Caroline Rossi

Christophe Parisse

Categories in the making : Assessing the role of semantics in the acquisition of noun and verb categories

Keywords: noun and verb categorization, semantic bootstrapping, semantic / perceptual categories, language acquisition

Little is known about what guides children in their acquisition of grammatical categories. This paper investigates how semantic knowledge could be involved in discovering these categories, thus confronting two competing hypotheses: are semantic categories innate, or are they developed in a piecemeal fashion? We tested for regular associations between basic semantic dimensions and the development of the founding categories of noun and verb. Six perceptually based semantic dimensions (Parisse and Poulain, 2010), shared by nouns and verbs but potentially distinctive, are coded in the productions of three children aged 1;06 to 2;06. Our results suggest that semantic dimensions do not offer an entry into the early differentiation of noun and verb categories.

INTRODUCTION

How do children manage to grasp the content and boundaries of grammatical categories? To the linguist, the distinction between noun and verb is not binary, but involves a complex set of notions [START_REF] Black | Noun-verb dissociations: a multi-faceted phenomenon[END_REF] including semantic classes, phonological properties, as well as grammatical notions such as the distributional and morphological properties of words (e.g. [START_REF] Bassano | Early development of nouns and verbs in French: Exploring the interface between lexicon and grammar[END_REF] for a discussion of these properties in French).

But are all these notions involved in early child language, where verbs can be used rather early on, and are not mistaken for nouns?

(1) Madeleine 2;04.15: line 2,432.

*CHI: On lit sur mon lit ! [Let's read on my bed!]

Here Madeleine uses and clearly distinguishes two homophones: the verb lit (read), and the noun lit (bed) probably without even noticing the phonological similarity, /li/. However, as one-to two-year-olds utter their first words, they start by producing forms that are devoid of syntactic marking, or incorporate ambiguous marks. The syntactic difference between nouns and verbs is made obvious only gradually [START_REF] Veneziano | The changing status of 'filler syllables' on the way to grammatical morphemes[END_REF]Veneziano and Parisse, 2010). If children do not -or only very partially-rely on syntactic information to acquire grammatical categories (Pinker, 1994: 384), the first cues that they use could then be semantic ones. Macnamara (1984:126) suggests that 'grammatical categories have an intuitive basis, namely a semantic one' which could eventually 'help the child get syntax acquisition started' (Pinker, 1994:385).

Pinker thus posits that children have innate knowledge of the semantic properties of nouns and verbs, allowing them to relate elements of the world to what are syntactic nouns and verbs in their input. In other words, children would rely on 'certain contingencies between perceptual categories and syntactic categories, mediated by built-in semantic categories' (Pinker, 1994:385) to discover the abstract principles of syntax.

Focusing on the categories of nouns and verbs, Pinker suggests that children have innate knowledge of a relation between the noun word class and reference to a person or thing on the one hand, and between the verb word class and such semantic categories as 'action' or 'change of state' on the other hand (Pinker 1984:41).

Another approach to language development has appeared over the past twenty years, rooted in Braine's initial work (1963): the so-called usage-based and construction-based approach (see [START_REF] Tomasello | Constructing a language: a usage-based theory of language acquisition[END_REF][START_REF] Goldberg | Constructions at work. The nature of generalization in language[END_REF]. This approach holds that linguistic knowledge is based on constructions, i.e. form-function associations with varying degrees of openness. The most specific constructions correspond to words of the lexicon and lexical collocations, whereas open ones correspond to grammatical rules such as subject/verb agreement (in languages in which this applies). A key feature in this approach is that categorical knowledge and construction openness (or generalisation) develop gradually in a piecemeal fashion (see [START_REF] Tomasello | Do young children have adult syntactic competence?[END_REF]. The approach also derives from functionalism, where all linguistic knowledge is construed as form-function associations. In line with such a framework, we hypothesize that semantic categories must develop gradually and in harmony with form categories -a proposal which would go against both Macnamara's and Pinker's.

The goal of the present paper is to test the hypothesis of a gradual development of semantics against the traditional innate-based approach of Macnamara and Pinker. In order to do that, we investigate the acquisition of the basic verb/noun opposition, which is central to the argument of innate semantic knowledge and has been the object of much discussion and emphasis in works about early language acquisition (e.g. [START_REF] Bassano | Early development of nouns and verbs in French: Exploring the interface between lexicon and grammar[END_REF][START_REF] Black | Noun-verb dissociations: a multi-faceted phenomenon[END_REF][START_REF] Gentner | Why nouns are learned before verbs: Linguistic relativity versus natural partitioning[END_REF][START_REF] Gentner | Why verbs are hard to learn[END_REF]. More specifically, we examine the possibility that early but gradual emergence of semantic organization could either parallel or interact with the emergence of noun and verb categories.

In line with usage-based accounts of the acquisition of nouns and verbs, we do not assume that children have existing categories that they are specifying on a semantic basis. Rather, we would like to see whether semantics could be used as a wedge in the discovery and shaping of those categories. Those claims are tested thanks to fine-grained semantic analyses.

We start from six basic semantic dimensions (Parisse and Poulin, 2010) that may be used to define and contrast nouns and verbs, and check their distribution in the first productions of nouns and verbs in three Frenchspeaking children: Antoine, Madeleine, and Théophile, from age 1;06 to 2;06, to see whether any dimension -or set of dimensions-is (or becomes) distinctive in profiling categories of nouns and verbs.

SETTING THE STAGE: INSIGHTS INTO THE EMERGENCE OF CATEGORIES

This paper focuses on the semantic interpretation of young children's production. 'Semantics' here refers to the regularities and generalisations of linguistic functions, whereas 'syntax' captures the regularities and generalisations of linguistic forms. The word 'lexicon' refers to elements produced by children that are considered close enough to an adult lexical target to be identified (and coded). Importantly, the term 'grammatical categories' is here used to refer to categories that are defined by both syntactic and semantic information. Our 'semantic dimensions' provide labels for the semantic properties of words or word groups produced by a child: not all of them are binary, as they correspond to sets of values in a semantic domain. For example, a dimension such as 'distance' can be reduced, medium, or long; a dimension such as 'number' can be singular, plural, or uncountable. Finally, we will use the term 'universal' to refer to semantic dimensions that can be found in all languages, either as grammaticalized forms, or as lexical or compound lexical constructions: e.g. 'distance'.

Categories in the making

In keeping with developmental approaches of categorization, one major problem addressed in this study is whether noun and verb categories are different in early child vs. adult speech. Clark (1982:395) examines the possibility that young children might overextend nouns for lack of the appropriate, specific verbs, e.g. using car and foot instead of drive and kick, respectively. But as she notices it, this would make children rather difficult to understand. Strikingly enough, nouns and verbs are not mistaken for one another in child speech, thus lending support to [START_REF] Pinker | Language Learnability and Language Development[END_REF] claim that children differentiate nouns and verbs very early, (i.e. at least by age two), a claim that was recently confirmed experimentally by Veneziano et al. (2010). The latter study showed that even in the case of polycategorical units, children were not misled by the similarity in form and meaning, and that by age two, 50% of the children displayed full knowledge of the syntactic information associated with the noun/verb distinction.

Another explanation for children's early differentiation of nouns and verbs, however, is that noun and verb semantics may form distinctive sets that children rely on, even in the absence of morpho-syntactic marking (e.g. in example (1) above). This assumption is the theoretical starting point of the present study. It does not amount to denying the use of syntactic information. Rather, it implies that semantic properties could be used as earlier and more fundamental cues in the acquisition of noun and verb categories.

Rethinking semantic bootstrapping with a usage-based model

According to [START_REF] Pinker | Language Learnability and Language Development[END_REF][START_REF] Pinker | How could a child use verb syntax to learn verb semantics[END_REF] semantic bootstrapping hypothesis, the 'discovery' of noun and verb categories depends on word meaning, which is established early on, possibly through the child's interactions with her material and human environment, and their regulation by innate rules.

Pinker's proposal might be correct, but to date, 'naturalistic analyses of early child language have failed to find any evidence for innate linking rules' (Tomasello, 2000:233). Besides, what naturalistic analyses may help us understand is precisely what Pinker's hypothesis remains 'agnostic' about (Pinker, 1994:186), namely: how children's semantic knowledge is achieved. We suggest that basic and perceptually-based regularities in children's productions might help us bridge this gap. It should be noted however that such a proposal does not merely complement Pinker's view: its usage-based starting point goes against it. Indeed, either the child's grammar is usage-based, or she has innate principles to rely on.

According to a non-nativist point of view, children do not discover syntax on the basis of the innate principles of a 'universal grammar' or 'language acquisition device' [START_REF] Chomsky | Aspects of the theory of syntax[END_REF][START_REF] Chomsky | Knowledge of Language[END_REF], but through repeated use in distinctive constructions [START_REF] Tomasello | Constructing a language: a usage-based theory of language acquisition[END_REF][START_REF] Goldberg | Patterns of experience in patterns of language[END_REF]2003). Construction grammar's perspective on syntax and semantics differs from the Chomskyan model in that it adopts a 'what you see is what you get' approach to syntactic form, and then looks for pairings of forms (i.e. constructions, including morphemes or words) with semantic or discourse function (Goldberg, 2003: 219). There is no such thing as a bootstrapping problem for usage-based models, since they do not see adult linguistic competence as an abstract set of rules. Instead, they provide us with both a 'more child-accessible' account of the adult endpoint, and 'new ways of thinking about how children learn and construct abstract cognitive entities' (Tomasello, 2000: 235).

On these grounds, a number of arguments have been produced to justify the gradual emergence of forms, but little work has been conducted into first categories of meaning (or functions) and how they are related to forms. [START_REF] Tomasello | Do young children have adult syntactic competence?[END_REF][START_REF] Tomasello | Constructing a language: a usage-based theory of language acquisition[END_REF] argues that children learn the category of nouns earlier than the category of verbs and that this takes place in a piecemeal fashion. What is unclear is the status of semantics at each developmental step. Indeed, the semantics of early words is difficult to delineate [START_REF] Bloom | Language development from two to three[END_REF], and verb semantics is even harder to grasp [START_REF] Gleitman | The structural sources of verb meanings[END_REF][START_REF] Golinkoff | Lexical principles can be extended to the acquisition of verbs[END_REF] -thus accounting for their later acquisition [START_REF] Gentner | Why verbs are hard to learn[END_REF].

Contrasting semantic dimensions for nouns vs. verbs

A major difficulty in analyzing semantic differentiation between noun and verb is avoiding the use of semantic dimensions that are so specific to the properties of nouns and verbs that they only reject explanation to another level. Thus the proposal that nouns are often linked to physical objects and verbs to actions gives no clue as to how children learn to distinguish (or discover the difference between) objects and actions.

To avoid this pitfall, we chose to use semantic dimensions that are shared by nouns and verbs, and that can be identified by the child by means of simple perception (i.e. children do not need to create cognitive categories to identify these dimensions), while potentially applying them differently to nouns and verbs. An example of such a dimension is movement. Any word used by a child, and which refers to something in the child's surrounding, can be judged as referring to something that is moving or to something that is immobile. This measure is absolutely independent from the form of the word produced by the child (whether it is a noun, a verb or other), but if indeed nouns protypically refer to objects and verbs to actions (as suggested by [START_REF] Pinker | Language Learnability and Language Development[END_REF]Croft, 2001:89), then we should find more words that are nouns referring to static referents and more words that are verbs referring to mobile referents. If such a difference were to be found, it would suggest that children are following prototypical use of nouns and verbs and could use prototypical values to distinguish the two categories from a semantic point of view. Such a difference should appear early if Pinker's hypothesis is valid, but should appear only gradually if our hypothesis is valid.

Beside motion, the other semantic dimensions that we use below are: number, concreteness, animacy, distance, and determination (see section 3.1 for details on the coding principles for each dimension). In many languages (including French), number applies to nouns and verbs alike. However, as shown by [START_REF] Macnamara | Names for things: A study in human learning[END_REF], number is closely related to sortals (i.e. concepts underlying the logical work of identifying and individuating count nouns), which Macnamara puts forward to explain how children learn the very concept of object. Besides, reference to count vs. mass elements is understood on the basis of objects rather than actions, so that reference to multiple or massive elements is more likely to develop with nouns before applying to verbs. Thus we should expect a tendency in young children to refer to number with nouns more than with verbs.

Because of their inherently relational content [START_REF] Gentner | Why nouns are learned before verbs: Linguistic relativity versus natural partitioning[END_REF][START_REF] Black | Noun-verb dissociations: a multi-faceted phenomenon[END_REF], verbs have been described as more abstract than object labels (Bird, Lambon Ralph, Patterson, & Hodges, 2000;Breedin, Saffran, & Coslett, 1994. In Vigliocco et al., 2004), thus concreteness should be biased towards nouns.

Animacy applies to both nouns and verbs, but just like motion, the dimension should apply more often to verbs in young children's universe, where the concept of animacy most probably develops in a piecemeal fashion (Rakison, 2005:187), and is primarily based on a notion of selfinitiated vs. caused motion [START_REF] Gelman | Children's Causal Explanations of Animate and Inanimate Motion[END_REF]. Also, reference to inanimate beings could be a characteristic of first nouns, if they prototypically refer to physical objects, but not of first (presumably egocentric) verbs.

Three levels were used to code distance: touching (proximal), visible (distal) or absent (distal + invisible). Although all three values can be grammaticalized on both nouns and verbs (at least in some languages, see [START_REF] Payne | Describing morphosyntax: a guide for field linguists[END_REF], distance (and especially visible vs. absent) is more closely associated with objects. For example, distance anchors definite vs. indefinite reference on nouns (see [START_REF] Maratsos | The use of definite and indefinite reference in young children: An experimental study of semantic acquisition[END_REF]. It is also at the root of the concept of object permanence. Finally, objects are more likely to be in the child's hands or close to him when he talks about them. However, the first actions named have also been analysed as egocentric, i.e. as referring to the child's own action, so that differences here might be toned down.

Similar

arguments might be used for determination (specificity/genericity), which, although it applies to nouns and verbs alike, should be more fundamental for nouns (Macnamara, 1984: 144-156). What is more, if first nouns are known to be specific and context-bound, this is not necessarily true for first verbs. Indeed, the importance of "light verbs" such as go, do, make or give [START_REF] Ninio | Language and the learning curve: a new theory of syntactic development[END_REF], which are used to refer to broad categories of actions in early productions, suggests that generic uses could be characteristic of early verbs, as opposed to nouns.

Goal of the paper

The goal of this paper is to see whether there is a gradual emergence of semantic differences between words in children's speech that are considered as nouns or verbs by adults, before the children acquire syntactic categories.

Alternatives would be that such differences exist from the start, or do not exist at all.

In order to test our hypothesis, we have defined six basic sets of semantic dimensions, and coded early nouns and verbs for those six dimensions in three longitudinal studies of French-speaking children. As explained above, all six dimensions apply to both nouns and verbs.

However, they should be used differentially in child language. Thus, if children do develop a semantic difference in profiling nouns and verbs, this difference should be seen in the preferential association of one dimension to nouns while another dimension would be preferred with verbs. The details of our coding system is explained and discussed in the next section.

METHODS

We coded all first linguistic productions in the longitudinal corpora of three French-speaking children (Antoine, Madeleine and Théophile), providing we could attribute them an adult target that was either a noun or a verb. Insert Table 1 here.

Coding principles

Coding was done according to the adult target, i.e. to the word, grammatical category and semantic dimension that an adult (the person doing the coding) would attribute to the child, if he were interacting with her as a conversational partner. We relied on mothers' (or observers') interpretations insofar as they matched the coder's decision, which was also based on the general situation and context of the interaction. Thus, our coding categories were devised to take into account all the elements that children may rely on as they discover language: perceptual elements as well as adult interpretations and feedback. According to our hypotheses, whether the child does indeed master the categories of nouns and verbs does not matterin fact, we are suggesting that he does not master them at first, and that these categories only gradually become productive. This, however, does not 2 Table 1 includes mean values and detailed values at the first and last point studied. These values show significant individual differences, based on mean length of utterance in words (MLU) and number of words produced -but see description of the corpus (this issue) for a more in-depth presentation of the children and definitions of these measures.

preclude the use of adult categories as independent variables (Bates et al., 1994: 90). Although we rely of the adults' interpretation, we have adopted the child's point of view, as understood by the interlocutor, on the general situation and discursive context: semantic dimensions were determined according to a combination of environmental clues that could be taken as representative of what the child is operating with -namely, social pragmatic, environmental and linguistic cues (Tomasello and Bates, 2001: 283).

All the relevant items were coded for syntactic category (noun vs.

verb), and according to six semantic dimensions defined below:

1) Distance from the child: we coded whether or not the referent of the noun or verb was in the child's hands, and if the action was performed by the child (touch); if it was visible (audible) to the child; or absent.

2) Number: without taking into account grammatical number, we looked at whether referent(s) was (or were) single -multiplemassive.

3) Concreteness: we coded for concrete versus abstract reference.

4) Animacy: for both nouns and verbs, referents could be either animate or inanimate.

5) Motion: nouns could refer to static or mobile elements, as could verbs with the dynamic vs. static events they referred to.

6) Determination: if one item, action or process was singled out by the child, it was coded as specific; if the noun or verb referred to a whole category, i.e. a series of instances considered together, it was coded as generic.

Insert Table 2 here.

A 'not coded' choice was available for all semantic dimensions, and was considered an important coding option. It was used whenever no other choice was possible, for lack of information, because it did not apply to the situation, or when we could not decide, thus avoiding the pitfalls of overinterpretation as much as possible. Inter-coder reliability, measured using Cohen's kappa, was 0.91 for distance, 0.89 for number, 0.95 for concreteness, 0.83 for animacy, 0.79 for motion, and 0.82 for specificity.

The values were high enough to confirm that the coding categories were valid and operational.

Coding perceptual categories in early child speech

The distinction of verb-versus satellite-framed languages in [START_REF] Talmy | Lexicalization patterns: semantic structure in lexical forms[END_REF]'s typology of motion events clearly show that verb semantics may vary considerably from one language to the other: verbs may conflate different aspects of a scene, and as a result, lexically-based semantic universals will be hard to find. What is more, early referential processes are not as strictly tied to the conventions of language as they are in adult speech: children are indeed likely to use, for example, abstract terms with concrete reference.

Because of the variability of referential processes in child speech, interpretation is bound to rely mostly on perceptual factors, which are at the heart of early lexical acquisition [START_REF] Golinkoff | How toddlers begin to learn verbs[END_REF][START_REF] Pruden | The birth of words: ten-month-olds learn words through perceptual salience[END_REF]. That is why our coding categories are essentially perceptual.

Coding problems and decisions

As shown by the general description (section 3.1), coding criteria were rather straightforward. Some items, however, proved more difficult to code than others. We had to make additional decisions in order to rule out the items that included very little information, and to adjust to the characteristics of child language.

Nominal and verbal constructions

First of all, it should be mentioned that we coded nominal and verbal constructions rather than bare nouns and verbs, so as to take into account the (at least partially) unanalysed chunks or amalgams [START_REF] Peters | The Units of Language Acquisition[END_REF][START_REF] Aksu-Koç | Acquisition of Turkish[END_REF] often found in early productions. This implies, for example, that we considered determiners together with nouns and auxiliaries together with verbs -both elements often being produced as filler syllables (i.e. syllables that do not directly correspond to conventional, adult markers, but are first signs of morpho-syntactic organisation -see e.g. [START_REF] Veneziano | The changing status of 'filler syllables' on the way to grammatical morphemes[END_REF].

In first combinations and multi-word utterances, however, we focused on each specific item rather than on the overall meaning of the utterance, so that within the same utterance, two different items could be coded differently. For example, a generic action could be performed with a specific object or vice versa, as shown in our coding of Madeleine's utterance presented in Table 2. In the same way, the use of two different verbs or nouns in the same utterance was more likely to be distinctive than repetitive.

Animates

Young children's understanding of animacy has been shown to be only partial to start with: we therefore based our decisions primarily on our knowledge of each child's development. For instance, one child, Madeleine, showed early understanding (at 2;02) that the characters in a story were animates, i.e. that they had feelings and could be understood in just the same way as the people she was regularly interacting with. In this session, she thus offers to lend her bunny to one of the characters in the story who looks sad [START_REF] Morgenstern | What can child language tell us about prepositions? A contrastive corpus-based study of cognitive and social-pragmatic factors[END_REF]. From 2;02 onwards, we therefore coded the characters she named in a story as animates. When it was less obvious, however, i.e. when children were only naming the characters upon request, those nouns were not coded for animacy. Moreover, [START_REF] Striano | Social and object support for early symbolic play[END_REF] have shown that before two years of age, children have symbolic skills with gestures but not with objects. Here again, we used our knowledge of the children's development, together with a close consideration of context, to decide on the coding. Thus for example, when children talked about toys figuring people or animals, but which were manipulated and put into containers, the referents were not coded as animates.

On the other hand, in our coding of animacy for verbs, we took into account the (most often implicit) agent, patient or instrument that the verb applied to: their semantics was considered as an integral part of verbal constructions.

Motion

Our coding of motion based on perceptual clues led us to code the dimension of motion for many nouns. Surprising as it may seem, this finding is perfectly in keeping with infants' predominant interest for things in motion (Slater 1989: 59;[START_REF] Casasola | Precursors to verb learning: infant's understanding of motion events[END_REF][START_REF] Pulverman | Conceptual foundations for verb learning: celebrating the event[END_REF].

Here again, however, some of our coding decisions are more obvious than others. For nominal constructions, the difference between static and mobile referents very much corresponded to mobile figures vs. static grounds.

Designated objects, however, could be static when shown from a distance, and were more likely to be mobile when in the child's hands (i.e. manipulated objects). For instance when Madeleine (2;04) says: C'est pas avec l'aspirateur (you don't do this with the vacuum-cleaner), while putting it down, l'aspirateur (vacuum-cleaner) was straightforwardly coded as mobile. This also applies to cases where nouns were not used in single-word utterances -e.g. in (2) below:

(

RESULTS

Variations in the absolute number of lexical elements produced in the various sessions would have masked any interesting trend in the children's production. Therefore, all our analyses had to be done using percentages.

For Théophile, the number of lexical verbs coded was quite small (M. = 39.8 for nouns, M. = 5.5 for verbs). Even in the final session, he produced only 47 verbs for 124 nouns. So for Théophile, trends about verbs may not be highly significant and ought to be regarded with caution. For Antoine (M. = 52 for nouns, M. = 29 for verbs), the total number of lexical elements produced in the first three sessions (from age 1;05 to age 1;07) is low (less than 10), so caution with statistical analysis should apply. After age 1;09, at least 50 and up to more than 200 lexical elements could be analyzed, making our data more reliable. For Madeleine, the number of words analyzed is consistently large (M. = 246 for nouns, M. = 154 for verbs) and she never produced less than 30 verbs in one session -see Table 1 for more information about the children's production.

Differences across categories

The percentages of nouns and verbs produced for each of the semantic categories analyzed is presented in Table 3 for all six coded dimensions, including the cases when no categorization was possible. The difference between mean percentage of 'noun' and 'verb' items coded was significant 36% of the time (the pairs of gray cells in Table 3 correspond to significant differences between noun and verb). In only two cases, all three children had a similar significant tendency to prefer nouns for a specific semantic dimension: namely for 'distance=absent' and 'animacy=inanimate'.

Insert Table 3 here

Developmental trends

For some specific dimensions, there was a significant trend throughout all the sessions. Significant trends are presented with a *, **, or *** in Table 3.

Increasing trends are indicated by superscript, decreasing trends by subscript. Out of the 28 dimensions coded (3 dimensions for distance, 3 for number, 2 for abstractness, etc.), trends were significant 4 times (see cells with asterisks) for Antoine, 8 times for Madeleine, and only one time for

Théophile. For reasons of space, we cannot make an extensive description of all the observed tendencies here, so we will focus on the most salient ones and indicate whether they are increasing or decreasing. For Antoine, an increasing trend to use verbs with 'distance=visible' resulted from the absence of reference to visible elements with verbs in the first two sessions (they were only coded as 'touch'), whereas in all of the following sessions, verbs were used 61% of the time with visible reference. This could correspond to an initial egocentric conceptualization of actions [START_REF] Piaget | Le langage et la pensée chez l'enfant[END_REF][START_REF] Nelson | Concept, word, and sentence: Interrelations in acquisition and development[END_REF], which only gradually enables the child to distinguish and name the actions performed by others:

(3) Antoine (1;07): Assis [Sit.] -asking to be put on the chair (4) Antoine (1;09): Assis! [Sit!] -asking the observer to sit next to him.

However, the small number of verbs produced in the first two sessions calls for caution. A more interesting trend was found for 'motion=static': with nouns and verbs alike, there is a steady increase of reference to static elements, and a decrease in the percentage of reference to mobile elements.

For Madeleine, the most interesting trends were found with 'abstractness' for verbs, 'specificity' for nouns and verbs, and 'absence' for verbs. First, verbs tended to become more abstract and there were fewer concrete items as Madeleine got older. The most frequent abstract verb forms produced by Madeleine included: attends (wait), arrive (come), faut (must), veux (want), and sais (know). No such trend was found with nouns, which were, one the whole, neither more nor less concrete than verbs.

Secondly, both nouns and verbs tended to become less specific and more generic, although generic reference remains the unusual case. As for the third trend, verbs were not used to refer to absent situations before age two.

After two, nearly a third of all verbs were coded for absent reference.

Although the change happened rather suddenly, the trend was significant.

Here again, the evolution in verb use could be linked to the child's social and cognitive development and, for Madeleine more specifically, to the early development of narrative skills:

(5) Madeleine (2;04) : Elle est où Marie que j'avais vue ?

[Where is Mary that I saw earlier?]

Differences between children

The differences between children were measured using a t-test for each pair of children. Antoine and Théophile were the most similar with differences found only three times out of 40 (20 dimensions x 2 syntactic categories).

These differences applied to nouns only, in the following three dimensions:

'motion=not coded', 'distance=touch', and 'abstractness=concrete'. Antoine and Madeleine were very similar for verbs (one difference only for 'specificity=generic'), but quite dissimilar for nouns, with eight differences out of 20 (the largest difference for animacy and motion). Madeleine and Théophile were dissimilar ten times out of 40, but with an equal number of differences in nouns and verbs. Once again there were differences in animacy, but also in distance. No differences were found at all in the use of number: all three children overwhelmingly used singular reference.

DISCUSSION

Our results did not confirm any of the hypotheses presented in the introduction. On the whole, no clear semantic differences between nouns and verbs could be found, although we expected either a difference from the start, or a difference that would appear with language development.

According to the semantic bootstrapping hypothesis, perceptual semantic categorization should be guiding the acquisition process before syntax appeared in the children's productions. The assumption implies that, based on semantic dimensions, a clear line could be drawn between major categories such as verbs and nouns. In that case, semantic differences across categories should not change much with age, as syntax is considered to tune itself to semantics and not the other way around. There should be no significant variation between children either.

According to construction grammar, on the other hand, semantic categorization should be item-based and variable from one child to another.

There should be no semantic tendencies to begin with, and categorization should develop over time. Syntax and semantics should develop together.

Overall, our results did not support either of the theories. There were few clear differences in semantic dimensions which could help the children to differentiate nouns and verbs. Surprisingly, differences in abstractness were found for one child only. The main difference between nouns and verbs was found with the "animacy" dimension. Nearly all verbs were used with reference to animates (counted as such even when the agent, patient or instrument was only implied). On the contrary, nouns were used 30 to 50% of the time with an animate referent, so reference to animates turns out to be a negative criterion: in our children data, verbs are characterized by the fact that they are never inanimate. The only other systematic difference between nouns and verbs is for 'distance=absent' which is used more for nouns than for verbs. But this is not distinctive for most cases, i.e. when they refer to visible or touched elements. However, especially for Madeleine, it appears that the distance used with nouns is often wider than for verbs. This tends to change with age, which makes it a potential developing feature.

Our results do not support developmental theories on the acquisition of categories any more than nativist theories, since very few developmental tendencies were found, even though at age 2;06 all the children's MLUs were higher than two (or close to two for Théophile-see description of the children at the beginning of this issue). At an age when children are already able to produce clear syntactic regularities, one could expect that they should have developed semantic trends. In fact, only Madeleine, with an MLU higher than three, seems to get closer to such a point.

Intra-and inter-individual variability

The explanation for these inconclusive results might be found in the great variability of the semantic use of language. This can be exemplified by the use of 'motion=mobile' by Madeleine, as shown in Figure 1, with the large difference found between her massive use of verbs for mobile reference and the (more unusual) use of nouns for this reference. Even though substantial differences across sessions forbid us to draw reliable conclusions, it does seem possible that the first references to motion by Madeleine are both verbal and nominal, a trend which may be confirmed by the results for Antoine (Figure 2). Antoine appears to be similar to Madeleine before age 1;10. As his overall language development is not as quick as Madeleine's, he may still be at an early stage of semantic development towards the end of our analyses, which would account for this persisting ambiguity towards the end of our coded data.

Insert Figures 1 and2 here

Towards usage-based semantic universals

Our results, however, could also be the consequence of our understanding of usage-based semantics. The theory is still in the making (see for example [START_REF] Croft | Exemplar semantics[END_REF][START_REF] Cysouw | Semantic maps as metrics on meaning[END_REF], for proposals about such a theory) and leaves room for various interpretations. We focused on context, which is crucial to both usage-based theories and early language acquisition, in accordance with Cysouw's usage-based approach, which proposes to '[define] the meaning of a language-specific expression as the collection of all contexts in which the expression can be used ' (2010: 71). [START_REF] Haspelmath | Pre-established categories don't exist: consequences for language description and typology[END_REF] also denies the existence of predefined syntactic categories, but not that of semantic universals. He considers that although 'there is ample evidence that meaning, too, is conventional and varies across languages', we could still use 'low-level notions' across languages (2007: 127). In order to solve the riddle of cross-linguistic understanding, which may not be so different from parent-child understanding, Haspelmath suggests we take as semantic universals something which is very close to semantic exemplars: a notion yet to be generalized in cognitive grammar, but which is certainly compatible with the principles of usage-based linguistics. Our coding of semantic dimensions is therefore not 'usage-based' in that the dimensions were not derived from a semantic mapping of the meanings found in our data, but rather from our intuitive semantic analyses. Whether those intuitions actually match child-directed speech remains to be determined.

Fuzzy semantic categories

Finally, our results point to the necessity of reconsidering the relationship between syntax and semantics. Although both the principle of a form/function (or signifier/signified) relationship and the basic postulates of cognitive linguistics call for a close correspondence between syntax and semantics, this strict correspondence is not as intuitive as it may seem.

Indeed, the difference between nouns and verbs as syntactic elements is clear-cut in French (nouns are preceded by determiner and verbs by subject personal pronouns) and is functional very early in young French children, but the semantic difference is not as clear-cut, as evidenced by the fact that categorization inevitably produces fuzzy sets.

Our results evidence the need for a reevaluation of the relationship between syntax and semantics throughout development. Instead of semantics being more fundamental than syntax, the relationship between syntax and semantics itself could be the result of a developmental process.

In this case, only when a child has moved from items and one-word utterances to syntactic organization, and simultaneously evolved from itembased meaning to semantic organization (categorization), would she be able to use form-function properties and relationships to create (or understand) new linguistic properties, derive new meanings and walk the path towards full mastery of language.

CONCLUSIONS

Ultimately, the absence of clearly delineated semantic sets, at a stage when nouns and verbs are probably only superficially distinguished, should come as no surprise. Our results suggest that in her first predications, the child masters neither the semantics, nor the syntax of noun and verb categories.

The construction of these categories seems to be achieved only through the gradual mastery of more and more complex and detailed language. In the end, then, our results confirm the proposals of distributional grammar, as well as those of construction grammar: it is only thanks to the understanding of constructions and of the relations between different elements in the utterance that noun and verb categories can be grasped.

( 6 )

 6 Madeleine (2;06) : Euh ah hum Martine elle avait écrasé un peu mon doudou. [Erm oh huh Martine, she'd squashed my teddy a little.]

Figure 1 :

 1 Figure 1: Percentages of nouns and verbs used for mobile references by Madeleine

  2) Antoine 2;01: line 348 & sq., as he watches the water flow from

	the hose
	*OBS:	c'est l'eau ? [This is water, right?]
	*CHI:	oui. [Yes.]
	%pho:	wi
	*OBS:	et pourquoi qu'est-ce qu'elle fait ? [Why? What is it
	doing?]
	*OBS:	hein qu'est-ce qu'il se passe là ? [Look, what's
	happening here?]
	%com:	children playing in the yard can still be heard
	*CHI:	a@fs di@fs l'eau. [There's water.]
	%pho:	a di lo
	%xpol:	NOUN/eau/

/visible(audible)/uncountable/concrete/inanimate/mobile/specific/ We chose to code l'eau (water) as mobile because to the child, it does designate a mobile referent, and our coding of a single nominal element reflects the absence of clearly verbal elements in the child's utterance (in spite of the observer's deliberate prompts).

Table 1 :

 1 Data coded

		Child	Age	MLU	Number of	Number of	Number of
					Nouns	Verbs	Words
	Madeleine	Mean values	2;00	2.44	246	154	1129
	(7 sessions)	First session	1;06	1.38	194	30	390
		Last session	2;06	3.02	199	169	1397
	Antoine (12	Mean values	1;11	1.48	71	51	330
	sessions)	First session	1;06	1	3	0	23
		Last session	2;06	2.72	183	187	354
	Théophile	Mean values	1;11	1.60	41	10	323
	(13 sessions)	First session	1;05	1.05	4	0	59
		Last session	2;08	2.61	124	47	1297

Table 2 :

 2 Coding examples

		Children NOUN VERB NOUN VERB NOUN VERB NOUN VERB
			not coded	Absent	Touch	Visible
	Distance	Antoine Madeleine	8 1	4 23	22 34	6 12 **	16 24	38 45 *	54 41	42 * 21
		Théophile	0	1	15	2	28	50	57 **	47
			not coded	Plural	Singular	Uncountable
	Number	Antoine Madeleine	3 0	1 6	5 10 **	0 * 1	91 79	88 92	1 11	0 0
		Théophile	0	0	4	0	83	100	13	0
			not coded	Abstract	Concrete
	Abstractness	Antoine Madeleine	6 0	4 1	35 17	8 20 ***	59 83	78 79 ***
		Théophile	0	0	16	10	84	90
			not coded	Animate	Inanimate
	Animacy	Antoine Madeleine	3 0	0 0	41 27	87 97 *	56 73	2 2
		Théophile	0	0	53	100	47	0
			not coded	Mobile	Static
	Motion	Antoine Madeleine	45 33	2 19	42 26	78 65	13 * 42	10 * 16
		Théophile	15	0	60	79	26	21
			not coded	Generic	Specific
		Antoine	8	8	2	1	90	81
	Specificity	Madeleine	5	15	16 **	11	78 *	74
		Théophile	1	0	5	0	94	100

Table 3 :

 3 Mean percentages of dimensions coded for all categories.

1 This work was supported by two grants from the Agence Nationale pour la Recherche, France (ANR COLAJE and ANR POLYCAT).


Danh-Tai Hoang

H T Diep

Effect of Dipolar Interaction in Molecular Crystals

Keywords: numbers: 61, 30, Cz, 64, 70, M-: numbers: Valid appear here I

We investigate in this paper the ground state and the nature of the transition from an orientational ordered phase at low temperature to the disordered state at high temperature in a molecular crystal. Our model is a Potts model which takes into account the exchange interaction J between nearestneighbor molecules and a dipolar interaction between molecular axes in three dimensions. The dipolar interaction is characterized by two parameters: its amplitude D and the cutoff distance r c . If the molecular axis at a lattice site has three orientations, say the x, y or z axes, then when D = 0, the system is equivalent to the 3-state Potts model: the transition to the disordered phase is known to be of first order. When D = 0, the ground-state configuration is shown to be composed of two independent interpenetrating layered subsystems which form a sandwich whose periodicity depends on D and r c . We show by extensive Monte Carlo simulation with a histogram method that the phase transition remains of first order at relatively large values of r c .

INTRODUCTION

The nature of the transition from one phase to another is one of the most important problems in statistical physics and in various areas of materials science not limited to physics. Since the introduction of the renormalization group 1,[START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF] with new concepts based on the system symmetry and scale, the understanding of the nature of the phase transition in many systems becomes clear. However, there are complicated systems where that method encounters much of difficulties in application. One can mention frustrated systems where competing interactions cause highly degenerate ground states with instabilities [START_REF]Frustrated Spin Systems[END_REF] .

In this paper, we are interested in the phase transition in molecular crystals which has been a subject of intensive investigations since 40 years. This spectacular development was due to numerous applications of liquid crystals in daily life [START_REF] De Gennes | The Physics of Liquid Crystals[END_REF][START_REF] Chandrasekhar | Liquid Crystals[END_REF] . Liquid crystals are somewhere between solid and liquid states where molecules have some spatial orientations which, under some conditions, can order themselves into some structures such as nematic and smectic phases. Liquid crystal display uses properties of these ordered phases [START_REF] Fukuda | [END_REF]7 . In spite of the large number of applications using experimental findings, theoretical understanding in many points is still desirable. One of the most important aspects of the problem is the origin of layered structures observed in smectic ordering: in smectic phases the molecules are ordered layer by layer with periodicity. Experiments have discovered, for example, smectic phases of 3-layer, 4-layer, or 6-layer periodicity [8][9][10][11][12] . Different theories have been suggested to interpret these observations [13][14][15][16][17][18][19][20] . One of the unanswered questions is what is the origin of the long-period lay-ered structure in observed smectic phases? Hamaneh et al. 21 suggested that the effective long-range interaction is due to bend fluctuations of the smectic layers which may stabilize commensurate structures in particular the sixlayer phase. This suggestion is too qualitative to allow a clear understanding of the long-period layered structure in smectic phases. To our opinion, we should distinguish long-range correlation and long-range interaction. We know that long-range correlation can be a consequence of short-range interactions, not necessarily of long-range ones, as it is well known in theory of critical phenomena 2 . However, a long-range correlation does not imply a particular long-period configuration.

To look for a physical origin of long-period structures, we concentrate ourselves in the present paper to the effect of a dipolar interaction in a Potts model, in addition to an exchange interaction between nearest neighbors (NN). We suppose that each molecule has a molecular axis which can lie on one of the three principal directions. An example of such a molecule is the ammoniac molecule NH 3 . Without the dipolar interaction, the interaction between neighboring molecules gives rise to an orientational order of molecular axes at low temperature. In this situation, the system can be described by a 3-state Potts model in three dimensions (3D). The Potts model is very important in statistical physics. It is in fact a class of models each of which is defined by the number of states q that an individual particle or molecule can have. The interaction between two neighboring molecules is negative if they are in the same state, it is zero otherwise. Exact solutions are found for many Potts models in two dimensions (2D) [START_REF] Baxter | Exactly Solved Models in Statistical Physics[END_REF] . In 2D, the phase transition is of second order for q ≤ 4 and of first order for higher q. In 3D, a number of points are well understood. For example, the phase transition is of first order for q > 2. The molecular crystal described above undergoes therefore a first-order transition in the absence of a dipolar interaction.

The purpose of this paper is to investigate the effect of the dipolar interaction on the ground state (GS) structure and on the nature of the phase transition. The dipolar interaction is a long-range interaction which yields different GS structures depending on the shape of the sample as will be discussed in the next section. To carry out our purpose, we use a steepest descent method for the GS determination and the Monte Carlo (MC) simulation combined with the histogram technique to distinguish first-and second-order characters.

In section II, we show our model and analyze the GS. Results of MC simulations are shown and discussed in section III. Concluding remarks are given in section IV.

II. MODEL AND GROUND-STATE ANALYSIS

We consider a simple cubic lattice where each site is occupied by an axial molecule. The molecular axis can be in the x, y or z direction. Let us denote the orientation of the molecule at the lattice site i by a unit segment, not a vector, which can lie in the x, y or z direction. We attribute the Potts variable σ=1, 2 or 3 when it lies in the x, y or z axes, respectively, in the calculation. By the very nature of the model shown below, the results do not depend on these numbers as in the standard q-state Potts model.

We suppose that the interaction energy between two nearest molecules is -J (J > 0) if their axes are parallel, zero otherwise. With this hypothesis, the Hamiltonian is given by the following 3-state Potts model:

H = -J (i,j) δ(σ i , σ j ) (1)
where σ i is the 3-state Potts variable at the lattice site i and (i,j) is made over the nearest sites σ i and σ j . The dipolar interaction between Potts variables is written as

H d = D (i,j) { 1 r 3 ij -3 [S(σ i ) • u ij ][S(σ j ) • u ij ] r 3 ij }δ(σ i , σ j )
(2) where u ij is the vector of unit length connecting sites i and j, D a positive constant depending on the material, S(σ i ) is defined as the unit vector lying on the axis corresponding to the value of σ i . The sum is limited at some cutoff distance r c . Without the Kronecker condition and in the case of classical XY or Heisenberg spins, the dipolar interaction gives rise to spin configurations which depend on the sample shape. For example, in 2D or in rectangular slabs spins lie in the plane to minimize the system energy.

Let us first discuss about the GS in the Potts model introduced above. When D = 0 the GS is uniform with one orientation value, namely it is 3-fold degenerate. However, for a nonzero D, the ground state changes with varying r c . To determine the GS, we use the steepest descent method which works very well in systems with uniformly distributed interactions. This method is very simple [START_REF] Ngo | [END_REF]24 (i) we generate an initial configuration at random (ii) we calculate the local field created at a site by its neighbors using ( 1) and (2) (iii) we take the Potts variable to minimize its energy (i. e. align the "Potts spin" in its local field) (iv) we go to another site and repeat until all sites are visited: we say we make one sweep (v) we do a large number of sweeps per site until a good convergence is reached.

We show some examples in Fig. 1. For very small D, the GS is uniform for any r c as shown in Fig. 1a. For increasing D, the GS has layered structures with period p = 1 (alternate single layers), p = 2 (double layers), p = 3 (triple layers),... depending on r c and D.

We note that in Fig. 1 there are only two kinds of molecular orientations represented by two colors (on line) in spite of the fact that we have three possible orientations. Let us discuss about the single-layer structure shown in Fig. 1c. It is very important to note that each layer has no coupling with two nearest layers whose molecules lie on another axis. However, it is coupled to two next-nearest layers of the same color, namely the same molecular axis, lying within r c . In other words, the GS is composed of two interpenetrating "independent" subsystems. The same is true for other layered structures: in a double-layer structure molecules of the same orientation (same color on line) interact with each other but they are separated by a double layer of molecules of another orientation. To our knowledge, this kind of GS has never been found before. It may have important applications at macroscopic levels.

Note also that if the molecules of the single-layer structure are successively in x-oriented and y-oriented planes, then the stacking direction of these planes is the z direction. The molecules can choose the x and z orientations or the y and z directions. In those cases the stacking directions are respectively the y and x directions. These three possibilities are equivalent if the sample is cubic. In rectangular shapes, the stacking direction is along the smallest thickness.

In order to understand how such GS configurations found by the steepest descent method depend on D and r c , we have considered the structures 121212...(single-layer structure), 11221122... (2-layer structure), 111222111222... (3-layer structure) and carried out the calculations of the energy of a molecule σ i interacting with its neighbors σ j . The case where the configuration is uniform, i. e. there is only one kind of molecular orientation, say axis x, the dipolar energy of σ i is

E i = D j [ 1 r 3 ij -3 (u x ij ) 2 r 3 ij ] (3)
Note that with the use of the Potts model for the dipolar term in Eq. ( 2), the energy depends only on the axis, not on its direction as seen by the square term (u x ij ) 2 . In this sense, the model is suitable to describe axial, but non-directed, interacting molecules.

If we transform the sum in integral, the sum in the first term gives 4π ln r c (integrating from 1 to r c ), while the second term gives -4π ln r c , which cancels the first term. This is valid for r c larger than 1. Thus the dipolar energy E d = D[4π ln r c -4π ln r c ] is zero for the uniform configuration in 3D space in the framework of our model. The energy of the system comes from the short-range exchange term, Eq. ( 1).

The energy per site of other layered structures with periodicity p = 2 and 3 is numerically calculated and shown in Fig. 2 for D/J = 0.8 and 2. In each figure, the GS is the structure which corresponds to the lowest energy. One sees in Fig. 2a We summarize in Fig. 3 the different GS in the space (r c , D) with J = 1. We note that for a given D, for example D = 0.8, the GS configuration starts with uniform configuration then with period 1, 2, 3, ... for increasing r c . large D, uniform configuration is not possible at any r c . Long-period configurations are on the other hand favored at small D and large r c .

III. PHASE TRANSITION: RESULTS

We consider a sample size of N × N × N z where N and N z vary from 24 to 48 but N z can be different from N in order to detect the shape-dependence of the ground state.. The 3-state Potts model with NN exchange interaction J = 1 is used to describe the three molecular orientations. For the dipolar term, a cutoff distance r c is taken up to √ 10 3.16 lattice distance. At this value, each molecule has a dipolar interaction with 146 neighbors. Periodic boundary conditions in all directions are employed.

We have used the standard MC method 25 with the system size from 24 3 to 48 3 . The equilibrating time N 1 is about 10 6 MC steps per site, and the averaging time N 2 is between 10 6 and 10 7 MC steps per site. The averages of the internal energy < U > and the specific heat C V are defined by

U = < H + H d > (4) C V = U 2 -U 2 k B T 2 (5)
where < ... > indicates the thermal average taken over N 2 microscopic states at T . We define the Potts order parameter Q by

Q = [q max(Q 1 , Q 2 , ..., Q q ) -1]/(q -1) (6)
where Q n is the spatial average defined by n(n = 1, ..., q) being the value of the Potts variable at the site j. For q = 3, one has n = 1, 2, 3 representing respectively the molecular axis in the x, y and z directions. The susceptibility is defined by

Q n = j δ(σ j -n)/(N × N × N z ) (7)
0 0 0 0 1 1 1 1 2 (18) 0 0 0 1 1 1 1 1 3 (26) 0 0 0 1 1 1 1 1 2.0 (32) 0 0 0 2 1 1 1 1 5 (56) 0 0 2 2 2
χ = Q 2 -Q 2 k B T (8)
In MC simulations, we work at finite sizes, so for each size we have to determine the "pseudo" transition which corresponds in general to the maximum of the specific heat or of the susceptibility. The maxima of these quantities need not to be at the same temperature. Only at the infinite size, they should coincide. The theory of finite-size scaling [START_REF] Hohenberg | [END_REF][27][28] permits to deduce properties of a system at its thermodynamic limit. We have used in this work a size large enough to be close to the extrapolated bulk transition temperature.

We show first in Fig. 4 the energy per site E ≡< U > /(N × N × N z ) and the order parameter M =< Q > versus T , for several lattice sizes in the absence of the dipolar interaction, i. e. D = 0. As said earlier this case corresponds to the 3-state Potts model. We find a very sharp transition. In order to check the first-order nature of this transition, we used the histogram technique which is very efficient in detecting weak first-order transitions and in calculating critical exponents of secondorder transitions 27,28 . The main idea of this technique is to make an energy histogram at a temperature T 0 as close as possible to the transition temperature. Often, one has to try at several temperatures in the transition region. Using this histogram in the formulae of statistical physics for canonical distribution, one obtains energy histograms in a range of temperature around T 0 . In second-order transitions, these histograms are gaussian. They allows us to calculate averages of physical quantities as well as critical exponents using the finite-size scaling. In first-order transitions, the energy histogram shows a double-peak structure at large enough lattice sizes. Using this method, we have calculated the histogram shown in Fig. 5 for several lattice sizes. With increasing size, the two peaks are well separated with the dip going down to zero, indicating a tendency toward an energy discontinuity at the infinite size. The distance between the two peaks is the latent heat ∆E. We take into account now the dipolar interaction. In order to see its progressive effect, we perform simulations with r c varying between r c = √ 6 and r c = √ 10 and follow the change of the characteristics of the phase transition.

We show in Fig. 6 the energy and the order parameter versus T for D = 2, r c = √ 10. The transition is still very sharp but the strong first-order character diminishes. We show in Fig. 7 the energy histogram for several values of r c with D = 2. As seen the latent heat becomes small at r c = √ 10.

We observe that the latent heat ∆E diminishes with increasing r c . However, we cannot conclude that the first-order disappears at large r c . To check that point we need to go to larger r c which will take a huge CPU time because of the increasing number of neighbors (we recall that for r c = √ 10, we have 146 neighbors for each molecules). We observe from Fig. 7 that the latent heat does not change significantly with r c up to r c 2.85, meaning that the first order transition is dominated by the short-range 3-state Potts interaction. The latent heat decreases rather strongly afterward but we do not know if it tends to zero or not.

IV. CONCLUSION

We have studied in this paper a molecular crystal characterized by three possible orientations of the molecular axes. The model is described by a short-range 3-state Potts model and a dipolar Potts interaction. We have analyzed the ground state as functions of the dipolar interaction strength D and its cutoff distance r c . The ground state shows remarkable properties: it is composed of two interpenetrating independent subsystems with a layered structure whose periodicity depends on D and r c . To our knowledge, such a ground state has not been seen in uniformly interacting molecules. The stacking of independent ordered layers is reminiscent of a smectic ordering. Note that the model is applied to smectic phases where tilted angles are uniform: it can be of type A as supposed here or of type C if the molecular axis does not coincide with the x, y or z axis of the lattice. The coupling between adjacent different ordered planes is here strictly zero at T = 0, due to the Potts condition in the model. We have used the Monte Carlo histogram method to study the phase transition in this system. In the absence of the dipolar interaction, the model which is a 3-state Potts model is known to undergo a first-order phase transition from the orientational ordered phase to the disordered state. Upon the introduction of a dipolar interaction of sufficient strength into the Potts model, the ground state is broken into layers as described above. We have shown that the transition remains of first order as the cutoff distance is increased at least up to r c = √ 10.

  the following GS configurations with varying r c : -uniform GS: for 1 ≤ r c ≤ 1.3 (zone I) -single-layer structure: for 1.3 ≤ r c ≤ 1.8 (zone II) -double-layer structure: for 1.8 ≤ r c ≤ 3.65 (zone III) -triple-layer structure: for r c ≥ 3.65 (zone IV) For D/J = 2 there are only two possible GS with varying r c as shown in Fig. 2b.

FIG. 1 .

 1 FIG. 1. Ground state in the case where a) D/J = 0.4, r c = 2.3: uniform configuration ; b) D/J = 1, r c = 2.3: doublestructure; c) D/J = 2, rc = 2.3: single-layer structure; d) D/J = 0.4, rc = √ 10 3.16: triple-layer structure. See text for comments.

FIG. 2 .

 2 FIG. 2. Ground state energy versus rc in the case where (a) D/J = 0.8 (b) D/J = 2. Blue, black, red and green lines represent GS energy of the uniform, single-layer, double-layer and triple-layer structures, respectively. Zones I, II, II, IV indicate these respective different GS. See text for comments.

FIG. 4 .

 4 FIG.4. Internal energy per spin E (upper) and order parameter M =< Q > (lower) versus temperature T without dipolar interaction, for several sizes Nz = N = 24 (black circles), 36 (void circles), 48 (blue triangles, color on line), J = 1.

FIG. 6 .FIG. 7 .

 67 FIG. 6. Internal energy per spin E and order parameter M =< Q > versus T for D = 2, r c = √ 10 with several lattice sizes N z = N = 24 (black circles), 36 (void circles), 48 (blue triangles, color on line), and J = 1.


Emilie Morvant

Stéphane Ayache

Amaury Habrard
email: amaury.habrard@univ-st-etienne.fr

Sparse Domain Adaptation in a Good Similarity-Based Projection Space *

We address domain adaptation (DA) for binary classification in the challenging case where no target label is available. We propose an original approach that stands in a recent framework of Balcan et al. [START_REF] Balcan | Improved guarantees for learning via similarity functions[END_REF] allowing to learn linear classifiers in an explicit projection space based on good similarity functions that may be not symmetric and not positive semi-definite (PSD). Following the DA framework of Ben-David et al. [START_REF] Ben-David | A theory of learning from different domains[END_REF], our method looks for a relevant projection space where the source and target distributions tend to be close. This objective is achieved by the use of an additional regularizer motivated by the notion of algorithmic robustness proposed by Xu and Mannor [START_REF] Xu | Robustness and generalization[END_REF]. Our approach is formulated as a linear program with a 1-norm regularization leading to sparse models. We provide a theoretical analysis of this sparsity and a generalization bound. From a practical standpoint, to improve the efficiency of the method we propose an iterative version based on a reweighting scheme of the similarities to move closer the distributions in a new projection space. Hyperparameters and reweighting quality are controlled by a reverse validation process. The evaluation of our approach on a synthetic problem and real image annotation tasks shows good adaptation performances. This work will appear in "IEEE International Conference on Data Mining (ICDM) 2011" [START_REF] Morvant | Sparse domain adaptation in projection spaces based on good similarity functions[END_REF].

Notations

Let X ⊆ R d be the input space of dimension d and Y = {-1, +1} the label set. A domain is a probability distribution over X×Y . In a DA framework [START_REF] Ben-David | A theory of learning from different domains[END_REF][START_REF] Mansour | Domain adaptation: Learning bounds and algorithms[END_REF], we have a source domain represented by a distribution P S and a target domain represented by a somewhat different distribution P T . D S and D T are the respective marginal distributions over X. A learning algorithm is provided with a Labeled Source sample LS = {(x i , y i )} d l i=1 drawn i.i.d. from P S , and an unlabeled Target Sample T S = {x j } dt j=1 drawn i.i.d. from D T . Let h : X → Y be an hypothesis function. The expected source error of h over P S is the probability that h commits an error: err S (h) = E (x,y)∼P S L 01 h,(x, y) , where L 01 (h,(x, y))=1 if h(x) = y and zero otherwise, it is the 0-1 loss function. The target error err T over P T is defined in a similar way. ê rr S and ê rr T are the empirical errors. An hypothesis class H is a set of hypothesis. The DA objective is then to find a low target error hypothesis.

Domain Adaptation Framework

We consider the DA framework proposed by Ben-David et al. [START_REF] Ben-David | A theory of learning from different domains[END_REF] allowing us to upper bound the target error err T according to the source error and the divergence between the domain distributions,

∀h ∈ H, err T (h) ≤ err S (h) + 1 2 d H∆H (D S , D T ) + ν. (1)
The last term ν can be seen as a kind of adaptation ability measure of H for the DA problem considered and corresponds to the error of the best joint hypothesis over the two domains: ν = argmin h∈H err S (h) + err T (h). The second term d H∆H (D S , D T ) is called the H∆H-distance between the two domain marginal distributions. This measure is actually related to H and an interesting point is that when the VC-dimension of H is finite, we can estimate d H∆H from finite samples by looking for the best classifier able to separate LS from T S. This bound suggests that one possible solution for a good DA is to look for a relevant data projection space where both the H∆H-distance and the source error of a classifier are low (two aspects a priori necessary for a good DA [START_REF] Ben-David | Impossibility theorems for domain adaptation[END_REF]).

Learning with Good Similarity Functions

Instead of working on the implicit high dimensional projection space induced by classical SVM's kernels (that may be strongly limited by symmetry and PSD requirements), we investigate a more flexible and intuitive similarity-based representation proposed recently by Balcan et al. [START_REF] Balcan | Improved guarantees for learning via similarity functions[END_REF] for learning with a good similarity function fulfilling the following definition.

Definition 1 ([1]). A similarity function is any pairwise function

K : X × X → [-1, 1]. K is an ( ,γ,τ
)-good similarity function for a learning problem P if there exists a (random) indicator function R(x) defining a set of reasonable points such that the following conditions hold:

(i) A 1 -probability mass of examples (x, y) satisfy E (x ,y )∼P yy K(x, x )|R(x ) = 1 ≥ γ, (ii) Pr x [R(x ) = 1] ≥ τ .
Def.1 requires that a large proportion of examples is on average more similar, w.r.t the margin γ, to the reasonable points of the same class than to the reasonable points of the opposite class. It includes all valid kernels as well as some non-PSD similarities and is thus a generalization of kernels [START_REF] Balcan | Improved guarantees for learning via similarity functions[END_REF].

Given K an ( ,γ,τ )-good similarity function, LS a sample of d l labeled points and R a set ofenough -d u potential reasonable points (landmarks), the conditions of Balcan et al. are sufficient to learn a low-error linear binary classifier (a SF classifier) in a φ R -space defined by the mapping φ R , which projects a point in the explicit space of the similarities to the landmarks in R such that,

φ R : X → R du x → K(x, x 1 ), . . . , K(x, x du ) .
The low-error SF classifier h can be learned by solving the following linear problem in the φ R -space,

min α 1 d l d l i=1 L g, (x i , y i ) +λ α 1 , with L g, (xi, yi) = [1-yig(x)] + and g(x) = du j=1 αjK(x, x j ), (2)
where [1 -z] + = max(0, 1 -z) is the hinge loss. Finally, we have h(x) = sign[g(x)]. Solving (2) not only minimizes the expected source error but also defines a relevant projection space for a given problem, since landmarks associated with a null weight in the solution α will not be considered. In this work we propose to add a new regularization term on α in order to constrain the explicit φ R -space to move closer the two distributions and to tend to decrease the H∆H-distance.

Contribution for Domain Adaptation with Good Similarity Functions

The objective here is to define a regularizer that tends to make the source and target sample indistinguishable. For this purpose, we have investigated the algorithmic robustness notion proposed by Xu and Mannor [START_REF] Xu | Robustness and generalization[END_REF] based on the following property: "If a testing sample is similar to a training sample then the testing error is close to the training error". This can be formalized as follows: If for any test point close to a training point of the same class the deviation between the losses of each point is low for a learned model, then this model has some generalization guarantees (even if the robustness is true for only a subpart of the training sample). This result actually assumes that the test and train data are drawn from the same distribution and is thus not valid in a classical DA scenario.

However, we propose to adapt this principle for making the target sample similar to the source one which is coherent with the minimization of the divergence d H∆H : For any pair (x s ,x t ) of close source and target instances of the same class y, the deviation between the losses of x s and x t is low. By considering the hinge loss of (2), this leads us to the following term to minimize for such a pair: L(g, (x s , y)) -L(g, (x t , y))

≤ ( t φ R (x s ) -t φ R (x t )) diag(α)
      min α 1 d l d l i=1 L g, (x i , y i ) + λ α 1 + β (xs,xt)∈C ST ( t φ R (x s ) -t φ R (x t )) diag(α) 1 ,
with L g, (x i , y i ) = [1-y i g(x)] + and g(x) = du j=1 α j K(x, x j ).

(

) 3
This problem is defined with a 1-norm regularization leading generally to very sparse models. We have in fact proved in the following lemma that the sparsity of the obtained models depends also on a quantity B R = min x j ∈R max (xs,xt)∈C ST |K(x s , x j ) -K(x t , x j )| related to the deviation between coordinates in the considered φ R -space. In other words, when the domains are far from each other, i.e. the task is hard, B R tends to be high which can increase the sparsity.

Lemma 1. For any λ > 0, β > 0, any set

C ST s.t. B R > 0, if α * is optimal, then α * 1 ≤ 1 βB R + λ .
Moreover, according to the robustness framework [START_REF] Xu | Robustness and generalization[END_REF] applied on the source domain, and from the DA bound (1), we can prove the following generalization bound for the expected target domain error.

Theorem 1. Problem (3) defines a procedure (2M η , Nη βB R +λ ) robust on the source domain, where

N η = max xa,x b ∼D S , ρ(xa,x b )≤η t φ R (x a )-t φ R (x b ) ∞ , η > 0, M η is the η-covering number of X.
Thus for every h in the SF classifiers hypothesis class, for any δ > 0, with probability at least 1-δ,

err T (h) ≤ ê rr S (h) + N η βB R + λ + 4M η ln 2 + 2 ln 1 δ d l + 1 2 d H∆H (D S , D T ) + ν.
From a practical standpoint, a critical issue is the estimation of the hyperparameters λ and β and the definition of the pair set C ST which is difficult a priori since we do not have any target label information. We propose to tackle these problems with the help of a reverse validation method.

Reverse validation and Iterative procedure

We choose the different parameters of our method by following the principle of reverse validation of Zhong et al. [START_REF] Zhong | Cross validation framework to choose amongst models and datasets for transfer learning[END_REF]. This principle is illustrated on Fig. 1 and consists in learning a so-called reverse classifier h r , from the target sample sef-labeled by a classifier h (inferred with Pb. (3)). We evaluate ê rr S (h r ) on the source sample and heuristically ê rr T (h r ) on the self-labeled target sample (both by cross-validation). We then obtain an heuristic estimate of ν (of the DA bound (1)) for h r such that ν = ê rr S (h r ) + ê rr T (h r ). We select the parameters and pair set C ST minimizing ν. In this context, ν can be seen as a quality measure of the φ R -space found: If the two domains are sufficiently close and related then the reverse classifier should perform well on the source domain ( [START_REF] Bruzzone | Domain adaptation problems: A DASVM classification technique and a circular validation strategy[END_REF]).

However, considering all the possible pairs for C ST remains clearly intractable. In practice, we select only a limited number of examples for building C ST . We compensate the possible loss of information by an heuristic iterative procedure still allowing to move closer the two distributions. Suppose that at a given iteration l, with a similarity K l , we obtain new weights α l by solving (3). Our regularization term can actually be seen as a L1-distance in a new φ R l+1 -space:

( t φ R l (x s )-t φ R l (x t )) diag(α) 1 = ( t φ R l+1 (x s ) -t φ R l+1 (x t )) 1 . φ R l+1
corresponding to the mapping defined by the similarity K l+1 obtained from K l by a conditional reweighting to each landmark: ∀x j ∈ R, K l+1 (x, x j ) = α l j K l (x, x j ) (K l+1 do not need to be PSD nor symmetric according to Def.1). We then iterate the process in the new φ R l+1 -space and we stop at iteration l when νl+1 has reached a convergence point or has increased. The main steps of the recursive approach are summarized on Algorithm 1.

Experimental evaluation

Our method DASF has been evaluated on a toy problem and on real image annotation tasks and compared with SF method of Pb (2) and SVM with no adaptation, the semi-supervised Transductive-SVM (T-SVM) [START_REF] Joachims | Transductive inference for text classification using support vector machines[END_REF] and the iterative DA algorithm DASVM [START_REF] Bruzzone | Domain adaptation problems: A DASVM classification technique and a circular validation strategy[END_REF]. We used a Gaussian kernel for the Algorithm 1 DASF (Domain Adaptation with Similarity Functions) Input: similarity function K, set R, samples LS and T S Output:

classifier h DASF h 0 (•) ← sign 1 |R| |R| j=1 K(•, x j ) ; K 1 ← K ; l ← 1;
while The stopping criterion is not verified do α l ← Solve Pb. (3) with K l , C ST and hyperparameters being selected by reverse validation; K l+1 ← Update K l according to α l ; Update R; l + +; end while return h DASF (•) = sign

x j ∈R α l j K l (•, x j ) ; last three methods and a re-normalization of this kernel as a non PSD similarity function for SF and DASF (see [START_REF] Morvant | On the usefulness of similarity based projection spaces for transfer learning[END_REF]). All the averaged results show that DASF provide better and sparser models in general. As an illustration, Tab.1 gives results for a real image annotation task, where the source images are extracted from the PascalVOC'07 corpus and the target ones from the TrecVid'07 video corpus. Moreover, the iterative procedure always tend to decrease the distribution divergence with the iterations [START_REF] Morvant | Sparse domain adaptation in projection spaces based on good similarity functions[END_REF]. Among all the possible perspectives, we notably aim to consider some few target labels to help the search of a relevant projection space.

Figure 1 : 3 :

 13 Figure 1: The reverse validation. 1: Learning h with (3). 2: Auto-labeling the target sample with h. 3: Learning h r on the auto-labeled target sample with (2). 4: Evaluation of r on LS.

  ST of close source-target examples, we then propose to consider this term for all the pairs as an additional regularizer on α, weighted by a parameter β, to the Problem (2) of Balcan et al. Our global optimization problem (3) can be then formulated as the following linear program, 

[START_REF] Balcan | Improved guarantees for learning via similarity functions[END_REF]

, where t φ R (•) is the transposed vector of φ R (•) and diag(α) is the diagonal matrix with α as main diagonal. Given any pair set C

Table 1 :

 1 The results obtained on the TrecVid target domains according to the Fmeasure. AVG. corresponds to the averaged results.

	CONCEPT BOAT BUS CAR MONITOR PERSON PLANE AVG.
	SVM	0.56 0.25 0.43	0.19	0.52 0.32 0.38
	MODEL SIZE 351 476 1096	698	951	428	667
	SF	0.49 0.46 0.50	0.34	0.45 0.54 0.46
	MODEL SIZE 214 224 176	246	226	178	211
	T-SVM 0.56 0.48 0.52	0.37	0.46 0.61 0.50
	MODEL SIZE 498 535 631	741	1024 259	615
	DASVM 0.52 0.46 0.55 0.30	0.54 0.52 0.48
	MODEL SIZE 202 222 627	523	274	450	383
	DASF	0.57 0.49 0.55 0.42	0.57 0.66 0.54
	MODEL SIZE 120 130 254	151	19	7	113

Work partially supported by the ANR VideoSense project (ANR-09-CORD-026) and the PASCAL2 network of excellence.