Nga Nguyen

Laurent Vandenbroucke

Alfredo Hernandez

Tu Pham

Alain Beuchee

Alfredo Hernández

Alain Beuchée

Patrick Pladys
email: patrick.pladys@chu-rennes.fr

Early-onset neonatal sepsis is associated with a high heart rate during automatically selected stationary periods

Keywords: heart rate, heart rate variability, neonatal sepsis, newborn infant, signal processing

INTRODUCTION

Preventative measures have successfully decreased the incidence of early neonatalonset sepsis (EOS), but identifying neonates with clinical signs of sepsis and a high likelihood of EOS remains an important challenge [START_REF] Polin | Management of neonates with suspected or proven early-onset bacterial sepsis[END_REF]. These neonates are usually identified on the basis of perinatal risk factors and diagnostic tests that have a poor predictive accuracy, leading to inappropriate antibiotherapy decisions. There is therefore a need to develop new approaches to stratify the risk of EOS [START_REF] Polin | Management of neonates with suspected or proven early-onset bacterial sepsis[END_REF][START_REF] Bravi | Monitoring and identification of sepsis development through a composite measure of heart rate variability[END_REF][START_REF] Escobar | Stratification of risk of early-onset sepsis in newborns >/= 34 weeks' gestation[END_REF].

Heart rate variability (HRV) reflects the multiple interactions required for normal cardiovascular function. In sepsis, induced inflammation of all the components of the control/regulatory systems can be altered, including receptors, effectors, autonomic nervous system and central influences. Different approaches have been used to select the most appropriate HRV indices to diagnose or stratify the associated risk of sepsis. HRV is a useful non-invasive, real-time marker for the diagnosis of sepsis [START_REF] Bravi | Monitoring and identification of sepsis development through a composite measure of heart rate variability[END_REF][START_REF] Beuchee | Uncorrelated randomness of the heart rate is associated with sepsis in sick premature infants[END_REF][START_REF] Moorman | Mortality reduction by heart rate characteristic monitoring in very low birth weight neonates: a randomized trial[END_REF][START_REF] Stone | Abnormal heart rate characteristics before clinical diagnosis of necrotizing enterocolitis[END_REF]. Moorman et al proposed the use of selected HRV measurements to identify neonatal late-onset sepsis in premature infants. This approach resulted in a decrease in sepsis-related infant mortality in a randomised controlled trial [START_REF] Moorman | Mortality reduction by heart rate characteristic monitoring in very low birth weight neonates: a randomized trial[END_REF][START_REF] Fairchild | Septicemia mortality reduction in neonates in a heart rate characteristics monitoring trial[END_REF], but this was associated with an increase in the number of blood cultures and antibiotic treatments. A lack of specificity was also suggested in a monocentric retrospective study [START_REF] Coggins | Heart rate characteristic index monitoring for bloodstream infection in an NICU: a 3-year experience[END_REF].

In this study, that focused on a population with a high prevalence of EOS, we investigated possible changes in HRV patterns recorded in stationary conditions in full-term neonates during the early stages of EOS [START_REF] Ahmad | Continuous multiparameter heart rate variability analysis heralds onset of sepsis in adults[END_REF] in order to distinguish between infected and uninfected infants. We therefore compared patients with proven sepsis to a control group.

METHODS

Patients

We carried out a prospective, observational controlled study comparing full-term neonates considered to have proven sepsis with a control group in the Neonatology Department of the National Pediatric Hospital in Hanoi Vietnam, from November 2011 to July 2012. Informed consent was obtained from the parents of the infants. No additional blood examinations were performed for the study.

Eligible patients were non-asphyxiated full-term infants born at 37-41 weeks of gestation, with a birth weight of 2500-4000g, an Apgar score of more than six at five minutes, clinical signs suggestive of EOS and a C-reactive protein concentration of > 10 mg/L before seven days of age. They were all included in the study during the first hours following admission. The clinical signs were: an abnormal temperature of less than 36.5°C or more than 37.5°C, respiratory distress with respiratory rate of more than 60 breaths per minute or a Silverman score of more than three, an haemodynamic instability with persistent tachycardia (>160 bpm) or bradycardia (<100 bpm) or prolonged capillary refilling time (>4 seconds), neurologic signs including hypotonia, hypertonia, lethargy, convulsion, and gastrointestinal signs including feeding intolerance and abdominal distension. The exclusion criteria were: cardiac conduction abnormalities, ischaemic-hypoxic encephalopathy and intracranial haemorrhage. The infants were classified into two groups: the proven and non-proven sepsis groups. They were considered to have proven sepsis in cases of clinically symptomatic septicaemia with at least one positive blood culture, meningitis defined as neurologic signs associated with a white blood cell count ≥ 100/mm 3 in cerebrospinal fluid -with or without a positive culture -pneumonia defined as respiratory distress associated with an abnormal chest X ray or enterocolitis presenting with gastrointestinal signs associated with significant abnormalities on abdominal X ray. The control group of quasi-healthy, full-term infants was recruited during the study period from neonates hospitalised in the same hospital. They were eligible to be included in the control Accepted Article group if they were admitted for maternal indications following an uncomplicated Caesarean section without general anesthesia or for simple neonatal jaundice with a total bilirubinemia concentration of less than 200 μmol/L and with a C reactive protein concentration of less than 5 mg/L without phototherapy indication.

Study protocol

An anonymized 24-hour electrocardiogram recording was acquired soon after inclusion for all patients in a standardised environment -side position, minimal sound or tactile stimuli -with a sampling frequency of 1000 Hz using a SpiderView Holter, (ELA Medical, Montrouge France).

Clinical signs, blood cell counts and C-reactive protein concentration were recorded.

HRV analysis

Custom-built signal processing tools, designed with the Matlab software 6.0.0.42a, release 12 (The MathWorks Inc, Natick, Massachusetts, USA), were used, as previously described [START_REF] Beuchee | Uncorrelated randomness of the heart rate is associated with sepsis in sick premature infants[END_REF][START_REF] Beuchee | Influence of hypoxia and hypercapnia on sleep state-dependent heart rate variability behavior in newborn lambs[END_REF]11). Each 24-hour Holter recording was processed manually to extract the first segment of 180 minutes, corresponding to a period of low levels of ECG noise with no knowledge of the patient's characteristics, no access to clinical data or the group they belonged to. QRS complexes were detected automatically with a modified version of the Pan and Tompkins algorithm [START_REF] Pan | A real-time QRS detection algorithm[END_REF], including filter coefficients specifically adapted for neonates [START_REF] Altuve | Analysis of the QRS complex for apnea-bradycardia characterization in preterm infants[END_REF]. Each RR intervals time series -used to study the fluctuations in the interval between heart beats -that we obtained was checked manually and an automatic algorithm was applied to select the most stationary segments of five, 15 and 30 minutes duration, based on the criteria proposed by Gonzalez [START_REF] Gonzalez | Detection and sources of nonlinearity in the variability of cardiac R-R intervals and blood pressure in rats[END_REF]. The different time durations were tested to determine the optimal period duration to discriminate between groups.

Accepted Article

The HRV parameters studied were chosen by taking into account previous publications regarding the diagnosis of sepsis, which showed changes in heart rate, changes in the distribution of cardiac cycle lengths, decrease in the complexity/regularity, alterations in the fractal organisation and changes in the frequency domain analysis (LF/HF ratio) [START_REF] Bravi | Monitoring and identification of sepsis development through a composite measure of heart rate variability[END_REF][START_REF] Escobar | Stratification of risk of early-onset sepsis in newborns >/= 34 weeks' gestation[END_REF][START_REF] Beuchee | Uncorrelated randomness of the heart rate is associated with sepsis in sick premature infants[END_REF][START_REF] Fairchild | Septicemia mortality reduction in neonates in a heart rate characteristics monitoring trial[END_REF][START_REF] Gunther | Heart rate variability -a potential early marker of sub-acute post-stroke infections[END_REF]. The time-domain analysis of HRV was based on the extraction of the mean RR duration, the standard deviation (SD), which reflects the magnitude of global HRV, and the square root of the mean squared differences of successive RR intervals (rMSSD), which measures short-term beat-to-beat variability and mainly reflects parasympathetic control. The distribution of RR intervals was assessed by calculating the skewness and kurtosis of each RR series. We estimated the complexity and regularity of RR series by approximate entropy (ApEn) and sample entropy (SampEn) measurements ( 4). The RR series, resampled at 10 Hz, was also subjected to frequency domain analysis, through autoregressive estimation of the power spectrum and integration for the low-frequency (LF) 0.02-0.2 Hz, and high-frequency (HF) 0.2-2 Hz spectral bands. The LF/HF ratio was used as an index of sympathovagal balance. Poincaré plot analysis was then performed for further analyses of short-term (SD1) and long-term (SD2) HRV in the selected RR series. Acceleration capacity (AC) and deceleration capacity (DC) were computed as previously described [START_REF] Bauer | Deceleration capacity of heart rate as a predictor of mortality after myocardial infarction: cohort study[END_REF]. The scale invariance was tested through the detrended fluctuation analysis technique. The fluctuations were characterised by a self-similarity parameter (α) representing the long-range fractal correlation properties of the signal. We evaluated the fractal scaling exponent α1 from 4 to 40 beats, and α2 from 40 to 1,000 beats [START_REF] Beuchee | Uncorrelated randomness of the heart rate is associated with sepsis in sick premature infants[END_REF][START_REF] Beuchee | Influence of hypoxia and hypercapnia on sleep state-dependent heart rate variability behavior in newborn lambs[END_REF].

Statistical analysis

We used Statistica 10 software (Statsoft, Tulsa, Oklahoma, USA,) Comparisons between the proven sepsis and control groups were based on chi-square and Mann-Whitney U tests. An adjustment for multiple comparisons of 16 HRV parameters was performed using Benjamini-Yekutieli correction. Spearman's rank correlation coefficient was calculated for the pooled

Accepted Article

proven and non-proven sepsis groups, to assess associations between HRV indices, C reactive protein concentrations and blood leukocytes. The control group was not included in this because the haemogram analysis was not performed systematically. Results are expressed as medians (interquartile range). A p value of < 0.05 was considered statistically significant.

RESULTS

Clinical data

The clinical data is summarised in Table 1. During the study period, 186 neonates were admitted to the National Pediatric Hospital in Hanoi for suspected EOS. Of these, 61 were fullterm and 125 were premature infants. We did not include 21 full-term neonates due to parental refusal (n=10) or technical issues due to the recording system being unavailable or poor signal quality (n=11). A total of 40 full-term infants were included -14 in the proven-sepsis group and 26 in the non-proven sepsis group -and 31 of them had received their first dose of antibiotic before their blood collection. All the included neonates had blood cultures taken. The control group consisted of 11 quasi-healthy full-term neonates. There was no significant difference in terms of sex ratio, chronological age and birth weight between these groups and the control group. The clinical signs at inclusion are summarised in Table 1. The diagnoses in the proven sepsis group were three cases of Escherichia coli septicaemia, five cases of meningitisincluding four with Escherichia coli identified on the cerebrospinal fluid culturefive cases of pneumonia and one case of enterocolitis. One patient with meningitis died four days after birth.

HRV analysis

The values of HRV parameters obtained in the control group from five minutes stationary sequences with the different methods used are listed below and the analysis appears in Table 2. The results in time the domain analysis were mean 507 (463-522) msec, SD 26 (15-37) msec and rMSSD 13 (4-23) msec. The distribution of the RR intervals were kurtosis 3.7 (2.9-4.6) and Accepted Article skewness 0.30 (-0.13-0.71). The complexity and regularity analysis were ApEn 1.02 (0.82-1. [START_REF] Tran | Morbidity and mortality in hospitalised neonates in central Vietnam[END_REF] and SampEn 0.058 (0.033-0.069). The results of the Poincaré plot analysis were SD1 9.3 (2.7-16) msec and SD2 36 (21-51) msec. The results of the frequency domain analysis were LF 205 (18-284) msec 2 × 10 3 , HF 55 (4-115) msec 2 × 10 3 and LF/HF 3.7 [START_REF] Bravi | Monitoring and identification of sepsis development through a composite measure of heart rate variability[END_REF][START_REF] Escobar | Stratification of risk of early-onset sepsis in newborns >/= 34 weeks' gestation[END_REF][START_REF] Beuchee | Uncorrelated randomness of the heart rate is associated with sepsis in sick premature infants[END_REF][START_REF] Moorman | Mortality reduction by heart rate characteristic monitoring in very low birth weight neonates: a randomized trial[END_REF][START_REF] Stone | Abnormal heart rate characteristics before clinical diagnosis of necrotizing enterocolitis[END_REF]. The acceleration and deceleration capacities were AC: 1.33 (0.27-1.80) and DC: 1.44 (0.45-2.28). The results of the fractal analysis were α1 1.20 (1.08-1.29) and α2 0.90 (0.71-1. [START_REF] Tran | Morbidity and mortality in hospitalised neonates in central Vietnam[END_REF]).

The mean RR intervals, measured in five, 15 and 30 minutes on automatically selected stationary sequences, were lower in the proven sepsis group than in the other groups. The mean RR intervals were not modified by the duration of the automatically stationary sequence studied with a median of the standard deviation of seven msec [START_REF] Beuchee | Uncorrelated randomness of the heart rate is associated with sepsis in sick premature infants[END_REF][START_REF] Moorman | Mortality reduction by heart rate characteristic monitoring in very low birth weight neonates: a randomized trial[END_REF][START_REF] Stone | Abnormal heart rate characteristics before clinical diagnosis of necrotizing enterocolitis[END_REF][START_REF] Fairchild | Septicemia mortality reduction in neonates in a heart rate characteristics monitoring trial[END_REF][START_REF] Coggins | Heart rate characteristic index monitoring for bloodstream infection in an NICU: a 3-year experience[END_REF][START_REF] Ahmad | Continuous multiparameter heart rate variability analysis heralds onset of sepsis in adults[END_REF][START_REF] Beuchee | Influence of hypoxia and hypercapnia on sleep state-dependent heart rate variability behavior in newborn lambs[END_REF](11)[START_REF] Pan | A real-time QRS detection algorithm[END_REF][START_REF] Altuve | Analysis of the QRS complex for apnea-bradycardia characterization in preterm infants[END_REF][START_REF] Gonzalez | Detection and sources of nonlinearity in the variability of cardiac R-R intervals and blood pressure in rats[END_REF] between the three periods. The ApEn, assessed from five and 15 minutes, and the AC and DC, assessed from 30 minutes stationary sequences, were also lower in the proven sepsis group using the U test, but this difference disappeared after correction for multiple comparisons. The other HRV parameters didn't differ between groups.

The mean RR intervals were generally correlated (p<0.05) with leukocyte counts at admission (r = 0.47, 0.51 and 0.56 for five, 15 and 30 minutes sequences, p<0.01) but not with C-reactive protein values.

DISCUSSION

This study of Vietnamese full-term neonates showed that EOS was associated with: (i) a significant increase in the mean heart-rate measured during automatically selected stationary periods, (ii) non-significant changes in ApEn with a tendency to low values during the five and 15 minutes sequences and AC and DC with a tendency to low values on 30-minute sequence and (iii) no other changes in HRV.

Early-onset neonatal sepsis is a life-threatening disease and its incidence is generally estimated to be about one per 1,000 live births in developed countries [START_REF] Verani | Prevention of perinatal group B streptococcal disease-revised guidelines from CDC, 2010[END_REF]. This incidence varies from 6.5

Accepted Article

to 23 per 1,000 in Africa and from 7.1 to 38 per 1,000 in Asia (16 per 1,000 in Vietnam) [START_REF] Hoang | A systematic review of the burden of the neonatal mortality and morbidity in the ASEAN Region[END_REF][START_REF] Tran | Morbidity and mortality in hospitalised neonates in central Vietnam[END_REF][START_REF] Vergnano | Neonatal sepsis: an international perspective[END_REF].

It is therefore plausible that differences in the course and epidemiology of neonatal early onset sepsis exist between countries. However, it would have been difficult to carry out a study of this type in a country with a low incidence of EOS. Many infants included were born outside study hospital and it was therefore not possible to correlate the results with antenatal events and to estimate the influence of the delay from onset of sepsis signs on the results. Despite the strict environmental control that was applied to all patients during the ECG recording, we cannot exclude the possibility that some differences in the quality or quantity of interventions between the proven-sepsis and quasi-healthy groups may have had an impact on the HRV. As the number of patients was relatively low, with only 14 patients in the proven sepsis group, we could not reach formal conclusions about the approximate entropy or the AC and DC, which tended to differ between the groups but did not reach a significant p value. However, from the present results it appears unlikely that these parameters would have been discriminant enough to support a medical decision.

A study by Escobar et al stratified the early-onset sepsis risk in neonates ≥ 34 weeks gestation from a database that included 350 cases with early onset sepsis and 1,063 controls One of the vital signs that the authors identified was tachycardia, which was defined as a heart rate ≥ 160 bpm or mean cardiac cycle length ≤ 375 msec over a period of ≥ two hours, obtained from paper charts or inpatients electronic records during the first 24 hours after birth (3). Another study suggested that fetal tachycardia during the second stage of labour should be used as a sign of infection [START_REF] Schiano | Second-stage fetal tachycardia and neonatal infection[END_REF]. The present study evaluated mean heart rate on stationary sequences, which were defined by the stability of the mean duration and the variance of the RR duration.

These automatically selected sequences corresponded to periods with homogeneous HRV and, therefore, corresponded to periods with limited environmental influences, which were likely to reflect the patients' intrinsic condition. In this study we observed a higher mean heart rate in the proven sepsis values, of around 150 beats per minute, than in the control group, of around 120 Accepted Article beats per minute. Moreover, we found that mean RR intervals were correlated with leukocyte counts on admission. These results confirm the potential interest of an accurate evaluation of heart rate in the evaluation of early onset sepsis risk and suggest that the setting of a threshold could be optimised by the use of an automatic standardised method.

In this study the global HRV, measured by SD, as well as the short-term (rmssd, HF, SD1) and long-term (LF, SD2) fluctuations measured by conventional methods were not significantly modified following sepsis. This differs from other findings reported on adults or animals. In adults, a decrease in SD before the clinical diagnosis and treatment of sepsis has been reported [START_REF] Ahmad | Continuous multiparameter heart rate variability analysis heralds onset of sepsis in adults[END_REF][START_REF] Chen | Postresuscitation autonomic nervous modulation after cardiac arrest resembles that of severe sepsis[END_REF]. In fetal lambs, an increase in SD was observed after the injection of a lipopolysaccharide endotoxin [START_REF] Blad | ECG and heart rate variability changes in preterm and near-term fetal lamb following LPS exposure[END_REF]. Studies on paediatric patients, adult humans and animals have also resulted in different conclusions with regard to short-term and long-term variabilities. These discrepancies may be accounted for by many factors, including sepsis severity, time since disease onset, age, species and treatments. AC and DC were quantified from the method called phase rectified signal averaging, which is more resistant to artefacts and non-stationarities than the conventional methods. AC and DC are mainly considered to represent an integrated quantification of the sympathetic and vagal activities. Both AC and DC were found to be low in the 30-minute stationary periods recorded in the proven-sepsis group, but this result didn't persist after multiple comparisons correction. As the correction increased the risk of false negatives, and because we had a relatively low number of patients with confirmed EOS, we cannot exclude that a decrease in AC and DC did exist but did not reach the significant threshold.

Similarly, we observed a decrease in ApEn in the proven sepsis group, which was not confirmed after multiple comparisons correction. A low ApEn indicates a periodic behavior, namely a decrease in complexity that reflects a low level of adaptability, with the deactivation of regulation Accepted Article mechanisms within the cardiovascular system. This tendency in our study was in agreement with the decrease in ApEn reported in all studies of adult [START_REF] Ahmad | Continuous multiparameter heart rate variability analysis heralds onset of sepsis in adults[END_REF] or neonatal sepsis [START_REF] Beuchee | Uncorrelated randomness of the heart rate is associated with sepsis in sick premature infants[END_REF][START_REF] Lake | Sample entropy analysis of neonatal heart rate variability[END_REF] and by the results of experimental studies based on LPS endotoxin injections [START_REF] Gholami | Endotoxemia is associated with partial uncoupling of cardiac pacemaker from cholinergic neural control in rats[END_REF].

Sepsis has been reported to be associated with alterations in autonomic regulation, adaptability and organisation of HRV. The results of the present study suggest that EOS was associated with a high mean heart rate when it was automatically measured in stationary condition and with a tendency to low adaptability, namely low ApEn and low AC/DC. We think that the automatic standardised method proposed to measure mean heart rate could be a useful noninvasive approach for identifying neonates at high risk of EOS, particularly through integration in multivariate predictive models.

Acknowledgements:

We thank Dr Ha Dang for her valuable assistance in enrolling the neonates in the study and Matthieu Doyen for his help in the signal analysis.

Conflicts of Interest

The authors have no conflicts of interest to declare.

Finance


Hussein Sabra

Luc Trouche

COLLECTIVE DESIGN OF AN ONLINE MATH TEXTBOOK: WHEN INDIVIDUAL AND COLLECTIVE DOCUMENTATION WORKS MEET

This paper focuses on the documentation work in the mathematics teaching. We show up the individual and collective components of this documentation. We present a new theoretical framework, the documentational approach which seems adapted for studying this issue. We applied it on a particular project of the French teachers association Sésamath: the collective design of an online mathematics textbook. We present a methodology for observing both this collective project and the case of Anaïs, a teacher involved in this project. We study her professional interest in Sésamath, her didactical interactions with the other project members, and their effects on her professional knowledge.

INTRODUCTION

Digitalization deeply changes conditions of thinking and sharing knowledge at each level of the society [START_REF] Pédauque | Le document à la lumière du numérique[END_REF]. Its visible manifestations, in mathematics teaching (Hoyles & Lagrange 2009), are both the profusion of resources available via Internet and the diversification of technologies that could be used by the teachers (USB keys, Interactive whiteboard, calculators, software). These evolutions dramatically modify the conditions of professional practice (for preparing as well as for doing the teaching). They prompt new collective forms of teachers work: on the one hand digitalization makes potentially easier the sharing and exchange of resources between teachers, on the other hand, its complexity (dispersion of resources, rapid technological evolutions) makes necessary for teachers to help each other. These new forms of collective work among mathematics teachers change the conditions for their professional growth: teams, networks and communities appear as new opportunities for teachers to learn (Krainer & Woods 2008).

The most significant phenomenon of this trend, in France, is, for us, the appearing of Sésamath 1 , a French online association of mathematics teachers, aiming to provide mathematics teachers with free online resources. For achieving this goal, Sésamath develops collaborative work of teachers, around common projects [START_REF] Sabra | Entre monde du professeur et monde du collectif: réflexion sur la dynamique de l'association Sésamath[END_REF]. We should want to address, in this article, two questions: why do teachers engage in Sésamath? How do they articulate their work on resources for individual purpose and for Sésamath purpose?

In the following section, we introduce a new theoretical approach, which seems necessary for addressing these issues. In the third section, we present our experimental field and specify our questions. We set out our methodology in the fourth section. In the fifth section we display some preliminary results. In conclusion, we propose some new questions that this research raises.

THEORETICAL FRAMEWORK

Our research lies on a new approach to teachers' work with resources and professional development, already presented in CERME 6 [START_REF] Gueudet | Teaching resources and teachers' professional Development: towards a documentational approach of Didactics[END_REF] and developed further (Gueudet, Pepin & Trouche, to appear): the documentational approach of didactics.

It is built on a distinction between resources and documents, extending the one, introduced by the instrumental approach [START_REF] Rabardel | Les hommes et les technologies, approche cognitive des instruments contemporains[END_REF], between artefact and instrument. The choice of the word "resource", instead of artefact, aims at catching a great variety of things intervening in teachers work: a textbook, a piece of software, a student's worksheet, an Internet resource, a discussion with a colleague, etc. We call documentation work what a teacher needs to do for designing her teaching: looking for resources, integrating them in her personal resource system, implementing it in practice, sharing it with colleagues, renewing it taking into account various feedback etc.

A teacher draws on resource sets for her documentation work. A process of genesis (Fig. 1) takes place, producing what we call a document 2 , made of resources and a scheme (i.e. an invariant organization of the activity to perform a type of task -here a task for preparing and performing a given teaching). Each scheme encapsulated professional knowledge, both shaping teacher's activity and permanently reshaped by this activity. [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF] proposed a categorization of teacher's professional knowledge. We are particularly interested in one of these categories that [START_REF] Grossman | The Making of a Teacher[END_REF] developed further: the Pedagogical Content Knowledge (PCK), defined as knowledge that a teacher develops to help her students in their learning.

Figure 1. Schematic representation of a documentational genesis

The documentational genesis combines two interrelated processes: the instrumentalization (the teacher acting on resources), and the instrumentation process (resources supporting teacher's activity).

Taking into account teachers collective work leads us to articulate the documentational approach to the frame of the community of practice (Wenger 1998). A community of practice (CoP) is a human group presenting three main features: a mutual engagement of all its members, the active participation to the joint enterprise, and the reification of elements of practice (that is to say the production of things, results of the common practice, and recognized as a common wealth). This frame appears relevant for studying, instead of the interplay between a teacher and a set of resources (Fig. 1), the interplay between a group of teachers and sets of resources they are working on/with. We will call community documentation the process of reification carried out in teachers CoP.

OUR EXPERIMENTAL FIELD

Sésamath is a mathematics teachers association founded in 2001. Its kernel is constituted by about one hundred subscribers (mathematics teachers), sharing a set of principles inscribed in a charter 3 : common philosophy of public service, "mathematics for everybody"… Elected by this kernel, the Sésamath board regularly launches new projects groups for designing resources on a given theme, corresponding to teachers' special needs and interests (new subject in the curriculum, new textbook… see § 5.1, Fig. 3). These groups gather about five thousand teachers (a number of teachers belonging to several groups at the same time), working mainly at distance, via a platform and mailing lists. The group members can benefit from the assistance of employees and computer developers hired by the association (see fig. 3). All these groups present the features of CoP, of course not at the same level of development: the projects groups appear, at the beginning, as potential CoP [START_REF] Wenger | Cultivating communities of practice: a guide to managing knowledge[END_REF], while Sésamath kernel appears as a maturing one (ibidem). Sésamath, beyond its regular members, thus constitutes a constellation of CoP (ibidem), each of them sharing a same commitment for the principles of the association. A questionnaire [START_REF] Sabra | Entre monde du professeur et monde du collectif: réflexion sur la dynamique de l'association Sésamath[END_REF] The first item confirms the Sésamath members' interest for online resources and evidences the place, for them, of resources reuse (revealing the importance, for Sésamath work, of exchanging, combining, modifying his/her own existing resources). The question item reveals that the main motivation, for joining Sésamath, is the exchange of experience, which evidences Sésamath role for its members' professional development.

Going further in our research required to follow the documentation work of one Sésamath project group on the field. We have chosen the group (named in the following DT10), created in 2009, aiming to design a digital textbook for the beginning (grade 10) of the French high school. This choice was motivated by two reasons: 1) after having designed textbooks for the French college (grade 7 to 9), Sésamath thus addresses, for the first time, the more complex mathematics of the grade 10 level; 2) after having designed textbooks quite classical (pdf, OpenOffice files that teachers could freely download and modify) with complement (spreadsheets and interactive applications), Sesamath aimed, with this project, to create a new type of digital textbook, that a teacher could appropriate and adapt to her own needs. We hypothesised that these new challenges (mathematical and technical) could enrich the documentation work of this group, and thus make it more interesting for our research.

Answering our initial questions needed also following the complete documentation work (both for Sésamath and for their own classes) of some DT10 members. For this article, we have chosen to present Anaïs' case. Anaïs is particularly engaged in DT10). She is 57 years old. After 15 years of various occupations (including some years as a member of a commune in the countryside), she came back to the university, achieved her mathematics studies, and got a position as a mathematics teacher. She has now 18 years of teaching experience (15 years for grade 10). Why does Anaïs engage in the DT10 project? How does her professional knowledge interact with individual and community documentation?

METHODOLOGY

Observing the individual and collective documentation is a complex task. It requires to take into account several conditions: long time observing to highlight the regularities; individual and collective observing; observing in and out of classroom; following both teachers' activities and resources. In some methodology of CoP observation, the researcher is engaged in practices [START_REF] Jaworski | The practice of (University) mathematics teaching: mediational inquiry in a community of practice or an activity system[END_REF]). In our case, we have just observed the practice as an outsider.

The reflexive investigation for observing individual documentation

For the observing of the Anaïs documentation, we adopted a methodology designed by [START_REF] Gueudet | Des ressources aux documents, travail du professeur et genèses documentaires[END_REF]: the reflexive investigation. According to this methodology, the teacher is an essential actor in the collection of data. Among its methodological tools: interview at home, 'guided tour' (the teacher being the guide) and schematic representation of her resource system (SRRS); questionnaire about her vision of mathematics and of mathematics teaching; follow-up during several weeks including a logbook fulfilled by the teacher, collect of the resources, classroom observations. Anaïs became sick at the beginning of the follow-up, thus a direct classroom observation was impossible. We have adapted the methodology, in observing (and videotaping) Irvin, Anaïs's substitute in class, who actually used Anaïs resources, and discussing afterwards with Anaïs on Irvin uses of her resources. In this paper, we only exploit data from the Anaïs' interview at home, from her comments on Irvin's video, from her SRRS and from a set of resources she designed for the teaching of functions (according to the discussion in DT10).

Designing a methodology for observing community documentation

Extending this methodology lying on individual reflexive investigation, new methodological tools have been designed for fostering teachers' reflection on their collective practice. Among them, an agenda, fulfilled by some members of the CoP, chosen for the role that they had in the project (for example, Adam was selected for its role identified in the mailing list as an effective coordinator of DT10). These agenda aims to identify and analyze, from different points of view of different actors, the effects of the incidents (something unexpected needing to reformulate the common goal or to reorganize the community documentation) occurring throughout the common project. We have also asked DT10 members a schematic representation of collective interactions (SRCI) in the case of Sésamath, and collect the mathematical resources that they designed.

The methodology takes also advantage of natural data which offers the experimental field. We thus exploit: the Anaïs online notebook; the DT10 mailing list, which offers discussions about project organisation, as well about mathematical, didactical and epistemological aspect of the community documentation work. During the period of our follow-up (11 months), 627 messages have been exchanged via the mailing list, involving 27 members (including Adam, Anaïs, John, Ben, Pierre and Henry, § 5). Anaïs has authored 111 messages among the 627. We particularly exploit in this paper the DT10 thread of discussions concerning the "mathematics functions", linked to an incident: a curriculum change occurring in the midst of DT10 work, and provoking the more intensive discussion.

DATA ANALYSIS AND DISCUSSION

We firstly analyze Sésamath and Anaïs' documentation, and then present the didactical interactions between Anaïs and DT10.

Documentation work: DT10 and Anaïs

Most of Anaïs's resources (courses, exercises, homework) are digital, stored in an external hard disc. This digital form facilitates the sharing with other teachers via USB key or Internet. The Anaïs' resource system is strongly articulated with Sésamath resources (see Anaïs SRRS, Fig. 2): emails, students' sheets and other resources exchanged with Sésamath seem to have a major role in her documentation. This osmosis between Anaïs and Sésamath resources facilitates Anaïs' participation in DT10 work.

Anaïs's documentation is conditioned by three factors: "curriculum changing, institutional recommendations and classroom general level" (Anaïs' interview). For example, her teaching of "function" is strongly related to curriculum change. She said that she lived "two different spirits" of teaching functions: "it was, before, more guided and now the curriculum recommends open questions... so that students have more initiative" (Anaïs' interview). She seems very sensitive to the students' level: this year, I had a low level class. I will modify [the documents of the previous year] by adapting them to the level of my class. But maybe, the next year I will have good students, I will reuse the document in the present form" (Anaïs' interview). John: The curriculum has profoundly changed and we must rebuild the chaptering. This chaptering is very important because it is common to several Sésamath projects that involve the grade 10. Thus, I propose to do a general RESET.

Henry: We will create a general mailing list for all Sésamath projects linked to the secondary level as there issues intersect. We will thus share resources and general links that might be useful to all.

John: This list must first allow everyone to have a clear idea of what is done in different projects.

DT10 and Anaïs's documentation do not answer to the same constraints: DT10 has to move forward systematically with other high school projects, while Anaïs is attracted by the level of her pupils, particularly in the new curriculum spirit. But both documentations are conditioned by the institutional recommendations, which constitute the joint constraint for designing the resources.

Anaïs's interactions with DT10 are linked to her general interest for collective matters, and particularly in this project. Anaïs underlines, in her SRCI for Sésamath (Fig. 3), that each project is constituted by teachers which are interested the project's topic. More precisely, her participation resulted from a professional interest: participating in DT10, constitutes, for her, an occasion for searching ideas for her own classroom course. She confirms this idea in the interview, illustrating the productive aspects of instrumentalisation process ( § 2): "for example, for the chapter "Variations and Extrema of functions" [named in the following 2N2], I don't have constructed the lesson yet. In fact, I will transform it [the lesson designed in DT10] and I will reuse it for my classroom. I will adapt it for my classroom".

Figure 3. Anaïs's SRCI, our translation (see comments on project groups §3)

We analyze in the following section the didactical aspects of the interactions between Anaïs and DT10 members.

The didactical interaction between Anaïs and DT10 members

The didactical interactions between Anaïs and other DT10 members take place, mainly, through the mailing list of the project. Most of the didactical discussion, in the thread "Create textbook chapters", concerns the theme "mathematics functions". This discussion came under the PCK category "instructional strategies for teaching the functions", as it appears in the following extract of DT10 mailing list about the 2N2 chapter (evidencing Adam's role as coordinator, see § 4.2):

Ben: We need to cut up 2N2 chapter. What are your proposals? Pierre: … How far will we go with the concept of variations? My proposal: 1) the notion of functions variation (from a curve); 2) Maximum and minimum of a function, 3) graph and table of variation; 4) comparison of numbers. Anaïs: My proposals: 1) functions variations and graphical reading... 2) variations and calculations: square function ...inverse and linear functions, 3) Maximum and minimum of a function, 4) table of variation. Adam: I have a slight preference for Pierre's proposal ... because I think that the table of variation is simpler to understand than computations for comparing numbers ... I think that slowly increasing the difficulty is a good strategy.

Anaïs's chapter happens following another structure. She first introduces the concept from what she calls a "start-up activity", with the aim of presenting the semantic values of the "mathematics functions" terminology; then, "the course" where she presents the main definitions (and related examples); at the end, the "series of exercises", and "homework and assessment". This structure appears in her online notebook (table 2). In the thread of discussion "Extremum 2N2", that concerns designing an exercise for 2N2, Anaïs initiates didactical discussions about the terminology: Anaïs: f(x) is smaller than (or the image f(x) is smaller than?). It seems...

John:

For M being a maximum, it must be both an upper bound and an image by f. That is to say, for all x belonging to a given interval I, f(x) ≤4.5, then 4.5 is an upper bound. And it is a maximum if it is also an image of some x belonging to I.

Adam: I propose this formulation: for every real x belonging to I, f(x) is smaller than, or equal to, f(3), which equals 4.5.

Anaïs: Either like this: for every x in the interval I, f(x) ≤4.5 with 4.5=f(3). Also, when she discovered Irvin's video, she has particularly commented the language used by the teacher:

Anaïs: the expressions ... he says "the straight line 3x +1", or the straight line Y. What does that mean? He should say the line whose equation is y = 3x+1.

Me: why does it bother you? How this is a problem?

Anaïs: I think we should be highly accurate, even when we talk. Being vague or loosely makes fuzzy pupils' mind, especially when introducing new concepts... We remark a difference between DT10 (more precisely DT10 members taking in charge this question) and Anaïs in the instructional strategies for teaching function. DT10 constructs a "function" concept from very simple tools (reading of curves and table of values) then it moves to more complex tools like calculation and articulation between different types of representation (algebraic, graphical). Anaïs takes care, in her documentation, of the terminology of mathematical concept and its use. This gap is an opportunity for Anaïs to discuss her view with other DT10 members.

On another subject, the design of a problem-situation reveals a convergence between Anaïs and other DT10 members documentation: Anaïs fosters the place of conjecture and experimentation. She presents thus a resource (interview):

Anaïs: I like to ask them first to experiment with a calculator... try first with the calculator, guess the number of solutions ... for example f (x) = 0 they speculate with the calculator and ... Once they guess, after they do the proof by calculation. But I think it is important to first think about from free explorations.

The thread of discussion "test_2N2" reveals, within DT10, a shared point of view on problem-situation. The problems designed consist in modelling geometrical situations with a function, graphing this function with a calculator and elaborating conjectures before computing and searching for evidence.

CONCLUSION

We have presented in this paper a theoretical approach and a methodology aiming to analyze the didactical interactions at stake within a community documentation work. We have carried out this approach and this methodology in the case of a documentation project, DT10, of an online teachers association, focusing on Anaïs, a member of this association.

Anaïs and DT10 have a common interest in collaborating for designing resources and collectively facing incidents (like changes in the curriculum). Anaïs has also an individual interest linked to her documentation needs: discussing the mathematics terminology and language, and its values in the documentation work. When Anaïs perceives that didactical discussions are part of her interest, she participates strongly in the community documentation.

Three factors seem to stimulate the active participation of a teacher in the community documentation: osmosis between her resource system and the CoP resources; a gap between the teacher and the CoP in the strategies for teaching a subject; a shared interest in the subject of discussion that is the origin of gap. Following an episode of Anaïs's active participation in the community documentation, we identified an aspect of her professional knowledge: the introduction of a concept has to be based on a precise introduction of terminology and language associated, and its uses.

This study revealed a question that deserves more deepening: how identifying the effects of the community documentation on the teacher professional knowledge in the case of convergence between the individual and community documentation?

Figure 2 .

 2 Figure 2. Anaïs' SRRS, our translation and schema as close as possible of the original DT10 documentation is also sensitive to curriculum changes and keeps Sésamath resources as a general background support (particularly when an incident occurs), as it appears in the following extract of the DT10 mailing list:

Table 1 .

 1 , proposed in 2008 to 36 members of the Sésamath kernel during a training session 4 , gave some answers (Table1) to our first question ( §1): why do teachers engage in Sésamath? Questionnaire to Sésamath, two questions (among 39) and their answers

	What are, for your teaching, your sources of	Online resources (33/36); Resources that you
	documentation?	have developed in previous years (35/36)
	What are your professional reasons for	Training (16/36); Exchange of experience
	engaging in Sésamath?	(23/36); Exchange of resources (13/36).

Table 2 .

 2 An extract of the Anaïs' online notebook

	Monday 21/09/2009 Constructing, from a sheet of paper, a box whose volume is
		the bigger one
	Friday 25/09/2009	Definitions: set of a function definition, variable, image, and
		antecedent. Table of values and graph using a calculator.
	Monday 28/09/2009 Area of a rectangle with a given perimeter.
		Table of values and graph of reference functions (that are
		inverse function, square function, square root function…)
	Thursday October 1 st Different ways of saying that f (a) = b
	Friday 2/10/2009	Exercises on image of an interval; first assessment.

This training session was organized by both Sésamath and the National Institute for Pedagogical Research.


Hélène Jaudoin
email: jaudoin@enssat.fr

Pierre Colomb
email: colomb@isima.fr

Olivier Pivert
email: pivert@enssat.fr

Ranking approximate query rewritings based on views

This paper considers the problem of rewriting queries using views by means of a tolerant method. The approach proposed is based on an approximate matching between the constraints from the query and those from the views, in the case where both the query and the views contain arithmetical constraints expressed as intervals. In such a context, the answers obtained are not certain anymore but only more or less probable. An algorithm which retrieves the top-k rewritings of a given query is described. Experimentations are reported, which show that the extra cost induced by the approximate nature of the rewriting process is perfectly acceptable.

Introduction

An integration system provides a uniform query interface to distributed data sources on a network. The problem of answering queries in integration systems has been intensively studied during the last decade [START_REF] Halevy | Answering queries using views: A survey[END_REF]. In particular, it has been shown that the semantics of queries in such a setting can be formalized in terms of certain answers [START_REF] Abiteboul | Complexity of answering queries using materialized views[END_REF]. Intuitively, a certain answer to a query Q over a global schema -i.e., a uniform interface -with respect to a set of source instances is an answer to Q in any database over that interface that is consistent with the source instances. The problem of answering queries in integration systems can be formalized as the problem of computing all the certain answers to the queries. A technique to effectively computing the certain answers to a query in Local-As-View based integration systems (i.e., where the data sources are defined as views over the global schema) is to reduce this problem into that of rewriting queries using views. Given a user query expressed over the global schema, the data sources that are relevant to answer the query are selected by means of a rewriting algorithm that allows to reformulate the user query into an equivalent or maximally contained query whose definition refers only to views. Any such query rewriting must satisfy all the constraints conveyed by a given query Q in order to return only correct answers to Q.

However, for integration systems in open environments -like the web where data sources are autonomous -, it can be very problematic to find views that totally satisfy the domain constraints imposed by the query. This raises the question of relaxing the notion of a mapping between the value domains of the views and those from the query. As an example, let us consider a query Q that aims to retrieve names of persons whose age is in the interval [28,38] and two views V 1 and V 2 such that: V 1 supplies names of persons whose age is in [25,35] and V 2 supplies names of persons whose age is in [36,46]. The views V 1 and V 2 both have an interval constraint on attribute age. The two intervals are not included in that of the query, thus the mappings between the two intervals and that of the query are only partial (imperfect). Moreover, since V 1 and V 2 only supply names of persons, selection on attribute age is impossible, hence V 1 and V 2 cannot be used to get certain answers to Q. In such a context, rewriting algorithms based on the certain answer semantics fail to reformulate the query, thus to provide the user with any answer.

In this paper, we assume an integration system based on a SuperPeer architecture [START_REF] Bellahsene | Querying distributed data in a super-peer based architecture[END_REF] where each SuperPeer acts as a mediation system based on a Local-As-View (LAV) approach. Each SuperPeer stores a global schema and a set of views describing queries that can be performed on data sources connected to the SuperPeer. Queries are submitted to a given SuperPeer that rewrites them in terms of its views. We consider that views and queries are expressed in the formal setting of Datalog and that they involve interval constraints. The idea we advocate is to exploit approximate mappings between the constraints from the views and those from the queries in order to compute approximate query rewritings. Any such rewriting Q is associated with a score between 0 and 1 which reflects the probability for a tuple returned by Q to satisfy the initial query Q. Consequently, the rewriting mechanism is not based on the notion of certain answers anymore, but on that of probable answers. Since approximate mappings between queries and views are obviously more frequent than strict mappings, the number of possible approximate rewritings may be huge. To cope with this situation, we propose an algorithm which generates only the top-k rewritings to a given query. This algorithm has been implemented and some experimentations on real data have been conducted. They show that the extra cost induced by the approximate nature of the approach (with respect to regular query rewriting algorithms) is perfectly acceptable.

Even though the notion of gradedness is central to the approach proposed here, it is important to emphasize that the problem tackled is essentially different from the issue of rewriting top-k queries using views dealt with, e.g., in [START_REF] Das | Answering top-k queries using views[END_REF]. In the work presented here, only regular (i.e., Boolean) queries are considered and the goal is not to compute a set of graded tuples, as in [START_REF] Das | Answering top-k queries using views[END_REF], but rather to compute the most satisfactory sets of answers, corresponding to the best rewritings of the query. In other words, we are not interested in rewriting top-k queries, but rather in computing the top-k rewritings of regular queries. Moreover, although the problem of rewriting queries using views in the presence of interval constraints is well known and the corresponding rewriting language has been shown to be a union of conjunctive queries with semi-interval constraints [START_REF] Pottinger | A scalable algorithm for answering queries using views[END_REF][START_REF] Afrati | Rewriting queries using views in the presence of arithmetic comparisons[END_REF], our contribution is to tackle this problem by means of a tolerant method. The purpose of our approach also differs from [START_REF] Dong | Data integration with uncertainty[END_REF] which deals with the problem of rewriting queries in the presence of imprecise view descriptions. In our approach, imprecision results from the approximative nature of the mappings considered, it does not concern the description of the data.

The remainder of the paper is organized as follows. Section 2 introduces the general principle of our approach and defines the notion of an approximate rewriting while Section 3 describes the algorithm aimed to compute the top-k approximate rewritings of a user query. Section 4 presents some experimentations and notably deals with the performance aspect, whereas Section 5 concludes the paper.

2 Approximate rewriting using views

General objective

We study the problem of ranking query rewritings based on views in the setting of conjunctive queries with simple interval constraints X i ∈ [a, b], where X i is an attribute and a and b are constants. Such a query is an expression of the form:

Q(Y ) : -r 1 (Y 1 ), . . . , r n (Y n ), C 1 , . . . , C m
where Q, r 1 , ..., r n are predicate names, and Y , Y 1 ,..., Y n are sets of variables, i.e., sets of attributes names, and Y ⊆ ∪ n j=1 Y j . The atoms r 1 (Y 1 ), . . . , r n (Y n ) are the subgoals of the query where r 1 , . . . , r n denote relations of the global schema. The atom Q(Y ) is called the head of the query and represents its result. Each C i , i ∈ [1, m] is an interval constraint represented hereafter by X i ∈ I Xi where X i belongs to ∪ n j=1 Y j and I Xi denotes an interval [a, b]. As an example, let us consider a global schema S made of the relations P erson(ssn, name, f irstname), Child(ssn, Cf irstname, Cage) and Emp(ssn, job, sal), and a query Q on S aimed to retrieve the social security number, name and first name of every person who has a child between 22 and 35 years old and a salary between 1200 and 2300 euros. Q can be expressed as: Q(nss, name) : -P erson(nss, name, f irstname), Child(nss, Cf irstname, Cage), Emp(nss, job, sal), Cage

∈ A Q , sal ∈ S Q with A Q = [22, 35] and S Q = [1200, 2300].
A view is a named query which uses relations from the global schema S. The idea we advocate in this paper is to reformulate a given query Q into expressions based on views which satisfy the interval constraints attached to Q as well as possible. Every such approximate rewriting is associated with a degree α ∈ ]0, 1] which expresses the probability for a tuple that it returns to be a certain answer to Q.

Definition 1 (Approximate rewriting). Let Q be a query, Q the query Q without its interval constraints and V a set of view definitions. Query Q is an approximate rewriting of Q using

V if: -Q is a conjunction of views from V, -Q Q, and -Q α Q, with α ∈ ]0, 1].
In this definition, Q α Q means that Q is approximately contained in Q at a degree α, according to the tolerant inclusion which will be defined in the next subsection. An approximate rewriting is then a regular rewriting in terms of views of a query when interval constraints are omitted. However when interval constraints are considered, the approximate rewriting approximates the interval constraints of the query. In addition to certain answers, such approximate rewritings provide answers that are likely to satisfy the initial query. Since a degree is associated with every approximate rewriting, it is possible to order them. Therefore, instead of computing all the possible approximate rewritings of Qwhich can be costly and not very useful for the user -, we focus in the following on the problem of computing only a meaningful subset of them: either the top-k ones (k being an integer) or those whose associated degree is over a given threshold α min ∈ ]0, 1]. The computation of such approximate rewritings requires two main steps: i) the research of candidate approximate rewritings, i.e., the regular query rewritings of the query when interval constraints are omitted, and ii) the assessment of the candidate approximate rewritings. As to the first step, we can adapt regular algorithms aimed at rewriting queries using views such as, for example, the MiniCon [START_REF] Pottinger | A scalable algorithm for answering queries using views[END_REF]. However, one must define a new algorithm in order to assess and rank approximate rewritings of a given query.

Concretely, the assessment of an approximate rewriting is based on the proportion of answers that it returns which satisfy the constraints from the user's query. Hereafter, we give a more formal definition of the notion of tolerant inclusion between two intervals that founds the semantics of the approximate rewriting approach.

Tolerant inclusion

Let I Q and I Q be two intervals. We define the tolerant inclusion of I Q in I Q in the following way:

deg(I Q ⊆ tol I Q ) = α = | I Q ∩ I Q | | I Q | (1)
where |I| denotes the cardinality of I.

Let I Q = [a, b], I Q = [c, d],
and

I Q ∩I Q = [e, f ]
, and let us assume that the attribute considered is encoded by numbers with n decimals (n ≥ 0), we get:

deg(I Q ⊆ tol I Q ) = (f -e) × 10 n + 1 (d -c) × 10 n + 1 .
Obviously, this degree corresponds to the proportion of elements from I Q which are in I Q ∩ I Q when the distribution of the values over the domain is uniform. Equation 1 can be straightforwardly adapted to the case where one has available some histogram or distribution function describing the value distribution over the domain. One uses the distribution function F the following way. Assume again that

I Q = [a, b], I Q = [c, d], and I Q ∩ I Q = [e, f ]. The degree deg(I Q ⊆ tol I Q ) is given by: α = x ∈ [e, f ] F (x) x ∈ [c, d] F (x)
.

(2)

From tolerant inclusion to approximate rewritings

In this section, we exploit the notion of a tolerant inclusion to make query rewriting using views a graded process. We consider a user query Q expressed in terms of a global schema S and a candidate approximate rewriting Q based on views, also expressed in terms of S, such that Q Q. Taking the interval constraints into account leads to computing tolerant inclusion degrees between intervals involved in Q and Q according to the principle described above, then to aggregating these degrees if several intervals are involved. The final degree obtained can be seen as the satisfaction degree attached to the rewriting as a whole. Due to the semantics of Equations 1 and 2, it expresses the probability for an answer returned by the approximate rewriting Q to be an answer to the initial query Q.

First, let us consider the situation where Q has a constraint on a single attribute X. It is worth noting that when Q has no interval constraint on X, the active domain of X is used as the interval constraint on X in Q . When Q has only one interval constraint, two cases must be considered: a) the constraint is attached to a distinguished variable in Q , i.e., X appears in the head of at least one of the views defining Q and, b) the constraint is attached to an existential variable in Q , i.e., X does not appear in the head of the views from Q . The following proposition states how the satisfaction degree attached to Q can be obtained in each case. Proposition 1. Let Q be a query and Q a conjunction of views such that Q Q. Let X be the single attribute on which Q has an interval constraint. Let I Q and I Q be the associated intervals in Q and Q respectively (if I Q does not explicitly exist, the active domain of the corresponding attribute is used). Let α be the degree of tolerant containment of

Q in Q (Q α Q). a) if X is a distinguished variable of a view from Q then I Q is added to Q and • if I Q ∩ I Q = ∅ then α = 0 • else α = 1. b) if X is an existential variable in the views from Q then α = deg(I Q ⊆ tol I Q ).
Consequently, the degree attached to an approximate rewriting is either 0, 1 or the inclusion degree between I Q and I Q . When α is 0, Q does not provide any probable answer and is not considered an approximate rewriting. The proof of this proposition is given hereafter.

-Case a): As X is a distinguished variable of at least one view from Q , there may exist a join over X between some views from Q . Moreover, one or several views from Q may have an interval constraint on X. Let {I 1 , . . . , I k } be the set of those intervals and t an answer returned by Q . Then, the projection of t on X belongs to the interval

I Q = I 1 ∩ . . . ∩ I k .
As X is a distinguished variable of at least one view from Q , one can apply a condition on X and notably, the interval constraint I Q , as done in [START_REF] Pottinger | A scalable algorithm for answering queries using views[END_REF], in addition to

I Q = I 1 ∩ . . . ∩ I k .
Therefore, two subcases must be considered. Either I Q ∩ I Q is empty and Q does not provide any probable answer to Q: then degree 0 is attached to Q ; or I Q ∩ I Q is not empty and Q with the constraint I Q ∩ I Q , returns only certain answers to Q since only the "compatible part" of the constraints was kept: then degree 1 is attached to Q . -Case b): X is an existential variable in all the views from Q . As no selection and no join on X are possible ( [START_REF] Pottinger | A scalable algorithm for answering queries using views[END_REF]), only one view of Q covers the subgoals of Q involving X. Therefore, only this view has an interval constraint I Q on X.

Let t be an answer returned by Q . Then the projection of t on X belongs to I Q . However I Q may not be contained in I Q . In this case, the satisfaction degree of Q with respect to Q must be based on a tolerant inclusion between I Q and I Q (cf. Equations 1 and 2).

In accordance with Equations 1 and 2, the degree obtained denotes the probability that the projection on X of an answer to Q be in I Q , knowing that the projection of answers to Q on attribute X is in the interval I Q . More generally, the degree obtained with a such procedure specifies the probability for answers to Q to be certain answers to Q. When constraints are associated with distinguished variables, query constraints are added to the rewritings. Therefore, either we get no correct answer and α is 0 or we get only certain answers and α is 1.

The following example illustrates the former case.

Example 1. Let us consider the query: Q(N, A, S) : -M an(N, A), Salary(A, S), A ∈ I A Q = [28, 38] and the two views V 6 and V 7 defined as: Let us now consider the general case where Q involves constraints on several attributes. In this case, the problem is to compute the probability for an answer returned by Q to be an answer to Q. Proposition 2. Let Q be a query and Q be a view-based expression such that

V 6 (N 1 , A 1 ) : -M an(N 1 , A 1 ), A 1 ∈ I A1 = [25, 40] V 7 (A 2 , S 2 ) : -Salary(A 2 , S 2 ), A 2 ∈ I A2 = [27, 45]. A possible rewriting of Q(N, A, S) is Q (N, A, S) : -V 6 (N, A), V 7 (A,
Q Q. Let C Q be the set of interval constraints -assumed to be independent - in Q such that |C Q | = n. Let α i be the degree attached to Q for any constraint c i ∈ C Q , obtained with the criteria of Proposition 1. The degree α of tolerant containment of Q in Q (Q α Q) is given by: α = n i=1 α i
Proof. Under the independence assumption of the interval constraints, if α i denotes the probability that an answer to Q satisfies constraint c i from Q, then the probability α that it satisfies all of the constraints from Q is the product of the α i 's.

The following example illustrates the computation of the overall degree attached to a given rewriting in such a complex case.

Example 3. Let us consider the following views:

V 1 (ssn 1 , n 1 ) : -P erson(ssn 1 , n 1 , f 1 ), V 2 (ssn 2 ) : -Child(ssn 2 , cf 2 , a 2 ), a 2 ∈ [20, 40], V 3 (ssn 3 ) : -Emp(ssn 3 , j 3 , s 3 ), s 3 ∈ [2100, 3400], V 5 (
Q 1 (nss, nom) : -V 1 (nss, nom), V 2 (nss), V 3 (nss) and Q 2 (nss, nom) : -V 1 (nss, nom), V 5 (nss).
Assuming that the attributes are encoded by integers, the degree attached to:

-the rewriting V 1 of the first subgoal of Q is α 1 = 1

An algorithm for computing the top-k rewritings

Instead of computing all the approximate rewritings of a query and then ranking them, we propose to adapt a well-known regular query rewriting algorithm, namely Minicon [START_REF] Pottinger | A scalable algorithm for answering queries using views[END_REF], in order to directly generate the k best rewritings ranked in decreasing order of their degree. MiniCon computes the maximally-contained rewriting of a given query Q, i.e., that which provides all the certain answers to Q, in the case where queries and views involve semi-interval constraints. The first step of this algorithm consists in enumerating all the so-called MiniCon descriptions (denoted by MCD's) of Q. Intuitively, the existence of an MCD associated with a view denotes that the view covers a subset of the query subgoals. The second step is devoted to the computation of MCD combinations which cover every subgoal in Q and which are pairwise disjoint w.r.t. the covered subgoals.

When tolerance comes into play, the first step of the MiniCon algorithm must be slightly modified in order to associate a degree α M with every MCD M of a given view V . First, the regular MiniCon procedure f ormM CD(V, M ) is used to check if an MCD can be created with V . If it is the case, the degree to be associated with M is computed on the basis of Proposition 1 or 2.

As to the second step of the MiniCon algorithm, it can be modeled by a hypergraph [START_REF] Berge | Hypergraphs[END_REF] where query subgoals correspond to vertices and each set of query subgoals covered by an MCD corresponds to an edge. Any satisfactory combination of MCDs corresponds to an exact cover of this hypergraph. This type of object is strongly related with the notion of a minimal transversal [START_REF] Eiter | Identifying the minimal transversals of a hypergraph and related problems[END_REF][START_REF] Eiter | Computational aspects of monotone dualization: A brief survey[END_REF] in hypergraph theory. Even though determining whether a given hypergraph possesses at least one exact cover is an NP-Complete problem [START_REF] Karp | Reducibility among combinatorial problems[END_REF], some scalable algorithms exist for computing all the exact covers of a hypergraph [START_REF] Knuth | Dancing links[END_REF]. Such algorithms can be used to get an efficient implementation of MiniCon. Algorithm 1 presented below belongs to this family; it generates either the k best rewritings of a query, or all the rewritings whose degree is over a given threshold α min . The rewritings are generated in descending order of their degree. Therefore, the k best rewritings of a query are those k first produced.

Our algorithm maintains an ordered list of partial rewriting of the considered query. A partial rewriting is a set of MCDs, pairwise disjoint, that does not cover all the subgoals of the query.

Algorithm 1 ComputeRW(X, M)

Require: M the set of weighted MCDs Ensure:

The set RW (Q, V) of rewritings of Q using V 1: RW (Q, V) = ∅ 2
: P artialRW = M 3: while P artialRW = ∅ do 4: M = P artialRW.max(); 5:

P artialRW = P artialRW \ M ; 6: if M covers Q then 7: RW (Q, V) = RW (Q, V) ∪ M ; 8: else 9:
M.cover(M ); 10:

for m ∈ M do 11:

if M.α * m.α ≥ αmin then 12:

P artialRW.Add(M + m); 13:

end if 14:

end for 15:

M.uncover(M ); 16:

end if 17: end while Given a partial rewriting M and an MCD m covering others subgoals, we denote by (M + m) the (partial) rewriting corresponding to the concatenation of M and m. Moreover, we denote by M.α the degree of a partial rewriting M . Let us remind than according to previous propositions, the degree of a concatenation (M + m).α is given by the product of M.α and m.α. Algorithm 1 takes as an input the set M of weighted MCDs returned by the first step, stored in a data structure which offers the following operations:

-cover(M ) deletes all the MCDs sharing a subgoal with M , -uncover(M ) re-inserts the MCDs sharing a subgoal with M previously deleted by cover(M ).

When we use a MCD M in a partial rewriting, we must forbid the use of all MCDs intersecting M . Indeed, every subgoal of the query must be covered only once. The operation cover() is used to do so. On the other hand, the purpose of the function uncover() is to re-allow the use of an MCD in the rest of process. Notice that M.cover(X).uncover(X) = M and the use of uncover(X) always follows the use of an operation cover(). This data structure can be implemented very efficiently using Dancing Links [START_REF] Knuth | Dancing links[END_REF], which are a structure based on circular doubly-linked lists to implement binary matrix. It makes it possible to implement the operations cover and uncover with a complexity linear in the number of deleted/inserted MCDs. The ordered generation of the rewritings is based on the property that the degree of a partial rewriting M can only decrease when it is combined with another MCD m. This property ensues from the fact that the degrees handled are probabilities, therefore real numbers between 0 and 1.

Algorithm 1 considers the partial rewritings in descending order of their degree and tries to complete them so as to generate the set RW (Q, V) of rewritings of the query Q considered. It first considers the partial rewritings made of a single MCD (line 2). Then, as long as there are still partial rewritings which may be completed, it considers the partial rewriting X of maximal degree (line 4), checks if it is a rewriting of Q (line 6), and tries to complete it if necessary (lines 9 -15). For doing so, one first removes from M all the MCDs which share a subgoal with X, then one tries to form a (partial) rewriting with each of the remaining MCDs. Finally, the MCDs previously removed are reinserted before the next iteration.

Experimentations

The approach has been implemented in a prototype using Java SE 5, and Post-greSQL 8.3 databases. We performed experimentations using sources of agricultural data consisting of 310 tables containing about 600 Mb of data. Twenty queries provided by real users were processed. In this experimental context, all data sources are related to the same application area. Such a context is convenient to find exact rewritings, and consequently certain answers (which explains the large number of certain answers retrieved). The following results show the evolution of the average number of tuples corresponding to certain and probable answers to queries.

The first experimentation shows the evolution of the number of answers when the threshold α min changes. Here the parameter k is fixed to infinity, i.e., all the rewritings are generated. Notice that the number of certain answers remains constant. Indeed, all exact rewritings are computed for any value of α min . On the other hand, the number of probable answers decreases when the threshold increases. When α min equals 1, only certain answers are provided.

In the second experimentation, the threshold is set to 0 but k, the number of best rewritings sought for, must be specified. When k decreases, the number of probable answers decreases too, which shows that only the best rewritings are generated. When k is less than 10, the number of certain answers decreases too. Indeed, for some queries, a number of rewritings less than 10 is not always enough to compute all the certain answers.

The prototype shows very good performances: the execution time is about 2 seconds in the worst case, and the performances of the approximate rewriting approach are comparable to that of the classical MiniCon algorithm (in the worst case, the overhead is only 10%). As mentioned above, the use of appropriate data structures makes our algorithm highly scalable: it can efficiently manage the extra MCDs induced by the approximate nature of the rewriting process.

Conclusion

This paper deals with the problem of getting ranked approximate rewritings in order to process only the best ones, thus reducing the processing cost of a query in a decentralized database context. We have proposed an approach to obtain such rewritings on the basis of interval constraints involved in the views and the query. This approach attaches a degree ∈ [0, 1] to each rewriting Q of a query Q, which corresponds to the probability for a tuple returned by Q to be an answer to Q. We have adapted the well-known MiniCon algorithm to get only the best rewritings, reducing thereby the combinatorics relative to the potentially high number of MCDs. Experiments show that the extra cost induced by the approximate nature of the matching is perfectly acceptable, while a significant number of additional answers is obtained.

These experiments also made it possible to estimate -in a specific context -the amount of additional answers provided by this approach with respect to the number of certain answers. However, it is worth noticing that the answers produced may not satisfy all of the criteria from the initial user query. So as to make the user aware of this state of fact, a simple solution consists in associating a degree with each answer: that attached to the approximate rewriting which produced the answer in question. Concerning this aspect, it would be interesting to complete the experimentations by some tests involving real users so as to assess their overall satisfaction regarding the non-certain answers that they are provided with.

Generally speaking, the approach described here applies to a particular context, namely the Web where exact mappings between the descriptions of data sources and a given user query are not always possible. In such a context, the answers to a user query can only be computed on the basis of the information which is available, even if it is pervaded with uncertainty. Flexible rewriting approaches allow to provide users with answers where classical rewriting algorithms based on the semantics of certain answers fail. The idea of computing probable answers is not exactly new since it has been used for open integration contexts in the works by Dalvi and Suciu [START_REF] Dalvi | Answering queries from statistics and probabilistic views[END_REF] and by Das Sarma et al. [START_REF] Sarma | Bootstrapping pay-as-you-go data integration systems[END_REF]. The nature of the probability degree associated with the answers returned in our approach is different from that considered in these works, though. Indeed, in [START_REF] Dalvi | Answering queries from statistics and probabilistic views[END_REF] it indicates the probability of existence of the answers produced, while in [START_REF] Sarma | Bootstrapping pay-as-you-go data integration systems[END_REF] it ensues from the uncertainty pervading the mappings between several data sources.

In terms of perspectives, we plan to investigate other, non-probabilistic, semantics to approximate query rewriting using views, based on different visions of query relaxation.

  ssn 5 ) : -Child(ssn 5 , cf 5 , a 5 ), Emp(ssn 5 , j 5 , s 5 ), a 5 ∈ [18, 38], s 5 ∈ [1000, 2400]. None of these views has any constraint on distinguished variables. Let us now consider the query: Q(ssn, name) : -P erson(ssn, name, f name), Child(ssn, Cf irstname, Cage), Emp(ssn, job, sal), Cage ∈ [22, 35], sal ∈ [1200, 2300] and the approximate rewritings:

Fig. 1 .Fig. 2 .

 12 Fig. 1. Impact of αmin on the number of answers

  S)where attribute A appears in the head of V 6 and V 7 , i.e., A is a distinguished variable in V 6 and V 7 . The final constraint applying to Q over attribute A would then be the interval[27, 40]. However, since A is a distinguished variable, it can be restricted by a selection and the only possible rewriting is Q (N, A, S) : Cage does not apear in the head of V 8 and consequently is existential in V 8 . If we ignore the interval constraints, Q is contained in Q. On the other hand, the presence of interval constraints leads to discard Q if a Boolean matching is performed. Using a tolerant matching, the "containment degree" of Q in Q is based on Equation1. It is the extent to which interval[20, 40] is included in[22, 35]. If one assumes that attribute age is encoded by integers, one gets the inclusion degree α = (35-22) * 10 0 +1 (40-20) * 10 0 +1 = 14 21 . Hence: Q 0.67 Q.

	Example 2. Let us consider the query Q:
	Q(ssn, name) : -P erson(ssn, name, f name), Child(ssn, Cf irstname, Cage),
	Cage ∈ [22, 35]
	and Q the candidate rewriting of Q:
	Q (ssn, name) : -V 8 (ssn, name)
	where
	V 8 (ssn, name) : -P erson(ssn, name, f name), Child(ssn, Cf irstname, Cage),
	Emp(ssn, job, sal), Cage ∈ [20, 40].
	The attribute
	-V 6 (N, A), V 7 (A, S), A ∈ [28, 38]. The interval constraint added to Q is that
	from Q. The degree attached to Q (N, A, S) is then 1 since the intersection
	between [27, 40] and [28, 38] is not empty: Q	1 Q.
	The example below illustrates Case b) of Proposition 1.

  as there is no interval constraints for this subgoal.-the approximate rewriting V 2 of the second subgoal of Q is computed from the interval constraints on age. It is α 2 = (35-22)+1 (40-20)+1 = 14/21 = 0.67. -the approximate rewriting V 3 of the third subgoal is computed from the interval constraints on salary. It is α 3 = (2300-2100)+1 (3400-2100)+1 = 201/1301 = 0.15. -the approximate rewriting V 5 of the second and the third subgoals of Q is computed from interval constraints on age and salary. It is:

	α 4 = (35-22)+1 (38-18)+1 * (2300-1200)+1 (2400-1000)+1 = 14/21 * 1101/1401 = 0.52.

Therefore, Q 1 (nss, nom) gets the degree 1 * 0.67 * 0.15 = 0.1 while Q 2 (nss, nom) gets 1 * 0.52 (Q 1 0.1 Q and Q 2 0.52 Q). The tuples issued from Q 1 are not likely to satisfy the query while those from Q 2 have a probability over 50%.

Acknowledgement. Work partially funded by the ANR Research Grant ANR-05-MMSA-0007.