Arnault Lassin

Lionel Mercury

Mohamed Azaroual

Geochemistry of Capillary Hydrogeochemical Systems in Arid Environments

In arid environments, porous media are unsaturated with water which is submitted to capillary constraints. The present chapter focuses on the geochemical impacts of such physical constraints and how theoretical analysis can help interpreting field or laboratory observations. The basic principles of capillary geochemistry suggest that the fate of contaminants, either organic or inorganic, can be significantly impacted in terms of reactive mass transfer in addition to flow and transport processes. All these mechanisms are closely interconnected, what makes the description of the behavior of such systems very complicated. An important work still has to be done in order to achieve such a goal: a number of mechanisms are not taken into account in the current state of development of the capillary geochemistry, namely mechanisms that occur in the thinnest confining geometries, like disjoining pressures, surface forces, etc.

INTRODUCTION

Porous media containing water in dry or arid contexts are only partially saturated, which means that water is strongly bounded to the solid matrix either in a capillary state or as thin hydration films on mineral surfaces. These particular features modify the thermo-physical properties of the confined water compared to the free bulk water (see previous chapters). This, in turn, leads to dramatic modification of the geochemical behavior of waterrock-gas systems compared to what is classically observed in saturated porous media. Then questions arise: i) how much different and specific are these behaviors? ii) What are their effective roles on mass and heat transfers from the soil surface to the underlying aquifers and on the water quality of these aquifers? iii) In particular, what can be their role in the fate of pollutants and pesticides? iv) Could they also play a role in the response to the global warming and the fate of greenhouse-effect gases?

In order to interpret observations made on the field or in the laboratory ( § 2), theoretical developments have been performed ( § 3) for dealing with such systems and for starting bringing some tentative answers to questions above ( § 4).

OBSERVATIONS FROM FIELD AND LABORATORY EXPERIMENTS

The first major field observations leading to the idea that the peculiar state of water in unsaturated porous media could shift chemical equilibria were performed by Yves Tardy and his collaborators [START_REF] Tardy | Geochemistry of laterites. Stability of Al-goethite, Al-hematite and Fe 3+ -kaolinite in bauxites and ferricretes. An approach to the mechanism of concretion formation[END_REF][START_REF] Trolard | The stabilities of gibbsite, boehmite, aluminous goethites and aluminous hematites in bauxites, ferricretes and laterites as a function of water activity, temperature and particle size[END_REF][START_REF] Tardy | Activité de l'eau et déplacement des équilibres gibbsite-kaolinite dans les profils latéritiques[END_REF]. These authors pointed out the coexistence of local mineralogical heterogeneities in old lateritic profiles, where thermodynamic equilibrium was expected. More precisely, they observed that hydrated minerals (like gibbsite or goethite) preferentially precipitate in large pores while anhydrous or less hydrated minerals (e.g. kaolinite, hematite) rather formed in smaller pores.

Recently, specific experiments have been performed in order to explore more deeply the underlying mechanisms. Bouzid et al. [START_REF] Bouzid | In-pore tensile stress by drying-induced capillary bridges inside porous material[END_REF] observed the formation of differently shaped salt crystals in the pores of polycarbonate filters, depending on the ambient relative humidity (RH) and the water repartition/dynamics in the porosity. Using NMR technics, Rijniers et al. [START_REF] Rijniers | Salt crystallization as damage mechanism in porous building materials--a nuclear magnetic resonance study[END_REF] measured the solubility of sodium carbonate in a natural porous medium. They observed that this solubility varied as a function of pore size on model porous materials of known pore sizes. The authors recorded that the solubility of sodium carbonate increased with decreasing pore size while sodium sulfate was not sensitive to pore size. Lassin et al. [START_REF] Lassin | Equilibrium partial pressure of CO 2 in the Callovo-Oxfordian argillite as a function of relative humidity[END_REF] monitored the partial pressure of CO 2 (pCO 2 ) at equilibrium with samples of an argillaceous rock at various RH. They observed that the equilibrium pCO 2 was decreasing with decreasing RH.

All these observations indicate that capillary and/or confined porous media exhibit specific complex chemical behaviors compared to bulk water-mineral-gas systems. Therefore, in parallel to these experimental works, theoretical developments have been performed, as described in the following for capillary systems.

THEORETICAL BACKGROUND

Thermodynamics of chemical reactivity in capillary systems

The chemical behavior of capillary systems has been studied for more than ten years, and the theoretical framework is now well established for calculating the standard thermodynamic properties of chemical compounds submitted to capillary pressure [START_REF] Mercury | Negative pressure of stretched liquid water. Geochemistry of soil capillaries[END_REF][START_REF] Mercury | Thermodynamic properties of solutions in metastable systems under negative or positive pressures[END_REF][START_REF] Lassin | Geochemistry of unsaturated soil systems: aqueous speciation and solubility of minerals and gases in capillary solutions[END_REF]. Briefly, the Young-Laplace and Kelvin laws imply that pore water at physical equilibrium with a given relative humidity is subjected to a pressure lower than in the surrounding gas phase. For sufficiently low relative humidity, the internal pressure of waterbelow the gaswater meniscuscan be negative. The corresponding impact on the thermodynamic properties of water, dissolved species and minerals, and therefore on operant chemical reactions can be calculated using the Thermo-ZNS calculation code [START_REF] Lassin | Geochemistry of unsaturated soil systems: aqueous speciation and solubility of minerals and gases in capillary solutions[END_REF]. The latter includes the equations of state (EoS) of water and aqueous species after Wagner and Pruss [START_REF] Wagner | The IAPWS formulation 1995 for the thermodynamic properties of ordinary substance for general and scientific use[END_REF] and Helgeson et al. [START_REF] Helgeson | Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600°C and 5 kb[END_REF]12], respectively. The extended use of these equations to capillary systems and negative pressures can be discussed (see Chapter XX [12]). Their limit of validity, in terms of pressure, is not known and specific studies would be necessary to check or improve existing equations.

However, using the state-of-the-art EoS, one can show that capillary constraints can significantly impact water thermodynamic and electrostatic properties like the dielectric constant, ɛ, and its derivatives with respect to pressure and temperature. As an illustration, the Born function Z (= -1/ɛ) is plotted on Fig. 1-a as a function of temperature, at different pressures. This affects the thermodynamic properties of aqueous species too, which depend on temperature and pressure, but also on the dielectric properties of the solvent, namely water [START_REF] Helgeson | Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600°C and 5 kb[END_REF]. The apparent standard Gibbs free energy of Na + , SO 4 = and Cl -are plotted on Fig. 1-b,c,d, respectively, as a function of pressure, at different temperatures. They show very different behaviors from one aqueous species to another, and from positive to negative pressures. Since, for every chemical reaction, the mass action law involves activity of aqueous species rather than concentration, another parameter related to aqueous species may be affected by capillary pressure. This is the activity coefficient by the means of the Debye-Hückel coefficients A  and B  [START_REF] Lassin | Geochemistry of unsaturated soil systems: aqueous speciation and solubility of minerals and gases in capillary solutions[END_REF] which are functions of water density and dielectric constant, and of temperature.

Consequently, the thermodynamic impact of capillary pressure on homogeneous chemical equilibria (i.e. in water) cannot be foreseen easily. Obviously, this also concerns heterogeneous equilibria (i.e. mineral-water and gas-water systems). Furthermore, in addition to the single pressure effect, the chemistry of capillary systems offers specific features called the anisobar contexts (i.e., where phases can undergo different pressures). They apply particularly to gas-aqueous solution equilibria since a gaseous compound cannot undergo negative pressures whereas its aqueous equivalent undergoes the internal capillary water pressure. A similar situation can be envisioned for minerals initially present in the rock matrix. They are submitted to the local pressure constraint when the porosity is saturated with water and the same pressure applies to these minerals during desaturation regardless of water content. On the other hand, if the capillary aqueous solution is supersaturated with respect to a mineral, the latter should precipitate at the capillary water pressure. Some evidence for this isobar/anisobar situation was obtained experimentally [START_REF] Bouzid | In-pore tensile stress by drying-induced capillary bridges inside porous material[END_REF] as explained in § 4. According to the theoretical developments described above, thermodynamic properties and thus solubility of a mineral differ according to the isobar or anisobar context. As a general trend, anisobar conditions are less favorable to mineral stability than isobaric ones, which is physically consistent. The difference between these two contexts depends on the volumetric properties of the mineral. In particular, hydrated mineral phases are generally more affected by pressure than anhydrous ones. On the other hand, gases are expected to be more soluble in capillary conditions.

Thermodynamic database

In order to simulate the behavior of hydro-geochemical systems, we use the well-known PhreeqC-2 geochemical calculation code [START_REF] Parkhurst | User's guide to PHREEQC (version 2)[END_REF]. It allows calculating the aqueous speciation of solutions, saturation indices of minerals, partial pressures of gases, amounts of precipitated or dissolved minerals, etc. The use of this code at temperatures and pressures far from the liquid-vapor saturation curve is possible if appropriately adapted thermodynamic databases are available. Such databases including both isobar and anisobar mineral dissolution/precipitation reactions were generated using the Thermo-ZNS code indicated above. The reference thermodynamic database implemented in Thermo-ZNS is the Thermoddem database [START_REF] Blanc | Thermoddem: A geochemical database focused on low temperature water/rock interactions and waste materials[END_REF] which provides information on critically selected standard thermodynamic properties of chemical compounds and reactions necessary for their calculation at various temperatures and pressures.

INTERPRETATIONS

Once the theoretical framework is defined, experimental observations evoked previously can be interpreted quite simply. In this section we focus on experiments from Bouzid et al. [START_REF] Bouzid | In-pore tensile stress by drying-induced capillary bridges inside porous material[END_REF] and from Lassin et al. [START_REF] Lassin | Equilibrium partial pressure of CO 2 in the Callovo-Oxfordian argillite as a function of relative humidity[END_REF]. In the former work, the pores of a polycarbonate filter were initially filled with a NaCl aqueous solution slightly undersaturated with respect to halite. During the progressive drying of the filter, different stages were observed and, applying the principles of capillary geochemistry, Bouzid et al. [START_REF] Bouzid | In-pore tensile stress by drying-induced capillary bridges inside porous material[END_REF] could provide the interpretation summarized hereafter (Fig. 2).

a-Precipitation of (cubic) halite while porosity is still saturated with water. Part of the volume of water which has evaporated is occupied by the salt. In some cases the corners of the precipitated salt can touch the pore wall, thus modifying the pore geometry. b-As evaporation proceeds, de-saturation occurs and capillary water goes in the angular spaces created between the pore wall and the salt. A typical anisobar context thus takes place between water and solids. The less soluble and most supple solid (the polycarbonate pore wall) deforms while the most soluble and less supple solid (halite) dissolves, its destabilization being predicted by thermodynamics. The corners of the crystal are eroded.

a) b) c) d)

c-On the contrary, the isobar polymorph of halite, probably a nano-crystalline or an amorphous halite, is stabilized and thus massively precipitates under the capillary meniscus before water has totally evaporated.

All of these successive stages are fully consistent with the theoretical interpretation of the geochemical mechanisms in capillary context. They cannot be explained by the classical geochemistry applied to watersaturated porous systems.

The second series of experiments [START_REF] Lassin | Equilibrium partial pressure of CO 2 in the Callovo-Oxfordian argillite as a function of relative humidity[END_REF] recorded the CO 2 partial pressure at equilibrium with a clay rock in a closed system at various relative humidity values. The rock (the Callovo-Oxfordian argillite) and its pore water are well known. A geochemical model has been developed, involving mineral dissolution/precipitation and cation exchange reactions at thermodynamic equilibrium [16], and is used as a reference for describing this water-rock system. In particular, this model calculates the correct in situ pCO 2 . By simply applying the principles of capillary geochemistry and without any parameter adjustment, Lassin et al. [START_REF] Lassin | Equilibrium partial pressure of CO 2 in the Callovo-Oxfordian argillite as a function of relative humidity[END_REF] could reproduce with a good accuracy the observed decrease of the equilibrium pCO 2 with decreasing relative humidity, from 8 down to less than 1 mbar and from 100 down to 50%, respectively. The authors took into account the evaporation of water from porosity and the consequent increase in salinity. They considered that anisobar alumino-silicates could only dissolve according to the time scale of the experiments (from 6 to 18 months). On the other hand, more reactive minerals, like carbonates, were initially present as anisobar phases and could evolve to their isobar polymorphs. Here again, the observed equilibrium pCO 2 could be reproduced only if principles of capillary geochemistry were applied.

CONCLUDING REMARKS

These results were obtained in very different porous hydro-geochemical systems thus illustrating the wide applicability of the capillary geochemistry approach. Consequently, a new interpretation of the observations of Rijniers et al. [START_REF] Rijniers | Salt crystallization as damage mechanism in porous building materials--a nuclear magnetic resonance study[END_REF], amongst others, could be done in the light of the principles of capillary geochemistry. Applications to other domains can therefore be envisioned quite straightly. They are related to industrial activities like CO 2 (or more generally: non-wetting phases) geological storage as well as nuclear waste repository in argillaceous rocks. They are also related to natural environments which have been impacted by human activity, in particular for studying the fate of inorganic and organic pollutants in arid environments.

Results above also illustrate the significant impact that capillary pressure can have on geochemical equilibria in arid environments. However, the obvious question to ask concerns the volume of water involved and the consequences on the mass/heat transfers through the porous medium. According to Pettenati et al. [START_REF] Pettenati | Capillary geochemistry in non-saturated zone of soils. Water content and geochemical signatures[END_REF] significant amounts of water can remain held in the porosity for low relative humidity, depending on the pore network structure and petrophysical properties of the soil.

On the other hand, the different chemical compounds present in the pore solution can play a significant role on the air-solution interfacial tension and on the wettability of the pore walls depending on their affinity to interfaces and on their concentration in water. This may in turn affect the water content versus relative humidity relationship, but also the relative permeabilities in multiphase systems as well as the hysteresis and memory effects, etc., and thus the mass/heat transfers.

The principles on which capillary geochemistry is based are fairly consistent with approaches focused on other processes like crystallization pressure (see chapter YY [18]) and solution pressure phenomena (see chapter XX [19]). In such processes, anisobar conditions are also encountered in confined porous media, with strong relationships with pore water chemistry and shapes of interfaces. These similarities suggest that a single and general formalism able to describe all the above systems can be developed. An important work remains to achieve such a goal since a number of mechanisms are not taken into account in the current state of development of the capillary geochemistry. In particular, this approach only deals with bulk properties of capillary water. It does not take yet into account mechanisms that occur in the thinnest confining geometries, like disjoining pressures or surface forces (see chapters ZZ [20] and VV [21]). Works are under progress in this direction (see chapters of [22], [23], [24]).

Figure 1 : 4 =,

 14 Figure 1: temperature and pressure dependence of various thermodynamic and electrostatic properties of water and aqueous compounds: a) Born function Z of water, and apparent standard Gibbs free energy b) of Na + , c) of SO 4 = , and d) of Cl -.

Figure 2 :

 2 Figure 2: Successive reactive steps during the evaporation of a NaCl brine from the porosity of a polycarbonate filter and calculated solubility product of halite as a function of capillary pressure, both in the isobar and the anisobar contexts. a) Precipitation of well crystallized cubic halite; b) establishment of capillary conditions and dissolution of the cubic (anisobar) halite angles; c) precipitation of isobar poorly crystallized halite under the capillary meniscus. Modified after [4].

ACKNOWLEDGMENTS

This work has received support from the French Agency for Research (Agence Nationale de la Recherche, ANR) through the grant CONGE BLAN-61001 and ANR-10-LABX-100-01.


Christine Vaillant

Dr Guillaume Leurent
email: guillaume.leurent@chu-rennes.fr

Ronan Garlantezec

Christophe Thebault

Raphael Martins

Emilie Bot

Isabelle Coudert

Dominique Boulmier

Hervé Le Breton

Marc Bedossa

Drs Vaillant

Leurent

Coronary angioplasty is

Keywords: cardiac arrest, mild therapeutic hypothermia, prognostic factors, coronary angioplasty, neurologic outcome

Dear editor,

The prognosis of out-of-hospital cardiac arrest (OHCA) is still poor. Because of its neuro-protective benefits, mild therapeutic hypothermia (MTH) is recommended for comatose adult patients with return of spontaneous circulation (ROSC) after OHCA [START_REF] Neumar | Post-cardiac arrest syndrome: epidemiology, pathophysiology, treatment, and prognostication. A consensus statement from the International Liaison Committee on Resuscitation (American Heart Association, Australian and New Zealand Council on Resuscitation, European Resuscitation Council, Heart and Stroke Foundation of Canada, InterAmerican Heart Foundation, Resuscitation Council of Asia, and the Resuscitation Council of Southern Africa); the American Heart Association Emergency Cardiovascular Care Committee; the Council on Cardiovascular Surgery and Anesthesia; the Council on Cardiopulmonary, Perioperative, and Critical Care; the Council on Clinical Cardiology; and the Stroke Council[END_REF][START_REF] Peberdy | Part 9: post-cardiac arrest care: 2010 American Heart Association Guidelines for Cardiopulmonary Resuscitation and Emergency Cardiovascular Care[END_REF]. However, in the era of modern management of OHCA, including MTH, prognostic factors of neurologic outcome for those patients are still unclear.

METHODS:

We performed a single-center retrospective review of all the patients admitted to our Intensive Cardiac Care Unit (ICCU) for a non-traumatic OHCA after successful resuscitation, from January 2008 to December 2011. Patients were excluded when OHCA was due to obvious non-cardiac cause and when MTH was not induced, regardless of the reason, such as response to verbal commands after ROSC. Coronary angiography was performed on all patients, except in the case of obvious noncoronary cause (e.g.: known non-ischemic cardiopathy). MTH was initiated as soon as possible, without deferring any required treatments.

The primary outcome was a favorable neurological outcome in living patients at hospital discharge, defined as a Pittsburgh cerebral performance category score (CPC) of 1 or 2 (respectively good recovery or moderate disability). Poor outcomes were category 3, 4 and 5 (respectively severe disability, vegetative state or death) [START_REF] Jennett | Assessment of outcome after severe brain damage[END_REF][START_REF] Abramson | Impact evaluation in resuscitation research: discussion of clinical trials[END_REF].

Univariate analysis was performed using the Pearson-Khi square test to identify the risk elements associated with survival. Prognostic factors that were found to be associated with a p value ≤0.20 in the univariate analysis were eligible to be tested in the multivariate model. Multiple imputation was performed to take missing data into account [START_REF] Ibrahim | Missing Data in Clinical Studies: Issues and Methods[END_REF]. Logistic regression was performed using a stepwise procedure. The final model included only covariates associated with a p value ≤0.05. Odds ratios (OR) and their confidence interval (95% CI) were presented. All calculations were performed using SAS 9.1 (SAS Institute, Cary, NC) and R (http://R-project.org) software.

RESULTS: Seventy patients who had been admitted for a resuscitated OHCA and treated by MTH in our ICCU constituted our study group (Table 1). The mean age was 59 ±7 years and the men-towomen ratio was 7:1. The OHCA was witnessed in 62 patients (88%), with a bystander cardiopulmonary resuscitation (CPR) in 35 (50%). "No-flow" mean-delay was 4.2±5 minutes (median:

2) and "low-flow" was 22.6±16 (median: 15). The initial rhythm was shockable in 56 patients (80%).

The number of electric shocks was 3.5±3 and 3.5±5 mg epinephrine was administered. The electrocardiogram (ECG) performed at the waning of the resuscitation was normal in 17 patients (24%). Biomarkers on admission were: arterial pH: 7.28±0.1, lactate: 3.7±2 mmol/L, creatinine: 112±41 µmol/L.

Good outcome n=35

Poor

DISCUSSION

In this population, the survival rate was 51% and a good neurological outcome was found in 50% of the patients on hospital discharge, in accordance with literature [START_REF] Yokoyama | Impact of therapeutic hypothermia in the treatment of patients with out-of-hospital cardiac arrest from the J-PULSE-HYPO study registry[END_REF]. This confirms the favorable impact of short delays and the presence of a witness in OHCA. Given the crucial importance of the time from collapse to CPR, efforts should be made to promote the culture of resuscitation not only among health professionals, but also among the general population [START_REF] Nielsen | Engaging a whole community in resuscitation[END_REF].

Otherwise, more than 80% of resuscitated patients underwent coronary angiography, in agreement with the fact that approximately 80% of sudden cardiac deaths are attributed to CAD [START_REF] Chugh | Current burden of sudden cardiac death: multiple source surveillance versus retrospective death certificate-based review in a large U.S. community[END_REF]. Overall, it remains difficult to identify which patients will benefit from early angiography and/or PCI after OHCA, as usual assessment criterion (chest pain and ST elevation) may be poor predictors of ACS in OHCA patients. Indeed, ST-segment elevation has poor accuracy in the setting of post resuscitation [START_REF] Edgren | The presenting ECG pattern in survivors of cardiac arrest and its relation to the subsequent long-term survival. Brain Resuscitation Clinical Trial I Study Group[END_REF]: reported sensitivity and specificity of ST-elevation to detect the presence of an underlying acute coronary lesion are 50% and 88%, respectively [START_REF] Lellouche | Sudden cardiac arrest: ECG repolarization after resuscitation[END_REF].This led to the recommendations that emergent coronary angiography may be reasonable in all OHCA patients, regardless of the initial post-ROSC ECG [START_REF] Peberdy | Part 9: post-cardiac arrest care: 2010 American Heart Association Guidelines for Cardiopulmonary Resuscitation and Emergency Cardiovascular Care[END_REF]. Furthermore, only PCI and MTH showed in the last decade a positive influence on long term prognosis following OHCA [START_REF] Kern | Optimal treatment of patients surviving out-of-hospital cardiac arrest[END_REF][START_REF] Dumas | Long-term prognosis following resuscitation from out of hospital cardiac arrest: role of percutaneous coronary intervention and therapeutic hypothermia[END_REF].

Finally, none of our patients cumulating the four independent prognostic factors of good neurologic outcome (witnessed OHCA, ROSC<25min, admission creatinine <104µmol/l and PCI) had a poor neurological outcome, even though the small number of patients and the selected population of the study preclude the development of prediction models.

There are several obvious limitations in this study: this is a single-center, retrospective and observational study, thus subject to confounders and selection bias.

CONCLUSION

Half of the patients hospitalized in our ICCU had a good neurological outcome after a resuscitated OHCA and treatment with MTH. Multivariate analysis showed 4 predictive independent factors of a good neurological outcome: a witnessed OHCA, ROSC<25min, admission creatinine <104µmol/l and a coronary angioplasty. These findings confirm the importance of public awareness for resuscitation care, and the importance of using emergency coronary angiography, in order to improve the neurological outcome.

outcome n=35 p value Characteristics and medical history

	Qualitative data are expressed as counts and percentages.
	CPR: Cardio-Pulmonary Resuscitation; ROSC: Return Of Spontaneous Circulation; VF: Ventricular
	Fibrillation/ VT: Ventricular tachycardia; PCI: Purcutaneous Coronary Intervention.
	Male gender	31 (89)	29 (82)	0.73
	Age	55.6 ±12	63.1 ± 15	0.018
	Diabete mellitus	3 (8)	6 (17)	0.47
	Current smoker	13 (37)	12 (34)	0.80
	Hyperlipidemia	17 (49)	19 (54)	0.63
	Hypertension	8 (23)	19 (54)	0.013
	Known cardiopathy The neurological outcome on hospital discharge was favorable ("good outcome" group) for 35 (50%) 17(49) 24 (69) 0.08
	Cardiac arrest history patients (CPC 1: 26 patients; CPC 2: 9 patients). The "poor-outcome" group included 35 (50%)
	Chest pain as 1 st symptom Location : public place patients (CPC 3: 1 patient, CPC 5: 34 patients).	15 (43) 15 (43)	12 (34) 14 (40)	0.62 0.23
	Time of occurrence (8am-8pm)	28 (80)	21 (60)	0.12
	Witnessed arrest	35 (100)	27 (77)	0.003
	Bystander CPR	21 (60)	14 (40)	0.47
	No-flow mean delay (min)	2.5 ± 3	6 ± 8	0.017
	Low-flow mean delay (min)	17 ±13	30 ±18	0.007
	Time between collapse and ROSC	20 ±16	37 ±16	0.005
	VF/VT as initial rhythm	29 (83)	27 (77)	0.55
	Initial treatment
	Number of electric shocks	3.4 ± 1	4.4 ± 3	0.68
	Dose of epinephrine (mg) Antiarrhythmic drugs Final diagnosis of OHCA etiology was coronary artery disease for 53 patients (75%), including 37 2.6 ± 2 15 (43) 4.9 ± 7 9 (25) 0.012 0.62
	Attempted PCI Successful PCI (52%) of ACS, with similar prevalence in both groups (60% in the "good outcome" group vs 46%; 21 (60) 20 (57) 7 (20) 5 (14) 0.002 0.006
	ECG Patterns p=0.3). The remaining causes of OHCA were other cardiomyopathies in 12 patients (17%) (dilated in
	ST-segment elevation Unspecific ST or T patterns Unknown Left Bundle Branch Block Normal ECG 5, hypertrophic in 1, valvulopathy in 5, and channelopathy in 1 patient ). No etiology was found in 5 14(40) 9 (26) 4 (11) 8 (22) 12 (34) 0.8 11 (31) 0.7 9 (25) 0.7 3 (9) 0.8 patients (7%).
	Blood biomarkers on admission Multivariate analysis identified 4 independent factors which are significantly associated with
	Arterial pH Lactate (mmol/L) Potassium (mmol/L) I Troponin (ng/mL) Creatinine (µmol/L) Glucose (mmol/L) neurological prognosis: a witnessed OHCA (100% in the "good outcome" group vs 77%, OR=0.18 7.30 ±0.1 2.7 ±2 4.0 ±1 8.2 ±25 103 ±39 12 ±6 0.09 7.27 ±0.1 4.7 ±4 15 ± 6 0.61 121 ±41 0.013 6.1 ±19 0.92 3.9 ±1 0.006 0.42 [0.05-0.80]), an attempted PCI (68 vs 27%, OR=0.11 [0.02-0.51]), a ROSC delay ≥ 25min (23 vs 66%

All pre-hospital delays (no-and low-flows) were shorter in the "good outcome" group. Regardless of the outcome, post resuscitation ECG in the global population showed ST-segment changes in 75% of patients, including a ST-segment elevation in 26 patients (37%). The predictive positive and negative values of the latter associated with an acute coronary syndrome (ACS) were 96% and 41% respectively, whereas sensitivity and specificity were 49% and 95% respectively. OR=6.05

[1.58-23.23

] and a creatinine blood level ≥104 µmol/L at hospital admission (31 vs 68% OR=4.62

[1.23-17.39

].


Franc ¸ois Tavard

Antoine Simon

Alfredo I Hernández

Julián Betancur

Erwan Donal

Christophe Leclercq

Mireille Garreau

Spatio-Temporal Registration of Electro-Anatomical Mappings with Functional Data for CRT Optimization

This work is based on the registration and fusion of multimodality data for cardiac resynchronization therapy (CRT) optimization. The goal is to provide local descriptors of the electro-mechanical (EM) coupling of the heart, using anatomical, electrical and mechanical data acquired from CT-scan volumes (CT), speckle tracking echocardiography (STE) and electro-anatomical mapping (EAM). In previous works, we have presented registration methods to place EAM, CT and STE in the same geometrical reference system.

In order to extract EM descriptors, data resulting from EAM and STE should be synchronized. This is done according to the ECGs acquired during both acquisitions. However, the number of ECG derivations as well as the cardiac frequencies differ in both acquisitions. In this work, the selection of the most appropriate ECG derivation acquired with EAM is based on the computation of the cross-correlation between the three ECGs from EAM and the one from the STE acquisition. A non-linear mapping (using dynamic time warping) is estimated between the two signals to handle the differences in cardiac frequencies. This whole process enables to represent useful descriptors for CRT resulting from EAM, STE and CT data in the same spatio-temporal reference system. Results are presented on three patients undergoing CRT.

Introduction

Cardiac Resynchronization Therapy (CRT) has been accepted as a therapeutic option in heart failure patients who remain highly symptomatic despite optimized medical treatment. However, 30% of the patients do not respond to the therapy [START_REF] Morgan | Lead positioning for cardiac resynchronization therapy: techniques and priorities[END_REF]. Therefore, imaging modalities including echocardiography or magnetic resonance imaging, aiming for mechanical dyssynchrony estimation, have been proposed to improve patient selection criteria and lead place-ment [START_REF] Morgan | Lead positioning for cardiac resynchronization therapy: techniques and priorities[END_REF][START_REF] Chung | Results of the predictors of response to CRT trial[END_REF]. The most challenging tasks to carry out remain both the identification of the most effective pacing sites and the guidance of the left ventricular (LV) lead implantation using a venous access.

Our goal is to better plan the placement of CRT leads using anatomical, functional and electrical data acquired with cardiac Computed Tomography (CT) imaging, Speckle Tracking Echocadiography (STE) and Electro-Anatomical Maps (EAM) respectively 1 . In previous works, we proposed to solve geometrical registration problems between CT and EAM data by a semi-interactive method [START_REF] Tavard | Data fusion of left ventricle electroanatomical mapping and multislice computerized tomography[END_REF] rather than mainly used simultaneous acquisitions [START_REF] Rhode | Modeling and registration for electrophysiology procedures based on three-dimensional imaging[END_REF] and between CT and STE data by geometrical method using the entire cardiac cycle [START_REF] Tavard | Fusion of electro-anatomical mapping and speckle tracking echography for the characterization of local electro-mechanical delays in CRT optimization[END_REF] rather than landmark methods [START_REF] Huang | Dynamic 2D ultrasound and 3D CT image registration of the beating heart[END_REF]. These two methods give a common 3D space for the three modalities.

We here propose to combine STE and EAM to compute local electro-mechanical delays (local EMDs). To achieve this goal, strain signals from STE and electrical data from EAM are to be temporally registered. For this purpose the ECGs acquired during both acquisitions are considered. However, the acquisitions in both modalities are quite different in term of derivation and present non-linear differences in cardiac frequencies. We propose a method to synchronize STE and EAM data based on cross-correlation and dynamic time warping (DTW). This paper first describes the mechanical and electrical data considered. Secondly, we give the outlines of our registration methods. The temporal registration of EAM and STE is then detailed step by step, leading to a discussion on the local EM coupling extracted for three patients and illustrated by quantitative 12 segments bullseye maps. Finally conclusions and future works are proposed.

Global strategy and data processing

The global strategy is to define a global spatial and temporal reference to integrate EAM, STE and CT before applying fusion process to extract local EM descriptors (Fig. 1). We give below a brief description of the acquired data and the proposed registration methods.

Minimum of strain. STE are acquired on 2 and 4chamber apical views over a cardiac cycle, along with a single ECG derivation. On each view, the strain is computed on 6 anatomical segments (giving 12 segments in total). The mechanical information used is the delay from the beginning of the QRS to the minimum of strain, which matches the maximum of contraction of the considered segment. We will refer to this measure as "global electromechanical delays" (global EMDs) by opposition to local EMDs.

Local activation time and voltage. During the EAM acquisition, a moving probe scans the endocardium wall and a signal of the electrical activity or electrogram (EGM) is acquired on sparse measurement points, along with three derivations of the ECG (DI, V1, aVF). From EGMs, the electrical informations extracted are: (i) the local activation times (LAT) i.e. the time from a fixed temporal reference (usually sinusal activation) to the maximum of the electrical activity. LAT represent the delay to reach local activation; (ii) the voltage i.e. the signal's peak-topeak amplitude. Voltage characterizes the amount of myocardium cells locally contracting.

Spatial registration methods have been proposed in our previous works. The registration of EAM and CT images is based on the automatic alignment of apex and great axis of surfaces extracted from both acquisitions, and on the manual rotation of the EAM surface around its great axis. LAT acquired by EAM are then projected on a LV model surface [START_REF] Tavard | Data fusion of left ventricle electroanatomical mapping and multislice computerized tomography[END_REF] (modelled from CT segmented surfaces [START_REF] Fleureau | Assessment of global cardiac function in MSCT imaging using fuzzy connectedness segmentation[END_REF]). A validation of this step has been done by checking that the stimulation lead position (given by angiography) matches the first activated point (given by EAM).

We have proposed a registration method for STE and CT based on a geometrical metric and a priori consideration [START_REF] Tavard | Fusion of electro-anatomical mapping and speckle tracking echography for the characterization of local electro-mechanical delays in CRT optimization[END_REF]. A second method has been recently proposed for this registration which relies on a Fourier contour description and on Dynamic Time Warping (DTW) [START_REF] Tavard | Dynamic registration of cardiac US and CT data using fourier descriptors and dynamic time warping[END_REF][START_REF] Itakura | Minimum prediction residual principle applied to speech recognition[END_REF] to handle the non-linearities between both temporal bases.

Extraction of local electro-mechanical delays

In this section we describe the two step method for temporal registration (signals synchronization and time warping) and the computation of local EMDs. DTW enables to measure similarities between two sequences which may vary in time or dynamics. The sequences are then warped non-linearly in the time dimension. For this purpose, DTW is based on the search for a warping function F = {(i 1 , j 1 ), (i 2 , j 2 ), . . . , (i N , j N )} which is a sequence of N couples of (t EAM , t ST E ) such that a distance D(F ) is minimized. This distance is defined to best match both ECGs signals along the cardiac cycle:

D(F ) = N n=1 d(i n , j n ).w(i n , j n ), where d(i n , j n ) = d(t ST Ei , t EAMj ) = dECG ST E i dt ST E - dECG EAM j dt EAM
and w(i n , j n ) is a weight (with a classical symmetric form) introduced to compensate the variation of the path length:

w(i n , j n ) = (i n -i n-1 + (j n -j n-1
)). By using the first derivative of the ECG signals, priority for time warping is given to the ECG shapes and not its values. Some classical restrictions on the warping function are added in order to preserve the structures of both cardiac cycles: monotony, continuity, boundary conditions (i 1 = 1, i N = t ST E n , j 1 = 1 and j N = t EAM m ) and slope constraint (to favour a (+1, +1) direction in the time space see Fig. 3-Top). Moreover, the search domain has been restricted using a combination of commonly used Itakura's parallelogram and Sakoe's ribbon (Fig. 3-Bottom). Finally, the correspondence between both time sequences is obtained by finding the minimum path D(F ) through dynamic programming.

Definition of local EMDs

4.

Results and discussion

Local EMDs are computed on the 12 segments on which strain is measured. In this study, we present results of EMD for three patients P1, P2 and P3. For patient P2, only the 4-chamber view was available, reducing the number of segments to six. Results are presented on 12 segments bullseye (Fig. 5) and described below.

Local activation time. First activated segments are located on the septal wall and the latest activated segments on the lateral or anterior wall. The entire electrical activation of the LV is propagated in approximately 60 ms for patients P1 and P2 and in about 40 ms for patient P3.

Global electro-mechanical delays. Since the reference here is the ECG, those delays are a combination of (i) activation time delays inter-segments (the delays generated by the electrical propagation) and (ii) the electrical and mechanical delays intra-segments (the delays generated by the local electrical impulse, the mechanical response and contraction). For these patients, the shortest global EMD are found on the septal wall which is not always the case for patients suffering from ventricular asynchronism. For the other segments, global EMD are greater (between 100 and 550 ms) and even greater than the systolic time for patient P3 (the maximum of contraction occurs after the aorta valve closure).

Local electro-mechanical delays. Unlike global EMD which combine inter and intra-segment delays, local EMD only consider the intra-segment electrical and mechanical delays, therefore better describing the local behaviour of the LV wall. By analysing local EMD and local activation time, patient P2 stands out compared to patients P1 and P3: (i) patient P2: the local EMD appears quite homogeneous with low values. These low local EMDs could show a good electrical propagation in the corresponding segments and / or a good contraction response to the electrical stimuli. The lateral medium and lateral basal segments show a lack of inter-segment electrical propagation with high value activation times (local activation time map). In a CRT scheme, this is an important fact: if those segment were to be stimulated, the mechanical response should give an efficient contraction. (ii) patients P1 and P3: the local EMD are of higher values, this is also an important fact, for it is likely that those segments do not offer a good contraction, even with an earlier electrical stimulation.

Voltage maps. Compared to the voltage map for patient P2 for which every value is above 2mV (showing a nice recruitment of cardiac muscular cells), the maps for patients P1 and P3 show low voltage segments. This is a complementary information to local EMDs even if local EMDs maps are more reliable for acquisition technique reasons.

Patient response to CRT. Presented results show the importance of combining local electrical data to STE mechanical data. The goal of CRT is to correct inter-segments electrical delays considering an efficient intra-segment activation. This intra-segment activation is characterized by local EMDs.

In our CRT scheme, we look for a predictive response of the patients to the probe implantation. The actual quality of the response to CRT is given by clinical follow-up for these patients:

• Patient P1 went from a 18% left ventricle ejection fraction (LVEF) to 25% after implantation giving a slightly better condition.

• Patient P2 respond to CRT with a 60% LVEF after CRT implantation (against 35% before) with no longer signs of asynchronism.

• The response to CRT of patient P3 is moderate. His LVEF went from under 20% to 35%. The high response to CRT of patient P2 is matching the analysis made with local EMDs. The fact that the latest segment is being stimulated explains the high delay correction and the synchronism of the LV after implantation.

Conclusion

We have proposed a new method for the fusion of electrical (from EAM) and mechanical (from STE) data on an anatomical LV surface (here computed from CT-scan). To In CRT the understanding of the local EM coupling of the LV is necessary to better localize the implantation probe site. We show the importance of EAM, a modality offering a local access to electrical information. Even if this modality cannot be used easily in routine acquisition, other ways to compute local electro-mechanical delays, for instance from surface ECGs [START_REF] Berenfeld | Toward discerning the mechanisms of atrial fibrillation from surface electrocardiogram and spectral analysis[END_REF], should be a key to CRT optimization.

Figure 1 .

 1 Figure 1. Global strategy for the fusion of anatomical, electrical and mechanical data.

  ECGs acquired in both modalities are used to synchronize the strain signals from STE and the EGMs from EAM. First the maximum of cross-correlation gives us the ECG derivation from EAM best suiting the ECG from STE. Then we use this maximum to synchronize both ECGs giving a unique time reference as shown on Fig. 2.

Figure 2 .

 2 Figure 2. Top: ECG signals acquired in EAM (three derivations) and STE. Bottom: EAM ECG derivation choice (in this example V1) and synchronization by maximum of correlation. Time warping. Both time spaces t ST E and t EAM will be defined by t ST E1 . . . t ST En and t EAM1 . . . t EAMm .DTW enables to measure similarities between two sequences which may vary in time or dynamics. The sequences are then warped non-linearly in the time dimension. For this purpose, DTW is based on the search for a warping function F = {(i 1 , j 1 ), (i 2 , j 2 ), . . . , (i N , j N )} which is a sequence of N couples of (t EAM , t ST E ) such that a distance D(F ) is minimized. This distance is defined to best match both ECGs signals along the cardiac cycle: D(F ) =

Figure 3 .

 3 Figure 3. Left: DTW between two signals U and V . Right: restriction domain on the search space.

Figure 4 .

 4 Figure 4. Localization of anatomical segments on a 12 segments bullseye.

  With both modalities in the same 3D+t space, we are able to reach local EM coupling. By opposition to global EMDs, we propose a new indicator of the EM coupling: local EMD. It is defined as the time from local activation t LAT to local minimum of strain t strain . It represents the time δ EM D from the electrical stimuli to the maximum of contraction. δ EM D = t strain -t LAT where t LAT ∈ [t ST E1 . . . t ST En ] (after DTW is applied).


Wudage Melisachew
email: melisachew.chekol@inria.fr

Mehwish Chekol
email: mehwish.alam@inria.fr

Amedeo Alam

Napoli

A Study on the Correspondence between FCA and ELI Ontologies

The description logic EL has been used to support ontology design in various domains, and especially in biology and medicine. EL is known for its efficient reasoning and query answering capabilities. By contrast, ontology design and query answering can be supported and guided within an FCA framework. Accordingly, in this paper, we propose a formal transformation of ELI (an extension of EL with inverse roles) ontologies into an FCA framework, i.e. KELI, and we provide a formal characterization of this transformation. Then we show that SPARQL query answering over ELI ontologies can be reduced to lattice query answering over KELI concept lattices. This simplifies the query answering task and shows that some basic semantic web tasks can be improved when considered from an FCA perspective.

Introduction

Relying on Semantic Web (SW) languages and principles, several ontologies have been created in various domains, especially, in biology and medicine. In addition to that, since the conception of linked data publishing principles, over 295 linked (open) datasets have been produced 1 . Querying these data is mainly done through the W3C recommended query language SPARQL 2 .

In parallel, knowledge discovery in data represented by means of objects and their properties can be done using formal concept analysis (FCA) [START_REF] Ganter | Formal Concept Analysis[END_REF]. Concept lattices can reveal hidden relations within data and can be used for organizing and classifying data. A survey of the benefits of FCA to SW and vice versa has been proposed in [START_REF] Sertkaya | A survey on how description logic ontologies benefit from FCA[END_REF]. As mentioned in that paper, a few of these benefits ranges from knowledge discovery, ontology completion, to computing subsumption hierarchy of least common subsumers. Additionally, studies in [START_REF] D'aquin | Extracting relevant questions to an RDF dataset using formal concept analysis[END_REF] and [START_REF] Kirchberg | Formal concept discovery in semantic web data[END_REF] are based on FCA for managing SW data while finite models of description logics (as EL) are explored in [START_REF] Baader | A finite basis for the set of el-implications holding in a finite model[END_REF][START_REF] Baader | Exploring finite models in the description logic EL gfp[END_REF]. All these studies propose methods to use FCA in the analysis of SW data. Nevertheless, none of them offer a precise way of representing SW data within a formal context. We deem it necessary to provide mathematically founded methods to formalize the representation and the analysis of SW data.

In this work, we focus particularly on ELI (an extension of EL with inverse roles) ontologies. EL is one of OWL 2 profiles (OWL 2 EL). In fact, OWL 2 EL is used mainly for designing large biomedical ontologies such as SNOMED-CT 3 , and the NCI thesaurus 4 . A common feature of these ontologies is that they possess large concept hierarchies that can be queried with SPARQL. Answering SPARQL queries is done by binding variables of the query into terms of the queried ontology. However, including inferred data in the query answers requires either a reasoner to infer all implicit data or query rewriting using regular expression patterns (that enable navigation in a hierarchy) [START_REF] Glimm | Using SPARQL with RDFS and OWL entailment. Reasoning Web[END_REF]. The latter obliges the user to know the nuts and bolts of SPARQL. To overcome these difficulties, we reduce SPARQL query answering in ELI ontologies into query answering in concept lattices along with the transformation of the queried ontology into a formal context. Querying a concept lattice appears to be a less complex task than using SPARQ. Further, the lattice organization, i.e., partial ordering, can help understanding the relations between data and visualization of SW data.

Overall, in this paper, we work towards (i) a formal characterization of the translation of ontologies into a formal context, (ii) minimizing the difficulty of SPARQL query answering over ontologies into LQL (Lattice Query Language) query answering over concept lattices, and finally (iii) providing organization of SPARQL query answers with the use of concept lattices.

Outline: after presenting the basics of ELI, SPARQL and FCA ( §2), we show how to transform ELI ontologies into formal contexts ( §3). We then present a query language for concept lattices called LQL ( §4). Therefore, SPARQL query answering over ELI ontologies can be reduced to LQL query answering over K ELI concept lattices ( §5). Finally, we present the related works ( §6) along with a summary of concluding remarks ( §7).

Preliminaries

In this section, we provide a very brief and intuitive introduction of the description logic ELI and FCA. For a detailed discussion, we refer the readers to [START_REF] Hayes | RDF semantics[END_REF][START_REF] Baader | Pushing the EL envelope[END_REF][START_REF] Ganter | Formal Concept Analysis[END_REF][START_REF] Carpineto | Concept data analysis: Theory and applications[END_REF].

In ELI, classes are inductively defined from a set N C of class names, a set N R of role names, and a set N I of individual names (N C , N R , and N I are finite), using the constructors: ⊤, C ⊓ D, and ∃R.C are classes. Where C and D refer to classes, R refers to a role name or its inverse R -, and in the assertion C(a), a refers to an individual. In this paper, we consider ∃R.C classes with C ∈ N C , i.e, C is an atomic concept in the expression of ∃R.C. The TBox of a EL knowledge base contains a set of class inclusion axioms such as C ⊑ D. The ABox contains class and role assertions: C(a) and R(a, b). The semantics of ELI is broadly discussed in [START_REF] Baader | Pushing the EL envelope[END_REF]. The semantics of ELI-classes is defined in terms of an interpretation I = (∆ I , . I ). The domain ∆ I is a non-empty set of individuals and the the interpretation function . I maps each class name A ∈ N C to a subset C I of ∆ I , each role name R ∈ N R to a binary relation R I on ∆ I , and each individual name a ∈ N I to an individual a I ∈ ∆ I . The extension of . I to arbitrary class descriptions is defined inductively [START_REF] Baader | Pushing the EL envelope[END_REF].

SPARQL is a W3C recommended query language [START_REF] Prud'hommeaux | SPARQL query language for RDF[END_REF] based on simple graph patterns. It allows variables to be bound to components in the queried graph. In addition, operators akin to relational joins, unions, left outer joins, selections, and projections can be combined to build more expressive queries. Queries are formed from query patterns which in turn are defined inductively from path patterns, i.e., tuple t ∈ UBV × e × UBLV, with V a set of variables disjoint from UBL (URIs, Blank nodes and Literals -are used to identify values such as strings, integers and dates.), and e is regular path expression. Path patterns grouped together using operators AND (.) and UNION form query patterns. Definition 1. A query pattern q is inductively defined as:

q ::= UBV × e × UBLV | q 1 . q 2 | {q 1 } UNION {q 2 } e ::= ǫ | U | V | e 1 /e 2 | e 1 e 2 | e + | e *
A SPARQL SELECT query can be formed according to the following syntax: SELECT W FROM O WHERE {q}. The FROM clause identifies the queried ontology O on which the query will be evaluated, WHERE contains a query pattern q that the query answers should satisfy and SELECT singles out the answer variables W ∈ V from the query pattern. For this work, we consider only AND and UNION SPARQL queries.

A formal context represents data using objects, attributes, and their relationships. Formally, it is a triple K = (G, M, I) where G a set of objects, M a set of attributes, and I ⊆ G × M is a relation. A derivation operator ( ′ ) is used to compute formal concepts of a context. Given a set of objects, the operator derives common attributes of these objects and vice versa. A set of formal concepts ordered with the set inclusion relation form a concept lattice [START_REF] Carpineto | Concept data analysis: Theory and applications[END_REF].

In the next section, we show the transformation of ELI ontologies into formal contexts.

Transforming ELI Ontologies into Formal Contexts

In the following, we introduce some terms and notions that we use. Materialization (closure) refers to computing the deductive closure of an ontology (alternatively, making all implicitly stored data explicit by using inference) [START_REF] Horst | Completeness, decidability and complexity of entailment for RDF schema and a semantic extension involving the OWL vocabulary[END_REF]. Ontology completion [START_REF] Baader | Completing description logic knowledge bases using formal concept analysis[END_REF] -refers to computing the closure of the ontology and adding additional instances by following class inclusions in the TBox (for instance, if Actor is a subclass of Artist and the instance Tom is an Actor, add another instance who is not an Actor but is an Artist. In this case, if an instance is not known, one can use anonymous resource to identify the unknown instance). Loss of semantics-the transformation of an ontology into a formal context results in loss of semantics if the context mixes TBox (schema axioms) and ABox (instance) data and if the concept lattice obtained from the formal context does not maintain the class hierarchy. Before presenting how a ELI ontology can be transformed into a formal context, we motivate our approach with an example. Starting with Example 1, one can ask whether it is possible to obtain a formal context from the ontology O while maintaining its semantics. The problem here is that DLs and FCA work on different assumptions, i.e, while DL languages are based on the open world assumption (OWA), FCA relies on the closed world assumption (CWA). The former permits to specify only known data whereas the later demands all data should be explicitly specified. To slightly close the gap between these two worlds:

Motivation

one can generate the formal context from the closure of the ontology. However, this approach fails when it is not possible to compute the closure of the ontology as this is the case for ontologies created from a DL language equipped with negation and disjunction constructs5 , and before transforming the ontology into a formal context, complete the ontology. A drawback of the second approach is that it adds unnecessary data, consequently, giving unwanted results when querying. To this end, our main objective is to come up with an approach which transforms an ontology into a formal context while maintaining the semantics. Accordingly, a formal context corresponding to the ontology in Example 1 has an associated lattice that looks like the one in Figure 1a. From this onwards, when we speak of this lattice, we refer to it as the target lattice. The target lattice maintains the semantics because: the class hierarchy of the ontology (TBox) is the same as that of the lattice, and the instance (ABox) and schema (TBox) part of the ontology are treated separately as discussed in Section 3.2.

In the following, we provide various formal contexts associated with the ontology in Example 1. For the sake of readability, we shorten concept and individual names as: ActorsFromNewYork (AFNY), FilmProducer (Prd), Actor (Act), Artist (Art), Person (Per), and tomCruise (tC). Consider the following transformations:

Naive approach: materialized ABox AFNY Prd Act Art Per

tC x x x x x
This formal context is obtained from the materialized ABox of the ontology. It does not include subclass relations as they can be acquired using attribute exploration [START_REF] Ganter | Formal Concept Analysis[END_REF]. But unfortunately, the resulting lattice considers all the attributes to be equivalent, implying loss of semantics, as it can be seen from the lattice in Figure 2b.

Direct approach: materialized ABox and TBox ). The concept lattice of this formal context is shown in Figure 2a. As it can be seen, it does not maintain the semantics because the concept hierarchy is different from that of our target lattice (in Figure 1a). In other words, the concept hierarchy of the concept lattice is different from that of the ontology (in Example 1). Everything is mixed: attributes are also objects and vice versa. Obviously, it is possible to find several other ways of transforming an ontology into a formal context. To avoid any semantic loss, we propose an another approach, where we separately manage the transformation of ABox and TBox assertions. In FCA, attribute exploration is used to discover implicit knowledge. In that, given a concept lattice, a domain expert is asked a series of questions to produce implications that correspond to DL like inclusion axioms. Ontologies contain instance and schema data, where the latter is similar to implications of concept lattices. Hence, when transforming, individuals in the ABox to become objects and concept names to become attributes, besides, assertions in the ABox are transformed into relations. Additionally, class inclusions of the TBox become background implications. The overall transformation pro-cedure leads to a formal context with respect to existing knowledge (this is also known as background implications according to [START_REF] Ganter | Attribute exploration with background knowledge[END_REF]). This procedure is formally described in definition 2.

Proposal

To

K AFNY Prd Act Art Per tC x L = { AFNY → Act, Prd → Art, Act → Art, Art → Per }
Construction of concept lattices: there are several algorithms that can compute concept lattices associated with a formal context. Some of these are discussed in the literature [START_REF] Ganter | Formal Concept Analysis[END_REF][START_REF] Carpineto | Concept data analysis: Theory and applications[END_REF] and have also been implemented. They work on an empty implication base. Thus, most are not suitable for contexts with background implications. In [START_REF] Ganter | Attribute exploration with background knowledge[END_REF], the author provides an algorithm for attribute exploration with background implications. This technique can be employed for our purpose. As a result, the concept lattice associated with the formal context and background implications of Example 2 is depicted in Figure 1b.

Next we show that concept lattices associated with ELI ontologies can be queried by LQL -lattice query language. SPARQL query answering over ELI ontologies can be considered as lattice query answering over K ELI concept lattices. To do this, we need to introduce a query language for concept lattices. Each node in a lattice can be seen as a query formed by a conjunction of: a concept intent and a concept extent. Intuitively, querying concept lattices amounts to fetching the objects given a set of attributes as query constants, alternatively, fetching the attributes given a set of objects as query constants or terms. Query terms can be connected using the logical operators: AND and OR to form a complex term. A term is either a set of objects called object term (OT) or a set of attributes called attribute term (AT). Based on the definitions of object and attribute terms, we introduce LQL queries. In this paper, we do not address the problem of negation in the query (and thus set difference).

Definition 4 (LQL -Lattice Query Language). Given an object term OT, an attribute term AT, and variables x, y ∈ V where V is a finite set of variables, an LQL query can take the following forms: q(y) = (OT, y); q(x) = (x, AT); q() = (OT, AT ) q(y), q(x), and q() do not necessarily correspond to formal concepts, only when OT and AT are closed sets. If OT is a closed set in q(y) = (OT, y), then y corresponds to the intent associated with OT. The same thing happens with x when AT is a closed set in q(x) = (x, AT). For evaluating x and y in every possible case we do the following:

if OT = {g}, then y = {g} ′ , i.e., all attributes that are associated with the object g.

-if OT = {g 1 } AND {g 2 }, then y = {g 1 , g 2 } ′ -if OT = {g 1 } OR {g 2 }, then y = {g 1 } ′ ∪ {g 2 } ′
Similarly, the evaluation of q(x) = (x, AT) is given as follows:

if AT = {m}, then x = {m} ′ , i.e., all objects that are associated with the attribute m.

-if AT = {m 1 } AND {m 2 }, then x = {m 1 , m 2 } ′ -if AT = {m 1 } OR {m 2 }, then x = {m 1 } ′ ∪ {m 2 } ′
Finally, the evaluation of q() = (OT, AT) is:

true if OT = AT ′ or AT = OT ′ and false otherwise. -For q 1 (x) = (x, {actor} AND {comedian} AND {writer}), we have x = {JerrySeinf eld}. -q 2 (x) = (x, {writer} OR {director}), we have x = {DavidSchwimmer, W illSmith, JerrySeinf eld}. -q 3 (y) = ({JustinT imberlake} AND {W illSmith}, y), we have y = {actor, musician}. -q 4 (y) = ({DavidSchwimmer} OR {JustinT imberlake}, y), we have y = {actor, director, musician}.

The complexity of answering LQL queries is polynomial in the size of the formal context, i.e., O|(G, M, I)|. The advantage of LQL over SPARQL is that, it allows to compute the least upper bound and greatest lower bound of query answers. We now present one important part of this work which is reducing SPARQL query answering over ELI ontologies into LQL query answering over K ELI concept lattices. query answering over concept lattices

Recently, SPARQL has been extended with different entailment regimes and regular path expressions 6 . The semantics of SPARQL relies on the definition of basic graph pattern matching that is built on top of simple entailment [START_REF] Glimm | Using SPARQL with RDFS and OWL entailment. Reasoning Web[END_REF]. However, it may be desirable to use SPARQL to query triples entailed from subclass, subproperty, range, domain, and other relations which can be represented using DL schema languages such as ELI. The SPARQL specification defines the results of queries based on simple entailment. The specification also presents a general parametrized definition of graph pattern matching that can be expanded to other entailments beyond simple entailment. Query answering under an entailment regime can be achieved via: (1) materialization (computing the deductive closure of the queried graph), (2) rewriting the queries using the schema, and

(3) hybrid (combining materialization and query rewriting) [START_REF] Glimm | Using SPARQL with RDFS and OWL entailment. Reasoning Web[END_REF]. SELECT ? x WHERE {? x a A r t i s t . }

Under simple entailment evaluation of a SPARQL query, the answers of Q over O is empty, i.e., Q(O) = ∅. For the reason that, simple entailment is based on graph matching which requires the variable ?x in the query to be bound with a term in the graph. Since there is no term where it can be bound to, the result is empty. However, under higher entailment regimes (such as the RDFS entailment [START_REF] Glimm | Using SPARQL with RDFS and OWL entailment. Reasoning Web[END_REF]) the result of Q is non-empty because inferred instances obtained through reasoning are taken into account for computing the answers. To get a non-empty answers for the above query, one can use one of the following approaches:

1. Materialization: involves all implicit data to be computed before the evaluation of the query. This can be done by using a DL reasoner. Consequently, in Example 1, the materialization of O is O ′ . Thus, the evaluation of

Q over O is Q ′ (O) = {tomCruise}. 2.
Query rewriting: is the task of converting a SPARQL query into one that involves schema axioms. It can be done using SPARQL property paths (a.k.a. regular path expressions). For instance, the above query can be rewritten as:

SELECT ? x WHERE {? x a /⊑ * A r t i s t . }

This query Q ′ selects all instances of Artist and that of its subclasses by navigating through the subclass relation (⊑ * ). The rewriting can be evaluated over O to obtain Q ′ (O) = {tomCruise}.

In summary, materialization requires a reasoner to expand the knowledge base, the complexity of this task depends on the type of the schema language. On the other hand, query rewriting requires modifying query patterns using SPARQL property paths. This also results in a further jump in the complexity of query answering.

As described above, unlike SPARQL query answering over ontologies, query answering over a concept lattice is relatively easier. Due to the fact that once the concept lattice is obtained from the ontology, LQL can be used to query the lattice. Consequently, alleviating those expensive tasks. The above SPARQL query can be converted into an LQL query as: q(x) = (x, Artist). The evaluation of this query over a concept lattice obtained from O is as expected

Q ′ (O) = {tomCruise}.
The complexity of SPARQL query answering over ELI ontologies is larger than that of LQL query answering over K ELI concept lattices. Since, the expressive power of SPARQL is superior than that of LQL. For ELI ontologies a query language like LQL is sufficient to retrieve individuals (or objects) and classes (or attributes).

Related work

To date, several studies have been carried out to assess the relevance and benefits of FCA for DL [START_REF] Baader | Completing description logic knowledge bases using formal concept analysis[END_REF][START_REF] Baader | A finite basis for the set of el-implications holding in a finite model[END_REF][START_REF] Baader | Exploring finite models in the description logic EL gfp[END_REF][START_REF] Sertkaya | A survey on how description logic ontologies benefit from FCA[END_REF][START_REF] D'aquin | Extracting relevant questions to an RDF dataset using formal concept analysis[END_REF][START_REF] Kirchberg | Formal concept discovery in semantic web data[END_REF]. Notably, the work in [START_REF] Sertkaya | A survey on how description logic ontologies benefit from FCA[END_REF] presents a survey on the advantageous of FCA for DL ontologies. Accordingly, some of the benefits that FCA can bring to the DL world include: knowledge discovery, extended subsumption hierarchy (of conjunctions of concepts) [START_REF]The Description Logic Handbook: Theory, Implementation, and Applications[END_REF], subsumption hierarchy of least common subsumers, exploring finite models [START_REF] Baader | A finite basis for the set of el-implications holding in a finite model[END_REF][START_REF] Baader | Exploring finite models in the description logic EL gfp[END_REF], role assertion analysis, supporting bottom-up construction and completion of ontologies. Since the survey, other studies, [START_REF] D'aquin | Extracting relevant questions to an RDF dataset using formal concept analysis[END_REF] and [START_REF] Kirchberg | Formal concept discovery in semantic web data[END_REF], have carried out experiments to characterize and analyse SW data using FCA tools. The former provides an entry point to a linked data using questions in a way that can be navigated. It gives a translation of an RDF graph into a formal context where the subject of an RDF triple becomes the object, a composition of the predicate and object of the triple becomes an attribute. The latter obliges the user to specify objects and attributes of a context. With that, it creates SPARQL queries to extract content from linked data in order to populate the formal context. Despite the fact that all these works have employed FCA techniques, to the best of our knowledge, none of them provide a formal and precise translation of ontologies into a formal context as we did here.

Conclusion

In this work, firstly, we have proposed a formal transformation of ELI ontologies into formal contexts. This enables to benefit from some advantages that FCA may offer to the DL world. Then we have shown that SPARQL query answering over ELI ontologies can be considered as lattice query answering over K ELI concept lattices. This alleviates some reasoning and query rewriting tasks that are required for SPARQL query answering. Moreover, even if there already exist substantial work relating DL, semantic web and FCA, there remains a lot of research work to be carried out. Such a research work is concerned with the correspondence between concept lattices from FCA and DL-based class hierarchies, query answering and information retrieval, and scalability as well. In addition, as FCA could benefit from DLbased reasoning capabilities, semantic web and DL-driven applications can take advantage of FCA-based ontology design, data analysis and knowledge discovery capabilities of FCA.

In the future, we plan to extend and experiment with the proposed approach. We will investigate how well it scales, given the size of ontologies.

Example 1 .

 1 Consider the following ELI ontology O = T , A :T = {ActorsFromNewYork ⊑ Actor, FilmProducer ⊑ Artist, Actor ⊑ Artist, Artist ⊑ Person} A = {tomCruise I ∈ ActorsFromNewYork I }In order to compare graphical representations of DL ontologies and their corresponding concept lattices, we represent O and its respective materialization O ′ as graphs as shown below: , dotted edges denote inferred instance and class subsumption relations.

  Target lattice associated with the ontology in Example 1. (b) Lattice associated with the context in Example 2.

Fig. 1 :

 1 Fig. 1: tomCruise (tC) and (.) are objects and the rest are attributes.

Fig. 2 :

 2 Fig. 2: Concept lattice associated with the ontology in Example 1.

-

  transform a ELI knowledge base KB = T , A into a formal context K = (G, M, I), the schema axioms in the TBox become background implications L((G, M, I)) and the ABox assertions become objects, attributes and relations. To elaborate, in K, individuals in the ABox constitute objects G, class names in the ABox and TBox yield attributes in M , and ABox assertions create relations between objects and attributes I ⊆ G × M . Here, we consider acyclic TBoxes so as to avoid class names becoming objects in a context. Definition 2 (Transforming ELI Ontologies into Formal Contexts). We define the transformation of KB = T , A into a formal context (G, M, I) thanks to a transformation function σ as follows: An axiom C ⊑ D in T corresponds to an implication in L((G, M, I)), i.e., the set of implications based on (G, M, I): C ⊑ D -→ C → D ∈ L((G, M, I)). -Concept expressions C (class name), ∃R.C, and ∃R -.C, correspond respectively to attributes C, ∃R.C, and ∃R -.C in M . -An individual a in A corresponds to an object a in G. -When a is an instance of C resp. ∃R.C, ∃R -.C, then (a, C) ∈ I resp. (a, ∃R.C) ∈ I, (a, ∃R -.C) ∈ I. -When a is related to b through R, then (a, ∃R.⊤) ∈ I and (b, ∃R -.⊤) ∈ I. Example 2. The translation of the ontology in Example 1 into a formal context K and its background implications L are shown below:

Definition 3 (

 3 Object and Attribute Terms). Given a formal context K = (G, M, I), an object term (OT) and an attribute term (AT) are defined inductively as:OT = {g} | OT 1 AND OT 2 | OT 1 OR OT 2 , where g ∈ G AT = {m} | AT 1 AND AT 2 | AT 1 OR AT 2 , where m ∈ MThe expression OT 1 AND OT 2 denotes the greatest lower bound (GLB) in the concept lattice B(G, M, I). The expression OT 1 OR OT 2 denotes the least upper bound (LUB) in B(G, M, I). Dually, the expression AT 1 AND AT 2 denotes the GLB in the concept lattice B(G, M, I) (keeping the orientation of B(G, M, I) based on the extents). Finally, the expression AT 1 OR AT 2 denotes the LUB in B(G, M, I).

Example 3 .

 3 Let us consider querying the concept lattice shown in Figure3.

Fig. 3 :

 3 Fig. 3: A concept lattice representing artists professions.

Example 4 .

 4 Let us consider the evaluation of the SPARQL query Q on the ontology O and O ′ of Example 1. Q = select all those who are artists.

http://www.ihtsdo.org/snomed-ct/

http://ncit.nci.nih.gov/

http://www.w3.org/TR/owl2-primer/

http://www.w3.org/TR/sparql11-query/


Stéphane Graham-Lengrand

Psyche: a proof-search engine based on sequent calculus with an LCF-style architecture

Psyche is a modular proof-search engine designed for either interactive or automated theorem proving, and aiming at two things: a high level of confidence about the output of the theorem proving process and the ability to apply and combine a wide range of techniques. It addresses the first aim by adopting and extending the LCF architecture to guarantee, using private types, not only the correctness but also the completeness of proof search. It addresses the second by offering a much more appropriate API than just the primitives corresponding to the inference rules of the logic in natural deduction: it uses instead a focused sequent calculus for polarised classical logic. Finally, Psyche features the ability to call decision procedures such as those used in Sat-Modulo-Theories solvers. We therefore illustrate Psyche by using it for SMT-solving.

1 Psyche in brief Psyche [START_REF]Psyche: the Proof-Search factorY for Collaborative HEuristics[END_REF], the Proof-Search factorY for Collaborative HEurisics, is a modular platform for automated or interactive theorem proving, built on an architecture (similar to LCF) where a small kernel interacts with plugins and decision procedures:

-The kernel is based on a proof-search engine à la Prolog, offering an API to perform incremental and goal-directed constructions of proof-trees in (a standard but carefully chosen) Sequent Calculus. -Psyche can produce proof objects (and print them in L A T E X format).

-Plugins can be programmed to drive the kernel, using its API, through the search space towards an answer provable or not provable; soundness of the answer only relies on the kernel via the use of a private type for answers (similar to LCF's theorem type). -Plugins can be interactive.

-Psyche offers a memoisation feature to help program efficient plugins.

-The kernel is parameterised by a procedure deciding the consistency of collections of literals with respect to a background theory, just as in SAT-modulotheories (SMT) solvers. The current version 1.5 of Psyche features a kernel designed for propositional logic modulo theories (same logic as that of DPLL(T ) used in SMT-solving), and decision procedures for the empty theory and Linear Rational Arithmetic (LRA). It is a program of about 4200 lines of OCaml 4.00, using hash-consing in most data structures for efficiency.

Motivation

Psyche's architecture is designed for the ambition of allowing various theorem proving techniques (generic or problem-specific) to collaborate on a common platform, whilst giving high confidence in the answers produced.

Interfacing the numerous techniques and tools available for theorem proving is legitimately receiving a lot of attention: Automated Theorem Provers, SAT/SMT-solvers, Proof assistants, etc. While trust is already an issue even for a tool running on its own, it becomes even more of an issue when different tools interact. Proof-checking is one way of addressing this, being permissive in the algorithms used for theorem proving, as long as they output some proof objects that can be checked. Another way is the LCF-style [START_REF] Gordon | Edinburgh LCF: a mechanized logic of computation[END_REF], where only a small kernel of primitives needs to be trusted, and anything smarter (e.g. the interaction between sophisticated techniques) boils down to calls to the primitives.

In the context of proof-checking, a natural way to interact with different (already implemented) techniques, is the black box approach, where an external tool is called and its output is converted back into a proof that can be checked by the system [START_REF] Armand | A modular integration of SAT/SMT solvers to Coq through proof witnesses[END_REF][START_REF] Besson | Modular SMT proofs for fast reflexive checking inside Coq[END_REF]. It is somewhat more surprising that, despite the highly programmable possibilities of the LCF architecture, the most successful integration of automated reasoning techniques in an LCF-based proof assistant such as Isabelle [START_REF]The Isabelle theorem prover[END_REF] seems to also use variants of the black box approach (as very impressively demonstrated by Sledgehammer) [START_REF] Weber | SMT solvers: New oracles for the HOL theorem prover[END_REF][START_REF] Paulson | Three years of experience with Sledgehammer, a practical link between automatic and interactive theorem provers[END_REF].

Psyche aims at producing answers that are correct by construction, not having to rely on proof-checking; it therefore adopts the LCF philosophy (although it can produce proof objects), also because having a simple trusted kernel is a convenient starting point for different techniques to collaborate. But the goal here is to open the black boxes and program their algorithms directly with calls to the kernel's API, as plugins for Psyche.

Such a deeper level of integration opens up the perspective of interleaving the use of different techniques: An external tool requires an input problem that it can entirely treat; but implementing the steps of its algorithm as small progressions in the search-space covered by the main system, allows more possibilities, such as running the technique up-to-a-point, where a switch to another technique may be appropriate (e.g. depending on newly generated goals).

The challenge is for the kernel to offer an appropriate API of proof-search or proof-construction primitives, to allow the efficient implementation of theorem proving techniques as plugins. Most LCF-style systems offer primitives corresponding to the inference rules of Natural deduction, or a Hilbert-style system. This is a very fine-grained level, that leaves most (if not all) of the work to the plugin and makes its implementation cumbersome: these formalisms are more about proof-construction than proof-search.

Psyche makes the choice of a bigger grain, and leaves to the kernel some real proof-search computation, but where no decision needs to be made. For this we use a focused sequent calculus LK p (T ) [START_REF] Farooque | Sequent calculi with procedure calls[END_REF][START_REF] Farooque | A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus[END_REF], which extends the logic programming paradigm to polarised classical logic modulo a background theory T . Polarities and Focusing [START_REF] Girard | Linear logic[END_REF][START_REF] Andreoli | Logic programming with focusing proofs in linear logic[END_REF] are tools that can be used to describe effective proof-search strategies in Sequent Calculus, hugely narrowing the search-space offered by Gentzen's original rules. In our case, they also specify a sensible division of labour between Psyche's kernel and Psyche's plugins, redesigning the standard LCF-style API.

Overview and general architecture

The kernel contains the mechanisms for exploring the proof-search space in a sound and complete way, taking into account branching and backtracking. It has no strategy regarding the order in which branches are explored, and this lack of intelligence makes its code rather short. If it reaches a proof, then that proof is correct by construction, and if the entire search space is explored and no proof is found, then the kernel correctly outputs that no proof exists.

The plugins then drive the kernel by specifying in which order the branches of the search space should be explored and to which depth, something that is expected to depend on the kind of problem that is being treated. The quality of the plugin is how fast it drives the kernel towards a answer Provable/NotProvable.

This already departs from the traditional LCF-style in that some actual proof-search computation is performed in the kernel, not just atomic steps of proof-construction:

prem 1 . . . prem n name conc
In traditional LCF, each inference rule of the logic (as on the right-hand side) will give rise to a primitive of the kernel's API: name: thm -> • • • -> thm -> thm In Psyche's kernel, such an inference rule will be wrapped in the kernel's unique API primitive: machine: statement -> output such that machine(conc) will trigger the recursive calls machine(prem_1),. . . , machine(prem_n).

Psyche's general architecture is illustrated by its main top-level call (slightly reworded for clarity):

Plugin.solve(Kernel.machine(Parser.parse input)) Psyche has a collection of parsers (currently one for DIMACS and one for SMT-Lib2) and calls the appropriate one on Psyche's input. The resulting abstract syntax tree is fed to the kernel's machine function that will initiate the search. This produces a value of type output that is given to the plugin to work with, so as to finally produce an answer provable or not provable. This could give the impression that the plugin performs computation after the kernel has finished its work, but this is not quite true, as illustrated by the nature of type output: type output = Final of answer | Fake of coin -> output which describes the kernel as a slot machine: when it is run, it outputs either a final answer provable or not provable or a fake output that represents unfinished computation: in order for computation to continue, the plugin needs to "insert another coin in the slot machine"; proof-search will then resume according to the inserted coin.

To summarise, the kernel performs proof-search as long as there is no decision to be made (on which backtrack may later be needed), and when it hits such a point, it stops and asks for another coin to indicate how to proceed next. The plugin drives the kernel in the exploration of the proof-search space by inserting carefully chosen coins, hoping that one day the machine will stop with the jackpot: a value of the form Final(...). Now while this architecture somewhat departs from LCF, it does share with it the distrust of anything outside the kernel: when concerned with the soundness of the answer (whichever it be), the plugin is here considered as an adversary, so Psyche defines the type answer as abstract, i.e. a private type that only the kernel can inhabit (just like the thm type of LCF). Psyche's type answer = Provable of statement*proof | NotProvable of statement can be read by the plugin and the top-level if need be, but cannot be inhabited by them. That way, a plugin cannot cheat about Psyche's answer: the worst it can do is to crash Psyche's runs. In Psyche as in traditional LCF, inhabitation of the abstract type (in case of Psyche, with a value of the form Provable(...)) explicitly or implicitly constructs a proof of the statement. But contrary to LCF, Psyche also gives guarantees when the output is not provable: it can only occur when the kernel has entirely explored the search-space unsuccessfully.

Psyche's Kernel

As described above, the kernel's API has the slot machine as its only primitive, controlled by the coins that are inserted in it. In order for efficient plugins to be conveniently programmed, the kernel's primitive needs to accept a rather expressive range of coins that can specify a smart exploration of the searchspace. This depends on the inference system that is used in the kernel for the incremental and bottom-up construction of proof-trees, and on identifying the inference rules that the kernel will perform automatically from those that will pause computation and prompt the plugin for new directions.

For this, our sequent calculus LK p (T ) [START_REF] Farooque | Sequent calculi with procedure calls[END_REF][START_REF] Farooque | A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus[END_REF] describing classical logic modulo a theory T uses polarities and focusing (see e.g. [START_REF] Andreoli | Logic programming with focusing proofs in linear logic[END_REF]). The connectives and literals of classical logic are tagged with polarities: + and -. Polarities do not affect the (classical) provability of formulae, but only the shape of proofs and therefore the structure of the proof-search space. Focusing is the phenomenon whereby the inference rules decomposing the connectives of the same polarity can be chained without losing completeness (thus narrowing the search-space), see [START_REF] Farooque | A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus[END_REF] for a full description. But in brief, focusing organises the proof-search process as an alternance between two kinds of phases: synchronous and asynchronous.

-An asynchronous phase decomposes the formulae of the sequent whose main connective is negative, using invertible inference rules (the premises are provable if the conclusion is): these represent no backtrack point in proof-search. -A synchronous phase starts with the selection of a positive formula: the formula and its sub-formulae are then decomposed recursively (before doing anything else in the sequent) as long as these remain positive. When these become negative, another asynchronous phase starts.

We use focusing to divide the labour between Psyche's kernel and plugins: The kernel applies the asynchronous steps automatically without any instruction from the plugin, and then stops and asks for another coin describing the next synchronous phase, where smart choices may have to be made (starting with the selection of the positive formula to work on). An important consequence of this division of labour is that every kernel call terminates, because the length of each phase is bounded by the size of the formula(e) being decomposed.

The choice of polarities on connectives and literals affects the kernel-plugin interaction. For instance the polarity of ∨ will determine whether it is decomposed

Γ ⊢ A i Γ ⊢ A 1 ∨ + A 2 Γ ⊢ A 1 , A 2 , Γ ′ Γ ⊢ A 1 ∨ -A 2 , Γ ′
automatically by the kernel (second rule, asynchronous) or with a smart choice by the plugin (first rule, synchronous):

The code of the kernel is rather small (575 lines) and purely functional. Continuation-Passing-Style (CPS) is used to minimise the use of the stack and provide a natural way to represent the progression of the kernel within the search space: the API function machine: statement->output actually wraps the real (tail-)recursive function search: statement->(output->'a)->'a with the identity continuation. Continuations are heavily used for branching and backtracking (e.g. when search applies a rule with several premises, it makes a recursive call on one of the branches and stacks up the others in the passed continuation; similarly when the plugin chooses to explore one branch, the kernel records the other ones -forcing in the end the entire exploration of the searchspace), and naturally implement a slot machine waiting for its coin.

Plugins

A plugin is an OCaml module of a fixed module type declaring a function solve: output->answer

(again, answer is for the plugin an abstract type). However, it is likey that the sophisticated strategies/heuristics that the plugin is meant to implement rely on some clever choice of data-structures for formulae, sets of formulae, sets of literals. So the plugin and the kernel have to agree on those three data-structures that are communicated both ways during the interaction. In Psyche 1.5, the kernel is parameterised by the data-structures, and the plugin provides them.

We first tested Psyche's architecture with a basic plugin Naive, which implements collections as lists and inserts the first available coin in the slot machine, whenever asked. This works fine for small tautologies, printable on a screen.

But the first real aim was to capture in Psyche some propositional SAT and SAT-Modulo-Theories solving techniques, making DPLL(T ) technology available in a generic proof-search framework like Psyche. For this we describe in [START_REF] Farooque | A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus[END_REF] how to see DPLL(T ), canonically expressed as a transition system [START_REF] Nieuwenhuis | Solving SAT and SAT Modulo Theories: From an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T)[END_REF], as a simple bottom-up proof-construction mechanism in LK p (T ). More practically, every rule of DPLL(T ) can be seen as the insertion of a particular coin in Psyche's slot machine. We implemented this as two different plugins for Psyche: DPLL_Pat and DPLL_WL. These remain toy plugins, because, although it is now clear from [START_REF] Farooque | A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus[END_REF] how to perform each rule of DPLL(T ) in Psyche, we still have to decide which rule to apply. This is where the two plugins differ: DPLL_Pat looks up the applicability of DPLL(T )'s rules by using Patricia tries to implement sets of clauses, while DPLL_WL looks it up using the technique of watched literals.

Just like DPLL(T )-based solvers are made efficient by using features such as backjumping and lemma learning, our plugins are made more efficient by the use of memoisation, which avoids re-doing, for some open branch, the same steps as those used in a previously completed branch. Psyche 1.5 therefore offers a memoisation module, to be used by plugins to record values of (the abstract) type answer. And the kernel's slot machine accepts from the plugin, as a special coin carrying such a value, "here is an already found answer that also applies to the current goal". The kernel accepts the value as closing the current branch (one way or another) without any proof-checking (since the abstract type ensures the value came as an earlier output of the kernel); it only checks that the value applies to the current goal. Now for a memoised answer Provable to be reusable as often as possible, it is useful to prune the provable sequent, just before it is tabled, from the formulae and literals that were not used in its proof. This is easy to do for the complete proofs of LK p (T ) (eager weakening are applied a posteriori by inspection of the inductive structure). Psyche's kernel actually performs the pruning on-the-fly whenever an inference is added to complete proofs, so that, whenever it outputs Final(sequent,proof), the sequent is already pruned. This is Psyche's way of performing conflict analysis, a key process of SMT-solving.

Conclusion and perspectives

Psyche is run from the command-line, taking as input the indicated file(s) or directory(ies) (or the standard input): psyche [OPTION]... [FILE/DIR]... Version 1.5 is distributed with a DIMACS parser, which we used to test Psyche on (propositional) SAT benchmarks, and an SMTLib2 parser (unmodified from the Alt-Ergo prover), which we used to test it on QF_LRA benchmarks (making use of the distributed simplex algorithm for LRA). The results are available on Psyche's website [START_REF]Psyche: the Proof-Search factorY for Collaborative HEuristics[END_REF]. Since Psyche has no ambition to beat state-of-the-art SAT-and SMT-solvers, it works well on small instances but its performance starts declining between 20Kb and 100Kb of input problem size (of course this is no appropriate measure of difficulty): There is no intrinsic problem of scalability, but the current plugins and decision procedures are illustrative toys (the heuristics for applying DPLL(T ) rules in the current plugins are still basic, and so is the decision procedure for LRA -e.g. it is not incremental). What we offer here is a platform and its modularity: anyone with better (or different) heuristics or decision procedures can simply write them as OCaml modules of our predefined module types, and Psyche will seamlessly run with them, keeping the same LCF-style guarantees. Moreover, nothing in Psyche's proof-engine relies on the input being sets of clauses, so Psyche might offer a convenient framework to generalise the known techniques for the satisfiability of formulae in clausal form.

The short-to-medium term plans are as follows:

-kernel: handle existential variables and propagate first-order unifiers through branching, construct and store proof-terms rather than whole proofs; -theories: improve the procedure for LRA (e.g. making it incremental) and implement other theories (Congruence Closure, LIA, bit vectors, etc); -plugins: implement a user-interactive plugin asking which coins to insert, improve the DPLL(T ) plugins to better handle non-clausal formulae, and implement other theorem proving techniques as plugins: analytic tableaux, clausal tableaux (e.g. connections) and resolution are all done in the theory.

In the long-term, we plan to investigate whether LK p (T ) may help mixing first-order reasoning with theories (i.e. investigate instantiations in presence of a theory), and Psyche's correctness in a proof assistant (since the functional kernel seems small enough and the plugins need not be certified).

Acknowledgements. Contributors of Psyche include S. Graham-Lengrand, A. Mahboubi, A. Bernadet, M. Farooque, Damien Rouhling and M. Vegreville.