Alain Couret
email: alain.couret@cemes.fr

Melissa Allen

Marcus Willi Rackel

Benjamin Galy

Jean- Philippe Monchoux

Volker Güther

Florian Pyczak

Pierre Sallot

Marc Thomas

Chemical heterogeneities in tungsten containing TiAl alloys processed by powder metallurgy

Keywords: Titanium aluminides, Powder atomisation, Spark Plasma Sintering, Tungsten segregation, Microstructure

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

During the last two decades, a number of investigations have demonstrated the interest of innovative powder metallurgy (PM) routes, such as Spark Plasma Sintering (SPS) or Additive Layer Manufacturing (ALM), to produce high quality TiAl alloys and/or sound components. Indeed, PM processing is generally known to result in grain size refinement as compared to cast processing, thus yielding significant strengthening for the considered alloys. However, at service temperatures, PM products can suffer from high creep rates in -TiAl alloys. In order to achieve a proper balance in mechanical properties, extensive work has been focused on alloy development, in particular with the addition of refractory elements such as niobium, molybdenum, tungsten, rhenium and tantalum.

Thence, the incorporation of these refractory elements, which promotes solidification by the  route, has emerged as an indispensable way of improving TiAl alloys to reach service temperatures superior to 700°C [START_REF] Appel | Gamma Titanium Aluminides : Science and Technology[END_REF]. In particular, the as-SPS IRIS alloy (Ti49.9Al48.0W2.0B0.1, composition in at.%) has been found to possess promising mechanical properties at high temperatures combined with a reasonable ductility at room temperature [START_REF] Couret | Procédé de fabrication d'une pièce en alliage TiA[END_REF][START_REF] Voisin | Mechanical properties of the IRIS TiAl alloy[END_REF]. The W addition was found to enhance mechanical properties through solid solution strengthening. Tungsten is also beneficial in terms of oxidation resistance [START_REF] Bacos | Oxidation Behavior of a Spark Plasma Sintered Ti-48Al-2W-0.1B Alloy at 800[END_REF]. Moreover, its low diffusion rate within the microstructure can guarantee an increased thermal stability at service temperatures for this family of W-bearing TiAl alloys [START_REF] Couret | Effect of long-term ageing on the properties of the W-containing IRIS-TiAl alloy[END_REF]. The originality of this alloy lies in the addition of only 2 at.% of W due to its very low diffusivity, which makes it possible to maintain a high Al content (47-48 at. %). Thus, the presence of 0 phase in the final microstructure, which is unfavourable at high temperatures and brittle at room temperature, is avoided while the presence of  phase is promoted. The challenge is to have a homogeneous distribution of W atoms in the  phase to ensure a homogeneous structural hardening of that phase and the alloy via the interaction of W atoms with dislocations moving by various deformation mechanisms [START_REF] Couret | On the high creep strength of the W containing IRIS-TiAl alloy at 850[END_REF].

One of the outstanding features of the SPS process lies in the homogeneous and texture-free microstructures, after densification. Moreover, the process is fast enough to prevent grain growth or long range migration of atomic species. In this alloy development work, pre-alloyed powders, i.e. with the expected chemical composition of the alloy, have been used.

A first batch of powder was supplied by CRUCIBLE/ATI companies (Pittsburgh, PA, USA) [START_REF] Yolton | Powder metallurgy processing of gamma titanium aluminide[END_REF] using the TGA technique (Titanium Gas Atomisation). More generally, TGA, as well as PIGA (Plasma Melting Induction Guiding Gas Atomization - [START_REF] Gerling | Power metallurgical processing of intermetallic gamma titanium aluminides[END_REF]) and VIGA (Vacuum Inert Gas Atomisation - [START_REF] Dunkley | Metalpowder atomisation methods for modern manufacturing[END_REF]), belong to the group of cold mould techniques [START_REF] Appel | Gamma Titanium Aluminides : Science and Technology[END_REF]. In this case, the alloy is fully melted, guided into the centre of a nozzle and then atomised with a stream of high pressure inert gas. The powder particles are very rapidly solidified from the homogeneous melt bath. Fig. 1a shows the microstructure at a low magnification of an IRIS alloy obtained after densification of this CRUCIBLE/ATI TGA powder, using a conventional SPS cycle with a dwell temperature of 1350°C. Some dark circular areas coming from powder particles that are depleted in heavy elements, can be observed in Fig. 1a.

The main alternative route to produce pre-alloyed TiAl powders is by using the crucible-free technique called EIGA (Electrode Induction melting Gas Atomisation - [START_REF] Gerling | Power metallurgical processing of intermetallic gamma titanium aluminides[END_REF]). In this case, the starting material is a pre-alloyed electrode, the end of which is machined into a cone shape. A conical induction coil, having the same shape as the tip of the electrode, is used to melt down the electrode, which is rotated to homogenise the temperature field at its tip and is slowly lowered into the induction coil as the fusion starts. The distance between the induction coil and the melting electrode is controlled to be constant. Drops of molten metal are falling through the gas-atomisation nozzle and are subsequently atomised. The absence of any contamination by a crucible is one of the main reasons, together with a reduced electric power consumption, large batches of powder capabilities and lower maintenance costs of the facility, why this process has attracted many industrial developments recently. Fig. 1b presents an IRIS TiAl alloy obtained after densification of an EIGA powder, using the same SPS parameters as for Fig. 1a. Some bright features can be seen, which reveal some microstructural and chemical heterogeneities.

The increasing propensity for structural heterogeneities in the EIGA-SPS-IRIS alloy compared to the TGA-SPS-IRIS one was unexpected since previous results with other TiAl alloys did show a rather robust EIGA procedure. For example, the EIGA-SPS route with the so-called GE composition (Ti-48Al-2Cr-2Nb) exhibited good homogeneity. In this context, the present authors have gathered their skills and means for further understanding and handling the problem of such structural heterogeneities in order to improve the EIGA production route applied to the IRIS alloy and, in a more general point of view, to TiAl alloys containing refractory elements.

Materials and methods

The IRIS alloys studied in this work were produced by a process in three successive steps: centrifugal casting of electrodes, powder atomisation and SPS densification.

Electrodes were produced at GfE-GmbH in three steps. More details about the GfE means and processes can be found in the paper of Güther et al. [START_REF] Güther | Metallurgical processing of titanium aluminides on industrial scale[END_REF]. First, electrodes were uniaxially compacted using Ti sponges, Aluminium, WAl and AlB master alloys. Second, small sized ingots (7kg) were processed by single VAR (Vacuum Arc Remelting). Third, VAR ingots were remelted for homogenisation in an induction skull melter and then centrifugally casted in cylindrical steel moulds with a tip at the bottom end. With this process, several electrodes with a diameter of 50 mm, typically from seven to nine, can be casted in a single pour. The critical issues are the control of the alloy chemical composition including the impurity level, as well as the homogeneity of the ingot. Two sets of electrodes, called GFE1 and GFE2 in the following, were processed. Their average compositions are given in Table 1. The aluminium contents are higher than scheduled due to a lower evaporation of this element than expected during the process. Aside from a 0.57at% difference in Al content, GFE2 mainly exhibits an extra 0.7at%Nb and a higher oxygen content with respect to GFE1. This excess in the oxygen content in GFE2 electrode is mainly due to the use of different master alloy batches.

Tab. 1. Chemical compositions of the electrodes GFE1 and GFE2. Electrodes of the two GFE1 and GFE2 compositions were atomized respectively at TLS Technik company (Bitterfeld -Wolfen, Germany) and Helmholtz-Zentrum Geesthacht (HZG, Geesthacht, Germany), by the EIGA process. At HZG,an electrode was atomised in six successive runs to get approximately powder batches representative of the material present around the locations L1 to L6 (Fig. 2). Two atomisation rates were alternatively used depending on the location: 16 kg/hour for locations L1, L3 and L5 and 45 kg/hour for L2, L4 and L6. Corresponding powder batches are denominated BS1, BR2, BS3, BR4, BS5 to BR6 respectively in the following (letters S and R stand for slow and rapid, respectively). On the one hand, the SPS material obtained by the atomisation at TLS of electrodes GFE1 will be used for mechanical tests (section 5). On the other hand, electrodes, powders and SPS billets from GFE2 will be used for microstructural and chemical analyses, to take advantage of the controlled atomization at HZG.

The SPS densifications of the EIGA powders were performed on the Dr Sinter Sumitomo 2080 apparatus (Sumitomo Coal Ming Co. Japan) implemented in the "Plateforme Nationale de

Frittage Flash /CNRS" in Toulouse, France. The usual SPS cycle presented in details elsewhere [START_REF] Voisin | Temperature control during Spark Plasma Sintering and application to up-scaling and complex shaping[END_REF] was used. The thermal cycle resulting from the application of a pulsed direct current consists of a temperature ramp at a rate of 100 K/min followed by a temperature plateau of of the temperature ramp. At the end of the plateau, both current and load are switched off.

Consistently with the high aluminium contents of the materials, the dwell temperatures used were relatively high, 1415°C and 1375°C for the GFE1 and GFE2 electrodes, respectively (these temperatures are the true temperatures of the material, i.e. the measured temperatures augmented by 60 K [START_REF] Voisin | Temperature control during Spark Plasma Sintering and application to up-scaling and complex shaping[END_REF]).

The microstructures are mainly analysed by scanning electron microscopy (SEM) in back scattered electron imaging (BSE) mode, so that the origin of the contrast is related to the local chemical composition of the material: the larger the atomic number, the brighter the image. The measurements of the chemical compositions were performed at various scale. At a macroscopic scale (~ 1cm 3 of material), ICP (Inductively Coupled Plasma) and IGA (Instrumental Gas Analysis) were used for the quantification of major and minor elements, respectively. Local chemical analyses of various elements were performed by SEM-EDX on the electrodes, powder particles and SPS samples. Each reported data represents an average of at least 10 local measurements. The experimental uncertainty on these EDX measurements is considered to be of the order of a few tenth of at. %. The SEM-EDX measurements were performed on areas of a few hundred square micrometers or locally using a SEM equipped with a field emission gun (FEG). In the following, our purpose is to focus on the three major alloying elements: Ti, Al and W, since the distribution of the other elements remain relatively stable at all stages of the production route.

Tensile tests were performed at a constant strain rate of 10 -4 s -1 at room temperature (CEMES) and 800°C (ONERA).

Materials after the three successive steps of the processing

Electrodes

Fig. 2a shows the macrostructure of one electrode which is composed of columnar grains having a radial growth from the surface in contact with the cold mold, and an equiaxed structure in the centre area, the size of which tends to increase with the distance from the tip of the electrode. Such a microstructure is typical for cast TiAl alloys, both for peritectic [START_REF] Güther | Metallurgical processing of titanium aluminides on industrial scale[END_REF] or solidified [START_REF] Grange | Influence of microstructure on tensile and creep properties of a new castable TiAl-based alloy[END_REF] alloys.

The chemical composition has been measured by ICP and IGA at six different locations along the length of the electrode (marked L1 to L6 in Fig. 2 a). It is found to be quite homogeneous all along the electrode length (Fig. 3).

Tab. 2. Chemical compositions at the four locations along the radius shown in

Powder particles

Fig. 4 shows the powder particles for batches BR2 and BS5 obtained with the GFE2 electrode.

Generally, the particles are spherical with a small number of satellites (Fig. 4 a and b). Their microstructure is made of a dendritic microstructure (Fig. 4 d). Such a dendritic microstructure is commonly observed in gas atomised TiAl powders and contains a large fraction of outequilibrium /2 phase [START_REF] Gerling | Power metallurgical processing of intermetallic gamma titanium aluminides[END_REF][START_REF] Habel | Gas atomized gamma-titanium aluminide based alloys -Processing, microstructure and mechanical properties[END_REF][START_REF] Jabbar | Microstructures and deformation mechanisms of a G4 TiAl alloy produced by spark plasma sintering[END_REF]. The microstructure exhibits some bright features, the size of which being about 20 µm (Fig. 4 c). These bright features show well-defined contours, often bordered by a grey band (Fig. 4 c). Such bright features are not present in every particles but if present, several of them could generally be observed in a single particle.

These micrographs seem to show that the number of bright features is higher in the powder atomised at a high feeding rate. To quantify this point, an image analysis post-processing method has been applied using the ImageJ software, as illustrated in With regard to the chemical composition of the SPS material, for the GFE2 alloy, Fig. 3 compares the composition of this material with that of the six samples taken from the electrode.

Little change is observed, except for an increase in oxygen content to 1200 ppm.

Tab. Indeed, these calculations predict enrichment of W in a bcc phase which is below 8% phase fraction even for a W content of 8 at.%. The other phases found are γ, α2 and β/β0 as established for TiAl alloys. In the areas away from the dendritic cores and in the interdendritic channels, the W content is in the order of 1.2 at. % and 0.8 at. %, respectively (data in black -right part of Fig. 6 a).

Microstructures and local chemical analyses

In the powder particles (Fig. 6 b), away from the W concentration a non equilibrium dendritic microstructure with a large amount of /2 phase is observed as usual in EIGA TiAl powders [START_REF] Jabbar | Identification of microstructural mechanisms during densification of a TiAl alloy by spark plasma sintering[END_REF]. Concerning the bright features, as indicated by the EBSD insert in Fig. 6 b, the brighter zones (yellow in the EBSD map) are of /0 phase and contain an amount of about 3 at. % of W (data in red in Fig. 6 b). The less bright peripheral zones are of 2 phase (purple in the EBSD map).

In the SPS material (Fig. 6 c), the bright features appear to be located in any part of the microstructure, independently from lamellar colonies or  borders which are surrounding these colonies and from the locally present phases. Now focussing on the W content, using a magnification roughly at the particle size scale for both SPS sample and powder particles (Fig. 6 b and c -data in green and orange), the W contents is in the order of 2 at. %. There slight increase of W (2.2 at. %-green zone -Fig. 6 c) in the SPS sample in the surrounding of the bright features is within our experimental uncertainty.

Besides, for magnifications at the scale of the bright features (~20 µm), some fairly clear correspondences can be pointed out. The bright features have about 3 at. % of W in both the particles and the SPS sample (data in red -Fig. 6 b andc). Moreover, when considering as a dendritic core 20 µm in size around the skeletons, the same W content of about 3 at. % is found (data in red -Fig. 6 a).

The zones between these bright features in the dendritic structure of particles and in the lamellar colonies of the SPS sample contain slightly less than 2.0 at. % of W in both cases (data in blue -Fig. 6 b andc). This indicates that a similar W content is obtained, everywhere microstructural heterogeneities are not present.

Mechanical properties

The effect of these chemical heterogeneities on the mechanical properties were evaluated by performing tensile tests using the SPS alloy processed with the powder atomised from the GFE1 electrode. Its microstructure (Fig. 7) is similar to those already presented with batches of powders atomised at low feeding rate (Fig. 5 c).

In Figure 8 a and b, tensile curves of this GFE1-IRIS alloy at room temperature and at 800°C are compared with those obtained with the heterogeneity-free TGA-SPS-IRIS alloy [START_REF] Voisin | Mechanical properties of the IRIS TiAl alloy[END_REF]. In In summary, these results underline a relatively small influence of the heterogeneities on the mechanical strength of the TiAl IRIS-SPS alloy, in particular for the dimensioning properties for applications, such as high temperature strength and room-temperature ductility.

Discussion

In this work, the microstructures and the local chemical compositions of the IRIS alloy obtained after the three successive steps (centrifugal casting, atomisation and SPS densification) have been studied. Local enrichments of W due to segregation have been found after each steps. In the atomised bar, the melting appears insufficient to homogenise the alloyed concentration of W. The atomisation step constraints these heterogeneities in some particles. The bright features observed in the powder after atomisation and in the SPS material after densification exhibit several similarities: (i) their average W content is close to 3 at. %; (ii) their characteristic dimension is about 20 µm, and (iii) in general, several of them are close to each other. Such similarities can be extended to the dendritic cores of the electrode considering they have similar size. These observations thus indicate that the W enrichment of the SPS samples originate from the W enrichment from the powder particles, themselves derived from the segregations observed in the dendritic cores of the electrode.

An attempt to explain the mechanism of the formation of those W enrichments is given below.

Starting from the electrode, as described in the W and Re containing G4 alloy [START_REF] Grange | Influence of microstructure on tensile and creep properties of a new castable TiAl-based alloy[END_REF] which similarly to IRIS solidifies through the  phase field,  skeletons result from the retention of primary  phase grown from liquid during the first solidification. Note that in these skeletons, the aluminium content is very low, which confirms that they are issued from the first solidified layers according to the description of Grange et al. [START_REF] Grange | Influence of microstructure on tensile and creep properties of a new castable TiAl-based alloy[END_REF]. They are thus probably not avoidable during solidification, even at very low solidification rate. In the electrode, these skeletons form interlinked networks, which explains the formation of several connected bright features in the particles.

During the EIGA atomisation, these first solidified parts of the electrode probably fall in the nozzle before a complete melting, thus leading to numerous and connected bright features in the particles. This results in homogeneous bright features in the particles with pronounced boundaries. In these zones corresponding to the dendritic cores, during the EIGA process the expansion of the  phase occurs, accompanied by a local homogenisation of the W content.

Their periphery is constituted of  phase consistently with a lower W content.

The increasing number of W enrichments with increasing feeding rate during atomisation, which reduces the time available to fully melt potential W-rich segregation in molten TiAl seems to support this interpretation. A FEM simulation of the electrode during melting was realised using Freefem++ software. The model was simplified to an axisymmetric problem and only thermal equilibrium was calculated. Boundary conditions were as follows:

 A Neumann condition on all the free surface modelling the heat exchange with the Arshielding atmosphere in the atomisation tower. The external temperature was set to 250°C and a heat transfer coefficient of 200 W/(m 2 *K) was used;

 A Dirichlet condition was applied on the conical-melted surface of the electrode, representing the coupling with the induction coil. A constant temperature of 1800°C was applied.

The speed was varied from 9.8*10 -4 m/s (16 kg/h) to 2.8*10 -3 m/s (45 kg/h), which corresponds to the melting of the electrode 20 mm in diameter. Figures 9 a and b represent the temperature field obtained in both cases. It appears that the higher the feeding rate, the shorter the melted area. Using the time equivalent, the temperature profile for a solid fraction at the centre of the electrode with time is given in figure 9 c.

For simplification, if full melting of the material is reached for temperature higher than 1500°C, the residence time of potential W-skeleton in molten TiAl is 3 s and 11 s for 45 and 16 kg/hour feeding rates, respectively. Clearly, the present results indicate that the residence time should be higher than 11 s to completely dissolve the W-skeleton and thus improve the homogeneity of the powder. This is consistent with results obtained on TGA, and VIGA powders where the material is fully molten and homogenised before atomisation, eliminating the first solidified areas and allowing more homogeneous powders.

During the SPS cycle, peripheral diffusion occurs resulting in diffuse contours for the bright areas in the SPS samples, but with an unchanged W content due to its low mobility in solid state.

Regarding areas without any W segregations, i.e. where the alloy was fully molten during the EIGA process, the dendritic structure of the particles with a high proportion of out-ofequilibrium hexagonal phase leads to the formation of a quasi-lamellar structure in the final material, which is composed of lamellar colonies and  borders [START_REF] Voisin | Obtaining of a fine nearlamellar microstructure in TiAl alloys by Spark Plasma Sintering[END_REF]. In these zones free of heterogeneities, measurements of aluminium and titanium contents lead to fluctuating values, the significance of which is still under debate. Among the three stages of the processing route, it is difficult to explain such variations which may be within our measurement uncertainty.

However, these variations may also result from local heterogeneities in the initial electrode depending on the position of the atomised material and/or relative to the dendritic microstructure, since little species migration is expected during the atomisation and the densification cycles.

Summary

In this work, the present authors have gathered to understand why an IRIS alloy densified by SPS with a powder atomised by the EIGA process exhibits chemical heterogeneities in the form of tungsten segregations. By tracing back these heterogeneities throughout the manufacturing process (centrifugal casting, atomisation and SPS densification), it was shown that these heterogeneities originated from the retention of the primary  phase in the dendritic cores present in the initial electrode. All along the process, the diffusion of tungsten atoms is too limited to achieve homogenisation of the final material.

Despite a dramatic appearance in scanning electron taken with backscattered electrons, the local W enrichment is only 1 at. %. in these heterogeneous zones, whereas in the rest of the SPS material, the W content is maintained at a little less than 2 at. %. This is the reason why mechanical properties at room and high temperatures have been found to be only slightly affected by the presence of these heterogeneities.

Figure captions

  Figs. 2 b to e show the microstructure variations along a radius, from the centre to surface. Along the radius, a dendritic microstructure is observed whatever the equiaxed or columnar microstructure. From the centre to the surface, a decrease in the dendrite size is noticed. The chemical composition at these four locations has been measured by EDX-SEM, integrating the whole area shown in the micrographs. The results (Tab. 2) indicate that tungsten is homogeneously distributed in the radial direction as well as along the longitudinal axis. However, a slight but significant depletion in titanium is measured at the edge of the electrode.

Fig. 4 eTab. 3 .Fig. 5 a

 435 Fig. 5 a shows the microstructure of the sample obtained with the powder batch BL3 (low

Fig. 6

 6 Fig. 6 displays the microstructures of the electrodes, powder particles and SPS materials at a

Fig. 8 a

 8 Fig. 8 a, the colour curves (red, blue and green) show tests which were carried out on specimens

Fig. 1 .

 1 Fig. 1. Microstructures of the TGA-SPS-IRIS (a) and GFE2-SPS-IRIS (b) alloys densified by

Fig. 2 .Fig. 3 .

 23 Fig. 2. Macrostructure (a) of the longitudinally cut symmetric area of the electrode and

Fig. 4 .

 4 Fig. 4. SEM and quantitative study of the powder particles. (a) and (b) micrographs of particles

Fig. 5 .

 5 Fig. 5. SPS material. (a) microstructure at low magnification (batch BS3); (b) binarized image

Fig. 6 .

 6 Fig. 6. Local chemical analyses by SEM-EDX of various elements of the microstructure of

Fig. 7 .Fig. 8 .

 78 Fig. 7. Microstructures of the GFE1-EIGA-SPS-IRIS alloy at a dwell temperature of 1365°C.

Fig. 9 .

 9 Fig. 9. FEM modelling of the electrode during the atomisation. (a) and (b) temperature

Fig. 1 .

 1 Fig. 1. Microstructures of the TGA-SPS-IRIS (a) and EIGA-SPS IRIS (b) alloys densified by SPS, using the same processing parameters. The micrographs were recorded in the same microscope using the same conditions.

Fig. 2 .

 2 Fig. 2. Macrostructure (a) of the longitudinally cut symmetric area of the ingot and microstructures (b to e) along a radius. L1 to L6 mark the positions at which the chemical measurements were performed.

Fig. 3 .

 3 Fig. 3. Measurements of the chemical position of the ingot at the six positions, L1 to L6.

Fig. 4 .

 4 Fig. 4. SEM and quantitative study of the powder particles. (a) and (b) micrographs of particles of batches BR2 and BS5 at a low magnification; (c) and (d) particles with and without bright zones at a higher magnification and (e) to (h) illustration of the quantitative measurements of the number of particles containing bright zones and of the surface fraction covered by the bright zones.

Fig. 5 .

 5 Fig. 5. SPS material. (a) microstructure at low magnification (batch BS3); (b) binarized image of (a) and (c) microstructure at high magnification (batch BS5).

Fig. 6 .

 6 Fig. 6. Local chemical analyses by SEM-EDX of various elements of the microstructure of SPS materials (a), powder particles (b) and ingot (c).

Fig. 7 .Fig. 8 .

 78 Fig. 7. Tensile curves at room temperature (a) and (b). Color curves correspond to the EIGA-SPS-IRIS-2 alloy and black ones to the TGA-SPS-IRIS alloy.

Fig. 9 .

 9 Fig. 9. FEM modelling of the electrode during the atomisation. (a) and (b) temperature distributions in the melted electrode at 16 kg/h and 45 kg/h, respectively; (c) evolution of the temperature with time for a solid particle at the centre of the electrode.

  Fig. 2.

		Ti	Al	W
	Location (b) -Centre	48.6	48.6	2.0
	Location (c) -Intermediate 1	48.6	48.7	1.9
	Location (d) -Intermediate 2	48.5	48.8	1.9
	Location (e) -Edge	47.6	49.6	1.9

  [START_REF] Bacos | Oxidation Behavior of a Spark Plasma Sintered Ti-48Al-2W-0.1B Alloy at 800[END_REF]. Surface fraction of bright areas in the SPS billets obtained by the densification of the six batches of powders.

	Batch	BS1	BR2	BS3	BR4	BS5	BR6
	Surface fraction	1.2	13.8	4.6	16.7	6.1	11.2
	Standard deviation	0.54	0.94	0.34	1.54	0.42	1.14

Acknowledgements

This study has been (partially) supported through the grant NanoX n° ANR-17-EURE-0009 in the framework of the « Programme des Investissements d'Avenir" through the project "ALTIAUTO".


Matthew J Kauffman
email: mkauffm1@uwyo.edu.

Francesca Cagnacci

Simon Chamaillé-Jammes

Mark Hebblewhite

J Grant

C Hopcraft

Jerod A Merkle

Thomas Mueller

Atle Mysterud

Wibke Peters

Christiane Roettger

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Luthando Dziba

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Claude Fischer

Stefano Focardi

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Chris Geremia

Benito A González

Anne Gunn

Elie Gurarie

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David Mallon

Pascal Marchand

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Ramiro J A Ovejero

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Antti Paasivaara

Craig Packer

Danila Panchenko

Luca Pedrotti

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Sonia Said

Albert Salemgareyev

Aleksandr Savchenko

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Erling Solberg

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Steffen Zuther

Mapping out a future for ungulate migrations

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Main Text:

Wildebeest, 1.3 million strong, plod steadily across the Serengeti plains, chasing the recent rains.

Hundreds of thousands of caribou crisscross the Arctic, their hooves pounding across the vast tundra. A mule deer doe pauses along her two-month march to nibble the green grass of spring before rejoining a well-worn trail through the sagebrush. The migrations of ungulates (hooved mammals) are some of the most impressive phenomena in the animal kingdom. And while millions of people have observed some of these journeys, such as the wildebeest migration, animal tracking studies are discovering new ungulate migrations that surprise scientists and the public alike.

Migration is a fundamental ecological process that promotes abundant herds [START_REF] Fryxell | Why are migratory ungulates so abundant?[END_REF], whose biomass cascades up and down terrestrial food webs. Migratory ungulates provide the prey base that maintains large carnivore and scavenger populations and underpins terrestrial biodiversity. When ungulates move in large aggregations, their hooves, feces and urine create conditions that facilitate unique biotic communities [START_REF] Geremia | Migrating bison engineer the green wave[END_REF]. The phenomenon of migration thus provides a wealth of ecosystem services, while also supporting indigenous people and local communities on many continents. More than just using them for food and clothing, humans have been culturally linked to and sustained by migratory ungulates for thousands of years.

Despite their importance, ungulate migrations are disappearing at an alarming rate (3). Without a strategic and collaborative effort, many of the world's great migrations will continue to be truncated, severed, or lost in the coming decades. Fortunately, an evidence-based alternative exists. A combination of indigenous knowledge, historical records, animal tracking datasets, and analytical tools can provide the information necessary to conserve ungulate migration. Through our collective efforts to study the world's ungulate migrations, we recognize an urgent need for a global inventory of known migrations, including those extant and recently lost.

New Technology, New Discoveries

Ungulates migrate in response to resources that wax and wane with the seasons. In the tropics, species like kob, eland, and elephant track rainfall across vast grasslands in search of food or water. In temperate systems, many ungulates move between mountains and plains to escape the deep snows of winter and feed on high-quality forage in summer. Across the Arctic, migratory caribou and reindeer move hundreds of kilometers from the boreal forests across the snowcovered tundra to aggregate on their northern calving grounds just as plants are greening up. The sheer numbers of caribou on the calving grounds dilutes the toll from their wolf, bear and wolverine predators [START_REF] Bergerud | Evolving perspectives on caribou population dynamics, have we got it right yet?[END_REF].

Powerful new methods have made it possible to map migrations with precision. Tracking devices that are small, affordable, and reliable are being deployed on many species, allowing movement data to be acquired hourly at ~10-meter resolution. Satellites can now provide remotely sensed data to characterize spring green-up, drought, or snow, at the same scale migrating animals make behavioral decisions. When animal tracks are overlaid on dynamic maps of the earth's seasonal resources they reveal diverse migratory behaviors, from long-distance movements across climatic gradients, to shorter elevational movements to access alpine habitats.

Recent discoveries span several continents. In 2014, a zebra migration was discovered that stretches 500 km across Namibia and Botswana, a new record for the species [START_REF] Naidoo | A newly discovered wildlife migration in Namibia and Botswana is the longest in Africa[END_REF]. In 2016, in the Greater Yellowstone Ecosystem, mule deer were found to travel as far as 400 km one way [START_REF] Sawyer | The extra mile: Ungulate migration distance alters the use of seasonal range and exposure to anthropogenic risk[END_REF].

On the Mongolian Steppe, gazelle were found in 2018 to explore an area roughly the size of Hungary (100,000 km 2 ) over their lifetime [START_REF] Nandintsetseg | Challenges in the conservation of wide-ranging nomadic species[END_REF]. Most recently, in Ethiopia's Gambella National Park, researchers discovered that white-eared kob migrate in an 860 km circuit connecting to the Boma-Bandingilo migration in South Sudan, extending the known migratory range [START_REF] Kassahun | Movement ecology and home ranges of White eared kob (Kobus kob leucotis) in the Boma-Gambella transboundary landscape between the Republic of South Sudan and Ethiopia[END_REF].

Altitudinal migrations are also being revealed across the world's mountain ranges. These short migrations allow animals to exploit highly variable habitats and derive functional benefits similar to long-distance migrations.

There is growing recognition that migratory behavior must be learned by mammals. In North America for example, migratory bighorn sheep and moose failed to migrate when first translocated into novel landscapes. Over multiple generations, however, individual populations gained knowledge to move and find forage, and they became more migratory [START_REF] Jesmer | Is ungulate migration culturally transmitted? Evidence for social learning from translocated animals[END_REF]. Sustained by spatial memory, migratory behavior may often exist as a type of animal culture that must be learned and transmitted between generations. The reliance on culture carries a stark warning for conservation, namely that the persistence of a given migration corridor depends on the survival of individuals that possess the knowledge to make it.

Migrations in Danger

The world's landscapes are changing rapidly, and unlike migratory birds, ungulates cannot fly over patches of unsuitable habitat. Instead, they must share the landscape with humans, making them especially vulnerable to land-use changes that impede their movements. Recent estimates project that we will build 25 million km of new roads across the globe by 2050 [START_REF] Laurance | A global strategy for road building[END_REF], with associated constraints on animal movement. The sheer scale of these developments casts a long shadow over the future of terrestrial migrations.

Impermeable barriers have long restricted the free movement of migratory herds. For example, nomadic movements of Mongolian gazelles are now constrained by railroads and border fences [START_REF] Nandintsetseg | Challenges in the conservation of wide-ranging nomadic species[END_REF]. In Botswana, veterinary fences built in the 1950s caused the death of hundreds of thousands of wildebeest. In the western U.S., highways have severed pronghorn and mule deer migrations [START_REF] Kauffman | Wild Migrations: Atlas of Wyoming's Ungulates[END_REF]. In Russia's Kola Peninsula, the construction of a railroad divided the wild reindeer population and eliminated the longest of the region's migrations. And in Europe, red deer must navigate a landscape shaped by centuries of human settlement, exploiting seasonal cycles of forage only in the most remote habitats.

Semipermeable barriers like unfenced roads, some fences, dwellings, and energy extraction are also increasing. Although animals may be able to pass through such impacted landscapes, their altered movements are often mismatched with patterns of rainfall or snowmelt. In the most extreme cases, migrations can be lost simply because the costs of navigating human-altered landscapes outweigh the benefits of migrating.

Climate change is an additional challenge. Many ungulates time their migrations to exploit patterns of plant green-up. Droughts are becoming commonplace, making it more difficult for animals to move in synchrony with green-up and access the best forage [START_REF] Aikens | Drought reshuffles plant phenology and reduces the foraging benefit of green-wave surfing for a migratory ungulate[END_REF]. Migratory ungulates are thus doubly challenged by a changing climate combined with barriers that prevent them from adjusting their movement tactics as conditions change. Protected areas rarely ease these challenges, because they remain too small to maintain the full sweep of resources needed to sustain most long-distance migrations.

Past landscape changes have taken their toll on ungulate migrations. Harris et al. (3) in 2009 described the loss of six out of 24 ungulate populations that make mass migrations. Historical accounts describe numerous migrations that no longer exist in the behavioral repertoire of current populations. The hundreds of thousands of Cape springbok that once traversed the Karoo landscape of South Africa were eradicated by fencing, disease, and hunting at the end of the 19 th century. In Kenya's Kajiado County, migrations of wildebeest, zebra and Thomson's gazelle have collapsed due to competition with livestock and massive land conversion. When we lost the tens of millions of bison that roamed North America, we lost the abundance necessary to sustain migration [START_REF] Geremia | Migrating bison engineer the green wave[END_REF]. Of the bison that now exist in small conservation herds, the Yellowstone population is one of a few that migrate.

When we lose the large herds of animals moving en masse across the landscape, we lose something else even more difficult to quantify. Such phenomena, the thundering of hooves kicking up dust as a herd moves across a vast landscape, remind us of the wildness that the Earth's habitats are still capable of supporting. Equally important are the cultural traditions and identity of many indigenous communities, such as the Inuit and Tlicho, whose livelihoods depend on migratory herds [START_REF] Cunsolo | You can never replace the caribou": Inuit experiences of ecological grief from caribou declines[END_REF].

The movements of ungulates across the globe are more diverse and behaviorally complex than previously recognized. To maintain this diversity of movement, conservation of corridors and landscape connectivity has taken on new urgency. Unfortunately, efforts by wildlife managers and conservationists are thwarted by a singular challenge: Most ungulate migrations have never been mapped in enough detail to guide effective conservation.

A Conservation Solution

New methods now allow corridors to be mapped in detail from tracking data [START_REF] Middleton | Conserving transboundary wildlife migrations: recent insights from the Greater Yellowstone Ecosystem[END_REF]. When maps are overlaid on real landscapes, barriers and other threats can be identified, pointing the way to effective solutions. Migration maps are increasingly being used to target fences for removal, site overpasses and underpasses, guide energy leasing, and prevent housing development from blocking corridors. For species with high fidelity to corridors, such as mule deer ( 6), conservation may be achieved by targeted planning within corridors. For species that make nomadic or variable migrations, such as red deer, Mongolian gazelle, khulan, and barren-ground caribou, broader-scale strategies are needed to map and conserve their annual migrations and seasonal ranges. Regardless, mapping efforts to date illustrate that empirical tracking data can effectively guide efforts to conserve migrations.

We also need to inventory and map migrations that have already been lost, which is possible through indigenous, local, and expert knowledge. Mapping lost migrations can provide an important benchmark from which we can push back shifting baselines. For example, after a veterinary fence in Botswana that had blocked movements for 36 years was removed, zebra were recorded in 2004 making a nearly 600-km round trip migration similar to the one described in reports of historical sightings [START_REF] Bartlam-Brooks | Will reconnecting ecosystems allow longdistance mammal migrations to resume? A case study of a zebra Equus burchelli migration in Botswana[END_REF]. In some South American rangelands, conservationists are hopeful that ranch abandonment will create opportunities to restore lost migrations of guanacos and huemul deer. We don't know which migrations can be restored, but mapping where they once existed is a necessary first step.

The Convention on Migratory Species (CMS) advanced conservation by formally recognizing animal migrations in 1979. Yet, tracking data have not existed in sufficient detail to map ungulate migrations until recently, which has constrained Parties to the Convention from developing concrete policies to protect migrations. We therefore propose a global atlas of ungulate migration, built from new and existing tracking studies. We envision a digital archive that translates tracking data into actionable migration maps that are standardized, in a central database, and publicly available. The atlas would be a resource for multiple stakeholders, such as indigenous peoples, local, regional, or national governments, and NGOs, to collaboratively manage shared landscapes to sustain known migrations while maintaining local livelihoods. Such an evidence-based effort is needed to assure that migrationsthe actual paths that herds have used for hundreds or thousands of yearsare incorporated into land management plans, environmental impact assessments, mitigation activities, and reserve designs. The atlas will also facilitate international coordination to protect transboundary movements and inform the development work of International Finance Institutions.

This will require dedicated collaboration on a global scale. Fortunately, tracking datasets are growing each year, and existing data can be analyzed now to begin an archive of detailed migration maps. Migrations that remain unmapped can be targeted for new field studies. To coordinate this effort, we have created the Global Initiative for Ungulate Migration (GIUM). The GIUM brings together scientists, conservationists and wildlife managers around the world to create a collaborative knowledge base of ungulate migrations, develop a global atlas, and spark new conservation and policy.

No doubt current and future generations will continue to be amazed by the annual treks of wildebeest across the Serengeti. We must now turn our research and conservation efforts to other species, such as the saiga and khulan in central Asia, the Arctic caribou and reindeer across North America and Eurasia, the moose in Scandinavia, and the guanaco and huemul in South America, to name just a few. Such an ambitious atlas brings hope that new migration maps can guide where we build roads, fences, and other infrastructure, and where we decide to forego such developments to safeguard the movement of animals. Through global collaboration, we can now assemble the knowledge, data, and tools necessary to maintain the wild and intact landscapes that migratory ungulates require and sustain the ecosystems supported by their abundance. productivity. This mass migration of herbivores supports a diverse predator and scavenger community that tracks their year-round movements.

Wildebeest migration

GPS locations for several wildebeest moving between the dry (orange) and wet season (green) illustrate the generalized migration route.

Lion movements

Despite being tied to their home ranges, lion prides follow the seasonal flux of ungulate prey (GPS locations illustrate the year-round track of one lion).

Vulture movements

Several species of vultures scavenge extensively on the carcasses of migratory wildebeest, synchronizing their movements to the dry season range from hundreds of kilometers away. The track of a lappet-faced vulture is shown, along with the more wide-ranging movements to the east of a white-backed vulture.

White-eared kob

In 2018, white-eared kob were discovered to make an 860-km migratory circuit between Ethiopia and South Sudan, extending the known migratory range [START_REF] Kassahun | Movement ecology and home ranges of White eared kob (Kobus kob leucotis) in the Boma-Gambella transboundary landscape between the Republic of South Sudan and Ethiopia[END_REF]. The herds traverse oil and gas conces-sions, hotspots of armed conflict, and commercial agricultural developments in the Boma-Gambella landscape.

Mongolian gazelle

On the Central Asian steppe, the nomadic movements of Mongolian gazelle are an order of magnitude larger than the region's protected areas.

Red deer, European Alps

In remote habitats, red deer are still able to fully exploit the abundance of seasonal forage by migrating across mountain ranges.

Red deer, semi-urban

Animals living near human population centers may try to maintain migration, but their movements are often constrained and truncated by human settlements.

African elephants

Migration tracking data is being used to evaluate proposed alternatives for an all-season road (yellow line) that would cut through the post-calving, summer and winter range of the Bathurst herd, which has declined 97.5% since 2000.

Fig. 1 .

 1 Fig. 1. Migration underpins terrestrial ecosystems. The annual migration of 1.3 million

Fig. 2 .

 2 Fig. 2. Ungulate migrations around the world. Animal tracking studies are being conducted

Fig. 3 .

 3 Fig. 3. A group of pronghorn pause during their spring migration to navigate a fence that bisects

Acknowledgments: We are grateful to Emilene Ostlind and Greg Nickerson for reading early drafts; Julia Olson for cartography; Ken Lay for early discussions about GIUM; Ralph Buij for vulture data; and the Government of the Northwest Territories for caribou data. Work in Kenya supported by the Director of the Directorate of Resource Surveys and Remote Sensing. Work in Zimbabwe supported by the Director General of the Zimbabwe Parks and Wildlife Management Authority, the Zone Atelier LTSER program, and H. Fritz. Red deer data provided by the Euromammals and Euroreddeer initiatives. Work of DP carried out under State Order №0218-2019-0080. Base maps sourced from ESRI, Natural Earth and Open Street Map. Funding: The GIUM is financially supported by the Knobloch Family Foundation, George B. Storer Foundation, and the Bole and Klingenstein Foundation. Field studies funded by European Union's Horizon 2020 research and innovation program #641918 (AfricanBioServices), French 'Agence Nationale de la Recherche' #ANR-16-CE02-0001-01, German Federal Ministry of Education and Research (MORESTEP, #01LC1710A and #01LC1820A), Hawk Mountain Sanctuary, NASA #NNX15AW71A, NSF LTREB #1556248, NSF DBI1915347, NSF DEB-9903416, NSF DEB-0343960, NSF DEB-1020479, The Peregrine Fund, Princeton University, and the Robert Bosch Foundation. Author contributions: MJK, FC, SCJ, MH, JGCH, JAM, TM, AM, WP and CR conceived of the study and wrote the original manuscript; MJK, SCJ, JGCH, TM, KA, JA, ND, CF, AG, CJK, SM, TAM, KAO, CP, HS, AT, and MZV collected the data; MJK, AS, JEM and EG designed the maps. All others reviewed and edited the manuscript.

Elephants leave Botswana and make a transboundary migration across unprotected areas to find permanent surface-water in their dry season range in Zimbabwe.

Wyoming mule deer

A 242-km mule deer migration was discovered in Wyoming in 2014, traversing a mix of public and private lands.

Barren-ground caribou

In the Northwest Territories and Nunavut, caribou range over vast areas of Arctic tundra and boreal forest before converging to give birth on herd-specific calving ranges near the coast.

A proposed railroad

The annual movements of Mongolian gazelle are bound by impermeable border fences with Russia and China. A proposed railroad threatens to further constrain their large nomadic movements.

Oil and gas development

Researchers analyzed tracking data [START_REF] Middleton | Conserving transboundary wildlife migrations: recent insights from the Greater Yellowstone Ecosystem[END_REF] to delineate a high-use corridor (dark blue) from individual routes (orange). The corridor was formally designated by State wildlife officials in 2016, allowing the corridor to be considered during recent oil and gas leasing decisions.

A proposed road


Bruno Yun
email: bruno.yun@abdn.ac.uk

Srdjan Vesic
email: vesic@cril.fr

Gradual Semantics for Weighted Bipolar SETAFs

Keywords:

Gradual semantics are now well-studied in the computational argumentation literature. In this paper, we argue that gradual semantics that can handle both bipolarity (e.g. attacks and supports) and sets of attacking arguments (i.e. several arguments together attacking an argument) might be useful in some contexts. We define the formal framework and properties for such semantics. We proceed by adapting, studying and implementing three well-known semantics from the bipolar gradual literature to this new framework.

Introduction

Recently, gradual argumentation semantics have drawn the attention of many scholars [START_REF] Matt | A Game-Theoretic Measure of Argument Strength for Abstract Argumentation[END_REF][START_REF] Leite | Social Abstract Argumentation[END_REF][START_REF] Pereira | Changing One's Mind: Erase or Rewind[END_REF][START_REF] Bonzon | A Comparative Study of Ranking-Based Semantics for Abstract Argumentation[END_REF][START_REF] Amgoud | Acceptability Semantics for Weighted Argumentation Frameworks[END_REF][START_REF] Potyka | Continuous dynamical systems for weighted bipolar argumentation[END_REF][START_REF] Mossakowski | Modular Semantics and Characteristics for Bipolar Weighted Argumentation Graphs[END_REF][START_REF] Amgoud | Weighted Bipolar Argumentation Graphs: Axioms and Semantics[END_REF]. Some of the existing approaches for gradual semantics allow for intrinsic weights of arguments [START_REF] Amgoud | Acceptability Semantics for Weighted Argumentation Frameworks[END_REF], some for intrinsic weights of attacks [START_REF] Amgoud | Gradual semantics accounting for varied-strength attacks[END_REF], some for sets of attacking arguments [START_REF] Yun | Ranking-based semantics for sets of attacking arguments[END_REF], and some for bipolarity, i.e. two types of interactions between argument are allowed: attacks and supports [START_REF] Cayrol | On the acceptability of arguments in bipolar argumentation frameworks[END_REF][START_REF] Amgoud | On bipolarity in argumentation frameworks[END_REF][START_REF] Potyka | Continuous dynamical systems for weighted bipolar argumentation[END_REF][START_REF] Amgoud | Weighted Bipolar Argumentation Graphs: Axioms and Semantics[END_REF]. One might need all of those features (weights on arguments and attacks, bipolarity and sets of attacking/supporting arguments) to model some situations in an intuitive manner, as illustrated by Figure 1, which is inspired from the example by Prakken [START_REF] Prakken | A study of accrual of arguments, with applications to evidential reasoning[END_REF]. We do not claim that our example cannot be modelled by simpler frameworks, but we believe that our framework allows to represent it in a natural way.

First, the intrinsic weights of the arguments might represent the confidence level of the source or the plausibility of the argument. For instance, the argument "hot" might have a greater intrinsic weight if the outside temperature is 40 degrees Celsius than if the outside temperature is 30 degrees. Second, the intrinsic weights on attacks represent how much arguments attack another argument. For instance, hot weather may be a good reason for cancelling a jogging whereas the rain might be less important. In Figure 1, this is represented by having a greater intrinsic weight on the attack coming from the argument "hot" (0.8) than "rain" (0.5).

Lastly, an agent might have different inclinations toward how the weather affects their jogging routine. For the sake of the example, let us model three possibilities. The simplest case is the one where both the hot weather and the rain are arguments against going for a jogging but there is no interaction between them (left part of Figure 1). The second case is the one where both the hot weather and the rain are still arguments against going for a jogging but the combination of the hot weather and the rain is a desirable reason to go for a jogging (center part of Figure 1). The third case is the one where the two factors (the hot weather and the rain) may interact together and represent a further reason against going for a jogging (right part of Figure 1). This latter case is modelled using an argumentation framework, known in the literature, under the name of set of attacking arguments framework (SETAF). Please note that to model all three possibilities, we also need supports (in addition to attacks) to model positive and negative effects of temperature and precipitation on jogging. The reader is invited to observe the intricate link with the idea of argument accrual, which is well studied in the literature [START_REF] Verheij | Accrual of arguments in defeasible argumentation[END_REF][START_REF] Prakken | A study of accrual of arguments, with applications to evidential reasoning[END_REF][START_REF] Bench-Capon | Justifying actions by accruing arguments[END_REF][START_REF] Lucero | Modelling argument accrual in possibilistic defeasible logic programming[END_REF][START_REF] Lucero | On the accrual of arguments in defeasible logic programming[END_REF][START_REF] Modgil | Integrating dialectical and accrual modes of argumentation[END_REF][START_REF] Prakken | Modelling accrual of arguments in ASPIC+[END_REF]. Note however, that arguments accrual was not studied for gradual semantics nor was adapted to sets of attacking argument frameworks.

The contributions of this paper are: (1) a new argumentation framework with sets of attacking/supporting arguments with intrinsic weights on relations, (2) the definition of properties for this framework and (3) the definition and characterisation of bipolar gradual semantics tailored for this framework.

The paper is structured as follows. We start by presenting the background notions and a list of desirable properties that a gradual semantics for weighted bipolar SETAFs should satisfy. We then show some links between those properties (e.g. some properties imply others). We continue by adapting three semantics from the literature (namely, Euler-based, DF-Quad and Sigmoid damped max-based semantics) to our framework and execute a formal study of their properties. We also provide a JAVA implementation for several semantics and extend the existing ASPARTIX format to the case of weighted bipolar SETAFs.

Formal Setting

We first extend the argumentation framework proposed by [START_REF] Nielsen | A Generalization of Dung's Abstract Framework for Argumentation: Arguing with Sets of Attacking Arguments[END_REF] with sets of supporting arguments which allow arguments to jointly support an argument. Definition 1. A weighted bipolar SETAF (WBSETAF) is AS = A, P, w , where A is a finite set of arguments, P ⊆ (2 A \ ∅) × A × {att, supp} is a set of collective interactions (each interaction being either an attack or a support), and w is a function from

A ∪ P to [0, 1].
The set of all WBSETAFs is denoted by X setaf . The set of non-maximal WB-SETAFs is defined as X <1 setaf = { A, P, w ∈ X setaf | for every b ∈ A, w(b) < 1} and the set of non-minimal WBSETAFs is defined as X >0 setaf = { A, P, w ∈ X setaf | for every b ∈ A, w(b) > 0}. (X, j, att) ∈ P means that there is an attack from X ∈ (2 A \ ∅) to j ∈ A whereas (X, j, supp) ∈ P means that there is a support from X to j. Please note that in our formal setting, we can have both an attack and a support from the same set of arguments toward an argument but all the properties in this paper still hold in the restricted case, where there is only one type of interaction (support or attack) between a set of arguments and an argument. If a ∈ A, we denote by w(a) the intrinsic weight of argument a representing its trustworthiness or the certainty degree of the argument's premises. For p ∈ P, we denote by w(p) the intrinsic weight of the attack/support p. The set of attackers of a ∈ A is Att(a) = {X ∈ 2 A | (X, a, att) ∈ P}. Likewise, the set of supports of a ∈ A is Supp(a) = {X ∈ 2 A | (X, a, supp) ∈ P}. The union of two WBSETAFs AS = A, P, w and AS = A , P , w s.t. A ∩ A = ∅ is AS ⊕ AS = A ∪ A , P ∪ P , w , where ∀a ∈ A ∪ A ∪ P ∪ P , w (a) = w(a), if a ∈ A ∪ P and w (a) = w (a), if a ∈ A ∪ P .

A path is a sequence of sets of arguments that are linked with either sets of attacking or supporting arguments. Definition 2 (Path). A sequence X 0 , X 1 , . . . , X n is called a path in AS = A, P, w iff ∀i ∈ {0, . . . , n -1}, ∃x ∈ X i+1 s.t. (X i , x, l) ∈ P, where l ∈ {att, supp}.

A path is significant w.r.t. a WBSETAF AS = A, P, w if it is, roughly speaking, composed of attacks/supports having strictly positive weights. Definition 3 (Significant path). A path X 0 , X 1 , . . . , X n in AS = A, P, w is significant if and only if ∀i ∈ {0, . . . , n -1}, ∃x ∈ X i+1 s.t. (X i , x, l) ∈ P and w((X i , x, l)) > 0, where l ∈ {att, supp}.

We can now define the notion of cycle. Definition 4 (Cycle). A path X 0 , X 1 , . . . , X n is called a cycle in AS = A, P, w iff ∃x ∈ X 0 s.t. (X n , x, l) ∈ P, where l ∈ {att, supp}.

A semantics is a function assigning an overall strength from [0, 1] to each argument of the WBSETAF. Definition 5 (Semantics). Let AS = A, P, w be a weighed bipolar SETAF. A semantics is a function σ transforming any AS ∈ X setaf into a function

σ AS : A → [0, 1].
When AS is clear from the context, we simply write σ(a) instead of σ AS (a). Definition 6 (Isomorphism). Let AS = A, P, w and AS = A , P , w be two WBSETAFs. We say that f is an isomorphism from A to A w.r.t. AS and AS iff all the following conditions are satisfied: (1

) ∀X ⊆ A, a ∈ A, z ∈ {att, supp}, (X, a, z) ∈ P iff (f (X), f (a), z) ∈ P , (2) ∀X ⊆ A, a ∈ A and z ∈ {att, supp} s.t. (X, a, z) ∈ P, w((X, a, z)) = w ((f (X), f (a), z)) and (3) ∀a ∈ A, w(a) = w (f (a))

Desirable Properties

We now extend the desirable properties that a semantics should satisfy [START_REF] Amgoud | Weighted Bipolar Argumentation Graphs: Axioms and Semantics[END_REF] for the more general case of WBSETAFs . In each set of attacking (resp. supporting) arguments, all the components are necessary. Thus, we follow an important intuition from the literature [START_REF] Yun | Ranking-based semantics for sets of attacking arguments[END_REF] (reflected in our new properties) which is that the strength of a set of arguments should be equal to the strength of the weakest argument. Of course, this is not the only intuition and other aggregation functions can be used without loss of generality.

The first property states that the overall strength returned by the semantics should only be computed based on the structural elements.

Property 1 (Anonymity).

A semantics σ satisfies anonymity iff for any two WB-SETAFs AS = A, P, w and AS = A , P , w , for any isomorphism f from A to A w.r.t. AS and AS , it holds that ∀a ∈ A, σ AS (a) = σ AS (f (a)).

The independence property states that overall strength of an argument should not be affected by any other arguments that are not connected to it.

Property 2 (Independence). A semantics σ satisfies independence iff for any two WBSETAFs

AS = A, P, w and AS = A , P , w s.t. A ∩ A = ∅, it holds that ∀a ∈ A, σ AS (a) = σ AS⊕AS (a).
The directionality property states that the overall strength of an argument should depend only on the significant paths directed to it.

Property 3 (Directionality).

A semantics σ satisfies directionality iff for any two WBSETAFs AS = A, P, w and AS = A, P , w s.t. all the following conditions are satisfied:

-∀y ∈ A ∪ P, w(y) = w (y), -∀x, b ∈ A, ∀X ⊆ A, if P = P ∪ {(X, b, z)}
, where z ∈ {att, supp} and there is no X ⊆ A s.t. b ∈ X and there is a significant path from X to x,

then σ AS (x) = σ AS (x).
The equivalence property states that the overall strength of an argument depends only on the overall strength of its direct attackers and supporters and the weight of the corresponding relations. The reinforcement property states that an argument becomes stronger if the quality of its attackers is reduced or the quality of its supporters is increased.

Property 5 (Reinforcement). A semantics σ satisfies reinforcement iff for every WBSETAF

AS = A, P, w , ∀C, C ⊆ 2 A , ∀a, b ∈ A and ∀X 1 , X 2 , Y 1 , Y 2 ⊆ A s.t. X i , Y i / ∈ C ∪ C
, if all the following conditions are satisfied:

-w(a) = w(b), Att(a) = C ∪ {X 1 } and Att(b) = C ∪ {Y 1 }, -Supp(a) = C ∪ {X 2 } and Supp(b) = C ∪ {Y 2 }, -min x1∈X1 σ AS (x 1 ) ≤ min y1∈Y1 σ AS (y 1 ), -w((X 1 , a, att)) ≤ w((Y 1 , a, att)), -min x2∈X2 σ AS (x 2 ) ≥ min y2∈Y2 σ AS (y 2 ), -w((X 2 , a, supp)) ≥ w((Y 2 , a, supp)), then σ AS (a) ≥ σ AS (b).
The stability property states that if an argument is neither attacked nor supported, then its overall strength should be equal to its intrinsic weight.

Property 6 (Stability). A semantics σ satisfies stability iff, for every WBSETAF

AS = A, P, w , for every argument a ∈ A, if Att(a) = Supp(a) = ∅, then σ AS (a) = w(a).
In the original work of [START_REF] Amgoud | Weighted Bipolar Argumentation Graphs: Axioms and Semantics[END_REF], it is assumed that the neutral overall strength is the minimum possible value, which is zero. The intuition is that the "worthless" arguments, i.e. those with the strength zero, do not contain any valuable information and thus should not impact the other arguments. In their initial work, [START_REF] Mossakowski | Bipolar Weighted Argumentation Graphs[END_REF] generalised the existing approaches by allowing for the neutral overall strength to be any value. In the later paper [START_REF] Mossakowski | Modular Semantics and Characteristics for Bipolar Weighted Argumentation Graphs[END_REF], the authors changed the neutral overall strength to be zero but left the overall strength value to be any interval. In our paper, we choose the overall strength of all semantics to be in [0, 1] to align with the literature standards. As a result, the neutral overall strength is chosen in the aforementioned interval.

The neutrality property states that the overall strength of an argument should not be affected by sets of attacking/supporting arguments containing arguments with an overall strength equal to the neutral overall strength. The first item of the neutrality property is necessary because if the semantics σ never assigns an overall strength β ∈ [0, 1] to an argument then neutrality is trivially satisfied by putting α = β (under the condition that whenever the intrinsic weight of an attack/support is equal to zero, it does not affect the overall strength of its target). If σ satisfies the neutrality property, α is called the neutral overall strength.

The monotonicity property states that if two arguments a and b have the same intrinsic weight and a is "less attacked" and "more supported" than b, then a should have a higher overall strength than b. The resilience property states that if the intrinsic weight of an argument is not 0 nor 1 then its overall strength cannot be 0 nor 1.

Property 9 (Resilience).

A semantics σ satisfies resilience iff, for every WB-SETAF AS = A, P, w , ∀a ∈ A, if 0 < w(a) < 1 then 0 < σ AS (a) < 1.

Franklin states that an attacker and a supporter of equal strength should counter-balance each other. Thus, the overall strength should remain unchanged. The strengthening property states that if an argument receives more supports than attacks, then its overall strength should increase. Roughly speaking, there are three cases: (1) there are more supporters than attackers, (2) the weakest strength of a set attack is strictly weaker than the weakest strength of the corresponding supporter, (3) the intrinsic weight of one attack is strictly weaker than the corresponding support.

Property 11 (Strengthening).

A semantics σ satisfies strengthening iff, for every WBSETAF AS = A, P, w , ∀a ∈ A, if w(a) < 1 and there exists an injective function f from Att(a) to Supp(a) s.t. ∀X ∈ Att(a), min

x∈X σ AS (x) ≤ min x ∈f (X)
σ AS (x ), ∀X ∈ Att(a), w((X, a, att)) ≤ w((f (X), a, supp)) and at least one of the following conditions is satisfied:

there exists X ∈ Supp(a)\{f (X) | X ∈ Att(a)} such that w((X , a, supp)) > 0 and min

x ∈X σ AS (x ) > 0,
there exists X ∈ Att(a) such that min

x∈X σ AS (x) < min x ∈f (X)
σ AS (x ) and w((f (X), a, supp)) > 0, or there exists X ∈ Att(a) such that w((X, a, att)) < w((f (X), a, supp)) and min

x ∈f (X)

σ AS (x ) > 0, then σ AS (a) > w(a).

Similarly, the weakening property states that if an argument receives less supports than attacks, then its overall strength should decrease.

Property 12 (Weakening).

A semantics σ satisfies weakening iff, for every WB-SETAF AS = A, P, w , ∀a ∈ A, if 0 < w(a) and there exists an injective function f from Supp(a) to Att(a) s.t. ∀X ∈ Supp(a), min

x∈X σ AS (x) ≤ min x ∈f (X) σ AS (x ),
∀X ∈ Supp(a), w((X, a, supp)) ≤ w((f (X), a, att)) and at least one of the following conditions is satisfied:

there exists X ∈ Att(a)\{f (X) | X ∈ Supp(a)} such that w((X , a, att)) > 0 and min

x ∈X σ AS (x ) > 0,
there exists X ∈ Supp(a) such that min

x∈X σ AS (x) < min x ∈f (X)
σ AS (x ) and w((f (X), a, att)) > 0, or there exists X ∈ Supp(a) such that w((X, a, supp)) < w((f (X), a, att)) and min

x ∈f (X)

σ AS (x ) > 0, then σ AS (a) < w(a).

The next property states that adding an attack to an argument should decrease its overall strength whereas adding a support should increase it. The idea of accumulation is similar to that of monotonicity except that monotonicity is defined on two arguments of the same graph whereas accumulation is defined on the "same" argument in the "copy" of the graph. Note that in the next section, we identify the conditions under which monotonicity and accumulation are equivalent (Corollary 1).

Property 13 (Accumulation).

A semantics σ satisfies accumulation iff, for every two WBSETAFs AS = A, P, w and AS = A, P , w , ∀a ∈ A, ∀X ∈ 2 A , if ∀p ∈ A ∪ P, we have w(p) = w (p), and

P = P ∪ {(X, a, z)} then (1) σ AS (a) ≤ σ AS (a) if z = att and (2) σ AS (a) ≥ σ AS (a) if z = supp.

Links Between Properties

We now study the link between the properties that we previously defined.

Proposition 1. Let σ be a semantics that satisfies independence, directionality, stability, accumulation. Then, σ satisfies monotonicity.

Proof. Let AS = A, P, w and a, b ∈ A s.t. the conditions of monotonicity are satisfied. The intuition of the next construction is the following. We will define a "copy" AS of AS. For each argument x, we create its copy x . Furthermore, we keep only the attacks from the direct attackers of a and b and only the supports from the direct supporters of a and b. All the other attacks and supports are forgotten. This is done to allow for the application of directionality when comparing AS and AS 1 (which will be defined later in the proof).

Let us define AS = A , P , w s.t. there exists a bijection f : Att(b) ∪ Supp(a) ∪ {a, b} → A defined as ∀x, f (x) = x and:

-A = {a , b }∪{x i | x i ∈ X and X ∈ Att(b)}∪{y i | y i ∈ Y and Y ∈ Supp(a)}. -P = {(f (X), f (a), att) such that (X, a, att) ∈ P} ∪ {(f (X), f (b), att) such that (X, b, att) ∈ P} ∪ {(f (Y ), f (a), supp) such that (Y, a, supp) ∈ P} ∪ {(f (Y ), f (b), supp) such that (Y, b, supp) ∈ P}. -Let us define w as follows: w (f (a)) = w(a), w (f (b)) = w(b), ∀x ∈ A \ {a , b }, w (x) = σ AS (x) and ∀(T, t, z) ∈ Att(b) ∪ Supp(a), let w ((f (T ), f (t), z)) = w((T, t, z)).
From stability, we know that ∀t ∈ A \ {a , b }, we have σ AS (t ) = w (t ) = σ AS (t). By using the independence and equivalence properties, we have that -Let AS 1 = A , P 1 , w 1 with P 1 = P \ {(X m , b , att)} and ∀p ∈ A ∪ P 1 , w 1 (p) = w (p). By accumulation, we have that σ AS 1 (b ) ≥ σ AS (b ). Moreover, by directionality, ∀t ∈ A , σ AS 1 (t) = σ AS (t). We repeat this procedure m -n times to remove all the attacks from Att(b ) \ Att(a ) to b . Let us call the resulting graph AS m-n . -Let us define AS m-n+1 = A , P m-n+1 , w m-n+1 with P m-n+1 = P m-n \ {(Y l , a , supp)} and ∀p ∈ A ∪ P m-n+1 , w m-n+1 (p) = w (p). By accumulation, we have that σ AS m-n+1 (a ) ≤ σ AS m-n (a ). Moreover, by directionality, ∀t ∈ A , σ AS m-n (t) = σ AS m-n+1 (t). We repeat this procedure l -k times to remove all the supports from Supp(a ) \ Supp(b ) to a . Let us call the resulting graph AS * .

By equivalence, we have that σ AS * (a) = σ AS * (b). Thus:

σ AS (b ) ≤ σ AS 1 (b ) ≤ • • • ≤ σ AS * (b ) = σ AS * (a ) ≤ • • • ≤ σ AS 1 (a ) ≤ σ AS (a )
Finally, since we have that σ AS (b) = σ AS (b ) and σ AS (a) = σ AS (a ), we conclude that σ AS (a) ≥ σ AS (b).

Regarding the previous result, the proposition holds for an arbitrary semantics; note also that if σ is a semantics that is defined only on acyclic graphs, we can show that σ satisfies monotonocity (on the class of acyclic graphs).

Proposition 2. Let σ be a semantics that satisfies anonymity, independence, directionality, equivalence and monotonicity on the class of acyclic graphs. Then, σ satisfies accumulation on the class of acyclic graphs.

Proposition 1 also holds for a semantics σ defined on acyclic graphs only. Hence, from that fact and Proposition 2, we conclude that accumulation and monotonicity are equivalent given anonymity, independence, directionality, stability and equivalence. We formally state this in the next corollary.

Corollary 1. Let σ be a semantics that satisfies anonymity, independence, directionality, stability and equivalence on the class of acyclic graphs. Then σ satisfies accumulation on the class of acyclic graphs iff it satisfies monotonicity on the class of acyclic graphs.

Gradual Semantics

As mentioned before, in each set of attacking (resp. supporting) arguments, all the components are necessary. Thus, removing one argument from the set would make the attack (resp. support) void. In this section, we generalise the semantics from the literature by considering the force of the set of attacking (resp. supporting) arguments to be the force of the weakest argument of the set. For simplicity, we also incorporate the intrinsic weight of attacks by means of the multiplication operator but the whole approach can be generalised with other aggregating methods. Moreover, the first two semantics (Euler-based and DF-Quad) do not always converge in the class of cyclic WBSETAFs. For this reason, as usual in the literature [START_REF] Amgoud | Weighted Bipolar Argumentation Graphs: Axioms and Semantics[END_REF][START_REF] Rago | Discontinuity-Free Decision Support with Quantitative Argumentation Debates[END_REF], we consider that the class of all acyclic WBSETAFs is still expressive enough to deserve attention. In the next definitions, we generalise the existing Euler-based [START_REF] Amgoud | Weighted Bipolar Argumentation Graphs: Axioms and Semantics[END_REF] and DF-Quad semantics [START_REF] Rago | Discontinuity-Free Decision Support with Quantitative Argumentation Debates[END_REF] for the new framework.

Definition 7 (Euler-based semantics). Let AS = A, P, w ∈ X setaf be an acyclic WBSETAF. ∀a ∈ A, we have:

σ EB AS (a) = 1 - 1 -w(a) 2 1 + w(a)e E(a)
where We also generalise the Sigmoid damped max-based gradual semantics [START_REF] Mossakowski | Modular Semantics and Characteristics for Bipolar Weighted Argumentation Graphs[END_REF] for this framework. For presentation purposes, we changed the original definition to match the other semantics by translating the range of the semantics from [-1, 1] to [0, 1]. We also changed the "top" operator from the original semantics to "max" because we wanted to take all the arguments into account for determining the overall strength. Note that the sets of attacking (resp. supporting) arguments with a minimum element having a score strictly inferior to 0.5 will increase (resp. decrease) the score of the attacked (resp. supported) argument. This effect is an intrinsic aspect of the original Sigmoid damped max-based semantics, which was not introduced by the generalisation we propose.

E(a) = Y -Z with Y = X∈Supp(a) min x∈X σ EB AS (x) • w((X, a, supp)) and Z = X∈Att(a) min x∈X σ EB AS (x) • w((X, a, att)). X setaf X >0 setaf X <1 setaf σ EB σ DF σ SD σ EB σ DF σ SD σ EB σ DF σ
Definition 9 (Sigmoid damped max-based semantics). Given the WB-SETAF AS = A, P, w ∈ X setaf and δ > 2. ∀a ∈ A, we have

σ SD AS (a) = f Y (a) δ + f -1 (w(a))
, where

Y (a) = max X∈Supp(a) min x∈X f -1 (σ SD AS (x)) • w((X, a, supp)) -max X∈Att(a) min x∈X f -1 (σ SD AS (x)) • w((X, a, att)) and f (x) = tanh(x)+1 2 .
In Table 1, we show the satisfaction of the properties by the aforementioned gradual semantics on the three classes X setaf , X <1 setaf and X >0 setaf . Due to space limitations, we do not show all the proofs in the paper although all the combinations were diligently proved. We now provide a brief overview of the property satisfaction on the general class X setaf .

The Euler-based semantics satisfies the greatest number of properties. The only two non-satisfied properties are strengthening and weakening. Note that the only case when an argument cannot be weakened (resp. strengthened) is the case of an argument that has the maximal (resp. minimal) intrinsic weight. It seems to us that in several reasonable application contexts, this behaviour of the Euler-based semantics can be seen as rational.

The DF-Quad semantics violates strengthening and weakening as well as Franklin and resilience. Again, this is not necessarily a fatal problem, since in some contexts it might not be a good idea to be able to cancel one positive argument with one negative argument of the same strength.

Finally, the Sigmoid damped max-based semantics violates even more properties, which is (we believe) linked to its simple idea to take into account only the strongest attacker and the strongest supporter and to ignore the others. Note that Sigmoid damped max-based semantics does not satisfy neutrality, even if we use the generalised version of the proposition that allows for any number to be the neutral overall strength. Similar to the original version by Mossakowski and Neuhaus [START_REF] Mossakowski | Bipolar Weighted Argumentation Graphs[END_REF], we could restrict our version of Sigmoid damped max-based semantics to arguments with intrinsic strengths above 0.5 only, which would lead to satisfying neutrality. Notice that every property is satisfied by Sigmoid damped max-based semantics on the class X setaf iff it is satisfied on X >0 setaf and on X <1 setaf . The reason is that the Sigmoid damped max-based semantics is only defined in the interval (0, 1). Interestingly, we also showed that the new DF-Quad semantics does not satisfy Franklin (see Example 1) and that the Euler-based semantics does not satisfy strengthening on the class of non-maximal graphs even in the acyclic case (see Example 2). Since neither of our examples uses set attacks, we conclude that the original versions of those two semantics do not satisfy the aforementioned properties, contrary to what was suggested by Amgoud and Ben-Naim [START_REF] Amgoud | Weighted Bipolar Argumentation Graphs: Axioms and Semantics[END_REF].

Discussion

In this paper, we studied gradual semantics for WBSETAFs. This is the first study of the properties that a gradual semantics should satisfy in the framework that allows for bipolar interactions (i.e. both attacks and supports), weights (on both arguments and attacks) and SETAFs (the possibility for several arguments to jointly attack or support an argument). We generalised twelve properties from the literature [START_REF] Amgoud | Weighted Bipolar Argumentation Graphs: Axioms and Semantics[END_REF][START_REF] Mossakowski | Modular Semantics and Characteristics for Bipolar Weighted Argumentation Graphs[END_REF] and introduced a new one called "accumulation". We proved some links between the properties, and generalised three semantics from the literature to be applicable with WBSETAFs. We conducted a formal evaluation of those semantics against the properties on three classes of argumentation graphs. We also provided an implementation of the three new semantics available at https://github.com/AnonymousConfsSubmissions/WBSETAFs. Please note that this implementation is based on an extension of the ASPARTIX format for WBSETAFs.

Fig. 1 .

 1 Fig. 1. Representation of the jogging example. The solid red lines represent sets of attacking arguments whereas the dashed lines represent sets of supporting arguments.

Property 4 (

 4 Equivalence). A semantics σ satisfies equivalence iff for every WB-SETAF AS = A, P, w and ∀a, b ∈ A, if all the following conditions are satisfied: w(a) = w(b), there exists a bijection f between Att(a) and Att(b) s.t. ∀X ∈ Att(a), min x∈X σ AS (x) = min x ∈f (X) σ AS (x ) and w((X, a, att)) = w((f (X), b, att)), there exists a bijection f between Supp(a) and Supp(b) s.t. ∀X ∈ Supp(a), min x∈X σ AS (x) = min x ∈f (X) σ AS (x ) and w((X, a, supp)) = w((f (X), b, supp)), then σ AS (a) = σ AS (b).

Property 7 (

 7 Neutrality). A semantics σ satisfies neutrality iff, there exists a unique α ∈ [0, 1] s.t. both of the following conditions are satisfied: there exists a WBSETAF AS = A, P, w and a ∈ A s.t. σ AS (a) = α, for every WBSETAF AS = A, P, w , for every argument a, b ∈ A s.t. w(a) = w(b), X ⊆ A s.t. Att(a) ⊆ Att(b), Supp(a) ⊆ Supp(b) and Att(a) ∪ Supp(a) ∪ {X} = Att(b) ∪ Supp(b), if min x∈X σ AS (x) = α or w((X, b, z)) = 0, where z ∈ {att, supp}, then σ AS (a) = σ AS (b).

Property 8 (

 8 Monotonicity). A semantics σ satisfies monotonicity iff, for every WBSETAF AS = A, P, w , ∀a, b ∈ A s.t. w(a) = w(b), Att(a) ⊆ Att(b) and ∀X ∈ Att(a), w((X, a, att)) ≤ w((X, b, att)), Supp(b) ⊆ Supp(a) and ∀X ∈ Supp(b), w((X, b, supp)) ≤ w((X, a, supp)), then σ AS (a) ≥ σ AS (b).

Property 10 (

 10 Franklin). A semantics σ satisfies Franklin iff, for every WBSETAF AS = A, P, w , ∀a, b ∈ A and X, Y ⊆ A, if all of the following conditions are satisfied: w(a) = w(b), w((X, a, att)) = w((Y, a, supp)), -Att(a) = Att(b) ∪ {X}, Supp(a) = Supp(b) ∪ {Y }, min x∈X σ AS (x) = min y∈Y σ AS (y), then σ AS (a) = σ AS (b).

  σ AS (a) = σ AS (a ) and σ AS (b) = σ AS (b ). Let us use the notation Att(a ) = {X 1 , . . . , X n }, Att(b ) = Att(a )∪{X n+1 , . . . , X m }, Supp(b ) = {Y 1 , . . . , Y k } and Supp(a ) = Supp(b ) ∪ {Y k+1 , . . . , Y l }. Let us now remove attacks from Att(b ) \ Att(a ) to b and supports from Supp(a ) \ Supp(b ) to a using the following procedures.

Definition 8 (

 8 DF-Quad semantics). Let AS = A, P, w ∈ X setaf be an acyclic WBSETAF. ∀a ∈ A, we have: if v s (a) = v a (a), σ DF AS (a) = w(a); else σ DF AS (a) = w(a) + (0.5 + vs(a)-va(a) 2•|vs(a)-va(a)| -w(a)) • |v s (a) -v a (a)|, where v a (a) = 1 -X∈Att(a) 1 -min x∈X σ DF AS (x) • w((X, a, att)) and v s (aAS (x) • w((X, a, supp)) . Please note that if Att(a) = ∅ then v a = 0. Similarly, v s = 0 if Supp(a) = ∅.

Example 1 . 7 Fig. 2 .

 172 Fig. 2. Counter-example of the satisfaction of Franklin by DF-Quad.

Table 1 .

 1 Satisfaction of the properties by the gradual semantics on acyclic graphs

	SD

* Srdjan Vesic was supported by "Responsible AI" ANR Chair in Artificial Intelligence, https://ia-responsable.eu/