Computing Contrast on Conceptual Spaces

Abstract

This paper provides an updated formalization of the operation of contrast, and shows that, by applying it on conceptual spaces, membership functions to categories as e.g. those captured by adjectives or directional relationships emerge as a natural by-product. Because the outcome of contrast depends not only on the objects contrasted (a target and a reference, as for instance a prototype), but also on the frame in which those are contained, it is argued that contrast enables a continuous contextualization, o ering a basis for \on the y" predication. This investigation is used for inferring general requirements for the application of contrast and its generalization, and for comparison with current practices in the conceptual space literature.

Keywords

• Contrast
• Conceptual Spaces
• Categorization
• Predication
• Contextualization

Introduction

In its original formulation, the theory of conceptual spaces [ 1 ] assumes a generally working association between regions and linguistic marks; because symbols are associated with sets of points, the theory can be seen as relying on an extensional semantics, in continuity with symbolic approaches. An alternative proposal, introduced in [ 2 ], starts from the observation that predication (i.e. the production of predicates about a certain object or situation) should follow principles of relevance, which, from a descriptive point of view, means determining an object by utilizing its distinctive features. Accepting this, predicates should be the result of contrast operations made on the y between conceptual objects. In a previous paper [ 3 ], starting from this basis, and representing conceptual objects as points in a conceptual space, we were able to explain, all in maintaining a geometric view of psychological spaces, the misalignments between theoretical properties and empirical observations of similarity judgments [ 4 ], concerning symmetry, triangle inequality, but also the overlooked minimality axiom and diagnosticity e ect. Motivated by this result, part of our research has been directed towards a more structured formalization of contrast, whose functional components were only sketched before. This paper aims to present preliminary results of this e ort and considerations about future developments.

The paper proceeds as follows. After a brief overview on the theory of conceptual spaces, the contrast mechanism is explained on a simple case issued from a mono-dimensional perceptual space (x 2). These results are then generalized to the multidimensional case, analyzing illustrative examples (x 3). The paper ends with a discussion, contextualizing our approach within the current practices in the conceptual space literature, highlighting some open questions (x 4). 1.1

Overview of the Theory of Conceptual Spaces theory

In the literature [ 1, 5 ], the introduction of conceptual space is motivated by the observation that the meaning of words can be represented as convex regions in a high-dimensional geometric space, whose dimensions correspond to cognitively primitive features. In practice, conceptual spaces are usually modeled as vector spaces (e.g. [6{8]). An object of a conceptual space is characterized by several qualities or attributes:

(q1; q2; : : : ; qn); 8i : qi 2 Qi where Qi are sets of possible values for each quality qi. Quality dimensions correspond to the ways in which two stimuli can be considered to be similar or di erent, depending on an ordering relation between them; they are usually modeled on R, R2, . . . , N; N2, etc. but proposals exist to process nominal dimensions (e.g. in [ 9, 10 ]). In agreement with the cognitive psychology literature, dimensions are organized in domains or sets of integral dimensions, i.e. dimensions that cannot be separated perceptually (e.g. for humans, the color dimensions hue-luminosity-saturation ). A conceptual space consists therefore of: C = D1

D2 : : :

Dm where each Di is a domain. As each Di consists of a set of qualities, the resulting structure is hierarchical. This representational infrastructure enables the distinction between objects, i.e. points of the space (used to represent exemplars and prototypes, i.e. exemplar-based and prototypical bodies of knowledge [ 9 ]), and concepts, de ned as regions of the space.

According to the original theory, natural properties emerge as convex regions within a domain [ 1 ] to guarantee betweenness among similar elements. Concepts generalize properties as weighted combinations of them, typically across multiple domains, and should satisfy the convexity constraint.3 Prototypes can be seen as centroids of concept (including property) regions. The other way around, the division of conceptual spaces into regions can be seen as the result of a competition between prototypes, that might be captured by e.g. Vorono tessellations ; this approach can be used for categorization. Following empirical evidence, the theory suggests to measure the distance between points in the conceptual spaces (i.e. distance between conceptual objects, exemplars or prototypes) through a weighted Manhattan metric of the intra-domain distances. 3 This is source of discussion: see for instance [ 11 ] about the consequent impossibility of capturing correlation geometrically, and the various arguments provided in [ 12 ]. 1.2

Restarting from Predication

For its insistence on lexical meaning, approached by the association of linguistic marks to regions, the original account of conceptual spaces can be seen as an extension of the symbolic approach, in the sense that it follows an extensional semantics. In contrast, the alternative proposal introduced in [ 2 ] considers that predication should follow principles of relevance, which, from a descriptive point of view, means describing an object by utilizing its distinctive features. Consequently, predicates are hypothesized to be the result of contrast operations made on the y between conceptual objects.

This change of view carries interesting innovations. First, whereas practically all other works rely on a global distance over all available dimensions (which potentially are in nite), the use of contrast does not necessarily require a holistic perspective, but restrains the focus to (adequately) distinctive dimensions. Second, working with contrast would allow for the most to bypass the problem of maintaining de nite regions, so as to require in principle only access to the representational level of points and some rough regional information. Third, it does not refer to average lexical meanings emerging from usage, but it is computed contingently with contrastors and prototypes grounded on the agent's own experience (the association of speci c symbols or anchoring [ 13, 14 ] remains deferred at social level). The present contribution aims to develop further this approach. 2

Contrast

Consider co ees served in a bar. Intuitively, whether one of these co ees is predicated as being \hot" or \cold" depends mostly on what the locutor expects of co ees served at bars (ignoring e ects due to her contingent state), rather than a speci c absolute temperature. In other words, the description of an object or exemplar o (e.g. a co ee served in a bar) results from its contrast with a certain prototype (of co ees served in bars) p. Naming the resulting output contrastor, we can utilize its categorization as a basis for predication, at least for such modi ers. Then, for instance, denoting provisionally the operation of contrast with , and categorization with , we would have: o p = c \hot"

Let us assume that conceptual spaces are vector spaces de ned on Rn; as for the moment we are focusing on the temperature dimension, we have n = 1. If both left and right operands were simple points, and contrast corresponded to vectorial di erence, c would be a free vector, maintaining the same scale of the input points. However, at least this last aspect is implausible, because \modifying" categorizations resulting from contrast might be applied to very di erent scales (cf. \small molecule" vs \small galaxy"). We need at least an adequate scaling. 2.1

Scaling

Because prototypes are associated with a certain concept region, we might consider some regional information as well. Following models dealing with imperfect spatial information such as the egg-yolk model (e.g. [ 15 ]), we refer to two boundaries: an internal one (containing the yolk ), computed as p , and external one (containing the egg ), computed as p , where p, the prototype point, is the centroid of the region (for simplicity, egg and yolk regions are here supposed to be symmetric). Elements falling within the yolk are typical, normal instances of the concept associated with the prototype; within the egg but not in the yolk are instances still associated with the same concept, but manifesting some distinctive characteristic; when elements are outside the egg, they are not directly associated with the concept.4 Partitioning Let us consider a perceptive space made just by one dimension (e.g. temperature), modeled as U = [ U ; U ]; this is a bounded space (plausible assumption in the context of nite cognitive resources), de ned by two boundaries of opposite polarity. The neutral element, as well as center of the space, is indexed by 0. We can denote U also as 0U , i.e. a region centered on 0 of radius U . This space may be naturally divided into partitions such as M2 = f[ U ; 0]; [0; U ]g = fU ; U +g, or M3 = f[ U ; 3U ]; [ 3U ; 3U ][ 3U ; U ]g, etc. Note that these constructions are made independently of any world semantics: they are just based on the perceptual possibilities given by the space. Centering Now, returning to our example of contrast, we aim to compare our target with typical exemplars of the concept, which, by construction, are contained in yolk region p .5 One way to do that is to perform a point-wise contrast. Let us denote the extensional descriptions of o and p respectively by O = fog and P = fx : jx pj g = p . Performing a point-wise vectorial di erence of the sets6 (here denoted with ) we have: O

P = fx y : x 2 O; y 2 P g = fo y : y 2 P g = f (y o) : y 2 P g = (P O)s where (X)s is the symmetric of X with respect to the origin. Observing that the inner operation is a translation, we have: O P = ( p fog)s = o p. 4 In principle, the distinction of and boundaries should reproduce the di erence in judgments about typicality and categorization observed in empirical settings [ 16 ]. 5 In other words, captures a minimal distance that brings an object outside the typicality zone centered on p, making it distinguishable. It can be used to express a sort of indeterminacy of the prototype, which will be propagated to the output contrastor. 6 Note that, in the general case, a point-wise di erence of two sets of points is a multi-set. Here, as O consists of a single point, it is a simple set.

target -ρ centering scaling

+1 +σ o-p -σ

+σ/ρ (o-p)/ρ -σ/ρ 0 reference (e.g. prototype) frame -1 representational container Scaling In order to exploit the full representational domain given by the perceptual space, contextualized for objects of the same category of the target, we might consider the egg region p , presumably containing all the exemplars of the related concept. For doing that, given U = [ U ; U ] = 0U , and denoting scaling with \ " ( x = x ), we need to use a scaling factor such that = U . Let U be arbitrarily 1. Thus, in extensional terms, the contrastor C obtained by contrasting an object o for its prototype p is given by: C = contrast(o; hp; ; i) = o p 1 = = (o p)= where hp; ; i speci es the region associated with the concept: the centroid, and the internal and external boundaries. Figure 1 illustrates its computation. 2.2

Categorization

At this point we need to settle upon a categorization method, i.e. how to select the category to which the contrastor belongs. When contrastors and categories are seen as extensional objects, a typical method would consist in nding the category with maximal overlap with the contrastor. Denoting the category label with r, and its extension with M (r), the method amounts to solve: arg rmax jC \ M (r)j (1) (2)

strength of M1

strength of M2 p-σ p p+σ (a) p (b) strength of M1 strength of M2

strength of M3 -σ p-ρ/3 +σ -σ p+ρ/3 +σ o o Thus, the category with the maximal overlap with the contrastor (i.e. with the maximal strength), if associated with a label, may be used to describe a distinctive feature of the object (with respect to the prototype).

Natural categories Let us consider as an example the partitions issued from the perceptual space, starting from the bipolar categorization captured by M2 = fM1; M2g. Using the de nition of contrastor with the strength equation given above, we obtain Figure 2a. A relevant point is o = p, the case in which our object is plainly prototypical, and then the contrastor is a region centered on zero, capturing the same surface on the two dual parts M1 and M2. If we interpret M1 as associated with \cold" and M2 to \hot", we could say that, according to this view, the co ee is as much cold as hot. However, this bipolar construction would fail to su ciently capture the graduality of judgment, in the sense that a co ee may also be \ok", condition which is usually not de ned as being hot and cold at the same time. Fortunately, to solve this we just need to add an additional category, as in Figure 2b with M3. Note that all partitions M2n+1 similarly built provide a neutral category. Evidently, we do not need necessarily to use such constructions, their use here is rather illustrative.

Choice of parameters In the previous sections the choice of the parameters was given qualitatively: captures the most typical exemplars, whereas in principle covers all exemplars belonging to the same category of objects O (here de ned on one dimension). A simple way to compute the radius would be: = maxo2O o

mino2O o One way to decide could be by relying on the standard deviation, thus introducing information concerning the relative frequencies amongst individuals. Alternatively, it could be calculated just as , but considering a core subset of O centered on p. Note that the de nitions given so far should be modi ed if we relax the assumption of symmetrical regions.

Adaptation The region parameters could be modi ed in two ways: { unsupervised : when an exemplar o, not within the current boundaries of the category of objects O, results to be more similar to the prototype p of O than to prototypes of other categories, it is then labeled as belonging to p, implying a rede nition of and ; { supervised : the user provides a new exemplar o explicitly labeled as O, going beyond the current boundaries of the category; and are then recomputed. In both cases, the process implies an e ect of relativization: providing more contrastive exemplars, those which were highly contrastive before become less contrastive. Conversely, if the number of maintained exemplars is bound, and pruning occurs on objects acquired more remotely in time{or, more plausibly, adaptation mostly concerns recent objects with an aggregated prototype{we can also observe a hardening e ect: the region will recenter around the most recent elements.

General application The previous speci cation has been introduced to contrast an exemplar, represented as a point, with a prototype, represented with a point and two boundaries. Additional cases of application may be imagined, such as contrasting two exemplars, or two prototypes (two regions). In order to handle them, we can generalize the previous formulation, considering that four elements play a role in practice: the target (e.g. o) and the reference (e.g. p ) are in the foreground; the frame (e.g. p) provides background in which the rst two are contextualized, and it is used to control the scaling with respect to the representational container (e.g. 10). In principle foreground inputs may be points or regions; on the other hand, frame and container cannot be points. Note that the frame in general could not be centered on the reference region, but it is plausible to require it to contain both the target and the reference.

One way to take into account regions with the previous method is to discretize them, or better, to consider points as regions of minimal granularity, similarly to what happens with digital images. In e ect, the assumption of limited cognitive resources implies not only the boundedness of the perceptual space, but also its nite granularity. In principle, we could apply a contrast as de ned in (1) for each point of the target region and then aggregate the results (see x 3.1). Alternatively, to avoid to specify aggregation, we could rescale the smallest region at stake (normally the target) so that it behaves like a point. Suppose that the container [ 1; 1] contains 2N grains, and so has granularity 1=N. Similarly, a certain frame with radius , assuming it exploits all the representation, has minimal granularity =N. Now suppose we have a region centered on o with radius ; expressed in grains, the region is long (2 N)= . To represent it as a single grain, all values should be divided by this value. Using this idea, we can reformulate contrast in terms of target, reference and frame regions as:

C = contrastR(ht; i ; hr; i ; hf; i) t contrast b 2 c

D r ; b 2 c; b c; b

E c (4) where b c is an approximation to the nearest integer value. Note that f , the center of the frame, plays only an indirect role, for the condition that the frame region should contain the target and reference.

Requirements for contrast So far, we assumed that objects, prototypes, etc. are speci ed on (a subset of) R, and that we can compute an algebraic di erence of two points, necessary for the \centering" step. The function of di erence here can be expressed as follows: it produces an object that, added (as inverse operation) to the reference point, reproduces (at least to a certain extent) the target. The \scaling" step can be seen instead as one of decontextualization from the magnitudes at stake for the type of objects given in inputs. These requirements could in principle be abstracted, in order to consider the contrast operator as a higher-level function over other types of representation: { to provide a relative order between inputs, the perceptual dimension could be represented for instance as a complete lattice, i.e. a partially ordered set for which all subsets have a supremum and an in mum; { the \di erence" between two objects should return an object, that, used as parameter with an \addition" operator, enables an adequate reconstruction of the target object from the reference. The presence of these speci c roles makes clear that we are not in front of the usual algebraic operations. Denoting them to highlight their asymmetric roles, we have: a

C b d such that b +B d a

This should hold in particular when a, b, d are points. { the \scaling" within the contrast operation should tune this parameter to be neutral from the frame within which the objects are placed.

The second requirement implies that most methods introduced in the literature relying on distance are unfortunately not su cient for de ning contrast, because we need an output that enables reversibility. To nd back the target point from the reference we require, in addition to distance, a sign for a single linear dimension, or a versor (unitary vector) for a multidimensional Euclidean vector space. In the general lattice case, additional parameters might be needed. 3

Multi-dimensional contrast

The previous section focused on objects speci ed by a single linear dimension. For practical uses, the previous result should be extended to the more general case of n dimensions. Suppose these dimensions to be perceptually independent : the sensory input coming from one of them cannot be speci ed (not even partially) by another one or any composition of the others. Objects are then described by a tuple of values like o = (o1; : : : ; on), where oi is the value perceived by sensory module i. Similarly, assume concepts to be formed as structures hp; ; i, where each item is de ned on multiple dimensions p = (p1; : : : ; pn), = ( 1; : : : ; n) and = ( 1; : : : ; n). We consider that a multidimensional contrast corresponds to the application of contrast on each dimension:

C = (C1; : : : ; Cn) = (contrast(o1; hp1; 1; 1i); : : : ; contrast(on; hpn; n; ni)) This output furnishes a contrastive description of the target object; each local contrast could be associated with a category with a certain strength, and the object could be described|with respect to its prototype|by the labels of the modifying categories with the greatest strengths. 3.1

Example of perceptual contrast: directional relationships

As a rst example of application of multidimensional contrast, we consider the description of directional relationships (e.g. \left-of") holding between visual objects. In this case, entities are perceptually de ned on a single perceptual domain. Let us consider an image space S N2 and two binary objects A (target) and B (reference). If A is a single point, we could in principle apply the same formula of point-region contrast (1), where the frame might be e.g. a circular region or bounding box containing A and B. By drawing all vectorial di erences between reference points and target point, we could visualize the contrastor as an image as well, representing in practice all translations that each point of B should perform to produce A.

When A consists of a set of points, we might consider discretization as in x 2.2, but in general this is not possible; considering for instance the case of A being a long, thin curve. In these cases, we can still apply the point-wise contrast iteratively; each of the points of A can be used to produce a contrastor image, and an aggregation of these images (e.g. a normalized sum) would produce a synthetic information about the (point-wise) modi cations to be performed to obtain (each point of) A from (each point of) B. In practice, this corresponds to accumulating the point-wise di erences during the iteration, to count them and to normalize their counts. More formally, the accumulation set is captured by: H(A; B)(z) = fa 2 A; b 2 B j a b = zg The normalized cardinality of this set corresponds to a histogram-like function, and, noting that H is the set resulting from the point-wise di erence of A and B, we override the previous notation A B:

A B(z) =

jH(A; B)(z)j maxw jH(A; B)(w)j (5) With this reformulation A B results to be a gradual representation in the general case. Contrast of regions can be rede ned accordingly, for cases in which discretization cannot be used:

C = contrastR(A; B; hf; i) = (A

B) 1 (6)

Then, similarly to the monodimensional case, we may consider natural partitions on the perceptual space; for instance by dividing a 2D container in four regions, we could obtain rough models of \left of", \right of", \above of" and \below to". By comparing the contrastor object with these models we can decide which predicate corresponds to the directional relationship(s) holding between the objects. Figure 3 reports an example of this application: the target is contrasted by the reference, obtaining a contrastor visual object, whose overlap is maximal with the \left of" region, therefore the target is described as \left of" the reference (c.f. [ 17 ]). 3.2

Example of conceptual contrast: fruit concepts

At this point, we want to investigate the use of contrast on a scenario involving multiple perceptual domains. Let us take the simple case utilized in [ 18 ], in which some concepts about fruits are speci ed as regions on three perceptually independent dimensions (normalized to [0; 1]): hue (a quality dimension of the color domain), roundness (of shape) and sweetness (of taste). The resulting conceptual space consists of three mono-dimensional domains. The conceptual regions are de ned as following: concept pear orange lemon granny smith apple (green type) apple (yellow type) apple (red type)

hue 0.50{0.70 0.80{0.90 0.70{0.80 0.55{0.60 0.50{0.80 0.65{0.85 0.70{1.00 0.40{0.60 0.90{1.00 0.45{0.55 0.70{0.80 0.65{0.80 0.65{0.80 0.65{0.80 0.35{0.45 0.60{0.70 0.00{0.10 0.35{0.45 0.35{0.50 0.40{0.55 0.45{0.60 hue Let us suppose we can obtain more abstract concepts by unifying these regions, as for instance the \apple" concept (including green, yellow and red types), or the fruit concept (or better, a partial version of it, as it includes only some fruits). Possible holes in the union are lled, implicitly assuming that it is possible for an object to be in that position. The prototype points for these new concepts could be computed in two ways: (a) as the center of the concept region (domaininduced prototype), or (b) as the average (i.e. weighted center or centroid) of the prototypes of the given sub-concepts (group-induced prototype). concept apple fruit

hue

At this point, we apply contrast to identify the most pertinent (here in the sense of discriminatory) features. As the application of contrast requires a frame, we can choose a more abstract concept containing all inputs to be contrasted. We could consider for instance the \fruit" concept for all concepts, or the \apple" concept for the three types of apple. These regions provide both the reference (a prototype point, e.g. the center, in this case) and the frame (the concept region). The following table reports for instance the centers of the contrastors obtained by contrasting fruit concepts as \apple" with respect to the \fruit" concept (using discretization and taking = 0:5 ):

The \roundness" and \sweetness" columns are easy to be interpreted. According to the given conceptual space, the orange is the sweetest fruit, the lemon the least. Pear and lemon are the least round, while orange is the most. Apples occupy almost neutral categories in all dimensions. The interpretation of \hue" is more complicated, because, by computing the di erence as an algebraic difference (i.e. interpreting angles in terms of rotation), we require to utilize the reference to identify a speci c color. In [ 3 ] we have considered an angular interpretation that in principle would solve this issue, but we have also acknowledged a problem with the scaling phase that, unfortunately, does not have a solution yet. Alternatively, we might consider to re-transform the hue dimension into the red-green-blue dimensions, which form a cartesian space, obtaining the three additional columns reported in the table above.7 From these we are able to infer that the pears are characterized by being the greenest fruits with respect to the color spectrum of the given fruits ; oranges by being the reddest (and least blue), lemons the most yellow (and least blue), whereas the concept of \apple" is not 7 To obtain color values that were plausible from those given as inputs, we computed h = [(1 hue) 360 30 ]=360 and then converted the hls tuple hh; 0:5; 1i to rgb. distinctive with respect to color, as there are red, green and yellow apples (its distinctiveness would increase if e.g. a truly blue fruit was included in the \fruit" concept.)

Intuitively, starting from a similar table, possibly including some regional information, we could de ne some weights reifying the salience of these qualities for the formation of a concept, by settling an adequate measure of their discriminatory power.

Finally, we observe that the results of contrast may be used also for associating the object to a category. For instance, within the frame of fruits, one can easily compute that \granny smith" is nearest to the concept of apple, or even more to the green type of apple. Intuitively, the perceptual independence hypothesis supports the use of the Manhattan distance. However, an additional ltering may be needed when many dimensions are present to consider only adequately distinctive dimensions. 3.3

Examples of conceptual merge: \red brick" vs \brick red"

To complete the operational cycle, we present a small example concerning the interpretation of simple linguistic expressions. Despite the richer spectrum illustrated in Gardenfors' books [ 1, 5 ], most works on conceptual spaces focus on the intersective type of concept composition, i.e. relying on conjunction of concepts, as with symbolic approaches; e.g. a \red brick" refers to an object that belongs to the class of \red" objects and to the class of \brick" objects.

An approach to predication based on contrast, instead, naturally implies an asymmetry between the roles of the modi er concept (\red") and the modi ed or reference concept (\brick") in the formation of the composed concept (\red brick"), in alignment with the modi er-head phenomenon observed in cognitive psychology [ 16 ]. We call merge, here provisionally denoted with +, the operation inverse to contrast, and revisit some of the examples brought by Gardenfors. Let us consider the two merges associated to \red brick" and \brick red". Suppose \red" is a label anchored to a concept de ned on the color dimension, and \brick" anchors to a multidimensional concept including a color dimension. As in [ 3 ], we consider a void value when a certain dimension is not applicable (an object does not have a certain quality, or a prototype cannot be formed on that dimension). Two intuitive properties of merge can be identi ed (note how merge is not commutative): { category consistency : dimensions which are not in the reference should not be added (for instance, we want a \brick red" to be a color):

(::; ; ::) + (::; a; ::) = (::; ; ::) { locality : merge applies modi er concepts locally, i.e. not modifying dimensions irrelevant to the modi er (e.g. a \red brick" is merely a brick more red than the prototype brick): (::; a; ::) + (::; ; ::) = (::; a; ::)

The general application of these properties makes however clear that additional cognitive mechanisms are at stake. For the rst property, consider predicates as \stone lion", or \stu ed gorilla". These are examples of merges that \break" the reference category. As there is no lion animal which is made of stone, a plausible repair would be \a stone object similar to a lion". Then, because the similarity between a stone object and a lion can be only constructed along the shape dimension (the only available comparison ground [ 3 ] between \stone object" and \lion"), we conceptualize a lion-like statue. For the second property, implicit correlations within dimensions do provide additional information; e.g. the \young" in \young man" does not seem to simply modify the age, leaving intact the other values: it seems that the merge on one dimension causes a recalibration of the prototype with this new information. Alternatively, this could be explained by assuming that a (sub)-concept \young man" is already available, and its prototype is activated by the merge (cf. the notion of lexical compounds [ 16 ]). Consider again the \red brick" case. For a layman, a red brick is simply a brick more red than the others; for an expert person, a red brick might be also e.g. more isolating than the average. An operationalization of this mechanism could explain part of the process of alignment of linguistic semantics with conceptual semantics. 4

Discussion

The present paper reports on ongoing research stemming from an alternative view on conceptual spaces, rooted on relevant predication [ 2 ]. This account insists on the importance of discriminatory aspects not only for individuation, but also for the formation of concepts. Recent, additional support for this hypothesis comes from cognitive studies in image recognition [ 19 ] proposing (predictive) recognition models based on internal discriminatory features such as the spatial organization of visual elements (cf. x 3.1). Not less importantly, as shown by the preliminary results presented here, by using contrast, the membership of an exemplar to a certain category|and therefore the possible consequent predication|is e ectively contextualized on the y, because the application of contrast requires always the intervention of background elements (conceptual frame and representational container), in addition to foreground elements (target and reference objects). By changing of pragmatic context, certain conceptual frames might become more accessible than others, and this would determine changes of interpretation for the same linguistic marks.

Additionally, the present proposal o ers a computational model naturally implementing modi er-head concept combination [ 16 ], which, in our model, can be seen as including the intersective case (cf. \red book" vs \red dog" [ 3 ]). Gardenfors [ 1, 5 ] presents an informal solution by referring to contrast classes, summarized by the concept combination rule [ 1 ]: \The combination CD of two concepts C and D is determined by letting the regions for the domains of C, conned to the contrast class de ned by D, replace the values of the corresponding regions for D." This means that the domains of the modi er (e.g. temperature for \hot") are scaled to compatible domains of the reference (e.g. temperature for \co ee"), determining a new region identifying the compound concept (\hot co ee"). This is very similar to the operations suggested here, but there are differences too. First, instead of maintaining de nite regions, we give priority to points and to rough regional information, in order to capture a sort of \intensity" of the modi cation. Second, by building upon contrast, conceptualization and conceptual composition become two sides of the same function, rather than a separate cognitive machinery. In short, the introduction of the operation of contrast might provide alternative foundations to the theory of conceptual spaces, and could be used to make explicit some of the internal mechanisms taken as given and to rely as less as possible to external parameters.

Evidently, many points still remain to be investigated. For instance, we aim to derive the salience of quality dimensions with respect to the formation of a certain concept from the conceptual structure, rather than being captured by externally given weights, and precisely by capturing the strength of their contrastive or discriminatory power. Thus, in addition to a re nement of the formalization and more detailed speci cations of the methods (e.g. for directional dimensions), we aim to better understand the intertwining between perceptual independence and statistical independence. Perceptual independence is an important assumption for the distribution of contrast along independent dimensions. Because we cannot express something about a certain dimension in any other way than saying something about that dimension, a measure to quantify an aggregate contrast seems to be intuitively captured by a Manhattan distance. However, in the case of arti cial devices, the internal con guration of sensors is not a consequence of evolutionary adaptation and may be the most various. For instance, cloned temperature sensors may receive the same information while being perceptually independent. Discovering a strong (or perfect) correlation has important consequences, in a way opposite to the introduction of new dimensions as e.g. children learn to do by separating volume from height of a container.

Furthermore, we would like to investigate in detail the convexity constraint, or more plausibly, the star-shaped constraint on concept regions; we expect it should naturally emerge from a categorization based on contrast. At a more fundamental level, while working on this paper, we acknowledged that by settling the functioning of contrast, conceptual spaces can be seen as emerging from contrastive functions. Stated di erently, by de ning contrast, we need also to de ne the inverse operation merge, the application of merge produces order relations between concepts, and the resulting lattice is in practice a conceptual space. Future investigations will target the formalization of this idea.

Finally, an additional track that we are currently studying concerns the use of the morphological operators erosion and dilation [ 20 ] for implementing approximate methods for contrast and merge, motivated by the observation that operators embedding non-linear functions as max and min carry computational advantages, and are cognitively more plausible than aggregations by average as those used in x 3.1.

References

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