Michael Freund
email: micfreund@gmail.com

On compound mixed concepts

Keywords: categorization, prototype theory, categorial membership, typicality, resemblance theory, concept, feature, relative clause

We extend to a certain class of compound concepts the binary model classically used for the representation of simple concepts. This class consists of concepts that are determined or modified by a single feature. The treatment of such a mixed composition shows the need to differentiate between exceptional and non exceptional modifiers. In the first case, typicality is easily retrieved from the components of the composition, while, in the second case, it is necessary to isolate the characteristic features of the initial concept that contradict the modifier. The distinction between exceptional and non exceptional modifiers plays a key role in the evaluation of resemblance with mixed compositions.

Introduction

The study of concept combination may be seen as the touchstone of concept theories. Indeed, while elementary concepts can be suitably represented through different models (see [START_REF] Barton | Defining features of natural kinds and artifacts[END_REF], [START_REF] Connolly | Why stereotypes don't even make good defaults[END_REF], [START_REF] Gärdenfors | Conceptual spaces as a framework for knowledge representation[END_REF], [START_REF] Osherson | On the adequacy of prototype theory as a theory of concepts[END_REF], [START_REF] Hampton | Conceptual combination: conjunction and negation of natural concepts[END_REF], [START_REF] Smith | Categories and concepts[END_REF], [START_REF] Smith | Structure and process in semantic memory: a featural model for semantic decisions[END_REF] or [START_REF] Murphy | The Big Book of Concepts[END_REF] for a general overview), the extension to compound concepts of such representations requires a robustness of the model that may be lacking in the original theory. Except for some particular cases, the basic notions linked with concept theories are not compositional. The evidence of some paradoxical results showed the difficulty to treat concept composition with the tools of set theory, and led to alternative approaches ( [START_REF] Sozzo | A quantum probability explanation in fock space for borderline contradictions[END_REF], [START_REF] Aerts | Quantum structure in cognition[END_REF], [START_REF] Franco | The conjunction fallacy and interference effects[END_REF], [START_REF] Wisniewski | When concepts combine[END_REF]).

In [START_REF] Freund | On categorial membership[END_REF] and [START_REF] Freund | Ordered models for concept representation[END_REF], we studied a certain family of concepts, which we called featured concepts. This family consists of concepts that can be described by a simple version of the binary model proposed by [START_REF] Smith | Structure and process in semantic memory: a featural model for semantic decisions[END_REF] and [START_REF] Smith | Categories and concepts[END_REF]. We showed that, for this class of concepts, the notions of categorial membership and typicality could be modeled by simple order relations. Such an order adequately translates the fact that an object may be more or less a member of a category, or that, inside this category, it appears as more or less typical. After having been a 'matter of degree' (see [START_REF] Rosch | Cognitive representations ofsemantic categories[END_REF]), categorial membership and typicality could be therefore treated as intuitive ordinal notions. The membership and the typicality orders that a concept induces among the objects of the universe were shown to provide a satisfactory solution, in the framework of featured concepts, to the problems classically linked with categorial membership, typicality, resemblance and concept induction.

One might think it odd to build a mathematical theory on items -e.g. concepts -for which there exists no mathematical definition. This apparent contradiction however disappears if one keeps in mind that we work on a representation of concepts and not on concepts themselves. As in the case of almost all theories of concept, we propose a mathematical model and use mathematical tools to investigate problems like categorial membership, typicality, concept composition and concept based induction. The solutions brought by this model can be then evaluated in the perspective of cognitive psychology.

The model built to acount for simple featured concepts easily extends to the combination of two such concepts, although the resulting concept is generally not a featured concept. An important side-result is the possibility of describing and explaining in our model the guppy effect as well as the conjunction fallacy (see [START_REF] Osherson | On the adequacy of prototype theory as a theory of concepts[END_REF]and [START_REF] Tversky | Extension versus intuitive reasoning: The conjunction fallacy in probability judgement[END_REF]), showing that our approach successfully models concept conjunction.

In our representation of compound concepts, a problem was however left aside, which is that of compound mixed concepts. These concepts result from the conjunction of a single attribute with a featured concept, e.g. a black-dog, an aquatic-bird, an English-physician. The class of compound mixed concepts deserves a study of its own because these concepts are omnipresent in every-day language as adjective-noun combinations. Furthermore, feature-concept combination seems simpler to investigate than concept-concept combination.

We shall devote this paper to the analysis of this kind of composed concepts.

It is clear that compound mixed concepts cannot receive a uniform treatment: it will be indeed necessary to distinguish between the simple determination of a concept, where the head feature can be integrated as a supplementary attribute of the concept (e.g. to-be-a-black-dog), and a modification of the concept, where the head feature is considered as exceptional because it stands in contradiction with one or several attributes of the original concept (e.g. to-be-a-silent-dog, to-be-a-walking-bird ). For each type of feature, we shall build a set of characteristic attributes of the resulting mixed concept. We shall see that, in the principal cases, this concept can be given the structure of a featured concept, to which we can apply our preceding results. We shall devote a particular attention to questions linked with resemblance. In particular, the presence of the head feature in resemblant objects will be debated, as it has no obvious solution: an animal that is said to resemble a black-dog is expected to be black itself, but a plant that resembles a tropicalflower need not have grown in the tropics.

Plan of the paper

In order to make this paper self-contained, we recall in section 2 the basic notions and results that are necessary to understand the questions evoked in the remainder of the paper. Section 3 introduces the notion of mixed concepts and explains the difference between exceptional and non-exceptional modifiers. In Section 4, after having recalled some results on simple concept resemblance, we examine the problem of resemblance relative to compound mixed concepts. We conclude in Section 5.

The proofs of the propositions and theorems are given in the Appendix.

2 Tools for categorization

Concepts, features and objects

The universe of discourse consists of a set of objects O that a human agent has at his disposal at a given time, a set of concepts C, and a set of features F. The set of concepts reflects the agent's knowledge of his environment and is used by the agent to build his reasoning process. For simplification, we shall adopt the original presentation of Frege [START_REF] Frege | Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens[END_REF] who assimilated concepts with one-place predicates. In this perspective, concepts will be generally introduced through the auxiliary to-be, followed by a noun: to-be-a-bird, tobe-a-vector-space, to-be-a-democracy. Objects are entities that are identified as such by the agent. The process of identification gives rise to a binary relation between concepts and objects. This relation is satisfied between an object x and a concept α when the agent considers that x falls under α, or that α applies to x. For example, one may say that 'Beethoven's opus 111 falls under the concept to-be-a-piano-sonata', or that 'The dog Tobby does not fall under the concept to-be-a-cat'.

An object that falls under a concept is said to be an exemplar or an instance of this concept. The set Ext α of exemplars of a concept α forms its associated category or extension.

Features are usually introduced in the form of a verb (e.g. to-fly), of the auxiliary to-have followed by a noun (e.g. to-have-a-beak), or of the auxiliary to-be followed by an adjective (e.g. to-be-tall ).

Features, like concepts, apply to the objects at hand but, contrary to concepts, they are context-sensitive: they borrow part of their significance from the concept to which they are attached. Properties like to-be-tall, tobe-rich or to-be-red take their full meaning only in a given context, that is when qualifying a well-defined entity. Even simple verbal forms like to-fly, torun, to-live-in-water, to-be-made-of-metal need a principal referent concept to fully seize the strength with which they apply to different items. The concept to which a feature applies may be seen itself as a contextual determination of this feature. To summarize, the meaning of a feature depends on the context where the feature is used, contrary to the meaning of a concept, which, for a given agent, exists by itself.

To account for the fact that we are dealing with a phenomenal representation of cognitive structures, we suggested in [START_REF] Freund | On categorial membership[END_REF] and [START_REF] Freund | Ordered models for concept representation[END_REF] to evaluate the applicability of a feature f to objects by a degree function δ f from O to a finite subset of [0, 1] (see [START_REF] Gärdenfors | Conceptual spaces-the geometry of thought[END_REF] for the diference between scientific and phenomenal representations).

The mathematical model we presented in [START_REF] Freund | Ordered models for concept representation[END_REF] and which we recall in this section describes the structure of concepts as well as the main relations that exist between concepts, like hierachy, resemblance or induction. We did not -and we shall not -examine the question of concept formation. Research in this domain may be found in [START_REF] Confalonieri | Computational aspects of concept invention[END_REF], [START_REF] Egré | Concept utility[END_REF], or [START_REF] Fauconnier | The way we think: Conceptual blending and the mind"s hidden complexities[END_REF]. The question of how agents acquire concepts will be therefore set aside. Our purpose is to investigate in a mathematical framework what it is, for an agent, to have a concept.

In the classical view of categorization, concepts are apprehended by their features: vertebrates that have beaks and feathers will be labelled birds, a coat is an outer garment with sleeves that opens down the front, and, (following Plato), a man is a featherless biped. Exemplar and Prototype theory evaluate categorial membership through the common features an item may have with a concept. Features also play a fundamental role in Formal Concept Analysis ( [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]), where, in a given context, a concept is defined by a pair (A, B), where A is a set of objects and B a set of features. Features may be also retrieved in the quality dimensions of the geometrical model of Gärdenfors ([17] and, more recently, [START_REF] Osta-Velez | Category-based induction in conceptual spaces[END_REF]), where a concept is characterized by a region in the space of its quality dimensions.

In this paper, we shall start from the binary model of [START_REF] Smith | Categories and concepts[END_REF] and [START_REF] Smith | Structure and process in semantic memory: a featural model for semantic decisions[END_REF]. There, a concept α is described by means of two auxiliary sets of features, the defining feature set and the characteristic set, which we shall detail now.

First of all, the defining feature set ∆(α) of a concept α gathers features that, in the agent's mind, are individually necessary and collectively sufficient to decide, in a given context, whether or not an item is an exemplar of α. Categorization relative to a concept is then evaluated through categorization relative to its defining features. For example, if, for some agent, the defining feature set associated with the concept to-be-a-bird consists of the four features to-be-warm-blooded, to-have-a-beak, to-have-feathers and to-have-wings, then, for this agent, the degree with which each of these features apply to a given item will determine its birdhood. Similarly, to quote an example of Putnam [START_REF] Hputnam | The meaning of meaning[END_REF], the meaning of the term tiger is to be searched by reference to the terms yellow feline, fierce, black stripes, and jungle: the defining features associated with the concept to-be-a-tiger consist of the properties tobe-a-yellow-feline, to-have-black-stripes, to-live-in-the-jungle and to-be-fierce. The defining set of a concept may also include as features the potential actions that are expected from instances of the concept and that our perception is aware of ( [START_REF] Noë | Action in Perception[END_REF]). Thus, to-drive-in-nails may be listed among the defining features of the concept to-be-a-hammer.

We have to underline that, given a concept, the choice of its associated defining feature set is purely subjective. For a same concept α, the set ∆(α) may vary from an agent to another, as it may vary over time for a same agent.

It may also be the case that, for some agent, a concept has a corresponding set of defining features, while being deprived of such a set for another agent.

Concepts defined by a conjunction of features form an exceptional subclass. Fodor for instance [START_REF] Fodor | Concepts: where cognitive science went wrong[END_REF] argued that there exists practically no examples of successful definition around. Without being so radical, it is clear that a great deal of concepts are deprived of any set of defining features: what list of attributes could be attached to concepts like to-be-a-game, to-be-a-lie, or to-be-a-heap ? However, it appears that the hypothesis of a defining set of features is justified for certain well-defined families of concepts: such is for instance the case for most nominal concepts, i.e. concepts that are conventionally defined, like to-be-a-mammal, to-be-a-theft or to-be-a-refugee. In particular, this remains true for scientific concepts, like to-be-a-vortex or tobe-a-square. Finally, it may happen that a concept, first grasped through its exemplars, is thereafter sharpened with the help of a set of defining features, like the concept to-be-a-bird, a natural kind concept that was revisited by naturalists and turned into the pseudo-nominal to-have-feathers + to-havea-beak + to-have-wings.

All the features that the agent has chosen to form the set ∆(α) do not necessarily have the same importance. For example, as far as birdhood is concerned, to-have-wings may be judged as a more important feature than to-have-a-beak. To account for this fact, it appears necessary to equip the set ∆(α) with a salience ordering > ∆(α) that reflects the relative importance of its elements. This is done in the classical models through a salience degree that is attached to every defining feature. Then, the degree with which a concept applies to an object is evaluated through the degree with which every defining feature applies to this object, together with the relative salience degree of this feature (for an example of such a calculation, see [START_REF] Hampton | Prototype models of concept representation[END_REF] or [START_REF] Hampton | Similarity-based categorisation: the development of prototype theory[END_REF]). In fact, as we argued in [START_REF] Freund | Ordered models for concept representation[END_REF], comparing the salience of the defining features is easier than assigning to each of them a numerical degree. For this reason, we shall only suppose that, in the agent's mind, the set ∆(α) is equipped with a strict partial order > ∆(α) which, in this paper, we shall suppose non empty. This salience order is, again, subjective, depending on the state of mind, at a given time, of a given agent. Moreover, it is independant of the relations that may exist between two features: an agent may consider that a feature f is more salient than a feature g even in the case where falling under f implies falling under g. Our model is supposed to reflect the agent's mind as it is, with its posible apparent contradictions.

The defining feature set associated with a concept is not enough to fully understand the link between this concept and the objects of the agent's universe. Indeed, all instances of a concept do not have the same status: some of them can be considered as typically representing this concept, while other ones may be viewed as exceptions among the concepts exemplars. Thus, a robin may be a judged as typical in the category of birds, contrary to an ostrich or a penguin.

To account for this difference among the elements of Ext α, the binary model adds to the defining feature set a second set of features, called the characteristic set of α, which will be denoted by χ(α). This set consists of the features that an exemplar of a concept should have to be qualified as typical of this concept. Thus, an example of a characteristic set relative to the concept to-be-a-bird could be {to-fly, to-sing, to-eat-seeds, to-live-in-the-trees}: any bird to which apply all these feature will be qualified as typical. Note that the sets ∆(α) and χ(α) are disjoint : contrary to a defining feature, a characteristic feature is not supposed to apply to each instance of a concept.

As was the case for the defining feature set, salience has to be taken into account among the set of characteristic features. This will be done by means of a non-empty strict partial order > χ(α) .

The above considerations lead to the following notion of featured concepts (see [START_REF] Freund | Ordered models for concept representation[END_REF], definition 1 and 5):

Definition 1 A featured concept α is a concept for which there exists a finite non-empty strictly ordered set of defining features (∆(α), > ∆(α) ) and a finite non-empty strictly ordered set of characteristic features (χ(α), > χ(α) ) that satisfy the following three properties:

1. ∆(α) ∩ χ(α) = ∅ 2.
For every defining or characteristic feature f , the corresponding applicability function δ f takes a finite number of values in [0, 1]

3. There exists at least one object z such that δ f (z) = 1 for all defining and characteristic features f .

The extension Ext α of α can be retrieved as the set of all objects z such that δ f (z) = 1 for all f in ∆(α).

The set T yp α of typical instances of α gathers all objects z such that δ f (z) = 1 for all f in ∆(α) ∪ χ(α). Condition 3 above guarantees that T yp α = ∅ for every featured concept α.

The impact of a featured concept in the binary model

A featured concept structures an agent's universe of discourse. In this section, we will show how such a structure can be represented by order relations among objects.

Induced orders

We first describe how to build an preorder relation in O from a finite ordered subset of features. This will enable us to build the membership and typicality orders of a concept from its defining and characteristic sets. Given a finite set of features X equipped with a non-empty strict partial order > X , we define on a subset T of O the relation X T by:

(1)

x X T y iff ∀f ∈ X : δ f (y) < δ f (x), ∃g ∈ X, g > X f : δ g (x) < δ g (y).
That is, we have x X T y if every feature of X that applies less to y than to x is dominated in X by a feature that applies more to y than to x.

It may be helpful to interpret this relation in the framework of Social Choice Theory: supposing that a certain number of voters have to decide between two candidates x and y, and supposing that the voters voices do not matter the same, the candidate y will be preferred over the candidate x if, for each voice in favor of x, there exists a more important voice in favor of y.

Lemma 1 The relation X

T is a preorder. For two elements x and y of T , one has simultaneously x X T y and y X T x iff, ∀f ∈ X, δ f (x) = δ f (y). The strict partial order ≺ X T associated with X T satisfies:

(2)

x ≺ X T y iff x X T y and ∃f ∈ X : δ f (x) < δ f (y).

We shall refer to the relation X T as to the order induced on T by X. Note that the X T -maximal elements z of T are those for which one has δ f (z) = 1 ∀f ∈ X. These elements satisfy x ≺ X T z for all elements x that are not X T -maximal.

The above construction leads to the two notions that govern concept representation.

The first one is the membership order µ α . It is induced on O by the defining set ∆(α), that is µ α = ∆(α) O

. For x and y in O, we have therefore

(3) x µ α y iff ∀f ∈ ∆(α) : δ f (y) < δ f (x), ∃g ∈ ∆(α), g > ∆(α) f : δ g (x) < δ g (y).
The following example was proposed in [START_REF] Freund | Ordered models for concept representation[END_REF] Example 1 Let α be the concept to-be-a-bird, and suppose that, from the point of view of a particular agent, its defining feature set in the context of living beings is the set {to-have-two-legs, to-lay-eggs, to-have-a-beak, tohave-wings}, equipped the salience order: to-have-a-beak > s to-lay-eggs > s to-have-two-legs, and to-have wings > s to-lay-eggs> s to-have-two-legs. Suppose for the sake of simplicity that, in the agent's mind, membership to any of these features is a two-valued function. Let s, m, t, b and d respectively stand for a sparrow, a mouse, a tortoise, a bat and a dragonfly. Then the induced membership order is determined by the following array :

two -legs lay -eggs beak wings sparrow mouse tortoise bat dragonfly One readily checks that d ≺ α s, m ≺ α t, and m ≺ α b. Note that one has b α d, since the concept to-have-two-legs under which the bat falls, contrary to the dragonfly, is dominated by the concept to-lay-eggs that applies to the dragonfly and not to the bat. On the other hand, one does not have d α b, as nothing compensates the fact that the dragonfly lays eggs and the bat does not. This yields b ≺ α d. Note also that the tortoise and the bat are incomparable: one has neither b α t, nor t α b.

The strict α-membership order therefore reads:

m b d t s c c c c c c
The maximal elements of µ α form the extension Ext α of α. One has x ≺ µ α z for every element z ∈ Ext α and x / ∈ Ext α.

The second order we shall deal with is the typicality order τ α . It is induced on Ext α by the characteristic set χ(α), that is τ α = χ(α) Ext α . For x and y in Ext α, we have then:

(4) x τ α y iff ∀f ∈ χ(α) : δ f (y) < δ f (x), ∃g ∈ χ(α), g > χ(α) f : δ g (x) < δ g (y).
The τ α -maximal elements of Ext α form the set T yp α of typical instances of α. One has x ≺ τ α z for every element z ∈ T yp α and x / ∈ T yp α. The sets Ext α and T yp α can be used to investigate the notion of subconcept. A subconcept of α is a concept β such that Ext β ⊆ Ext α. In [START_REF] Freund | Ordered models for concept representation[END_REF] we showed the necessity of refining this notion and isolating two particulalrly interesting kinds of subconcepts. The first one is that of smooth subconcept ([14] definition 11): a subconcept β of α is smooth in α if it satisfies T yp β ⊆ T yp α. For instance, to-be-a-white-bird is smooth in to-be-a-bird. The second one is that of a typical subconcept: β is a typical subconcept of α if Ext β ⊆ T yp α. Thus, to-be-a-robin is a typical subconcept of to-be-abird. These notions play an important role in concept-based induction. The interested reader may report to the chapter 7 of [START_REF] Freund | Ordered models for concept representation[END_REF].

Remark 1 It is interesting to compare the notions of salience and the resuting typicality order with the Typicality Criterion defined by Osta-Vélez and Gärdenfors in [START_REF] Osta-Velez | Nonmonotonic reasoning, expectations orderings and conceptual spaces[END_REF]. There, contrary to our approach, the authors define salience from typicalty: they propose a salience quasi order ≥ defined on χ(α) by f ≥ g iff ∀x : δ f (x) = 0, ∃y : δ g (y) = 0 : x τ α y. Note that f > χ(α) g implies f ≥ g, provided there exists an object y that does not fall under g but falls under all other characteristic features of α.

Given a feature f , let us denote by

Ext f the set of elements x ∈ O such that δ f (x) = 1. We have then clearly Ext α = f ∈∆(α) Ext f . One has similarly T yp α = Ext α ∩ f ∈χ(α) Ext f : an object x is a typical exemplar of a concept α if and only if δ f (x) = 1 ∀ f ∈ ∆(α) ∪ χ(α)
The knowledge of the membership and the typicality orders induced by a concept renders possible the process of categorization. Not only does it choose among the set O the objects that fall under the concept, and, among them, those that can be considered as typical, but it also establishes a relation among the environing objects, comparing their distance to the referred category. 1 Note however that two agents may agree on the membership and the typicality orders of a concept, while having in mind a different defining set or a different characteristic one. This observation leads to the following notion:

Definition 2 Two concepts α and β are equivalent, written α ≡ β, if they induce the same membership and typicality orders.

One has clearly Ext α = Ext β and T yp α = T yp β whenever α ≡ β, but the converse need not be true. For instance, we may have ∆(α) = ∆(β), while > ∆(α) = > ∆(β) . In this case, the extensions will agree, but the associated membership orders may not be the same.

Concept determination

Concept composition is a blending operation that yields a new concept from two components. The problem of how to form such a composition and of determining the properties inherited by the resulting concept has been at the origin of numerous articles (see in particular [START_REF] Osherson | On the adequacy of prototype theory as a theory of concepts[END_REF], [START_REF] Hampton | Overextension of conjunctive effects: Evidence for a unitary model of concept typicality and class inclusion[END_REF], [START_REF] Hampton | Similarity-based categorisation and fuzziness of natural categories[END_REF], [START_REF] Kamp | Prototype theory and compositionality[END_REF] and [START_REF] Wisniewski | When concepts combine[END_REF]).

We proposed on [START_REF] Freund | On the notion of concept I[END_REF] to introduce a determination connective that accounts for a specific kind of concept combination. This idea was developed in [START_REF] Freund | Ordered models for concept representation[END_REF].

The determination of a concept α by a concept β is possible whenever Extβ ∩ Extα = ∅. It yields a compound concept, denoted by β α, that corresponds to a noun-noun combination -like in to-be-a-pet-fish, or to-be-abarnyard-bird -and, more generally, to a relative clause that will be globally encapsulated by the concept β (e.g. to-be-an-American-who-lives-in-Paris). Typically, the determiner β becomes a simple feature of the compound concept β α. Its role can be considered as secondary, compared with that played by the principal concept α: to-be-a-woman becomes a feature of tobe-physician-that-is-a-woman. Unlike conjunction, concept determination is not commutative : to take a well-known example, the concept of games-thatare-sports differs from the concept of sports-that-are-games. It is important to keep in mind that only the intersective conceptual combinations are accounted for: we consider the determination of α by β in the only case where Ext α ∩ Ext β = ∅ (see [START_REF] Kamp | Prototype theory and compositionality[END_REF] for the distinction between intersective and nonintersective determiner). This shows that the determination connective is only a partial operator: given arbitrary α and β, it may be meaningless to build the concept β α. For instance, there is no sense in talking of a violin-bus or a wolf-frog. Such pseudo-concepts correspond to nothing, and no object, real or fictitious, can be thought of falling under them, contrary to imaginary concepts like a plane-car, or a Martian-teacher : these latter definitely have a non-empty extension, because we can imagine a car that flies and a teacher who comes from Mars.

Given two featured concepts α and β such that Ext α ∩ Ext β = ∅, the determination β α of α by β can be described by a membership order µ β α and a typicality order τ β α defined as follows:

(5) for x, y ∈ O, x µ β α y iff x µ α y and (x ≺ µ α y or x µ β y).

(6) for x, y ∈ Ext α ∩ Ext β, x τ β α y iff x τ α y and (x ≺ τ α y or x τ β y).
Note that both orders give priority to α over β.

The set of µ β α -maximal elements of O forms the extension Ext (β α) of the concept (β α). One has Ext (β α) = Ext α ∩ Ext β.

The set of τ β α -maximal elements of Ext (β α) forms the set T yp (β α) of typical instances of (β α).

Typicality is not compositional: typical pet-fish are not typical fish that are typical pets. However, in the particular case where T yp α ∩ T yp β = ∅, it can be shown that T yp (β α) = T yp α ∩ T yp β ([14], Theorem 2). In this case, β α has the structure of a featured concept with membership order equal to µ β α and typicality order equal to τ β α .

As shown in [START_REF] Freund | On categorial membership[END_REF], the above construction provides a faithfull interpretation of conjunction fallacy and is free of some shortcomings encountered in classical theories. Our task is now to extend this construction to the case where a feature is used as a concept modifier.

3 Features as concept determiners

The impact of a mixed compound concept

The representation of compound concepts proposed in the preceding section can be extended to the determination of a featured concept α by a single feature h. As we shall see, it is indeed possible to build an ordinal model that accounts for simple constructions of adjective-noun type.

Definition 3 Let α be a featured concept and h a feature. Suppose that Ext h ∩ Ext α = ∅. The membership order µ h α on O is defined by :

(7) x µ h α y iff x µ α y and x ≺ µ α y or δ h (x) ≤ δ h (y).
We denote by Ext (h α) the set of µ h α -maximal elements of O. It consists of the elements

x of O such that δ f (x) = 1 ∀f ∈ ∆(α) ∪ {h}. One has immediately Ext (h α) = Ext h ∩ Ext α.
We now turn to the definition of the typical order on the set Ext (h α). Note that if x and y are two elements of this set, one has δ h (x) = δ h (y) = 1. This justifies the following definition: Definition 4 The typicality order τ h α of the concept h α is the restriction to the set Ext h ∩ Ext α of the order τ α .

Using the notation paragraph 2.2.1, we have τ h α = χ(α)

Ext h∩Ext α .

Definition 5

The τ h α -maximal elements of Ext h ∩ Ext α form the set T yp (h α) of typical instances of the concept h α.

In the general case, there exists no link between T yp α and T yp (h α). As we shall see now, the nature of h, considered as a modifier of α, plays an important role in evaluating the typicality of T yp (h α).

Exceptional and non-exceptional modifiers

There exists an important distinction concerning the structural changes that the determination of a concept by a feature may bring to this concept. This question was evoked in [START_REF] Connolly | Why stereotypes don't even make good defaults[END_REF] and [START_REF] Jönsson | On prototypes as defaults (comment on Connolly[END_REF], where was studied the impact of a blank predicate on a predicate-noon combination, and in [START_REF] Smith | Conceptual combination with prototype concepts[END_REF] (thereafter developed in [START_REF] Smith | Combining prototypes: a selcetive modification model[END_REF]), where was operated a distinction between compatible, not compatible and neutral adjectives.

To give an idea of what is involved, consider for instance the concept tobe-a-bird. If we compose this concept with the feature to-be-black, we get the concept to-be-a-black-bird, whose typical instances are the typical birds that are black. But if we compose it with the feature to-live-on-water, we get as a result the concept to-be-an-aquatic-bird, whose typical instances have little to do with typical birds. The difference between these two examples is that, in the first example, the feature to-be-black does not stand in contradiction with the features that are specifically attributed to typical birds. In the second example, on the contrary, the feature to-live-on-water (or to-be-aquatic) stands in contradiction with the feature to-live-in-the-trees that an agent may consider as applying to all typical birds: for this agent, no typical bird lives on water.

This observation lead to the following definition, proposed in [START_REF] Freund | On the notion of concept I[END_REF]:

Definition 6 A feature h is exceptional for α if T yp α ∩ Ext h = ∅, that is if h applies to no typical instance of α. On the contrary, h is said to be non-exceptional for α if T yp α ∩ Ext h = ∅. Recalling that T yp α = Ext α ∩ ( f ∈χ(α) Ext f ), we see that a feature h is exceptional for α if and only if Ext h ∩ Ext α ∩ ( f ∈χ(α) Ext f ) = ∅.
An important result concerning the determination of a featured concept α by a non-exceptional feature h is given by the following:

Theorem 1 Let h be a feature that is not exceptional for α. Then one has T yp (h α) = Ext h ∩ T yp α. The concept h α is equivalent to a featured concept whose defining feature set ∆(h α) is equal to {h} ∪ ∆(α), and whose characteristic set χ(h α) is equal to χ(α) if h / ∈ χ(α), and to

χ(α) \ {h} if h ∈ χ(α).
By the first part of the theorem, we see that h α is a smooth subconcept of α. More precisely, the typical instances of (h α) are the typical instances of α to which applies the attribute h: thus, a typical black-bird is simply a typical bird that is black. In this particular case, typicality is compositional.

Example 2 Suppose that the defining fatures of the concept to-be-a-bird consist of the set {to-be-warm-blooded, to-have-a-beak, to-have-feathers, tohave-wings}, and that its characteristic set is equal to {to-fly, to-sing, to-livein-the-trees, to-eat-seeds}. Then the concept to-be-a-flying-bird is equivalent to a featured concept with defining feature set equal to {to-fly, to-be-warmblooded, to-have-a-beak, to-have-feathers, to-have-wings}, and the characteristic set is {to-sing, to-live-in-the-trees, to-eat-seeds}: the feature to-fly has simply moved from the characteristic set of to-be-a-bird to the defining set of to-be-a-flying-bird.

Theorem 1 has attractive consequences for a particular class of features: Definition 7 Given a featured concept α and a feature g, we say that g is a typical attribute of α if g applies to all typical elements of α, that is if

T yp α ⊆ Ext g.
By Theorem 1, we see that if h is not exceptional for α, a typical attribute for α remains a typical attribute for h α. Thus, to take a well-known example, knowing that ducks have webbed feet, the model predicts that quacking ducks have webbed feet.

It is interesting to translate this result in the framework of inference relations: given a concept α and a feature g, let us say that α generally implies g, or that α typically implies g, written α ∼ g, if g is a typical attribute of α. Then the meaning of a conditional like α ∼ ¬h is that h applies to no typical instance of α, that is that h is exceptional for α. It is therefore natural to write α ∼ ¬h to express the fact that h is not exceptional for α. Using this formalism, we see that Theorem 1 can be translated into the well-known rule of Rational Monotony (see [START_REF] Gärdenfors | Nonmonotonic inference based on expectations[END_REF], or [START_REF] Kraus | Nonmonotonic reasoning, preferential models and cumulative logics[END_REF] for a general presentation of preferential inference relations).

Rational Monotony

α ∼ g, α ∼ ¬h h α ∼ g
For example, knowing that birds generally live in the trees and that there exists typical birds that are black, our model predicts that black birds generally live in the trees.

It is clear that a typical attribute of α is non exceptional for α. Applying Theorem 1 in the case where h is a typical attribute of α, we get the interesting Corollary 1 For any typical attribute h of α, one has T yp(h α) = T yp α.

It follows from this that a feature g is a typical attribute of α if and only if g is a typical attribute of h α. Translating this result in the framework of non-monotonic inference relations enables us to retrieve two fundamental rules of preferential logic :

Cautious Monotony α ∼ h, α ∼ f h α ∼ f Cut α ∼ h, h α ∼ f α ∼ f
These two rules may be reinterpreted in the framework of Concept Based Induction. The first one provides a solution to the problem of Within-Category Induction: the subconcept h α inherits the typical attributes of α. Thus, knowing that birds generally eat seeds and birds generally fly, we may deduce that fying-birds generally eat seeds. The second one is an answer to the Over-Category induction problem: the concept α inherits the typical attributes of its subconcept h α. For instance, knowing that flying birds are generally small and that birds generally fly, we can conclude that birds are generally small.

As we shall see, the result of Theorem 1 can be extended to the exceptional case, provided this exceptionality is due to a single characteristic feature of α. Before examining this situation, we need to establish a technical result:

Proposition 1 Let h be a feature and k a characteristic feature of α such that Ext h ∩ Ext α ∩ Ext k = ∅ but Ext h ∩ Ext α ∩ ( f ∈χ(α)\{k} )Ext f = ∅. Suppose that δ k is a two-valued function, so that we have δ k : O → {0, 1}. Then one has T yp (h α) = Ext h ∩ Ext α ∩ ( f ∈χ(α)\{k} )Ext f .
It follows from this proposition that the typical elements of h α are easily determined, even in this exceptional case: they correspond to the instances of α to which h applies, and to which applies every characteristic feature of α different from k. For instance, if the characteristic set of to-be-a-dog consists of the features to-bark, to-run-after-sticks and to-gnaw-bones, we deduce from this proposition that typical silent dogs run after sticks and gnaw bones, although they do not bark. Definition 8 A feature h for which the hypotheses of proposition 1 are satisfied is said to be k-exceptional for α; the resulting concept h α will be called a k-exceptionally modified concept.

We shall speak of an elementary feature when referring to a feature whose applicability is given by a two-valued function.

We now prove that, as in the non-exceptional case, a k-exceptionally modified concept is equivalent to a featured concept:

Theorem 2 If k is an elementary feature, the k-exceptionally modified concept h α is equivalent to a featured concept whose defining feature set ∆(h α) is equal to ∆(α) ∪ {h} and characteristic set χ(h α) equal to χ(α) \ {k}.

Theorem 1 and Theorem 2 show that the family of featured concepts is stable under feature determination when the determiner contradicts at most one of the characteristic features of the original concept. This is an interesting result, as it renders possible the study of k-exceptionally modified concepts with the tools used for elementary featured concepts.

Resemblance to modified concepts

We now turn to an important problem of the categorization process, which concerns the link that exists between categorial membership, typicality and resemblance. The notion of resemblance (or similarity) plays a key role in most theories of concepts (see for instance [START_REF] Rosch | Cognitive representations ofsemantic categories[END_REF], [START_REF] Tversky | Extension versus intuitive reasoning: The conjunction fallacy in probability judgement[END_REF], [START_REF] Wittgenstein | Philosophical Investigations[END_REF], and more recently [START_REF] Osta-Velez | Category-based induction in conceptual spaces[END_REF] or [START_REF] Clark | Vector space models of lexical meaning[END_REF]).

We shall briefly recall the basis of our approach before examining how resemblance theory can be applied to the case of mixed compound concepts.

The stereotypical set as a tool for resemblance evaluation

As noted in [START_REF] Freund | Ordered models for concept representation[END_REF], resemblance may be considered as a binary relation between objects (Henry resembles his brother ), between concepts (the wolf resembles a dog), or between an object and a concept (Henry's dog resembles a wolf ). We shall only treat the two latter cases, in which the 'referent' of the comparison is a concept. We observed in [START_REF] Freund | Ordered models for concept representation[END_REF] that resemblance between an object and a concept has to be evaluated through the typical instances of this concept. Thus, looking at a bat, one may say it resembles a bird just because it has wings, flies, and has the size or the shape of a bird. Conversely, an animal may be declared not to resemble a bird if it does not resemble a typical bird, even though this animal is known to be a bird. For instance, looking at a penguin, an assertion like 'this animal does not resemble a bird' is perfectly justified. Resemblance first deals with the typical attributes of a concept.

Similarly to what was done for the notions of categorial membership and typicality, resemblance theory can be modeled through a qualitative approach, so that it is possible to compare the resemblance of two objects to a given concept. Such an ordinal model would account for judgments like Henry's dog resembles more a wolf than Jack's wolfhound. For this purpose, given a featured concept α, we could equip the set ∆(α) ∪ χ(α) with the salience order that extends the salience orders of ∆(α) and χ(α) and makes every element of χ(α) more salient than the elements of ∆(α). Then the corresponding resemblance order could be defined as the order

∆(α)∪χ(α) O induced on O by∆(α) ∪ χ(α).
However, the preeminence this order would give to hierarchy over number, together with the fact that ∆(α) ∪ χ(α) contains more elements that ∆(α) and χ(α), may pose a problem. For instance, if α is the concept to-be-a-bird, and if to-fly has maximal salience in χ(α), bird-resemblance will principally rest of the ability of an item to fly; consequently, bats will be more bird-resemblant than kiwis, although having feathers, singing, having a beak and building nests taken together should, at least compensate the fact that kiwis do not fly.

Another reason to consider changing our paradigm and letting aside the qualitative model is that it makes more sense to speak of a degree of resemblance than to speak of a membership or a typicality degree. For instance, to evaluate the α-membership degree of an item x, one has to take into account not only x and α, but also all the objects that stand between x and the category Ext α (see footnote 1). In contrast, to evaluate resemblance of an object with a concept, one has only to deal with the properties the concept shares with the object. It is a fact that, since the first work of Tversky ( [START_REF] Tversky | Features of similarity[END_REF]), it is exclusively with quantitative approaches that resemblance has been studied.

For these reasons, we chose in [START_REF] Freund | Ordered models for concept representation[END_REF] to work in a unitary model, more adapted than the binary model that was used in the representation of categorial membership and typicality. Thus, we proposed to attach with every featured concept its stereotypical set Π(α) (see [START_REF] Connolly | Why stereotypes don't even make good defaults[END_REF], [START_REF] Fodor | Concepts: a potboiler[END_REF], or [START_REF] Jönsson | On prototypes as defaults (comment on Connolly[END_REF]). This set consists of the union of the defining feature set ∆(α) and the characteristic set χ(α), on which we put a salience order that gives priority to the characteristic features over the defining ones.

More precisely, we equip the set Π(α) = ∆(α) ∪ χ(α) with the order > Π(α) that extends the orders > χ(α) on χ(α) and > ∆(α) on ∆(α) and satisfies furthermore f > Π(α) g for all f ∈ χ(α) and g ∈ ∆(α). This latter condition is justified by the fact that, as previously noted, resemblance with a concept is first resemblance with the typical instances of this concept.

This salience order on Π(α) yields, for each f ∈ Π(α), a salience degree s α (f ) that reflects the number of stereotypical features that f dominates. This degree is defined by s α (f ) = 1 + |{h ∈ Π(α); f > Π(α) h}|.

We define the weight w(α) of the concept α by w(α) = Σ f ∈Π(α) s α (f ), and we denote by (α) the number (α) = |∆(α)| w(α) . It is now possible to introduce the α-resemblance degree of an object x: Definition 9 The degree to which an object x resembles a featured concept

α is the number ρ α (x) = Σ f ∈ Π(α) s α (f )δ f (x) w(α)
.

It is immediate that ρ α (x) = 1 if and only if x ∈ T yp α. Note that one has ρ α (x) ≥ (α) for all objects x ∈ Ext α: an object that falls under a concept cannot be totally dissimilar to this concept.

Example 3 Suppose that the defining feature set associated with to-be-abird is the set {l = to-have-two-legs, e = to-lay-eggs, b =to-have-a-beak, w = to-have-wings} with salience order: b > ∆(α) e > ∆(α) l, and w > ∆(α) e > ∆(α) l. Suppose also that the corresponding characteristic set consists of the three features: n = to-build-nests, s = to-sing, and f = to-fly, with the order : f > χ(α) n, and f > χ(α) s. We have s α (l) = 1, s α (e) = 2, s α (b) = s α (w) = 3, s α (n) = s α (s) = 5 and s α (f ) = 7. This yields w(α) = 26 and (α) = 4/26. If the applicability of all features is given by elementary functions, the bird-resemblance degrees of a bat ba, a penguin pe, a kiwi k and a plane pl is given by ρ α (ba) = 11/26, ρ α (pe) = 9/26, ρ α (k) = 19/26 and ρ α (pl) = 10/26. Note that the bat, which is not a bird, is more birdresemblant than the penguin -which is a bird.

Remark 2

The notion of concept resemblance that results from Definition 8 is not based on the impact of a concept, that is on its induced membership and typicality orders. Rather, it rests on its Gestalt, giving priority to its defining and characteristic set. Note that resemblance degrees allow comparing the resemblance that an object may have with different concepts: it makes possible to express judgments like 'Henry's dog resembles more a fox than a wolf '.

The notion of resemblance between objects and concepts can be easily enlarged to that of resemblance between concepts. Indeed, the resemblance between two concepts is best evaluated as a resemblance of their typical instances: to say that a concept β resembles a concept α amounts to saying that every typical instance of β resembles α. This justifies the following definition:

Definition 10 The α-resemblance degree ρ α (β) of a featured concept β is the number M in x∈T yp β ρ α (x).

One has immediately ρ α (β) = 1 if and only if T yp β ⊆ T yp α.

The resemblance degree provides a similarity measure that can be used in Concept-Based Induction (see [START_REF] Osta-Velez | Category-based induction in conceptual spaces[END_REF] for an overview of different ways of modeling CBI and a geometrical approach of similarity). Note that the value of this resemblance degree is context-dependent: it rests on the features that, at a given time, an agent decides to consider as characteristic of a concept. For instance, the attributes of the concept to-be-a-piano are not the same if considered by an agent in the context of music or in the context of removal.

The following result shows that there exists a resemblance threshold : any object whose α-resemblance is large enough must fall, at least partially, under the characteristic features of α: Theorem 3 Let x be an object such that ρ α (x) > 1 -(α). Then one has δ g (x) > 0 for all elements g of Π(α) such that |∆(α)| ≤ s α (g).

In particular, we see by this theorem that an object that is sufficiently αresemblant will inherit every elementary characteristic feature of α. Thus, in Example 3, the resemblance threshold is equal to 22/26 and any item whose bird-resemblance degree is greater than 22/26 has, to some degree, the properties of flying, singing and building a nest... 2 From the construction of the order > Π(α) , it follows from Theorem 3 that the elementary characteristic features of α apply to any typical instance of a concept that sufficiently resembles α. This result can be translated in the framework of concept-based induction: using the formalism described in section 3.2, we can display the following similarity rule, valid when g is an elementary feature:

Similarity Rule ρ α (β)>1-(α), g ∈ χ(α)
β ∼ g 2 If the characteristic features of Example 3 are elementary features, noting that only birds fly, sing and build nests, an item with resemblance degree greater than 22/26 is necessarily a (typical) bird.

The case of compound mixed concepts

The results of the preceding section can be applied to evaluate resemblance to compound mixed concepts, either in the non-exceptional case, or in the case of a k-exceptionally modified concept. To do this, we can directly make use of the results of Theorem 1 and Theorem 2, assimilating h α with a featured concept and applying the results of the preceding section. This leads us to define the h α-resemblance degree of an object x by

ρ h α (x) = Σ f ∈ Π(h α) s h α (f )δ f (x) w(h α) .
We have then, as an analogue of theorem 3, the Corollary 2 Let x be an object such that δ ρ h α (x) > 1 -(h α), and g an elementary feature of Π(h α) such that |∆(h α)| ≤ s h α (g). Then g applies to x.

A particular attention has to be given to the state of the modifier h in the set Π(h α). Observe indeed that, whether h is a non-exceptional or a k-exceptional feature, it results from the proofs of Theorem 1 and Theorem 2 that the salience order > Π(h α) gives to h a salience s (h α) (h) that is always less or equal than |∆(α)|. By this we see that h does not generally satisfy the conditions of corollary 2, and nothing guarantees that this feature should apply to objects that are sufficiently h α-resemblant. This may lead to paradoxical results. For instance, our model built on the salience order > Π(h α) predicts that an animal that strongly resembles a bird should fly, but that this will not be the case for an animal that strongly resembles a flying bird ! Indeed, the most salient characteristic feature of α, f = to-fly, became the least salient defining feature of f α, and we have s

(h α) (f ) = 1.
This remark provides an important clue concerning the choice of the right salience order in Π(h α). If the feature h is not expected to apply to objects that are h α-resemblant, the original order > Π(h α) need not be changed. But if h is expected to apply to objects similar to h α, it is necessary to increase the salience degree of h to a number greater than |∆(h α)|.

Another problem may occur in the case of a modification of α by a feature that is part of χ(α): to take the above example, reducing to 1 the salience of f will make a bat more bird -resemblant (ρ α (b) = 11/26) than flying-bird resemblant (ρ h α (b) = 7/26). In any way, we expect that determining a concept by one of its characteristic features emphasizes its importance.

To remedy these shortcomings, in the case where h is expected to apply to h α-resemblant objects, we propose to consider the salience degree function s * h α that agrees with s h α on Π(h α) \ {h} and gives to h a salience s * h α (h) equal to |∆(h α)|. With these changes indeed, we will have in all cases s * h α (h) ≥ |∆(h α)|, so that Corollary 2 will apply to h, provided h is an elementary feature.

To give an idea of what is involved, consider the following examples:

1. α stands for to-be-a-dog and h for to-be-black 2. α stands for to-be-a-flower and h for to-be-striped 3. α stands for to-be-a-dog and h for to-be-young 4. α stands for to-be-a-flower and h for to-be-tropical.

In the first two cases, we expect the modifier to apply to any item similar to the compound concept: one would not say that an animal resembles a black dog unless this animal is black; similarly, one would not say that an object resembles a striped flower unless this object has stripes. On the contrary, in the two last examples, the modifier is not expected to apply to objects that are h α resemblant: an animal that is said to resemble a young dog need not be young itself, and an item may resemble a tropical flower without having grown in the tropics. It is difficult to decide which type of modified concept requires a model of the first or of the second kind. We conjecture that when the feature used to determine a concept has a simple 'physical' significance, directly related with one of our senses, strong resemblance to the modified concept implies the presence of this feature in any resemblant item. The situation is different when the modifier is more sophisticated. This may happen in particular when it shows, in the context of the principal concept, a certain number of proper characteristic attributes; it is these attributes, more than the feature itself, that are expected to be retrieved through resemblance. An animal that resembles a young dog has the behavior of a young dog, likely to be clumsy, cute and playful, a plant that resembles a tropical flower is expected to have large leaves and flourishing colors. When used in a comparison, such a feature may be considered as the conjunction of some undisplayed attributes. In the young dog example, these attributes would be to-be-clumsy, to-be-cute and to-be-playful. Then, (to-be-young-dog) may be assimilated to the simple concept (to-be-a-puppy).

More generally, if the modifier h shows, in the context of α, a certain number of charateristic attributes, this modifier may be assimilated with a featured concept. Then the composition h α can be operated as described in section 2.2.2. This shows that features with attributes deserve a special treatment. On one hand, they cannot be considered as simple features, because their applicability cannot be measured by a degree function. On the other hand, they cannot be considered as concepts because they have no significance outside of a precise context. This tends to show that our model finds its limits in the representation of such a kind of hybrid item.

Another point that deserves to be underlined is that, apart from its simple significance, the link between a feature and the concept it modifies has to be taken into account. An exceptional modifier has more weight in the modified concept than a non exceptional one: to-be-unfriendly is expected from an animal that resembles an unfriendly dog, although to-be-unfriendly, in the context of dogs, is an aggregate of several attributes.

From now on, we shall denote by Π * (h α) the set Π(h α) endowed with the salience order induced by s * h α , and by ρ * h α the corresponding degree function.

Comparing α and h α-resemblance

In this paragraph, we shall compare the α-resemblance and the h α-resemblance degree of an object. Making use of Definition 10, we shall also determine to what extent a concept α resembles its modified version h α, and consequently to what extent the modified version differs from the head concept.

The non-exceptional case

Given a non exceptional modifier h, our aim is to compare the resemblance degrees ρ α and ρ h α .

We need first to establish some technical results:

Lemma 2 The salience degrees s h α and s * h α are given by:

• In the case where h / ∈ ∆(α) ∪ χ(α):

s h α (h) = 1, s h α (f ) = 1 + s α (f ) ∀f ∈ Π(α) \ {h}, s * h α (h) = 1 + |∆(α)|, s * h α (f ) = s α (f ) ∀f ∈ Π(α) \ {h} • In the case where h ∈ ∆(α): s h α (f ) = s α (f ) ∀f ∈ Π(α), s * h α (h) = |∆(α)|, s * h α (f ) = s α (f ) ∀f ∈ Π(α) \ {h} • In the case where h ∈ χ(α): s h α (h) = 1, s h α (f ) = 1 + s α (f ) ∀f ∈ ∆(α), s h α (f ) = s α (f ) if f ∈ χ(α), f > χ(α) h, s h α (f ) = 1 + s α (f ) if f ∈ χ(α), f > χ(α) h, s * h α (f ) = s α (f ) ∀f ∈ Π(α)
. This result enables us determining the link between the weights w(α) and w(h α).

Proposition 2 One has w(h α) = 1 + w(α) + |Π(α)| if h / ∈ ∆(α) ∪ χ(α), w(h α) = w(α) if h ∈ ∆(α) if h ∈ χ(α), w(h α) = w(α) + 1 + |∆(α)| -s α (h) + |{f ∈ χ(α) \ {h}, f > χ(α) h}|.
The weights w(h α) being given as above, we can now compare the h αresemblance of an object and its α-resemblance:

Proposition 3 The h α resemblance degree ρ h α (x) of an element x ∈ O satisfies: w(h α)ρ h α (x) = Σ f ∈χ(α) δ f (x) + w(α)ρ α (x) + (1 + |∆(α)|)δ h (x) if h / ∈ Π(α) w(h α)ρ h α (x) = Σ f ∈Π(α) δ f (x) + w(α)ρ α (x) + δ h (x) if h / ∈ Π(α), ρ h α (x) = ρ α (x) if h ∈ ∆(α), and, if h ∈ χ(α), w(h α)ρ h α (x) = w(α)ρ α (x) -s α (h)δ h (x) + δ h (x) + Σ f ∈Π(α)\h,f > χ(α h δ f (x).
Example 4 In the context of birds, consider the feature h = to-be-european, and let us compute how much a bat ba resembles an european-bird. Since we do not expect an animal that resembles an european bird to be itself european, we use the salience function s h α (f ). We have then w(h α) = 34; ba falls under the three characteristic features to-have two-legs, to-have-wings, to-fly. We saw that w(α) = 26, and δ α (ba) = 11/26. This shows that δ h α (ba) = 14/34.

Applying Proposition 3 to the elements x of T yp α enables us to determine the h α resemblance of the concept α: Theorem 4 In the non exceptional case, given the order > Π(h α) on Π(h α), the h α resemblance of α is given by

ρ h α (α) = 1 -1 w(h α) if h / ∈ Π(α), ρ h α (α) = 1 if h ∈ Π(α).
Concerning the α-resemblance degree of h α, things are much simpler: indeed, given an element x of T yp (h α), one has x ∈ T yp α by Theorem 1, and therefore ρ α (x) = 1. It follows that ρ α (h α) = 1: a black dog maximally resembles a dog, although a dog does not maximally resemble a black dog.

The model with salience functions s h α (f ) is inadequate when the modifier h is expected to apply to items that are sufficiently h α-resemblant, and in particular when h is a characteristic feature of α. Suppose for example that, in the context of birds, h is the feature to-fly. Then a computation of the flying-bird -resemblance of a penguin through the functions s h α (f ) shows that its degree is equal to 13/26, and therefore greater than its bird -resemblance degree, which we found equal to 9/26. This paradox is a consequence of the fact that resemblance with a concept is, before all, resemblance with the typical instances of the concept. The feature to-fly, which became a defining feature of to-be-a-flying-bird, is no more a salient characteristic feature of this concept. As we shall see now, the use of the salience degree s * h α can remedy this situation.

Suppose that the salience on Π(h α) is given by the function s * h α . Then the analogue of Proposition 2 for the weight w * (h α) reads:

Proposition 4 One has w * (h α) = 1 + w(α) + |∆(α)| for h / ∈ ∆(α) ∪ χ(α), w * (h α) = w(α) -s α (h) + |∆(α)| if h ∈ ∆(α), w * (h α) = w(α) if h ∈ χ(α).
The link between ρ * h α (x) and ρ α (x) is then given by the Proposition 5 The h α-resemblance degree of an element x ∈ O satisfies:

w * (h α)ρ * h α (x) = Σ f ∈χ(α) δ f (x) + w(α)ρ α (x) + (1 + |∆(α)|)δ h (x) if h / ∈ Π(α), w * (h α)ρ * h α (x) = w(α)ρ α (x)-s α (h)δ h (x)-Σ f ∈∆(α),f > ∆(α) h δ f (x)+|∆(α)|δ h (x) when h ∈ ∆(α), and ρ * h α (x) = ρ α (x) if h ∈ χ(α).
The paradox evoked above desappears when salience is evaluated by the function s * h α : we find that the flying-bird -resemblance of a penguin becomes equal to its bird -resemblance.

Example 5 We take again the bird example and compute the resemblance of a bat with the concept to-be-a-white-bird. An animal resembling a white-bird is expected to be white, and we shall therefore use the salience s * h α . We have w * (h α) = 31, and it follows from the first line of the above proposition that ρ * h α (ba) = 12/31. A bat resembles less a white-bird than an european-bird.

From Proposition 5 we deduce the h α-resemblance of a featured concept α:

Theorem 5 In the non exceptional case, we have:

ρ * h α (α) = |χ(α)|+w(α) w * (h α) if h / ∈ Π(α), ρ * h α (α) = 1 if h ∈ Π(α).
Example 6 If h is the feature to-be-white and α stands for to-be-a-bird we have w * (h α) = 31 , |χ(α)| = 3 and w(α) = 26. Birds resemble white birds to the degree 29/31.

The exceptional case

In the case where h is k-exceptional, we recall that it follows from the proof of Theorem 2, that χ(h α) = χ(α) \ {k} and ∆(h α) = {h} ∪ ∆(α). The order in this latter set makes h less salient than any element of ∆(α). The salience order > Π(h α) is defined as in the case of simple featured concepts (see the begining of section 4.1). Similarly to what was done in the preceding paragraph, we will also consider the salience order s * Π(h α) , in which h is given a salience equal to 1 + |∆(α)|.

For k-exceptionally modified concepts, the analogue of Lemma 2 reads:

Lemma 3 If h is k-exceptional, the salience degrees in Π(h α) are given by s h α (h) = 1, s h α (f ) = 1 + s α (f ) ∀f ∈ Π(h α) \ {h}, f > Π(h α) k s h α (f ) = s α (f ) for all other elements of Π(h α) \ {h}. s * h α (h) = |∆(α)| + 1, s * h α (f ) = s α (f ) ∀f ∈ Π(α) \ {k}).
From this we draw the relation between the weight of h α and the weight of α.

Proposition 6 Suppose that h is k-exceptional for α. Then

w(h α) = w(α) + |∆(α)| + 1 -s α (k) + |{f ∈ χ(α) \ {k}, f > χ(α) k}|. w * (h α) = w(α) -s α (k) + 1 + |∆(α)|.
The two above results allow the calculation of the h α resemblance degrees when h is k-exceptional for α: Proposition 7 In the k-exceptional case, the h α-resemblance degrees of an object x are given by: Let us finally evaluate the resemblance between a concept and its modification by an exceptional feature: Proposition 8 The resemblance of a concept with its modification by a kexceptional feature is given by:

w(h α)ρ h α (x) = w(α)ρ α (x) + δ h (x) -s α (k)δ k (x) + Σ f ∈Π(α)\{k},f >k δ f (x) w * (h α)δ ρ * h α (x) = w(α)ρ α (x) -s α( k)δ k (x) + (∆(α) + 1)δ h (x).
ρ h α (α) = 1 -1 w(h α) δ ρ * h α (α) = 1 -(h α)
Contrary to what we saw in the non-exceptional case, the resemblance of a k-exceptionally modified concept with its original head concept is no longer equal to 1:

Proposition 9 One has ρ α (h α) = 1 -sα(k) w(α)
The α resemblance of h α is maximal when the relative salience of k in χ(α) is > χ(α) -minimal. This translates the fact that, due to the kexceptionality of h, k is no longer an element of the characteristic set of h α. For instance, if we think that to-live-in-the-trees is not a salient attribute of birds, we are ready to consider that typical aquatic birds are not really dissimilar to birds.

Conclusion

The model that we studied in [START_REF] Freund | Ordered models for concept representation[END_REF] to represent the basic notions linked with categorization can be carried over to mixed concept composition. It proposes a rigorous definition of notions that were shown to play an important role in concept study, like those of smooth subconcept, concept equivalence and concept resemblance, as well as the necessary distinction between exceptional and non-exceptional modfiers. It provides interesting results in the domain of concept composition, even in the exceptional case where the modifier stands in contradiction with some characteristic feature of the original concept. Problems linked with categorial membership, typicality, and resemblance can be then treated as in the elementary case, leading to interesting and non-trivial results. At the same time, such an ordinal model has the advantage of being intuitive and user-friendly, as it does not require the use of sophisticade mathematical tools.

However this study needs to be improved in several points. It appeared for instance that the notion of feature is not fully determined. In particular, we noticed an ambiguity concerning the treatment features should receive within the classical binary model. Apart from the limit case where the modifier has an hybrid status that stands half-way between a feature and a concept, two possible issues were considered, each leading to a different solution. In the first one, the modifier of a concept will apply to objects that sufficiently resemble the modified concept. In the second case, the modifier is not preserved by resemblance. In the end, it appears that the criteria on which we should base feature representation remain unclear at this stage, showing the necessity of refining our primitive model and investigate more closely the different ways a feature relates with a concept.

We are also aware that this model lacks experimental confirmations. Thus, we considered several times as granted some common sense statements. We claimed for instance that ordering the elements of a set provides a simpler and more accurate way of comparing them than attributing to each of them a degree number. This should be supported by experiments. Above all, the main results proposed by the model concerning membership ordering and resemblance evaluation definitely need to be confirmed by experiments. At this stage, this work can only be taken as an hypothetical construction.

A Proofs

Lemma 1 The relation X T is a preorder. For two elements x and y of T , one has simultaneously x X T y and y X T x iff, ∀f ∈ X, δ f (x) = δ f (y). The strict partial order ≺ X T associated with X T satisfies:

x ≺ X T y iff x X T y and ∃f ∈ X : δ f (x) < δ f (y).

Proof : For the first part of the lemma, the reflexivity is immediate, and we have only to prove transitivity. Let therefore x, y and z be three elements of T such that x T X y and y T X z; suppose that there exists f ∈ X such that δ f (z) < δ f (x). We have to show that there exists a feature g ∈ X, g more salient than f , such that δ g (x) < δ g (z). We consider two cases:

• Suppose first that δ f (x) ≤ δ f (y). Then we have δ f (z) < δ f (y), and there exists a feature g of X, g > X f , such that δ g (y) < δ g (z). We can suppose that g is maximally salient in X for this property (recall that X is finite). If δ g (x) ≤ δ g (y), we get δ g (x) < δ g (z) and we are done. Otherwise, we have δ g (y) < δ g (x). Since we supposed x T X y, this implies that there exists a feature h ∈ X, h more salient than g, such that δ h (x) < δ h (y). We cannot have δ h (z) < δ h (y) otherwise there would exist k ∈ X, k more salient than h, and therefore more salient than g, such that δ k (y) < δ k (z) contradicting the choice of g. We have therefore δ h (y) ≤ δ h (z), hence δ h (x) < δ h (z) as desired.

• Suppose now that δ f (y) < δ f (x). Then there exists k ∈ X, k more salient than f such that δ k (x) < δ k (y), and we can again suppose k maximally salient for this property. In the case where δ k (y) ≤ δ k (z), we get δ k (x) < δ k (z) and we are through. Otherwise, we have δ k (z) < δ k (y) and there exists g more salient than k such that δ g (y) < δ g (z). Let us show that δ g (x) ≤ δ g (y): if this were not the case, we would have δ g (y) < δ g (x), so that there would exist h more salient than g such that δ h (x) < δ h (y). But then h would be more salient than k, which is impossible. We have therefore δ g (x) ≤ δ g (y), hence δ g (x) < δ g (z), which completes the proof of the transitivity.

Concerning the second part of the lemma, we observe that one has x X T y and y X T x whenever δ f (x) = δ f (y) ∀f ∈ X. Conversely, suppose that one has x X T y and y X T x, and let us show that δ f (x) = δ f (y) for all f ∈ X. If this were not the case, there would exist a feature f of X such that δ f (x) = δ f (y), f being > X -maximal for this property. We have either δ f (x) < δ f (y), or δ f (y) < δ f (x). In the first case, the hypothesis y X T x implies the existence of a feature g of X, g > X f , such that δ g (y) < δ g (x), which contradicts the choice of f . In the second case, the hypothesis x X T y leads again to a contradiction.

For the last part of the lemma, by definition of the strict partial order associated with a preorder, we have x ≺ X T y iff x X T y and not y X T x. This latter condition is equivalent to the existence of a feature f of X that satisfies δ f (x) < δ f (y) and such that δ g (x) ≤ δ g (y) for all features g ∈ X, g > X f . The RHS of equation ( 1) then clearly follows from the LHS. If now the RHS is satisfied, that is if we have x X T y and ∃f ∈ X such that δ f (x) < δ f (y), then, by what precedes, we cannot have y X T x. This completes the proof of the lemma.

Theorem 1 Let h be a feature that is not exceptional for α. Then one has T yp (h α) = Ext h ∩ T yp α. The concept h α is equivalent to a featured concept whose defining feature set ∆(h α) is equal to {h} ∪ ∆(α), and whose characteristic set χ(h α) is equal to χ(α) if h / ∈ χ(α), and to χ(α) \ {h} if h ∈ χ(α).

Proof : Let z be an element of Ext h ∩ T yp α. Choose an element x in Ext h ∩ Ext α that is not in T yp α. We have then x ≺ τ α z. Since x and z are elements of Ext h ∩ Ext α, it follows from Definition 4 that x ≺ τ h α z. This shows that the τ h α -maximal elements of Ext h ∩ Ext α are the elements of Ext h ∩ T yp α.

We now prove the second part of the theorem. A) We suppose first that h / ∈ ∆(α) ∪ χ(α).

To build the membership order {h}∪∆(α) O

, we define on the set {h}∪∆(α) the salience order > {h}∪∆(α) that extends the salience order on ∆(α) and makes every element of ∆(α) more salient than h. We claim that one has x µ h α y if and only if x {h}∪∆(α) O y. Given two objects x and y such that x {h}∪∆(α) O y, let us indeed show that x µ h α y. If f ∈ ∆(α) is a feature such that δ f (y) < δ f (x), the hypothesis implies that there exists g ∈ {h} ∪ ∆(α), g more salient than f , such that δ g (x) < δ g (y). By the choice of the salience order in {h} ∪ ∆(α), we have necessarily g = h, so g ∈ ∆(α), and we have proven that x µ α y. If δ h (x) ≤ δ h (y), this implies x µ h α y. If, on the contrary, δ h (y) < δ h (x), by the hypothesis, this implies the existence of a feature k ∈ {h} ∪ ∆(α), k more salient than h, such that δ k (x) < δ k (y). We have then k ∈ ∆(α). By Lemma 1, this leads to x ≺ µ α y, from which follows x ≺ µ h α y. In any case, we have shown that x µ h α y whenever x {h}∪∆(α) O y. Conversely, suppose that one has x µ h α y. To show that x {h}∪∆(α) O y, we suppose that δ f (y) < δ f (x) for some feature f of {h} ∪ ∆(α). We have to prove that there exists g ∈ {h} ∪ ∆(α), g more salient than f , such that δ g (x) < δ g (y).

-If f ∈ ∆(α), the hypothesis x µ h α y, which implies x µ α y, shows the existence of a feature g ∈ ∆(α), more salient than f , such that δ g (x) < δ g (y).

-If f = h, it follows from the hypothesis x µ h α y that we have x ≺ µ α y. Then, there exists g ∈ ∆(α) such that δ g (x) < δ g (y). By the choice of the salience order on ∆(h α), g is more salient than h. In any case, we have shown that x The salience order on ∆(h α) is that of ∆(α), and we have µ α = {h}∪∆(α) O . Let us show that µ α = µ h α : We know that x µ h α y implies x µ α y. Conversely, suppose that we have x µ α y and let us show that x µ h α y. We have to prove that x ≺ µ α y or δ h (x) ≤ δ h (y). But if we have δ h (y) < δ h (x) there exists g > ∆(α) h such that δ g (x) < δ g (y). This together with the fact that x µ α y implies x ≺ µ α y as desired. As for the typical order, the proof of the equality τ h α = χ(h α)

Ext h∩Ext α mirrors that made in case A).

C) In the case where h ∈ χ(α).

We have h / ∈ ∆(α) and the proof that µ h α = {h}∪∆(α) O mirrors that made in A).

Concerning the typcality order, note first that that χ(α) = {h}, since we supposed the existence of a non-empty salience order on this set. We put on χ(α) \ {h} the restriction of the order > χ(α) . Given two element x and y of Ext h ∩ Ext α, we have to prove that x τ α y iff x χ(α)\{h}

Ext h∩Ext α y. Suppose first that x τ α y and let f be a feature of χ(α) \ {h} such that δ f (y) < δ f (x). Then f ∈ χ(α), and there exists g ∈ χ(α), g more salient than f , such that δ g (x) < δ g (y). We cannot have g = h because δ h (x) = δ h (y) = 1. We have then g ∈ χ(α) \ {h}, showing that x χ(α)\{h} Ext h∩Ext α y. Conversely, suppose that x χ(α)\{h} Ext h∩Ext α y. If δ f (y) < δ f (x) for some feature f ∈ χ(α), we have f = h by the choice of x and y, so f ∈ χ(α) \ {h}. It follows that there exixts g ∈ χ(α) \ {h}, g more salient than f , such that δ g (x) < δ g (y). We have therefore x τ α y as desired, and this completes the proof of the theorem. Proof : Let x be an element of T yp (h α), and suppose that there exists a characteristic feature g = k such that δ g (x) < 1. Take an element y of Ext h ∩ Ext α ∩ ( f ∈χ(α)\{k} )Ext f . We would have then x ≺ τ α y and therefore x ≺ τ h α y, which is impossible because, by the choice of x, x is τ h α -maximal in Ext (h α). We have therefore proven that all characteristic features g = k apply to the typical instances of h α. This shows that T yp

(h α) ⊆ Ext h ∩ Ext α ∩ ( f ∈χ(α)\{k} )Ext f . Conversely, given an element x ∈ Ext h ∩ Ext α ∩ ( f ∈χ(α)\{k} )Ext f , let us prove that x is τ h α -maximal in Ext (h α).
If this were not the case, there would exist y ∈ Ext (h α) such that x ≺ τ h α y, that is x ≺ τ α y. By the choice of x, this could be possible only if δ k (x) < δ k (y), that is δ k (y) = 1. But then, this would imply y ∈ Ext h ∩ Ext α ∩ Ext k = ∅, a contradiction. We have therefore shown that Ext h ∩ Ext α ∩ ( f ∈χ(α)\{k} )Ext f is a subset of T yp (h α), and this completes the proof of the proposition.

Theorem 2 If k is an elementary feature, the k-exceptionally modified concept h α is equivalent to a featured concept whose defining feature set ∆(h α) is equal to ∆(α) ∪ {h} and characteristic set χ(h α) equal to χ(α) \ {k}.

Proof : Since h is exceptional for α, we cannot have h ∈ ∆(α) ∪ χ(α). We attach to (h α) the defining set ∆(h α) = ∆(α) ∪ {h}, equipped with the order > ∆(α)∪{h} that extends the salience order of ∆(α) and makes h less salient than every element of ∆(α). Then the proof that

∆(h α) O = µ
h α is similar to that made in Theorem 1.

We equip the set χ(h α) with the restriction to χ(α) \ {k} of the order > χ(α) . Recalling that, on Ext (h α), the order τ h α is equal to τ α , we have to prove that, for x and y in Ext h∩Ext α, one has x τ α y iff x χ(α)\{k}

Ext h∩Ext α y. Suppose first that one has x τ α y, and let f be an element of χ(α) \ {k} such that δ f (y) < δ f (x). There exists g ∈ χ(α), g more salient than f , such that δ g (x) < δ g (y). If we had g = k, we would have δ k (x) < δ k (y), that is, since k is an elementary feature, δ k (x) = 0 and δ k (y) = 1. But then y would be an element of Ext h ∩ Extα ∩ Ext k, contradicting the k-exceptionality of h. This shows that g ∈ χ(α) \ {k}. Since g is more salient than f in this set, we have proven that x χ(α)\{k} Ext h∩Ext α y. Conversely, suppose that we have x χ(α)\{k} Ext h∩Ext α y for two elements of Ext h ∩ Ext α. We have to show that x τ α y. If f ∈ χ(α) is such that δ f (y) < δ f (x), the same reasoning that before shows that f = k, that is f ∈ χ(α) \ {k}. The conclusion follows.

Theorem 3 Let x be an object such that ρ α (x) > 1 -(α). Then one has δ g (x) > 0 for all elements g of Π(α) such that |∆(α)| ≤ s α (g).

Proof : Suppose that g is a stereotypical feature such that |∆(α)| ≤ s α (g). Proof : The proof directly follows from the preceding lemma. As an example we shall only examine the last case where h ∈ χ(α). We have w(h α) = Σ f ∈Π(h α) s h α (f ) = Σ f ∈χ(α)\{h} s h α (f ) + Σ f ∈∆(α) s h α (f ) + s h α (h) = |{f ∈ χ(α) \ {h}, f > χ(α) h}| + Σ f ∈χ(α)\{h} s α (f ) + |∆(α)| + Σ f ∈∆(α) s α (f ). The result follows, noting that w(α) = s α (h) + Σ f ∈χ(α)\{h} s α (f ).

w * (h α) = 1 + w(α) + |∆(α)| for h / ∈ ∆(α) ∪ χ(α), w * (h α) = w(α) -s α (h) + |∆(α)| if h ∈ ∆(α), w * (h α) = w(α) if h ∈ χ(α).

Proof : If h / ∈ ∆(α), the result is immediate from Lemma 2. Suppose h ∈ ∆(α). We have Σ χ(α) s * h α (f ) = Σ χ(α) s α (f ). Writing then the sum Σ f ∈∆(α) s * h α (f ) as Σ f ∈∆(α)\{h},f > Π(α) h s * h α (f )+Σ f ∈∆(α)\{h},f > Π(α)h s * h α (f )+s * h α (h), we apply the second part of Lemma 2. Proof : Apply Proposition 5 for x ∈ T yp α and take the minimum possible value: If h / ∈ Π(α), we have δ h (x) = 0. If h ∈ Π(α), we have δ h (x) = 1. This yields the result.

One may also remark that one has T yp (h α) = Ext h ∩ T yp α (Theorem 5): any typical instance of α that falls under h is a typical instance of h α... Proof : The first equality is a direct consequence of the preceding Lemma.

For the second equality, we use once more the fact that Σ f ∈Π(α)\{k} s α (f ) = w(α)ρ α -s α (k).

Proposition 7 In the k-exceptional case, the h α-resemblance degree of an object x is given by: w(h α)ρ h α (x) = w(α)ρ α (x) + δ h (x) -s α (k)δ k (x) + Σ f ∈Π(α)\{k},f >k δ f (x), w * (h α)δ ρ * h α (x) = w(α)ρ α (x) -s α (k)δ k (x) + ∆(α) + 1.

Proof : For the first equality; we have w(h α)ρ h α (x) = δ h (x) + Σ f ∈Π(h α)\{h} s h α (f )δ f (x). We then write Π(h α) \ {h} = {f ∈ Π(h α) \ {h}, f > Π(α) } ∪ {f ∈ Π(h α) \ {h}, f > Π(α) } and apply Lemma 2. The second equality directly follows from Lemma 3.

Proposition 8

The resemblance of a concept with its modification by a kexceptional feature is given by: ρ h α (α) = 1 -1 w(h α)

Example 7

 7 Let us compute how much a penguin pe resembles a walkingbird. By Proposition 6, we have w(h α) = 26 + 4 + 1 -7 + 2 = 26, and by Proposition 7 and Example 3, 26 • ρ h α (pe) = 26 • (9/26) + 1 -0 + 4 = 14, showing that ρ h α (pe) = 14/26. Then w * (h α) = 26 -7 + 14 = 24, so 24 • δ ρ * h α (pe) = 26 • 9/26 -0 + 4 + 1 = 14, which yields δ ρ * h α (pe) = 14/24.

  h∩Ext α , recall that, by definition 4, the orders τ h α and τ α agree on the set Ext h ∩ Ext α. Since h / ∈ χ(α), we have by hypothesis χ(h α) = χ(α). The order τ h α is therefore the restriction to Ext h ∩ Ext α of the order τ α , and thus equal to χ(h α) Ext h∩Ext α . B) In the case where h ∈ ∆(α).

Corollary 1 Proposition 1

 11 For any typical attribute g of α, one has T yp(g α) = T yp α.Proof : Immediate. Let h be a feature and k a characteristic feature of α such that Ext h ∩ Ext α ∩ Ext k = ∅ but Ext h ∩ Ext α ∩ ( f ∈χ(α)\{k} )Ext f = ∅.Suppose that δ k is a two-valued function, so that we have δ k : O → {0, 1}. Then one has T yp (h α) = Ext h ∩ Ext α ∩ ( f ∈χ(α)\{k} )Ext f .

Corollary 2 Lemma 2 Proposition 2

 222 From 0 ≤ s α (g) -|∆(α)|, we get Σ f ∈Π(α)\{g} s α (f ) ≤ Σ f ∈Π(α) s α (f ) -|∆(α)|. If δ g (x) = 0, we would have Σ f ∈ Π(α) s α (f )δ f (x) ≤ Σ f ∈Π(α) s α (f ) -|∆(α)|, and therefore ρ α (x) ≤ 1 -∆(α)| w(α) = 1 -(α), a contradiction. Let x be an object such that ρ h α (x) > 1 -(h α), and g an elementary feature of Π(h α) such that |∆(h α)| ≤ s h α (g). Then g applies to x.Proof : Immediate. The salience degrees s h α and s * h α are given by:• In the case where h / ∈ ∆(α) ∪ χ(α): s h α (h) = 1, s h α (f ) = 1 + s α (f ) ∀f ∈ Π(α) \ {h}, s * h α (h) = 1 + |∆(α)|, s * h α (f ) = s α (f ) ∀f ∈ Π(α) \ {h}• In the case where h ∈ ∆(α):s h α (f ) = s α (f ) ∀f ∈ Π(α), s * h α (h) = |∆(α)|, s * h α (f ) = s α (f ) ∀f ∈ Π(α) \ {h}• In the case where h ∈ χ(α):s h α (h) = 1, s h α (f ) = 1 + s α (f ) ∀f ∈ ∆(α), s h α (f ) = s α (f ) if f ∈ χ(α), f > χ(α) h, s h α (f ) = 1 + s α (f ) if f ∈ χ(α), f > χ(α) h, s * h α (f ) = s α (f ) ∀f ∈ Π(α). Proof : Straightforward from the definitions of s h α and s * h α . One has w(h α) = 1 + w(α) + |Π(α)| if h / ∈ ∆(α) ∪ χ(α), w(h α) = w(α) if h ∈ ∆(α), w(h α) = w(α)+1+|∆(α)|-s α (h)+|{f ∈ χ(α)\{h}, f > χ(α) h| if h ∈ χ(α).

Proposition 5 Theorem 5

 55 Inn the non-exdeptional case, the h α resemblance degree of an element x ∈ O satisfies:w * (h α)ρ * h α (x) = Σ f ∈χ(α) δ f (x) + w(α)ρ α (x) + (1 + |∆(α)|)δ h (x) if h / ∈ Π(α), w * (h α)ρ * h α (x) = w(α)ρ α (x) -s α (h)δ h (x) + |∆(α)|δ h (x) if h ∈ ∆(α), ρ * h α (x) = ρ α (x) if h ∈ χ(α). Proof : If h / ∈ Π(α), we write w * (h α)δ ρ * h α (x) = Σ χ(α) s * h α (f )δ f (x) + Σ ∆(α) s * h α (f )δ f (x) + s * h α (h)δ h (x), and apply the first part of Lemma 2.If h ∈ ∆(α), the second part of Lemma 2 yieldsw * (h α)ρ * h α (x) = Σ Π(α)\{h} s h α (f )δ f (x) + |∆(α)|δ h (x), that is w * (h α)ρ * h α (x) = w(α)ρ h α (x) -s α (h)δ h (x) + |∆(α)|δ h (x).If h ∈ χ(α), we write w * (h α)ρ * h α (x) = Σ Π(α) s h α (f )δ f (x) and conclude by Lemma 2, noting that w * (h α) = w(α). In the non exceptional case, the salience s * h α yields: ρ * h α (α) = |χ(α)|+w(α)w * (h α) if h / ∈ Π(α), ρ * h α (α) = 1 if h ∈ Π(α).

Lemma 3 Proposition 6

 36 If h is k-exceptional, the salience degrees in Π(h α) are given bys h α (h) = 1, s h α (f ) = 1 + s α (f ) ∀f ∈ Π(h α) \ {h}, f > Π(α) k, s h α (f ) = s α (f ) for all other elements of Π(h α) \ {h}, s * h α (h) = |∆(α)| + 1, s * h α (f ) = s α (f ) ∀f ∈ Π(α) \ {k}. Proof : Straightforward. Suppose that h is k-exceptional for α. Then w(h α) = w(α) + |∆(α)| + 1 -s α (k) + |{f ∈ χ(α) \ {k}, f > χ(α) k}|. w * (h α) = w(α) -s α (k) + 1 + |∆(α)|.

δ ρ * h α (α) = 1 -Proposition 9

 19 (h α) Proof : If z is an element of T yp α, it follows from Proposition 7 that w(h α)ρ h α (z) = w(α) -s α (k) + |{f ∈ Π(α) \ {k}, f > k}|. Writing {f ∈ Π(α) \ {k}, f > k} = {f ∈ χ(α) \ {k}, f > k} ∪ {f ∈ ∆(α)}, we get, by Proposition 6 w(h α)ρ h α (z) = w(h α) -1 whence the result.Similarly, we have by Proposition 7w ρ (h α)ρ h α (z) = w(α) -s α (k) = w ρ (h α) -(|∆(α) + 1)|. One has ρ α (h α) = 1 -sα(k)w(α) .Proof :For any z ∈ O, w(α)ρ α (h α) = Σ Π(α\{k} s α (f )δ f (z) + s α (k)δ k (z). If z ∈ T yp (h α), we have δ f (z) = 1∀ f ∈ Σ Π(α)\{k} and δ k (z) = 0 since h is k-exceptional.

The distance of an object x to a category Ext α may be defined as the maximal length of a chain x ≺ µ α x 1 ≺ µ α x

≺ µ α . . . ≺ µ α x n with x n ∈ Ext α.

Proposition 3 The h α resemblance degree ρ h α (x) of an element x ∈ O satisfies:

By Lemma 2, the first summand is equal to

We have

The result follows.

Theorem 4 If h is not exceptional for α, the h α resemblance of α is given by

Proof : By definition 10, we have ρ h α (α) = M in x∈T yp α ρ h α (x). If x is an element of T yp α, we have ρ α (x) = 1 and δ f (x) = 1 ∀ f ∈ Π(α). The result then follows from Proposition 3, noticing that M in x∈T ypα ρ h α (x) is reached when δ h (x) = 0.

If h ∈ ∆(α), one has ρ h α (α) = 1 from Proposition 2.

In the case where h ∈ χ(α), the result follows from Proposition 2 and Proposition 3. One may also directly apply Corollary 1 of Theorem 1.

Proposition 4 One has