of the electrodes thereafter. [START_REF] Adler | Factors Governing Oxygen Reduction in Solid Oxide Fuel Cell Cathodes[END_REF] This contribution was fitted at high temperature, where it is easily modeled, and fixed at lower temperature,

where it has negligible contribution to the overall ASR. The activation energy for the gas diffusion process at high temperature was in the order of 0.05 eV based on the few high temperature points where this contribution was easily refined. The two thermally activated processes showed very similar behavior with previously published results on La 0.2 Sr 0.8 MnO 3 /Bi 0.6 Er 0.4 O 3 composite electrodes. [START_REF] Lee | Interfacial Modification of La0.80Sr0.20MnO3-δ-Er0.4Bi0.6O3 Cathodes for High Performance Lower Temperature Solid Oxide Fuel Cells[END_REF][START_REF] Park | Enhancing Performance of La0.8Sr0.2MnO3-δ-Infiltrated Er0.4Bi1.6O3 Cathodes via Controlling Wettability and Catalyst Loading of the Precursor Solution for IT-SOFCs[END_REF] The lower frequency phenomenon was clearly the limiting step with an associated resistance about one order of magnitude larger than the higher frequency phenomenon. In our study as well, the high frequency element showed an increase of its activation energy as the temperature was lowered below 700 °C. Based on the very different frequency range, the limiting process with an activation energy of 1.38 eV was ascribed to the dissociative adsorption of oxygen on the surface of LSM and the subsequent surface migration of the ionic species generated. The second phenomenon, at high frequency, exhibiting a break in its activation energy at temperatures similar to the change of slope observed in the conductivity of ESB (Fig. S1-b) was assigned to bulk incorporation of oxo anions into ESB. [START_REF] Lee | Interfacial Modification of La0.80Sr0.20MnO3-δ-Er0.4Bi0.6O3 Cathodes for High Performance Lower Temperature Solid Oxide Fuel Cells[END_REF][START_REF] Park | Enhancing Performance of La0.8Sr0.2MnO3-δ-Infiltrated Er0.4Bi1.6O3 Cathodes via Controlling Wettability and Catalyst Loading of the Precursor Solution for IT-SOFCs[END_REF] Differently from La 0.85 Sr 0.15 MnO 3 , the high frequency bulk diffusion process was not observed for LSM compositions with x > 0.15, as exemplified here with La 0.6 Sr 0.4 MnO 3 (Fig. 5, red curves). The magnitude and the frequency range of the remaining constant phase element clearly show that the modeled phenomenon can be unambiguously attributed to the surface properties of the electrode. The ASR, corrected for gaseous diffusion, of all the composite electrodes under study are reported in Figure 6. Except for La 0.2 Sr 0.8 MnO 3 , all the compositions display fairly similar performance. At low temperature the two compositions La 0.6 Sr 0.4 MnO 3 and La 0.5 Sr 0.5 MnO 3 clearly stand out, mostly due to their lower activation energy. Interestingly, the increase in strontium concentration from x = 0.15 to x = 0.4 induces a ~14 % decrease in the energy required to dissociate the oxygen molecules and/or transfer the oxygen ions on the surface of the electrode. Lower activation energy is a key parameter to control when considering the reduction of the operating temperature. It resulted in our case in a polarization resistance at 500 °C almost twice as small for x = 0.4 than for x = 0.15 (see inset, Fig 6). Further increase in the strontium content, corresponding to composition containing a small amount of 4H-hexagonal perovskite as impurity, showed a rapid increase of both the polarization resistance and the activation energy. However, it is worth noting that the large decrease in the ASR of the La 0.2 Sr 0.8 MnO 3 for temperature higher than 750 °C was reversible and observed in all three thermal cycles the symmetric cell was subjected to (Fig. S5-a). High temperature diffraction failed to evidence any structural event at this temperature (Fig. S5-b) that could explain this anomaly. In particular, the proportion of 4H-hexagonal perovskite remained identical, confirming our initial assumption that this impurity has only marginal role in the overall performance of La 0.2 Sr 0.8 MnO 3 .

Discussion

The performance of La 1-x Sr x MnO 3 /ESB composite electrodes cannot be simply linked to the electronic conductivity of the LSM as evidenced by the fact that La 0.4 Sr 0.6 MnO 3 , the best electronic conductor of the series, is one of the lower performance material at reduced temperature. It is clear that the decrease in the polarization resistance observed for x = 0.4 -0.5 at low temperature is related to the lower activation energy of the surface processes involved during the catalytic dissociation of the O 2 molecules. This is in good agreement with studies on oxygen transfer by isotope exchange. Using tracer diffusion profile, the oxygen surface exchange coefficient of La 0.80 Sr 0.2 MnO 3 , 42 La 0.65 Sr 0.55 MnO 3 and La 0.5 Sr 0.5 MnO 3 [START_REF] De Souza | Oxygen Transport in La1-xSrxMn1-yCoyO3±δ Perovskites: Part II. Oxygen Surface Exchange[END_REF] were measured at 900 °C to be k* = 8×10 -9 , 5×10 -8 and 9×10 -8 cm s -1 respectively, showing that the efficiency of the oxygen dissociation increases with the Sr-content in these perovskites.

However, the segregation of SrO on the surface of perovskite materials is generally thought as the main limiting factor of perovskite materials. [START_REF] Druce | Surface Termination and Subsurface Restructuring of Perovskite-Based Solid Oxide Electrode Materials[END_REF][45][START_REF] Huber | In Situ Study of Activation and De-Activation of LSM Fuel Cell Cathodes -Electrochemistry and Surface Analysis of Thin-Film Electrodes[END_REF] The migration of strontium species on the surface of La 0.6 Sr 0.4 Co 0.2 Fe 0.8 O 3- (LSCF) was evidenced by Low-Energy Ion Scattering (LEIS), the driving force being the formation of a neutral Sr 2+ -O 2-surface layer while other cations, being trivalent, would result in a polar surface. [START_REF] Druce | Surface Termination and Subsurface Restructuring of Perovskite-Based Solid Oxide Electrode Materials[END_REF] A similar mechanism was expected for LSM based on reports of the surface enrichment in Sr 2+ measured by XPS. [START_REF] Huber | In Situ Study of Activation and De-Activation of LSM Fuel Cell Cathodes -Electrochemistry and Surface Analysis of Thin-Film Electrodes[END_REF][START_REF] Wu | X-Ray Photoelectron Spectroscopy of La0.5Sr0.5MnO3[END_REF][START_REF] Decorse | A Comparative Study of the Surface and Bulk Properties of Lanthanum-Strontium-Manganese Oxides La1-xSrxMnO3±δ as a Function of Sr-Content, Oxygen Potential and Temperature[END_REF][START_REF] Caillol | Investigation of Chemisorbed Oxygen, Surface Segregation and Effect of Post-Treatments on La0.8Sr0.2MnO3 Powder and Screen-Printed Layers for Solid Oxide Fuel Cell Cathodes[END_REF] Hence the enhanced catalytic properties that we observe by increasing the strontium content up to 0.4~0.5 in La 1-x Sr x MnO 3 is unexpected. Despite the significance of LSM electrodes, we could not find published LEIS spectra, the only technique able to selectively probe the top atomic layer, other than a brief confirmation of Sr-surface segregation presented at a conference. 50 We therefore used LEIS to look at the surface composition of La 0.85 Sr 0.15 MnO 3 and La 0.6 Sr 0.4 MnO 3 after a 72 h annealing at 800 °C in air, i.e. the temperature and atmosphere used for symmetric cells preparation. As shown in Figure 7, in contrast to previous assumptions, all three cations are present on the outermost surface in these conditions.

Additionally a small molybdenum contamination, probably due to the evaporation of the MoSi 2 heating elements, is visible, on the surface only, for x = 0.4. However, by looking at the evolution of the composition as a function of the depth, it is clear that the surface of these oxides is actually Sr-deficient.

While this is surprising considering the HT-SOFC literature, it was previously reported that low-temperature annealing of LSM resulted in the suppression of the strontium segregation, [START_REF] Caillol | Investigation of Chemisorbed Oxygen, Surface Segregation and Effect of Post-Treatments on La0.8Sr0.2MnO3 Powder and Screen-Printed Layers for Solid Oxide Fuel Cell Cathodes[END_REF] in good agreement with the performance and stability of LSM/ESB LT-SOFC electrodes. As shown in Figure 7-b, for both materials the Sr/La ratio stabilizes rapidly after removal of the first few atomic layers. In contrast, the evolution of the Mn/La ratio shows very different behavior. In the case of La 0.85 Sr 0.15 MnO 3 , the concentration of manganese species increases slowly with the depth, consequently the subsurface Mn/La ratio is almost equal to the surface level and a 50 % increase is observed once the bulk of the material is being analyzed. Conversely, La 0.6 Sr 0.4 MnO 3 exhibits a rapid increase of the Mn/La from the surface to the sub-surface level and only a 15 % increase once the bulk of the material is reached. This indicates a deficiency in manganese at the surface, compared to the bulk for both compositions but the deficiency impacts a much larger volume for the La rich composition.

In order to understand the evolution of the cationic ratios it is important to remember that La 1-x Sr x MnO 3 perovskites with high Mn 3+ content, i.e. lanthanum rich compositions, spontaneously oxidize on cooling, resulting in the formation of cationic vacancies rather than the insertion of interstitial oxygen. Because oxygen does not diffuse in LSM, this oxidation process implies that the outer surface of the grain gradually grows by capturing oxygen and stabilizing it into the structure thanks to the migration of cations from the bulk (Fig. S6). Our LEIS results indicate that the concentration of Mn-vacancies are significantly higher on the surface of the materials compare to the bulk while the A-site vacancies are much more homogeneously distributed throughout the materials. This distribution of the cationic vacancies is most probably related to difference in mobility between the A-site and B-site cations. While the former have a straightforward diffusion path through an O 4 square window, B-site cations have to hop from one octahedral site to the next going through two triangular O 3 windows. Because of its higher oxygen overstoichiometry, La 0.85 Sr 0.15 MnO 3+ ( = 0.09), exhibits a high concentration of Mn-vacancies reaching at least a couple of atomic layers deep. On the other hand, La 0.6 Sr 0.4 MnO 3+ ( = 0.03), needs to accommodate a reduced number of vacancies, restricting the vacancies to the outermost surface. Hence, mitigating the cationic vacancies using a higher Sr-content enables a higher Mn-content on the outermost surface and sub-surface, critical for oxygen dissociation and surface transport.

Finally, it is interesting to note that the high-frequency ionic contribution to the ASR of La 0.85 Sr 0.15 MnO 3 /ESB composite electrode is suppressed by even small amount of Sr 2+ . This contribution sets the x = 0.15 composition clearly aside from all the other, also pointing at the detrimental role of cationic vacancies.

Conclusion

We showed in this work that it is possible to increase the already promising performance of LSM/ESB composite electrodes through the optimization of the strontium content in La 1-x Sr x MnO 3 . By using composition with x = 0.4 -0.5, rather than the typical high temperature oxides with x = 0.15 -0.2, it is possible to accelerate both the oxygen surface and bulk diffusion phenomena taking place in the electrodes. In particular the significant decrease of the activation energy of the oxygen dissociation and surface migration offers a decrease of the ASR by approximatively 50 % at reduced temperature.

The limiting steps of the oxygen reduction reaction are confirmed to be located on the surface of the ESB/LSM electrodes and therefore the surface composition of the La 1-x Sr x MnO 3 material is expected to play a major role in the overall catalytic activity. We showed using LEIS that the important non-stoichiometry observed for La 0.85 Sr 0.15 MnO 3+ ( ~ 0.09) composition results in a high concentration of manganese vacancies on the surface and subsurface level. For this reason, Sr-rich compositions which are near-stoichiometric exhibit higher Mn concentration at the outermost surface which is likely the reason for the increased catalytic activity of La 1-x Sr x MnO 3 with x = 0.4 and 0.5. Contrary to what is expected from high temperature literature, we do not observe SrO segregation at the surface of LSM below 800 °C, even in the case of La 0.6 Sr 0.4 MnO 3 which is a good sign of the stability of the electrode. Further increase in the strontium content, i.e. for x = 0.6 -0.8, results in the formation of 4H-hexagonal perovskite and a rapid decay of the electrochemical performance.

It is not clear whether the impurity directly impedes the ORR or if it is a consequence of the change in the surface chemistry of the LSM particles. Results obtained by LEIS are preliminary, a more detailed study of the surface composition of the various strontium doped manganites will be undertaken to refine our understanding of the oxygen reduction reaction mechanism at reduced temperature. However, our work clearly highlights that the development of lower temperature SOFCs must take into account the flexibility offered by the reduced temperature range and the need to reinvestigate materials that were set aside for higher temperature applications due to their chemical or mechanical incompatibilities.

his PhD grant. Special thanks to Romain Jooris for his help with measurement and data handling.

  Influence of the strontium content on the performance of La1-xSrxMnO3/Bi1.5Er0.5O3 composite electrodes for low temperature Solid Oxide Fuel Cells Martin Pajot, Victor Duffort, Edouard Capoen, Anne-Sophie Mamede, Rose-Noelle Vannier To cite this version: Martin Pajot, Victor Duffort, Edouard Capoen, Anne-Sophie Mamede, Rose-Noelle Vannier. Influence of the strontium content on the performance of La1-xSrxMnO3/Bi1.5Er0.5O3 composite electrodes for low temperature Solid Oxide Fuel Cells. Journal of Power Sources, 2020, Journal of Power Sources, 450, pp.227649. 10.1016/j.jpowsour.2019.227649. hal-02428599

Figure 1 :

 1 Figure 1: a) PXRD patterns of La1-xSrxMnO3 showing good purity for x = 0.15 -0.5. For x = 0.6 and 0.8 a small contamination of about 2 and 6 wt.% respectively, of 4H-hexagonal perovskite evidenced by the * symbols. The upper right inset shows a magnification of the most intense peaks and b) pseudo-cubic cell parameters evolution as a function of the strontium content.

Figure 2 :

 2 Figure 2: Determination of the oxygen overstoichiometry by TGA of the various LSM compositions under argon atmosphere.

Figure 3 :

 3 Figure 3: Electrical conductivities as a function of the temperature measured under air for all the LSMs compositions studied

Bi 1 . 5

 15 Er 0.5 O 3 and La 0.85 Sr 0.15 MnO 3 and annealing the mixture at 800 °C for 24 h. No impurity is visible on the resulting diffraction pattern (Fig.4-a) demonstrating the excellent chemical compatibility of these two oxides up to at least 800 °C. This combination of materials was used to study the optimal electrode thickness, grafting temperature and ESB/LSM ratio. The optimized parameters were selected based on the data presented in FigureS3, resulting in symmetric cells with an electrode thickness of ~50-60 µm, composed of a ESB/LSM 50/50 wt.% ratio, grafted at 800 °C. Post mortem imaging of the cross-section of the symmetric cells (Fig.4-b) after impedance spectroscopy shows micron sized ESB particles well dispersed in a framework of slightly larger LSM grains. The electrode/electrolyte interface did not show any sign of delamination and no crack could be seen on the electrode despite the three thermal cycles from room temperature to 800 °C the symmetric cell was subjected to, showing promising mechanical compatibility. The stability of these electrodes is also reflected by the stability of their electrochemical impedance spectra during these three thermal cycles, at both ends of the temperature range considered (Fig.S4), supporting the absence of secondary phases growth.

Figure 4 :

 4 Figure 4: a) Rietveld refinement showing data points (red dots), calculated profile (black line) difference curve (blue line) and Bragg peak position of a mixture of La0.85Sr0.15MnO3 (upper green bars) and Bi1.5Er0.5O3 (lower

Figure 5 :

 5 Figure 5: a) EIS spectra of the electrode response of the La1-xSrxMnO3/ESB|ESB|La1-xSrxMnO3/ESB symmetric cells at 678 °C for x = 0.15 (upper pane) and x = 0.4 (lower pane); the continuous lines represent the spectrum calculated from the fit. b) Temperature dependence of the resistance (upper pane) and characteristic frequencies of the different phenomena observed for x = 0.15 (black) and x = 0.4 (red).

Figure 6 :

 6 Figure 6: Temperature dependence of the ASR (corrected for gaseous diffusion) of La 1-xSrxMnO3/ESB symmetric cells for x = 0.15, 0.3, 0.4, 0.5, 0.6, 0.8. Lower left inset represents the evolution of the ASR as a function of the chemical composition at a fixed temperature of 500 °C.

Figure 7 :

 7 Figure7: a)[START_REF] Jiang | Bismuth Oxide-Coated (La,Sr)MnO3 Cathodes for Intermediate Temperature Solid Oxide Fuel Cells with Yttria-Stabilized Zirconia Electrolytes[END_REF] Ne + @5keV LEIS spectra of La0.85Sr0.15MnO3 (upper pane) and La0.6Sr0.4MnO3 (lower pane) at three different ion doses corresponding to compositions at the surface (green), the sub-surface (red) and the bulk (blue) of the materials and b) evolution of the peak area of strontium (dots) and manganese (triangle) relative to the peak area of lanthanum as a function of the depth for La0.85Sr0.15MnO3 (black) and La0.6La0.4MnO3 (red).

Table 1 :

 1 Structural and physical parameters of the materials used in this study.

	Composition		Space Group	a (Å)	b (Å)	c (Å)	TECx10 6 (K -1 )	@500 °C (S cm -1 )	Average Mn oxidation state
	La0.85Sr0.15MnO3+ 0.09 R 3 c	5.53551(5)	13.3694(2 )	14.9	117	3.3
	La0.7Sr0.3MnO3+	0.02 R 3 c	5.5092(5)	13.371(1)	12.8	131	3.3
	La0.6Sr0.4MnO3+	0.03 R 3 c	5.4912(4)	13.361(1)	13.3	235	3.5
	La0.5Sr0.5MnO3+	< 0.01 Pnma 5.4650(3) 7.6917(4) 5.4825(3)	14.1	215	3.5
	La0.4Sr0.6MnO3+	<0.01 Pnma 5.467(2) 7.6732(1) 5.471(2)	13.6	474	3.6
	La0.2Sr0.8MnO3+	<0.01 Pnma 5.4561(1) 7.6369(1) 5.4327(1)	14.5	86	3.8
	Bi1.5Er0.5O3	-	Pm 3 m		5.4852(1)		16.5	2.02×10 -2	-

Acknowledgements

This work is supported by the French National Research Agency (ANR-18-CE05-0001) as well as the Region Haut de France. The Chevreul Institute (FR 2638), ULille, the Fonds Européen de Développement Régional (FEDER), CNRS, are acknowledged for the access to the various characterisation platforms. MP is grateful to ULille and Region Haut de France for