Nouredine Tamani
email: tamani@enssat.fr

Ludovic Liétard
email: ludovic.lietard@univ-rennes1.fr

Daniel Rocacher
email: rocacher@enssat.fr

Bipolarity in Flexible Querying of Information Systems Dedicated to Multimodal Transport Networks

Keywords: Flexible querying, complex preferences, fuzzy bipolar conditions, bipolar SQLf, fuzzy sets theory, fuzzy description logics, multimodal transportation

Flexible querying of information systems dedicated to multimodal transport networks aims to help user organizing his/her trips by promoting public transport networks (bus, subway, train, boat, plane, etc.). In this context, it is necessary to provide an integrated environment in which it is possible for the user to express queries with complex preferences so as to meet his/her expectations. Complex preferences are modeled by fuzzy bipolar conditions which associate negative and positive conditions. Queries involving such conditions are called bipolar queries. In our case, such queries are addressed to multimodal transport information systems, which are often made of several distributed and heterogeneous databases. Therefore, semantic aspects have to be taken into consideration in the querying process so that only the most relevant data is targeted to evaluate queries. We introduce then in this paper a new approach for flexible querying of information systems that combines a reasoning mechanism (fuzzy bipolar DLR-Lite) with a relational language of a high expressiveness (bipolar SQLf language).

I. INTRODUCTION AND MOTIVATION

Multimodal transport information system is intended to provide information called multimodal transport information (denoted MTI) which is about various modes and available transport networks and services to help travelers to plan and to organize their trips [START_REF] Predim | Plate-forme de recherche et d'expérimentation pour le développement de l'information multimodale[END_REF]. It covers information about transport modes, timetables, pricing, traffic analysis, etc.

User oriented services are developed to promote public transport networks for the planning of user's daily journeys. To fulfill user requirements, multimodal transport information systems must be endowed with powerful and complex functionalities. The issues of the heterogeneity and distribution of data in the transport field have also to be taken into consideration. Ontologies could be used to manage these aspects because they not only unify the domain vocabulary but they also provide reasoning mechanisms capable to extract implicit data from explicit one. Several ontologies have been developed for the multimodal transport domain (see [START_REF] Chen | Nimbletransit: Public transportation transit planning using semantic service composition schemes[END_REF], [START_REF] Lewis | Using ontology to integrate railway condition monitoring data[END_REF], [START_REF] Niaraki | Ontology based personalized route planning system using a multi-criteria decision making approach[END_REF], [START_REF] Obitko | Integration transportation ontologies using semantic web languages[END_REF], [START_REF] Saad | The flexible negotiation ontologybased knowledge management system: the transport ontology case study[END_REF]). The more complete one is Ontology Transportation Network (OTN) [START_REF] Lorenz | Ontology of transportation networks[END_REF], which allows the modeling of multimodal transport networks from different points of view such as spatial, temporal, services, etc.

Flexible querying systems allow users expressing preferences in their queries. These queries are addressed to regular relational databases and deliver a set of discriminated answers, which are ranked from the most to the least preferred element. Within the scope of flexible querying, the fuzzy set theory provides a general model for the interpretation of queries involving fuzzy predicates. It is also possible to consider fuzzy bipolar conditions to model preferences. Several interpretations are introduced for the evaluation of queries involving fuzzy bipolar conditions (see [START_REF]Bipolarity in flexible querying[END_REF], [START_REF] Dubois | Bipolarité dans un processus d'interrogation flexible[END_REF], [START_REF] Tré | Dealing with positive and negative query criteria in fuzzy database quering bipolar satisfaction degrees[END_REF], [START_REF] Zadrozny | Bipolar queries using various interpretations of logical connectives[END_REF], [START_REF]Bipolar queries: An approach and its various interpretations[END_REF]), and in this paper, we rely on the interpretation introduced by Dubois and Prade [START_REF]Bipolarity in flexible querying[END_REF], [START_REF] Dubois | Bipolarité dans un processus d'interrogation flexible[END_REF], in which a bipolar condition is made of two components: a constraint that is a mandatory condition and a wish that is an optional condition. More precisely, for the expression of user preferences, we rely on fuzzy bipolar conditions in which the constraint and the wish are defined by fuzzy sets. We define then a bipolar query as a query that involves bipolar conditions. A bipolar relational algebra is also proposed in [START_REF] Liétard | On the extension of sql to fuzzy bipolar conditions[END_REF], [START_REF] Bosc | Extending relational algebra to handle bipolarity[END_REF] and the algebraic operators (selection, projection, join, union, intersection) have been extended to fuzzy bipolar conditions.

We introduce in this paper a flexible querying approach of distributed and heterogeneous information systems, such as those developed for multimodal trip planning systems, which allows users to express complex preferences in their queries. In this context, the system has to deal with a large amount of data while taking into account efficiency in terms of quality of answers. Two aspects have been considered: (i) the expressivity in terms of complex preferences and query language and (ii) the efficiency of the data management. Our approach is then based on the combination of the Bipolar SQLf language (which is an extension of the SQLf language [START_REF] Bosc | SQLf: A relational database langage for fuzzy querying[END_REF], [START_REF] Bosc | Base de données -Gradualité et imprécision dans les bases de données Ensembles flous[END_REF] to fuzzy bipolar conditions, suitable to express complex preferences) with a reasoning system, built on a fuzzy bipolar DLR-Lite knowledge base (which is also an extension of the fuzzy DLR-Lite [START_REF] Colucci | Semantic Web Information Management[END_REF], [START_REF] Straccia | Softfacts : a top-k retrieval engine for a tractable description logic accessing relational databases[END_REF] to fuzzy bipolar conditions, used to target the most relevant subset of data to answer user queries).

The remainder of the paper is organized as follows. Section II reminds both fuzzy sets theory (which is the main model for fuzzy conditions) and the SQLf language (which is an extension of the SQL language to fuzzy conditions). In section III, fuzzy bipolar conditions and Bipolar SQLf language are respectively described. Section IV recalls the fuzzy DLR-Lite and introduces our contribution which consists in the extension of the fuzzy DLR-Lite to fuzzy bipolar conditions and Bipolar SQLf language. In section V, the fuzzy bipolar DLR-Lite is used to develop a sample application in the field of multimodal transportation, in order to show the improvement that could be gained from integrating fuzzy bipolar conditions into both fuzzy DLR-Lite and SQLf language. Section VI summarizes our contribution and draws some lines for future works.

II. FLEXIBLE QUERYING WITHIN THE SQLF LANGUAGE

In this section, we briefly introduce fuzzy sets theory, used to define fuzzy predicates such as fast, high, large, expensive, ..., and the SQLf language which is an extension of the SQL language to fuzzy conditions.

A. Fuzzy Sets Theory

The fuzzy sets theory is introduced by Zadeh [START_REF] Zadeh | Fuzzy sets[END_REF] to express the gradual membership of an element to a set. Formally, a fuzzy set F is defined on a referential U by a membership function µ F : U → [0, 1] such that µ F (x) denotes the membership grade of x in F .

In particular, µ F (x) = 1 denotes the full membership of x in F , µ F (x) = 0 expresses the absolute non-membership and when 0 < µ F (x) < 1, it reflects a partial membership (the closer to 1 µ F (x), the more x belongs to F ).

A fuzzy set generalizes a crisp set in which membership grades are in {0, 1}. If a fuzzy set is a discrete set then it is denoted F = {µ F (x 1 )/x 1 + ... + µ F (x n )/x n }, else it is characterized by its membership function, generally a trapezoidal function.

The union ∪ and the intersection ∩ operators are defined with a couple of (t-norm, t-conorm) such as (min, max). Let F , G be two fuzzy sets,

µ F ∪G (x) = max(µ F (x), µ G (x)), µ F ∩G (x) = min(µ F (x), µ G (x)), and the complement of F , noted F c , is defined by µ F c (x) = 1 -µ F (x).
The logical counterparts of ∩, ∪ and the complement are resp. ∧, ∨ and ¬. Other operators have also been defined such as fuzzy implications [START_REF] Bouchon-Meunier | Fuzzy sets in approximate reasoning and information systems, ser. The Handbook of fuzzy sets[END_REF]. We consider particularly Rimplications, which are based on the residual principle:

I R (a, b) = sup{z ∈ [0, 1] : T (a, z) ≤ b}, (1)
where I R (a, b) means a → R b and T is a triangular norm. R-implications satisfy properties of monotonicity, neutrality, exchange, identity and of ordering [START_REF] Bouchon-Meunier | Fuzzy sets in approximate reasoning and information systems, ser. The Handbook of fuzzy sets[END_REF]. For example, the Gödel fuzzy implication is an R-implication which is defined by the formula [START_REF] Bosc | SQLf: A relational database langage for fuzzy querying[END_REF], in which the used t-norm is min:

I Gö (a, b) = sup{z ∈ [0, 1] : min(a, z) ≤ b}, (2)
The Gödel fuzzy implication is also defined as follows:

I Gö (a, b) = 1 if a ≤ b, b otherwise.

B. The SQLf Language

The SQLf language [START_REF] Bosc | SQLf: A relational database langage for fuzzy querying[END_REF], [START_REF] Bosc | Base de données -Gradualité et imprécision dans les bases de données Ensembles flous[END_REF] is an extension of the SQL language to flexible querying by extending its statements to fuzzy conditions. When querying a regular relation R with such conditions, a fuzzy relation r (fuzzy set) is delivered. Each tuple t from r is attached with a degree of membership that expresses its level of satisfaction. The closer to 1 µ r (t), the more t is preferred.

Example 1: Let fast be a fuzzy predicate as defined in Fig. 1. A fuzzy relation fastJourney can be defined from the relation Journey (#Journey, source, destination, cost, duration) s.t. for each tuple t: µ F astJourney (t) = µ F ast (t.duration).

Table I is an example of extension of fastJourney. p

The SQLf language is based on the extension of the relational algebra operators to fuzzy conditions, defined as follows [START_REF] Bosc | SQLf: A relational database langage for fuzzy querying[END_REF]: let r, s be two fuzzy relations such that the schema of r (resp. s) is X (resp. Y ):

• Fuzzy projection: µ π(r,V ) (v) = max w µ r (vw), where V ⊆ X, v ∈ V and w ∈ (X -V ). • Fuzzy restriction: µ σ(r,p) (x) = min(µ r (x), µ p (x)), where p is a fuzzy predicate. • Fuzzy join: µ (r, s, θ, A, B) = min(µ r (x), µ s (y), µ θ (x.A, y.B)), where A and B are compatible sets of attributes such that A (resp. B) is a subset of X (resp. Y ) and x.A (resp. y.B) is the value of A in x (resp. B in y), and θ is either a crisp or a fuzzy binary operator (θ ∈ {=, ≈, <, >, much larger than, ...}).

Syntactic elements of the SQLf language:

The main form of an SQLf query (fuzzy restriction) is:

Select [distinct] [n|t|n,t] attributes From relations
Where fuzzy_cond; The above query returns a set of ranked tuples with their degree of satisfaction, where n is a limitation of the size of the returned set and t ∈]0, 1] is a minimal threshold of satisfaction.

Example 2: We consider a sample database which is consistent with the Transmodel standard [START_REF] Carles | Modèle conceptuel de données générique pour la représentation des réseaux intermodaux[END_REF]. It contains the following relations: Network, Line, Station, Link and LinkLine. We define Journey relation as Journey (#journey, #link, de-parture, arrival, source, destination, cost, duration, DepLink, ArrLink, modeLink, waitLink, durLink, costLink, comfLink).

For example, the query "find the 2 fastest journeys from Lannion to Paris" could be expressed in SQLf by: Select 2 #journey From Journey Where source='Lannion' and destination='Paris' and fast (duration);

The fuzzy condition fast(duration) delivers the fuzzy relation FastJourney (see table I) and the query returns the 2 best answers : journeys #12 and #13, attached respectively with the satisfaction degrees 1 and 0.66. p

The SQLf language allows more complex statements such as partitioning, nesting and division involving fuzzy relations.

III. FLEXIBLE QUERYING WITHIN BIPOLAR SQLF LANGUAGE

In this section, we briefly introduce fuzzy bipolar conditions and Bipolar SQLf language which is an extension of the SQLf language to fuzzy bipolar conditions.

A. Fuzzy Bipolar Conditions

A bipolar condition is an association of both negative condition (pole) and positive condition (pole). In this paper, a bipolar condition is made of two conditions defined on the same universe: i) a constraint c, which describes the set of acceptable elements, and ii) a wish w which defines the set of desired or wished elements. The negation of c is the set of rejected elements (non-acceptable elements). Since it is not coherent to wish a rejected element, the following property of coherence holds: w ⊆ c.

In addition, condition c is mandatory since an element which does not satisfy c is rejected (¬c is the negative condition), whereas w is optional because its non-satisfaction by an element does not mean its rejection (w is the positive condition). In this paper, a bipolar condition is noted (c, w) and means: "satisfy c and if possible satisfy w" [START_REF]Bipolarity in flexible querying[END_REF], [START_REF] Dubois | Bipolarité dans un processus d'interrogation flexible[END_REF].

If c and w are boolean conditions, the satisfaction with respect to (c, w) is a couple from {0, 1} 2 . When querying a database with such a condition, tuples satisfying the constraint and the wish are returned in priority. If such answers do not exist, the tuples satisfying only the constraint are delivered.

If c and w are fuzzy conditions (defined on the universe U ), the property of coherence becomes: ∀u ∈ U, µ w (u) ≤ µ c (u) and the satisfaction with respect to (c, w) is a couple of degrees from [0, 1] 2 . Each element u from U is then attached with a couple of grades (µ c (u), µ w (u)) that expresses the degree of its satisfaction respectively to the constraint c and the wish w.

When querying a relation R with a fuzzy bipolar condition, each tuple t from R is then attached with a couple of grades (µ c (t), µ w (t)) that expresses the degree of its satisfaction respectively to the constraint c and the wish w; and a so-called fuzzy bipolar relation is obtained. A tuple t is then denoted (µ c , µ w )/t. We assume that any tuple u such that µ c (u) = 0 does not belong to the fuzzy bipolar relation. In such a context, tuples cannot be ranked using an aggregation of µ c and µ w because the constraint and the wish are not commensurable. However, they can be ranked using the lexicographical order:

t 1 is preferred to t 2 , denoted t 1 t 2 , if and only if µ c (t 1 ) > µ c (t 2 ) or (µ c (t 1 ) = µ c (t 2 ) ∧ µ w (t 1 ) > µ w (t 2 )).
In this case, the satisfaction with respect to the constraint is firstly used to discriminate among answers (the constraint being mandatory). The satisfaction with respect to the wish being optional, it can only be used to discriminate among answers having the same evaluation with respect to the constraint. A total order is then obtained on µ c and µ w (with (1, 1) as the greatest element and (0, 0) as the least element).

Based on the lexicographical order, the lmin and lmax operators [START_REF] Liétard | On the definition of extended norms and co-norms to aggregate fuzzy bipolar conditions[END_REF], [START_REF] Bosc | Extending relational algebra to handle bipolarity[END_REF] are introduced in order to define the conjunction (resp. intersection) and the disjunction (resp. union) of bipolar conditions (resp. relations). They are defined as follows:

([0, 1] × [0, 1]) 2 → [0, 1] × [0, 1] ((µ, η), (µ , η )) → lmin((µ, η), (µ , η )) = (µ, η) if µ < µ ∨ (µ = µ ∧ η < η ), (µ , η ) else. ([0, 1] × [0, 1]) 2 → [0, 1] × [0, 1] ((µ, η), (µ , η )) → lmax((µ, η), (µ , η )) = (µ, η) if µ > µ ∨ (µ = µ ∧ η > η ), (µ , η ) else.
The lmin (resp. lmax) operator is commutative, associative, idempotent and monotonic. The couple of grades (1, 1) is the neutral (resp. absorbing) element of the operator lmin (resp. lmax) and the couple (0, 0) is the absorbing (resp. neutral) element of the operator lmin (resp. lmax).

Remark 1: Fuzzy bipolar conditions generalize fuzzy conditions since a fuzzy condition c can be rewritten (c, c) to express " satisfy c and if possible satisfy c". So, a fuzzy relation R is a particular case of a fuzzy bipolar relation such that ∀t ∈ R, µ R (t) = µ c (t) = µ w (t). It has been proven [START_REF] Liétard | On the definition of extended norms and co-norms to aggregate fuzzy bipolar conditions[END_REF], [START_REF] Bosc | Extending relational algebra to handle bipolarity[END_REF] that the lmin (resp. lmax) extends the t-norm min (resp. the t-co-norm max) to bipolarity.

B. Basis of Bipolar SQLf Language

Bipolar SQLf language is an extension to fuzzy bipolar conditions of the SQLf language. We introduce in this subsection the fuzzy bipolar extension proposed in [START_REF] Liétard | On the definition of extended norms and co-norms to aggregate fuzzy bipolar conditions[END_REF], [START_REF] Bosc | Extending relational algebra to handle bipolarity[END_REF] of the relational algebraic operators (the union, the intersection, the cartesian product, the projection, the selection and the join) of fuzzy bipolar relations. This extension is based on the couple of the extended t-norm and t-co-norm (lmin, lmax).

Let r and s be two fuzzy bipolar relations defined respectively by fuzzy bipolar conditions (c 1 , w 1 ) and (c 2 , w 2 ).

1) The intersection operator: The intersection of r and s is a fuzzy bipolar relation, in which each tuple t is attached with a couple of degrees (µ c (t), µ w (t)). It is defined as follows:

r ∩ s = {(µ c , µ w )/t|(µ c1 , µ w1 )/t ∈ r ∧ (µ c2 , µ w2 )/t ∈ s∧ (µ c , µ w ) = lmin((µ c1 (t), µ w1 (t)), (µ c2 (t), µ w2 (t)))}.
2) The union operator: The union of r and s is a fuzzy bipolar relation, in which each tuple t is attached with a couple of degrees (µ c (t), µ w (t)). It is defined as follows:

r ∪ s = {(µ c , µ w )/t|(µ c1 , µ w1 )/t ∈ r ∧ (µ c2 , µ w2 )/t ∈ s∧ (µ c , µ w ) = lmax((µ c1 (t), µ w1 (t)), (µ c2 (t), µ w2 (t)))}.
3) The cartesian product operator: The cartesian product of r and s is a fuzzy bipolar relation, in which each tuple t is attached with a couple (µ c (t), µ w (t)). It is defined as follows:

r⊗s = {(µ c , µ w )/t⊕t |(µ c1 , µ w1 )/t ∈ r∧(µ c2 , µ w2 )/t ∈ s ∧ (µ c , µ w ) = lmin((µ c1 (t), µ w1 (t)), (µ c2 (t ), µ w2 (t )))},
where ⊕ is the operator of concatenation of tuples.

4) The projection operator: The projection of distinct tuples on attributes a 1 , ..., a k from r is a fuzzy bipolar relation of elements a 1 , ..., a k , defined as follows:

π a1,...,a k (r) = {(µ c1 , µ w1 )/ a 1 , ..., a k | (µ c1 , µ w1 ) = lmax t∈r∧t[a1,...,a k ]= a1,...,a k ((µ c1 (t), µ w1 (t))),
where t[a 1 , ..., a k ] is the value of the tuple t on a 1 , ..., a k .

5) The selection operator: The selection of tuples from r, based on the fuzzy bipolar condition (c , w ) is defined as:

σ(r, (c , w )) = {(µ c , µ w )/t|(µ c1 , µ w1 )/t ∈ r∧ (µ c , µ w ) = lmin((µ c (t), µ w (t)), (µ c1 (t), µ w1 (t)))},
where µ c (t) (resp. µ w (t)) is the degree of satisfaction of t with respect to the constraint c (resp. the wish w ).

6) The join operator: As in both boolean and fuzzy cases, the join operator is defined in the context of bipolarity by the combination of a cartesian product with a selection operator.

Bipolar SQLf basic statements:

A Bipolar SQLf basic statement is a fuzzy bipolar selection (restriction) of the following main format:

Select [distinct] [n|(t 1 , t 2 )|n, (t 1 , t 2 )
] attributes From relations [as alias] Where fcond 1 and if possible fcond 2 ;

The above statement is a simple selection of attributes from fuzzy bipolar relations based on the fuzzy bipolar condition (fcond 1 , fcond 2 ). It is also possible to express an n-top query by positioning the optional integer value n, which delivers the n best tuples.

(t 1 , t 2 ) ∈ [0, 1] 2 , such that t 2 ≤ t 1 , is a minimal threshold of satisfaction.
Example 3: We reuse the database defined in the example 2. The query "Find journeys which are fast and if possible not expensive", can be expressed in Bipolar SQLf as:

Select #Journey From Journey as J Where fast(J.duration) and if possible not expensive(J.cost); Due to the coherence property of fuzzy bipolar conditions, the above bipolar query is interpreted as "Find journeys which are fast and if possible (fast and not expensive)". We define, for example, the fuzzy predicate expensive as depicted in Fig. 2. Its negation is defined as follows: ∀x ∈ R + , µ notExpensive (x) = 1 -µ Expensive (x). Based on the definition of fuzzy predicates fast and not expensive, the query is evaluated over the relational table Journey and delivers the fuzzy bipolar relation Journey (Fast, notExpensive) . Table II is an example of extension of Journey (Fast, notExpensive) , and the delivered tuples are ranked, using the lexicographical order, from the best journey to the worst one, depending on their satisfactions to both the constraint fast and the wish not expensive: (1, 0.40)/12, (0.66, 0.50)/13, (0.33, 0.33)/10. p

IV. FUZZY BIPOLAR DLR-LITE

In this section, we introduce the fuzzy DLR-Lite and our contribution which consists in its extension to bipolarity called fuzzy bipolar DLR-Lite.

A. The Fuzzy DLR-Lite

The flexible querying system developed in [START_REF] Colucci | Semantic Web Information Management[END_REF], [START_REF] Straccia | Softfacts : a top-k retrieval engine for a tractable description logic accessing relational databases[END_REF] is based on a Description Logic extended to m-ary Relations, noted DLR-lite [START_REF] Calvanese | Data complexity of query answering in description logics[END_REF]. The knowledge base K of the fuzzy DLR-Lite consists of three components (O, F, A) defined as follows:

1) The facts component F: It contains extensions of relations of the form R(c 1 , ..., c m )[s], where R is an m-ary relation, c i,i=1,...,m are constants and s ∈]0, 1] is a degree of membership of tuple c 1 , ..., c m in the relation R.

2

) The ontology component O: the ontology defines the relevant abstract concepts of the application domaine by means of projection and inclusion axioms:

a) The inclusion axiom: The inclusion axiom has the form

(R 1 R 2 ... R l R r )[s]
, where l ≥ 1, R i,i=1,...,l are relations of the same arity and s ∈]0, 1] is the truth value of the axiom. This axiom states that if c is an instance of R i,i=1,...,l to degree s i,i=1,...,l then c is an instance of R r to degree at least s ⊗ s 1 ⊗ ... ⊗ s l , where ⊗ is a t-norm.

b) The projection axioms: They allow defining new concepts by extracting columns from other relations. Two kinds of projection axioms are defined: the simple projection and the restricted projection. The former is of the form ∃[i 1 , ..., i k ]R that expresses the projection of the relation R on the columns i 1 , ..., i k and the latter is of the form ∃[i 1 , ..., i k ]R.(Cond 1 ... Cond h ) which restricts the projection of ∃[i 1 , ..., i k ]R according to conditions Cond i,i=1...h of the form ([i]θv), where θ ∈ {<, ≤, =, =, >, ≥} and v is a value.

3) The abstraction component A: It forms the set of statements which connect concepts and relations, defined in the ontology component, to physical relational tables from a database. Simple and complex abstraction statements are defined as follows:

a) Simple abstraction statement: 4) The query language: A query in a fuzzy DLR-Lite knowledge base querying system consists of a conjunctive query with a scoring function to rank between answers. The query has the following form:

It has the form R 1 -→ R 2 (c 1 [t 1 ], ..., c k [t k ])[c s ]
q(x)[s] head ← ∃yR 1 (z 1 )[s 1 ], ..., R l (z l )[s l ], body OrderBy(s = f (s 1 , ..., s l )) scoring f unction , ( 3
)
where q is an m-ary relation, every R i is either an m i -ary relation or a concrete predicate of the form of (zθv), where θ ∈ {<, ≤, =, =, >, ≥} and v is a value of a datatype, x (resp. y) is a vector of distinguished (resp. non distinguished) variables, z i is a tuple of constants or variables in x or y, f is a scoring function which combines scores s 1 , ..., s l and s is a degree of satisfaction of a tuple x to the query q.

B. Extension of the Fuzzy DLR-Lite to Bipolarity

The bipolarity is integrated into the fuzzy DLR-Lite knowledge base in order to provide means to define and to handle fuzzy bipolar concepts and relations. For that purpose, we propose the extension of the fuzzy DLR-Lite to fuzzy bipolar conditions as described in what follows.

1) Extension of the facts component F: Facts in our case are delivered from fuzzy bipolar relations. A fact has the following form: R(c 1 , ..., c n )[s c , s w ], where c i,i=1,...,n are constants which form the tuple u = c 1 , ..., c n , R is a fuzzy bipolar relation defined by the fuzzy bipolar condition (c, w) and s c (resp. s w ) is the grade of satisfaction of the tuple u = c 1 , ..., c n with respect to the constraint c (resp. to the wish w). The consistency condition holds for each element u (s w ≤ s c ) and every tuple u s.t. s c (u) = 0 is discarded.

2) Extension of the abstraction component A: Simple and complex abstraction rules are both extended to fuzzy bipolar conditions as follows:

a) The simple abstraction rule:

It has the form R 1 -→ R 2 (c 1 , ..., c n )[s c , s w ],
where R 1 is a relation or concept (in the sense of the fuzzy DLR-Lite); R 2 is an m-ary fuzzy bipolar relation from a database, c 1 , ..., c n , with n ≤ m, are columns from R 2 and (s c , s w ) is the couple of grades of satisfaction attached to the tuple c 1 , ..., c n with respect to the fuzzy bipolar condition (c, w) which is used to define R 2 .

b) The complex abstraction rule: It has the form R 1 -→ (c 1 , ..., c n )[s c , s w ].sqlfb, where sqlfb is a Bipolar SQLf query which delivers n-ary tuples c 1 , ..., c n with their corresponding couples of grades (s c , s w ).

3) Extension of the ontology component O: The ontology component defines intersection axioms ( ), simple and conditional projection axioms and subsumption (inclusion) axioms ( ). They are extended to bipolarity as follows:

a) The intersection axiom:

It is denoted R 1 R 2 ,
where R 1 and R 2 are fuzzy bipolar relations defined resp. by fuzzy bipolar conditions (c 1 , w 1 ) and (c 2 , w 2 ). It is interpreted as a fuzzy bipolar relation R in which each tuple u is attached with a couple of degrees (s c (u), s w (u)), defined as follows:

(R 1 R 2 ) I ⇔ ∀u, (s c (u), s w (u)) = lmin((µ c1 (u), µ w1 (u)), (µ c2 (u), µ w2 (u))) (4) b) The simple projection axiom: It is denoted ∃[i 1 , ..., i k ]R.
It corresponds to the projection of columns i 1 , ..., i k from the fuzzy bipolar relation R of arity n such that n ≥ k. This projection extracts new facts from a database or from facts already defined in F.

This rule is interpreted as a bipolar projection of distinct elements of fields i 1 , ..., i k from the fuzzy bipolar relation R defined by the fuzzy bipolar condition (c, w). This axiom delivers a fuzzy bipolar relation made of the projected columns and its interpretation attaches a couple of grades for each retrieved tuples u = i 1 , ..., i k as follows: d) The subsumption (inclusion) axiom: This axiom is denoted (R l R r )[n 1 , n 2 ], where R l and R r are fuzzy bipolar concepts (relations) of the same arity defined respectively by fuzzy bipolar conditions (c l , w l ) and (c r , w r ).

∀u, (∃[i 1 , ..., i k ]R)(u) I = (s c (u), s w (u)) = lmax t∈R∧t[i1,...,i k ]=u ((µ c (t), µ w (t))) (5) c) The restricted projection axiom: It is denoted ∃[i 1 , ..., i k ]R.(Cond 1 ... Cond h ), where Cond i,i=1...h = ([i]θv) are conditions of selection of fields with θ ∈ {<, ≤ , =, =, >, ≥}, [i]
The interpretation of this axiom is based on the strengthened membership-based interpretation of fuzzy bipolar conditions. Let R be a fuzzy bipolar relation defined by a fuzzy bipolar condition (c, w). R is seen as a particular fuzzy set of desired elements such that each element u holds two grades:

• µ c (u) is a basic grade of membership to the fuzzy set,

• µ w (u) is a strengthening grade of membership that make distinguishable tuples having the same basic grade. In other terms, for tuples having the same basic grade, more a tuple satisfies the wish more it belongs to the fuzzy set. In this context, it becomes possible to extend the interpretation of the inclusion axiom to bipolarity as follows:

((R l R r )[n 1 , n 2 ]) I = ∀x ∈ R l , lmin((µ c l (x), µ w l (x)), (n 1 , n 2 )) ≤ lex (µ cr (x), µ wr (x)) (7)
where ≤ lex is the operator s. is interpreted as the minimal threshold of inclusion of R l in R r , and it processed by the following formula:

((R l R r )[n 1 , n 2 ]) I ⇔ lmin x∈R l ((if x ∈ R l then x ∈ R r ) I ) = (n 1 , n 2 ) (8)
The formula [START_REF] Colucci | Semantic Web Information Management[END_REF] is the extension to fuzzy bipolar conditions of the formula [START_REF] Dubois | Bipolarité dans un processus d'interrogation flexible[END_REF], which is introduced in [START_REF] Tamani | Extension d'une ontologie floue pour l'interrogation flexible d'une base de données embarquée[END_REF] to compute the truth value of inclusion axioms in the case of fuzzy DLR-Lite.

((R l R r )[n]) I ⇔ min x (R l (x) → Gö R r (x)) = n, (9)
where → Gö is the Gödel fuzzy implication.

The formula (if x ∈ R l then x ∈ R r ) I defines a kind of implication for bipolar elements. We propose the extension of the formula (2) to bipolarity to define an extended bipolar R-implication operator. The formula [START_REF]Bipolarity in flexible querying[END_REF] is, then, proposed:

I R ((a c , a w ), (b c , b w )) = lmax{(z, z ) ∈ [0, 1] 2 : z ≤ z∧ lmin((a c , a w ), (z, z )) ≤ lex (b c , b w )}. (10)
Finally, we obtain the following interpretation for (if ( x ∈ R l ) then (x ∈ R r )) in the bipolar context:

(if ( x ∈ R l ) then (x ∈ R r )) I = (1, 1) if (µ c l (x), µ w l (x)) ≤ lex (µ cr (x), µ wr (x)), (µ cr (x), µ wr (x)), else. (11)
It is easy to prove that formula [START_REF] Lewis | Using ontology to integrate railway condition monitoring data[END_REF] verifies properties of Rimplications extended to bipolarity (monotonicity, neutrality, exchange, identity and ordering).

It is also easy to show that formula [START_REF] Lewis | Using ontology to integrate railway condition monitoring data[END_REF] is compatible with formula [START_REF]Bipolarity in flexible querying[END_REF] and it defines an extension of the Gödel fuzzy implication to fuzzy bipolar conditions. Indeed, in the formula [START_REF]Bipolarity in flexible querying[END_REF] 4) Extension of the query language to bipolarity: A query addressed to a fuzzy bipolar DLR-Lite knowledge base consists in a conjunctive query which is extended to bipolarity. The main form of such queries is:

q(x)[s c , s w ] head ← ∃yR 1 (z 1 )[s c1 , s w1 ], ..., R l (z l )[s c l , s w l ], body OrderBy((s c , s w ) = f ((s c1 , s w1 ), ..., (s c l , s w l ))) scoring f unction . (12)
Symbols q, x, y, R i and z i are the same as those used in the fuzzy case (formula (3)). The main form of conjunctive queries are extended to bipolarity to handle couples of degrees expressed at each atom of the body. These couples of degrees are finally aggregated by the bipolar scoring function f , which can be simplified to the lmin operator, due to the conjunctive nature of the query. [s c , s w ] corresponds to couples of degrees (s c , s w ) returned by f and attached to the resulting tuples. Syntactically, couples of the form [1, 1] could be omitted.

V. FUZZY BIPOLAR DLR-LITE APPROACH FOR FLEXIBLE QUERYING: A SAMPLE APPLICATION

We show the relevance of the fuzzy bipolar DLR-Lite based approach through the design and the querying of a sample application in the field of multimodal transportation networks.

The relational database previously introduced in the example 2 is reused. For each table from the database, a relation is defined in the ontology component and an abstraction statement is associated to it in the abstraction component. In addition, we define the following relations in the ontology:

Step ≡ ∃ [START_REF] Bosc | SQLf: A relational database langage for fuzzy querying[END_REF][START_REF] Bosc | Extending relational algebra to handle bipolarity[END_REF][START_REF] Bouchon-Meunier | Fuzzy sets in approximate reasoning and information systems, ser. The Handbook of fuzzy sets[END_REF][START_REF] Calvanese | Data complexity of query answering in description logics[END_REF][START_REF] Carles | Modèle conceptuel de données générique pour la représentation des réseaux intermodaux[END_REF]Journey is a simple projection of Journey relation on columns #link, source, destination, departure and arrival.

AreaStation ≡ ∃ [START_REF] Bosc | Base de données -Gradualité et imprécision dans les bases de données Ensembles flous[END_REF][START_REF] Bosc | SQLf: A relational database langage for fuzzy querying[END_REF][START_REF] Bosc | Extending relational algebra to handle bipolarity[END_REF][START_REF] Bouchon-Meunier | Fuzzy sets in approximate reasoning and information systems, ser. The Handbook of fuzzy sets[END_REF]Station defines stations such as railway station and airport. It is a simple projection of Station relation on columns #station, designation, city and type.

Let RailwayStation, noted RailStation, and Airport be two concepts, extracted from the AreaStation by the following restricted projection axioms:

• RailStation ≡ ∃[1]AreaStation.(([3] = "Railway")) • Airport ≡ ∃[1]AreaStation.(([3] = "Airport"))
Let PreferredRailStation be a fuzzy bipolar concept defined by the fuzzy bipolar condition (Large, NearCenter) that corresponds to railway stations which are large and if possible (large ∧ located near to the town center). This concept could be expressed by the following abstraction rule:

PreferredRailStation → (#station).Select #station From RailStation Where LargeRail(#station) and if possible NearCenter(#station); where LargeRail (resp. NearCenter) is the membership function of the fuzzy predicate LargeStation (resp. near b) Bipolar SQLf generator: It performs user queries transformation (relevant substitutions based on the inclusion axioms) using the fuzzy bipolar DLR-Lite knowledge base and generates equivalents bipolar SQLf queries to be evaluated by the right side of the system. c) Fuzzy bipolar DLR-Lite knowledge base: it defines the knowledge base of the system (relations, facts, axioms).

The right side deals with the mediation issues for the combination of data fetched from remote databases, and processes bipolar SQLf queries received from the bipolar SQLf generator. It is composed by the following modules: d) Mediation: It evaluates the received bipolar SQLf queries by retrieving the needed data from relevant remote databases, and by translating queries using the right vocabulary in compliance with a domain ontology (taxonomy).

e) Taxonomy: It is based on the Ontology Transportation Networks (OTN). Depending on the remote information system to query, this module provides the Mediation module with the right vocabulary to use in the query translation process.

VI. CONCLUSION AND FUTURE WORKS

To be able to express complex preferences in user queries, addressed to distributed and heterogeneous information systems, we have proposed a new flexible querying approach in which the fuzzy DLR-Lite is extended to both fuzzy bipolar conditions and Bipolar SQLf language.

We showed through an application the relevance of this approach in the field of the multimodal transportation networks.The bipolar fuzzy DLR-Lite can go beyond relational tables by inferring implicit knowledge and by summarizing data, while improving widely the expressiveness of the system. So, it becomes possible to express complex preferences of the form "c and if possible w" and to handle them within a reasoning system to answer user queries.

A software architecture for such a querying system has also been proposed. The common formal framework (fuzzy sets theory) of both Bipolar SQLf language and fuzzy bipolar DLR-Lite allows their combination in a consistent way.

An implementation of the system is in progress. It is based on algorithms for query evaluation which are introduced in [START_REF] Straccia | Softfacts : a top-k retrieval engine for a tractable description logic accessing relational databases[END_REF]. As future works, we plan the study of performances of our approach in terms of response time and quality of answers, i.e. in which extent answers meet user expectations.

Fig. 1 :

 1 Fig. 1: The membership function of the fuzzy predicate fast.

Fig. 2 :

 2 Fig. 2: The membership function of the fuzzy predicate expensive.

  is a column number corresponding to a field from the fuzzy bipolar relation R, defined by the fuzzy bipolar condition (c, w), v is a value and is the operator and. This rule corresponds to a combination of a bipolar projection and a bipolar selection of distinct tuples from R. It delivers a fuzzy bipolar relation formed by the retrieved tuples u = i 1 , ..., i k . The attached couples of grades are defined by the formula (5) applied on the retrieved tuples limited to those satisfying the boolean condition Cond = (Cond 1 ... Cond h ). ∀u, (∃[i1, ..., i k ]R.(Cond1 ... Cond h ))(u) I = (sc(u), sw(u)) = lmax t∈R∧t[i 1 ,...,i k ]=u ((µc(t), µw(t))), if Cond I (u) = 1 (6)

  t. (a, b) ≤ lex (a , b ) ⇔ lmin((a, b), (a , b )) = (a, b). The couple of degrees (n 1 , n 2 )

  , when (a c , a w ) ≤ (b c , b w ), any valid couple of degrees (z, z ) from [0, 1] 2 satisfies lmin((a c , a w ), (z, z )) ≤ (b c , b w ) and the greatest couple of degrees is (1, 1). When (a c , a w ) > (b c , b w ), couples of degrees which satisfy lmin((a c , a w ), (z, z )) ≤ (b c , b w ) are (z, z ) such that (z, z ) ≤ (b c , b w ); then, the greatest couple of degrees (z, z ) corresponds to (b c , b w ).

Fig. 3 :

 3 Fig. 3: Software architecture of the flexible querying system.

TABLE I :

 I Extension of the fuzzy relation fastJourney.

	#Journey cost ($) duration (h) ... µ FastJourney
	12	80	2	...	1
	13	75	3	...	0.66
	10	65	4	...	0.33

TABLE II :

 II Fuzzy bipolar relation Journey (Fast, notExpensive) .

	#Journey cost ($) duration (h) µ Fast µ FastJourney ∧
					µ notExpensive
	12	80	2	1	0.40
	13	75	3	0.66	0.50
	10	65	4	0.33	0.33

  which states that the k-ary relation R 1 of the ontology component is mapped into the projection on columns c 1 of type t 1 , c 2 of type t 2 , ..., and on c k of type t k of the m-ary table R 2 . The score of these tuples is provided by the column c s . b) Complex abstraction statement: It has the form R 1 -→ (t 1 , ..., t k )[c s ].sql, where R 1 is a k-ary relation of the ontology component and sql is an SQL query which returns the k-ary tuples t 1 , ..., t k ranked in decreasing order of scores provided by column c s .

ACKNOWLEDGMENTS

We warmly thank the Brittany region, the department of Côtesd'Armor and the National Agency for Research (AOC Ref. ANR-08-CORD-009) for financing this work.

We can enrich the system with data from web services or other information systems. For example, we query a web service about restaurants, from which we define the fuzzy bipolar relation hasPreferredRestaurant, which corresponds to places which have good and if possible cheap restaurants as railway stations, hotels, airports, shopping centers (see table IV). Instead of saving the table IV on the facts component, we focus only on data related to our transport system. For efficiency reason, we represent them as data summary using the two following bipolar inclusion axioms:

where (n 1 , n 2 ) (resp. (n 3 , n 4 )) is the minimal couple of degrees of inclusion of PreferredRailStation (resp. PreferredAirport) in the set of hasPreferredRestaurant. These couples of degrees are processed based on formulas [START_REF] Chen | Nimbletransit: Public transportation transit planning using semantic service composition schemes[END_REF], ( 8) and ( 11):

(n 1 , n 2 ) = lmin((0.5, 0.4) → (0.6, 0.6),

(1, 0.7) → (0.8, 0.7), (0.6, 0.5) → (0.8, 0.3)) = lmin((1, 1), (0.8, 0.7), (1, 1)) = (0.8, 0.7). (n 3 , n 4 ) = lmin((1, 0.8) → (0.7, 0.6), (0.5, 0.3) → (0.5, 0.4), (0.4, 0.4) → (0.5, 0.3)) = lmin((0.7, 0.6), (1, 1), (1, 1)) = (0.7, 0.6).

Even if we have no restaurant relation in the facts component and without querying any external database, the system can answer queries about journeys which involve preferred restaurants, such as "go from Lannion to Paris in 2 steps with a preferred restaurant at the midway", which is expressed by: q(x 1 , x 2 )[s c , s w ] ← ∃a, b, c, d, e, f, g, h, i. AreaStation(a, e, "Lannion"), Step(x 1 , a, c, d, g),

Step(x 2 , c, b, d, h), AreaStation(b, f, "P aris"), hasPreferredRestaurant(c, i)[s c1 , s w1 ], OrderBy((s c , s w ) = (s c1 , s w1 )).

To evaluate this query, the atom hasPreferredRestaurant is substituted by the left-hand side of each inclusion axiom in which it appears in its right-hand side. Each substitution generates a new query. Two subqueries are, then, obtained: b,c,d,e,f,g,h.

AreaStation(a, e, "Lannion"), Step(x 1 , a, c, d, g),

Step(x 2 , c, b, d, h), AreaStation(b, f, "P aris"),

OrderBy((s c , s w ) = lmin((s 1 , s 2 ), (0.8, 0.7))).

AreaStation(a, e, "Lannion"), Step(x 1 , a, c, d, g)

Step(x 2 , c, b, d, h), AreaStation(b, f, "P aris"),

OrderBy((s c , s w ) = lmin((s 3 , s 4 ), (0.7, 0.6))).

The scoring function in both q 1 and q 2 takes into consideration the inclusion axiom used in the substitution process. Both q 1 and q 2 are to be translated into Bipolar SQLf queries, in order to be evaluated over the facts component.

For example, query q 1 (x 1 , x 2 ) is expressed as follows: Select R1.#step, R2.#step From Step as R1, Step as R2 Where R1.destination = R2.source and R1.source in (Select #station From Station Where city = 'Lannion') and R2.destination in (Select #station From Station Where city = 'Paris') and R2.Source in (Select #station From PreferredRailStation);

The answers are ranked depending on couples of degrees delivered by lmin((s 1 , s 2 ), (0.8, 0.7)), where (s 1 , s 2 ) is the couple of degrees attached to each R2.Station corresponding to a preferred rail station. The query q 2 (x 1 , x 2 ) is similarly processed and the obtained sets of answers are merged, while respecting the lexicographical order.

The Main Architecture of the System:

For implementation purpose, we propose a software architecture which consists of a combination of a reasoning system (based on the fuzzy bipolar DLR-Lite) and a flexible information system (based on Bipolar SQLf).

As depicted in Fig. 3, the system is of two parts. The left side consists of the following modules: a) Query interpreter: It provides the user with interface that receives queries and returns answers.


Marie Leroy-Collombel

Rising grammatical awareness in a French-speaking child from 18 to 36 months: uses and misuses of possession markers

Keywords: possession, possessive determiners, language acquisition, nonstandard forms

 calls 'emergent categories'. We focus our longitudinal study on a French speaking child's use of possession markers (Anaé, Paris Corpus), and her creative nonstandard productions (la poupée de moi for ma poupée/my doll). We provide a detailed analysis of the ways in which she moves between a global strategy thanks to which she locates, identifies and uses whole blocks or constructions without analyzing them, and a more analytic strategy that parallels her progressive mastery of the semantic and syntactic complexity of grammatical morphemes.

ISSUES AT STAKE AND STATE OF THE ART

Theoretical framework -Construction grammar

Young children learn the set of linguistic conventions used around them through interactions with members of their community. These conventions consist of hundreds of thousands of words and constructions, including the abstract constructional patterns that speakers of each language have grammaticalized.

Scientists have differed in their accounts of this acquisition process (see introduction to this special issue). In more recent years, a number of studies in developmental psychology, linguistics and cognitive science have suggested that children have at their disposal very powerful learning mechanisms integrated with social-cognitive skills that enable them to learn from the input. Theories representing this viewpoint come mainly from cognitive or functional linguistics and take a usage-based approach to language development. Such approaches take language structure to emerge from language use (Langacker 1987;Croft 1991;Goldberg 1995, Givon 1995[START_REF] Bybee | Morphology: A study of the relation between meaning and form[END_REF][START_REF] Tomasello | Constructing a language[END_REF]. On this view, adult linguistic competence is viewed as based on concrete pieces of language that undergo generalizations. Children are seen as using meaning and function to acquire linguistic constructions in context. Crucially, linguistic competence does not correspond to the mastery of a formal grammar based on semantically empty rules, but rather on the mastery of an inventory of meaningful linguistic constructions that each pair meaning with form. For Tomasello (2003), toward the beginning of the acquisition process these constructions are dominated by isolated lexically specific 'item-based' constructions.

Acquiring grammar is viewed as the gradual generalization of such itembased constructions into more abstract patterns based on communicative function. More concretely, children first extract tangible elements (words, morphemes, unanalyzed strings, constructions) by identifying their communicative function; then, based on recurring items with similar functions across various situations in the input, they create more or less abstract categories and constructions.

In this study of the acquisition of constructions marking possession, we will illustrate how at the beginning of acquisition, some of the children's productions do not resemble adult language. Rather, the actual structures and representations involved in children's productions seem to change across different stages of the acquisition process. The main factors affecting language development in our view are 1) communicative intention, 2) frequency and saliency in the input, but also 3) children's very own affective, social and practical concerns. Those three main factors come into play at various degrees according to the specific linguistic item that is being acquired.

The development of the marking of possession serves as an especially interesting locus for a study of the way these factors intermingle. As shown by Marcel [START_REF] Cohen | Sur les langages successifs de l'enfant, in Mélanges linguistiques offerts à M. J. Vendryes par ses amis et ses élèves[END_REF], children seem to construct their own transitory systems in the course of language acquisition. Since the beginning of the 20 th century, a number of scientists have tried to explain the mismatches between the input and some of the children's peculiar productions. [START_REF] Karmiloff-Smith | From meta-processes to conscious access-Evidence from children's metalinguistic and repair data[END_REF] makes a theoretical proposal in terms of continual revision of form-function mappings illustrated by examples of how the child's system does not directly reflect the characteristics of the input.

Children produce a number of nonstandard forms at the beginning of the learning process [START_REF] Rondal | Développement du langage oral[END_REF]; these are typical of what [START_REF] Clark | Emergent categories in first language acquisition[END_REF] has named 'emergent categories'. Some of those forms illustrate how children are able to move beyond frozen expressions thanks to productive analyses of the input, and might create non-adult constructions in the process of learning form-function pairings.

In the course of this study of a French-speaking child's forms used to mark possession, we will analyze her 'errors' or nonstandard forms, which have been referred to as 'barbarisms' by [START_REF] Egger | Observations et réflexions sur le développement de l'intelligence et du langage chez l'enfant[END_REF] or 'incorrect forms' by Buhler (1926). We view these forms as revealing signs of her cognitive processing of the grammatical categories and the semantic features at play in the marking of possession.

Expressing possession in French

Even though possession is quite a stable notion cross-culturally, and all languages seem to have means to encode that notion linguistically, there is no universal linguistic structure common to all possessive constructions. These constructions often express other concepts, and various forms used to express other concepts might be borrowed to express possession [START_REF] Heine | Possession : Cognitive sources, forces, and grammaticalization[END_REF]. It is therefore quite difficult to analyze the use of possessive constructions in a large corpus (especially spontaneous adult-child conversations), since automated searches are not easy when the forms are diverse unless the notion of possession is specifically tagged.

Possession in French may be marked in many ways, either locally in nominal expressions (attributive possession) or more globally in the utterance using a predicate, and particularly the verb avoir / to have (predicative possession). In this paper, we will focus on attributive possession. Expressions involving mere juxtaposition are much less frequent in French than in English (as in New York buildings). Just as in English, however, French has not grammaticalized the distinction between inalienable (inherent) possession (son nez -her nose) and alienable (noninherent) possession (sa voiture -her car). (sa voiture -her car). If the information is not known (new), the speaker will use [N1 de N2] (la voiture de Martial -Martial's car). A third grammatical marking of possession is possible, with the use of an anaphoric possessive pronoun (la sienne -hers), when not only the possessor, but also the possessed object, are given information in the discourse context, though this pattern is less frequent in adult data.

For first and second-person possessors, the construction used is [Possessive Determiner + Noun] as in ma maison (my house), ton sac (your bag), since the reference to the possessor is constructed deictically. An emphasis can be expressed with the addition of [à + tonic pronoun] to the [possessive determiner + noun] structure, as in ton papa à toi (your own dad). This phrase can also stand by itself: the answer to questions such as c'est à qui ? (whose is it?) can be à moi (it's mine).

In French, as opposed to English, agreement of possessive determiners marks both the possessed object and the possessor. These determiners must agree in number and gender with the noun they determine (referring to the possessed object/s) and in person with the possessor(s).

According to [START_REF] Rondal | Développement du langage oral[END_REF], their use is guided by an economy principle that reduces expressions such as 'X belongs to Y' or 'X is part of Y' to [Poss Det + X]. The choice of a possessive determiner is the result of four combined operations: 1) the selection of personal reference according to the interpersonal context of the verbal exchange (first person when the speaker is the possessor, second person when the interlocutor is the possessor, third person when the possessor is neither the speaker nor the interlocutor); 2) the selection of the agreement in gender and number according to the gender and number of the possessed object; 3) the selection of the agreement in number according to the number of the possessor (singular or plural); 4) the initial phoneme of the possessed element (in son amie, the possessive determiner is masculine because the noun begins with a vowel).

Possessive determiners can therefore be analyzed as an amalgam of several grammatical and phonological features. They seem to be simple markers on the surface but are quite complex at different levels.

How children acquire the marking of possession

Children start talking about possession from an early age (Brown, 1973), for it plays a central role in their daily lives [START_REF] Eisenbeiss | Learning to Encode Possession[END_REF]. [START_REF] Slobin | Cross linguistic evidence for the Language-Making Capacity[END_REF] suggests that they start out with a higher-level category of location that includes possession. But [START_REF] Bowerman | What shapes children's grammars?[END_REF] argues that nothing proves which of location or possession is more basic. Children might focus on either of these semantic relations at different times.

Since French subdivides the semantic notion of possession into several constructions according to the various factors identified above (discourse status, person, gender, number), children learning French often produce 'errors' while in the process of sorting out these factors [START_REF] Clark | Emergent categories in first language acquisition[END_REF]. More precisely, children have a preference for seeking consistent ways to mark certain distinctions as they create their own form-function pairings [START_REF] Slobin | Cross linguistic evidence for the Language-Making Capacity[END_REF][START_REF] Budwig | A Developmental-Functionalist Approach to Child Language[END_REF]. They may thus pick a single construction to express all instances of a category they mean to target. This may lead them to overgeneralization errors before they acquire all the conventional forms for that category and learn how to express finer distinctions. Besides, constructions might be acquired in stages with omission of certain grammatical markers. This leads to paratactic uses such as chemise papa (shirt daddy) instead of la chemise de papa (daddy's shirt).

According to [START_REF] Rondal | Développement du langage oral[END_REF], French-speaking children don't use possessive determiners at first; they express possession in three different The construction [N + à moi] is quite frequent in some registers of French used in families from the working class [START_REF] Grégoire | L'apprentissage du langage, les deux premières années[END_REF], but the child in the data we have collected is not surrounded by that kind of input.

Those uses are not standard in her environment and could be considered to express possession as an emergent category (a single construction generalized to nonstandard uses). However, since children are so sensitive In this paper, we examine the developmental pattern in our longitudinal data and compare it to the relatively rare mentions of the acquisition of possessive markers in the literature, as well as to the marking of possession in the child's maternal input. We mostly focus on the nonstandard forms used by the child1 and on her acquisition of the conventional standard alternatives.

DATA, METHOD AND RESULTS

Data and Method

This paper is focused on Anaé's longitudinal data (see description of the Paris corpus, this issue). We used the three types of data at our disposal to study Anaé's linguistic development: 1) transcriptions in CHAT format from 1;06 to 4;0 linked to video; 2) the mother's diary from birth to 4;00; 3) detailed coding of eleven one-hour sessions from 1;07 to 2;11.

We coded all utterances that contained a construction marking possession. In the child's language, we found the following standard constructions embedded in her utterances:

-[Poss Det + N]: possessive determiner + Noun (sa maison -her house) -[N1 de N2]: Noun 1 + preposition de + Noun 2 (la maison de Dora -Dora's house)

We also found the following constructions in isolation:

-C'est à + tonic pronoun (c'est à moi -it's mine) -A + tonic pronoun (à moi -mine).

Very few of these constructions are used in the second and third person.

Nonstandard constructions were also coded:

-Le N de moi / Le N à moi
This construction was only produced in the first person.

The coding was done using Excel based on the following features: 1) Dataset, 2) Speaker, 3) The whole utterance, 4) Presence of a possessive structure (Yes/No), 5) Noun determined, 6) Possessive structure, 7) Concrete possessive phrase, 8) Personal reference of the possessor.

Table 1: Example of Coding

Examples such as la poche de ma maman (my mummy's pocket) were coded twice, once for the [N1 de N2] structure, once for the [Poss Det + N structure] (ma maman -my mummy).

Distribution of the various structures in the data

Possessive structures in the input

Our coding enabled us to count the number of occurrences of each construction in the input, in order to compare mother and child's uses.

Overall, we found 271 occurrences of possessive constructions in the child's verbal productions and 552 in the mother's input in the recordings between 1;07 and 2;11. In (1), the mother introduces the family relationship between Sacha and the two discourse objects mentioned in the first part of the utterance. The relation constructed with [N1 de N2] enables the mother to clarify who the discourse-objects are, and to locate them according to a discourse-object whose identification seems to be easier for Anaé. Sacha is a friend she knows very well, but she probably does not remember the names of his parents.

(2) -Anaé 2;05 MOT: le bébé kangourou, il est dans la poche de sa maman. [the baby kangaroo is in his mummy's pocket].

In (2), there are two possessive constructions in the same utterance 1) la poche de sa maman (his mummy's Pocket) (N1 de N2) and 2) sa maman (his mummy) (Poss Det + N). In the first construction, the relationship between poche (pocket) and maman (mummy) is presented for the first time 2 The translation is literal in order to maintain the syntax of the French utterance. in discourse. It is new information and gives the location of 'bébé kangourou'. The second construction is made possible because the discourse-object has been mentioned in the first part of the utterance and is pointed back to in the second part with the third-person possessive determiner sa (his), which also plays an anaphoric function.

In 6 occurrences of the mother's 367 uses of the construction with a possessive determiner (0.01%), she adds the structure [à + tonic pronoun], which expresses a contrastive value.

(3) -Anaé 2;03

The mother is moving Anaé's toy horse. The other horses belong to her brothers.

MOT: Voilà ton cheval à toi qui arrive. [look, your own horse is coming]

The marking of possession in the mother's input is therefore quite straightforward and follows the rules of the French grammatical system, with some rare occurrences of a construction specific to spoken French to mark contrastive possession.

Possessive structures in the child's data

As shown in figure 2, possession is marked quite early in Anaé's data with a variety of forms. There is only one case of a paratactic juxtaposition of two nouns, which are linked semantically by possession in the data. At 1;09, Anaé holds out a plate to her mother (who is setting the table) and says c'est l'assiette (pause) Ael (It's Ael's plate, Ael is one of her elder brothers).

Juxtaposition therefore is not a privileged means to express possession, even though the child does juxtapose nouns for other types of relations (carte Dora meaning a card displaying a picture of the character Dora; sirop pêche instead of 'sirop de pêche', meaning a peach-flavored syrup).

The child therefore mainly expresses possession with overt forms and in the first person (there is only one instance of à toi -yours, and one instance of à elle -hers -throughout the corpus with a possessive sense).

She uses the construction à moi and [N + à moi] a lot until 2;03. This construction is probably derived from the massive use of à moi in the input (by her mother, but also those by her two elder brothers) in situations of turn-taking (where 'à moi' means 'my turn'). We did not consider those uses to be marking possession, even though these recurrent situations are strongly related to the marking of identity (and otherness) expressed in possessive relations. Example (4) illustrates the nonstandard possessive use of [N + à moi].

(4) -Anaé 2;00: c'est la puzzle à moi [it's my puzzle]. The recording at 2;00 is particularly interesting to study the blossoming of several constructions through Anaé's production of various nonstandard forms. In (4), on top of the nonstandard use of [N + à moi], the gender error (la puzzle instead of 'le puzzle') may be a sign that Anaé is constructing her utterances without always rigidly replicating the input and makes the wrong choice in a paradigm of determiners. Anaé proves to be particularly prone to gender errors (Leroy-Collombel, 2010) such as un fleur (a flower), le poupée (the doll), le poule (the chicken), la ballon (the ball), une oiseau (a bird) but with non-systematic gender assignment.

Once possessive determiners are acquired (at 2;02, when the child's MLU is 3.0), they become dominant, as is the case in the input. Anaé's uses of possessive constructions are therefore quite similar to Brown's description of the use of possessive constructions in English. Genitive constructions and possessive determiners are stabilized in the course of what Brown refers to as 'stage 3', (between MLU 2.5 and 3) (Brown, 1973).

However a few occurrences of nonstandard forms such as [N + de moi] are also produced such as example ( 5) and ( 6).

(5) -Anaé 2;08: le caddie de moi il était bougé aussi. [my shopping cart was also moved].

(6) -Anaé 2;08 ça c'est mon truc de moi [that's my own thing].

In example (6,) Anaé superimposes the [Poss Det + N] structure (mon truc -my thing) and her nonstandard [N + de moi] structure used to emphasize possession. This might also demonstrate that she has not entirely analyzed the role of mon (my) as being both a masculine determiner and a marker of first-person possession, so that she resorts to a second grammatical marker to express her intended meaning. There are two occurrences of such a phenomenon: one at 2;08 and one at 2;11.

The [N1 de N2] construction emerges at 2;02 and is used for all first mentions with a third-person possessor. The possessor can eventually (but rarely) be the child herself, in constructions such as la tête de Anaé (Anaé's head). As shown by Caët (this issue), the child sometimes uses her own name to designate herself in specific pragmatic conditions, which creates a certain distance between herself as a speaker and herself as a discourseobject. This particular case is interesting, because when she produces this utterance, she is playing a game where she is alternatively hiding and showing her head. This use of her own name in a possessive construction enables her to take on the other's perspective and might also, as [START_REF] Deutsch | Form and function in the development of possessives[END_REF] note for English, mark less 'pragmatic control' as opposed to the use of my. At 2;02, [N1 de N2] alternates with [Poss Det + N] with a very clear advantage for the latter -92.3% in the whole longitudinal data. There are 5.8% [N1 de N2] constructions and 1.9% à moi and c'est à moi productions in isolation. The nonstandard productions are quite frequent up to 2;00 and then become sporadic in the rest of the data.

It is interesting to note that when Anaé starts making a productive use of possessive determiners at 2;02, out of the 17 different nouns she produces with those determiners, 10 are also used in other contexts either bare (with no determiner), either with filler syllables or other determiners. The possessive determiner may therefore very well be chosen in a paradigm of alternative determiners, and not learned with the noun as a frozen expression or a 'noun island'.

(7) -2;02 mon bébé / le bébé / ce bébé / [e]3 bébé (my baby / the baby / this baby / [e] baby) mon chien / le chien / ce chien (my dog / the dog / this dog)

In the same recording, the mother uses five different determiners (ton, le, un, les and ces) with bébé and seven different determiners (le, mon, ce, son, un, des and les) with chien. It is therefore interesting to note that at 2;02, similarly to her mother, the child already has a paradigm of determiners at her disposal for a number of nouns. She has moved beyond frozen expressions (a noun always used with the same determiner) and shows productivity and a sense of grammatical categories in her use of determiners in nominal expressions.

Possessive determiners among nominal determination markers

In the mother's data (figure 3), definite articles are used predominantly (with an average of 53%), then come indefinite articles (29%), possessive determiners (14.6%) and very few demonstrative determiners (3%).

Although the distribution and number of determiners depends on context (for example, there are more indefinite articles when looking at picture books and labeling objects), the number of possessive determiners (solid grey line in Figure 4) remains relatively constant throughout the data.

Figure 4 Even at the end of the data, as shown in Figure 4, the child uses fewer tokens than her mother (73 tokens at 1;10 when the mother uses 482, and, at 3;01, 146 tokens for the child and 229 for the mother). The child uses filler syllables up to the age of 2;03. As soon as they emerge at the age of 2;0, she predominantly and constantly uses definite articles, just like her mother (53.4% of all her determiners throughout the data are definite articles). The number of indefinite articles increases at 2;05, a session in which they are more numerous than definite articles (possibly due to the prevalence of a labeling activity). Possessive determiners start being used a little bit later and stabilize at 2;02 with an average of 12%, which is quite similar to adult use. Demonstrative determiners are very infrequent and only start being used as of 2;05 (1% of overall use of determiners). The child's profile is therefore very similar to her mother's. However, she has a tendency to use infrequent forms even less often than her mother, which is quite typical of that age range [START_REF] Morgenstern | A la source du futur : premières formes verbales dans les productions spontanées de deux enfants français de 18 mois à 3 ans[END_REF].

Personal reference of possessors in the data

In the mother's data, as shown in Figure 5, there are very few mentions of first-person possessors (7.2%). Second-person references are used most (54.4% overall). She therefore speaks mostly of her child as possessor.

Many of the third-person references (which total 38.4%) are made during games with various dolls and figurines, or during book reading, but family members and friends are also mentioned. Very few references in first, second and third person are plural possessive determiners (16 occurrences out of 680 nouns, which represent 0.02%).

Figure 6

The distribution is very different in the child's data (figure 6). There is a majority of reference to self as possessor (45.3%), but mostly in the first half of the data. Reference to the addressee as possessor is quite low (5.8%) and a parallel could be made with reference to the addressee as grammatical subject (see Caët, this issue). The rate of reference to third-person possessors is quite similar to the rate found in the input: both mother and daughter refer a lot to toys and characters in books, friends and family.

There were no uses of plural possessive determiners before three4 .

As far as personal reference of possessives is concerned, the child thus does not directly replicate the frequency of the actual forms observed in the input. However, it should be clear that both the mother and the child refer predominantly to the child as possessor. A pragmatic feature -focus on the child in adult-child interaction -therefore influences the person of expressed possessors. Personal reference can thus be considered 'usage-based' in the sense that it reflects the typical pragmatic experience in which possession is expressed in interaction.

FORMS WITH NONSTANDARD USES

During the first half of the data, possession in the child's data is mainly expressed by nonstandard constructions. As shown in graph 2, the use of à moi in [Noun + à moi] is quite significant between 1;07 and 2;03. This construction enables the child to mark the possessive function even before she has the whole range of possessive constructions of the target adult language at her disposal. She has co-opted this particular form for marking the possessive function before producing all the conventional forms. As characterized by Slobin (1973: 184): 'New forms first express old functions, and new functions are first expressed by old forms'. The early use of this construction may demonstrate how important the semantic notion of possession is for children. The nonstandard use of the form à moi represents 50% of Anaé's productions of possessive marking from 1;10 to 2;00, as shown in figure 2.

We hypothesize that this particular usage might be connected to the general tendency French children have around two years old, to overuse the word moi. At 2;00, 39% of Anaé's productions of possessive constructions are either à moi or c'est à moi. When added to the 50% uses of nonstandard [N + à moi], we find 89% uses of moi in possessive constructions.

We counted the uses of moi in the whole data and found an increase at 2;00 as shown in Figure 7 (60% more uses than at 1;09 and at 2;01).

Figure 7

The term moi is found quite frequently in the data. At two years old, it is used in over 10% of Anaé's productions, and is the third most frequent word at 1;9, 1;10 and 2;00. In the subsequent recordings, moi is no longer one of the 10 most frequent words.

Since the overuse of moi is so important at the age of 2;00, we focused our analyses on this recording. We note that Anaé expresses a great amount of contrastive agency, to explain either that she is capable of doing things by herself, or that she wants to do them as opposed to others. Our hypothesis is that the overuse of the term moi has 'contaminated' the marking of possession. Our coding (Figure 2) shows that out of 24 constructions marking possession at 2;00, 22 contain à moi and two are instances of the [Pos Det + Noun] structure (mon jeu; mon papa -my game; my daddy).

Moreover, out of 60 occurrences of moi, 26 are used in à moi constructions, of which 22 mark possession. We took a closer look at these uses of à moi and noticed that 14 nonstandard forms combined a noun with à moi and would have been replaced by a possessive determiner in adult language.

(8) -Anaé 2;00 CHI : oh y'a puzzle à moi (when she sees the puzzle on the À moi is not only used to mark the possessor of the discourse-object with great pragmatic force [START_REF] Deutsch | Form and function in the development of possessives[END_REF]), but it is also extended to other functions, as shown in the following examples.

(9) -Anaé 2;00 CHI : le loup a pas mangé à moi (she puts her hands on her chest). [ the wolf has not eaten 'of me'].

The patient should be expressed by an object pronoun, me or m' in front of a vowel (le loup ne m'a pas mangée -the wolf has not eaten me).

(10) -Anaé 2;00.

CHI : [se]

[a]6 donné à moi [it has given to me].

The à moi construction is used here instead of the object pronoun (m'a donné -given to me), marking the beneficiary. Anaé's production brings the underlying grammatical structure to the surface as in clusters such as de le chien (of the dog) instead of the amalgam 'du' du chien.

(11) -Anaé 2;00 CHI : c'est [a] cadeau à moi [it's a gift of me]

In this example, Anaé uses à moi instead of pour moi (for me): her comment is produced just when the observer is about to give her gift, so she is the beneficiary in this example as well.

In their study of acquisition of French as a second language, Chaudenson, Véronique and Vally (1986) also noticed this overuse of à moi to mark possession. According to Chaudenson (2004), the use of the post-posed personal pronoun demonstrates that the child creates her own rule, based on two types of utterances in the input: 1) possessive constructions with a proper name (le X de Y); 2) isolated utterances such as c'est à moi.

By blending these two structures, the child produces X de/à moi. [START_REF] Grégoire | L'apprentissage du langage, les deux premières années[END_REF] linked his son's production of [Noun + à moi] to a more popular use that the child could have overheard around him (such as la voiture à son frère -the car of his brother -instead of 'de son frère'). In the case of Anaé, we don't find [definite article + noun + à + tonic pronoun] in the input, but there are a few constructions such as ma maman à moi (my mummy of me) composed of the following elements: [possessive determiner + noun + à + tonic pronoun] with a contrastive value. There is of course a high frequency of constructions with à moi or à toi, mostly for turn-taking in her social environment (she has two elder brothers). The child shows a clear preference for the [à moi] construction at 2;00 and overgeneralizes the form to other functions.

Around two, the child therefore seems to construct her own system in order to mark the possessive relation, based in particular on the production of à moi (meaning either 'my turn' or 'mine'). She treats that form as a frozen expression, which serves the purpose of making a reference to herself in a variety of contexts and with a certain flexibility in its pragmatic and syntactic functions.

CONSTRUCTING ADULT STANDARD FORMS

Determiners

As Anaé makes sporadic uses of [Noun + de moi] up to 2;11. At 2;08, an interesting occurrence of mon truc de moi (my thing of me) shows that she might not consider possessive determiners to be semantically strong enough to mark nominal determination, reference to a first-person possessor, masculine gender and possession all at once and therefore needs to resort to the nonstandard de moi to emphasize the identity of the possessor. This type of production, which is not found in adult language (though it may be derived from the [N1 de N2] construction), demonstrates how the little girl ascribes functions to forms that she has picked up in her analysis of the input. These types of unusual uses can be seen as 'emergent categories' [START_REF] Clark | Emergent categories in first language acquisition[END_REF] As illustrated in graphs 5 and 6, the distribution of first-, second-and third-person reference differs dramatically between Anaé and her mother, with very high percentages of first-person possessive determiners in Anaé's data.

It is also interesting to note that Anaé starts using possessive pronouns (le mien, le tien, le sien-mine, yours, his) later on, after the end of the data coded for this paper, in the mother's diary. A pronoun such as le mien (mine), for example when referring to one's dog, amalgamates reference to the noun (with its gender and number) and to the possessive determiner (with its person). We found examples in the diary such as j'ai trouvé ton mien (I found your mine) at 3;0 as she speaks about her brother's toy or

The [N1 de N2] construction

The 'correct' use of the [N1 de N2] construction appears only at 2;02, at the same time as the productive use of possessive determiners. At 2;01, there is one nonstandard production in the data with the preposition 'à' instead of the preposition 'de' (les épaules à papa -the shoulders of daddy). At 2;02, the session in which the possessive determiner becomes predominant, there are two occurrences (la tête de monsieur -the head of mister and la tête de Anaé -the head of Anaé). The child designates herself with her name instead of the deictic determiner ma, and treats the reference to herself in the same way as the reference to 'monsieur'.

Her uses are more numerous and diversified as of 2;05. Her utterances are like building blocks made of several different pieces (at 2;05 dans le lit de moi -in the bed of me; at 2 ;06 le bébé de ours brun -the baby of brown bear; les poils de le bébé -the hair of the baby; at 2;11 des petites poils de Omer -the little hairs of Omer). The construction [N1 de N2] is therefore an excellent locus to observe the various nonstandard forms a child can create in the process of acquiring language. Since it is made of several blocks (nouns, determiners, a preposition), there is ample opportunity for creativity on the part of the child.

CONCLUSION

Our analysis of the emergence and development of possession markers in Anaé's longitudinal data illustrates a dynamic constructive process, which evolves as the child acquires the means to express all the semantic features they amalgamate. Her increasing capacity to analyze the input seems to guide her usage. She assembles pieces of various structures without having full control over the complexity of each grammatical marker or each construction. Anaé uses two complementary strategies. On some occasions, she behaves as a 'lumper' and produces target forms learned as whole frozen expressions, which she either uses correctly or in inadequate ways (for example, she overgeneralizes the expression à moi by assigning it various functions). But sometimes she behaves as a 'splitter' who analyzes each little form in trying to put them together but somehow misses the target (as in de le bébé instead of du bébé or the too overmarking in mon truc de moi -my thing of me -when mon truc would have been sufficient). As linguists, we are especially thankful for those variations and deviations, which give us some insights into her cognitive and linguistic processing. 1;07 1;10 2;00 2;01 2;02 2;03 2;05 2;06 2;08 2;09 2;11 N de N poss det 1;07 1;10 2;00 2;01 2;02 2;03 2;05 2;06 2;08 2;09 2;11 1;10 2;00 2;01 2;02 2;03 2;05 2;08 2;09 2;11 1;10 2;00 2;01 2;02 2;03 2;05 2;08 2;09 2;11 1;07 1;10 2;00 2;01 2;02 2;03 2;05 2;06 2;08 2;09 2;11 1;07 1;10 2;00 2;01 2;02 2;03 2;05 2;06 2;08 2;09 2;11 1 ;06 1 ;07 1 ;09 1 ;10 2 ;0 2 ;01 2 ;2 2 ;03 2 ;04 2 ;05 2 ;06 2 ;08 2 ;09 2 ;10 2 ;11

  For third-person possessors, there are two competing structures in French: [Noun1 + preposition de + Noun 2] (le sac de maman -mummy's bag) and [Possessive Determiner + Noun] (son sac -her bag). The selection of the construction depends on the discourse status of the possessor. If the possessor has been mentioned (given), the speaker will use [Poss Det + N]

  à) + possessor: à moi (mine) -(article) + possessed object + (preposition à/de) + possessor: la balle de Dédé (Dédé's ball) -possessor + possessed object: papa bic (daddy pen)

  to the input, these non-standard [N + à moi] constructions are transitory and are soon replaced by the adult forms [N1 de N2] and [Poss Det + N].

Figure 1

 1 Figure1

  and are transitional forms children use while in the process of acquiring the target linguistic forms. In acquiring their language, children must attend to all grammatical distinctions relevant in their own language and might not employ them multifunctionally as shown for instance in Karmiloff-Smith (1979)'s work on the acquisition of determiners. When several grammatical features are marked by the same term, this could lead to some confusion about which features are marked by what grammatical element, as shown by Anaé when she says mon truc de moi (my thing of me).

  votre mien (your mine) referring to her parents' bed. In these examples, she seems to use the first-person pronoun with only one function (possession), instead of amalgamating person and possession. She therefore adds another grammatical marker to indicate person.

Figure 1 :

 1 Figure 1: Distribution of possessive constructions in the input

Figure 2 :

 2 Figure 2: Distribution of possessive constructions in the child's

  Figure 3: Percentage of occurrences and types of determiners

  Figure 4: Percentage of occurrences and types of determiners

  Figure 5: Personal reference of possessors in the input

Figure 6 :

 6 Figure 6: Personal reference of possessors in the child's productions

Figure 7 :

 7 Figure 7: Percentage of utterances containing moi

  table) [oh there's puzzle of me 5 ] CHI : c'est le loup à moi (as she picks up a piece of the puzzle) [it's wolf of me]

CHI : le chapeau à moi [the hat of me] CHI : c'est le tapis à moi [it's the rug of me] CHI : c'est la vache à moi [it's the cow of me] CHI : c'est le cadeau à moi [it's the present of me] CHI : ah les cadeaux à moi [oh, the presents of me] CHI : c'est l'anniversaire à moi [ it's the birthday of me]

  noted earlier, possessive determiners in adult French mark possession, gender, number and person in one grammatical form. This amalgam of several features might make it difficult for children to handle those unstressed monosyllabic grammatical words. Anaé's analysis of gender marking is quite visible when she makes repairs of her own productions, as in le son son sa queue (the, his, her tail) or il met son son sa robe (he's putting on his his, her dress) at 2;02 and Timothé i veut pas me prêter mon v/, ton v/ le vélo de Timothé (Timothé doesn't want to lend me my, your,

	amalgamate several functions in one term. At 2;00, there are two 'correct'
	occurrences of first-person possessive determiners (mon jeu -my game, and
	mon papa -my dad). At 2;01 Anaé starts using a third-person possessive
	determiner (sa maman -his mummy) but it is only at 2;02 that possessive
	determiners are used very productively, and ma, mon, mes take on the
	function that was marked by her nonstandard use of [Noun + à moi], which
	totally disappears from the data. At 2;02, there are 27 occurrences of
	possessive determiners with 17 different nouns. Some nouns appear with
	third-and first-person possessive determiners, both singular and plural (mes
	doudous, son doudou).
	Timothé's bike) at 2;10.
	Possessive determiners are first used at 1;08 in frozen expressions
	(tatête -yourhead -to designate the toy's head is not yet analyzed as two
	separate elements and the wrong person, second instead of third is used). At
	1;10, there is a very interesting use of donne ma main (give my hand) when
	she means 'give me your hand' or 'hold my hand' as if ma was used to
	amalgamate the role of patient (instead of using moi) and the possessive
	relation. During the same session, she also makes the opposite mismatch,
	saying toi chaussure (you shoe) to her father as she tries to take his shoes
	off, instead of 'ta chaussure' (your shoe). The handling of those
	grammatical markers thus seems to be quite confusing, since they

A short version of a study on nonstandard forms (gender and possessive marking) is presented inLeroy--Collombel (2010).

In [e bébé], the phoneme [e] is what we call a « filler syllable » (Peters 2001).Filler syllables are unglossable syllables that children incorporate in front of verbs or nouns, which will progressively be replaced by determiners, auxiliaries and pronouns.

Later in the data, Anaé starts using idiosyncrasic plural forms such as noutre instead of 'notre', which is based on the subject pronoun form 'nous'.

We used 'of me' to translate à moi here even though English speaking children would not use that expression.

We chose to transcribe [se] and[a] in phonetics in order to avoid interpreting the child's production in terms of grammatical categories.

In contradiction to the poverty of stimulus argument (Chomsky1959)

We would like to thank Nancy Chang for insightful comments, suggestions and thorough editing of this paper, Martine Sekali and the CoLaJE team for all their feed-back during the research process and our anonymous reviewers for their rich comments. Any errors and other shortcomings are, of course, our own responsibility.