Henri Prade
email: henri.prade@irit.fr

POSSIBILITY THEORY IN DECISION SUPPORT SYTEMS

This paper provides a summary of recent research trends, to which the author has taken part, in order to develop tools for flexible case-based querying, evaluation and information search. The fuzzy set and possibility theory framework offers a unified treatment of these issues. Other related research directions are also briefly mentioned.

INTRODUCTION

The interest of fuzzy set techniques in fuzzy information engineering has been recently emphasized [START_REF] Dubois | Fuzzy Information Engineering: A Guided Tour of Applications[END_REF]Dubois and Prade, 1999). Three main components have been distinguished in information engineering, which aim respectively at i) putting the information in an improved format by enhancing, structuring or summarizing it; ii) making it available for the user, either by retrieving it, or by obtaining it through reasoning machineries applied to the available knowledge and factual information; iii) exploiting the information in solving fuzzy constraint satisfaction or optimization problems. The usefulness in information engineering of fuzzy sets and possibility theory [START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF], greatly relies on their ability to provide formats for representing gradual notions such as similarity, preference, and uncertainty which play a basic role in information handling (Dubois and Prade, 1997).

In order to help a user to prepare or to make a decision, information engineering systems, and more particularly decision support systems should offer a variety of functionalities, such as: -selecting items on the basis of user's preferences; -estimating plausible values, or providing recommendations by reasoning from cases or general knowledge; -merging, or summarizing information.

In the following, some fuzzy set and possibility theory-based techniques, to the development of which the author has taken part, and which provides such functionalities are briefly surveyed. The next two sections deal respectively with flexible and case-based querying. Then, a possibilistic approach to case-based prediction and decision on the one hand, and data summarization using fuzzy cardinalities on the other hand are the topics of the last two main sections. The conclusion briefly points out some researches on related issues and an experimental work implementing the above-mentioned ideas.

FLEXIBLE QUERIES

One of the main benefits of allowing for flexible queries is to take into account user's preferences. This has been advocated at length in the fuzzy database literature (e.g., Bosc and Kacprzyk, 1995;[START_REF] Petry | Fuzzy Databases: Principles and Applications[END_REF]; see [START_REF] Bosc | Some approaches for relational databases flexible querying[END_REF] for a unified presentation in the fuzzy set framework of existing proposals for handling flexible queries.

More precisely, the interest of fuzzy queries is twofold: -i) Indeed fuzzy set membership functions [START_REF] Zadeh | Fuzzy sets[END_REF]) are a convenient tool for modeling user's preference profiles. For instance, "he/she is looking for an apartment which is not too expensive and not too far from downtown". In such a case, there does not exist a definite threshold for which the price becomes suddenly too high, but rather we have to differentiate between prices which are perfectly acceptable for the user, and other prices, somewhat higher, which are still more or less acceptable (especially if the apartment is close to downtown). Obviously, the meaning of vague predicate expressions like "not too expensive" is context/user dependent, rather than universal. It reflects the own point of view of the user. So the user's intended meaning of such linguistic expressions has to be elicitated. On ordered attribute domains, it can be easily done in terms of areas of (non-ambiguous) values which are found fully acceptable, or which are fully rejected by the user, the other areas corresponding to gradual transitions of the possibility levels.

-ii) Fuzzy queries, by expressing user's preferences, provide the necessary information in order to rank-order the answers contained in the database according to the degree to which they satisfy the query. It contributes to avoid empty sets of answers when the queries are too restrictive, as well as large sets of answers without any ordering when queries are too permissive.

The large panoply of fuzzy set connectives can capture the different user's attitude concerning the way the criteria present in his/her query compensate or not. Moreover in a given query, some part of the request may be less important to fulfill; this leads to the need for weighted connectives. Thus, an assessment of the importance of the (soft) constraints involved in a request can be taken into account. Especially, using a min-based conjunction, several types of importance weighting, having different semantics can be used; see, e.g., [START_REF] Dubois | Extended divisions for flexible queries in relational databases[END_REF]. Besides, min-based aggregation may lead to a ranking of retrieved items that is insufficiently discriminating because it relies on the comparison of the least satisfied requirements. Indeed min-based aggregation is often criticized for its lack of sensitivity to the change of the arguments. However refinements of the min operator have been proposed, namely the discrimin and the leximin orderings (Dubois et al., 1997), which have not this drawback while remaining compatible with a qualitative handling of the satisfaction levels.

A basic issue underlying flexible querying, and more generally decision problems, is the modeling of preferences. The possibilistic logic framework enables the expression of preferences in various formats, which are indirect ways of specifying a possibility distribution over a referential of possible choices. Indeed, preferences can be expressed in terms of prioritized goals, of sets of possible choices reaching the same satisfaction level, or in terms of conditional statements referring to contexts; see (Benferhat et al., 2001).

More recently, the interest of a bidirectional possibilistic logic setting for distinguishing between positive preferences and negative preferences has been emphasized (Benferhat et al., 2002). Both types of preferences may be a matter of degree, but negative preferences refer to values, to choices, which are rejected by the user (this defines by complementarity what is acceptable for him/her), while positive preferences refer to wishes to be satisfied if possible. Taking a solution in the set of non-rejected possible choices is compulsory, but preference should be given, in the previous set, to solutions having the highest positive appraisal. These ideas can be applied to flexible queries also. For example, imagine that the user is looking for an apartment which is on the one hand "not too expensive and not too far from downtown" (this corresponds to the negative preferences since an apartment which is too expensive, or, too far should be rejected), and on the other hand "preferably on the first floor and quite large" (these are the positive preferences). Positive and negative preferences should be handled separately. Obviously positive preferences should be consistent with acceptable, i.e. non-rejected choices. This means that the fuzzy set of possible choices satisfying the positive requirements should be revised in order to having it included in the fuzzy set of acceptable choices. For example, an expensive first floor apartment has not to be retrieved for the user. Note also that the requirements issued from negative preferences have to be combined conjunctively (they are constraints in essence). By contrast, positive requirements are more liable to be combined disjunctively, using leximax for instance (since it is obviously better to satisfy several positive requirements rather than one), or to be combined by means of compensatory operators.

QUERYING BY EXAMPLES

The user is not always able to easily express his request even in a flexible way. First, it may be more convenient for him to express what he is looking for from examples. Second, since he may have absolutely no a priori knowledge of the amount of retrievable documents (for a given request), it may be useful if the system is able to guide him about what exists, by incrementally building queries from examples. Querying based on examples, for eliciting user's preferences, can provide the necessary information for building a query. Thus, the user may say to what extent a few examples of items are representative of what he is looking for by using some (finite) similarity scale. Then, relevance of stored items should be evaluated in terms of their similarity with respect to these examples and may be counter-examples. We are then close to fuzzy case-based reasoning.

Suppose examples as well as counter-examples are described in terms of precisely known attribute values. These attributes are assumed to be relevant and sufficient for describing their main features from a user's point of view. Then a request looking for the items, which are similar to, at least one example (w.r.t. all the attributes) and which are dissimilar to all the counter-examples (each time w.r.t. at least one attribute). Moreover we can take into account the extent to which each example or counter-example is highly representative, and the importance of the (relevant) attributes. Thus, the more similar an item t is w.r.t. (at least) a representative example and the more dissimilar t is w.r.t. all representative counter-examples, the more possible t is eligible for the user. This evaluation might be also improved by requiring the similarity of t with most examples only (see, e.g., [START_REF] Dubois | Weighted fuzzy pattern matching[END_REF] for the introduction of a soft quantifier in a weighted min expression).

Suppose for simplicity that all the importance and representativeness weights are equal, the items t are then rank-ordered in terms of the function m(t) = min[max j min i S i (a i (t), a ij ), min k max i 1-S i (a i (t) ,b ik )],

where a i (t) is the value of attribute i for item t, a ij is the value of attribute i for example j, b ik the value of attribute i for counter-example k, where S i is a fuzzy similarity relation on the attribute domain of i (S i is supposed to be reflexive and symmetrical). Clearly, S i ( . , a ij ) defines a fuzzy set of values close to a ij , while 1-S i ( . , b ik ) defines a fuzzy set of values not similar to b ik , for each attribute i (it may be replaced by a smaller fuzzy set of values significantly different from b ik ). Thus this expression also provides the starting point for generating a description of the documents which are looked for. This description may then be used for interface needs with the user. Indeed, letting t unspecified in the expression, it defines a compound fuzzy set expression, which can be logically analysed. Besides, we may think of hybrid queries made of examples and of classical (fuzzy) restrictions expressing what attribute values are undesirable. See [START_REF] Dubois | Fuzzy logic techniques in multimedia database queyring: A preliminary investigation of the potentials[END_REF] for more details and discussions.

CASE-BASED PREDICTION AND DECISION

Case-based reasoning, in general, assumes the following implicit principle: "similar situations may give similar outcomes". Thus, a similarity relation S between problem descriptions or situations, and a similarity measure T between outcomes are needed. This implicit CBRprinciple can be expressed in the framework of fuzzy rules as, "the more similar are the values of the situation attributes in the sense of S, the more possible the similarity of the values of the outcome attributes in the sense of T" [START_REF] Dubois | Fuzzy set modelling in case-based reasoning[END_REF]. Given a situation s 0 associated to an unknown outcome t 0 and a current case (s, t), this principle enables us to conclude on the possibility of t 0 being equal to a value similar to t. This acknowledges the fact that, often in practice, a database may contain cases which are rather similar with respect to the problem description attributes, but which may be distinct with respect to outcome attribute(s). This emphasizes that case-based reasoning can only lead to cautious conclusions. This can be modeled in terms of the possibility rule [START_REF] Dubois | What are fuzzy rules and how to use them[END_REF] "the more similar s and s 0 , the more possible t and t 0 are similar". Then the fuzzy set of possible values t' for t 0 with respect to case (s, t) is given by π t 0 (t') = min(S(s, s 0 ), T(t, t')).

As it can be seen, what is obtained is the fuzzy set T(t, . ) of values t' which are T-similar to t, whose possibility level is upper bounded by the global degree S(s, s 0 ) of similarity of s and s 0 . The max-based aggregation of the various contributions obtained from the comparison with each case (s, t) in the memory M of cases, acknowledges the fact that each new comparison may suggest new possible values for t 0 . Thus, we obtain the following fuzzy set E s 0 of possible values t' for t 0 : E s 0 (t') = max (s, t)∈M min(S(s, s 0 ), T(t, t')).

Note that this latter expression can be put in parallel with the evaluation of a flexible query. Indeed here a situation is described in terms of a collection of attributes, and here we are looking for cases, which are constrained to be similar to s 0 w.r.t. the attributes. But rather than exhibiting the fuzzy set of cases similar to s 0 , we produce the fuzzy set of the fuzzy sets of values similar to the t-attribute values of these cases (a level 2 fuzzy set reduced here to an ordinary fuzzy set by performing the union of the fuzzy sets of similar values). See [START_REF] Dubois | Flexible control of case-based prediction in the framework of possibility theory[END_REF] for more details and a comparative discussion with other approaches to instance-based prediction.

The above framework can be adapted into a case-based decision machinery. If the memory of cases stores a set of triples of the form (situation, decision, result), and the space of situations is equipped with a fuzzy similarity relation, then a good decision in a situation s 0 is a decision which led to good results in all the situations similar to s 0 in the past. The requirement on 'all the situations' may be relaxed into 'most situations', or even into 'at least one situation', moving from a pessimistic to an optimistic attitude; see (Dubois and Prade, 2001). This is the possibilistic counterpart of a proposal originally made by [START_REF] Gilboa | Case-based decision theory[END_REF] in a sum/product setting.

The expression of preferences is not the only motivation underlying the use of fuzzy sets. The use of a fuzzy label may be rather a way for avoiding to refer to a precise category with arbitrary boundaries. In an evaluation process, this may enable us to check if a result does not vary too much with the slightly different possible crisp interpretations of a considered fuzzy label, and if there is some brittleness of the result with respect to these different interpretations. This idea applies, for instance, in the following data summarization method from a database, in terms of fuzzy association rules [START_REF] Bosc | Fuzzy summarization of data using fuzzy cardinalities[END_REF].

A set of a (small) number of attributes of interest is supposed to be chosen. The idea is to figure out to what extent one of the attributes (say the "output attribute" B) is influenced by the others (called "input attributes" A i ). Each of the input attribute domain are supposed to be equipped with a fuzzy partition made of a relatively small number (2 to 5) of fuzzy sets A ij or B k respectively. These fuzzy partitions are supposed to be meaningful for the user and/or to be in agreement with the way the attribute values are scattered. For a given tuple (A 1j ,..., A nj, B k ), we compute the fuzzy cardinalities of the set of items of the database which are "A 1j and ... and A nj and B k " , and of the set of items which are "A 1j and ... and A nj ". From which the fuzzy-valued confidence and support of the fuzzy association rule "if x is A 1j and ... and A nj then x is B k " can be easily obtained. In order to obtain interesting association rules, the B k are not chosen beforehand, but on the basis of the possibilistic case-based reasoning machinery presented in Section 4, which after a small adaptation, can provide the fuzzy set F of possible output values for a class of situations described in terms of labels A ij of the fuzzy partitions of the input attributes. Then B k can be taken as F, or of a subpart of F if F is too large or not convex.

Thus, what can be obtained is a set of fuzzy association rules summarizing a database, in terms of the labels of the fuzzy partitions of the input attribute domains, and in terms of fuzzy sets of the output attribute domains computed by means of the possibilistic case-based prediction method. The interest of computing fuzzy cardinalities (and thus fuzzy confidence and fuzzy support degrees) is to figure out if the confidence in a considered fuzzy association rule varies, or not, with the understanding of the involved fuzzy labels.

When querying a database, the user may have intricate goals in mind: retrieving items corresponding to some requirements, referring to similar cases, using stored data for prediction purposes, trying to figure out range of attribute values for classes of items, looking for rules summarizing the data in order to have a better understanding of the contents of the database. In all these operations flexibility can be introduced with profit. The ideas briefly surveyed in this overview are illustrated on the website http://www.irit.fr/PRETI using a database describing houses to be rent; see also [START_REF] De Calmes | A fuzzy approach to flexible case-based querying: Methodology and experimentation[END_REF].

This overview has not covered some other issues which may be of interest in decision support systems, and where the fuzzy set and the possibility theory framework offers valuable techniques, especially, for reasoning with fuzzy rules from general knowledge and examples [START_REF] Ughetto | Implicative and conjunctive fuzzy rules-A tool for reasoning from knowledge and examples[END_REF], or for merging information (Dubois, Prade and Yager, 1999), and for fusing preferences (Benferhat et al., 2002).

Besides, there has been an increasing interest for recommender systems [START_REF] Resnick | Recommender systems[END_REF] in the last past years. Then fuzzy set-based techniques appear to be promising for providing case-based recommendations, subject to flexible constraints, to a user [START_REF] Yager | Fuzzy logic methods in recommender systems[END_REF][START_REF] Perny | Preference-based search and machine learning for collaborative filtering: the "Film-Conseil" movie recommender system[END_REF][START_REF] Hüllermeier | Case-based recommendation and decision support[END_REF].

Acknowledgements

The ideas presented here have greatly benefited from joint discussions and works with Salem Benferhat, Patrick Bosc, Martine de Calmes, Didier Dubois, Eyke Hüllermeier, Souhila Kaci, Olivier Pivert, Laurent Ughetto, and Florence Sèdes.


J Vincent
email: vincent@laplace.univ-tlse.fr

P Borderies

J R Poirier
email: poirier@laplace.univ-tlse.fr

V Gobin

Long-Range Three-Dimensional Ground Wave Propagation Modeling Over Flat, Irregular Terrain

Keywords: Ground wave propagation, hybridization, lowfrequency wave, problem of Sommerfeld, three-dimesional (3-D) FDTD

Ground wave propagation of low-frequency (LF) electromagnetic waves is well known for the canonical case of a flat, azimuthally homogeneous, soil. Moreover, it is well investigated for realistic propagation problems, some of which include Earth's curvature, irregular and lossy terrain profiles, and mixed paths. In this paper, kinds of environment are considered in the far-field of the radiating antenna with an original methodology. It rests on the use of hybridization of finite-difference in time-domain (FDTD) with the direct numerical integration of Sommerfeld-type integrals which is also presented and validated. This hybridization uses domain decomposition and Huygens' surfaces in a stratified dielectric medium in three-dimensional (3-D) space. The approach is validated with respect to a reference solution both in the canonical flat ground case and in a hilly one.

I. INTRODUCTION

U SUALLY the electromagnetic problem of low-frequency (LF) radiating antennas is decomposed into two parts: 1) the near-field radiation emitted by the antenna and 2) the propagation of this near-field in the far range for the ground wave [START_REF] Deminco | Propagation prediction techniques and antenna modeling (150 to 1705 kHz) for Intelligent Transportation Systems (ITS) broadcast applications[END_REF]. LF electromagnetic field propagation over the ground is well known for the canonical case of a flat homogeneous soil [START_REF] Deminco | Propagation prediction techniques and antenna modeling (150 to 1705 kHz) for Intelligent Transportation Systems (ITS) broadcast applications[END_REF], [START_REF] Norton | The propagation of radio waves over the surface of the earth and in the upper atmosphere[END_REF], and in the case of a stratified one [START_REF] Wait | Electromagnetic Waves in Stratified Media[END_REF] through approximate analytical approaches which are widely used. An appropriate numerical method for solving the propagation problem over long distances is the solution of parabolic equations (PE) above stratified media [START_REF] Levy | Parabolic Equation Methods for Electromagnetic Wave Propagation[END_REF], [START_REF] Ozgun | PETOOL: MATLAB-based one-way and two-way split-step parabolic equation tool for radiowave propagation over variable terrain[END_REF]. Three-dimensional (3-D) vector PE approach exists [START_REF] Janaswamy | Path loss prediction in the presence of buildings on flat terrain: A 3D vector parabolic equation approach[END_REF], but PE method is mostly used in two dimensions [START_REF] Akleman | A novel MoM and SSEP-based groundwave propagation field-strength prediction simulator[END_REF] and it is not a full-wave method [START_REF] Fabbro | Apparent radar cross section of a large target illuminated by a surface wave above the sea[END_REF]. Perturbation methods using the compensation theorem treat typical forms of radial heterogeneity such as mixed paths and irregular ground [START_REF] Maclean | Radiowave Propagation Over Ground[END_REF], [START_REF] Apayadin | MATLAB-based FEM-Parabolic equation tool for path-loss calculations along multi-mixed-terrain paths[END_REF].

However, these techniques do not take into account the 3-D topography near the propagation path. Then, there is still a need of rigorous numerical computation of the radiated field when the distant irregularity does not correspond to the domain of validity of the usual methods or when there is a need of a reference solution for the interaction of the ground wave with a very general discontinuity.

Finite-difference in time-domain (FDTD) method is a wellknown method used for computational electromagnetics [START_REF] Taflove | The Finite-Difference Time Domain Method[END_REF]. Among its advantages is the ability to treat very general geometries involving heterogeneous complex media in volume and surface [START_REF] Chanal | Homogenization and scattering from heterogeneous media based on finite-differencetime-domain Monte Carlo computations[END_REF]. In [START_REF] Akleman | Realistic surface modeling for a finitedifference time-domain wave propagator[END_REF], a time-domain wave propagator based on two-dimensional (2-D) FDTD computes the electromagnetic field propagation over Earth's surface. In [START_REF] Zhou | LF ground-wave propagation over irregular terrain[END_REF] and [START_REF] Zhou | Application in low-frequency ground-wave propagation of parallel FDTD based on GPU[END_REF], bi-dimensional FDTD is applied to obtain the electric field both in the near and the far-field over an irregular terrain, with the constraint of a radial cylindrical geometry. FDTD in three dimensions has been applied to study the near-field radiation of wire antennas in a complex environment [START_REF] Vincent | Simulations and effects of natural environments on low frequency antennas with 3D FDTD method[END_REF]. However, FDTD is not applicable in three dimensions at large distance because of the huge amount of computational resources which are required and also because of the numerical problems arising at very long distances (numerical dispersion, precision, boundary conditions, etc.).

In this paper, an original hybrid approach using the FDTD method is proposed. It rests on the use of the direct numerical integration of Sommerfeld's integrals, which is also presented, and its hybridization with 3-D FDTD applied to the surroundings of the irregularity under consideration, yielding an approach free of the former numerical problems. This hybridization technique aims at treating complex kinds of environment located in the far-field.

Section II is dedicated to the implementation of the hybridization in the 3-D FDTD method. First the Sommerfeld's half-space problem is solved with a numerical integration along the positive real axis using an adaptive algorithm called NISP (numerical integration of Sommerfeld's problem). Then, the new hybrid NISP-3-D FDTD method is presented. The validation and illustration of the method are presented in Section III.

II. PRINCIPLES OF HYBRIDIZATION

A. Sommerfeld Half-Space Problem

In [START_REF] Sommerfeld | Propagation of waves in wireless telegraphy[END_REF], Sommerfeld computed the radiation of an infinitesimal vertical electric current element I = I 0 dlz located at a height h above a homogeneous dielectric half-space. The XOY plane separates the two media, the medium 1 is free space in the upper half-space with constants µ 0 ,ǫ 0 and soil is the medium 2 in the lower half-space with constants µ 2 = µ 0 ,ǫ 2 , and σ 2 . The geometry is presented in Fig. 1.

Using Hertz vector potentials and the boundary conditions at height z =0, one obtains in (1) the exact formulas of all components of the electromagnetic field E and H in both media 1 and 2 [START_REF] Robin | Rayonnement d'un Dipôle Électrique en Présence de Deux Milieux Séparés par un Plan[END_REF]. The wavenumbers in free space k 0 and soil k 2 are given by k i =[-jωµ i (σ i + jωǫ i )] 1/2 . [See [START_REF] Deminco | Propagation prediction techniques and antenna modeling (150 to 1705 kHz) for Intelligent Transportation Systems (ITS) broadcast applications[END_REF], shown at the bottom of the page.]

The free space components in (1) are expressed as the sum of three terms: the first represents a wave traveling directly from the current element I 0 dl; the second can be seen as radiated from a negative image current in the ground at -h; and the third is related to a surface wave term. The distance between the image current at distance h below the plane and the observation point is R ′ =[r 2 +(z + h) 2 ] 1/2 . J 0 and J 1 are Bessel functions of the first kind for orders 0 and 1, respectively [START_REF] Waston | A Treatise on the Theory of Bessel Functions[END_REF].

H 2 φ (r, z)= k 2 2 I 0 dl 2π +∞ 0 J 1 (λr) λ 2 dλe z √ λ 2 -k 2 2 -h √ λ 2 -k 2 0 k 2 2 λ 2 -k 2 0 + k 2 0 λ 2 -k 2 2 E 2 ρ (r, z)= jωµ 0 I 0 dl 2π +∞ 0 J 1 (λr) λ 2 -k 2 2 λ 2 dλe z √ λ 2 -k 2 2 -h √ λ 2 -k 2 0 k 2 2 λ 2 -k 2 0 + k 2 0 λ 2 -k 2 2 E 2 z (r, z)=- jωµ 0 I 0 dl 2π +∞ 0 J 0 (λr) λ 3 dλe z √ λ 2 -k 2 2 -h √ λ 2 -k 2 0 k 2 2 λ 2 -k 2 0 + k 2 0 λ 2 -k 2 2
(1)

H 1 φ (r, z)= I 0 dl 4π ⎡ ⎣ - ∂ ∂r e -jk0R R + ∂ ∂r e -jk0R ′ R ′ + +∞ 0 J 1 (λr) 2k 2 2 λ 2 dλe -(z+h) √ λ 2 -k 2 0 k 2 2 λ 2 -k 2 0 + k 2 0 λ 2 -k 2 2 ⎤ ⎦ E 1 ρ (r, z)=- jωµ 0 I 0 dl 4πK 2 0 ⎡ ⎣ ∂ 2 ∂z∂r e -jk0R R - ∂ 2 ∂z∂r e -jk0R ′ R ′ + +∞ 0 J 1 (λr) λ 2 -k 2 0 2k 2 2 λ 2 dλ e -(z+h) √ λ 2 -k 2 0 k 2 2 λ 2 -k 2 0 + k 2 0 λ 2 -k 2 2 ⎤ ⎦ E 1 ρ (r, z)=- jωµ 0 I 0 dl 4πK 2 0 ⎡ ⎣ ∂ 2 ∂Z 2 + k 2 0 e -jk0R R - ∂ 2 ∂Z 2 + k 2 0 e -jk0R ′ R ′ + +∞ 0 J 0 (λr) 2k 2 2 λ 3 dλ e -(z+h) √ λ 2 -k 2 0 k 2 2 λ 2 -k 2 0 + k 2 0 λ 2 -k 2 2 ⎤ ⎦ .

B. Numerical Integration

The integrands in all components in both media are the product of three factors. The first element is the quotient whose denominator is

(k 2 0 (λ 2 -k 2 2 ) 1/2 + k 2 2 (λ 2 -k 2 0 ) 1/2
). An oscillating term depending on the distance r of the observation point is created by the spherical Bessel functions J 0 and J 1 . Finally, there is an exponential attenuation factor depending on the height of the dipole h and the observation point z. The system has a pair of complex poles given by s

2 = k 2 0 k 2 2 /(k 2 2 + k 2 0
). There is no singularity on the integration path and the integrand is convergent but very oscillating along the positive real axis. It can be computed by a numerical integration with a small error. To determine the best strategy, the behavior of the integrand functions is detailed. In the following, the z-directed electric field integral in ( 2) is only studied but the approach is identical for the other five components of the electromagnetic field

N (r, z)= +∞ 0 n (λ, r, z) dλ = +∞ 0 J 0 (λr) 2k 2 2 λ 3 dλe -(z+h) √ λ 2 -k 2 0 k 2 2 λ 2 -k 2 0 + k 2 0 λ 2 -k 2 2 . (2)
For a current element located at h =25m above a medium wet soil (ǫ r =15,σ =10 -2 S•m -1 ) and an observation point located at z =1m and r =1km, the real and imaginary values of the integrand are plotted in Fig. 2 as a function of the integration parameter λ. The four main parts of the integrand are bounded by the vertical lines. The first interval value is small; the function is regular and has low oscillations. The second is the most important contributor because the function is less regular. In fact, it contains a singularity which corresponds to the positive real part of the complex pole s. The third part is longer to evaluate than the previous two because both real and imaginary values of the integrand oscillate due to the Bessel function. The last part is dominated by the exponential attenuation term and may not be calculated because it is negligible despite the oscillating behavior. These intervals have different properties and numerical integration does not have the same convergence behavior. Integration functions are available in many software products; however, changes in the integration scheme are complicated; it is therefore difficult to implement an optimal strategy.

Indeed the previous analysis shows that different low-and high-order integration methods have to be used for each part of the integrand. Third degree Newton-Cotes method is chosen for parts one and four integrations and Gauss-Legendre quadrature with more than ten points is chosen for the integrations of parts two and three.

An adaptive algorithm for numerical integration is realized to ensure a good compromise between computational time and numerical accuracy [START_REF] Quarteroni | Numerical mathematics[END_REF]. The algorithm performs the integration independently on the four intervals with minimal discretization ∆ min. At each iteration, the step is halved for a new integration until the fixed convergence or the maximum number of iterations is reached. The four integration intervals are heuristically defined as follows:

I 1 =[0;0.9 •ℜe (s)] ; I 2 =[0.9 •ℜe (s);1.1 •ℜe (s)] I 3 =[1.1 •ℜe (s);λ η ];I 4 =[λ η ;3• λ η ] (3)
where ℜe (s) is the real part of complex pole s and λ η is determined from 1% of the maximal value of the integrand and the exponential attenuation factor given in the following:

k 0 ≤ λ η = 1 (z + h) • ln 100 max {n (λ, r, z)} . (4)
The algorithm NISP has been tested for vertical component E z radiated at 100 kHz by a dl =25m long current element located at h =25m over different soils, along the direction of propagation r at height z =1m. The soil is characterized with the dielectric constant ǫ r and the conductivity σ, which depend on the humidity, see Table I, and can be found in [START_REF] Hippel | Dielectric Materials and Applications[END_REF].

The minimal discretization of NISP is set to ∆ min = 2π/(10r) because of the 2π-periodic Bessel function. The convergence is computed with an absolute relative error, its value is fixed to cv =10 -3 , and the maximal number of iteration is max =4. Fig. 3 shows the comparison between two different methods: the numerical integration and compared with the analytical formulas given by Norton [START_REF] Norton | The propagation of radio waves over the surface of the earth and in the upper atmosphere[END_REF]. We can see that there is a good agreement between these two approaches on the upper half-space.

Consequently, NISP may be used to compute the radiated field whatever the distance beyond a wavelength. It has to be noted that it yields the radiated field also in the ground: it is then relevant to initialize the hybrid method presented in the next section.

C. Hybrid NISP-3-D FDTD Method

The main objective of this section is to present a method able to compute numerically the effects of a perturbation in/on the soil at large distance from the radiating antenna. It rests on the hybridization of the approach presented in Section II-B and the 3-D FDTD algorithm. The scattered-field/total-field (SF/TF) formulation has been first proposed in [START_REF] Mur | Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations[END_REF]: along a Huygens' surface surrounding the scatterer, the incident plane wave is introduced on the six faces with the goal of dividing the computational domain into a total-field zone and a scattered-field one. In [START_REF] Reineix | Time domain modelling techniques improvement for new EMC problems[END_REF] and [START_REF] Mouysset | A multidomain decomposition method to solve electromagnetic scattering problems in time domain[END_REF], still in free space, the incident plane wave is substituted by the field radiated by a remote source, leading to multidomain approaches. In [START_REF] Demarest | FDTD modeling of scatterers in stratified media[END_REF], again with an incident plane wave, free space is substituted by a stratified medium. As a simple example, the SF/TF one-dimensional (1-D) formalism is represented in Fig. 4. Considering the total electromagnetic field as a sum of the incident and the scattered electromagnetic field (5), one can write ( 6) and ( 7): the nth time step of the FDTD scheme applied on the whole Huygens' surface located at grid point i S in a dielectric medium (µ 0 ,ǫ = ǫ r ǫ 0 ,σ) with space step ∆x and time step ∆t. All components of the electromagnetic field on the whole 3-D Huygens' surface are easily deduced from ( 5)-( 7), shown at the bottom of the page.

In this paper, the incident field is the ground wave created by the remote source which is computed by NISP and the computation is done in a medium which is stratified.

The computational domain includes soil and free space above, both of them bounded by adapted absorbing boundaries. NISP algorithm allows to identify the values of the electromagnetic field at any point in space in the frequency domain: it is used to compute at all times the incident field both in free space and in the soil along the six faces of the Huygens' surface with a sine excitation at the frequency of interest. It is expected that when the soil is flat, the total field is the incident one: the unperturbed ground wave is retrieved. When there is some irregularity either in the soil composition or on the relief (Fig. 5), then a scattered field is generated.

E TOT = E INC + E SCAT , H TOT = H INC + H SCAT (5) E TOT n +1 i S = 2ε -σ∆t 2ǫ + σ∆t E TOT n i S + 2∆t (2ε + σ∆t)∆x H TOT n + 1 2 i S + 1 2 -H SCAT n + 1 2 i S -1 2 - 2∆t (2ε + σ∆t)∆x H INC n + 1 2 i S -1 2 (6)
H SCAT n + 1 2 i S -1 2 = H SCAT n -1 2 i S -1 2 + ∆t µ 0 ∆x E TOT n i S -E SCAT n i S -1 - ∆t µ 0 ∆x E INC n i S (7)

III. RESULTS

A. Validation of the Method

In order to validate the method, various scenarios were implemented. First the hybridization technique is used with a homogeneous flat ground (medium wet) and compared with the analytical formulas given by Norton [START_REF] Norton | The propagation of radio waves over the surface of the earth and in the upper atmosphere[END_REF]. The antenna radiating at f = 100 kHz is dl =25m long and at height h =25m. The electromagnetic field is computed at height z =76m along the distance r from 1 to 100 km with several reduced domains. The computation space for each hybridization implementation is a 100 cells side cube with the mesh size ∆x =∆y = ∆z =75m and the ground surface (z =0m) is located at the middle of the z-directed cells. Thus the total computational volume is 7500 m × 7500 m × 2500 m. The volume inside the Huygens' surfaces is a 60 cell side cube centered in the domain which means that 60 × 60 × 30 cells mesh the ground inside the volume. Uniaxial-PML is used to terminate the 3-D computational domain [START_REF] Gedney | An anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices[END_REF] and the NISP parameters are set to ∆ min =2π/(10r) and max =4. Fig. 6 shows the results of the electromagnetic field computations. It validates the method when applied to an infinite flat ground since it agrees with the reference method for a wide range of distances to the source.

Then the hybridization technique in the vicinity of the source is compared to 3-D FDTD method including the source when both methods are relevant (FDTD is here the reference solution since the distance of observation is short enough). A 250-m high hill is located at 3.4 km of the infinitesimal antenna previously described, over a medium wet soil. It is modeled with a stack of 10 dielectric blocks which are all 25 m high (Fig. 7). Horizontally, the largest block is 1.5 km × 3 km; the second is 1.35 km × 3 km, up to the top which is 0.15 km × 3 km. For the FDTD, the current element and the hill are computed in a 240 cells side cube with ∆x =∆y =75m and ∆z =25m. The previous monopole antenna is centered inside the computed volume. The same geometry of the hill and the space steps are used with a 100 cells side cube for the hybridization. The volume inside the Huygens' surfaces is still a 60 cell side cube centered in the domain. The normalized magnitude of the electric field at constant elevation 175 m over the soil and the hill is displayed in Fig. 8. Both methods are in agreement better than 0.3 dB and a relative error of 3.39%, which demonstrates the accuracy of the hybridization technique when an obstacle is present.

B. Study of Ground Wave Propagation With Obstacles

To complete the study of obstacles along the ground wave propagation path, several natural elements are remotely located from the current element. The FDTD parameters, the source, the ground, the distance, and the size of the Huygens' surface are the same in all these computations. The current element at f = 100 kHz is dl =25m long and at height h =25m. The FDTD mesh for each hybridization implementation is a 100 cells side cube with the mesh size ∆x =∆y =75m and ∆z =25m; the ground surface (z =0m) is located at the middle of the z-directed cells. The volume inside the Huygens' surface is a 60 cells side cube and is located at 23 km from the source. Since the volume bounded by the Huygens' surfaces is the same for all the following studies, the incident electromagnetic field computed by NISP is stored in a file. Thus, only the nature of the obstacle changes between each computation which will save time. The chosen obstacles in this section are a hill, a valley, a lake, and a forest.

First, the geometry of the hill previously used for the validation is translated along x-axis (Fig. 7). Thus the total electric field around the same hill, remotely located at 24.5 km from the current element is computed in the XOZ vertical plane at y =0 (Fig. 9) and XOY horizontal plane at z =25m (Fig. 10). Then these values of the total electric field in the air are compared to the case of a flat homogeneous soil with (8) applied in the XOZ vertical plane at y =0and z>0

η = |E natural element | dBV•m -1 -|E f lat soil | dBV•m -1 . (8)
Results are shown in Fig. 11, in which a field enhancement brought about by the natural element is determined by the positive values of η and conversely a decrease by a negative value. On the top of the hill, a 3-dB field enhancement is observed and some interferences appear in front of it (1.5 dB). Further than the obstacle, after a shaded area there is no visible effect beyond the distance of 27 km as η ≈ 0 dB. Now other natural elements such as valley, a lake, and a forest are computed at the same distance of 23 km from the current element. The dimensions of the valley are the same as those of previous hill, the dielectric constants and the z-axis are reversed in order to build the hollow in the ground. In Fig. 12, contrasts are opposite to the previous case. The lake is a square-shaped dielectric block with dielectric constants (ǫ r = 80; σ =5• 10 -2 S•m -1 ). It is located at the surface of the ground and its dimensions are 1.5-km large and 100-m deep (Fig. 13).

The forest is a 25-m height square-shaped dielectric block with dielectric constants [START_REF] Tewari | Evaluation of relative permittivity and conductivity of forest slab from experimentally measured data on lateral wave attenuation constant[END_REF] (ǫ r =1.065; σ =10 -3 S•m -1 ). It is located at the surface of the ground and its dimensions are 1.5-km large. Edge enhancements are presented in Fig. 14.

The results shown do not exceed 3 dB of contrast in all cases. One can conclude from these observations that all natural elements have an effect on the electromagnetic field which is in their vicinity: from 0.3 dB with the lake, 1 dB with the forest, to of these obstacles does not affect the long-range ground wave propagation.

IV. CONCLUSION

In this paper, a method to study the propagation of ground wave at low frequencies with the presence of an irregular halfspace has been studied with a 3-D approach.

First, the NISP algorithm was proposed to compute the electromagnetic field emitted by a vertical current element over an infinite flat ground and validated for the relevant values of soils parameters.

Then, an hybrid NISP-3-D FDTD method was introduced to compute the field in the vicinity of remote irregularities and validated both in the case of an infinite half-space and in the presence of a hill. This study includes the exploitation of this technique to treat various kinds of ground irregularities with emphasis on the three dimensions disturbance effects. These irregularities are computed over simplified models, located at 24.5 km from the current element and computed with the hybridization technique. The obtained values of the total electric field are compared to the case of a flat homogeneous soil. The observed natural elements create a field enhancement up to 3 dB in its vicinity but they do not affect the ground wave propagation beyond them.

Fig. 1 .

 1 Fig. 1. Vertical z-directed current element I 0 dl at height h above a planar lossy ground with electrical parameters (µ 2 ,ǫ 2 ,σ 2 ).

Fig. 2 .

 2 Fig. 2. Real (top) and imaginary (bottom) parts of the integrand in (2) as a function of λ, with a medium wet soil, h =25m,z =1m, and r =1km. The four main parts are bounded by the three vertical lines.

Fig. 3 .

 3 Fig. 3. Comparison of the vertical electric field component in Sommerfeld's problem solved with two different methods: NISP and numerical approximation of Norton. A dl =25m long current element at frequency f = 100 kHz is at height h =25m above different soils, the magnitude of the vertical electric field is computed a long distance r at height z =1m.

Fig. 4 .

 4 Fig. 4. SF/TF representation for 1-D FDTD. The Huygens' surface separates the computational space into regions: a region where the total field is considered and the scattered field is considered in the other one.

Fig. 5 .

 5 Fig. 5. NISP-3-D FDTD Hybridization: a vertical z-directed current element source I 0 dl creates the electric and magnetic fields (E inc ,H inc ) which are incorporated as incident wave sources in the SF/TF FDTD formulation.

Fig. 6 .

 6 Fig. 6. Solution comparison between the NISP-3-D FDTD hybridization and Norton's analytic formulas. The electromagnetic field along the distance r from 1 to 100 km at height z =75m is emitted by a 25-m long vertical current element located at height h =25m along distance r at height z =1m.

Fig. 7 .

 7 Fig. 7. 3-D representation of the computed hill with the hybridization. A stack of 10 dielectric blocks with different widths are located on the flat ground.

Fig. 8 .

 8 Fig. 8. Solution comparison between the NISP-3-D FDTD hybridization and the 3-D FDTD in this vicinity of the source and at a constant elevation 175 m over the soil. A hill is located at 3.4 km, electric field values are plotted in dBV•m -1 unit.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Magnitude of the electric field, computed in the XOZ vertical plane, with the hybridization at 23 km of the antenna. A hill is located at 24.5 km and the values are plotted in dBV•m -1 .

Fig. 12 .

 12 Fig. 12. Difference between the value of the electric field in the air with the valley and the homogeneous flat soil. It is computed with (5) in the XOZ vertical plane at z>0 at 23 km of the antenna. The natural element is located at 24.5 km and the values of η are plotted in dB.

Fig. 13 .

 13 Fig. 13. Difference between the values of the electric field in the air with a lake and the homogeneous flat soil. They are computed with (5) in the XOZ vertical plane at z>0 at 23 km of the antenna. The natural element is located at 24.5 km and the values of η are plotted in dB.

Fig. 14 .

 14 Fig. 14. Difference between the value of the electric field in the air with the forest and the homogeneous flat soil. It is computed with (5) in the XOZ vertical plane at z>0 at 23 km of the antenna. The natural element is located at 24.5 km and the values of η are plotted in dB.

TABLE I DIECLECTRIC

 I CONSTANTS AND CONDUCTIVITY FOR DIFFERENT SOILS

dB with the hill and the valley. A little further on the propagation path of the ground wave, η approaches the zero value for all natural elements, the electric field is the same as the case of a flat homogeneous half-plane. Therefore, the presence


Jia Luo

Rui Xue

Didier Jinglu Hu

El Baz

Combating the Infodemic: A Chinese Infodemic Dataset for Misinformation Identification

  L'archive ouverte pluridisciplinaire


Didier El Baz

Bilal Fakih

Romeo Sanchez Nigenda

Vincent Boyer

Parallel best-first search algorithms for planning problems on multi-core processors

published or not. The documents may come   L'archive ouverte pluridisciplinaire


Frederic Serin
email: frederic.serin@univ-lehavre.fr

Katerine Romeo
email: katerine.romeo@univ-rouen.fr

Towards Accessible Complete Plain Text to General Public Reader

In this article, we analyze the difficulties to access spatial information and propose some solutions. Our objective is two-fold: to create any type of document easily, and to make it accessible and intelligible for all audiences, blind and sighted people. Digital document accessibility, which in addition to readability also includes understanding and appropriation, should be improved. We propose some solutions to enhance the efficiency of screen readers for difficult 2D objects such as graphics and tables. Linear reading can be facilitated by inserting additional tags and attributes proposed by HTML5 or WAI-ARIA. Thanks to a simple language with lightweight tags, we offer the possibility to automatically generate more accessible documents without specialized knowledge of these functionalities. From the plain text document, HTML and SVG codes are generated. To complete the formatted document, a description file is produced in parallel to accompany the understanding of complex documents -UML diagrams or tables -and also to present the structure of the complete document.

Introduction

This article presents some solutions of accessible document writing for people with visual impairments in order to communicate with sighted audiences [START_REF] Altinier | Accessibilité Web[END_REF][START_REF] Romeo | Accessibilité des sites web pour les personnes présentant une incapacité visuelle[END_REF]. It innovates by the fact that it does not place the visually impaired person (VIP) as a reader but as an author of documents. We also wanted to make this project attractive to other audiences: students learning to design schemes, diagrams and Web developers sometimes bewildered by the difficulty of making their work accessible.

We propose to enhance the level of accessibility for digital documents [START_REF] El Andaloussi | L'accessibilité numérique, entre nécessité et opportunité, Une obligation légale vis-à-vis des citoyens, un levier stratégique pour les acteurs[END_REF][START_REF] Murphy | Accessibility of graphics in technical documentation for the cognitive and visually impaired[END_REF]. We suggest with our approach, not to increase the number of constraints proposed by the WCAG [START_REF]Web Content Accessibility Guidelines (WCAG) 2[END_REF], but to decrease the quantity of rules that an editor must have in mind.

Our approach has the particularity to be developed for a blind teacher which expresses the wish to be able to communicate with well-sighted students. Our project, which is progressing according to an iterative method, differs by taking into account a proactive participation of the readers [START_REF] Van Der Perk | A guide for scientific writing[END_REF].

If you are a sighted person, you have about a 36% chance of not having a VIP among your 200 readers. But, if you are a VIP, the probability to have at least 199 sighted readers is close to 100%. This example uses the probabilities of a binomial law with a prevalence of 0.5% and it is based on statistics provided by the WHO [7]. Each generated document must be accessible but also must bring to students added value for their understanding.

In the first section, we present how a simple text could be written by blinds. This text requires additional information for a complete perception. Indeed, the visual rendering is not always identical to what is heard from the automatic reading by a screen reader.

The inclusive text editor for tabular structure and diagram description has been already implemented. Table representation will be presented in the second section.

Finally, we shortly speak about diagram conception, needed enhancement, and we conclude with perspectives of development.

Linear Composition

Digital documents are made accessible mainly through the constant improvement of screen readers [8]. As with listening, the vocal restitution of any document is linear [START_REF] Romeo | Accessibilité aux informations textuelles et visuelles sur les sites web pour les personnes avec une déficience visuelle[END_REF]. So, it is important to understand any complex structure [START_REF] Sutton | A Guide to Making Documents Accessible to People Who Are Blind or Visually Impaired[END_REF][START_REF] Leuthold | Beyond web content accessibility guidelines: Design of enhanced text user interfaces for blind internet users[END_REF].

We create the documents using a generic basic tool: a text editor. We follow the principle KISS: keep it simple and intuitive. If a basic literary text is relatively simple to understand, it is quite different for a more complex structure. We have drawn inspiration from languages such as Markdown and PlantUML [START_REF] Vanderheiden | Fundamental Principles and Priority Setting for Universal Usability[END_REF]13] to extend the functionalities of the basic text editor in order to facilitate the creation of documents and allow their universal use. We want to maintain, as much as possible, the principle of least surprise, therefore a lightweight markup language with simple, non-obtrusive tags is used.

Our "tag text editor" is designed to be easy to write and, for the source text, also easy to understand through TTS (Text-To-Speech); indeed, a special care has been taken to ensure that the tags are easily identified by screen readers. Without knowledge of dedicated languages (such as HTML and SVG), the final output (generated file) meets the requirements of readability and visibility for all audiences. These languages are widespread for the diffusion of information and permit to include many possibilities to make accessible documents [START_REF] Romeo | Accessibilité des sites web pour les personnes présentant une incapacité visuelle[END_REF][START_REF] El Andaloussi | L'accessibilité numérique, entre nécessité et opportunité, Une obligation légale vis-à-vis des citoyens, un levier stratégique pour les acteurs[END_REF].

We have developed our software, called Latitude, using design patterns from literature [START_REF] Gamma | Design Patterns : Elements of Reusable Object-Oriented Software[END_REF] and in particular: Factory, Builder, and Chain of Responsibility. Latitude (abbreviated to Lttd in the following diagram) is the short name for Light and Accessible Text Including Tags and Using Universal Design. The user creates a simple text file, a plain text enriched with lightweight tags. This raw text is processed in the Latitude factory (Lttd Fty). From this point of entry, the structured and accessible HTML document is generated as well as a document description file. In addition, for diagrams, the definition elements are sent to a second factory (Diagram Fty). This second factory generates an SVG file for the schemas as well as a description document allowing a textual description.

According the target audience, blinds and students, the description texts are very useful. This is why we would like to draw your attention to these complementary documents. They contain statistical elements (size, complexity), warnings such as typographical elements (paragraphs or sentences that are too long) and suggestions for improvement.

For texts or tables, it will give indications such as size and reading complexity (Flesch score [START_REF] Flesch | The Art of Plain Talk[END_REF]), point out titles, items or paragraphs that are too long, make suggestions on structuring into sections and subsections. For a class diagram, it can give indications on the structure, its complexity, vocalize as a reader would the various component elements and indicate overlaps or problematic positioning.

The description file therefore helps blind people, who do not have a global perception of the document, but also students who want to benefit from an indicative feedback.

Such approach is disruptive with classic approaches to the accessibility: it takes it the other way round. Our extended editor enables VIP writers to produce a document that meets the expectations of both, VIP and (well-) sighted readers; it is an inclusive text editor.

The proposed approach may be attractive for other audiences: students learning to design inclusive text with graphics, web designers who must design accessible sites, etc. [START_REF] Romeo | Accessibilité des sites web pour les personnes présentant une incapacité visuelle[END_REF] The proposed approach gives insight in ways to make data presentations more ergonomic and easy to understand; moreover, it gives the opportunity to discover new ways to data and knowledge acquisition, and new ways for learning (inclusive) strategies.

Text Composition

We have drawn inspiration from languages such as Markdown to extend the functionalities of the basic text editor in order to facilitate the creation of documents and allow their universal use. We want to maintain, as much as possible the principle of least surprise, therefore a lightweight markup language with simple, non-obtrusive tags is used.

A document of a relatively large size is made up of sections that are themselves divided into subsections. Each section is identified by a title. In many languages, these titles are indicated by specific tags. These are not well identified by a screen reader. For example, we encounter brackets, exclamations points, dashes, etc. This is why we are introducing a symbol that stands out from the usual. We propose the equal sign (=), and we reverse the usage that the number of symbols increases with the depth of the nested sections. As we respect the typographical advice, we only offer four levels of nesting (division). The first level, equivalent to the most important title "h1" of HTML will be represented by five equal signs. The last level (the least important) will be composed with only two equals.

The advantage of this choice is double: the equal sign is well identified by vocal synthesis and the user can easily navigate with respect to the levels with the search function. The text created is enriched by light markup. The latter makes it possible to provide informative elements on the formatting such as the presence of a title, a word placed in italics, etc. The presence of the tags must guarantee that the legibility of the original text is maintained.

Each section is composed of a title, paragraphs and possibly, subsections. We automatically study if these sections are well balanced inside the document and indicate to the author what are the shortest or the longest ones. We suggest, if necessary, to add sections and subdivisions [START_REF] Van Der Perk | A guide for scientific writing[END_REF]. The paragraphs are decomposed in sentences, each of them in words and syllables. This process permits to apply the Flesch score and to measure the readability of the document [START_REF] Flesch | The Art of Plain Talk[END_REF][START_REF] Flesch | How to test Readibility[END_REF].

The balance between sections and the determination of readability are very useful guidelines for blind authors. They are also useful for novice writers [START_REF] Ciampa | The effects of an online reading program on grade 1 students' engagement and comprehension strategy use[END_REF].

The perception of a good distribution of sections is not arithmetic but geometric. Therefore, we calculate the number of sections and subsections desired according to the overall size of the document. We indicate the imbalances found between sections.

To calculate the Flesch score, we need to be able to extract not only the words but also the syllables from a sentence [START_REF] Georges | Lisibilité et compréhension[END_REF]. We can thus indicate the readability of the text, but we can also point out the complexity of a paragraph, the inappropriate length of a title or an item in a list.

Creation of Tables

We present how to create a table and its visual and accessible restitution. A table consists of cells, each cell being situated at the junction of a column and a row [START_REF] Duval | Comment analyser le fonctionnement représentationnel des tableaux et leur diversité ?[END_REF].

In HTML, a table consists of four important notions: cells and among them, headings, rows, and caption.

With the progressive evolution of HTML, the WAI initiative [20], and the efficiency of screen readers, the tables are relatively accessible. But this accessibility is relative. It is limited, at the base, to the vocalization by the screen reader of the contents and the indication of the positioning of the cell: the number of its line and that of its column. The headings of the columns are indicated if they are correctly filled in [START_REF] Duval | Comment analyser le fonctionnement représentationnel des tableaux et leur diversité ?[END_REF]. If the table is a bit complex, i.e. if there are: several header lines with column groupings, cell merges, row headers, then the voice description is impossible without the introduction of specific tags and attributes proposed by the WAI-ARIA [20]. An example of a complex table with merged headings is given in table 1. Current languages such as Wikitext for example, only allow these indications by writing these attributes in the tables created. However, they are very difficult to implement and, moreover, they are not known to the general public [21]. This public is precisely the one that proposes the tables. It is for this reason that we have developed a code analyzer that automatically inserts all the tags necessary to make any type of table accessible.

Our parser scans the lines defining a table and automatically detects the separator. This frees the user from a limited choice. We allow the use of three commonly used cell separators: pipe, tabulation, and semicolon. The pipe has our preference for its perfect vocalization in the plain text by the screen reader. Nevertheless, we allow tabulation for its common use by certain languages and writers. The semicolon is useful whenever interaction is possible with a CSV-formatted document.

With our proposed solution, it is no longer useful for the writer to provide tags to allow a complete description of a cell with its row or column headings. If cells are merged, if a header gathers several columns or if headers are placed at the beginning of a row, a table not using the ARIA recommendations will not allow voice output. However, the voice output is part of the information of the contents and structure of the table. We automatically process the data so that this information can be entered. When browsing the table, we create an object model to detect the headers that inform each cell. We then place the ARIA tags correctly so that the screen reader reads all the information that will help to navigate through the table.

Latitude creates an object-oriented image of a table, composed of typed cells: header and body. The groups of cells (fusion) define automatically references (id) in attributes. This id is added to the reference, headers, encapsulated into the td html tag. Different other attributes like role and scope are systematically included permitting accessibility [START_REF] Romeo | Tableaux accessibles dans les documents numériques[END_REF].

For the user, it is no longer necessary to explicitly specify the headers: just leave a blank line to avoid placing a header. In the same way, the first cell of each row is the row header; a blank cell allows not specifying it. Finally, you must specify several separators in a row to merge cells. To create complex headers on several lines, a separator, following many dashes, visually and audibly separates head and body. According to our syntax, the table title is situated just after the dedicated __table tag. This possibility, indicated in the description file in case of omission, will be transformed as information elements of the html caption.

As to draw bar charts or pie charts the data is sometimes grouped in a table, we used the same format to make these diagrams. The first line of the table allows you to define the labels while we allow up to three lines of data that will allow this information to be presented on the same bar chart or with one pie chart per line, sized to be placed side by side in the same figure. In both cases, a calculation is automatically performed to correctly dimension the information. For the bars, it is necessary to estimate the height differences, in order to balance as well as possible and to allow the best visibility. A scale is determined automatically. For the choice of colors, a library is proposed by default, which allows distinguishing the contiguous zones by following the recommendations of accessibility in particular, those of contrast and color blindness.

Simple and complex tables are formatted to be accessible with the tags proposed by W3C. Their design remains, for the creator, as simple as possible. It is not required to know the attributes or behaviors of screen readers [START_REF] Murphy | Accessibility of graphics in technical documentation for the cognitive and visually impaired[END_REF].

Accessible Diagrams for All

Our product allows easy integration of diagrams and graphics that we know can be made accessible with SVG [START_REF] Altmanninger | Dynamically Generated Scalable Vector Graphics (SVG) for Barrier-Free Web-Applications[END_REF], vector graphics structured by XML. We are inspired by PlantUML [13]. We add some complementary aims: readable code, accessible SVG use of ARIA attributes addition of a document given a textual description of the properties of the diagram.

Our first need was the easiness to ensure readability of the source code by a screen reader. The syntax must be understood by blinds throughout the vocalization [START_REF] Luque | On the Inclusion of Blind People in UML e-Learning Activities[END_REF]. Second, the description proposes a good rendering for visualization. Any sighted person must see what he does. Third, to facilitate the learning of the elements of the code, we have ensured consistency among the various proposed UML diagrams: class, object, sequence and activity. In Figure 2, an excerpt from the object model used by Latitude, is given: There are three inheritance trees, designing the structure of the builder pattern used by Latitude. Each Block is either the main document or a section. Finally, in parallel with the generation of the diagram, our software creates a description document. This document analysis offers an alternative description, given information about the structure like: number of various elements, sentence explaining links (associations, messages), size and positions… All this information is given in the form of sentences that are clearly stated by the screen reader. If this annex is an invaluable aid for the blind, this document proves to be interesting for learning.

Conclusions and Perspectives

Our project is incremental. With each request, we add the expected functionalities. The primary target audience is the blind, who must be able to communicate with a normal audience, integrating appreciated and necessary visual elements [START_REF] Beech | Assisting Students with Disabilities[END_REF]. On the other hand, we make a tool that is also useful to students because they have to formalize their diagrams and get the phrased description of their realization. Finally, we show that it is possible to enrich web page design tools by making them accessible without having to deal with an impressive number of constraints. This can be an alternative solution to allow everyone to access websites. We need to improve automatic text analysis. To do this, we need to get closer to typographers, authors and educational experts.

For tables, we are in the process of adding the ability to merge cells vertically, and to allow content formatting: centered or aligned, automatically detecting whether columns or rows contain numbers for example. But the multiplicity of situations requires extreme caution in this area. Also, we aim to introduce a score, equivalent to the Flesch one, to indicate the complexity of a table. The proposed library of diagrams must be enhanced offering various functionalities. We add them as and when requested. Students are the best evaluators: they need to take ownership of the opportunities. At the frontier between tables and graphs, we plan to introduce curves drawn from data. Already, we can read a table and draw a bar chart or pie chart on demand. This is not trivial, it is necessary to be able to make a judicious choice of colors according to accessibility constraints and, also, to define an appropriate scale for the best possible visibility.

Fig. 1

 1 Fig. 1 Text document creation process

Fig. 2

 2 Fig. 2 An example of the object model coded in SVG, generated by Latitude (in depth description in text above).Each individual is a part of a division. A scheme represents a diagram; it is a particular individual, coded in SVG. There are many other classes in our model, but this diagram gives a glimpse of our achievement.We can generate these diagrams either directly by calling a specific class or by inserting the code into a document containing, for example, text. Our original characteristics are the automatic generation of a SVG graphic including description tags for linear browsing, and including this diagram in a captioned figure in the global document. To enable this accessibility, we have used all the possibilities offered by the ARIA attributes.

Table 1

 1 An example of complex table

		Main Cities
	Country			Residents (millions)
		Capital Metropole
	Algeria	Alger	34
	Australia Canberra	Sydney	42
	Belgium	Bruxelles	12
	Brazil	Brasilia	Sao Paulo	110

  The Principal and Article classes inherit from Block. Each Block can itself be composed of Articles. Within each Block we find individuals and, optionally, other Blocks. The instances of Individual correspond to different kinds of HTML elements: Paragraph, Table, Figure… These subclasses inherit from Individual and structure the code. The class Scheme is linked to another external class called Graphics. The instances of a subclass of Graphics, translate different UML diagrams and generate the SVG accessible code.