Mohua Banerjee

Didier Dubois

Lluis Godo

Possibilistic vs. Relational Semantics for Logics of Incomplete Information

Keywords: Modal logic, possibility theory, epistemic logic, rough sets

This paper proposes an extension of the MEL logic to a language containing modal formulae of depth 0 or 1 only. MEL is a logic of incomplete information where an agent can express both beliefs and explicitly ignored facts, that only uses modal formulae of depth 1, and no objective ones. The extended logic, called MEL + has the same axioms as, and is in some sense equivalent to, S5 with a restricted language, but with the same expressive power. The semantics is not based on Kripke models with equivalence relations, but on pairs made of an interpretation (representing the real state of facts) and a non-empty set of possible interpretations (representing an epistemic state). Soundness and completeness are established. We provide a rationale for using our approach when an agent reasons about what is known of the epistemic state of another agent and compares it with what is known about the real world. Our approach can be viewed as an alternative to the basic epistemic logic not concerned with introspection. We discuss the difference with S5 used as a logic for rough sets, and the similarity with some previous non-monotonic logics of knowledge is highlighted.

Introduction

In the recent past [START_REF] Banerjee | A Simple Modal Logic for Reasoning about Revealed Beliefs[END_REF][START_REF] Banerjee | A simple logic for reasoning about incomplete knowledge[END_REF], some efforts have been made to relate possibility theory and modal logic under the simplest possible syntactic and semantic framework. The resulting language of this logic called MEL is a fragment of KD where no objective formula (not preceded by a modality) appears, and where modalities cannot be nested. Models of MEL are simply non-empty subsets of interpretations of some standard propositional language. They represent the possible epistemic states of some agent. Then the necessity modality represents belief, and an agent believes a proposition if and only if the latter is true in all propositional interpretations compatible with the agent's epistemic state. This logic has axioms K and D, and another axiom saying that the agent believes all tautologies. Each epistemic state can be interpreted as a Boolean possibility distribution, and it can be shown that the necessity modality is a Boolean necessity measure. This attempt to relate possibility theory and modal logic contrasts with other previous connections between these notions, involving more elaborate constructions [START_REF] Del Cerro | A modal analysis of possibility theory[END_REF][START_REF] Hájek | A qualitative fuzzy possibilistic logic[END_REF][START_REF] Boutilier | Modal logics for qualitative possibility theory[END_REF][START_REF] Hájek | On modal logics for qualitative possibility in a fuzzy setting[END_REF].

The MEL logic has a language which is a fragment of the language of the logic S5 often used as an epistemic logic. Its semantics does not use accessibility relations explicitly, and is much simpler. Moreover one cannot express the introspective axioms 4 and 5 in MEL as the nesting of modalities is not possible in its language. Nevertheless, two issues are worth investigating in connection with MEL:

-Is it possible to extend the language of MEL to objective non-modal formulae, while preserving the same style of semantics as MEL ? -Given that standard epistemic logics such as S5 rely on accessibility (equivalence) relations, what is the connection between these two semantics ?

The aim of this paper is to provide an answer to both questions. In the next section, a brief presentation of the logic MEL is recalled. Then an extension of the MEL language to objective formulae and the corresponding extension of the MEL semantics is proposed. Finally we discuss the usual Kripke style semantics of S5 and compare them to our semantics, which we consider more natural than the relational semantics based on indiscernible possible worlds according to an equivalence relation. We claim that the latter semantics is more fit to rough sets and can account for the idea of forgetting [START_REF] Lang | A forgetting-based approach[END_REF]. Moreover, we show that our approach comes closer to some knowledge logics proposed in the early 1990's, in the area of non-monotonic reasoning.

MEL, A Simple Epistemic Logic

The usual truth values true (1) and false (0) assigned to propositions are of ontological nature (which means that they are part of the definition of what we call proposition), whereas assigning to a proposition a value whose meaning is expressed by the word unknown sounds like having an epistemic nature: it reveals a knowledge state according to which the truth value of a proposition (in the usual Boolean sense) in a given situation is out of reach (for instance one cannot compute it, either by lack of computing power, or due to a sheer lack of information). It corresponds to an epistemic state for an agent that can neither assert the truth of a Boolean proposition nor its falsity. Admitting that the concept of "unknown" refers to a knowledge state rather than to an ontic truth value, we may keep the logic Boolean and add to its syntax the capability of stating that we ignore the truth value (1 or 0) of propositions. The natural framework to syntactically encode statements about knowledge states of propositional logic (PL) statements is modal logic, and in particular, the logic KD. Nevertheless, if one only wants to reason about e.g. the beliefs of another agent, a very limited fragment of this language is needed. The logic MEL [START_REF] Banerjee | A Simple Modal Logic for Reasoning about Revealed Beliefs[END_REF][START_REF] Banerjee | A simple logic for reasoning about incomplete knowledge[END_REF] was defined for that purpose.

Let us consider L to be a standard propositional language built up from a finite set of propositional variables V = {p 1 , . . . , p k } along with the Boolean connectives of conjunction and negation ¬. As usual, a disjunction ϕ ∨ ψ stands for ¬(¬ϕ ∧ ¬ψ) and an implication ϕ → ψ stands for ¬ϕ ∨ ψ. Further we use ⊤ to denote ϕ ∨ ¬ϕ, and ⊥ to denote ¬⊤. Let us consider another propositional language L ✷ whose set of propositional variables is of the form V ✷ = {✷ϕ | ϕ ∈ L} to which the classical connectives can be applied. It is endowed with a modality operator expressing certainty, that encapsulates formulae in L. In other words

L ✷ = {✷α : α ∈ L} | ¬Φ | Φ ∧ Ψ .
MEL is a logic on the language L ✷ and with the following semantics. Let Ω be the set of classical interpretations for the propositional language L, i.e. Ω consists of the set of mappings w : L → {0, 1} conforming to the rules of classical propositional logic. For a propositional formula ϕ ∈ L we will denote by M od(ϕ) the set of w ∈ Ω such that w(ϕ) = 1. Models (or interpretations) for MEL correspond to epistemic states, which are simply subsets ∅ = E ⊆ Ω. The truth-evaluation rules of formulas of L ✷ in a given epistemic model E is defined as follows:

-E |= ✷ϕ if E ⊆ M od(ϕ) -E |= ¬Φ if E |= Φ -E |= Φ ∧ Ψ if E |= Φ and E |= Ψ
Note that contrary to what is usual in modal logic, we do not evaluate modal formulas on particular interpretations of langage L because modal formulas in MEL do not refer to the actual world.

The notion of logical consequence is defined as usual Γ |= Φ if, for every epistemic model E, E |= Φ whenever E |= Ψ for all Ψ ∈ Γ .

MEL can be axiomatized in a rather simple way (see [START_REF] Banerjee | A simple logic for reasoning about incomplete knowledge[END_REF]). The following are a possible set of axioms for MEL in the language of L ✷ :

(PL) Axioms of PL for L ✷ -formulas (K) ✷(ϕ → ψ) → (✷ϕ → ✷ψ) (D) ✷ϕ → ✸ϕ (Nec) ✷ϕ, for each ϕ ∈ L that is a PL tautology, i.e. if M od(ϕ) = Ω.
The only inference rule is modus ponens. The corresponding notion of proof, denoted by ⊢, is defined as usual from the above set of axioms and modus ponens.

This set of axioms provides a sound and complete axiomatization of MEL, that is, it holds that, for any set of MEL formulas Γ ∪ {ϕ}, Γ |= ϕ iff Γ ⊢ ϕ. This is not surprizing: MEL is just a standard propositional logic with additional axioms, whose propositional variables are the formulas of another propositional logic, and whose interpretations are subsets of interpretations of the latter.

Notice that MEL also actually captures the L ✷ -fragment of the normal modal logics KD, hence of other logics, such as the well-known logics KD45 and S5, also commonly referred to as the logics of belief and knowledge, respectively. However, they are obtained from KD with axioms (called 4, 5, T) that cannot be expressed in the MEL language L ✷ . and Someone's Beliefs

In this section we extend the language of MEL to allow dealing with not only subjective formulas that express an agent's beliefs, but also objective formulas (i.e. non-modal formulas) that express propositions that hold true in the actual world (whatever it might be). The extended language will be denoted by L + ✷ , and it thus contains both propositional and modal formulas. It exactly corresponds to the non-nested fragment of the language of usual modal logic.

Language, Axioms and Semantics

More precisely, the language L + ✷ of MEL + extends L ✷ and is defined by the following formation rules:

-If ϕ ∈ L then ϕ, ✷ϕ ∈ L + ✷ -If Φ, Ψ ∈ L + ✷ then ¬Φ, Φ ∧ Ψ ∈ L + ✷
✸ϕ is defined as an abbreviation of ¬✷¬ϕ. Note that L ⊆ L + ✷ and that in L + ✷ there are no formulas with nested modalities. Semantics for MEL + are given now by "pointed" MEL epistemic models, i.e. by structures (w, E), where w ∈ Ω and ∅ = E ⊆ Ω. The truth-evaluation rules of formulas of L + ✷ in a given structure (w, E) is defined as follows:

-(w, E) |= ϕ if w(ϕ) = 1, in case ϕ ∈ L -(w, E) |= ✷ϕ if E ⊆ M od(ϕ)
usual rules for ¬ and ∧ Logical consequence, as usual: Γ |= Φ if, for every structure (w, E), (w, E) |= Φ whenever (w, E) |= Ψ for all Ψ ∈ Γ .

The following are the axioms for MEL + in the language of

L + ✷ : (PL) Axioms of propositional logic (K) ✷(ϕ → ψ) → (✷ϕ → ✷ψ) (D) ✷ϕ → ✸ϕ (Nec) ✷ϕ, for each ϕ ∈ L that is a PL tautology, i.e. if M od(ϕ) = Ω.
The only inference rule is modus ponens. 1

Completeness

In what follows, we will denote by ⊢ P L the notion of proof of classical propositional language on the language L + ✷ taking all ✷-formulas as new propositional variables. We will further let Γ ∪ {Φ} be a set of L + ✷ -formulas. We need first to recall the following lemma [START_REF] Banerjee | A simple logic for reasoning about incomplete knowledge[END_REF].

Lemma 1. Γ ⊢ Φ iff Γ ∪ {✷ϕ | ⊢ P L ϕ} ∪ {instances of axioms (K), (D) and (Nec) } ⊢ P L Φ Theorem 1. Γ ⊢ Φ iff Γ |= Φ.
Proof. From left to right is easy, as usual. For the converse direction, assume Γ ⊢ Φ. By the preceding lemma and the completeness of PL, there exists a propositional evaluation v on the whole language L + ✷ (taking ✷-formulas as genuine propositional variables) such that v(Ψ ) = 1 for all Ψ ∈ Γ ∪ {✷ϕ | ⊢ P L ϕ} ∪ {instances of axioms (K) and (D)} but v(Φ) = 0. We have to build a structure (w, E) that it is a model of Γ but not of Φ. So, we take (w, E) as follows:

w is defined as the restriction of v to L, i.e. w(ϕ) = v(ϕ) for all ϕ ∈ L.

-E = {M od(ϕ) | v(✷ϕ) = 1}
Note that, since by assumption v(✸⊤) = 1, E = ∅. Then the last step is to show that, for every

Ψ ∈ L + ✷ , v(Ψ ) = 1 iff (w, E) |= Ψ.
We prove this by induction. The case Ψ being a non-modal formula from L is clear, since in that case w(Ψ ) = v(Ψ ). The interesting case is when Ψ = ✷ψ. Then we have:

(i) If v(✷ψ) = 1 then, by definition of E, E ⊆ M od(ψ), and hence (w, E) |= ✷ψ. (ii) Conversely, if E ⊆ M od(ψ), then there must exist γ such that v(✷γ) = 1
and M od(γ) ⊆ M od(ψ). Hence this means that γ → ψ is a PL theorem, and hence we have first, by the necessitation axiom, that v(✷(γ → ψ)) = 1, and thus v(✷γ) ≤ v(✷ψ) holds as well by axiom (K), and therefore v(✷ψ) = 1 holds as well.

As a consequence, we have that (w, E) |= Ψ for all Ψ ∈ Γ and (w, E) |= Φ.

Remark 1. Notice that if the notion of logical consequence |= is reduced to considering only structures (w, E) such that w ∈ E, then one should add the following well-known axiom (T): ✷ϕ → ϕ to keep completeness.

4 Relating MEL and MEL + to KD45 and S5

Recall the normal modal systems KD, KD4, KD45 and S5 (see e.g. [START_REF] Chellas | Modal Logic: An Introduction[END_REF] for details). Proof. The proof is very similar to that of Proposition 1.

Corollary 2. Let ϕ a formula from L + ✷ . Then MEL + ⊢ ϕ iff L ⊢ ϕ for L ∈ {KD4, KD45}.
Note that this corollary does not hold for L = S5, indeed, ✷ϕ → ϕ is an axiom of S5 that is not provable in MEL + .

Let us call MEL ++ the extension of MEL + with the axiom (T): ✷ϕ → ϕ. Then notice that an easy adaptation of the proof of completeness theorem for MEL + proves that MEL ++ is complete with respect to the class of reflexive pointed epistemic models (w, E) where w ∈ E.

Proposition 3. Let ϕ a formula from L + ✷ . Then, MEL ++ ⊢ ϕ iff S5 ⊢ ϕ.
The proof easily follows from that of Proposition 2 by taking into account that one has to deal with reflexive epistemic models.

Moreover, by recalling the well-known result that any formula of KD45 and S5 is logically equivalent to another formula without nested modalities, we can formulate the following stronger relationships.

Proposition 4. The following conditions hold true:

-For any arbitrary modal formula ϕ, there is a formula

ϕ ′ ∈ L + ✷ such that KD45 ⊢ ϕ iff MEL + ⊢ ϕ ′ .
-For any arbitrary modal formula ϕ, there is a formula

ϕ ′ ∈ L + ✷ such that S5 ⊢ ϕ iff MEL ++ ⊢ ϕ ′ .
Remark 2. Recently, Petruszczak [START_REF] Petruszczak | Simplified Kripke style semantics for modal logics K5, KB5 and KD45[END_REF] indicated that simplified Kripke frames could indeed be used for the semantics of systems K45, KB5 and KD45, using subsets of propositional valuations in place of relations, as we proposed. He proves it by constructing specific accessibility relations equivalent to such subsets, as in [START_REF] Banerjee | A Simple Modal Logic for Reasoning about Revealed Beliefs[END_REF] for MEL, while the completeness proof in [START_REF] Banerjee | A simple logic for reasoning about incomplete knowledge[END_REF] and here is direct.

MEL + vs. Other Logics of Incomplete Information

The language MEL is supposed to encode the following situation [START_REF] Banerjee | A Simple Modal Logic for Reasoning about Revealed Beliefs[END_REF]. There are two agents, one of which, say A, reasons about some beliefs possessed by another agent B, the former is aware of, on the basis of the testimony of the latter. Namely, A partially knows what the other agent believes. A belief base in MEL typically contains the testimony of agent B, namely propositions agent B believes (✷α), some that he explicitly does not know (✸α ∧ ✸¬α), and finally some other propositions that agent A is aware the agent B knows the truthvalue of, without guessing which (✷α ∨ ✷¬α). The logic MEL enables agent A to infer more beliefs agent B possesses but did not reveal. Such a (meta-)belief base for agent A is equivalent to a set of possible epistemic states for agent B. In MEL + , agent A is allowed to add what is known about the real world in the form of standard propositions. So α ∧ ✷¬α means that agent A considers α is true, while he knows that agent B believes it is false. Under this set-up, a MEL + model (w, E) is interpreted as the fact that A envisages the real world to be w and the epistemic state of B to be E. If A considers that B's beliefs are always correct, the former can assume axiom T is valid, thus he reasons in MEL ++ to strengthen his own knowledge of the real world. Alternatively, A may mistrust B and may wish to take advantage of knowing wrong beliefs of A.

Epistemic Logic and Accessibility Relations: A Critique

In contrast, usual semantics of S5 [START_REF] Halpern | Reasoning About Knowledge[END_REF] consider the epistemic state of an agent is modelled by an equivalence relation R on a set of possible worlds W . 2 The statement wRw ′ reads "world w ′ is accessible from w". The world w ′ is said to be an epistemic or doxastic alternative to world w for the agent, depending on whether knowledge or belief is the considered attitude. There are various attempts to make sense of this relation, such that the agent cannot distinguish w from w ′ , or w ′ is a possible state of affairs from the point of view of what the agent knows in w, etc. The underlying idea seems to be that "the set of worlds considered possible by an agent depends on his or her informational resources at that instant" (Stanford Encyc. Philos.).

However this view, which seems to be shared by many scholars, is not so easy to grasp. Interpreting "accessible worlds" as worlds compatible with the agent epistemic state, we can assume that the epistemic state of the agent depends on his or her informational resources at that instant (in MEL this possibility is not considered). But it is not clear that the agent is aware of his own informational resources to the point of articulating them in the same language as the one he uses to speak about the current states of affairs. If w stands for an objective state of facts, it may not include the particulars of the agent. The epistemic state of the agent depends on many hidden internal features of the agent but his current observations about the actual world w refer to something external, the agent is focused on. It is not clear that the vocabulary used to describe the actual world w is rich enough to also account for the inner state (of health or informational resources) of the agent that holds some beliefs about w. For instance suppose the agent has incomplete information on the outcome of coin flipping round after the toss: this epistemic state will generally not depend on the outcome of the toss (like "if the result is heads then he knows it, otherwise he does not know").

In a nutshell, while an equivalence class of R represents context-dependent knowledge of the agent, it is not clear that this contextual dependence is part of the agent's knowledge about himself, let alone about another agent. So in the epistemic logic approach the accessibility relation seems to be a circular notion, where possible worlds seem to include the description of the agent mental circumstances as well as the description of his epistemic state regarding the problem he considers. This view may make sense when introspection is the main issue (the agent being partially unaware of his own knowledge), but it seems to be at odds with the problem of an agent reflecting about other agents knowledge, as in the set-up for MEL and MEL + . As the set-up for MEL is not introspective, this relational semantics looks like a questionable artifact for this logic, where we assume agents are aware of their own knowledge and lack of knowledge. Note that if R = W × W one can only distinguish between tautologies (i.e. ✷φ where φ is a PL tautology), contradictions and contingent modal propositions.

One may extend the MEL set-up by considering a separate set of possible mental dispositions S corresponding to "informational resources" (due to specific situations or circumstances) an agent can access at a particular moment. On the other hand, W encodes the question the agent is concerned with at that moment; it pertains to the outside world, so S ∩ W = ∅ The accessibility relation R is relating S to W , namely E = R(s) ⊆ W is the epistemic state of the agent when his mental disposition is s. Note that, under this view, there is no point of R being an equivalence relation. But this extension assumes that the set S of mental dispositions of the agent is known and observable by another agent.

Comparison to Logics of Rough Sets

The semantics of S5 in terms of equivalence relations [START_REF] Chellas | Modal Logic: An Introduction[END_REF] makes it in fact the natural logical setting for rough sets [START_REF] Orlowska | A logic of indiscernibility relations[END_REF]. Pawlak's rough sets [START_REF] Pawlak | Rudiments of rough sets[END_REF] are based on the notion of approximation spaces (W, R), where R, called the indiscernibility relation, is an equivalence relation on the domain W of discourse. The premise is that due to lack of complete information about the objects in the domain, it is likely that many of the objects are indistinguishable from each other. This is patent in information systems I := (W, At, V At , f ), where At is a set of attributes, V At a set of values for the attributes in At, and f : The lower and upper approximations of a subset X of the domain with respect to R are defined as X := {x ∈ W : R(x) ⊆ X}; X := {x ∈ W : R(x) ∩ X = ∅}.

W × At → V
Sets with identical approximations are said to be roughly equal, and for any X, the collection of all subsets of the domain roughly equal to X is termed a rough set by Pawlak in [START_REF] Pawlak | Rudiments of rough sets[END_REF]. Any logic of rough sets thus has an essential modal nature, the necessity and possibility operators capturing the lower and upper approximations in the rough set semantics respectively. In fact, for any S5 Kripke model M = (W, e, R), one may observe that e(✷ϕ) = e(ϕ) and e(✸ϕ) = e(ϕ), where e(ϕ) := {w ∈ W : e(w, ϕ) = 1}. However, unlike MEL or MEL + , rough set logics make use of the full modal language, that is, nested modalities are allowed. For instance one would use a formula such as ✸(p ∧ ✷q) to refer to a set (X ∩ Y ).

This approach can easily be extended to rough set models based on a relation that is not necessarily an equivalence one [START_REF] Yao | Generalization of rough sets using modal logic[END_REF][START_REF] Yao | A review of rough set models[END_REF]. These logics remain modal, and use nested modalities. Indeed, it is well known in modal logic [START_REF] Chellas | Modal Logic: An Introduction[END_REF] that, once fixed the basic axioms (PL) and (K), then each additional modal axioms corresponds a different property of the accessibility relation.

Comparison to the Logic of Minimal Belief and Negation as

Failure (MBNF)

In [START_REF] Lifschitz | Minimal belief and negation as failure[END_REF] Lifschitz defines a simplified version of Lin and Shoham logic [START_REF] Lin | A logic of knowledge and justified assumptions[END_REF] of minimal knowledge and justified assumptions. His (nonmonotonic) logic, in the propositional version, contains two modal operators, one for minimal belief B and another for negation as failure not. For positive formulas, i.e. formulas not containing not, the (monotonic) semantics is very similar to that of MEL + : semantics are given by structures (I, S), where I is an interpretation of propositional variables (or equivalently a set of atoms) and S a set of interpretations. The author writes that,"intuitively, I represents 'the real world', and S the set of 'possible worlds' ". As in MEL + , for a nonmodal formula ϕ, a structure (I, S) satisfies the formula Bϕ whenever each I ′ ∈ S satisfies ϕ. Therefore MBNF structures (I, S) exactly correspond to pointed epistemic MEL + models. The nonmonotonic semantics of MBNF is defined as to capture the notion of minimal belief. A model of a theory T (set of positive formulas) is a structure (I, S) such that it makes true all the formulas of T and it is maximal in the sense that there is no other structure (I ′ , S ′ ) making the formulas of T true and such that S S ′ . For instance, the only model of Bϕ in this semantics is (I, M od(ϕ)), while the models of Bϕ ∨ Bψ are (I, M od(ϕ)) and (I, M od(ψ)). Then the corresponding notion of (nonmonotonic) consequence relation is defined accordingly. For instance one has Bp |= ¬Bq but {Bp, Bq} |= ¬Bq.

Actually, MBNF models for formulas of the language of MEL (resp. MEL + ) correspond to the minimum specific epistemic models in MEL (reps. pointed epistemic models in MEL + ) in the sense of possibilistic logic.

Conclusion

In this paper, we argue that the usual semantics of epistemic logics based on accessibility relations is not very natural when the purpose is to reason about beliefs or incomplete knowledge entertained by an external agent, and introspection is ruled out. To this end, we have shown that the fragment MEL + (resp. MEL ++ ) of the KD45 (resp. S5) logic, the richest of doxastic (resp. epistemic) logics, involving modal formulas of depth 0 or 1 can have simplified semantics that are more intuitive than equivalence relations, while the latter make sense for capturing rough sets. The connection between MEL + and MBNF clearly suggests the former has more to do with logic programming than to the mainstream modal logic tradition, as already noticed with generalized possibilistic logic (a multimodal extension of MEL), that can encode answer-set programming [START_REF] Dubois | Stable Models in Generalized Possibilistic Logic[END_REF].

Proposition 1 .Corollary 1 .Proposition 2 .

 112 Let ϕ a formula from L ✷ . Then MEL ⊢ ϕ iff KD ⊢ ϕ.Proof. Assume KD ⊢ ϕ, then there is a serial Kripke model (W, e, R) and w ∈ W such that e(w, ϕ) = 0. Since ϕ does not contain nested modal operators, e(w, ϕ) only depends on the truth-evaluations of subformulas of ϕ at all the worlds from R(w) = {w ′ | wRw ′ }, which is non-empty since R is serial. We can assume R(w) ⊆ Ω. Define, for each w ′ ∈ R(w), the propositional evaluation v w ′ (.) := e(w ′ , .), and the epistemic modelE w := {v w ′ : w ′ ∈ R(w)}. Then E w |= ϕ,if and only if e(w, ϕ) = 1. Hence MEL ⊢ ϕ. Conversely, assume MEL ⊢ ϕ, then there is an epistemic model E such that E |= ϕ. Now, consider the Kripke model M = (E, e, R) where R = E × E with e(v, •) = v(•) for every v ∈ E. M is clearly a serial model. The fact that E |= ϕ implies that e(w, ϕ) = 0 for some w ∈ E, and hence M |= ϕ, and by completeness, KD ⊢ ϕ. Let ϕ a formula from L ✷ . Then MEL ⊢ ϕ iff L ⊢ ϕ for L ∈ {KD4, KD45, S5}. Now let us consider the case of MEL + . Then we have even stronger relationships to KD45 and S5. Let ϕ a formula from L + ✷ . Then MEL + ⊢ ϕ iff KD ⊢ ϕ.

  At a function assigning values for attributes to objects of the domain. I then induces an indiscernibility relation R corresponding to every subset B ⊆ At: xRy, if and only if f (x, b) = f (y, b), for all b ∈ B.

An equivalent presentation could be to replace (Nec) by the usual Necessitation rule in modal logics, but restricted to tautologies of propositional logic: if ϕ ∈ L is a theorem, derive ✷ϕ.

See[START_REF] Fagin | An internal semantics for modal logic[END_REF] for an alternative semantics that makes the internal structure of possible worlds more explicit, encoding both objective facts and agents' mental states in a possible world. We thank one reviewer for pointing this out to us.

Acknowledgments. Godo acknowledges partial support by the Spanish MINECO project EdeTRI TIN2012-39348-C02-01.


Martin Giurfa
email: martin.giurfa@univ-tlse3.fr

Aurore Avarguès-Weber

Mathieu Lihoreau

Guillaume Isabel

Information transfer beyond the waggle dance: observational learning in bees and flies

Keywords: bumblebees, Drosophila, social cognition, social learning, visual cognition

Social information transfer is part of the success of animal societies and has been documented in a variety of taxa, from slime molds to humans. In invertebrates, the historical research focus has been on the specialized signals shaped by selection to convey information, such as the honeybee waggle dance. However, growing evidence shows that invertebrates also commonly glean critical information about their environment by observing others. For instance, a bumblebee's choice between novel flower species is influenced by the observation of the foraging choices of more experienced conspecifics. Recent studies suggest that these seemingly complex learning abilities can be explained in terms of simple associative learning, whereby individuals learn to associate social cues (conditioned stimuli) to reward cues (unconditioned stimuli). Here, we review the behavioral evidence of observational learning both in bees and Drosophila. We discuss the validity of associative accounts of observational learning and the potential neural circuits mediating visual social learning in these model species to define future research avenues for studying the neurobiology of social cognition in miniature brains.

Introduction

The waggle dance of the honey bee constitutes one of the most sophisticated systems known for information transfer about profitable food sources [START_REF] Frisch | The Dance Language and Orientation of Bees[END_REF]. A successful forager performs a stereotyped behavior within the hive, originally described as a "dance, " which conveys information about the direction and distance from the hive of an exploited food source [START_REF] Riley | The flight paths of honeybees recruited by the waggle dance[END_REF][START_REF] Grüter | The honeybee waggle dance: can we follow the steps?[END_REF][START_REF] Seeley | Honeybee Democracy[END_REF]. This dance "language" is a case of true communication [START_REF] Markl | Manipulation, modulation, information, cognition: some of the riddles of communication[END_REF] whereby the sender explicitly transfers information to the receivers in order to modify their behavior. Yet, many animals use social information to learn about their environment simply by attending the behavior of others [START_REF] Danchin | Public information: from nosy neighbors to cultural evolution[END_REF][START_REF] Galef | Social learning in animals: empirical studies and theoretical models[END_REF]. This type of social learning differs from true communication in that the demonstrator does not explicitly attempt to modify the receiver's behavior. In this review we will focus on social learning based on visual observation of a demonstrator's behavior [START_REF] Zentall | Perspectives on observational learning in animals[END_REF].

Invertebrates provide paradigmatic examples of observational learning. For instance, Octopus observers that are allowed to watch conditioned Octopus demonstrators choosing one of two different colored objects presented simultaneously, consistently select the same object as the demonstrators did [START_REF] Fiorito | Observational learning in Octopus vulgaris[END_REF]. Other fascinating cases are found among insects. These animals exhibit developed learning capacities and accessible miniature nervous systems, thereby constituting ideal organisms for dissecting the neural and molecular bases of learning [START_REF] Giurfa | Cognition with few neurons: higher-order learning in insects[END_REF]. Model species, such as the honey bee Apis mellifera (Menzel, 1999;[START_REF] Giurfa | Behavioral and neural analysis of associative learning in the honeybee: a taste from the magic well[END_REF][START_REF] Galizia | Honeybee Neurobiology and Behaviour[END_REF], and the fruit fly Drosophila melanogaster [START_REF] Heisenberg | Mushroom body memoir: from maps to models[END_REF][START_REF] Davis | Olfactory memory formation in Drosophila: from molecular to systems neuroscience[END_REF] Guven-Ozkan and [START_REF] Guven-Ozkan | Functional neuroanatomy of Drosophila olfactory memory formation[END_REF] have been extremely useful for pioneer studies on the mechanisms of learning and memory.

Here we focus on social bees (Hymenoptera) and Drosophila, in which observational learning has been documented. Our goal, beyond various excellent reviews on the topic of social learning in insects (e.g., Leadbeater and Chittka, 2007b;[START_REF] Dukas | Evolutionary biology of insect learning[END_REF][START_REF] Grüter | Insights from insects about adaptive social information use[END_REF][START_REF] Leadbeater | What evolves in the evolution of social learning?[END_REF] is to provide a mechanistic view of these complex behaviors. It has recently been suggested that social learning can emerge from simple associations between a relevant stimulus (unconditioned stimulus, US), such as a food reward or a predator threat, and a conspecific's presence or behavior (conditioned stimulus, CS), which is not different from individual learning of non-social cues (Leadbeater and Chittka, 2007b;[START_REF] Avarguès-Weber | Visual cognition in social insects[END_REF][START_REF] Heyes | What's social about social learning?[END_REF][START_REF] Giurfa | Social learning in insects: a higher-order capacity? Front[END_REF]. Using this idea, we discuss the nature of learning associations and the neural circuits potentially involved in insect observational learning.

Observational Learning in Bees Behavioral Evidences

During their foraging activities, bees need to exploit multiple floral resources whose reward levels change rapidly and unpredictably [START_REF] Heinrich | Resource heterogeneity and patterns of movement in foraging bumblebees[END_REF][START_REF] Heinrich | Bumblebee Economics[END_REF][START_REF] Goulson | Bumblebees: Behaviour, Ecology, and Conservation[END_REF][START_REF] Lihoreau | Radar tracking and motion-sensitive cameras on flowers reveal the development of pollinator multi-destination routes over large spatial scales[END_REF]. A forager's choice of plant species is guided by unlearned preferences and learned information about current reward levels gained through individual sampling [START_REF] Raine | Adaptation, genetic drift, pleiotropy, and history in the evolution of bee foraging behavior[END_REF]. As many pollinators often work concurrently in a meadow, information acquired individually can be complemented by social information [START_REF] Grüter | Insights from insects about adaptive social information use[END_REF], but also by information gained inside the nest through communication and food exchange [START_REF] Biesmeijer | The use of waggle dance information by honey bees throughout their foraging careers[END_REF][START_REF] Arenas | Floral scents experienced within the colony affect long-term foraging preferences in honeybees[END_REF]. It has long been known that during foraging, bees are attracted to visibly occupied flowers [e.g., bumblebees [START_REF] Brian | Differences in the flowers visited by four species of bumblebees and their causes[END_REF]; stingless bees [START_REF] Slaa | The use of field-based social information in eusocial foragers: local enhancement among nestmates and heterospecifics in stingless bees[END_REF]; honey bees [START_REF] Frisch | The Dance Language and Orientation of Bees[END_REF], suggesting that they learn to exploit food resources by copying the choices of other bees [START_REF] Romanes | Mental Evolution in Animals[END_REF]]. Recent studies with bumblebees have shown that individuals can indeed glean information from watching other foragers, and change accordingly their floral choices [START_REF] Leadbeater | What evolves in the evolution of social learning?[END_REF]Chittka, 2005, 2007a;[START_REF] Worden | Flower choice copying in bumblebees[END_REF][START_REF] Kawaguchi | Contrasting responses of bumble bees to feeding conspecifics on their familiar and unfamiliar flowers[END_REF][START_REF] Baude | Conspecifics as informers and competitors: an experimental study in foraging bumble-bees[END_REF]Avarguès-Weber and Chittka, 2014a,b), their choice of location [START_REF] Leadbeater | What evolves in the evolution of social learning?[END_REF]Chittka, 2005, 2009;[START_REF] Kawaguchi | Do bumble bees save time when choosing novel flowers by following conspecifics?[END_REF][START_REF] Baude | Inadvertent social information in foraging bumblebees: effects of flower distribution and implications for pollination[END_REF][START_REF] Dawson | Conspecific and heterospecific information use in bumblebees[END_REF][START_REF] Plowright | Bumblebees occupy: when foragers do and do not use the presence of others to first find food[END_REF] and their handling strategies [START_REF] Leadbeater | Social transmission of nectarrobbing behaviour in bumble-bees[END_REF][START_REF] Goulson | Social learning drives handedness in nectar-robbing bumblebees[END_REF][START_REF] Mirwan | Social learning in bumblebees (Bombus impatiens): worker bumblebees learn to manipulate and forage at artificial flowers by observation and communication within the colony[END_REF].

In particular, when bees observe the floral choices of conspecific demonstrators from behind a transparent screen (Figure 1A), they land more often on the flower type chosen by demonstrators in tests where the demonstrators are absent, than compared to non-observing controls (conspecifics separated from demonstrators by an opaque screen) [START_REF] Worden | Flower choice copying in bumblebees[END_REF]. Similar results are obtained with artificial demonstrators (inanimate model bees made of resin), thus indicating that visual cues associated with the presence of conspecifics are sufficient to promote social acquisition of flower preferences [START_REF] Worden | Flower choice copying in bumblebees[END_REF][START_REF] Dawson | Learning by observation emerges from simple associations in an insect model[END_REF][START_REF] Avarguès-Weber | Observational conditioning in flower choice copying by bumblebees Bombus terrestris: influence of distance and demonstrator movement[END_REF]).

An Associative Account

The fact that bees acquire flower preferences by observing foragers through a screen [START_REF] Worden | Flower choice copying in bumblebees[END_REF][START_REF] Dawson | Learning by observation emerges from simple associations in an insect model[END_REF][START_REF] Avarguès-Weber | Observational conditioning in flower choice copying by bumblebees Bombus terrestris: influence of distance and demonstrator movement[END_REF]) is incompatible with a simple associative hypothesis in which foragers associate profitable flowers (US) with the presence of the conspecifics (CS). In this case, the demonstrators are no longer present when the observer makes its choice, which implies that the positive value of conspecifics has been transferred to the associated flowers themselves. An explanation based on associative learning is still possible but in the form of a phenomenon termed second-order conditioning [START_REF] Pavlov | Conditioned Reflexes: An Investigation of the Physiological Activity of the Cerebral Cortex[END_REF]. Under second-order conditioning, bees learn a positive association between a conspecific (CS1) and a food reward (US), due to past-shared foraging experience on the same resources. When observing a conspecific feeding from an unknown flower, the close association between the CS1 (conspecifics) and the flower induces the bee to consider flower cues themselves as indicative of reward (CS2). Such a mechanism would lead to a socially acquired preference for all flowers sharing the same characteristics and not only for the occupied flowers (Leadbeater and Chittka, 2007b;[START_REF] Dawson | Learning by observation emerges from simple associations in an insect model[END_REF].

If observational learning for new flower preferences is the consequence of a second-order conditioning, then impeding or modifying the first association should alter flower preference. In bumblebees, this hypothesis is supported by the fact that naive bees with no previous social foraging experience tend to ignore the choices of conspecifics in their foraging decision [START_REF] Dawson | Learning by observation emerges from simple associations in an insect model[END_REF]Avarguès-Weber and Chittka, 2014a), suggesting that there is a decisive role of prior associations between social cues and a reward. Additionally, the preference for socially demonstrated flowers can be reversed into avoidance if the tested bees are allowed to form an association between the conspecifics and a bitter aversive solution beforehand [START_REF] Dawson | Learning by observation emerges from simple associations in an insect model[END_REF].

The associative learning hypothesis also predicts that nonsocial cues should promote social-like learning behavior given that they have been previously associated with rewarding flowers. However, bumblebees follow different flower choice strategies when social cues (model bees) or non-social cues (wooden white blocks) are used as indicators of reward (Avarguès-Weber and Chittka, 2014a). If they have learned that the white blocks are present on a rewarded flower color, they will choose afterwards a different color, only if it displays the presence of the blocks. Unoccupied flowers presenting that same color will not be chosen. This behavior can be explained in simple associative terms as the blocks were previously associated with reward. The situation is different if bees have learned to forage on a flower color on which bees were present. In this case, they will choose afterwards a novel Drosophila (virgin female) can observe interactions between two pairs of demonstrator males (dusted with color pigments) and females through a screen. The observer is first presented a colored male (e.g., green) mating with a sexually receptive female. The pair of demonstrators is then replaced by a male of a different color (e.g., pink) and a non-sexually receptive female. Observational learning is successful if the observer preferentially copulates with the male of the same color as the demonstrator did, when subsequently given the choice between new males of each color [START_REF] Mery | Public versus personal information for mate copying in an invertebrate[END_REF].

color if dummy bees are present on it but, in addition, they will extend their choice to unoccupied flowers with that same color. Thus, in the latter case, the bees' choice is not restricted to the flowers occupied by a model bee but includes all flowers presenting the same characteristics (Avarguès-Weber and Chittka, 2014a). The difference between these two scenarios may reside in the fact that foragers gather experience in the field in which conspecifics, contrary to wooden blocks, are not only predictive of reward but also mobile. This mobility may allow transferring the choice form the occupied flower to the unoccupied flower as long as both share the same color. This strategy may be advantageous: in many typical flower species, it might not be adaptive for a pollinator to visit a flower that is currently being drained by another visitor. Instead, it would be more useful to steer toward unvisited flowers of the same plant species where the visitation activities of others indicate that the flower type is profitable. Therefore, specific mechanisms might have evolved to promote efficient social information use in flower foraging, suggesting that social and asocial learning are dissociated.

It is, however, also possible that there is no special role of conspecifics in the flower generalization pattern observed, specifically when demonstrators indicate rewarding flowers. Bees that never got the chance to see live conspecifics within or outside the nest and were only familiarized with model static bees pinned on flowers show a pattern of choices similar to that of bees familiarized and tested with non-social cues (Avarguès-Weber and Chittka, 2014a). By contrast, bees exposed to live bees experienced that the socially indicated flower species will sometimes be occupied by demonstrators and sometimes not, occasionally in rapid succession, and the situation might change while the observer is on the flower. Thus, observers will get exposure to mobile demonstrators physically dissociated from the flowers, and this in turn may favor future generalization to unoccupied but socially indicated flowers. The possibility that non-social moving objects could generate a social-like flower choice pattern remains to be tested. Alternatively, social learning specificity may require a familiarization phase with live conspecifics to learn to associate the conspecific's chemical signature acquired within the nest [START_REF] Krasnec | Eusocial evolution and the recognition systems in social insects[END_REF], and with the species visual characteristics.

Relying on the choices of others is not always an adaptive strategy [START_REF] Giraldeau | Potential disadvantages of using socially acquired information[END_REF][START_REF] Laland | Social learning strategies[END_REF][START_REF] Rieucau | Exploring the costs and benefits of social information use: an appraisal of current experimental evidence[END_REF][START_REF] Grüter | Insights from insects about adaptive social information use[END_REF]. In the most extreme case of a population always favoring social learning over individual sampling, an ecological dead end would be quickly reached with some resources being overexploited while others are left unexplored. From the colony perspective, keeping enough individual information acquisition is essential for social learning behavior to remain beneficial [START_REF] Giraldeau | Potential disadvantages of using socially acquired information[END_REF][START_REF] Rieucau | Exploring the costs and benefits of social information use: an appraisal of current experimental evidence[END_REF][START_REF] Grüter | Insights from insects about adaptive social information use[END_REF]. Bumblebees present restrictions in the use of social over individual learning that are consistent with the theory. Indeed, the response to social cues is flexible, depending on the context of the observation [START_REF] Leadbeater | A new mode of information transfer in foraging bumblebees?[END_REF][START_REF] Kawaguchi | Contrasting responses of bumble bees to feeding conspecifics on their familiar and unfamiliar flowers[END_REF][START_REF] Baude | Conspecifics as informers and competitors: an experimental study in foraging bumble-bees[END_REF]. In a field study, B. diversus foragers were given a choice between two inflorescences attached to a stick ("interview bouquet"), one of which was occupied by a conspecific (freshly killed bee pinned on flowers). While bees preferred occupied inflorescences when they were presented two unfamiliar flower species, they avoided conspecifics when confronted with flower species found in their daily environment [START_REF] Kawaguchi | Contrasting responses of bumble bees to feeding conspecifics on their familiar and unfamiliar flowers[END_REF]. Presumably, this conditional use of social information enables bees to maximize their foraging efficiency when searching for novel food items while minimizing the costs of competition when they know resource locations [START_REF] Laland | Social learning strategies[END_REF][START_REF] Dall | Information and its use by animals in evolutionary ecology[END_REF]. Competition level is also reduced by another flexible usage of social information as B. terrestris foragers do not follow the preferences of demonstrators when the conspecifics density on the flowers patch is too high [START_REF] Baude | Conspecifics as informers and competitors: an experimental study in foraging bumble-bees[END_REF][START_REF] Plowright | Bumblebees occupy: when foragers do and do not use the presence of others to first find food[END_REF].

All these results suggest that observational social learning in bumblebees is the consequence of simple associative processes and specific enhanced attention toward conspecifics as cue providers (stimulus enhancement) and/or places where these conspecifics can be seen (local enhancement) [START_REF] Zentall | Imitation: definitions, evidence, and mechanisms[END_REF][START_REF] Leadbeater | What evolves in the evolution of social learning?[END_REF]. An intricate interplay between evolutionary adaptation to attend to conspecific cues, individual experience with such cues and their contingencies with salient aspects of the environment is probably at hand to generate the observed complexity of observational social learning.

Observational Learning in Drosophila

Behavioral Evidence

Although considerable knowledge on insect observational learning comes from research on bumblebees, visual social learning has also been described in a non-social species, the fruit fly D. melanogaster. In this species, females learn the quality of potential mating partners by observing their success with other females [START_REF] Mery | Public versus personal information for mate copying in an invertebrate[END_REF] (Figure 1B). This capacity was shown in experiments in which two artificial male phenotypes were produced by dusting flies with green or pink pigments [START_REF] Mery | Public versus personal information for mate copying in an invertebrate[END_REF]. An observer (virgin) female was placed in a glass tube where she could see demonstrator males and females through a colorless screen. In the first observation phase, the demonstrator male (e.g., green) successfully mated with the demonstrator female. In the second phase, a male of another color (e.g., pink) was paired with a non-receptive female, thus leading to unsuccessful copulation attempts by the male. When the observer female was later presented with two males (green and pink) simultaneously, she preferentially mated with the male of the color that was associated with a successful copulation (e.g., green) (Figure 1B). This effect disappeared when the observers could not directly observe the demonstrator flies [START_REF] Mery | Public versus personal information for mate copying in an invertebrate[END_REF]. This example shows that observational learning is not restricted to social insects. Rather, it seems to be a general capacity issued from the insects' faculty to learn associations in their environment. Observational learning in Drosophila could also be interpreted as a special case of associative visual learning. It is possible that the vision of a female copulating with a male acts as a biologically relevant reinforcement to be associated with the male color (CS). Under this hypothesis, observer flies should learn to associate a male color phenotype with a successful mating signal. Later, when confronted with males of different phenotypes, observers would preferentially choose the learned color based on a simple associative memory. Visual associative learning has been extensively documented in Drosophila in an individual context [START_REF] Heisenberg | Flexibility in a single behavioral variable of Drosophila[END_REF][START_REF] Foucaud | Use of spatial information and search strategies in a water maze analog in Drosophila melanogaster[END_REF][START_REF] Schnaitmann | Appetitive and aversive visual learning in freely moving Drosophila[END_REF][START_REF] Ofstad | Visual place learning in Drosophila melanogaster[END_REF][START_REF] Vogt | Shared mushroom body circuits underlie visual and olfactory memories in Drosophila[END_REF] so that transferring this capacity to a mating, observational context is plausible.

Genetic and Molecular Basis

The discovery of mate choice copying in a main model organism holds considerable promises to unravel the genetic and molecular substrates of observational learning in insects, an approach that is currently not possible in bees. While such analysis has not been conducted yet, recent studies have begun to identify the neural substrates of Drosophila visual learning that may also be involved in observational learning in particular if the associative learning hypothesis is verified.

Different forms of visual learning are mediated by the central complex (CX). This neuropil is located between the protocerebral brain hemispheres and comprises four interconnected regions: the fan-shaped body, the ellipsoid body, the protocerebral bridge and the paired noduli (Figure 2). It receives information from visual processing neuropils (lamina, lobula, medulla) connected to each compound eye, and whose learning-dependent plasticity has not been explored until now. The implication of the CX in visual recognition was first demonstrated using a flight simulator, in which a fly whose head is attached to a torque meter controls the position of visual patterns on the walls of a circular arena with its flight direction [START_REF] Heisenberg | Flexibility in a single behavioral variable of Drosophila[END_REF]. Using this approach, flies can be trained to learn to avoid visual cues (such as colors and geometric forms, CS) due to their association with an aversive stimulus (a heat beam, US). The sequence of CS and US stimuli can either be controlled by the fly itself (operant training) or by the experimenter (Pavlovian training) [START_REF] Brembs | The operant and the classical in conditioned orientation of Drosophila melanogaster at the flight simulator[END_REF]. Memory mutants lacking the Rutabaga (Rut) protein-a type 1 Ca2+/Calmodulin-dependent adenylyl cyclase that produces cAMP-display impaired operant and Pavlovian visual learning, indicating that Rut plays a decisive role in the US/CS association, probably as a coincidence detector of the visual CS and the heat US [START_REF] Liu | Distinct memory traces for two visual features in the Drosophila brain[END_REF]. By using the UAS/GAL4 system to differentially express Rut in specific subsets of cerebral neurons, it has been shown that the discrimination of visual patterns of different elevations or orientations requires two different groups of neurons extending branches in the fanshaped body, respectively the F5 and F1 neurons [START_REF] Liu | Distinct memory traces for two visual features in the Drosophila brain[END_REF]. Another subset of large field neurons located in the ellipsoid body (the ring neurons R2 and R4m) are also involved in recognition of several pattern features through excitatory and inhibitory visual subfields [START_REF] Pan | Differential roles of the fan-shaped body and the ellipsoid body in Drosophila visual pattern memory[END_REF][START_REF] Seelig | Feature detection and orientation tuning in the Drosophila central complex[END_REF] (Figure 2). Taken together, these results demonstrate the implication of the CX in visual learning and memory through dynamic interactions between the ellipsoid-body and the fan-shaped body.

Recent studies also point toward a contribution of the mushroom bodies (MBs) in visual memories. The MBs are central brain structures involved in olfactory learning and memory [START_REF] Davis | Olfactory memory formation in Drosophila: from molecular to systems neuroscience[END_REF], courtship [START_REF] Mcbride | Mushroom body ablation impairs short-term memory and long-term memory of courtship conditioning in Drosophila melanogaster[END_REF], locomotion [START_REF] Martin | Mushroom bodies suppress locomotor activity in Drosophila melanogaster[END_REF], and sleep [START_REF] Joiner | Sleep in Drosophila is regulated by adult mushroom bodies[END_REF][START_REF] Pitman | A dynamic role for the mushroom bodies in promoting sleep in Drosophila[END_REF], among others. Despite the absence of obvious anatomical connections between the optic lobes to the MBs [START_REF] Barth | Vision affects mushroom bodies and central complex in Drosophila melanogaster[END_REF][START_REF] Otsuna | Systematic analysis of the visual projection neurons of Drosophila melanogaster. I. Lobula-specific pathways[END_REF][START_REF] Mu | Optic glomeruli and their inputs in Drosophila share an organizational ground pattern with the antennal lobes[END_REF], the volume of MB calyces (dendrites) changes with light regime, suggesting that MBs are involved in visual information processing [START_REF] Barth | Vision affects mushroom bodies and central complex in Drosophila melanogaster[END_REF]. Indeed, it has been shown that MBs are required in visual context generalization [START_REF] Liu | Context generalization in Drosophila visual learning requires the mushroom bodies[END_REF] and could stabilize visual memories against context changes [START_REF] Brembs | Context and occasion setting in Drosophila visual learning[END_REF]. Interestingly, the MBs (γ neurons) seem also necessary for the memorization of simple associations between color stimuli and a sugar reward or with an electric shock [START_REF] Vogt | Shared mushroom body circuits underlie visual and olfactory memories in Drosophila[END_REF]. Presumably, the implication of the CX or the MBs might be dependent of the locomotion state (flying vs. walking) as flies were trained in a flight simulator in one case [START_REF] Liu | Distinct memory traces for two visual features in the Drosophila brain[END_REF][START_REF] Pan | Differential roles of the fan-shaped body and the ellipsoid body in Drosophila visual pattern memory[END_REF] vs. a walking plate in the other [START_REF] Vogt | Shared mushroom body circuits underlie visual and olfactory memories in Drosophila[END_REF]. Locomotor activity is known to affect the activity of octopamine neurons and the behavioral response to CO 2 [START_REF] Suver | Octopamine neurons mediate flight-induced modulation of visual processing in Drosophila[END_REF][START_REF] Wasserman | Drosophila tracks carbon dioxide in flight[END_REF], and thus possibly modifies neural pathways involved in visual information memorization [START_REF] Kottler | Taking a new look at how flies learn[END_REF][START_REF] Vogt | Shared mushroom body circuits underlie visual and olfactory memories in Drosophila[END_REF]. Additionally, walking activity has no direct effect on the activity of ring neurons of the CX while flying activity significantly decreases their responses to visual stimuli [START_REF] Seelig | Feature detection and orientation tuning in the Drosophila central complex[END_REF].

Importantly, the MBs and their associate dopaminergic signaling are also involved in visual attention in the form of visual tracking of a moving bar [START_REF] Xi | Mushroom bodies modulate salience-based selective fixation behavior in Drosophila[END_REF][START_REF] Van Swinderen | Shared visual attention and memory systems in the Drosophila brain[END_REF]. They may consequently mediate the specific attentional state elicited by social visual cues during an observational learning task.

Conclusion

Placing social learning within the conceptual framework of associative learning is an appealing approach for explaining seemingly complex behavior in insects with pinhead-sized brains. However, bumblebee studies are beginning to suggest that observational learning by insects does not only reflect visual associative learning but also involves attentional processing of social cues as information providers.

In parallel, the neurogenetic approaches well mastered in Drosophila hold considerable promises in revealing the neural basis of such complex behavior. Future investigations may target the CX and MBs as the potential neuronal structures involved, given their implication in visual learning and attention.

The popularity of bees and fruit flies as models for visual cognition research associated with the abundance of genomic information available make them ideal study systems to explore the genetic, molecular, neuronal, and behavioral basis of visual social learning, a major challenge on the way of understanding the evolutionary relationships between animal brains, cognitive capacities and their social environment (Lihoreau et al., 2012a).

FIGURE 1 |

 1 FIGURE 1 | Experimental set-ups used to study observational learning in insects. (A) An observer bumblebee can observe a vertical array of two unfamiliar flower types (e.g., green or orange) through a transparent screen. Demonstrators (e.g., living bees, dead bees or model bees) are associated to one flower type (e.g., green). Observational learning is successful if the observer bee preferentially visits flowers of the same type as the demonstrators did, when subsequently tested alone (Worden and Papaj, 2005; Avarguès-Weber and Chittka, 2014b). (B) An observer

FIGURE 2 |

 2 FIGURE 2 | Neurobiological structures involved in visual learning in the Drosophila brain. (A) A schematic diagram of the head of D. melanogaster revealing several major neuropiles: the lamina (La) and medulla (Me) involved in visual processing, the antennal lobes (AL) involved in olfactory processing and the mushroom bodies (MB) and the central complex (CX) involved, among other functions, in visual learning. Subdivisions of the central complex: the protocerebral bridge (pb;

Frontiers in Ecology and Evolution | www.frontiersin.org

March 2015 | Volume 3 | Article 24

Acknowledgments M.G. thanks the Institut Universitaire de France, the Human Frontier Science Program (HFSP) and the French National Research Agency (ANR) for generous support. M.L. was funded by an IDEX-University of Toulouse Starting Grant and a Research Grant from the Foundation Fyssen. A.A-W. thanks the l'Oréal Foundation for support. All authors acknowledge the support of the French Research Council (CNRS) and the University of Toulouse.

Conflict of Interest Statement:

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2015 Avarguès-Weber, Lihoreau, Isabel and Giurfa. This is an openaccess article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.


Didier Dubois
email: dubois@irit.fr

Henri Prade
email: prade@irit.fr

Agnès Rico
email: agnes.rico@univ-lyon1.fr

On the informational comparison of qualitative fuzzy measures

Keywords: Fuzzy measures, possibility theory, qualitative reasoning, information content

Fuzzy measures or capacities are the most general representation of uncertainty functions. However, this general class has been little explored from the point of view of its information content, when degrees of uncertainty are not supposed to be numerical, and belong to a finite qualitative scale, except in the case of possibility or necessity measures. The thrust of the paper is to define an ordering relation on the set of qualitative capacities expressing the idea that one is more informative than another, in agreement with the possibilistic notion of relative specificity. To this aim, we show that the class of qualitative capacities can be partitioned into equivalence classes of functions containing the same amount of information. They only differ by the underlying epistemic attitude such as pessimism or optimism. A meaningful information ordering between capacities can be defined on the basis of the most pessimistic (resp. optimistic) representatives of their equivalence classes. It is shown that, while qualitative capacities bear strong similarities to belief functions, such an analogy can be misleading when it comes to information content.

Introduction

Qualitative fuzzy measures (or q-capacities) are monotonic set-functions on a finite set with a range in a finite totally ordered set. They are instrumental in the representation of uncertainty of events, or yet, of the relative weights of groups of criteria in multicriteria evaluation, in the non-numerical environment, when likelihood or value scales are just totally or partially ordered sets, e.g. complete lattices [START_REF] Dubois | The use of the discrete Sugeno integral in decision-making: a survey[END_REF]. An important issue to be clarified, if q-capacites are to be seriously considered as a tool for representing uncertainty, is the one of information content, and more generally the comparison of q-capacities in terms of their relative information content.

Important special cases of q-capacities are possibility and necessity measures [START_REF] Dubois | Qualitative possibility functions and integrals[END_REF]. For the latter set-functions, there is an abundant literature concerning information comparison, based on the notion of relative specificity [START_REF] Yager | On the specificity of a possibility distribution[END_REF][START_REF] Dubois | Qualitative possibility functions and integrals[END_REF]. Namely, a possibility (or a necessity) measure is entirely characterized by a possibility distribution over the elementary events, that defines a fuzzy set, and information comparison is carried out using fuzzy set inclusion, whereby a fuzzy set is more specific than another one, if and only if the former is a subset of the latter. However, there is almost no literature on the informational comparison of q-capacities in the general case. Some authors [START_REF] Marichal | Entropy of discrete fuzzy measures[END_REF] have proposed notions of entropy for discrete capacities that evaluate the diversity of the coefficients appearing in the set-function. Our aim is rather to extend the specificity ordering to general q-capacities.

To do so, one may get inspiration from the theory of belief functions, where several informational orderings of various strength have been defined [START_REF] Yager | Entropy and specificity in a mathematical theory of evidence[END_REF][START_REF] Dubois | A set-theoretic view of belief functions -Logical operations and approximation by fuzzy sets[END_REF]. A belief function is a ∞-monotone set-function, that can be defined by means of a probability distribution over a power set (its Moebius transform), the degree of belief of an event summing the probability weights of all subevents that imply it. A belief function is then more informative than another if the former is eventwise greater than the latter. Another stronger definition, called specialisation, is based on the inclusion of focal sets. In the qualitative setting, it is also possible to compare q-capacities in an eventwise way. Besides, a qualitative counterpart of the Moebius transform can be defined, and a qualitative form of specialisation can be defined as well [START_REF] Prade | Possibilistic Evidence[END_REF]. However, when indistinctly applied to q-capacites (like possibility and necessity measures), these definitions, even if formally welldefined, seem to baffle intuition.

In this paper, we try to provide a meaningful notion of information comparison between capacities, noticing that q-capacities differ not only by their information content, but also by the range of attitudes towards uncertainty they may encode. For instance, based on a given possibility distribution, the possibility measure is optimistic and the necessity measure is pessimistic [START_REF] Dubois | Decision theoretic foundations of qualitative possibility theory[END_REF], but one may argue they have the same information content.

Framework and notations

We consider a finite set (of states, criteria, etc.) S = {s 1 , • • • , s n } and a finite totally ordered scale L with top 1 and bottom 0. Let min denote the minimum, max the maximum. Moreover L is equipped with an involutive negation ν. A q-capacity is a mapping γ : 2 S → L such that γ(∅) = 0, γ(S) = 1, and if A ⊆ B then γ(A) ≤ γ(B). When L = {0, 1}, γ is called a Boolean capacity.

A special case of capacity is a possibility measure defined with a possibility distribution π : S → L. The possibility measure is defined by Π(A) = max s∈A π(s). The value π(s) is understood as the possibility that s be the actual state of the world. Precise information corresponds to the situation where ∃s * , π(s * ) = 1 and ∀s = s * , π(s) = 0. Complete ignorance is represented by the vacuous possibility distribution π ? such that ∀s ∈ S, π ? (s) = 1. A possibility distribution π is more specific than another possibility distribution ρ if ∀s ∈ S, π(s) ≤ ρ(s) [START_REF] Yager | On the specificity of a possibility distribution[END_REF][START_REF] Dubois | Qualitative possibility functions and integrals[END_REF]. This definition makes perfect sense since the set of possible values represented by π is smaller, hence more precise, than the one represented by ρ.

The q-capacity conjugate of γ, denoted by γ c , is defined using the involutive negation ν by γ c (A) = ν(γ(A c )) ∀A ⊆ S, where A c is the complement of the set A. In particular, the conjugate of a possibility measure is a necessity measure which is of the form N (A) = ν(max s ∈A π(s)).

The inner qualitative Moebius transform of γ is a mapping γ # : 2 S → L defined by γ # (E) = γ(E) if γ(E) > max B⊂E γ(B) and 0 otherwise [START_REF] Grabisch | On the representation of k-decomposable measures[END_REF][START_REF] Mesiar | k-order pan-discrete fuzzy measures[END_REF]. It contains the minimal information that is sufficient to reconstruct γ as γ(A) = max E⊆A γ # (E). Let F γ = {E, γ # (E) > 0} denote the family of so-called focal sets associated to γ. Note that the inner qualitative Moebius transform of γ is strictly monotonic with inclusion on F γ . The inner qualitative Moebius transforms of a possibility measure Π coincides with its possibility distribution π (focal sets are singletons) while the focal sets of a necessity measure N are nested (the cuts of π).

3 Can we transfer quantitative definitions of information comparison to the qualitative setting?

When comparing capacities γ and δ, the inequality γ ≤ δ is not always expressing something relevant about how informative γ is with respect to δ. Indeed, for instance if γ is the vacuous possibility function Π ? (A) = 1 if A = ∅ and δ is the vacuous necessity function N ? (A) = 0 if A = Ω, we have Π ? > N ? . However, they have exactly the same information content since based on the vacuous possibility distribution assigning 1 to all elements of S and they are maximally uniformative among other possibility and necessity functions.

In the numerical setting, information comparison relations exist especially in the setting of belief functions. A belief function on S is defined by means of a probability distribution m over 2 S \ {∅} as

Bel(A) = E,E⊆A m(E), ∀A ⊆ S. (1)
The conjugate functions are plausibility measures defined by P l(A) = 1 -Bel(A c ). Bel(A) = 1 expresses full certainty of A, since then P l(A c ) = 0 expresses that the complement A c is impossible.

There are several definitions of information comparison for belief functions. A belief function Bel 1 is said to be more informative (in the wide sense) than Bel 2 if ∀A ⊆ S, Bel 1 (A) ≥ Bel 2 (A). This is due to the fact that Bel 1 assigns greater degrees of certainty to events (while Bel(A) = 0 expresses no information). In terms of imprecise probabilities, it is equivalent to have the inclusion {P : P ≥ Bel 1 } ⊆ {P : P ≥ Bel 2 }. And notice that for plausibility functions the inequality is reversed (P l 1 (A) ≤ P l 2 (A)).

Another information comparison method is based on specialisation, that relies on the mass assignments : m 1 is a specialization of m 2 (denoted by m 1 ⊑ s m 2 ) if and only if there exists a joint mass x(A, B) with marginals m 1 and m 2 , such that There is a formal analogy between q-capacities and belief functions, since γ(A) = max E,E⊆A γ # (E), whereby γ # plays the role of a mass assignment, and turns into max. This is why it is then tempting to consider γ as more informative than δ whenever ∀A ⊆ S, γ(A) ≥ δ(A). This is all the more natural as the following result, adapted from [START_REF] Prade | Possibilistic Evidence[END_REF], holds:

x(A, B) = 0 whenever A B, A ∈ F 1 , B ∈ F 2 .
Proposition 1. ∀A ⊆ S, γ(A) ≥ δ(A) if and only if ∀F ∈ F δ , ∃E ∈ F γ s.t. E ⊆ F , γ # (E) ≥ δ # (F ). Proof: γ(A) ≥ δ(A) writes max E,E⊆A γ # (E) ≥ max F,F ⊆A δ # (F ). Suppose A = F is a focal set of δ. Then the latter is equivalent to max E,E⊆F γ # (E) ≥ δ # (F ), hence to ∀F ∈ F δ , ∃E ∈ F γ s.t. E ⊆ F and γ # (E) ≥ δ # (F ).
Conversely, suppose the latter holds. If δ(A) = 0 then the result is obvious. If

δ(A) = 0 hence let us consider F the focal element included in A such that δ(A) = δ # (F ). There exists E a focal element of γ included in F such that γ # (E) ≥ δ(A). We have E ⊆ F ⊆ A so γ(A) ≥ γ # (E) ≥ δ(A). The condition ∀F ∈ F δ , ∃E ∈ F γ s.t. E ⊆ F , γ # (E) ≥ δ # (F )
is clearly a qualitative rendering of the specialisation relation. It formally means that for any focal set F of δ there is a more important and more precise focal subset of γ, that explains the domination of γ over δ. Here, the two definitions of informational comparison are equivalent, which departs from the quantitative case.

However, this result is misleading. Neither the values of their lower Moebius transforms γ # and δ # , nor the size of focal sets A, B with γ # (A) > 0 and δ # (B) > 0 tell us much on their information content. For instance, focal sets of the vacuous possibility measure Π ? are all singletons, and the unique focal set of N ? is the whole set S. Viewing γ as a counterpart to belief functions w.r.t. γ # is thus not appropriate.

In such a context this article focuses on a basic question: When does a qcapacity represent the idea of certainty like belief functions, when does it represent plausibility (the conjugate of a belief functions)? In other words, when it is uncertainty-averse or pessimistic ? When it is uncertainty-prone or optimistic ?

4 Optimistic and pessimistic q-capacities Given a possibility distribution π the corresponding Π is optimistic in the sense that Π({s}) = 1 as soon as s is fully possible (among other states); and its corresponding conjugate N = Π c is pessimistic, in particular N ({s}) = 0, ∀s ∈ S as soon as two distinct elements in S are fully possible. More generally, N (A) ≤ Π(A), ∀A ⊆ S and more specifically N (A) > 0 implies Π(A) = 1. Finally, N (A) = 1 expresses full certainty while Π(A) = 1 just expresses a lack of surprise for the occurrence of A. Likewise, belief functions are pessimistic while their conjugate plausibility functions, which rely on the same mass function, are optimistic.

The above considerations motivate the following definition.

Definition 1. A q-capacity γ is said to be pessimistic (resp. optimistic

) if γ ≤ γ c (resp. if γ ≥ γ c ).
This definition first appears in [START_REF] Dubois | The use of the discrete Sugeno integral in decision-making: a survey[END_REF] where a pessimistic (resp. optimistic) capacity is called uncertainty-averse (resp: uncertainty-prone). It is easy to see that:

a q-capacity may be neither pessimistic nor optimistic. There may exist A, B such that γ(A) < γ c (A) (pessimistic for A), and γ(B) > γ c (B) (optimistic for B). -a q-capacity may be both : γ = γ c is possible. For instance with Boolean q-capacities (L = {0, 1}) on a space with 2n + 1 elements, γ n (A) = 1 if |A| > n and 0 otherwise.

To check if a capacity is pessimistic, it is enough to check the property γ(A) ≤ γ c (A) for focal sets. For if A is not focal, then there is a focal set E contained in A such that γ(A) = γ(E), and it is clear that if γ(E) ≤ γ c (E) then γ(A) ≤ γ c (E) ≤ γ c (A), since A ⊆ E. This remark helps checking the pessimism of γ.

Proposition 2. The following properties hold:

-If γ is pessimistic then γ(A) = 1 implies γ(A c ) = 0. For Boolean capacities, γ is pessimistic if and only if min(γ(A), γ(A c )) = 0. -If γ is optimistic then γ(A) = 0 implies γ(A c ) = 1. For Boolean capacities, γ is optimistic if and only if max(γ(A), γ(A c )) = 1.
Proof: Let us suppose γ pessimistic and γ(A) = 1. Then γ c (A) = 1, hence γ(A c ) = 0. Conversely in the binary case, either γ(A) = 0 and then γ(A) ≤ γ c (A), or γ(A c ) = 0 then γ c (A) = 1 and γ(A) ≤ γ c (A).

The proof for the optimistic case is similar.

Note that we find a result proved in [START_REF] Chateauneuf | Rico: Modeling attitudes toward uncertainty through the use of Sugeno integral[END_REF]. In that paper, a preference relation between acts represented by functions is called strongly pessimistic (resp. optimistic) if and only if it is represented by a Sugeno integral with respect to a necessity measure (resp. possibility measure). In that paper, contrary to ours, a preference relation is called pessimistic (resp. optimistic ) the relation is represented with a Sugeno integral with respect to capacity γ such that min(γ(A), γ(A c )) = 0 (resp. max(γ(A), γ(A c )) = 1)3 . Here we shall respectively call such capacities strictly pessimistic (resp. optimistic). Strictly pessimistic capacities are indeed special cases of pessimistic ones: A,min(γ(A),γ(A c )) = 0 then it is pessimistic.

Proposition 3. If a capacity is such that for all subsets

Proof: Suppose γ is not pessimistic. Then ∃A, γ(A) > ν(γ(A c )). Hence γ(A) > 0, but then by assumption, γ(A c ) = 0, hence γ(A) > 1, which is impossible.

.

Considering a capacity γ, for each α > 0 ∈ L we can define a Boolean capacity γ α called its α-cut, as follows: for all A ⊆ S, γ α (A) is equal to 1 if γ(A) ≥ α and 0 otherwise. Then, γ = max α∈L min(α, γ α ), F γ = ∪ α∈L F (γ α ) , and we can show:

Proposition 4. A capacity is strictly pessimistic if and only if ∀α > 0 ∈ L, γ α is pessimistic.

Proof: : If ∃A, α > 0 : min(γ(A), γ(A c )) = α, then γ α (A) = γ α (A c ) = 1 hence by Proposition 2, γ α is not pessimistic. The converse is obvious.

We can moreover describe the topological features of families of focal sets of pessimistic q-capacities: Proposition 5. γ is a pessimistic q-capacity if and only if any two focal sets E and F that have empty intersection satisfy γ # (E) ≤ ν(γ # (F )).

Proof:

γ ≤ γ c if and only if ∀E ∈ F γ , γ # (E) ≤ ν(γ(E c )). But γ(E c ) = max F :F ∩E=∅ γ # (F ). So γ # (E) ≤ min F :F ∩E=∅ ν(γ # (F )).
The condition γ # (F ) > ν(γ # (E)) that prevents disjoint focal sets means that weights γ # (F ), γ # (E) are both high enough. In particular, the focal sets E of a pessimistic γ such that γ # (E) > ν(γ # (E)) (i.e., weights of focal sets in the upper part of the value scale) intersect pairwisely. All focal sets intersect the focal set with weight 1. And if the focal sets of a q-capacity all pairwisely intersect, then the q-capacity is pessimistic. This property is characteristic for Boolean capacities: γ is a pessimistic Boolean capacity if and only if the intersection of two focal sets is not empty.

In the non-Boolean case, one may have disjoint focal sets with small enough weights inside or overlapping focal sets with high weights.

Example 1. Consider a q-capacity with 4 focal sets, E, F, G 1 , G 2 such that F ⊆ E, G i ∩ E = ∅ for i = 1, 2, and F, G 1 , G 2 are disjoint (Fig. 1). Assume γ For optimistic capacities one gets dual results:

# (E) = 1, γ # (F ) = α > ν(α), and γ # (G i ) = β i < ν(α). Then it is easy to check that γ is pessimistic, since γ c (E) = 1, γ c (F ) = ν(max(β 1 , β 2 )) > α, γ c (G i ) = ν(α) > β i .
Proposition 6. Let γ be an optimistic q-capacity. If A is a focal set such that γ(A) = 1 then A c contains a focal set of γ. Proof: Let A be a focal set of γ. γ(A) = α entails γ(A c ) ≥ ν(α) > 0.
So for each A such that γ # (A) = 1 we can find another focal set B such that A∩B = ∅. It means that the focal sets of optimistic capacities tend to be disjoint while those of pessimistic capacities tend to overlap. The precise description of focal sets of optimistic capacities can benefit from the knowledge of the focal sets of pessimistic ones, as explained in [START_REF] Dubois | Rico: Qualitative Capacities as Imprecise Possibilities[END_REF].

equality. If γ * = δ * it means that for all subsets A of S, the sets of values {γ(A), γ(A c )} and {δ(A), δ(A c )} are equal. So for each event A we must decide the attitude in front of uncertainty : pessimistic if we assign the least value, optimistic if we assign the greatest one (while respecting monotonicity). If δ ∈ C ≈ (γ) is a pessimistic q-capacity, then min(δ, δ c ) = δ = γ * so γ * is the unique pessimistic q-capacity in C ≈ (γ). Similarly one obtains that γ * is the unique optimistic q-capacity in C ≈ (γ). As a consequence, we must compare q-capacities in terms of their information content via a comparison of their equivalence classes, which justifies the following definition : Definition 5. A q-capacity γ is said to be to be more informative than a qcapacity δ in the wide sense if and only if γ * ≥ δ * .

Likewise we can compare two q-capacities in terms of their relative pessimism in front of uncertainty by means of another relation: Definition 6. A q-capacity γ is said to be to be less pessimistic than a q-capacity δ in the wide sense if and only if {A :

γ(A) = γ * (A)} ⊆ {A : δ(A) = δ * (A)}.
These definitions completely disentangle the two aspects carried by q-capacities: namely the attitude in front of uncertainty and the information content, in agreement with possibilistic specificity.

Particular cases

We consider in fact extreme cases of q-capacities.

Complete ignorance

We consider the q-capacity defined by γ(S) = 1 and γ(A) = 0 otherwise. This is nothing but the vacuous necessity measure N ? whose conjugate is Π ? . Clearly, N ? is pessimistic and and it is less informative than any capacity, since [N ? (A), Π ? (A)] = [0, 1] for A = S, ∅.

Precise information For each element s i in S we define the following q-

capacity σ i (A) = 1 if s i ∈ A 0 otherwise
. It represents the precise information x = s i .

Note that σ i = σ c i is self-conjugate, so we have σ i * = σ * i = σ i . Some obvious remarks concerning this familly of q-capacities:

-If we consider two q-capacities σ i and σ j with s i = s j then neither of the two is more informative than the other.

σ i and σ j contain the same information if and only if

s i = s j . -C ≈ (σ i ) = {σ i }, which is obvious since it is self-conjugate.
-There is no pessimistic q-capacity γ = σ i such that γ ≥ σ i . Indeed σ i is selfconjugate and [σ i * (A), σ * i (A)] reduces to 0 or 1, i.e., cannot contain the nontrivial intervals [γ * (A), γ * (A)]. For each capacity γ, either the information content of γ and σ i are incomparable or γ is less informative than σ i .

Self-conjugate q-capacities In the numerical setting the most well-known self-conjugate capacities are probability measures that cannot be defined here, but for the precise q-capacities σ i . Self-conjugate capacities σ are such that ∀A ⊆ S, σ(A) = ν(σ(A c )) = σ c (A). They are at the same time pessimistic and optimistic. It is then obvious that they are maximally specific: there is no capacity that is more informative than any σ since the intervals [σ * (A), σ * (A)] are reduced to points, hence cannot contain [γ * (A), γ * (A)] for any γ = σ. A remarkable subclass of self-conjugate capacities are symmetric ones, where γ

(A) = α |A| only depend on the cardinality of A, α |A| = ν(α |S\A| ) ≥ α |S\A| if |A| ≥ |S|/2.
They are completely defined by a strictly increasing sequence of n = |S| coefficients α i such that α i = ν(α n-i+1 ). Of course, the precise q-capacities are a subclass of self-conjugate ones since if σ # ({s i }) = 1 then F σ = {{s i }}, i.e. σ = σ i .

Conclusion

This paper has tried to define the information content of a capacity when the value scale is purely ordinal. We have exploited the fact that a qualitative capacity accounts for both uncertainty and attitude in front of it, as pointed out in [START_REF] Dubois | The use of the discrete Sugeno integral in decision-making: a survey[END_REF][START_REF] Yager | Measures of assurance and opportunity in modeling uncertain information[END_REF]. We propose a mathematical definition of information content comparison irrespective of the attitude toward uncertainty, and a comparison of capacities in terms of pessimism, irrespective of how much uncertainty they express.

Some issues remain open, and especially the structure of the set of focal sets of optimistic capacities. Informally, the focal sets of a pessimistic capacity tend to overlap each other, while the focal sets of an optimistic capacity tend to be disjoint. And the larger the focal sets, the less informative the pessimistic capacity. A strict form of pessimism is when all focal sets intersect pairwisely; maximal pessimism is obtained for nested focal sets (necessity measures). Alternatively, maximal pessimism is obtained for disjoint focal sets (possibility measures); then the smaller the focal sets the less informative the optimistic capacity. It would also be interesting to deepen the analogy and the differences between the qualitative and the quantitative settings for the modeling of uncertainty aversion. These results may be useful to clarify the potential use of qualitative capacities in various areas where aggregation functions are useful, especially in multicriteria analysis and information fusion. For instance, in a multiple source problem, q-capacities can be useful to merge possibility distributions [START_REF] Assaghir | Numerical information fusion: lattice of answers with supporting arguments[END_REF]; focal sets of the q-capacity can then be understood as possibly antagonistic points of view on information.

Fig. 1 .

 1 Fig. 1. Focal sets of a pessimistic q-capacity

Proposition 7 .

 7 The equivalence class C ≈ (γ) of γ is upper bounded by γ * and lower-bounded by γ * . Proof: min(γ * , γ c * ) = γ * so γ * ∈ C ≈ (γ) and γ * ≥ min δ∈C≈(γ) δ. Moreover for all δ ∈ C ≈ (γ) we have δ ≥ min(δ, δ c ) = δ * = γ * which entails min δ∈C≈(γ) δ ≥ γ * . So min δ∈C≈(γ) δ = γ * . Similarly, we can prove that max δ∈C≈(γ) δ = γ * .

  It expresses inclusion betweenfocal sets of Bel 1 and Bel 2 , that is sets E with m i (E) > 0. It can be checked that m 1 ⊑ s m 2 implies that Bel 1 is more informative (in the wide sense) than Bel 2 , but not conversely.

In the case of Boolean functions, Sugeno integral reduces to a capacity.

Based on the above considerations, it makes sense to consider the following definition: Definition 2. If γ 1 and γ 2 are two pessimistic capacities then γ 1 is more informative than γ 2 if and only if γ 1 ≥ γ 2 (eventwise). Indeed, if a pessimistic q-capacity γ 1 assigns a high value to event A, then it will assign a very small value to its complement. In particular γ 1 (A) = 1 implies γ 1 (A c ) = 0 (since ν(γ 1 (A c )) = 1), so that this kind of capacity expresses the idea of certainty, while its conjugate expresses the notion of plausibility [START_REF] Dubois | Formal representations of uncertainty[END_REF] and satisfies the opposite statement. Hence if γ 1 systematically assigns a certainty value higher than or equal to the one assigned by γ 2 to events, the former provides more information than the latter.

Note that by construction,

In fact γ 1 and its conjugate contain the same amount of information but differ by the attitude toward uncertainty. As pointed out in [START_REF] Dubois | Formal representations of uncertainty[END_REF], the width of the interval [γ 1 (A), γ c 1 (A)] reflects the quantity of ignorance regarding event A, namely, for any pessimistic capacity

1 (A) = 1 and γ 1 (A) = 0 expresses total ignorance regarding A.

Note that if the optimistic capacities are two possibility measures, this notion of relative information content reduces to the specificity ordering: Π 1 is more informative than Π 2 if and only if Π 1 ≤ Π 2 if and only if π 1 is more specific than π 2 , i.e. π 1 ≤ π 2 .

Given a capacity γ one can derive its pessimistic and optimistic versions (respectively called assurance and opportunity functions by Yager [START_REF] Yager | Measures of assurance and opportunity in modeling uncertain information[END_REF]): Definition 3. The pessimistic version of a q-capacity γ is γ * (A) = min(γ(A), γ c (A)) and its optimistic version is γ * (A) = max(γ(A), γ c (A)).

Note that γ * and γ * are both q-capacities (monotonic increasing with inclusion), and, by construction, γ * is the ν-conjugate of γ * (i.e. γ * = γ c * ) So each has the same information content as the other.

In such a context we can introduce a relation ≈ between q-capacities expressing the idea of containing the same amount of information :