The question of the logical modelling of natural language is concerned with providing a formal framework, which enables representing and reasoning about utterances in natural language. The body of work in this area is organised around two different hypotheses, which yield significantly different notions of what the object of study is. Each of those two hypotheses is based on a different side of Logic: the proof-theoretic hypothesis, and the model-theoretic one. The proof-theoretic hypothesis, on one hand, considers that natural language can be modelled as a formal language. It sets the syntax of the observed natural language at the syntax level of the modelling formal language. All the works based on Generative Grammar rely on this hypothesis. The model-theoretic hypothesis, on the other hand, considers that natural language, along all its dimensions including Syntax, must be modelled through the semantic level of Logic. Underlying are two fairly different notions of what natural language is -and is not, and what should -or not -be modelled.

We introduce and compare the main characteristics of both the Proof-Theoretic and the Model-Theoretic paradigms. We argue that representing the linguistic description of an utterance solely through a hierarchical syntactic structure is severely restrictive. We show evidence of these restrictions through the study of specific problems. We then argue that a Model-Theoretic representation of Syntax does not show those restrictions, and provides a more informative linguistic description than Proof-Theoretic Syntax. We give an overview of a specific framework for MTS, called Property Grammar (PG), which we use to illustrate our point. We show, in particular, how to rely on a graph as linguistic representation, in order to address various language problems.

MODEL THEORY FOR MODELLING NATURAL LANGUAGE

The Proof-Theoretic and the Model-Theoretic approaches to modelling Natural Language differ in scope of modelling, i.e. with regard to the observations being modelled. They also differ with regard to the nature of the linguistic knowledge being captured and represented.

Proof-Theoretic Syntax With the proof-theoretic one, natural language is defined as the set L (G)1 of all the strings licensed by the grammar G. What is meant here by licensed is proven: all the strings in L (G) are those, and only those, which can be proven by a set of production rules from G. The proof itself captures the whole linguistic knowledge about any string s in L (G), in the form of a tree representation: the parse tree -or syntactic structure. Thus, a parse tree for the sentence s is merely a graphic (in the sense of the Graph Theory) representation of the proof that s ∈ L (G). This isomorphism between linguistic structure and proof of membership has strong consequences on the modelling scope, since anything that can not be proven can not be represented either. Therefore, and assuming that a grammar G is available, which captures all the observed canonical linguistic phenomena of human language, the set of all the objects being modelled under the proof-theoretic hypothesis is limited to the set of the grammatical strings, and only these ones. All ungrammatical strings are just out of scope, and no knowledge can be represented about them. This notion is very restrictive, for it does not account for the extreme variability of language usages, including non-canonical or even ill-formed productions.

Furthermore, the linguistic description being represented is heavily driven by Syntax, and does not (or rarely) account for other linguistic dimensions. Even when it does, the information on syntax is required in order for information on other dimensions to be represented. Recent works, on the contrary, propose to consider all different sources of information as interacting at the same level (see, in particular, works around Construction Grammar, e.g. [START_REF] Goldberg | Constructions: A new theroretical approach to language[END_REF], [START_REF] Sag | Sign-Based Construction Grammar: An Informal Synopsis[END_REF]).

Model-Theoretic Syntax

The model-theoretic notion of language, on the other hand, does not show those limitations in scope. Here, no assumption is made as to which utterances ought to be described -hence covered by the model, and which ones ought not to be. All observed utterances get a linguistic description of their structure, irrespective of their well-formedness. The description takes the form of a set of grammar statements, each of them being either verified or not by the structure. All those grammar statements together, instantiated for a given utterance, constitute a constraint network. The grammaticality of an utterance is then defined with respect to this description: the utterance is grammatical if and only if all the statements are verified. Meanwhile, descriptions which would include failing statements may still exist -they are simply not deemed grammatical. Hence scopewise, any observed utterance may get a linguistic description.

Another incentive of a model-theoretic representation of grammar is that it offers the possibility to interpret in different ways the knowledge at stake. For instance, in this chapter, we present two different (though not opposed) perspectives on what the representation of a linguistic structure should be like: the constructive perspective, and the descriptive perspective.

The Constructive Perspective on MTS The constructive perspective considers that the constraint network instantiated for an utterance is self-sufficient for describing the linguistic knowledge about this utterance. The network comes in replacement of the conventional linguistic phrase and dependency structures. Yet, if required, those structures may be recovered by induction from the network. This perspective also makes use of the MT grammar for the parsing process, which, therefore, solely relies on the constraint grammar for building up the constraint network through an inference process. Section 1.3 elaborates further on various aspects of such a perspective on MTS.

The Descriptive Perspective on MTS The descriptive perspective sees the constraint network as a complement of the conventional parse structure (phrase or dependency). The parse structure merely serves the purpose of the structure, in the sense of the Model Theory in Logic. According to it, the objects of study are models of theories expressed in a formal metalangage, where a theory is a set of statements specified in a formal language, and a model of a theory is a structure, which satisfies all the statements of that theory. Section 1.4 elaborates further on the Descriptive perspective on MTS.

Either way (constructive or descriptive), the MT representation, inclusive of the constraint network, is much more informative than the sole parse structure, and allows for more exact reasoning on the corresponding utterance. The next two sections introduce those two perspectives.

THE CONSTRUCTIVE PERSPECTIVE: A CONSTRAINT NETWORK FOR REPRESENTING AND PROCESSING THE

LINGUISTIC STRUCTURE

The generative conception of grammar relies on the derivation process which, in turn, depends on a hierarchical representation of syntactic information. However, several works have shown the limits of such a representation. From a generative point-of-view, parsing an input corresponds to finding a set of derivation rules, which makes it possible to generate the surface realisation of this input. This conception of grammar relies then on a specific view of what language is: the set of surface forms that can be generated by the grammar. This conception is very restrictive for several reasons. One is the extreme variability of language usages, including non canonical or even ill-formed productions. Another is the fact that this view is purely syntactically driven: only syntax is taken into account here and when other sources of information such as prosody are considered (which is rarely the case), they are considered as "complementary" to syntax, giving syntax a preeminent position. Recent works propose on the contrary to consider all different sources of information as interacting at the same level (see in particular works around Construction Grammar, [START_REF] Goldberg | Constructions: A new theroretical approach to language[END_REF], [START_REF] Sag | Sign-Based Construction Grammar: An Informal Synopsis[END_REF]).

Even though many linguistic theories now challenge this way of considering the relationship between language and grammar, most of them remain more or less based on the generative framework, in the sense that what we can call the context-free backbone still occupies a central position. Model-Theoretic Syntax [START_REF] Pullum | Model-Theoretic Syntax Foundations -Linguistic Aspects[END_REF], [START_REF] Pullum | The evolution of model-theoretic frameworks in linguistics[END_REF] proposes a paradigmatic shift, making it possible to escape from this framework.

This section presents the main characteristics of these different conceptions of syntax. It describes more precisely the specific problems coming from the hierarchical conception of syntax, showing how it can constitute a severe limitation for linguistic description. We propose then an overview of a specific MTS framework, called Property Grammars, following this requirements. We precise formally the status of the constraints we use, and how, in this approach, a syntactic description comes to a graph. We explain in particular how it is possible to take advantage of such a representation in order to shift from the classical tree domain to a graph one.

GENERATIVE-ENUMERATIVE VS. MODEL-THEORETIC SYNTAX

There are two different approaches in logic: one is purely syntactic and only uses the form of the formulae in order to demonstrate a theorem, the second is semantic and relies on formulae interpretation. The same distinction also holds for natural language syntax. A first approach (the syntactic one in logic) consists in studying sentence well-formedness. The problem consists there in finding a structure adequate to the input. In this case, grammatical information is then represented by means of a set a rules, the syntactic structure representing the set of rules used during the parse. An alternative approach consists in studying directly the linguistic properties of the sentence, instead of building a structure. [START_REF] Pullum | On the Distinction Between Model-Theoretic and Generative-Enumerative Syntactic Frameworks[END_REF] call these approaches respectively Generative Enumerative Syntax (GES) and Model-Theoretic Syntax (MTS). The first corresponds to the generative theories, it has been extensively experimented. The latter still remains less studied and only a few works belong to this paradigm .

One of the reasons is that generativity has been for years almost the unique view for formal syntax and it is difficult to move from this conception to a different one. In particular, one of the problem comes from the fact that all approaches, even those in the second perspective, still rely on a hierarchical (tree-like) representation of syntactic information.

The generative conception of syntax relies on a particular relation between grammar and language: a specific mechanism, derivation, makes it possible to generate a language from a grammar. This basic mechanism can be completed with other devices (transformations, moves, feature propagation, etc.) but in all cases constitute the core of all generative approaches. In such case, grammaticality consists in finding a set of derivations between the start symbol of the grammar and the sentence to be parsed. As a side effect, a derivation step coming to a local tree, it is possible to build a syntactic structure, represented by a tree. It is then possible to reduce in a certain sense the question of grammaticality to the possibility of building a tree. This reminder seems to be trivial, but it is important to mesure its consequences. The first is that grammaticality is reduced, as it has been noticed in [START_REF] Chomsky | The Logical Structure of Linguistic Theory[END_REF], to a boolean value: true when a tree can be built, false otherwise. This is a very restrictive view of grammaticality, as it also has been noticed in [START_REF] Chomsky | The Logical Structure of Linguistic Theory[END_REF] (without proposing a solution), which forbids a finer conception, capable of representing in particular a grammaticality scale (also called gradience, see [START_REF] Keller | Gradience in Grammar -Experimental and Computational Aspects of Degrees of Grammaticality[END_REF] or Aarts ( 2007)).

This generative conception of syntax is characterized as being enumerative (see [START_REF] Pullum | On the Distinction Between Model-Theoretic and Generative-Enumerative Syntactic Frameworks[END_REF] in the sense that derivation can be seen as an enumeration process, generating all possible structures and selecting them by means of extra constraints (as it is typically the case in the Optimality Theory, see [START_REF] Prince | Optimality Theory: Constraint Interaction in Generatire Grammar[END_REF]).

Model Theoretic Syntax proposes an alternative view [START_REF] Blackburn | Talking about trees[END_REF], [START_REF] Cornell | Model theoretic syntax[END_REF], [START_REF] Pullum | On the Distinction Between Model-Theoretic and Generative-Enumerative Syntactic Frameworks[END_REF]). In this conception, a grammar is a set of assessments, the problem consists in finding a model into a domain.

From a logical perspective, generative approaches rely on a syntactic conception in the sense that parsing consists in applying rules depending on the form of the structures generated at each step. For example, a nonterminal is replaced with a set of constituents. On the opposite, model-theoretic approaches rely on a semantic view in which parsing is based on the interpretation (the truth values) of the statements of the grammar. A grammar in MTS is a set of statements or, formally speaking, formulae. Each formula describes a linguistic property; its interpretation consists in finding whether this statement is true or false for a given set of values (the universe of the theory in logical terms). When a set of values satisfies all assessments of the grammar (in other words when the interpretation of all the formulae for this set of values is true), then this set is said to be a model.

As far as syntax is concerned, formulae indicate relations between categories or, more precisely, between descriptions of categories. These descriptions correspond to the specification of a variable associated with several properties: they can be seen as formulae. For example, given K a set of categories, a description of a nominative noun comes to the formula:

∃x[cat(x, N) ∧ gend(x, masc)]
(1.1)

A category can be described by a more or less precise description, according to the number of conjuncts. A grammatical statement is a more complex formula, adding to the categories descriptions other relations. For example, a statement indicating that a determiner is unique within a noun phrase comes to the formula:

[cat(x, Det) ∧ cat(y, Det) → x ≈ y] (1.2)
Parsing Concretely, when parsing a given input, a set of categories is instantiated, making it possible to interpret all the atomic predicates corresponding to the features (category, gender, number, etc.), making it possible to interpret in turn the complex predicates formed by the grammar statements. In this perspective, we say that an instantiated category is a value and finding a model consists in finding a set of values satisfying all the grammatical statements. For example, the set of words "the book" makes it possible to instantiate two categories with labels Det and N (these labels representing the conjunction of features). Intuitively, we can say that the set of values {Det, N} is a model for the NP.

Various parsing strategies have been implemented in line with that approach. Balfourier et al. (2002) implements an incremental strategy, where the choice of a suitable assignment relies on a heuristic of shortest possible span. VanRullen ( 2005) implements a multi-graph representation of the constraint network, and a strategy, which allows different granularities of solution, from chunks to deep parses. [START_REF] Prost | Modelling Syntactic Gradience with Loose Constraintbased Parsing[END_REF] revisit the CKY parsing algorithm, based on dynamic programming techniques, in order to optimise the proportion of properties (instead of probabilities) violated by the solution parse. More recently, [START_REF] Duchier | Property grammar parsing seen as a constraint optimization problem[END_REF] explore the possibility to model the parsing problem according to a model-theoretic grammar as a Constraint Optimisation Problem.

Finding a model is, then, completely different from deriving a structure. As stressed by [START_REF] Pullum | On the Distinction Between Model-Theoretic and Generative-Enumerative Syntactic Frameworks[END_REF], instead of enumerating a set of expressions, an MTS grammar simply states conditions on these expressions.

Model-Theoretic Syntax (herafter MTS) moves then from a classical tree domain to a graph domain for the representation of syntactic information. We show how constraints can be an answer to this problem: first, they can represent all kinds of syntactic information and second, they constitute a system, where all constraints are at the same level and evaluated independently from each others (no order is enforced on the constraints for evaluation).

GENERATIVITY AND HIERARCHICAL STRUCTURES

Geoffrey Pullum, during a lecture at ESSLLI in 2003, explained that "Model Theoretic Syntax is not Generative Enumerative Syntax with constraints". In other words, constraints are not to be considered only as a control device (in the DCG sense for example) but have to be part of the theory. Some theories (in particular HPSG) try to integrate this aspect. But it remains an issue both for theoretical and technical reasons. The problem comes in particular from the fact usually, dominance relation plays a specific role in the representation of syntactic information: dominance structures have first to be built before verifying other kinds of constraints. This is a problem when no such hierarchical relations can be identified. Moreover, we know since GPSG that dominance constitutes only a part of syntactic information to be represented in phrase-structure approaches, not necessarily to be considered as more important than others.

Syntactic information is usually defined, especially in generative approaches, over tree domains. This is due to the central role played by the notion of dominance, and more precisely by the relation existing between the head and its direct ancestor. In theories like HPSG (see [START_REF] Sag | Syntactic Theory. A Formal Introduction[END_REF]), even though no rules are used (they are replaced with abstract schemata), this hierarchical organization remains at the core of the system. As a consequence, constraints in HPSG can be evaluated provided that a tree can be built: features can be propagated and categories can be instantiated only when the hierarchical skeleton is known. This means that one type of information, dominance, plays a specific role in the syntactic description.

However, in many cases, a representation in terms of tree is not adapted or even not possible. The following example illustrate this situation. It present the case of a cleft element adjunct of two coordinated verbs. Many other examples can be given, illustrating this problem: it is not always possible to give a connected structure on the basis of syntactic relations. Moreover, when adding other kinds of relations, the structure is not anymore a tree. This conception has direct consequences on the notion of grammaticalness. First, building a tree being a pre-requisite, nothing can be said about the input when this operation fails. This is the main problem with generative approaches that can only indicate whether or not an input is grammatical, but do not explain the existence of levels of grammaticality (the gradience phenomenon, see [START_REF] Keller | Gradience in Grammar -Experimental and Computational Aspects of Degrees of Grammaticality[END_REF], [START_REF] Pullum | On the Distinction Between Model-Theoretic and Generative-Enumerative Syntactic Frameworks[END_REF]).

A second consequence concerns the nature of linguistic information, that is typically spread over different domains (prosody, syntax, pragmatics, and related domains such as gestures, etc.). An input, in order to be interpreted, does not necessarily need to receive a complete syntactic structure. The interpretation rather consists in bringing together pieces of information coming from these different domains. This means that interpreting an input requires to take into account all the different domains and their interaction, rather than building a structure for each of them and then calculating their interface. In this perspective, no specific relation plays a more important role than others. This is also true within domains: as for syntax, the different properties presented in the previous section has to be evaluated independently from the others.

THE PROPERTY GRAMMAR FRAMEWORK

A seminal idea in GPSG (see [START_REF] Gazdar | The Logic of Typed Feature Structures[END_REF]) was to dissociate the representation of different types of syntactic information: dominance and linear precedence (forming the ID/LP formalism), but also some other kinds of information stipulated in terms of cooccurrence restriction. This proposal is not only interesting in terms of syntactic knowledge representation (making it possible to factorize rules, for example), but also theoretically. Remind that one of the main differences between GES and MTS frameworks lies in the relation between grammar and language: MTS approaches try to characterize an input starting from available information, with no need to "overanalyze", to re-build (or infer) information that is not accessible from the input. For example, GES techniques have to build a connex and ordered structure, representing the generation of the input. On the opposite, nothing in MTS imposes to build a structure covering the input, which makes it possible for example to deal with partial or heterogeneous information. Property Grammars (see Blache ( 2005)) systematizes the GPSG proposal in specifying these different types. More precisely, they propose to represent separately the following properties:

• Constituency: set of all the possible elements of a construction

• Uniqueness: constituents that cannot be repeated within a construction

• Linearity: linear order

• Obligation: set of obligatory constituents, one of them (exclusively to the others) being realized.

• Requirement: obligatory cooccurrence between constituents within a construction

• Exclusion: impossible cooccurrence between constituents within a construction

This list is not closed and other types of information can be added. For example, dependency (syntactico-semantic relation between a governor and a complement), or adjacency (juxtaposition of two elements). We focus in this paper on the 6 basic relations indicated above. These relations makes it possible to represent most of the syntactic information. We call these relations "properties", they can also be considered as constraints on the structure.

We adopt in the remaining of this paper the following notations: x, y (lower case) represent individual variables; X,Y (upper case) are set variables. We note C(x) the set of individual variables in the domain assigned to the category C (see Backofen et al. (1995) for more precise definitions). We use the set of binary predicates for immediate domination ( ), linear precedence (≺) and equality (≈).

Let us now define more precisely the different properties. The first one (constituency) implements the classical immediate dominance relation. The others can be defined as follow:

• Const(A, B) : (∀x, y)[(A(x) ∧ B(y) → x y]
This is the classical definition of constituency, represented by the dominance relation: a category B is constituent of A stipulates that there is a dominance relation between the corresponding nodes.

• Uniq(A) : (∀x, y)[A(x) ∧ A(y) → x ≈ y]
If one node of category A is realized, there cannot exists other nodes with the same category A. Uniqueness stipulates constituents that cannot be repeated in a given construction.

• Prec(A, B) : (∀x, y)[(A(x) ∧ B(y) → y ≺ x)]
This is the linear precedence relation as proposed in GPSG. If the nodes x and y are realized, then y cannot precedes x

• Oblig(A) : (∃x)(∀y)[A(x) ∧ A(y) → x ≈ y]
There exists a node x of category A and there is no other node y of the same category. An obligatory category is realized exactly once.

• Req(A, B) : (∀x, y)[A(x) → B(y)]

If a node x of category A is realized, a node y of category B has too. This relation implements cooccurrence, in the same way as GPSG does.

• Excl(A, B) :

(∀x)( ∃y)[A(x) ∧ B(y)]
When x exists, there cannot exist a sibling y. This is the exclusion relation between two constituents.

What is interesting in this representation of syntactic information is that all relations are represented independently form each others. They all are assessment in the MTS sense, and they can be evaluated separately (which fits well with the non-holistic view of grammar information proposed by Pullum). In other words there is no need to assign the dominance relation a specific role: this is one information among others, what is meaningful is the interaction between these relations. More precisely, a set of categories can lead to a well-formed structure when all these assessments are satisfied, altogether. We do not need first to build a structure relying on dominance and then to verify other kind of information represented by the rest of the relations. In other words, in this approach, "MTS is not GES with constraints" [START_REF] Pullum | Model-Theoretic Syntax Foundations -Linguistic Aspects[END_REF]).

Parsing with Property Grammar Concretely, when taking into consideration a set of categories (an assignment), building the syntactic structure comes to evaluate the constraint system for this specific assignment, in order to infer new categories and build up the parse structure. The result of the evaluation indicates whether or not the assignment corresponds to a wellformed list of constituents. For example, given two nodes x and y, if they only verify a precedence relation, nothing else can be said. But when several other properties such as requirement, uniqueness, constituency are also satisfied, the assignment {x, y} becomes a model for an upper-level category. For example, if we have x and y such that Det(x) and N(y), this assignment verifies precedence, uniqueness, constituency and requirement properties. This set of properties makes it possible to characterize a NP. At the opposite, if we take x and y such that Det(x) and Adv(y), no constraint involving both constituents belong to the system: they do not constitue a model, and no new category can be inferred.

In terms of representation, unlike the classical approaches, syntactic information is not represented by means of a tree (see [START_REF] Huddleston | The Cambridge Grammar of the English Language[END_REF]), but with a directed labelled graph. Nodes are categories and edges represent constraints over the categories: dominance, precedence, requirement, etc. A non-lexical category is described by a set of constraints, that are relations between its constituents. It is possible to take under consideration only one type of property (in other words one type of relation): this comes to extract a subgraph from the total one. For example, one can consider only constituency properties. In this case, the corresponding subgraph of dominance relations is (generally) a tree. But what is needed to describe precisely an input is the entire set of relations.

In the following, we represent the properties with the set of relations noted ⇒ (requirement), ⊗ (exclusion), • (uniqueness), (constituency), ↑ (obligation), ≺ (precedence). A Property Grammar graph (noted PGgraph) is a tuple of the form:

G = W, ⇒, ⊗, •, , ↑, ≺, θ
in which W is the set of nodes, θ the set of terminal nodes. A model is a pair G,V where V is a function from W to Pow(W). We describe the use of such graphs in section 1.5.

THE DESCRIPTIVE PERSPECTIVE: A CONSTRAINT NETWORK FOR COMPLETING THE LINGUISTIC STRUCTURE

According to the model-theoretic hypothesis, human language is represented on the semantic level of Logic: the objects of study are models of theories expressed in a formal language, where a theory is a set of statements specified in a formal meta-language, and a model of a theory is a structure, which satisfies all the statements of that theory. Hence, applied to natural language:

• a theory is a set of grammar statements, specified by a conjunction Φ = i φ i , where every atom φ i is a logical formula, which puts elements of the structure in a relationship ;

• a structure is a linguistic parse structure (e.g. phrase structure, dependency structure, or both).

A grammar is, then, a conjunctive formula, parameterised by the structure, and a theory is an instance of the grammar for a given structure. For instance, for a domain of phrase structures, the φ i are relations, which hold on constituents (e.g. In a Noun Phrase in English, the Determiner precedes the Noun). [START_REF] Duchier | A model-theoretic framework for grammaticality judgements[END_REF] formulate a Model-Theoretic semantics for Property Grammar along these lines.

Model checking

We first give a few definitions, in order to fix the notations in use in the following.

Let S be a set of words in the target language, and L a set of labels, which denote morpho-syntactic categories; a lexicon is then a subset V ⊂ L × S (which implicitly assumes that the terminals are POS-tagged words). Let P L be the set of all the possible properties on L ; a PG grammar Φ is specified by a pair (P G , L G ), with P G ⊆ P L .

Let τ : s be a (phrase structure) tree decorated with labels in L , and whose surface realisation is the string of words s; let Φ s be an instantiation of Φ for τ : s; τ : s is a model for Φ s iff τ : s makes Φ s true. We denote by τ : s |= Φ s the satisfaction of Φ s by τ : s. The instantiation Φ s is also called the constraint network associated with τ : s for the grammar Φ.

Definition 1 (Grammaticality). τ :s is grammatical with respect to the grammar Φ iff τ : s |= Φ s Since Φ s = i φ s i , Definition 1 means that every instance of property φ s i of Φ s for the sentence s must be satisfied for s to be deemed grammatical with respect to the grammar Φ.

The model checking process involves:

• instantiating the grammar Φ for the parse tree τ : s,

• building up the corresponding constraint network Φ s , and

• checking the truth of every atom φ s i .

Processing-wise, the existence of a linguistic structure is required prior to checking it against a grammar -which involves constructing the constraint network. Figure 1.4 exemplifies a phrase structure deemed ungrammatical through model checking.

Under such an interpretation, the linguistic structure plays the role of a semantic object, which makes a theorie true (in which case the structure is deemed a model of the theory), or not. That is, the parse structure is the object, which makes the grammar (as a conjunction of statements) true, or not. Here, the conventional parsing process, seen as the generation process of the parse structure, is kept separate from the model checking process.

Language cover As far as knowledge representation is concerned, a formal framework for MTS shows the following properties:

1. the independence of the grammar statements allows for the definition of non-classical satisfaction, whereby a given structure only partially satisfies a subset of Φ, and violates its complement;

(NP (NC quatorze))) (PUNC ,) (COORD (CC soit) (NP (DET la) (NC moitié)))) (PUNC ,) (VN (V ont)) (NP (DET un) (NC financement) (Srel (NP (PROREL qui)) (VN (ADV n') (V est) (AdP (ADV toujours) (ADV pas)) (VPP assuré)))) (COORD (CC et) (Sint (NP (NC dont) (DET le) (NC calendrier)) (VN (ADV n') (V est) (ADV pas_encore) (VPP arrêté)))) (PUNC .)))
Figure 1.1: Example of parse deemed ungrammatical through model checking 2. how structures are generated is not specified in the grammar.

Property 1 means that any structure may be represented in the framework, whether grammatically satisfactory or not. The syntax of non-canonical language can, thus, be modelled by a structure, which only loosely meets the grammar. Property 2 means that processing-wise, the generation of candidate-model structures is formally distinct from the grammar check. It opens all kinds of perspectives with regard to the parsing process. We have already seen in section 1.3 that the entire parsing process, including the generation of the structure, may be driven by the PG grammar itself. Another option, in line with the model theory, is to consider that the generation of the candidate structures is not of concern to the theory. In this case, different options are available.

Generation of model-candidates Since a model-theoretic representation is independent from any processing aspects regarding the generation of candidate structures, the strategy for generating them may be conceived separately from model checking. Although an inference process may be designed in order to construct structures on the sole basis of the constraint grammar (see the Constructive perspective 1.3, and for instance [START_REF] Maruyama | Structural Disambiguation with Constraint Propagation[END_REF]Balfourier et al. (2002); [START_REF] Prost | Modelling Syntactic Gradience with Loose Constraintbased Parsing[END_REF]; [START_REF] Duchier | Property grammar parsing seen as a constraint optimization problem[END_REF] for parsing strategies), nothing makes it compulsory. It is, therefore, possible, for instance, to check likely structures generated by a probabilistic parser against the MT grammar.

Note, as well, that the type of linguistic structure concerned by a Model-Theoretic representation may take different forms, depending on the formal framework in use. The seminal work of [START_REF] Maruyama | Structural Disambiguation with Constraint Propagation[END_REF], for instance, is concerned with dependency structure, while Optimality Theory [START_REF] Prince | Optimality Theory: Constraint Interaction in Generatire Grammar[END_REF] is more used for describing phonological structures. As for Property Grammar, the framework is essentially used with phrase structures, though a few works rely on it for multimodal annotation, or biological sequences analysis.

Structure enrichment for solving language problems A major incentive of the descriptive perspective on MTS, and PG in particular, stands in its potential alliance with other structures and processes in order to address a variety of language problems. Grammaticality judgement is one of those. The problem is encountered in contexts such as Statistical Machine Translation [START_REF] Zwarts | Choosing the right translation: A syntactically informed classification approach[END_REF], Summarisation [START_REF] Wan | Towards Statistical Paraphrase Generation: Preliminary Evaluations of Grammaticality[END_REF], or second language learning [START_REF] Wong | Exploiting Parse Structures for Native Language Identification[END_REF], where the grammaticality of a candidate solution is a non-trivial decision problem. A common approach to address it is to train a statistical classifier [START_REF] Foster | Adapting a wsj-trained parser to grammatically noisy text[END_REF]; [START_REF] Wong | Exploiting Parse Structures for Native Language Identification[END_REF]; [START_REF] Wagner | Detecting Grammatical Errors with Treebank-Induced, Probabilistic Parsers[END_REF] in order to determine the grammaticality of a candidate around a threshold of likelyhood. The use of a constraint network, associated with the linguistic structure, alleviates the decision through exact model checking (Prost, forthcoming).

The graph-based representation also shares properties with the graph structure of semantic networks, such as Lafourcade's 2008. The combination of the two is expected to open avenues of research with respect to the Syntax-Semantics interface. The solving of problems concerned with both dimensions, such as Grammar error detection, should be eased by the resulting enriched structure [START_REF] Prost | Pairing Model-Theoretic Syntax and Semantic Network for Writing Assistance[END_REF].

GRAMMATICALITY JUDGMENT

Classically, syntactic information is usually represented in terms of decorated ordered trees (see [START_REF] Blackburn | Talking about trees[END_REF], [START_REF] Blackburn | Linguistics, logic and finite trees[END_REF]). In this approach, tree admissibility relies on a distinction between dominance relation (that gives the structure) and other constraints on the tree such as precedence, cooccurrence restriction, etc. In our view, all relations has to be at the same level. In other words, dominance does not play a specific role: cooccurrence restriction for example can be expressed and evaluated independently from dominance. This means that each property represents a relation between nodes, dominance being one of them. When taking into consideration the entire set of relations, the structure is then a graph, not a tree. More precisely, each property specifies a set of relations between nodes: precedence relations, cooccurrence relations, dominance relations, etc. It can be the case that the dominance subset of relations (a subgraph of the graph of relations), is a tree, but this can be considered as a side effect. No constraint for example stipulates a connexity restriction on the dominance subgraph.

In PG, a grammar is then conceived as a constraint system, corresponding to a set of properties as defined above. Parsing an input consists in finding a model satisfying all the properties (or more precisely, the properties involving the categories of an assignment). In this case, the input is said to be grammatical, its description being the set of such properties. However, it is also possible to find models that satisfy partially the system. This means that some constraints can be violated. If so, the input is not grammatical, but the set of satisfied and violated properties still constitute a good description. We call such set a characterization. This notion replaces that of grammaticalness (which is a particular case of characterization in which no property is violated).

The following example (figure 1b) illustrates the case of an assignment A={NP, Det, Adj, N}. All properties are satisfied, each relation forms an labeled edge, the set of relations being a graph. A phrase is characterized when it is connected to a graph of properties. NP y y % %

Det ≺ 3 3 ⇒ 7 7 ≺ & & N ↑ U U Ad j ≺ 8 8 Fig. 1a S y y % % NP x x & & ≺ / / V P % % ⇒ v v Det ≺ 3 3 ⇒ 7 7 ≺ & & N ↑ U U Aux ≺ 6 6 V ↑ q q Ad j ≺ 8 8
Fig. 1b Figure 1b shows a more complete graph, corresponding to an entire sentence. Again, no relation in this graph plays a specific role. The information comes from the fact that this set of categories are linked by several relations. The set of relations forms a description: it tells us that linearity, requirement, obligation, constituency properties are satisfied, they characterize an S. Theoretically, each node can be connected to any other node. Nothing forbids for example to represent a relation of some semantic type between the adjective and the verb nodes. By another way, when taking from this graph constituency relations only, we obtain a dominance tree: S NP VP Finally, insofar as a property can be satisfied or violated in a characterization, we have to label relations with their type and their interpretation (true or false, represented + or -). The following example presents a graph for the assignment A={NP, Adj, Det, N}, in which the determiner has been realized after the adjective.

NP

+ y y + $ $ + Ad j ≺+ 3 3 ≺-& & N ↑+ U U Det ≺+ 8 8 ⇒+ I I
In this graph, all constraints but the precedence between Det and Adj have been satisfied, the corresponding relations being labeled with +.

As a side effect, representing information in this way also constitues a possibility to rank the inputs according to a grammaticalness evaluation. We present in this section how to use characterizations in order to quantify such information. The idea (see Blache et al. (2006)) consists in analyzing the form of the graph and its density, taking into account the interpretation of the relations. More precisely the method consists in calculating an index from the cardinality of P + and P -, (respectively the set of satisfied and violated properties). Let's call N + and N -the cardinality of these sets. The first indication that can be obtained is the ratio of satisfied properties with respect to the total number of evaluated properties E. This index is called the Satisfaction ratio, calculated as SR = N + E . Going further, it is also possible to give an account of the coverage of the assignment by means of the ratio of evaluated properties with respect to the total number of properties T describing the category in the grammar. This coefficient is called Completeness coefficient: CC = E T . A Precision Index can to its turn be proposed, integrating these two previous information: PI = SR+CC 2 . Finally, a general index can be proposed, taking into consideration the different indexes of all the constituents. For example, a phrase containing only well-formed constituents has to be assigned a higher value than one containing ill-formed ones. This is done by means of the Grammaticalness Index, d being the number of embedded constructions C i : if d = 0 then GI = PI, else GI = PI × ∑ d i=1 GI(C i ) d

. In reality, this different figures need to be balanced with other kind of information. For example, we can take into consideration the relative importance of constraint types in weighting them. Also, the influence of SR and CC over the global index can be modified by means of coefficients.

This possibility of giving a quantified estimation of grammaticalness directly comes from the possibility of representing syntactic information in a fully contraint-based manner, that has been made possible thanks to the MTS view of grammar.

CONCLUSION

The representation of syntactic information by means of constraints, as described in this paper, shows several advantages. First, it provides an elegant computational framework for MTS, where derivation does not play any role. In the Constructive perspective on MTS, the shift from generative to model-based conception of natural language syntax then becomes con-crete: constraint satisfaction completely replaces derivation. This evolution becomes possible provided that we abandon a strict hierarchical representation of syntax in which dominance plays a central role.

As a consequence, such fully constraint-based approach offers the possibility to replace ordered trees domain with that of constraint graphs. This is not only a matter of representation, but has deep consequences on theory itself: different types of information is represented by different relations, all of them being at the same level.

The Descriptive perspective on MTS, where the constraint networks complements and enriches the conventional lingustic struture, also shows interesting properties. Through the provision of a finer-grained description of the linguistic properties of a sentence than the sole parse structure, it alleviates the address of various language problems. Grammaticality judgement, for instance, or the interaction between the Syntax and Semantics dimensions, should benefit from such a graph-based representation.

The Property Grammar framework described in this paper represents the possibility of an actual MTS implementation, in which constraints are not only a control layer over the structure, but represent the structure itself: MTS is not GES plus constraints, provided that dominance is not represented separately from other information. BIBLIOGRAPHY Aarts, B. (2007). Syntactic gradience: the nature of grammatical indeterminacy. Oxford University Press.

Backofen, R., Rogers, J., and Vijay-Shanker, K. (1995). A first-order axiomatization of the theory of finite trees. Journal of Logic, Language, and Information, 4(1).

Balfourier, J.-M., Blache, P., and Rullen, T. V. (2002)

  Arrows in this figure shows in what sense the tree fails in representing the distribution of the cleft element onto the conjuncts. Moreover, there also exists other kinds of relations, for example the obligatory cooccurrence in French between "c'est" and "que".The second example, presented in the following structure, illustrates the fact that in many cases, it is not possible to specify clearly what kind of syntactic relation exists between different parts of the structure: Le piano les doigts ça a beaucoup d'importance The piano the fingers it has a lot of immportance This example illustrates a multiple detachment construction. In this case, detached element are not directly connected by classical syntactic relations to the rest of the structure: the two relations indicated by arrows are dependencies at the discourse level (plus an anaphoric relation).

  . From Shallow to Deep Parsing Using Constraint Satisfaction. In Proc. of the 6th Int'l Conference on Computational Linguistics (COLING 2002). Blache, P. (2005). Property grammars: A fully constraint-based theory. In H. C. et al., editor, Constraint Solving and Language Processing, volume LNAI 3438. Springer. Blache, P., Hemforth, B., and Rauzy, S. (2006). Acceptability prediction by means of grammaticality quantification. In Proceedings of the 21st International Conference on Computational Linguistics and 44th Annual

More formally, the language L (G) is usually defined as the n-uple L,C, S, G , where L is a lexicon (terminal vocabulary), C is a set of morpho-syntactic categories (non-terminal vocabulary), S ∈ C is a start symbol (for labelling the tree root), and G (the grammar) is a set of production rules.


M B Kébré

F Cherblanc

F Ouédraogo

F Jamin

B Naon

F Zougmoré

J.-C Bénet

Water flow in soil at small water contents: A simple approach to estimate the relative hydraulic conductivity in sandy soil

Keywords: soil hydraulic property, soil water flow, soil water profile, modelling, capillary flow, film flow, relative hydraulic conductivity. Highlights

The unsaturated soil hydraulic properties are key properties in the fields of soil science and civil engineering. Because of their strong dependence on water content, there are considerable experimental and numerical difficulties in their determination, specifically in the dry range. This situation is encountered regularly in arid and semi-arid regions. The models commonly used for predicting unsaturated hydraulic conductivity function rely on pore bundle concepts that account for capillary flow only and neglect film flow. Furthermore, the assumption of a local equilibrium between liquid water and its vapour is no longer valid at small water contents. Thus, with classical approaches, the experimental identification of hydraulic characteristics can fail at small water contents. To emphasize the weakness of capillary models, soil column experiments have been carried out with two sandy soils from Burkina Faso. Special care was taken to prevent any transport processes that are not directly related to liquid transport. Data from profiles of water content were introduced into an inverse numerical procedure to identify the coefficients of a new relative hydraulic conductivity function. Our results show that this simple approach is suitable for the analysis of flow processes at small water contents. It provides a simple, robust and inexpensive method to identify the properties of unsaturated conductivity function that account for capillary and film flows.

Introduction

The description of water flow processes in soil has been the focus of intensive research in the fields of agronomy, hydrology and geomechanics. If the situation associated with temperate regions where water content varies from intermediate to large seems to be well understood, the case for dry soil in arid and semi-arid zones still requires further investigation. The development of alternative modelling approaches for water resources management dedicated to arid soil is of prior importance for sub-Saharan countries such as Burkina Faso (Bénet et al., 2009;[START_REF] Ouédraogo | Water transfer in soil at low water content. Is the local equilibrium assumption still appropriate[END_REF].

As water content decreases, hygroscopic effects become predominant and adsorptive forces prevail over capillary ones [START_REF] Or | Liquid retention and interfacial area in variably saturated porous media: upscaling from single pore to sample scale model[END_REF][START_REF] Tokunaga | Hydraulic properties of adsorbed water films in unsaturated porous media[END_REF]. Liquid water is present in the form of adsorbed layers around soil particles and its thermodynamic state can differ markedly from bulk water. Therefore, the modelling of transport phenomena at small water contents requires a specific description. In particular, it has been emphasized that vapour diffusion can contribute considerably to water transport [START_REF] Saito | Numerical analysis of coupled water, vapor and heat transport in the vadose zone[END_REF], which leads to reconsidering the classical assumption of a local equilibrium [START_REF] Smits | An evaluation of models of bare soil evaporation formulated with different land surface boundary conditions and assumptions[END_REF]Ouédraogo et al., 2013;[START_REF] Massman | A non-equilibrium model for soil heating and moisture transport during extreme surface heating[END_REF]. In such situations, liquid filtration, vapour diffusion and phase change are simultaneous transport processes that must be taken into account to describe global water exchanges accurately in natural soil.

In particular, the physical processes that govern liquid water flow in the small water content range are still in question. In general, numerical modelling based on the Richards' equation requires the complete determination of its hydraulic properties, i.e. the hydraulic conductivity and water retention functions. Numerous attempts have been made to derive theoretically the relative hydraulic conductivity function directly from the water retention curve based mainly on capillary bundle models [START_REF] Burdine | Relative permeability calculation size distribution data[END_REF][START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF]. Although fairly accurate at large water contents where water is held mainly by capillary forces, these descriptions generally underestimate the permeability at small water contents [START_REF] Sakai | Water and vapor movement with condensation and evaporation in a sandy column[END_REF]. The van Genuchten-Mualem permeability function is appropriate for intermediate water contents (w > 0.04 kg kg -1 ), which highlights the shortcomings of capillary bundle models to predict the permeability function in this dry range of the soil (Kébré et al., 2013a). Some theoretical extensions to the full range of water content have been proposed such as a segmented soil water retention function with a non-zero value of residual water content, [START_REF] Zhang | Relative permeability for multiphase flow for oven-dry to full saturation conditions[END_REF].

One challenge to derive closed form equations for the soil hydraulic conductivity function is that it should be mathematically simple, continuously differentiable and physically consistent in the full range of soil water content [START_REF] Peters | Simple consistent models for water retention and hydraulic conductivity in the complete moisture range[END_REF][START_REF] Peters | Simple consistent models for water retention and hydraulic conductivity in the complete moisture range[END_REF][START_REF] Iden | Improving prediction of hydraulic conductivity by constraining capillary bundle models to a maximum pore size[END_REF]Raneem et al., 2016).

During the past decade, several authors have emphasized the major contribution of flow in thin liquid films, which are likely to become important at small water contents [START_REF] Or | Liquid retention and interfacial area in variably saturated porous media: upscaling from single pore to sample scale model[END_REF][START_REF] Tuller | Hydraulic conductivity of variably saturated porous media -Film and corner flow in angular pore space[END_REF]Peters & Durner, 2008b;[START_REF] Tokunaga | Hydraulic properties of adsorbed water films in unsaturated porous media[END_REF][START_REF] Lebeau | A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media[END_REF].

To address this issue, [START_REF] Tuller | Hydraulic conductivity of variably saturated porous media -Film and corner flow in angular pore space[END_REF] proposed a mathematically complex model of unsaturated hydraulic conductivity resulting from film and corner flow. An empirical formulation has been proposed by Peters & Durner (2008b) to extend classical relative hydraulic conductivity functions to take film flow into account. Although easily integrated in the usual numerical models, this approach has been used successfully to fit previous experimental data better. Alternatively, some theoretical approaches have been developed for direct formulations of film flow dynamics [START_REF] Tokunaga | Hydraulic properties of adsorbed water films in unsaturated porous media[END_REF][START_REF] Lebeau | A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Wang | A new theoretical model accounting for film flow in unsaturated porous media[END_REF]. The starting point is to partition water retention into capillary and adsorptive components to introduce the macroscopic contribution of thin film flow. Without any empirical coefficients, these models rely on physical properties that could be difficult to identify such as the Hamaker constant. More recently, correction factors have been proposed to reproduce experimental measures [START_REF] Zhang | Soil water retention and relative permeability for conditions from oven-dry to full saturation[END_REF].

In parallel to these theoretical descriptions, various experimental methods have been proposed to determine the relative hydraulic conductivity function. Multi-step outflow (MSO) [START_REF] Eching | Unsaturated hydraulic conductivity from transient multi-step outflow and soil water pressure data[END_REF][START_REF] Schneider | Novel evaporation experiment to determine soil hydraulic properties[END_REF][START_REF] Nasta | Simplified multistep outflow method to estimate unsaturated hydraulic functions for coarse-textured soils[END_REF], multi-step inflow (MSI) [START_REF] Fujimaki | A multi-step inflow method for estimating hydraulic conductivity at low pressure under the wetting process[END_REF], evaporation (Peters & Durner, 2008a;[START_REF] Schindler | Evaporation method for measuring unsaturated hydraulic properties of soils: Extending the measurement range[END_REF] or condensation [START_REF] Sakai | Water and vapor movement with condensation and evaporation in a sandy column[END_REF] methods provide experimental data to be used in the inverse numerical approach. In general, the use of tensiometers to measure the water potential at different depths sets a lower limit to the applicable water potential of about 100 J kg -1 . Indeed, if suction falls below the air-entry value of the tensiometer or below the water vapour pressure, discrepancies in the measurements are expected [START_REF] Schneider | Novel evaporation experiment to determine soil hydraulic properties[END_REF]. These experimental difficulties would explain why most of the improved hydraulic functions proposed in the literature have been validated against the same set of experimental data published more than 20 years ago (Peters & Durner, 2008b;[START_REF] Lebeau | A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media[END_REF][START_REF] Zhang | Soil water retention and relative permeability for conditions from oven-dry to full saturation[END_REF].

In the above mentioned methods, the water flux is estimated globally through the continuous weighing of samples, and combining the contributions from liquid water and vapour. If the role of vapour diffusion is negligible in the intermediate moisture range, its relative contribution to water transport becomes significant at small water contents. Thus, recent research has explicitly introduced an additional term to describe the unsaturated hydraulic conductivity, K, as the sum of capillary K cap , film K film and vapour parts K vap [START_REF] Peters | Revisiting the simplified evaporation method: Identification of hydraulic functions considering vapor, film and corner flow[END_REF][START_REF] Rudiyanto | A complete soil hydraulic model accounting for capillary and adsorptive water retention, capillary and film conductivity, and hysteresis[END_REF]. This representation, however, assumes implicitly a local equilibrium between liquid and gaseous phases, whereas recent research questions the appropriateness of this hypothesis [START_REF] Smits | An evaluation of models of bare soil evaporation formulated with different land surface boundary conditions and assumptions[END_REF][START_REF] Ouédraogo | Water transfer in soil at low water content. Is the local equilibrium assumption still appropriate[END_REF][START_REF] Massman | A non-equilibrium model for soil heating and moisture transport during extreme surface heating[END_REF]. In the case of dry soil, non-equilibrium between vapour and liquid water has been shown experimentally to require non-equilibrium modelling of water transport. Therefore, with standard methods such as the evaporation method, overestimation of liquid water flux might lead to an incorrect determination of relative hydraulic conductivity. Furthermore, the extrapolation of hydraulic functions towards the dry range can lead to numerical instabilities [START_REF] Schelle | Water retention characteristics of soils over the whole moisture range: a comparison of laboratory methods[END_REF].

The aim of this paper is to propose a simple procedure to identify the hydraulic conductivity function in the dry range that accounts for both capillary and film flows. To be appropriate for non-equilibrium description, liquid flow processes must be identified separately from other transport mechanisms. We did a new kind of soil column experiment without the use of tensiometers on two kinds of sandy soil sampled in Burkina Faso, a sub-Saharan country.

Profiles of the evolution of water content from the experiment were then used as data in a numerical inverse procedure to characterize the relative hydraulic conductivity of these sandy soils. This approach is a simple, robust and cheap characterization procedure. It provides data to overcome the shortcomings of the use of soil water potential in capillary bundle models to compute the hydraulic conductivity function in dry soil.

Materials and methods

Basic soil properties

Two upper layers of a natural Arenosol (IUSS Working Group WRB, 2006) from Nasso, Burkina Faso, (latitude 11.210733°, longitude -4.420386°) were investigated. The first soil sample, denoted NH1, was taken from the top layer of the soil (0-30-cm depth) and the second one, denoted NH2, comes from a greater depth (30-70-cm depth) in the same profile.

The main morphological properties of the two samples are summarized in Table 1. According to the International Society of Soil Science (ISSS) classification, the textures of these samples are sand for NH1 and loamy sand for NH2, which correspond globally to sandy soil. Figure 1 shows their particle-size distributions; the silty and clayey fractions are almost negligible.

Physical analysis was done by dry sieving after washing (France standard NF P94-056, 1996) for particles larger than 80 μm and sedimentation analysis (France standard NF P94-057, 1996) for smaller particles. Bulk density, ρs, of each sample was measured by the core sampler method on undisturbed soil (France standard NF X31-501, 1992). If the soil particle density is assumed to be ρs * = 2650 kg m -3 , the saturated gravimetric water content, wsat, can be determined. The saturated hydraulic conductivity Ksat was measured with a variable head permeameter (France standard NF X30-441, 2008).

Soil water retention curve

Classical experimental devices were used to acquire data to characterize the soil water retention of the two sandy soil samples: (i) pressure plate apparatus at constant temperature (30°C) for large water contents that correspond to the funicular state and (ii) saturated salt solutions for small water contents in the hygroscopic state. For both methods, at equilibrium, gravimetric water content, w, was determined by weighing after oven-drying at 105°C for 48 hours. The results and discussion about water retention function modelling, especially at small water contents, are presented in Kébré et al. (2013b).

Various potentials have been used to characterize the thermodynamic state of water such as suction, matric potential, pressure head, relative humidity, water activity and so on. Following the ideas put forward by [START_REF] Bénet | Measurement of the chemical potential of a liquid in porous media[END_REF] and [START_REF] Ouédraogo | Water transfer in soil at low water content. Is the local equilibrium assumption still appropriate[END_REF], water content is plotted as a function of the absolute value of the water chemical potential rather than pressure head or suction. Indeed, a variety of micro-scale processes governs soil-water interactions and liquid pressure cannot be defined at small water contents. From its thermodynamic definition, the chemical potential can describe the state of water consistently over the whole range of water content, from saturation to dryness. It can be seen as a unifying concept that is well-defined within a thermodynamic framework.

The models used in our research are given below:

 VG80 (van Genuchten, 1980)   r el s a t r = = 1 m n ww S ww       
(1)  FX94 [START_REF] Fredlund | Equation for the soil-water characteristic curve[END_REF]

                            (3)
In these equations, Se is the effective saturation, wr (kg kg -1 ) and wsat (kg kg -1 ) are the residual and saturated water content, respectively, wa (kg kg -1 ) is a curve-fitting parameter that represents the soil water content at l  = 10 J kg -1 , 0  (J kg -1 ) is the chemical potential at oven dryness, usually taken as 10 6 J kg -1 , r  (J kg -1 ) is a parameter used to adjust the lower portion of the curve, i  (J kg -1 ) is a fitting parameter closely related to the air-entry value for the soil and  (kg J -1 ), n and m are fitting parameters.

Laboratory column experiments

As mentioned above, to obtain experimental data that enables us to determine the hydraulic conductivity at small water contents remains important, but measurements of water content profiles are a reliable alternative (Vereeken, 2008). Therefore, the objective was to determine the profiles of gravimetric water content in a homogeneous soil column subjected to a water potential gradient.

A robust and inexpensive experimental approach is proposed. Two portions of wet soil were prepared and stored for 24 hours to homogenize the water content prior to compaction. Each portion of soil was prepared at the desired initial water content to reach the field-measured dry density given in Table 1, i.e. water content in the upper half of the column was w = 0.06 kg kg -1 , whereas it was w = 0.02 kg kg -1 in the lower part. Several soil columns made of PVC tubes (20-cm high and 7.5-cm diameter) were prepared with the portions of soil and were compacted in with a standardized Proctor hammer (Controlab, Paris, France). Finally, both column ends were hermetically sealed to avoid dehydration so that the average water content remained constant.

During one experiment, six soil columns were prepared with identical initial conditions and placed in a thermo-regulated chamber (Figure 2). Because a destructive method was used to determine the water content profile, one soil column was removed from the thermo-regulated chamber at given time intervals. The soil was then extracted from the PVC tube and sliced into several 5-mm thick samples. The water content of each slice was measured by differential weighing between the wet and dry sample after 48 hours of oven-drying at 105°C.

The experiment resulted in a redistribution of water from the upper to the lower part of the column because of the moisture gradient. The water activity was almost one between water contents of 0.02 kg kg -1 and 0.06 kg kg -1 , (Figure 3). Thus, vapour pressure is uniform over the soil column and is equal to the saturated vapour pressure, which prevents phase change and vapour diffusion. Therefore, the only water transfer process that can occur is liquid flow governed by capillary and gravity effects.

Relative hydraulic conductivity

Several attempts have been made to predict the relative hydraulic conductivity function theoretically. Most statistical models conceptualize the pore space as either bundles of cylindrical tubes of uniform size or assemblies of cylindrical capillary tubes of various sizes [START_REF] Lebeau | A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media[END_REF]. The most widely used formulations are those of [START_REF] Burdine | Relative permeability calculation size distribution data[END_REF] and [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF]: Similarly, models of the soil-water retention curve proposed by [START_REF] Fredlund | Equation for the soil-water characteristic curve[END_REF] and

     
Fayer & Simmons (1995) (Equations ( 2) and (3), respectively) did not lead to a closed-form equation, and a numerical solution is required to determine the relative hydraulic conductivity function with Equation (4). We denote the relative hydraulic conductivity models computed from [START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF] integral formulation in conjunction with the [START_REF] Fredlund | Equation for the soil-water characteristic curve[END_REF] and [START_REF] Fayer | Modified soil water retention functions for all matric suctions[END_REF] water retention models as FX94-M and FS95-M, respectively.

Theoretical and experimental investigations suggest that water transport processes that take place at large water contents (low chemical potential) are quite different from those that occur at small water contents (high chemical potential) [START_REF] Or | Liquid retention and interfacial area in variably saturated porous media: upscaling from single pore to sample scale model[END_REF]Peters & Durner, 2008b;[START_REF] Tokunaga | Hydraulic properties of adsorbed water films in unsaturated porous media[END_REF][START_REF] Lebeau | A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media[END_REF]. In dry soil, the water phase is distributed in thin films that do not behave as bulk water.

The pioneering research of [START_REF] Or | Liquid retention and interfacial area in variably saturated porous media: upscaling from single pore to sample scale model[END_REF] to 10 -2 , which suggests that this coefficient should be seen more appropriately as a physical characteristics rather than as a weighting coefficient.

Similarly, [START_REF] Lebeau | A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media[END_REF] describe the relative effect of film flow based on a partitioning function that depends only on the water content. In contrast to using a weighting coefficient, their formulation combines the capillary and film flow models without requiring any additional fitting parameter and is based on a three-parameter water retention function derived from [START_REF] Campbell | Prediction of hydraulic properties of soils using particle-size distribution and bulk density data[END_REF] and [START_REF] Kosugi | General model for unsaturated hydraulic conductivity for soils with lognormal pore-size distribution[END_REF]. as proposed by Peters & Durner (2008b), a weighting function is not required because each term rapidly becomes negligible with respect to the other as water content varies. Therefore, we proposed to use Equation ( 6) directly without any weighting coefficient. With a simple power function for an empirical approach to describe relative film flow (Peters & Durner (2008b), the 'unconstrained new function' for relative hydraulic conductivity is the following:

2 1 r e e e e ( ) ( ) 1 1 m m k S S S S              , ( 7
)
which introduces two fitting coefficients,  and  . This formulation has the advantage of two independent terms to describe capillary and film flow that are not coupled through a weighting factor. In general, it improves identification of the coefficients in inverse analysis.

Computational approach

Experimental conditions ensure that liquid flow occurs only in the soil columns. The numerical modelling is based, therefore, on the classical Richards equation. Following [START_REF] Bénet | Measurement of the chemical potential of a liquid in porous media[END_REF] and [START_REF] Ouédraogo | Water transfer in soil at low water content. Is the local equilibrium assumption still appropriate[END_REF], the mass balance equation is written in term of gravimetric water content and the gradient of chemical potential is taken as the controlling force of the liquid flow:

ll s a t r s 0 g ww K k g t z w z                    (8)
where g (m s -2 ) is the gravity acceleration, r k is the relative hydraulic conductivity function, s a t K (m s -1 ) is the hydraulic conductivity at saturation. The mass density of water l  is taken to be 1000 kg m -3 .

This equation is discretized based on a finite-volume formulation with a one-dimensional regular mesh where the unknown, w, is at the centre of the grid blocks. A first-order upstream scheme is used to describe the convective term. Temporal discretization is done by a fully implicit scheme to obtain unconditional numerical stability. To handle non-linearities, a

Newton-Raphson method ensures accurate convergence with a moderate time step. This numerical modelling was validated with HYDRUS-1D software (Simunek et al., 2008) by comparing the evolution of saturation profiles under various conditions.

To identify the coefficients of the relative hydraulic conductivity function (Equation ( 7)), the inverse analysis was based on numerical evolution of the water content profiles by Equation ( 8). The criterion to be minimized is based on the root mean square error (RMSE), with the difference between experimental and numerical data obtained from the water content profiles at four successive time steps measured from the soil column experiments:

    2 * 11 1 R M S E , , nm i j i j ij w z t w z t nm      , (9)
where * w and w denote the experimental and predicted water contents, respectively, at depth zi and time tj, n = 38 is the number of observation depths and m = 4 is the number of profiles.

The numerical minimization procedure was based on the Nelder-Mead downhill simplex method or Levenberg-Marquardt algorithm. Both approaches lead to identical coefficients, which emphasizes the robustness of this inverse analysis.

Results and discussion

The sorption isotherm and water retention curves are given in Figures 3 and4 for the two sandy soil samples under investigation. The values of the coefficients of Equations ( 1) to (3) are given in Table 2, and the corresponding retention curves are represented in Figure 4(a,b).

Because we aim to predict the liquid water flow process at small water contents, the concept of residual water content is meaningless and we assume wr = 0 in Equations ( 1) and ( 7).

Globally, a good agreement between experimental and predicted data is achieved with the three soil water retention models. However, at very small water contents, the model proposed by [START_REF] Fredlund | Equation for the soil-water characteristic curve[END_REF] leads to the best fit to experimental data.

With identical boundary and initial conditions, the water content profiles predicted with Equation ( 8) for sample NH1 are compared to experimental data in Figure 5. Kébré et al. (2013a), have already shown that the use of the van Genuchten-Mualem model (Equation ( 5))

for the relative hydraulic conductivity, VG80-M, in Equation ( 8) leads to good agreement between predicted and experimental water content profiles for intermediate water contents (w > 0.04 kg kg -1 ), whereas there are marked discrepancies at small water contents. Even the flexibility introduced by taking independent values of m and n [START_REF] Van Genuchten | On describing and predicting the hydrualic properties of unsaturated soils[END_REF] for Equation (4), the related hydraulic conductivity function, does not enable the prediction of water content profiles that match experimental profiles at the small water content range. Although the retention curve models from [START_REF] Fredlund | Equation for the soil-water characteristic curve[END_REF]Xing (1994), FX94, and[START_REF] Fayer | Modified soil water retention functions for all matric suctions[END_REF][START_REF] Fayer | Modified soil water retention functions for all matric suctions[END_REF], FS95, fit best to the experimental soil-water retention data at small water contents (Kébré et al., 2013b), their related relative hydraulic conductivity, FX94-M and FS95-M, used to predict water content profiles from Equation (8) lead to unsatisfactory descriptions of flow processes (see Figure 5b) because the relative hydraulic conductivity is overestimated.

Even if the VG80-M model gives better results than FX94-M and FS95-M, the three approaches fail in the range of small water contents. The results show that statistical models based on the capillary bundle concept are not appropriate for predicting the hydraulic conductivity at very small water contents. The drawback is that these models have assumed that the liquid thermodynamic state is identical over the full range of water contents and that a single concept can be used to describe water transport from saturation to dryness [START_REF] Lebeau | A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media[END_REF]. This shortcoming can be attributed to the neglect of film and corner flows that are probably the dominant flow process in the dry range (Peters & Durner, 2008b).

Coefficients for the relative hydraulic conductivity (Equation ( 7)) identified from soil column experiments with the inverse process are given in Table 3 for the two sandy soil samples, NH1 and NH2. Comparisons between measured and predicted water content profiles are shown in Figure 6, whereas the unsaturated conductivity functions (K = Ksat kr) are plotted as a function of chemical potential in Figure 7. The results from VG80-M and the model proposed by [START_REF] Lebeau | A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media[END_REF] are also given for comparison. It emphasizes the film correction for small water contents (high chemical potential) where it diverges from the VG80-M description, and the unsaturated conductivity is several orders of magnitude larger. There are some discrepancies, however. First, at the interface between the upper and lower parts of the soil columns, the sharp change in water content is not described correctly by numerical predictions. Because of the method of preparation of the soil columns, the bond between the two parts does not seem to be correct because water continuity was not ensured at the beginning of experiment. This might result in local variation in soil properties and introduce an additional hydraulic resistance. Clearly, numerical prediction tends to soften this water content front. Second, the error increases mainly with depth and time because the water has then crossed more heterogeneities [START_REF] Lambot | Laboratory evaluation of a hydrodynamic inverse modeling method based on water content data[END_REF]. Indeed, an important cause of the discrepancies originates from the initial conditions because it is impossible to make the conditions in the six soil columns (replication scheme) exactly the same. This introduces a random bias that can impair the inverse analysis.

The evolution of unsaturated hydraulic conductivity as a function of chemical potential (Figure 7(a,b)) focuses on the two flow mechanisms, capillary flow for low chemical potential with a fast decreasing slope and thin film flow for large chemical potential with a slight decreasing slope. The transition between both states corresponds to the point where adsorptive forces predominate with respect to capillary forces [START_REF] Lebeau | A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media[END_REF]. This transition point for soil sample NH1 corresponds to a chemical potential of 100 J kg -1 and a water content of about 0.04 kg kg -1 . This corroborates previous investigations which have identified that film flow processes start to prevail when soil water potential becomes larger than 100 J kg -1 in various soil textures (sand, loam, silt) [START_REF] Tuller | Hydraulic conductivity of variably saturated porous media -Film and corner flow in angular pore space[END_REF]Peters & Durner, 2008b;[START_REF] Zhang | Soil water retention and relative permeability for conditions from oven-dry to full saturation[END_REF].

Figure 6(a,b) and Table 3 shows that the results are slightly better for the more sandy sample, NH1, than for the loamy sand, NH2, when the coefficient of determination R 2 and the RMSE are compared. The two soil samples differ mainly in the amount of clay (Figure 1). This difference is emphasized through the sorption behaviour because NH2 is more hygroscopic than NH1 (Figure 3). The smaller water content (w = 0.02 kg kg -1 ) imposed at the bottom of the soil columns is somewhat above the hygroscopic domain for the NH1 sample, but this is clearly not the case for the sample NH2.

Conclusions

The term for film flow (a simple empirical relation) introduced in the relative hydraulic conductivity function clearly improves the prediction of water content in the sandy soil samples by increasing water transport in the dry range. We have shown a reasonable agreement between experimental and predicted water content profiles, which underlines the strength of this approach. Therefore, the residual water concept becomes meaningless. Our approach provides a simple and robust experimental method to identify the relative hydraulic conductivity without the requirement of any sophisticated or expensive device. We still observed discrepancies, however, in particular when the clay fraction become non-negligible.

Further investigations are required to address this issue with the experimental approach described above.

For developing nations, especially in regions that suffer drought, the implementation and the popularization of such a simple and cheap approach would contribute to efforts to manage land and water resources effectively for sustained agricultural production. 7)). Plotted on a logarithmic scale.

  soil-water retention curve, q,  and  are specific constants (q= 2,  = 2,  = 1 in[START_REF] Burdine | Relative permeability calculation size distribution data[END_REF] and q= 0.5,  = 1,  = 2 in[START_REF] Mualem | A new model for predicting the hydraulic conductivity of unsaturated porous media[END_REF]).van Genuchten (1980) substituted his model of the soil-water characteristic curve (Equation (1)) in the above statistical model, and derived a closed-form expression for the relative hydraulic conductivity function. One of the most popular models used in soil science is the van Genuchten-Mualem model (denoted by VG80-M) given by: mn  is the shape parameter from Equation (1).

  [START_REF] Tuller | Hydraulic conductivity of variably saturated porous media -Film and corner flow in angular pore space[END_REF] provides a set of equations that are derived from hydrodynamic considerations based on the assumptions of a specific pore geometry to account for capillary and film flows. Their mathematically complex model is not easily associated with one of the classical descriptions of the soil-water retention curve. To simplify, the relative hydraulic conductivity function for unsaturated soil can be described as a linear superposition of the contributions of capillary flow, on the authors.To represent the relative contribution of film flow,Peters & Durner (2008b) introduced a weighting coefficient, . The values of  identified from experimental data range from 10 -7

Figure 6

 6 Figure6(a,b) shows the strength of the experimental and numerical approaches for inverse identification of the relative hydraulic conductivity for the sandy soil under investigation.Profiles of predicted water content are close to those of measured data and cohere to the features of global shape. A comparison of Figures5(a) and 6(a) shows that introduction of the film flow correction clearly improves the description of water transport at small water contents (w < 0.04 kg kg -1 ) without inducing marked modifications at intermediate water content (w > 0.04 kg kg -1 ). The simple power function introduced in Equation (7) is adequate to account for film flow accurately.

Figure 4

 4 Figure 4 Experimental and theoretical soil-water retention curves for samples (a) NH1 and (b) NH2.

Figure 5

 5 Figure 5 Profiles of gravimetric water content along soil columns: a comparison between experimental results and numerical prediction based on the models: (a) VG80-M, (b) FX94-M and FS95-M for soil sample NH1.

Figure 6

 6 Figure 6 Profiles of gravimetric water content along soil columns: a comparison between experimental results and numerical predictions based on the description proposed in Equation (7) for soil samples (a) NH1 and (b) NH2.

Figure 7

 7 Figure 7 Unsaturated hydraulic conductivities as a function of the chemical potential for soil samples (a) NH1 and (b) NH2: a comparison between the results from VG80-M, LK10 and this research (result of Equation (7)). Plotted on a logarithmic scale.

Table 2

 2 Evaluation of parameter values from Equations (1) to (3) for the water retention curve with the statistics R 2 and RMSE for the accuracy of prediction.

	Sample	wa	α/	n	m	µr	µi	R 2	RMSE
	and	/kg kg -1	/kg J -1			/J kg -1	/J kg -1	/%	/kg kg -1
	model
	NH1
	VG80	-	0.9127	1.392	1 -1/n	-	-	99.30	0.004171
	FX94	-	-	1.648	0.9318	50.03	2.086	99.84	0.003319
	FS95	0.03377 0.4375	1.615	0.5069	-	-	99.79	0.003843
	NH2
	VG80	-	1.902	1.297	1 -1/n	-	-	99.52	0.005225
	FX94	-	-	1.431	0.7436	59.42	1.067	99.69	0.004231
	FS95	0.04664 1.544	3.536	0.1271	-	-	99.69	0.004231

Table 3

 3 Coefficients from Equation (7) for relative hydraulic conductivity and evaluation of the accuracy of prediction of gravimetric water content profiles with R 2 and RMSE

		β	γ	R 2 /	RMSE /
				%	kg kg -1
	NH1	2.405× 10 -6	1.981	97.84	0.002141
	NH2	5.490× 10 -4	4.210	94.37	0.003215

Acknowledgements

This work was done within a collaborative framework between the universities of Ouagadougou and Montpellier, and was funded by the French Ministry of Foreign affairs through the French Embassy at Ouagadougou.