Benoit Gaudou
email: benoit.gaudou@irit.fr

Dominique Longin
email: dominique.longin@irit.fr

Emiliano Lorini
email: emiliano.lorini@irit.fr

Luca Tummolini
email: luca.tummolini@istc.cnr.it

Anchoring Institutions in Agents' Attitudes: Towards a Logical Framework for Autonomous MAS

Keywords: I.2.4 [Knowledge Representation Formalisms and Methods]: Modal logic, I.2.11 [Distributed Artificial Intelligence]: Intelligent agents, I.2.0 [General]: Philosophical foundations Theory Normative systems, logics for agent systems, modal logic. 1

HAL is

these problems is to devise artificial institutions ([23, 9]). Following the classical work of Douglass North artificial institutions are usually conceived as human-like: "the rules of the game in a society or the humanly devised constraints that structure agents' interaction" ( [24, p. 3]). With this model in mind, AI practitioners have interpreted their task as that of advancing logical or computational frameworks to represent institutions, while leaving to the agents' autonomy the decision whether to comply or not with the specified rules ( [START_REF] Ågotnes | On the logic of normative systems[END_REF][START_REF] Conte | Autonomous norm acceptance[END_REF]). This approach, however, has at least three strong limitations. First of all, the institutions are not only constraints but also 'enablements' ( [START_REF] Searle | The Construction of Social Reality[END_REF]): new possibilities of actions (i.e. institutional actions like paying, marrying, promising etc.) are possible when an institution is in place. Secondly, artificial institutions are usually inspired by human legal ones which, however, are only a small part of the institutionalized human interactions. Moreover, to work effectively, legal institutions should interact with informal ones. 1 Finally, and more importantly, institutions should be constructed by the agents themselves and not imposed from the outside.

More precisely, while it is a widely shared that, in order to face complex and dynamical problems, the individual agents must be autonomous, less emphasis is devoted to the fact that the multiagent systems (MAS) themselves (for exactly the same reasons) should be conceived and designed to be autonomous. In fact, etymologically, autonomous means self-binding ('auto' and 'nomos'), and an autonomous MAS is the vision of an artificial society that is able to create, maintain, and eventually change its own institutions by itself, without the intervention of the external designer in this process.

This challenge is also strongly tied to the new trend of designing self-organizing MASs but, in contrast to many efforts in the area, we are after a notion of self-organization that is amenable for, and can make profit of, more complex cognitive agents (i.e. BDI-like; see [START_REF] Conte | Cognitive and social action[END_REF] for the general approach). In fact, quoting North again [21, p. 77]: "Only because institutions are anchored in peoples minds do they ever become behaviorally relevant. The elucidation of the internal aspect is the crucial step in adequately explaining the emergence, evolution, and effects of institutions (emphasis added)."

In this paper, we aim to provide a logical framework for the specification of autonomous MASs, that is, MASs whose agents are ca- 1 Following [START_REF] North | Institutions, Institutional Change, and Economic Performance[END_REF], we consider informal such institutions as social norms and social practices (i.e. promise). In his seminal book North explicitly states the relevance of this informal layer but still this component is widely neglected in the MAS literature. On the importance of informal normative relations in social contexts see [START_REF] Castelfranchi | Formalising the informal? Dynamic social order, bottom-up social control, and spontaneous normative relations[END_REF].

pable of creating and maintaining their institutions by themselves (Section 3). The focus of this contribution is on modeling social or informal institutions, rather than legal ones. Social institutions are the basic structures of a society on top of which more complex legal ones are constructed. By social or informal institutions, we refer to rule-governed social practices in which no member with 'special' powers is introduced. 2 More specifically, we will introduce the notion of an agent's acceptance of a proposition qua group member in a given institutional context (Section 2), and we will study its interaction with different notions such that of common belief and private belief (Section 4). On the basis of these attitudes qua group members, we will specify how a group can create and maintain normative and institutional facts which hold only in an attitude-dependent way. That is, it is up to the agents, and not to the external designer, to support such facts (Section 5). We will compare our proposal with related logical works on the issues of collective belief and institutions (Section 6). In conclusion we will identify directions for future work on the basis of our framework (Section 7). Anchoring institutions, and their facts, in agents' minds is just the first step towards a more complete characterization of the "internal aspect" of normative systems and towards the vision of autonomous MASs.

ACCEPTANCE QUA GROUP MEMBER

Although in this paper the notion of acceptance qua group member is a primitive (i.e. it is not analyzed in more specific mental attitudes), some conceptual clarification is needed because of the crucial role it plays in explaining the maintenance of social institutions. Whereas beliefs have been studied for decades [START_REF] Hintikka | Knowledge and Belief[END_REF] as representative of doxastic mental states, acceptances have only been examined since [START_REF] Stalnaker | Inquiry[END_REF] and [START_REF] Cohen | An essay on belief and acceptance[END_REF] while studying the nature of argument premises or reformulating Moore's paradox [START_REF] Cohen | An essay on belief and acceptance[END_REF]. If a belief that p is an attitude constitutively aimed at the truth of p, an acceptance is the output of "a decision to treat p as true in one's utterances and actions" [START_REF] Hakli | Group beliefs and the distinction between belief and acceptance[END_REF] without being necessarily connected to the actual truth of the proposition. In order to better specify this distinction, it has been suggested [START_REF] Hakli | Group beliefs and the distinction between belief and acceptance[END_REF] that while beliefs are not subject to the agent's will, acceptances are voluntary; while beliefs aim at truth, acceptance are sensitive to pragmatic considerations; while beliefs are shaped by evidence, acceptances need not be; while beliefs come in degrees, acceptances are qualitative; finally, while beliefs are context-independent, acceptance depends on context.

For the aims of this paper we are particularly interested in the last feature, namely the fact that acceptances can be context-dependent. In fact, one can decide (say for prudential reasons) to reason and act by "accepting" the truth of a proposition in a specific context, and possibly rejecting the very same proposition in a different one. Although, usually, this aspect of the acceptance state is studied in private contexts (i.e. when an agent, in order not to take too many risks, accepts that the total cost of her house restructuring will be beyond her reasonable expectations; see [START_REF] Bratman | Practical reasoning and acceptance in context[END_REF]), we will explore the role of this attitude in institutional contexts. Institutional contexts are rule-governed social practices on the background of which the agents reason. For example, take the case of a game like Clue. The institutional context is the rule-governed social practice which the agents conform to in order to be competent players.

On the background of such contexts, we are interested in the explicit mental states (the acceptances) that can be formally captured. In the context of Clue, for instance, an agent accepts that something has happened (see Example 3) qua player of Clue. The state of acceptance qua group member in an institutional context is the kind of acceptance one is committed to when one is "functioning as a group member" [START_REF] Tuomela | The Philosophy of Sociality[END_REF]. Although space restrictions prevent a full analysis of this notion, it is important to stress that we consider this attitude as one that is held by an agent. Nevertheless, there are specific consequences deriving from the agent's functioning as a group member: e.g. the acceptance of a proposition qua group member is always a public fact (see Section 4.1).

THE LOGIC OF ACCEPTANCE

Syntax

The syntactic primitives of our logic AL (Acceptance Logic) are the following: -a finite set of n > 0 agents AGT = {1, 2, ..., n}; -a nonempty finite set of atomic actions ACT = {a, b, ...}; -a set of atomic formulas ATM = {p, q, ...}; -a finite set of labels denoting institutional contexts INST = {inst1, inst2, ..., instm}; -a symbol λ denoting the private context. For notational convenience we note 2 AGT ⋆ = 2 AGT \ {∅} the set of all non empty subsets of agents, ∆1 = C:x|C ∈ 2 AGT ⋆ , x ∈ INST the set of all couples of non empty subsets of agents and institutional contexts, ∆2 = {i:λ|i ∈ AGT } the set of all couples of single agents and private context, and i:x for {i} :x. Finally, ∆ = ∆1 ∪ ∆2.

The language LAL is defined as the smallest superset of ATM such that: if ϕ, ψ ∈ LAL, i ∈ AGT and C:x ∈ ∆ then ¬ϕ, ϕ ∨ ψ and [C:x] ϕ ∈ LAL. The classical boolean connectives ∧, →, ↔, ⊤ (tautology) and ⊥ (contradiction) are defined from ∨ and ¬ in the usual manner.

Formula [C:x] ϕ has to be read "the agents in C accept that ϕ while functioning as group members in the institutional context x". EXAMPLE 1. [C:Greenpeace] protectEarth is read "the agents in C accept that the mission of Greenpeace is to protect the Earth while functioning as activists in the context of Greenpeace" and [i:Catholic] PopeInfallibility is read "the agent i accepts that the Pope is infallible while functioning as a Catholic in the context of the Catholic Church".

For C:x ∈ ∆1 : [C:x] ⊥ has to be read "agents in C are not functioning as group members in the institutional context x" because we assume that functioning as a group member is, at least in this minimal sense, a rational activity; conversely, ¬ [C:x] ⊥ has to be read "agents in C are functioning as group members in the institutional context x"; ¬ [C:x] ⊥ ∧ [C:x] ϕ stands for "agents in C are functioning as group members in the context x and they accept that ϕ while functioning as group members" or simply "agents in C accept that ϕ qua group members in the institutional context x" which, for us, is tantamount to "The group C accepts that ϕ in the institutional context x" (i.e. group acceptance). Similarly, the formula ¬ [C:x] ϕ has to be read "agents in C are functioning as group members in the institutional context x and they do not accept that ϕ while functioning as group members in x" or simply "agents in C do not accept that ϕ qua group members in x" (i.e. "The group C does not accept that ϕ in the institutional context x). EXAMPLE 2. ¬ [{i, j} :Europe] ⊥ ∧ [{i, j} :Europe] EuroMea-nsOfExchange stands for "i and j accept qua Europeans that the Euro is the official means of exchange in the context of Europe", whereas ¬ [{i, j} :Europe] DollarMeansOfExchange stands for "i and j qua Europeans do not accept that dollar is the official means of exchange".

Modal operators of the form [i:λ] correspond to standard dox-astic operators. 3 Hence a formula [i:λ] ϕ has to be read "agent i believes that ϕ".

Semantics

We use a standard possible worlds semantics and a model is a triple M = W, A , V where:

• W is a set of possible worlds;

• A : ∆ -→ (W -→ 2 W ) associates each C:x ∈ ∆ and possible world w with the set AC:x(w) of possible worlds accepted by the group C in w, where agents in C are functioning as group members in the institutional context x;

• V : W -→ 2 ATM is a truth assignment which associates each world w with the set V (w) of atomic propositions true in w.

The rules defining the truth conditions of formulas of our logic are inductively defined as follows.

• M, w |= p iff p ∈ V (w);

• M, w |= ¬ϕ iff not M, w |= ϕ; • M, w |= ϕ ∨ ψ iff M, w |= ϕ or M, w |= ψ; • M, w |= [C:x] ϕ iff for all w ′ ∈ W , if w ′ ∈ AC:x(w)
then M, w ′ |= ϕ.

Axiomatization

The axiom system of LAL is made of all tautologies of propositional calculus and, the axioms and rules of inference of the basic normal modal logic for every operator [C:x] where C:x ∈ ∆. That is, we have all K-theorems for every C:x ∈ ∆.

Moreover, we suppose that given a set of agents C, all B ⊆ C have access to all the facts that are accepted (or that are not accepted) by agents in C while functioning as group members in the institutional context x. In particular, we suppose the following relations between the acceptances of the group members with respect to the institutional contexts: if agents in C (do not) accept that ϕ while functioning as group members in the institutional context x then for every subset B of C and institutional context y while functioning as group members in the institutional context y, agents in B accept that agents in C (do not) accept that ϕ while functioning as group members in the institutional context x. Furthermore, we suppose the following relations the acceptance qua group and individual beliefs: if agents in C not) accept that ϕ functioning as group members in the institutional context x then, for every agent i in C, we have that i believes that agents in C (do not) accept that ϕ while functioning as group members in the institutional context x. Finally we suppose standard properties of introspection for beliefs: if agent i believes that ϕ then he believes that he believes that ϕ; if agent i does not believe that ϕ then he believes that he does not believe that ϕ. Such properties are captured by the following two axiom schemas. For every C:x, B:y ∈ ∆, if B ⊆ C then:

[C:x] ϕ → [B:y] [C:x] ϕ 4 [C :x],[B :y] ¬ [C:x] ϕ → [B:y] ¬ [C:x] ϕ 5 [C :x],[B :y]
S1 if w ′ ∈ AB:y(w) then AC:x(w ′ ) = AC:x(w)
We also suppose that if agents in C accept that ϕ qua group members in the institutional context x then, for every subset B of C, it holds that agents in B accept ϕ qua group members in the institutional context x. This means that things accepted by the agents in a set C (qua group members) with respect to a certain institutional context x are also accepted by agents in all C's subsets with respect to the same context x. Formally, for every C:x, B:x ∈ ∆, if B ⊆ C then:

Inc [C :x],[B :x] ¬ [C:x] ⊥ ∧ [C:x] ϕ → ¬ [B:x] ⊥ ∧ [B:x] ϕ EXAMPLE 3.
Imagine three agents i, j, k that,qua players accept, in the context of Clue, that someone called Mrs. Red, has been killed: ¬ [{i, j, k} :Clue] ⊥∧[{i, j, k} :Clue] killedMrsRed . This implies that also the two agents i, j qua players accept that someone called Mrs. Red has been killed in that context:

¬ [{i, j} :Clue] ⊥ ∧ [{i, j} :Clue] killedMrsRed . Axiom Inc [C :x],[B :x] has the following semantic characteriza- tion. For every w ∈ W , C:x, B:x ∈ ∆, if B ⊆ C then: if AC:x(w) = ∅ then AB:x(w) = ∅
and AB:x(w) ⊆ AC:x(w) S2

As far as operators of type [i:λ] for beliefs are concerned, we suppose that an agent cannot believe contradictions. Formally, for every i:λ ∈ ∆2:

D [i:λ] ¬([i:λ] ϕ ∧ [i:λ] ¬ϕ)
which corresponds to the following standard property of seriality.

For every w ∈ W and i:λ ∈ ∆2 we have:

S3 A i:λ (w) = ∅
Thus, every doxastic operator and we write ⊢AL ϕ iff formula ϕ is a theorem of AL. Moreover, let M be a model such that M = W, A,V as defined in Section 3.2 and satisfying the semantic constraints S1-S3 given above. We write |=AL ϕ iff formula ϕ is valid in all AL models, i.e. M, w |= ϕ for every AL model M and world w in M. Finally, we say that a formula ϕ is satisfiable if there exists an AL model M and a world w in M such that M, w |= ϕ.

GROUP ACCEPTANCE PROPERTIES

The public nature of group acceptance

In Section 3.1, we have analyzed the notion of group acceptance as the set of the acceptances of all the agents in the group while functioning as group members. This notion of acceptance qua group member however must not be confused with (nor reduced to) that of a private mental attitude. On the contrary we claim that group acceptances are always public so much that it is part of the concept of functioning as a group member that all the agents commonly believe that one is functioning in this way. In MAS literature, an operator to express common belief is given (see for instance [START_REF] Fagin | Reasoning about Knowledge[END_REF]). The notion of common belief can be built on the concept of individual belief and on a particular kind of distributed belief of the form "every agent in C believes that ϕ". The former concept is expressed in our logic by operators of type [i:λ]. The latter concept is formally expressed by operators of type EC where a formula EC ϕ is defined as follows:

EC ϕ def = i∈C [i:λ] ϕ
Given a set of agents C ⊆ AGT , formula CB C ϕ is meant to stand for "there is common belief in C that ϕ", that is, "everyone in C believes that ϕ, everyone in C believes that everyone in C believes that ϕ, everyone in C believes that everyone in C believes that everyone in C believes that ϕ, and so on".

If E 1 C ϕ denotes EC ϕ and E k C ϕ denotes EC (E k-1 C
ϕ), we can define CB C ϕ as follows:

CB C ϕ def = k>0 E k C ϕ
With the aim of making the public nature of group acceptance explicit, the following theorem highlights the relationship between our notion of group acceptance (i.e. acceptance by each of the agents qua group members) expressed by operator of type [C:x] and the concept of common belief.

THEOREM. For any C:x ∈ ∆:

(1) ⊢AL [C:x] ϕ ↔ CB C [C:x] ϕ PROOF. Direction → can be proved from proving that ∀k > 0, [C:x] ϕ → E k C [C:x] by induction on k: • [C:x] ϕ → EC [C:x] ϕ (case k = 1) • From [C:x] ϕ → E k C [C:x] ϕ infer [C:x] ϕ → E k+1 C [C:x] ϕ (inductive case)
To prove the case k = 1, we just apply Axiom 4 Let us prove the inductive case. We suppose that

[C:x] ϕ → E k C [C:x] ϕ. By rule of necessitation on every [i:λ], we infer i∈C [i:λ] ([C:x] ϕ → E k C [C:x] ϕ) which is (by definition of EC ) equivalent to: EC ([C:x] ϕ → E k C [C:x] ϕ). Thus from the latter, case k = 1 and definition of E k+1 C we can deduce that [C:x] ϕ → E k+1 C [C:x] ϕ. This is enough to prove that [C:x] ϕ → E k C [C:x] ϕ (for k > 0) is a theorem. We can thus infer that k>0 ([C:x] ϕ → E k C [C:x] ϕ) holds. By stan- dard modal principles, k>0 ([C:x] ϕ → E k C [C:x] ϕ) implies [C:x] ϕ → k>0 E k C [C:x] ϕ which is equivalent to [C:x] ϕ → CB C [C:x] ϕ.
We leave to the reader the proof of ← direction of the theorem.

2

According to Theorem 1, the agents in C accept that ϕ while functioning as group members in the institutional context x if and only if there is common belief in C that they accept that ϕ while functioning as group members in the institutional context x. Hence, accepting a proposition while functioning as a group member is always a public fact which is out in the open and that is used by all the members to reason about each other in an institutional context.

Group acceptance and individual beliefs

As far as the relationship between acceptances qua members and individual beliefs is concerned, it has to be noted that

¬ [C:x] ⊥ ∧ [C:x] ϕ ∧ i∈C [i:λ]
¬ϕ where C:x ∈ ∆1 is satisfiable in our logic. This means that the attitudes privately endorsed by the agents and those entertained qua group members can dione can privately disbelieve that which is accepted while functioning as a group member. EXAMPLE 4. Consider the discursive dilemma as elaborated in [START_REF] Pettit | Deliberative democracy and the discursive dilemma[END_REF] in which a three-member court has to make a judgment on whether a defendant is liable for a breach of contract. If one assumes that the group accepts the majority rule to decide on the issue, it might happen that each judge can privately believe that the group ought to accept a certain conclusion ( e.g. that the defendant is liable), while each is forced to accept the opposite qua group member ( i.e. qua judge).

ATTITUDE-DEPENDENT FACTS

Normative and institutional facts are a class of facts that are typical of institutional contexts [START_REF] Searle | The Construction of Social Reality[END_REF]. Such facts have the peculiar feature of being dependent on the agents' attitudes in a way that we are now in the position to specify in detail. More precisely it has been noted that these facts are characterized at least by two features [START_REF] Lagerspetz | The opposite mirrors[END_REF][START_REF] Searle | The Construction of Social Reality[END_REF][START_REF] Tuomela | The Philosophy of Social Practices: A Collective Acceptance View[END_REF].

• Performativity: an attitude of certain type shared by a group of agents towards a normative or an institutional fact may contribute to the truth of a sentence describing the fact.

• Reflexivity: if a sentence describing a normative or an institutional fact is true, the relevant attitude is present. EXAMPLE 5. If the agents qua group members accept that a certain of paper as money (an institutional fact), then, in the appropriate context, this piece of paper is money for that group (performativity). At the same time, if it is true that a certain piece of paper is money for a group, then the agents qua group members accept the piece of paper as money (reflexivity).

In order to represent in AL these kind of facts, we need first to the concept of truth with respect to an institutional context in way that respects these two principles.

Truth in an institutional context

We formalize the notion of truth w.r.t. to a certain institutional context with the operator [x]. A formula [x] ϕ is read "within the institutional context x, it is the case that ϕ". Here we suppose that "within the institutional context x it is the case that ϕ" if and only if "for every set of agents C, the agents in C that ϕ while functioning as group members in the institutional context x". Formally, for x ∈ INST :

[x] ϕ def = C∈2 AGT ⋆ [C:x] ϕ
It is straightforward to prove that [x] are normal modal operators. Given the previous analysis, a fact is true w.r.t. an institutional context if and only if such fact is accepted by all the agents while they function as group members (hence the performativity and the reflexivity principles are maintained). Moreover, following Theorem 1, this group acceptance is the object of a common belief.

At this point, it might be objected that there are facts which are true in an institutional context but only "special" group members in the institution are aware of them. For instance, there are laws in every country which known only by the specialists of the domain (lawyers, judges, members of the parliament, etc.). Aren't these facts true notwithstanding that many group members are not aware of them?

In order to resist to this objection recall that, at this stage, our model applies to the basic informal institutions of a society. Relative to this restriction, the proposed assumption is justified because, w.r.t. these institutions, there is no other special institutional contexts in which the agents have the power to create and eliminate institutional facts characterizing the institution itself (i.e. nobody has the power to change the rules for promising). It is in fact peculiar of legal (formal) institutions to create such a specialized meta-context in which the agents have special powers to interpret and modify the institution itself. Given the aims of this paper, we leave this special case for future work.

Finally, the following abbreviation is defined:

[Univ ] ϕ def = x∈INST [x] ϕ
which stands for is universally accepted as true".

Contextual conditionals

From the concept of truth with respect to an institutional context a notion of contextual conditional can be defined. A contextual conditional is a material implication of the form ϕ → ψ in scope of an operator [x]. A contextual conditional is a local one, that is, a conditional that is not universally valid while it is accepted by the group members in a specific institutional context. More precisely, we exclude the situation in which [Univ ] (ϕ → ψ) is true.

EXAMPLE 6. Let consider the institutional context of gestural lan-

There exists a contextual conditional in this language according to which, the nodding gesture "counts as" an endorsement of what the speaker is suggesting. This conditional is formally expressed by the construction [gesture] (nodding → yes). It is clear that this kind of conditional is not universally valid ( e.g. in a difcultural context the same gesture may express exactly the opposite fact). Thus, ¬ [Univ ] (nodding → yes) holds.

More generally, for every x ∈ INST we define the following abbreviation:

ϕ x ⊲ ψ def = [x] (ϕ → ψ) ∧ ¬ [Univ ] (ϕ → ψ) ϕ x
⊲ ψ stands for "in the institutional context x, if ϕ then ψ". Although space restrictions prevent from presenting and discussing all relevant properties of our construction ϕ x ⊲ ψ, it is interesting to note that ϕ x ⊲ ψ satisfies some intuitive properties of count-as conditionals as isolated in [START_REF] Jones | A formal characterization institutionalised power[END_REF].

THEOREM. For every x ∈ INST :

From ⊢AL (ϕ2 ↔ ϕ3) infer ⊢AL (ϕ1

x ⊲ ϕ2 ↔ ϕ1 x ⊲ ϕ3) (2) From ⊢AL (ϕ1 ↔ ϕ3) infer ⊢AL (ϕ1 x ⊲ ϕ2 ↔ ϕ3 x ⊲ ϕ2) (3) ⊢AL (ϕ1 x ⊲ ϕ2 ∧ ϕ1 x ⊲ ϕ3) → (ϕ1 x ⊲ (ϕ2 ∧ ϕ3)) (4) ⊢AL (ϕ1 x ⊲ ϕ2 ∧ ϕ3 x ⊲ ϕ2) → ((ϕ1 ∨ ϕ3) x ⊲ ϕ2) (5) ⊢AL (ϕ1 x ⊲ ϕ2 ∧ (ϕ1 ∧ ϕ2) x ⊲ ϕ3) → (ϕ1 x ⊲ ϕ3) (6)
PROOF. We only provide a proof of Theorem 6 as an example. This theorem expresses a property of cumulative transitivity (cut). The other theorems and rules of inference can be proved straightforwardly by definition of ϕ ⊲ ψ) the axioms and rules of inference of AL.

ϕ1 x ⊲ ϕ2 ∧ (ϕ1 ∧ ϕ2) x ⊲ ϕ3 implies [x] (ϕ1 → ϕ2) and [x] ((ϕ1 ∧ ϕ2) → ϕ3) which in turn imply [x] (ϕ1 → ϕ3) (by fact that [x] is normal. Moreover, ϕ1 x ⊲ ϕ2 ∧ (ϕ1 ∧ ϕ2) x ⊲ ϕ3 implies ¬ [Univ ] ((ϕ1 ∧ ϕ2) → ϕ3) which is equivalent to ¬ [Univ ] (¬ϕ1 ∨ ¬ϕ2 ∨ ϕ3).
It is straightforward to prove that operator [Univ ] is also a normal modal operator (space restrictions prevent from giving the proof here). Therefore, ¬ [Univ ] (¬ϕ1 ∨ ¬ϕ2 ∨ ϕ3) implies ¬ [Univ ] (¬ϕ1 ∨ ϕ3) (by the fact that [Univ ] is normal) which in turn is equivalent to ¬ [Univ ] (ϕ1 → ϕ3).

Moreover, we can easily show that our concept of contextual conditional does not satisfy reflexivity, transitivity and weakening of the antecedent, that is, the following three formulas are not valid: andϕ1 x ⊲ ϕ2 → (ϕ1 ∧ ϕ3)

ϕ x ⊲ ϕ, (ϕ1 x ⊲ ϕ2 ∧ ϕ2 x ⊲ ϕ3) → ϕ1 x ⊲ ϕ3,
x ⊲ ϕ2. As discussed in section 6.2 our notion of contextual conditional is similar to the notion of proper classificatory rule given in [START_REF] Grossi | Classificatory aspects of counts-as: An analysis in modal logic[END_REF]. 4

Normative facts

While contextual conditionals are useful to understand the notion of institutional facts, they are not sufficient for a more precise characterization. In fact, as noted in [START_REF] Searle | The Construction of Social Reality[END_REF], institutional facts are always connected to a deontic dimension that up to now is still missing.

In our perspective, a contextual conditional ϕ

x ⊲ ψ can be to represent an institutional fact if and only if the term ψ in the contextual conditional is a fact to which a certain number of obligations and permissions are associated within the institutional context x. In this sense, is an institutional fact with respect to the institutional context x. EXAMPLE 7."Being eighteen years old counts as being of age" is a constitutive rule accepted by a set of agents qua citizens in Italy and "being of age" is an institutional fact with respect to this context. Moreover, to such an institutional fact a certain number of permissions and obligations are associated ( e.g. in Italy if you are of age you have the permission to vote and the obligation to fulfill the military duties). In this sense the constitutive rule "being eighteen years old counts as being of age" connects the institutional fact "being of age" with the brute fact "being eighteen years old" which is a fact intrinsically connected to certain normative facts.

In order to capture this core feature, our logic AL can be appropriately extended by introducing a violation atom V as in Anderson's reduction of deontic logic to alethic logic [START_REF] Anderson | A reduction of deontic logic to alethic modal logic[END_REF] and in dynamic deontic logic [START_REF] Meyer | A different approach to deontic logic: Deontic logic viewed as a variant of dynamic logic[END_REF]. By means of this new formal construct we can specify normative facts (i.e. what it is obligatory and permitted) in a way that respect their being also a kind of attitude-depend fact holding relative to certain attitudes and in a specific institutional context. As far as obligations are concerned, we say that "ϕ is something obligatory within the institutional context x" (noted O(ϕ, x)) if and only if "¬ϕ → V is a contextual conditional in the institutional context x" more specifically, "¬ϕ counts as a violation within the institutional context x". Formally:

O(ϕ, x) def = ¬ϕ x ⊲ V .
As far as permission are concerned we say that "ϕ is something permitted within the institutional context x" (noted P (ϕ, x)) if and only if ¬ϕ is not obligatory within the institutional context x. Formally:

P (ϕ, x) def = ¬O(¬ϕ, x).
Formulas of type O(ϕ, x) and P (ϕ, x) can be conceived as particular instances of so-called regulative rules, that is, rules which specify the ideal behavior of agents in terms of permissions, obligations, and prohibitions. We refer to these rules as normative facts. 5EXAMPLE 8. The formula O(driveCar → ¬RightSide, UK ) is a normative fact in the UK within whose context it is obligatory to drive on the left side of the street ( i.e. "driving a car on the right side of the street counts as violation in UK").

Again, it is important to stress the fact that normative facts, by being represented with a contextual conditional are attitudedependent facts and are intrinsically connected with the acceptance of all the agents qua group members in a specific institutional context.

Institutional facts and constitutive rules

We are now in the position to formalize what an institutional fact is. Let 2 L AL ⋆ = 2 L AL \ {∅} the set of non empty subsets of LAL. From the previous construction ϕ

x ⊲ ψ it is straightforward to come up with a formal characterization of the concept of such a fact. Formally, for every x ∈ INST and ΣO, ΣP ∈ 2 L AL ⋆ :

InstFact Σ O ,Σ P x (ϕ) def = σ∈Σ O O(ϕ → σ, x) ∧ σ ′ ∈Σ P P (ϕ ∧ σ ′ , x) InstFact Σ O ,Σ P
x (ϕ) stands for "ϕ is an institutional fact within the institutional context x characterized by the set of obligations ΣO and the set of permissions ΣP ". EXAMPLE 9. The formula InstFact {military},{vote} Italy (toBeOfAge) stands for "being of age is an institutional fact in the context of Italy and is characterized by the permission to vote in the political and the obligation to fulfill the military duties". 6From the concept of institutional fact we can also formalize the concept of constitutive rule. To this aim, we must make explicit the fact that the term ψ in ϕ x ⊲ ψ an institutional fact to which a set of obligations and a set of permissions are associated. Formally, for every x ∈ INST and ΣO, ΣP ∈ 2 L AL ⋆ :

ConstRule Σ O ,Σ P x (ϕ, ψ) def = ϕ x ⊲ ψ ∧ InstFact Σ O ,Σ P x (ψ) ConstRule Σ O ,Σ P
x (ϕ, ψ) stands for "ϕ counts as ψ is a constitutive rule of institution x where ψ is an institutional fact within the institutional context x characterized by the set of obligations ΣO and the set of permissions ΣP ". EXAMPLE 10.The formula ConstRule {military},{vote} Italy (eighteen, toBeOfAge) stands for "being eighteen years old counts as being of age is a constitutive rule in the context of Italy and being of age is an institutional fact characterized by the permission to vote in the political elections and the obligation to fulfill the military duties". In this sense ConstRule {military},{vote} Italy (eighteen, toBeOfAge) is a specific kind of contextual conditional in which the connection between the institutional fact toBeOfAge and the brute fact eighteen is established. A number of normative facts consisting in obligations and permissions pertain to the institutional fact toBeOfAge, namely O(toBeOfAge → military, Italy) and P (toBeOfAge ∧ vote, Italy).

RELATED WORKS

Link between AL and the G logic

A logic of what is publicly grounded in a group has been introduced in [START_REF] Gaudou | Grounding and the expression of belief[END_REF]: GC ϕ means that "it is publicly grounded for group C that ϕ is true". When C is reduced to a singleton G {i} is identified with the belief à la Hintikka [START_REF] Hintikka | Knowledge and Belief[END_REF]. We can show that it can be viewed as an operator of group belief (in the Gilbert's sense [12] 7 ). In this view, group belief is rational (DG C ), public for every subgroup (SR+ and SR-) and it has been formed by the joint acceptance of all members (WR and CG). Its axiomatics is thus the following one:

(DG C ) GC ϕ → ¬GC ¬ϕ (SR+) GC ϕ → G C ′ GC ϕ, C ′ ⊆ C (SR-) ¬GC ϕ → G C ′ ¬GC ϕ, C ′ ⊆ C (WR) GC ϕ → GC G C ′ ϕ, C ′ ⊆ C and ϕ objective. 8 (CG)
( i∈C GC Giϕ) → GC ϕ of group belief and group acceptance seem to be very close. Thus the idea of expressing the G operator in AL appears intuitive because AL is more expressive, with the notion of context lacking in the G logic. We show in the sequel that AL can subsume the G's logic.

Contrary to our framework, G operator does not take into account various institutional contexts, what expresses that it considers (implicitly) only one. Thus formally we have: GC ϕ ≡ [C:xC ] ϕ, where xC is the only institution whereby C is concerned and where x {i} ≡ λ.

We need both to examine and compare axiomatics. Axioms 4 cannot be expressed in the grounding logic 9 . An axiom such as: GC ϕ → GBϕ, would be too strong because we consider that belief of a subgroup is not related to the uppergroup beliefs (and in particular group belief is totally independent of individual group members beliefs). Some axioms lack in AL to represent the G operators. In particular the axiom D [i:λ] should be generalized to [C:xC ] that agents in C are de facto functioning as group members in the context xC . Moreover axioms (WR) and (CG) express that a group belief is established by a consensus of expressed opinion. They do not have a counterpart in the AL logic, because we are only concerned here by properties of acceptance (not by its formation). (WR) and (CG) could be translated directly.

These three additional axioms are due to the features of the particular context xC : they represent the strong link existing between xC and C. We can note that theorem 1 is also a theorem of the grounding logic. In the sequel, we explore interactions between GC defined as [C:xC ] and general acceptance [C:x], which produces mixed theorems. 7 The proof is based on common features shared by both notions. They are public, commonly believed when established and formed by consensus (i.e. by acceptance of every member). 8 An objective formula is a formula that is not equivalent to a formula under the scope of Gi operator, for each member i of C. This restriction is due to the fact that, by asserting proposition ϕ, an agent expresses that he believes ϕ and thus this belief is automatically grounded for the group thank to public actions hypotheses. 9 Except under his tautological and uninformative form where From previous theorem, we can also prove that:

B = C. As ¬ [C:x] ⊥ → ¬ [B:x] ⊥ (with B ⊆ C)
[C:xC ] ϕ → [C:xC ] [B:xC ] ϕ, with B ⊆ C.
This theorem extends the previous one: ϕ is collectively believed, every subgroup accept it in the context xC (by former theorem), but this acceptance is also collectively believed. This theorem is in fact quite close to axiom (WR) in the grounding logic.

Related works on normative systems

Because of interesting formal similarities and given the space restrictions, we will just compare AL with [START_REF] Grossi | Classificatory aspects of counts-as: An analysis in modal logic[END_REF] in which a modal logic for the formalization of count-as assertions and the specification of normative systems has been proposed. This logic is based on a set of modal operators [x] * where the index x is in a set of indexes C0. 10 An index x is supposed to denote a certain institutional context (or normative system). Operators [x] * are similar to our operators [x] defined in Section 5.1. A formula [x] * approximately stands for "in the institutional context/normative system x it is the case that ϕ". An operator [u] * is also used for denoting facts which universally hold. The set C = C0 ∪ {u} is given by adding index u to the set of indexes C0. Differently from our logic where the contextual operator [x] is built on the notion of group acceptance, in Grossi's logic the contextual operator [x] * is given as a primitive operator. Operators [x] * and [u] * are exploited in Grossi's logic to define contextual conditionals called proper classificatory rules noted by ϕ ⇒ cl+ i ψ which is an abbreviation of [x] * (ϕ → ψ) ∧ ¬ [u] * (ϕ → ψ) and is meant to stand for "ϕ counts as ψ in the normative system x". The construction ϕ ⇒ cl+ i ψ is similar to our ϕ x ⊲ ψ. 11 Operator [u] * is S5 and the logic is supposed to satisfy the following additional principles. For any x, y ∈ C:

1. [x] * ϕ → [y] * [x] * ϕ 3. [u] * ϕ → [x] * ϕ 2. ¬ [x] * ϕ → [y] * ¬ [x] * ϕ 4. [u] * ϕ → ϕ
According to the both principles 1. and 2., truth and falsehood in institutional contexts/normative systems are absolute because they remain invariant even if evaluated from another institutional context/normative system. This means that every normative system y has full access to all facts which are true in a different normative system x. These two principles are in our view criticizable because they rely on the very counter-intuitive assumption that all facts true in an institutional context are public to all other institutional contexts. But, what does it mean that a fact is known by an institution? Our aim here is to show that such an assumption can be disambiguated in our logical framework. Concerning the principle 4., it says that: if ϕ universally holds then ϕ is true. This principle is also criticizable in our opinion. For instance, during the 7th-6th century BC people believed that the earth was flat. But it has never been the case that earth was/is/will be flat.

More generally, if we suppose that: Axioms 4

[x] ϕ → [y] [x] ϕ • ¬ [x] ϕ → [y] ¬ [x] ϕ • [Univ ] ϕ → [x] ϕ • [Univ ] satisfies

CONCLUSION

Let's take stock. We have started the paper by raising the challenge of autonomy at the level of MASs, so that they will be able to bind themselves in ways that further the achievement of collective goods in dynamic and uncertain environments as human societies do.

As a first step to meet this challenge, we have proposed the AL logic in which the agents' attitudes qua group members can be analyzed. Given the properties of a demystified notion of group acceptance in an institutional context, we have provided an analysis of the kind of attitude-dependent facts typical of institutions. In particular, we have introduced a notion of obligation and permission with respect to an institutional context (i.e. so-called normative facts). Then, we have defined institutional facts. In our perspective an institutional fact within the institutional context x is a fact to which a number of obligations and permissions are (contextually) associated. Finally, we have formalized the concept of constitutive rule, that is, a rule which is responsible for the connection between an institutional fact and a brute physical fact. In our view, a constitutive rule is a rule of type "ϕ counts as ψ in the institutional context x" where ψ denotes an institutional fact within the institutional context x. While such rules are usually defined from the external perspective of a normative system or institution, we have, once again, anchored these rules in the agents' attitudes.

Although the present model is focused on the neglected layer of informal institutions, it still lacks sufficient expressiveness to represent the phenomenon of "institutionalized power" [START_REF] Jones | A formal characterization institutionalised power[END_REF] which is, of course, crucial also within this kind of institutions. In order to cope with limitation, in future work, we will expand AL with Propositional Dynamic Logic (PDL) in order to be able to talk about actions within our language. Moreover, a first kind of dynamics will be studied in which agents, qua group members in specific institutional contexts, will be able to create new institutional facts. Given the way we have modeled such facts, the agents will update and revise their own deontic commitments accordingly. This extension will further give the opportunity for a foundation of artificial legal institutions and for their connections with informal ones. In fact the "basic norm" [START_REF] Kelsen | Pure Theory of Law[END_REF], i.e. the basic informal institution that provides the validity of legal systems, will be represented on the model of the other informal institutions. Representing the "basic norm" is in fact the crucial step for making it possible for a MAS to create and maintain by itself a legal system that is acknowledged as valid from the agents themselves.

The long term project is then to provide a three-layered model in which legal institutions, social institutions, and the socio-cognitive relations between the agents dynamically interact in order to enable institutional change and adaptation.

Axioms 4 [

 4 C :x],[B :y] and 5 [C :x],[B :y] together correspond to the following semantic property of Kripke models. For every w ∈ W and C:x, B:y ∈ ∆, if B ⊆ C then:

  [i:λ] is KD45 (Indeed, besides satisfying Axiom D, it also satisfies Axioms 4 and 5 as particular instances of Axioms 4 [C :x],[B :y] and 5 [C :x],[B :y] where C = B = {i} and x = y = λ.) We call AL (Acceptance Logic) the logic axiomatized by the four principles 4 [C :x],[B :y] , 5 [C :x],[B :y] , Inc [C :x],[B :x] , D [i:λ]

  [C :x],[B :y] with B:y = i:λ for each i ∈ C, which implies that [C:x] → i∈C [i:λ] [C:x] ϕ. The latter the case k = 1 by definition of EC .

  [C :x],[B :y] and 5 [C :x],[B :y] generalizations of the (SR+) and (SR-) for contexts xC and xB instead of x and y. They represent the public nature of both notions. Axiom Inc [C :x],[B :x]

  [C :x],[B :y] and 5 [C :x],[B :y] studied in Section 3.3 are also valid for B C; the T axiom is valid for [Univ ] operator (in a similar way of the previous principle 4.); and the following translations of Grossi's operators [x] * and [u] * into our logic AL are given tr([x] * ϕ) = [x] ϕ • tr([u] * ϕ) = [Univ ] ϕ, we can prove that the translations into AL of all Grossi's axioms are AL theorems. This is shown by the following theorem. THEOREM. Suppose that: i) [Univ ] ϕ → ϕ is valid, and that for every C:x, B:y ∈ ∆, ii) [C:x] ϕ → [B:y] [C:x] ϕ and iii) ¬ [C:x] ϕ → [B:y] ¬ [C:x] ϕ are valid in AL. Thus, the following properties can be inferred in AL: •

  all Axioms and rules of inference of the system S5 PROOF. We only provide a proof of the last item of the theorem. The other items can be proved in a similar way. First of all [Univ ] is a normal modal operator by definition as a conjunction of normal modal operators [x]. We have property T [Univ ] by Hypothesis i). We only need to prove that 4 [Univ ] and 5 [Univ ] can be inferred from the hypotheses. From Hypothesis ii) we can deduce that[C:x] ϕ → B:y∈∆ [B:y] [C:x] ϕ which is equivalent (by definition of [Univ ] ϕ) to [C:x] ϕ → [Univ ] [C:x] ϕ, which entails C:x∈∆ [C:x] ϕ → C:x∈∆ [Univ ] [C:x] ϕ, which is equivalent to [Univ ] ϕ → [Univ ] [Univ ] ϕ (i.e. 4 [Univ ] ). 5 [Univ ] (i.e. ¬ [Univ ] ϕ → [Univ ] ¬ [Univ ] ϕ) can be inferred from Hypothesis iii) in a similar way.

  is a theorem of AL (the proof can be easily built from Inc [C :x],[B :x] ), we have: ¬ [C:xC ] ⊥ → ¬ [B:xC ] ⊥, with B ⊆ C, which means that every agent in subsets of C are also functioning as group members in the context xC . Moreover, as ¬ [C:xC ] ⊥ is valid, Axiom Inc [C :x],[B :x] is reduced to the following theorem: [C:xC ] ϕ → [B:xC ] ϕ, with B ⊆ C. Thus if ϕ is a belief of the group C, every subgroup accept it in the context of xC ; there is a group acceptance on what is collectively believed. For example, if it is collectively believed by the activists that the aim of Greenpeace is to protect the Earth then in the context of Greenpeace every subgroup must accept it. This does not implies anything about subgroup and individual beliefs.

  The relevant question is: under what additional assumptions formulas[x] ϕ → [y] [x] ϕ and ¬ [x] ϕ → [y] ¬ [x]ϕ can be inferred in our logic? On the one hand, it is easy to prove that the principles given in Section 3.3 are not sufficient to infer such formulas. Indeed, formulas[x] ϕ ∧ ¬ [y] [x] ϕ and ¬ [x] ϕ ∧ ¬ [y] ¬ [x] ϕare satisfiable in AL. On the other hand, it is straightforward to show that: if Axioms 4 [C :x],[B :y] and 5 [C :x],[B :y] are weakened by supposing that they also hold for B C, then formulas[x] ϕ → [y] [x] ϕ and ¬ [x] ϕ → [y] ¬ [x]ϕ can be inferred. This means that in our logic Grossi's properties can be derived under the assumption that, given two arbitrary sets of agents B and C, agents in B has access to all facts that agents in C accept (do not accept), while functioning as group members in a certain institutional context x. That is, given arbitrary set of agents C, if agents in C that ϕ while functioning as a group members in the institutional context x then this fact is public in such a way that all other agents outside C accept that agents in C that ϕ while functioning as group members in the institutional context x.

It is in fact specific to legal institutions to have specialized agents empowered to change the institution itself on behalf of everybody else (see Section 5.1).

For the sake of compactness we prefer to adopt this non-standard notation for doxastic operators.

We refer to[START_REF] Grossi | Classificatory aspects of counts-as: An analysis in modal logic[END_REF] for interesting arguments concerning why proper classificatory rules should not necessarily satisfy reflexivity, transitivity and weakening of the antecedent.

The distinction between regulative rule and constitutive rule has been emphasized by Searle[START_REF] Searle | The Construction of Social Reality[END_REF] and then modelled in logic by several authors. For an example see[START_REF] Boella | Regulative and constitutive norms in normative multiagent systems[END_REF].

[START_REF] Cohen | An essay on belief and acceptance[END_REF] A more precise formulation of this example needs a representation of the right relation which is, however, beyond the scope of this article. See[START_REF] Makinson | On the formal representation of rights relations[END_REF] for more details.

Here we use the notation [x] * in order to distinguish Grossi's operators from our operators [x].

[START_REF] Gaudou | Grounding and the expression of belief[END_REF] The author distinguishes proper classificatory rules from mere classificatory rules and constitutive rules. Differently from classificatory rules, proper classificatory rules are rules which would not hold without the normative system/institution stating them. In[START_REF] Grossi | Counts-as: Classification or constitution? An answer using modal logic[END_REF] a further distinction between classificatory rules and constitutive rules is given.

ACKNOWLEDGMENTS

Emiliano Lorini is financed by the project ForTrust (Social Trust Analysis and Formalization) funded by the french Agence Nationale de la Recherche (ANR). Luca Tummolini is supported by the project SOCCOP (The Social and Mental Dynamics of Cooperation), funded by the ESF.


Jean-François Letard

H Daubric

HERVÉ DAUBRIC b Christophe Cantin

Janis Kliava

Yacine Bouhedja

Olivier Nguyen

Olivier Kahn

Jean-François Létard

Yacine A Bouhedja't

Electron paramagnetic resonance and LIESST effect in spin-crossover cis-bis(thiocyanato)bis[N-(2-pyridylmethylene

Keywords: Electron Paramagnetic Resonance, Spin-crossover, Bistability

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

ln iron(II) coordination compounds spin conversion from a low-spin (LS, S = 0) state to a high-spin (HS, S = 2) state has been the subject of many studies during the last three decades. Spin transition can be induced by a change of temperature or pressure and or by light irradiation. 1 Recently, the use of such materials as molecular-based memory devices and displays has been suggested. 2 In fact, several iron(ll) derivatives show a bistability : the LS to HS transition occurs at a higher temperature than the HS to LS transition, forming a thermal hysteresis loop. In this event, a number of questions must be answered, concerning structural modifications accompanying the spin transition. Recently. we have synthesized strongly cooperative spin-crossover assemblies of mononuclear molecules, based on strong interactions through the stacking of aromatic rings: Along this line, we already reported two mononuclear spin-crossover derivatives ; i) cis bis( thiocyanato )bis(N-2 • -pyridy lmethy lene )-4-(phenylethyny !)aniline iron(ll), Fe(PM-PEA) i (NCSh, with a very large hysteresis as well as relatively high temperatures of transition: T 11 J = 194 K, T 1 ,iÎ = 231 K" and ii) cis-bis( thiocyanato }bis(N-2 • -pyridylmethylene }-4-(aminobiphenyl) iron(ll). Fe(PM-BiA) i (NCS) 2 which undergoes an unusually abrupt transition with a very narrow and sharp hysteresis: T 112 ,I, = 168 K, T 112 t = 173 K. 4 The crystal structure of Fe(PM-BiA)z(NCS) 1 solved both at room temperature (HS state) and at 140 K (LS state) shows that the space group is not changed during the spin transition (orthorombic Pccn) and that the main structural feature is the reorganization of the FeN ,, core toward a more regular octahedron in the LS state.••" Recently, we have reported the LIESST (Light lnduced Ex.cited Spin State-Trapping) in Fe(PM-BiA) i (NCS) 2 recorded at 10 K within a SQUID cavity. 4 A slow rate of quantum mechanical tunneling from the metastable HS state to the LS state bas been observed and the lîmit temperature was estimated around 80 K. This behavior îs correlated with an unusually large change in Fe-N(organic ligand) bond lengths in the course of the spin transition.

Meanwhile for Fe(PM-BiA) 2 (NCS) 2 depending on the synthetic method, a second phase can be obtained, exhibiting a complete and graduai spin transition with T11 2,!, = 205 K and T 1 nt = 209 K. 5 Up to now, single crystal of phase II has not be obtained. A powder diffraction showed that at room temperature the best cell was obtained with a monoclinic phase (P2/c). 4 In the present work, we report the EPR spectra of the phases I and II. A direct observation of spin transitions in iron(II) compounds by EPR is difficult, if not impossible, since the Fe z+ ion is "EPR silent" not only in the LS diamagnetic state, but, in practice, also in its HS paramagnetic state because of a very short spin-lattice relaxation time and/or a very large zero field splitting. [START_REF] Servant | First Asia-Pacific EPR/ESR Symposium[END_REF] The spin transition experienced by the Fe i+ ions can however be detected by EPR in an indirect way, from an anal y sis of modifications of EPR spectra of some foreign paramagnetic ions doped in the spin-transition compound. Up to now, two foreign paramagnetic ions have been systematically used in the EPR studies of iron(II) spin crossover compounds, viz., Cu 2 + (3d 9 We also present the magnetic properties of Fe(PM-BiA) i (NCS) 2 doped with Cu l+ , Zn 2 • and Mn 2 •. The LIESST effect recorded at 10 K within a SQUID cavity for two naturally doped samples (phases I and Il) and an intentionally doped with 21.6 % of manganese sample are compared. In fact, doping spin transition compounds with foreign species can modify their characteristics (transition temperatures, hysteresis loop, critical temperature limit of LIESST effect. .. ), which can be very interesting for potential applications.

EXPERIMENTAL SECTION

The phases I and II of Fe(PM-BiA)/NCS)2 were found to be naturally doped with manganese ions. In ail cases, the y11s (T) curves were deduced from the magnetic susceptibility measurements as X M T/((X M T)m], where X M T is the molar magnetic susceptibility at a temperature T and (x M T\ 1 r is the high-temperature limit of X M T. The YHs (T) curves ofintentionally doped compounds are similar to those of phase II of Fe(PM-BiA)z(NCS) i . The spin transition is complete and graduai, and successive thermal cycles do not modify the thermal hysteresis loop.

EPR spectra

We only present the EPR spectra of the Mn 2 ' ions due to the fact that i) both phases (1 and Il) of Fe(PM-BiA) i (NCSh show typical spectra and ii) the investigation of intentionally doped compound with Cu 2+ reveals only some residual traces in X-and Q-bands at low temperature where the iron center is in LS state. The EPR spectra in X-band (v = 9.317 GHz) and in Q-band (v = 34 GHz) of phase l are presented in Figure 2 as a function of temperature. The EPR spectra change slightly from room temperature down to 174 K, then a pronounced change occurs in the region 174 -168 K. At lower temperature EPR spectra remain alrnost constant. ln the warming mode a sirnilar behavior is observed: from 100 to 171 K the spectra do not appreciably change, then around 171 -177 K some transformations occur and at higher temperature the EPR spectra remain nearly identical.

For the phase II, EPR spectra have been observed only in Q-band (figure 3). The absence of X-band EPR spectra seems to indicate a very low concentration of Mn 2 ' ions. The EPR spectra of Mn 2 • ions in a powder system can be described by the orthorhombic spin Hamiltonian (eq. 1), where g ,., g, = 2.0023, S = 5/2 and / = 5/2 . The g-factor and the hyperfine structure (hfs) constant A for Mn 2+ are isotropie with a good accuracy. For the g = 2.0 hfs transitions are hroken down, so that each fs multiplet, besides six "allowed" hfs components, m=±S/2,±3/2,±1/2, !:.m=O, features a number of "forbidden" hfs components with t:.m=±l,±2, ... ,±5.

(eq. l)

In X-band the EPR spectra of phase I (figure 2 The improvement of resolution in the Q-band spectra can be explained by the form of the expression of the resonance magnetic field B, for a central fs transition (-1/2,mJ..+(+l/2,m+6mJ (eq. 2) calculated to third order in perturbation theory. 15 16 , 1 J ' 8A ' 1 j l g/JB, =g/3B 0 (m, t:.m)+ gf) B , (l,îl--21-'1 )-(2m+6m) (g/J B ,) 2 (91,ll--2PI)

4A' (eq. 2)

-(2m+ 6m)(l + t:.m) (g/1B, ) ' u where A , ' A 1

gf)8 0 (m.t:.m) = hv-(2m+ t:.m) 2 + [(2rn + 6m)" + (17 -t:.m)" -324] Sg/JB, ' ' A } -[4(m+ t:.rn) +4rn -(2m+t:.m)(65+ 1346m)J 16{g/JB,) 2 l..tl 2 = sin" .9[(D + E cos2q,) ! cos ! 9 + E ! sin 2 2q,] I l = \[Dsin 2 .9-E(I + cos 2 .9)cos2q,f + 4E 2 cos 2 .9sin 2 2q,) / 16 CT= [D(3cos 2 .9-l)-3Esin 2 .9cos2q,]/2

The width of the resonance features in a powder EPR spectrum is mainly due to angular variations of their resonance fields. In passing from X-to Q bands, the central lines of fs transition are narrowed approximately in proportion to the ratio of resonanoe frequencies of the two bands. The increase nf the lateral pans resolution of the X-band (figure 2) in the HS state (T .J, = 174 K) in regard to the LS state (Tt = 174 K), can not be explain by a simple line broadening caused by spin-spin interactions between Mn" and paramagnetic HS Fe' -ions or by a fast spin-lattice relaxation of the latter ions. It seems that the key factors to explain such transformations could be an indication of a structural transformation in the environment of the Mn'' during the spin transition of the Fe'' ion.

In order to get a more quantitative insight in this transformation, simulations of the EPR spectra in the LS and HS states of Fe'+ have been carried out.

The laboratory-developed simulation program takes into account ail "allowed" fs and both "allowed" and "forbidden" hfs components. Their intensities are determined using the Bir's method,' 0 with W the transition intensity averaged over ail directions of the microwave magnetic field and F the lineshape, with an orientation-dependent linewidth l!JJ, including broadening due to spin-lattice and spin-spin interactions as well as to static disorder" (eq. 3). The notations :J and ip are the polar and azimuthal angles of the directing magnetic field B with the axes of the spin Hamiltonian (eq. 1) and v is the frequency of the microwave magnetic field.

Figure 5 presents a comparison between experimental and best-fit computer-simulated EPR spectra for the naturally doped sarnple in phase I. The best-fit EPR parameters ti.1r Mn:-ions in Q-and X-bands are listed in Table 1 for the two phases and for the intentionally doped compound. The good accuracy of the fils between the experimental and the compute simulated EPR spectra and the similar parameters obtain by the fit of the Q and X-bands provide to us some confidence in the description of the EPR spectra.

Table l. Best-fit EPR parameters for Mn 2 + ions in Q-and X-band. The numbers in parenthesis are the standard errors.

(LS) (HS) (LS) (HS) X-band Q-band Q-hand Q-band Q-band Q-band Q-band K 2.000 (
E(I0--1 c111•l1 55 5) 162(5) 418(5) 415(10) 310(5) 385( 10) 310(5)
,l / = EID)

113 1/3 1125 1/4 1/4 1/4 1/4
From Table 1, it is seen that the g and A values remain constant during the spin transition, respectively 2.0 ± 0.003 and -82 ± 1 10 -4 cm• 1 , whereas the fs parameters (D and E) change significantly. The calculated ratio À ( = EID) shows that for the phase 1, the Mn 2 ' ions are subject to maximal degree in rhombic distortion (À"" l/3) when the Fe i+ center is in LS state, whereas a quasi-axial symmetry (À"' ! / 25) is obtain when Fe 2

• is in HS state. The axial fs parameter D in the HS state almost three times larger than in the LS state of Fe 2 + reveals a strong axial distortion arising at the Mn'+ sites in 1he HS state of Fe" ions. For the phase II and the intentionally doped sample. the À-parameter (1/4) of the HS state indicates more rhombic distortions of Mn'+ ions than for 1he phase l (l/25). The structures of the phases I and Il are very different. The EPR spectra show that in the course of the spin conversion of the iron(Il) centers the environment of the manganese ions are strongly modified. Two reasons for such modifications can be envisaged; i) some

LIESST effeçt

The LIESST experiments were carried out on the powder sample of the Fe(PM-BiAMNCS). in phases I and li, as well as for the Fe 021 µ11o. 784 (PM BiA)z(NCS) 2 . A typical experiment is shown in Figure 6. The sample is first slowly cooled from ca. 100 K down to 10 K, and the X:MÎ versus T curve confinns that Fe 2 • ions are in LS state. At 10 K, the sample is then irradiated (mode ON) with a red light (À.= 647.1 -676.4 nm) for one hour with an intensity of 50 m W; XM T was found to increase rapidly, then to reach a value of ca. 0.7 cm 3 K mol" 1 , indicating a partial conversion("" 20 %) of the LS state into the HS state, according to the well-known LIESST process. 17 Without further irradiation (mode OFF), the temperature was slowly increased, and the temperature dependence of XM T was recorded. XM T remains nearly constant as T increases from IO K up to 70 K. The XM T value at 70 K is 0.61 cm 3 K mor 1 , corresponding to about 17 % of HS molecules, then drops rapidly as the temperature is further increased, and reaches a value close to zero around 80 K. The critical LIESST temperature may then be defined as Tue,st "" 80 K for the Fe(PM-BiA) i (NCS) 2 in phase 1. For Fe(PM-BiA)i (NCS)z in phase Il and Fe 0 _ 21 µ11o. 784 (PM-BiA)z(NCS) 2 the LIESST effect has also been observed at 10 K. The increase of the temperature reveals a critical value for the observation of " § (Light lnduced Thennal Hysteresis).' Figure 7 shows the magnetic behavior of f-e(PM-BiA),(NCS) i in phases I and li and Fe,u 16 Mn,, 784 (PM-BiA)iNCS) 2 . This figure shows that for the three compounds a LITH effect is observed. The temperature dependence of XM T under irradiation (in mode ON) is rather similar to that in mode OFF when the temperature is increased from 10 K to l 00 K ( figures 7 and 6). The tempcrature dcpendence of XM T when cooling the sample in mode ON reveals a new type of hysteresis. For instance for the phase L the HS molar fraction remains close to zero as T is lowered down to 60 K. then progressively incrcascs as T is further lowered down to 10 K. and eventually reachcs 20 %. lt follows that the two curves in mode ON in the warming and cooling regime. respectively. show a bistability.

Figure 1 .

 1 Figure 1. Fe(PM-BiA)z(NCS)z in(□ a □) phase I and (o o o) phase II and Fe 1 .,M,(PM-BiA)z(NCS) 2 with (a) (♦ ♦ ♦) x = 0.01 and ( + + +) x = 0.21 of Mn/Fe; (b) (♦ ♦ ♦) x = 0.1 of Zn/Fe and (c) (♦ ♦ ♦) x = 0.001 of Cu/Fe.

�Figure 2

 2 Figure 2. Xand Q-bands EPR spectra of the naturally doped Fe(PM BiA) 2 (NCS) 2 in phase I in the warming (TÎ) and cooling (T -1,) modes.

  From room temperature down to 130 K a continuous transformation of the EPR spectra have been recorded ; at 290 K mT) the spectra are constituted of six broad bands, whereas at 130 K some slight lines structure appear. The EPR spectra recorded for intentionally doped compound Fe 0 99Mn 0 m in X-and in Q-bands are shown in figure 4.

Figure 3 .Figure 4

 34 Figure 3. Q-band EPR spectra of the naturally doped Fe(PM BiA) i (NCS) 2 in phase II in the warming (T'Î) mode.

  centres the condition I D!, JEl, I AI « gf)B holds. The presence of the fine structure (fs) terms in equation 1 gives rise to five "allowed" fs multiplets (M =±5121..+(M =±3121, (M = ±3/ 2I ..+ (M = ±1/2I and (M =-1121..+(M =+1121. Meanwhile, for comparable absolute values offs and hfs parameters, the selection rules governing the intensities of various

  ) are resolved only for hfs components of the central fs multiplet, (M=-1121..+(M=+l/21 at low temperature where the Fe 2+ ions are in LS state, whereas in the HS state of Fe 2 • the resolution of the hfs components is lost. In Q-band a quite well resolved EPR spectrum is observed already at room temperature (figs. 2-4).

  spin-spin interactions between Mn 2+ (S = 5/2) foreign ions and Fe 2 • ions in HS state and ii) some structural change in the course of the spin transition. The former reason can be excluded in a first hypothesis due to the fact that in the crystal structure bath in HS and LS states of Fe(PM-BiA) i (NCS) 2 the metal-metal distances are around lO A. 4 • 6 Thus, we can conclude that any structural changes accompanying the spin transition are sure short-range ones (inside the Fe 2 • complex) but also long-range ones involving the whole structure. Such modifications are in contrast with the previous results obtained for triazole-based compounds were only a rather moderate line broadening bas been observed in the HS state of Fe l+ .

  Figure s. Experimental and computer simulated EPR spectra for the narurally doped phase I Fe(PM BiA) i (NCS),.

Figure 6 .

 6 Figure 6. X M T versus T plots for Fe(PM-BiA) i (NCS) 2 in phase I ; ( ◊ ◊ . ◊ ) without irradiation, ( o o o) under irradiation at 10 K and(•••) under irradiation.

Figure 7 .

 7 Figure 7. Magnetic properties recorded under light irradiation for (□ D □) in phase I. (o o o) in phase Il and (■ ■ ■) for Fe 0 216MI\, 784 (PM-BiA) 2 (NCS) 2 •

784 C J8 H 28 N 6 S 2 C. 66.29: H, 4.10; N, 12.21; S, 9.32; Fe, 6.65; Mn, 1.44. Found: C, 66.23; N, 12.27; Fe, 6.83; Mn, 1.

	Elemental analysis performed by the Service Central
	d'Analyse (CNRS) in Vernaison (France) has shown a doping level of ca.
	0.1 % Mn/Fe. Besides. intentionally doped compounds with 1 % and
	21.6 % of Mn/Fe were prepared. Anal. Calcd. for Fe 0 _ 99 Mn.-,niC J8 H 18 N 0 S i : C. 66.28; H. 4.10: N. 12.20: S. 9.31: Fe. 8.03: Mn. 0.08. Found: C. 66.33; N.

12

.17: Fe. 7.65; Mn. 0.10: Anal. Calcd. for Fe 0216 M1',

For the phase II and Fe,12 10 M11o 78 /PM-BiAl i (NCS) 2 , the time dependence can be studied only until 40 K (figure 9). The relaxation curves can be satisfactory filled with an Arrhenius law. lt is well established that the HS ➔ LS relaxation for spin transition compounds in the solid state is strongly influenced by cooperative effects. 19 Hauser demonstrated by optical spectroscopy that in the highly diluted single crystal [Fe,Zn 1 .x(ptz) 6 ](BF4)z (x = 0.1) the YHs versus T curves followed first order kinetics. whereas for pure [Fe(ptz) 6 ](BF 4 h compound sigmoidal relaxation curves were found. Such curves can be The result recorded for the Fe(PM-BiAh(NCS) i is in perfect agreement with Hauser's observations: a sigmoidal relaxation is measured for the phase l, which presents an abrupt spin transition (highly cooperative system), whereas an cxponential behavior is observed for the phase Il. where a graduai spin conversion occurs (weakly coopcrativc system). Ncvertheless, according to the previous work performed on 1Fe(ptz) ( ,)(BF 4 h and [Fe,Zn 1 _.(ptz) 6 ](BF 4 )i, a faster HS ➔ LS relaxation rate is expected for the phase I due to the sigmoidal hehavior. Our findings point out that the relaxation rate strongly depends on the crystal structure.

CONCLUSIONS

Phase I of Fe(PM-BiA),(NCS) 1 exhibits an exceptionally abrupt thermally induced spin transition, whereas phase II and doped compounds show graduai conversions. The EPR spectra reveals that the structure of the two phases of Fe(PM-BiA} i (NCS) 2 are totally different. A pronounced change in the LIESST behavior is also measured between the two phases. The rate of relaxation for the phase I is slower than for the phase Il. Recently, we have suggested that the slow rate of relaxation of the phase I was due to an unusually large change in Fe-N bond lengths between LS and HS states. The LIESST data indicate that the modification of the FeN 6 core is more pronounced for the phase II than for the phase I.