Julian Tang
email: jtang@srk.co.uk

Eric Oelkers
email: e.oelkers@ucl.ac.uk

Julien Declercq
email: jdeclercq@srk.co.uk

Rob Bowell
email: rbowell@srk.co.uk

Effects of pH on arsenic mineralogy and stability in Poldice Valley, Cornwall, United Kingdom

Keywords: Arsenic, mining, weathering, scorodite, thermodynamic, mineralogy, environmental contamination, oxygenated

Many abandoned mine sites in Cornwall, UK, are characterised by elevated concentrations of arsenic (As), which can cause contamination of surrounding soil and water resources. These sites have important historical value that requires access to be maintained, despite exposure of humans to toxins that may lead to health issues including hyperpigmentation keratosis (including skin cancers) and liver fibrosis. The abandoned mine tailings at Wheal Maid has been assessed for As-bearing mineralogy and stability taking into account the public footpaths made by the local council to areas of potential contamination.

To assess the potential risk associated with these mine sites, the As concentration in waters along the tailings dam and Carnon River have been measured and range up to 3.6 ppm, which is 2 orders of magnitude above the WHO guideline value of 0.01 ppm for drinking water.

Samples of water, rocks and soils from the mine tailings ponds and the Carnon River were analysed using Inductively Coupled Plasma -Optical Emission Spectrometry (ICP-OES) and Inductively Coupled Plasma -Mass Spectrometry (ICP-MS) to determine the concentration of individual elements in each sample followed by mineral identification using X-Ray Diffraction (XRD). Mineralogical evaluation indicated that the majority of mine tailings consist of clay-rich rocks, with few associated As-bearing minerals. Scorodite (FeAsO4.2H2O) is observed in the mine tailings pond and appears critical to the As distribution and storage in this surface environment. Using the analysed water chemistry, a modified version of PHREEQC is used to calculate the saturation index of scorodite as a function of pH conditions. The strong variation of the solubility of this mineral with pH and oxidation state highlights potential risks for using scorodite for As fixation and storage.

INTRODUCTION

Background

Human exposure to high arsenic (As) levels can lead to numerous health issues. These include hyperpigmentation keratosis (including skin cancers) and liver fibrosis [START_REF] Atsdr | What are the Physiologic Effects of Aresnic Exposure? Agency for Toxic Substances and Disease Research[END_REF].

Arsenic may form toxic dust that can rest on garden soils and footpaths that young children may ingest via hand to mouth [START_REF] Bowell | Geochemical assessment of aresnic toxicity in mine site along the proposed Mineral Tramway Project, Camborne, Cornwall. Geochemistry, Exploration[END_REF]. Arsenic is a known toxin [START_REF] Pascoe | Bioavailability of metals and arsenic to small mammals at a mining site-contaminated wetland[END_REF][START_REF] Lorenzana | Bioavailability of Arsenic and Lead in Environmental Substrates[END_REF][START_REF] Farago | Health aspects of human exposure to high arsenic concentrations in soil in south-west England[END_REF][START_REF] Kenyon | Tissue distribution and urinary excretion of inorganic arsenic and its methylated metabolites in C57BL6 mice following subchronic exposure to As (V) in drinking water[END_REF][START_REF] Pinyayev | Preabsorptive metabolism of sodium As(V) by anaerobic microbiota of mouse cecum forms a variety of methylated and thiolates Asals[END_REF] and numerous studies have demonstrated higher As uptake in children [START_REF] Eligali | Evaluation of exposure of young children to arsenic contamination in South West England[END_REF]. A

preliminary study of inorganic As and methylated metabolites in urine from adults and children in Cornish mining villages showed elevated urinary As contents [START_REF] Farmer | Assessment of occupational exposure to inorganic arsenic based on urinary concentrations and speciation of arsenic[END_REF].

High As concentrations are commonly found associated with the desired ore products during the mining process and are typically discarded with the rest of the mine wastes [START_REF] Craw | The characterization of Arsenic in Mine Waste[END_REF].

Arsenic concentrations at Earth surface conditions are largely controlled by As-bearing minerals through dissolution, precipitation, incorporation, sorption and desorption [START_REF] Fendolf | Arsenate and chromate retention mechanisms on goethite. 1. Surface structure[END_REF][START_REF] Ford | Rates of hydrous ferric oxide crystallization and the influence on coprecipitated arsenate[END_REF][START_REF] Bostick | Arsenite sorption on troilite (FeS) and pyrite (FeS2)[END_REF][START_REF] Amirbahman | Kinetics of sorption and abiotic oxidation of arsenic (III) by aquifer materials[END_REF][START_REF] Benzerara | Nanoscale study of As biomineralization in an acid mine drainage system[END_REF][START_REF] Deditius | A proposed new type of arsenian pyrite: composition, nanostructure and geological significance[END_REF][START_REF] Catalano | Effect of aqueous Fe(II) on arsenate sorption on goethite and hematite[END_REF][START_REF] Li | Differential pair distribution function study of the structure of arsenate adsorbed on nanocrystalline gamma-Alumina[END_REF][START_REF] Zhu | Secondary minerals of weathered orpiment-realgar-bearing tailings in Shimen carbonate-type realgar mine, Changde, Central China[END_REF][START_REF] Karimian | Antimony and arsenic behavior during Fe(II)-induced transformation of jarosite[END_REF][START_REF] Soest | Sorption of arsenic to biogenic iron (oxyhydr) oxides produced in circumneutral environments[END_REF][START_REF] Catelani | Arsenic uptake in bacterial calcite[END_REF]. Arsenic is similar to many chalcophile elements in that it is commonly released by sulphide oxidation. Arsenic is modified by various biogeochemical processes, and attenuated by adsorption and co-precipitation with Fe-minerals, clays, and organic matter [START_REF] Wen | Molecular Evidence of Arsenic Mobility Linked to Biodegradable Organic Matter[END_REF][START_REF] Schittich | Investigating Fluorescent Organic-Matter Composition as a Key PRedictor for Arsenic Mobility in Groundwater Aquifers[END_REF][START_REF] Bauer | Mobilization of arsenic by dissolved organic matter from iron oxides, soils and sediments[END_REF][START_REF] Redman | Natural Organic Matter Affects Arsenic Speciation and Sorption onto Hematite[END_REF][START_REF] Kalbitz | Mobilization of heavy metals and arsenic in polluted wetland soils and its dependence on dissolved organic matter[END_REF][START_REF] Wang | Effect of natural organic matter on arsenic release from soils and sediments into groundwater[END_REF][START_REF] Pallier | Effect of organic matter on arsenic removal during coagulation/flocculation treatment[END_REF][START_REF] Dobran | Arsenic speciation and mobilization in CCA-contaminated soils: Influence of organic matter content[END_REF]. It is a non-metal/metalloid and is a member of the Group VB elements (P, As, Sb, Bi)

in the periodic table. In natural waters (CCME, 2001), the predominant form of As is arsenate (As +5 ) and may be also found as arsenite (As +3 ) in a reduced environment. The common Asbearing compounds can be grouped into arsenate, arsenide, oxide, and sulphide and sulphosalts. The precipitation of the Fe-arsenate scorodite, FeAsO4.2H2O, is commonly found in oxidised zones of As-bearing ore deposits [START_REF] Rieros | Arsenic disposal practices in the metallurgical industry[END_REF] and understanding how it transforms over time is key to reduce health risks.

Mining for copper and tin in Cornwall began at least by the early eighteenth century [START_REF] Hunt | Arsenic speciation and distribution in the Carnon estuary following the acute discharge of contaminated water from a disused mine[END_REF]. Towards the end of the nineteenth century, Cornwall dominated copper, tin and arsenic production globally [START_REF] Buckley | The Story of Mining in Cornwall[END_REF]. Prolonged periods of mining without environmental regulations have had a significant impact to the surroundings [START_REF] Farago | Health aspects of human exposure to high arsenic concentrations in soil in south-west England[END_REF][START_REF] Camm | Characterization of a mining related arsenic contaminated site, Cornwall, UK[END_REF][START_REF] Bowell | Geochemical assessment of aresnic toxicity in mine site along the proposed Mineral Tramway Project, Camborne, Cornwall. Geochemistry, Exploration[END_REF]. One such mine, the Wheal Maid is located in the Poldice Valley in central Cornwall (Figure 1). This site is of particular interest due to its popularity for outdoor recreation, the lack in reclamation of mine dumps, and its prominence in the production of As. Mine waste and derelict As kilns are still present at the site [START_REF] Camm | Characterization of a mining related arsenic contaminated site, Cornwall, UK[END_REF]. Runoff from the workings flow through tributaries approximately 2 km before entering the Carnon River to the southeast.

The objective this study are to: (1) identify all potentially harmful contaminants and Asbearing minerals present at the site, (2) assess and evaluate the most common As-bearing minerals in an oxygenated mine tailings dam, (3) collect and analyse water, rock and soil samples from Wheal Maid and along the Carnon River, and (4) determine the minerals controlling As mobility and the effects of varying pH conditions. Thermodynamic and crystallographic data collected in (2), combined into an internally consistent thermodynamic database, can help to predict and explain the occurrence and assemblages of arsenates and associated minerals.

1.2

Geology and Conceptualisation

The Wheal Maid mineralization consisted of hydrothermal veins of cassiterite-sulphidequartz-fluorite-clay within a Devonian aged Mylor Slate Formation. The mineralization is associated with Carboniferous-Permian magmatism that provided the heat source that developed a hydrothermal cell under the area leading to metal leaching from shales and deposition along faults in the Mylor formation [START_REF] Hill | Falmouth[END_REF].

Due to the increasing challenges of deep mining 'the Great County Adit' was constructed to remove water starting in 1748 and reaching the Poldice Valley 12 years later [START_REF] Buckley | The Story of Mining in Cornwall[END_REF]. Water from the mines is discharged more than 60 km (40 miles) away into the Carnon River. The estimated depth of the adit beneath the Poldice Mine is 510 meters. An engine (Taylor's engine house) worked in association with gravity in the adit to pump water out of the mine towards to the river mouth. Runoff from the mines caused a decline in biodiversity in the river system [START_REF] Neal | The water quality of the River Carnon, west Cornwall, November 1992 to March 1994: the impacts of Wheal Jane discharges[END_REF] [START_REF] Bowell | Geochemical assessment of aresnic toxicity in mine site along the proposed Mineral Tramway Project, Camborne, Cornwall. Geochemistry, Exploration[END_REF].

A conceptual model for the Wheal Maid mine tailings facility (Figure 2) was developed from a review of previous studies mentioned above and observation of the general topography and the following assumptions are made through discussions regarding the flow of water onsite. The assumptions are as follow:

1. Water falls onto the tailings facility from rainfall;

2. Water from rainfall enters the Carnon River through surface runoff and infiltration of the tailings facility;

3. Water flows downstream moving from tailings pond 1 to tailings pond 3 and enters the Carnon River through a controlled water barrier;

4. Water in the tailings pond infiltrates only a minor amount of water into the deep "Great Adit" groundwater system and mixes with the tailings water leaving the facility at the controlled water barrier; and 5. Surface runoff is neglected in the model as it is not quantified, and the amount is deemed negligible compared to the flows from the tailings ponds and the "Great Adit" system.

PREVIOUS STUDIES

In a study focusing on the potential for As contamination on the Mineral Tramway and mining sites in the Gwennap and Camborne mining districts, sediments were found to contain elevated bioavailable metalloids and metals [START_REF] Buckley | The Story of Mining in Cornwall[END_REF]. The source of this material is primarily related to past mining activity but also reflects naturally elevated background levels associated with the mineralisation [START_REF] Bowell | Geochemical assessment of aresnic toxicity in mine site along the proposed Mineral Tramway Project, Camborne, Cornwall. Geochemistry, Exploration[END_REF]. A further study [START_REF] Neal | The water quality of the River Carnon, west Cornwall, November 1992 to March 1994: the impacts of Wheal Jane discharges[END_REF] measured daily water quality in the River Carnon.

Arsenic related minerals and solubilities

The Camborne and Gwennap mining district contains a considerable array of supergene minerals. For the purpose of this study, several As-bearing minerals have been studied on their environmental occurrence and solubility understanding. Known As-bearing minerals in the environment stated in [START_REF] Bowell | Geochemical assessment of aresnic toxicity in mine site along the proposed Mineral Tramway Project, Camborne, Cornwall. Geochemistry, Exploration[END_REF] included scorodite and liroconite. Other

As-bearing minerals are described in Supplementary Information A to give a wider perspective of what minerals can be dominant in a mine tailings system which include: arseniosiderite, geminite, haidingerite, pushcharovskite, yukonite and zýkaite.

Scorodite

Scorodite (FeAsO4.2H2O) water interaction is critical to As distribution and storage in surface environments (Zhu, et al., 2019). Scorodite has a low solubility in a low pH environment and is a common secondary As-bearing mineral in acidic Fe(III)-As(V)-H2O systems, such as As-contaminated soil and mine wastes [START_REF] Drahota | Secondary arsenic minerals in the environment: a review[END_REF].

Understanding of iron arsenates in the natural environment are still ambiguous but past studies have clearly demonstrated the presence of scorodite from mine waters [START_REF] Bowell | Assessment of Scorodite Precipitation from Mine Waters[END_REF].

The low solubility of scorodite has motivated many studies to suggest the precipitation of this mineral as As fixation strategy in the mining industry (Zhu, et al., 2019). A large number of studies have reported scorodite solubility over the past six decades (Table 2). Reported values of Ksp consistent with the reaction

𝐹𝑒𝐴𝑠𝑂 ! . 2𝐻 " 𝑂 = 𝐹𝑒 #$ + 𝐴𝑠𝑂 ! #% + 2𝐻 " 𝑂 [1]
vary by five orders of magnitude (10 -20 ± 0.58 and 10 -25.81±0.1 ). These inconsistencies were addressed by (Zhu, et al., 2019) and originated from 1. The effect of crystal size on solubility (natural scorodite crystals are mostly nano-sized, but laboratory synthesised crystals can rarely replicate natural scorodite dimensions); Depending on the product precipitated, if goethite precipitated instead of ferrihydrite (curve C), the predicted arsenic release is overestimated at pH > 2.5 but closely matches measured As concentrations collected from acid mining drainage waters at pH 3 stated in Zhu, et al. (2019).

Additional data points showing acid mining drainage (AMD) solutions were measured by Zhu, et al. (2019) and closely matches curve C. This is taken into account in this study with a modified PHREEQC database which is explained further in section 3.5.

Liroconite

Variability in the chemical composition and crystal chemical properties of secondary copper minerals lends them for the understanding of the formation and evolution of oxidation zones of ore deposits (Magalhães et al., 1988, Williams, 1990). Among others, copper arsenates are also common and widely distributed minerals in such oxidation zones (e.g. Števko et al., 2017;Southwood et al., 2020) where they form through weathering of sulfidic minerals in low pH environments. Copper arsenates are capable of taking up and releasing not only copper and arsenic but also other elements such as lead, zinc, and selenium (e.g. Ingwersen, 1990).

Liroconite was named by Wilhelm Karl von Haidinger in 1825 (Mohs and Haidinger, 1825), and before this it was known as octahedral arsenate of copper (de Bournon, 1801). One of the first descriptions of the crystal structure of liroconite was in 1962 as Cu2Al[((AsP)O4)(OH4)].4H2O by Giuseppetti et al. (1962).

Natural liroconite is found in the type locality Wheal Gorland at St. Day, Cornwall, UK (Plumhoff et al., 2020). The mine Wheal Gorland is part of the Camborne-Redruth mining district which is situated in Cornwall, the south-western part of Great Britain, approximately 90 km west of Plymouth. The study used a tabular liroconite crystal of 0.080 mm by 0.065 mm by 0.045 mm for the acid-solution calorimetry and relaxation calorimetry, giving a solubility log Ksp of -4.92 from reaction [2].

Cu2Al(AsO4).4H2O + 4H + = 2Cu 2+ + Al 3+ + AsO4 3-+8H2O [2]

METHODOLOGY

World Health Organisation Guideline

Water samples representative of the mine tailings and the Carnon River were collected, and characterised following guidelines (

Field Methods

Water sampling

Water samples were collected in June 2018 at each of the three tailings ponds and 400 metres along the Carnon River (Figure 1). A summary of the quantity of water samples collected is shown in Table 3. Sampling teflon vials were prepared in the lab using ultra-pure deionised and distilled water (to 18.2 MΩ cm resistivity at +25 o C) produced by the VWR Ultrapure water system (VWR, 2019), P series, to prevent contamination from bacterial growth. Each vial was filled with ultra-pure water that were removed and dried thoroughly 12 hours before sampling. At each sampling location, water were extracted using a 0.2 μm Whatman syringe filter and two drops of 2% aqueous nitric acid (HNO3) were added (dwi, 2005). The pH was measured on site while collecting each sample using a Mettler Toledo 5-easy plus pH probe. The pH probe was calibrated on-site using three buffer solutions, pH 4.0, pH 7.0 and pH 9.21. Detection limit was not applicable. Duplicate samples were collected at each of the tailings ponds.

Rock/Soil sampling

Rocks and soils were sampled in June 2018 at five sites from high to low topography to determine the leachability of material from these solids (the locations are shown in Figure 4).

Table 3 summarises the quantity of samples collected and analysed in the study. The top 10 to 20 cm of the surface material was removed prior to collecting these samples. This method was adopted to avoid solids that may have been affected by surface chemical weathering.

Samples were then retrieved and stored in separately labelled transparent sample bags (dimensions: 76 x 83 mm). Sampled sediments greater than 0.5 cm in size were considered as rocks, and those less than 0.5 cm as soil. Preliminary elemental analysis was completed using a portable X-Ray Florescent (p-XRF) analyser (Figure 5). The instrument used was the Olympus Delta XRF Analyser. Calibration was completed using three certified reference materials which include a blank SiO2, NIST2710a and NIST2711a reference materials given by the provider. A map showing the locations of the p-XRF samples can be found in Supplementary Information B.

Sample Analysis

Water sample analysis

The collected water samples were stored in a refrigerator prior to analysis. 18 samples were analysed using inductively coupled plasma optical emission spectroscopy (ICP-OES)

and the inductively coupled plasma mass spectroscopy (ICP-MS) for the major and minor elements respectively. The analysis was completed on a Varian 720 ICP-OES (axial configuration) equipped with autosampler and a Bruker M90 ICP-MS equipped with autosampler. 3 duplicates and 5 blank samples were analysed simultaneously to assess the reproducibility. The detection limits specified in Table 5 have been removed from the subsequent calculations. Certified reference material TMDA-70 (fortified lake water) was used during the analysis. The duplicate and blank samples together with the TMDA-70 were used to calculate the uncertainties of each analysis and check accuracy and reproducibility. The uncertainty of all elemental analyses were found to be less than +/-10% when evaluated with TMDA-70. Detailed data processing method can be found in Supplementary Information C.

X-ray diffraction

Samples for diffraction analysis were prepared by side-filling the powder, against a groundglass surface, into a 1.5 mm wide sample holder and consolidating it by gentle tapping. X-ray powder diffraction data were then collected with Cobalt (Co) Potassium (K) alpha (α) radiation, using a PANalytical X'Pert Pro diffractometer, with Bragg-Brentano para-focusing reflection geometry. The X-ray peaks produced are evaluated using the database provided by the PANalytical HighScore Plus software with the ICDD (International Centre for Diffraction Data)

database. The minimum tip width for the searched peaks is 0.01, which corresponded to a relative error of 1%, the counting statistical error. A more detailed methodology can be found in Supplementary Information D.

Geochemical Calculations

Thermodynamic calculations in this study were computed with PHREEQC (version 3.5.0.14000) together with a modified version of the WATEQ4F.v4 database [START_REF] Parkhurst | Description of Input and Examples for PHREEQC Version 3 -A Computer Program for Speciation, Batch-Reaction, One[END_REF]. This database was selected for this study because it includes the all of elements considered for this study of water quality in a mine tailings environment. The solubility constant of scorodite in the database was updated from 10 -20.25 to 10 -24.5 . This scorodite solubility constant value concurs with the value from that published in literature (Zhu, et al., 2019), as pure scorodite does not occur in the natural environment. As discussed in section 2, a solubility constant of 10 -24.5 closely matches with measured As concentrations collected from acid mining drainage waters in low acidic pH waters. Thus, the modified value for the scorodite + goethite system is more fitting for an acid mine drainage environment.

The pH of the mine drainage waters was measured in-situ but the temperature and pe at the site were not. Samples were taken during June, hence, calculations were performed assuming an average temperature of 25°C. The pe was assumed to be in equilibrium [START_REF] Soest | Sorption of arsenic to biogenic iron (oxyhydr) oxides produced in circumneutral environments[END_REF] with the atmosphere such that:

• pe = -pH + 20.85
The objective of the geochemical calculations is to verify site observations with the analysed data and to validate the thermodynamic data available in the literature, as detailed above.

Additions to PHREEQC database

Nordstrom et al., 2014 described the PHREEQC thermodynamic database as an overdetermined network in a theoretical sense, but also contains large uncertainties. This inconsistency led to the idea of evaluating previous literature on how solubility measurements were made, in return to limit the use of solubility products measured where large impurities were present. [START_REF] Wagman | The NBS tables of chemical thermodynamic properties: selected values for inorganic and C1 and C2 organic substances in SI units[END_REF] pointed out that typically two methods have been used to evaluate thermodynamic data, the sequential method, and the simultaneous fit method. In this study, a mixture of both techniques have been used and the solubility product values added to the PHREEQC WATEQ4F.v4 database are shown in Table 4. The details on how the updated thermodynamics of the As-bearing minerals are found in section 2.1 and in Supplementary Information A.

RESULTS

Mineralogy of the Mine Site

Mineralogical samples collected from the Wheal Maid (Figure 4) were found to be similar in composition confirming a similar style of mineralisation. Whole-sample XRD demonstrated that the predominant mineralogy for sand, silt and clay fractions are quartz, muscovite and chlorite. The XRD patterns produced are presented in Supplementary Information E.

Cassiterite, dickite, lizardite (Mg3(Si2O5)(OH)4), tourmaline are also present. These occurrences depend on local geology and weathering of nearby outcrops. Of the samples taken, the products identified by XRD include jarosite), goethite and pyrite (FeS2). Table 6 summarises the major and minor minerals. The major and minor minerals are characterised by the abundance of clusters in which the XRD identifies. Minor rutile (TiO2), arsenopyrite (FeAsS), arsenian pyrite (Fe(As,S)S), löllingite (FeAs2), tennantite-tetrahedrite (Cu10(Zn,Ag,Fe)2(As,Sb)4S13) have previously been reported at these sites, but were not observed in this study [START_REF] Bowell | Geochemical assessment of aresnic toxicity in mine site along the proposed Mineral Tramway Project, Camborne, Cornwall. Geochemistry, Exploration[END_REF].

Bulk Geochemistry

Geochemistry of rocks and soils

The arsenic concentrations in the soil samples range from 0.001 pp, to 1 ppm as measured using the p-XRF (Figure 5). The results of the other metal concentrations such as Al, Fe and K can be found in Supplementary Information F. The concentration of all these parameters tend to increase with decreasing elevation. The highest concentrations of As and K were found at the lowest elevation (sites 23 to 25). This is an indication the metals are being leached. Some data-points lack measurements which are most likely due to concentrations lower than the detection limit.

Hydrogeochemistry in the vicinity of the mine site

To assess background levels of As in surface water, three tailings ponds and five river water samples were collected and analysed using ICP-OES and ICP-MS. The concentrations are compared with the WHO drinking water guideline values. In the Carnon River, arsenic concentrations were 1 -2 orders of magnitude above the 0.01 ppm WHO water quality limits despite the pH being circum-neutral (pH 6-7). The pH of the waters were made near to neutral pH by the dilution of tailings pond waters with groundwater.

Several elements including Cr, Cu and Ni had concentrations below the WHO limits in the river 345 water, but Cd and Pb concentrations remained to be above the drinking water guidelines.

DISCUSSION

Mass balance calculation

Taking account of the conceptual model in Section 1.2, water flows downstream from the tailings pond and enters the Carnon River through a controlled water barrier. The dilution of tailings pond water with subsurface water was a strategy created by previous land remediation efforts. The flow rate of the waters from the tailings pond and the underground source were not measure, hence, the mixing ratio of these waters was estimated through onsite observation of the quantity of water flowing from each source entering the river. The estimated mixing ratio of water entering the Carnon River comprises of 15% tailings pond water and 85% subsurface underground adit-water. This mixing ratio is understood to be representative as the majority of water entering the main river was observed to be from the underground source.

Geochemical Calculations

The water chemistry at the monitoring points delineated in section 4.2 was used as inputs (Table 7) to the PHREEQC geochemical model and the saturation index (SI) of minerals was calculated. The SI of a mineral indicates the "tendency" of the water to want to precipitate the mineral, if the SI is at or above 0 (the solubility equilibrium, it indicates that the water wants to precipitate the mineral. The SI of the mine runoff waters before and after dilution with respect to scorodite is shown in Figure 7A which indicates that the SI ranges from -2.75 to +2.5.

Scorodite is supersaturated in the second tailings pond water and in the marshland of the river which suggests that precipitation of the mineral is likely and can be a factor controlling As mobility in the waters.

Using the provided mixing ratios described in section 5.1, the water chemistry at the monitoring points were mass balanced and used as inputs to the PHREEQC geochemical model to produce the SI of scorodite downstream of the mixing zone (Figure 7B). The four orange bars represents the four monitoring points where predictions are made. Three out of four predictions are noticeably similar to the site measurement. The constrains placed on the water calculations involved changing pH (+/-2 s.u. from pH 7) shown in Figure 7C. The bar chart suggests that changing pH alters the saturation index of scorodite significantly in the waters.

With an increase of pH units (more alkaline), the water becomes heavily under-saturated in terms of the As-bearing mineral. However, in a more acidic environment (lower pH) the solubility of As is little affected.

The result of the mass balance indicate that an increase of pH leads to the waters becoming increasingly undersaturated with respect to scorodite. The SI of other As minerals have been studied but all minerals are shown to be heavily undersaturated in the waters, hence, scorodite is the most likely mineral to be controlling As concentration in the water. Thus, it may be expected that increasing the pH could lead to the enhanced concentration and mobility of As in the water. Due to the decades of infiltration of mine tailing water into the groundwater system, the dilution mechanism used previously does not prove to be effective as the water chemistry from the groundwater outlet is very similar to those found in pond C of the mine tailings delineated in Figure 6.

Environmental Consequences

Sampling of surface soil and water at Wheal Maid identified the presence of elevated As content related to the mining activities and re-distribution of that As in response to chemical weathering and surface runoff. This is shown from the leaching of metals to the lower elevations delineated in Figure 5. The mining activity and formation of the tailings dam in the nineteenth century led to faster weathering rates of the clay-rich shale and hornfels country rock through the disturbance of the natural geology. The high abundance of clay minerals found in the sampled rocks (Table 6) may be a contributing factor in the increased As after the weathering processes.

Elevated As concentration was found in the soil and water samples of the Wheal Maid. This is similar to the high As concentrations found in the overburden at other mines such as Higher Condurrow and Wheal Fortune [START_REF] Bowell | Geochemical assessment of aresnic toxicity in mine site along the proposed Mineral Tramway Project, Camborne, Cornwall. Geochemistry, Exploration[END_REF]. In such systems, oxidized iron can form secondary As(+V) mineral phases or minerals that absorb As. This is observed in the SI values of the PHREEQC output using site measured chemistry. The mass calculation concludes that scorodite may be a controlling factor of the high As in the water, as the SI values of scorodite through the monitoring points lie very close to the equilibrium line (Figure 7). Therefore, slight changes in the saturation index of scorodite could result in major changes in the concentration of As in the water. Efforts were made to find other common As-bearing minerals in a mining environment such as: arseniosiderite, geminite, and others, but all the SI values were extremely negative, hence, were not investigated further.

CONCLUSIONS

Wheal Maid mine tailings is integrated along a wider mining network known as 'the Great County Adit' that contains elevated concentrations of numerous potentially toxic metal(loid)s. The source of these metal(loid)s is related to past mine waste and disturbance of clay-rich country rocks leading to faster weathering rates from placing waste products in mine tailings sites. Arsenic concentrations in waters along the tailings dam and Carnon River range up to 3.6 mg/L which is more than 2 orders of magnitude above the WHO guideline value of 0.01 mg/L for drinking water.

This study indicated that the soil and water of mine tailings at Wheal Maid has the potential of releasing As and precipitating As-bearing minerals. The sampling programme captures information for the site during a relatively the summer period but could be improved by monthly sampling to monitor seasonal fluctuations in one year. The SI of scorodite was near to equilibrium indicating precipitation of the mineral is likely, however, it doesn't conclude that it is the sole mineral controlling As concentrations. This study highlights the need to better understand As-bearing minerals and slight environmental changes such as increase alkalinity in waters may cause serious consequences in an industrial setting. Water 18

In the three mine tailings ponds and 400 m along the Carnon River. Duplicate samples were made in each of the three mine tailings ponds. Cassiterite (SnO2)

FIGURE CAPTIONS

2 .

 2 The formation of secondary Fe-containing minerals during incongruent dissolution experiments; and 3. Mis-interpretation of certain solution chemistry effects.These factors were taken into consideration byZhu, et al. (2019) who synthesised scorodite crystal ranging in sizes from 10 to 30 μm as measured using scanning electron microphotography. This size range is similar to those found in nature. Scorodite dissolution was investigated at pH 1.15 over the temperature range of 50 -90 o C and measurement were extrapolated to 25 o C to obtain thermodynamic properties at room temperature. The estimated scorodite solubility of 10 -25.81±0.1 specified byZhu, et al. (2019) agreed with past literature data at low pH conditions but is one to two orders of magnitude smaller than previous values measured as pH > 4.0 -4.5 as shown in curve A in Figure3. When ferrihydrite is allowed to precipitate at pH > 4.0 consistent with incongruent dissolution (curve B), however, the match between calculated and measured solubilities are significantly improved, suggesting that previous studies may have provoked incongruent dissolution through iron phase precipitation.

Figure 6

 6 delineates the As chemistry in each of the monitoring points. The accumulation of metals in the tailings ponds results in highly acidic waters (pH 2-3) with As concentrations 2 orders of magnitude above the WHO drinking water guideline value (0.01 mg/L). Other elements with concentrations exceeding the WHO drinking water guideline values are Cd, Cr, Cu, Ni and Pb.

Solid

  highest to the lowest elevation points

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Map of St Day, Cornwall, South-west England and the location of Wheal Maid, Poldice Valley with annotated water sampling localities including the mine tailings pond 1 -3 and along the Carnon River

Figure 4 :Figure 5 :Figure 6 :

 456 Figure 4: Localities of rock and soil sampling and the elevation difference created using ArcGIS

Figure 7 :FIGURESFigure 1 :Figure 3 :Figure 4 :Figure 5 :

 71345 Figure 7: Saturation index (SI) of scorodite along the mine drainage pathway. (A) SI using measured chemistry; (B) SI comparison of measured (blue) and mass balanced (orange); (C) SI change varying pH +/-2 from natural conditions

Figure 6 :

 6 Figure 6: Arsenic concentration in the water compared to WHO guidelines (WHO, 2011). The location names are as follow: (A) Tailings Pond A; (B) Tailings Pond B; (C) Tailings Pond C; (D) Carnon River.

Figure 7 :

 7 Figure 7: Saturation index of Scorodite along mine (A) Measured water quality data (B) Comparison of measured (blue) and mass balanced (orange) (C) SI change varying pH +/-2 from natural conditions of mass balanced values

  Table1) where available of the World Health Organization's Guidelines for Drinking-water Quality (WHO, Guidelines for drinking-water quality, fourth edition, 2011). Many national authorities have followed the WHO guideline value of 0.01 mg/L (10 ug/L) for As; only one out of 104 countries set a value below this value[START_REF] Who | A global overview of national regulations and standards for drinking-water quality[END_REF]. The guideline values adapted from WHO's "A global overview of national regulations and standards for drinking water quality" for As, B, Ba, Cd, Cr, Cu, Ni and Pb are deemed appropriate for this study because these values have been obtained and validated against other members of the WHO International Network of Drinking-water Regulators (RegNet).

TABLE CAPTIONS Table 1 :

 CAPTIONS1 WHO Drinking Water Quality Guideline Values (WHO, 2011)

Table 2 :

 2 Summary of scorodite solubility data available in the literature

Table 3 :

 3 Summary of the quantities, location and purpose of samples taken

Table 4 :

 4 Solubility data added to the PHREEQC database

Table 5 :

 5 Water quality analysis limit of detection

Table 6 :

 6 Summary of rock mineralogical composition from XRD analysis (chemical formula)

Table 7 :

 7 Water Chemistry Inputs into PHREEQC TABLES

Table 1 :

 1 WHO Drinking Water Quality Guideline Values (WHO, 2011)

		WHO Drinking Water
	Element	Quality Guideline
		Values (ppm)
	As	0.01
	B	2.4
	Ba	0.7
	Cd	0.003
	Cr	0.05
	Cu	2
	Ni	0.07
	Pb	0.01

Table 2 :

 2 Summary of scorodite solubility data available in the literature

	Reference	Medium used	Temperature	pH	pH Final	pH	Logarithm of
			( o C)	Initial	Average	Range	the Solubility
				Average			Constant
	Chukhlantsev 1956	Sulfuric Acid (H2SO4)	20	N/A	2.31	1.9 -	-20 ± 0.58
		Nitric Acid (HNO3)				2.95
	Nishimura & Tozawa	Sulfuric Acid (H2SO4)	25	N/A	2.45	1 -4	-21.1
	1978
	Robins 1987	Perchloric Acid	25	N/A	1.47	0 -2	-22.3
		(HClO4)
		Sulfuric Acid (H2SO4)
		Nitric Acid (HNO3)
	(Dove & Rimstidt,	Hydrochloric Acid	25	5.57	6.02	5.53 -	-21.7 ± 0.5
	1985)	(HCl)				6.36
		Sodium Hydroxide
		(NaOH)
	(Krause & Ettel,	Sulfuric Acid (H2SO4)	23	1.58	1.84	0.97 -
	1988)					2.00
	(Krause & Ettel,	Sodium Hydroxide	23	From	From	3.45 -	Overall average
	1988) using	(NaOH)		3.45 to	2.55 to	7.92	-24.41 ± 0.15
	Incongruent	Sulfuric Acid (H2SO4)		12.36	7.92
	dissolution
	(Zhu, et al., 2019)	Hydrochloric Acid	50-90	1.5			-25.81
		(HCl)	(extrapolated
			to 25)

Table 3 :

 3 Summary of the quantities, location and purpose of samples taken

	Type of Sample Quantities	Location Description

Table 5 :

 5 Water quality analysis limit of detection

	Limit of Detection Table
	Parameters	Units Concentration
	Ag	mg/L	1.9E-06
	Al	mg/L	0.06
	As	mg/L	2.9E-04
	B	mg/L	0.07
	Ba	mg/L	0.02
	Be	mg/L	0.004
	Ca	mg/L	0.06
	Cd	mg/L	0.04
	Co	mg/L	0.01
	Cr	mg/L	0.01
	Cu	mg/L	0.02
	Fe	mg/L	0.002
	K	mg/L	0.002
	Li	mg/L	5.0E-06
	Mg	mg/L	0.001
	Mn	mg/L	1.2E-04
	Mo	mg/L	0.04
	Na	mg/L	4.3E-06
	Ni	mg/L	0.01
	Pb	mg/L	0.30
	V	mg/L	0.001
	Zn	mg/L	0.03

Table 6 :

 6 Summary of rock mineralogical composition from XRD analysis (chemical formula)

	Rock Sample	Major Composition	Accessory Minerals
	Location
	1	Quartz (SiO2)
		Chlorite ((Mg,Fe,Li)6AlSi3O10(OH)8)
		Muscovite (KAl2(AlSi3O10)(OH)2)
	2	Quartz (SiO2)	Kaolinite (Al₂Si₂O₅(OH)₄)
		Chlorite ((Mg,Fe,Li)6AlSi3O10(OH)8)
		Muscovite (KAl2(AlSi3O10)(OH)2)
	3	Quartz (SiO2)	Pyrite Jarosite (KFe3(SO4)2(OH)6)
		Chlorite ((Mg,Fe,Li)6AlSi3O10(OH)8)
		Muscovite (KAl2(AlSi3O10)(OH)2)
	4	Quartz (SiO2)	Dickite (Al2Si2O5(OH)4)
		Muscovite (KAl2(AlSi3O10)(OH)2)	Jarosite (KFe 3+ 3(OH)6(SO4)2

Table 7 :

 7 Water Chemistry Inputs into PHREEQC

							Locations
	Parameters	Units	Pond A	Pond B	Pond C	Mixing Zone 1	Mixing Zone 2	Mixing Zone 3	River 1	River 2	River 3
	pH	s.u.	7.84	2.56	2.57	2.71	8	6.67	6.8	7.16	7.28
	Ag	mg/L	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003
	Al	mg/L	0.02	0.33	7.83	12.2	0.05	0.27	0.42	0.92	0.51
	As	mg/L	0.002	0.268	0.001	0.008	0.004	0.002	0.003	0.035	0.002
	B	mg/L	0.026	0.070	0.150	0.120	0.021	0.029	0.029	0.040	0.026
	Ba	mg/L	0.0003	0.0001	0.0004	0.0002	0.0002	0.0003	0.0007	0.0005	0.0004
	Be	mg/L	0.0001	0.0002	0.0013	0.0010	0.0001	0.0001	0.0000	0.0001	0.0001
	Ca	mg/L	4.49	20.0	5.35	3.02	11.3	12.4	11.9	18.1	12.4
	Cd	mg/L	0.0003	0.083	0.066	0.076	0.0002	0.0001	0.0003	0.0053	0.0007
	Cl	mg/L	13.4	13.7	17.2	16.2	20.7	21.6	21.5	18.5	21.8
	Co	mg/L	0.0003	0.103	0.184	0.197	0.003	0.004	0.006	0.004	0.005
	Cr	mg/L	0.0012	0.03	0.03	0.04	0.0009	0.0006	0.0014	0.0013	0.0014
	Cu	mg/L	0.03	9.62	9.88	10.0	0.15	0.10	0.17	0.20	0.19
	Fe	mg/L	0.043	0	1.73	1.71	0.21	0.18	0.53	0	0.62
	K	mg/L	1.72	0.64	1.16	0.60	4.04	4.18	4.13	0.69	4.18
	Li	mg/L	0.0025	0.2062	0.4686	0.4902	0.0033	0.0052	0.0063	0.0051	0.0065
	Mg	mg/L	1.99	11.4	15.4	3.17	3.43	3.76	3.72	4.47	3.85
	Mn	mg/L	0.05	3.02	7.97	7.66	0.04	0.08	0.09	0.56	0.12
	Mo	mg/L	0.003	0.003	0.003	0.002	0.003	0.001	0.003	0.003	0.003
	Na	mg/L	8.70	8.87	11.2	10.5	13.4	14.0	14.0	12.0	14.1
	Ni	mg/L	0.006	0.198	0.324	0.332	0.004	0.002	0.009	0.009	0.005
	Pb	mg/L	0.002	0.066	0.019	0.022	0.002	0.014	0.005	0.005	0.004
	Ru	mg/L	8.93E-06	6.30E-06	3.08E-06	3.10E-06	0	3.30E-06	2.60E-06	3.10E-06	2.10E-06
	S(6) as
	CaCO3	mg/L	38.5	77.0	77.0	38.5	38.5	38.5	38.5	38.5	38.5
	V	mg/L	1.01E-05	2.08E-04	3.94E-05	6.47E-05	1.00E-05	9.20E-06	3.03E-05	2.54E-05	1.06E-04
	Zn	mg/L	0.048	0.273	0.530	0.505	0.098	0.331	0.411	0.059	0.444

Taylor's Engine

Supplementary Information A

Other As-bearing minerals Ca-Fe arsenates such as yukonite and arseniosiderite are important constituents of mine wastes resulting from the processing of some gold and base metal ores and neutralisation of acidic effluents (Paktunc et al., 2015). Both of them minerals occur as secondary alteration products in relation to the oxidation of arsenopyrite and arsenian pyrite and as discrete grains in mine tailings and contaminated soils. Although the understanding of the conditions promoting their formation and dissolution is limited, the study by Paktunc et al., 2015 found both minerals readily form from solutions with highly variable compositions at a wide pH range from slightly acidic to alkaline conditions. A mixture of synthesis and calorimetric experiments were carried out to form arseniosiderite and yukonite. It was found that the formation of arseniosiderite is kinetically controlled with faster development of crystallinity at neutral to slightly acidic pH and slower kinetics under alkaline conditions. The resultant log Ksp values for arseniosiderite and yukonite are -21.6 and 6.88 respectively.

Geminite and pushcharovskite are Cu arsenates and are commonly distributed minerals in oxidised ore deposits. Geminite was reported as a new species by Sarp and Perroud (1990) on material from Cap Garonne (Var, France) as Cu(AsO3OH)(H2O). Pushcharovskite was described later by Sarp and Sanz-Gysler (1997) on material from the same site with the same ideal chemical composition and space group P-1. They concluded that pushcharovskite is a polymorph of geminite and structurally related to yvonite (Cu.AsO3OH)(H2O).H2O (Plumhoff et al., 2020). Their thermodynamic properties were determined by a combination of acid-solution

Water data analysis

Results were produced in an Excel Spreadsheet format. For determination of reproducibility, the limit of detections (LODs) was calculated using the blank and the standard solution data.

Comparison of the LODs with the field sample data-set found a majority of values were above the LOD. The average LOD from the ICP-AES and ICP-MS were under 0.1 ug/L, indicating high precision measurements were made for most elements. In addition, the ICP-MS data produced and adapted in this study were found to have differences of lower than 15% compared against the certified water reference material (TMDA 70). The resulting data were plotted in respect of the distance from the mine tailings pond and along the Carnon RiverError! Reference source not found..

Supplementary Information D

XRD Analysis Methodology

Samples were prepared by side-filling the powder, against a ground-glass surface, into a rectangular sample holder 1.5 mm deep and consolidating it by gentle tapping. X-ray powder diffraction data were then collected with Cobalt (Co) Potassium (K) alpha (α) radiation, using a PANalytical X'Pert Pro diffractometer, with Bragg-Brentano para-focusing reflection geometry. This instrument is equipped with a Ge(1 1 1) Johansson geometry focusing monochromator, producing a CoKα1 incident beam. The X-ray tube was operated at 40 kV and 30 mA. Variable-width divergence and anti-scatter slits were used in the incident-beam and diffracted-beam optics, together with a 15 mm wide beam mask in the incident beam, so as to illuminate a constant 12 x 15 mm area of the sample; 0.04 radian Soller slits were present in both the incident and diffracted beams to reduce the axial divergences. The X-ray detector was an "X'Celerator" position-sensitive detector; this device covers an angular range in 2θ of ±1.061 degrees, with an effective fixed step size of 0.0167 degrees. The minimum tip width for the searched peaks is 0.01, which corresponded to a relative error of 1%, the counting statistical error. The PANalytical HighScore Plus software was then used to convert the data to fixed-divergence slit geometry, determine the peak positions and intensities and determine the phases present in the sample by comparison with the ICDD (International Centre for Diffraction Data) database.

Supplementary Information E


Vivien Beuselinck
email: vivien@beuselinck.fr

Jérôme Delobelle
email: jerome.delobelle@u-paris.fr

Srdjan Vesic
email: vesic@cril.fr

On Restricting the Impact of Self-Attacking Arguments in Gradual Semantics

Keywords: Abstract argumentation, Gradual semantics, Self-attack. 4

The issue of how a semantics should deal with self-attacking arguments was always a subject of debate amongst argumentation scholars. A consensus exists for extension-based semantics because those arguments are always rejected (as soon as the semantics in question respect conflict-freeness). In case of gradual semantics, the question is more complex, since other criteria are taken into account. A way to check the impact of these arguments is to use the principles (i.e. desirable properties to be satisfied by a semantics) from the literature. Principles like Self-Contradiction and Strong Self-Contradiction prescribe how to deal with self-attacking arguments. We show that they are incompatible with the well-known Equivalence principle (which is satisfied by almost all the existing gradual semantics), as well as with some other principles (e.g. Counting). This incompatibility was not studied until now and the class of semantics satisfying Self-Contradiction is under-explored. In the present paper, we explore that class of semantics. We show links and incompatibilities between several principles. We define a semantics that satisfies (Strong) Self-Contradiction and a maximal number of compatible principles. We introduce an iterative algorithm to calculate our semantics and prove that it always converges. We also provide a characterisation of our semantics. Finally, we experimentally show that our semantics is computationally efficient.

Introduction

Theory of computational argumentation allows to model exchange of arguments and conflicts between them. Although in most cases a conflict occurs between two arguments, sometimes an argument may conflict with itself. Such an argument is called a self-attacking argument. Discussion on how to deal with self-attacking arguments is often indirectly included in the problems of dealing with odd-length cycles, because a self-attack is the smallest odd-length cycle. However, in contrast to greater odd-length cycles, the presence of a self-attack is due to inconsistency in an argument itself.

In order to reason in presence of these arguments, several methods have been defined in abstract argumentation by proposing to deal with them directly [START_REF] Bodanza | Two approaches to the problems of self-attacking arguments and general odd-length cycles of attack[END_REF][START_REF] Baumann | Revisiting the foundations of abstract argumentation -semantics based on weak admissibility and weak defense[END_REF][START_REF] Baumann | Comparing weak admissibility semantics to their Dung-style counterparts -reduct, modularization, and strong equivalence in abstract argumentation[END_REF][START_REF] Dauphin | A principle-based analysis of weakly admissible semantics[END_REF] or indirectly [START_REF] Baroni | Solving semantic problems with odd-length cycles in argumentation[END_REF]. These methods essentially concern extension-based semantics. In the context of ranking-based and gradual argumentation semantics [START_REF] Amgoud | Ranking-based semantics for argumentation frameworks[END_REF][START_REF] Amgoud | Gradual semantics accounting for varied-strength attacks[END_REF], little research was conducted to find out how self-attacking arguments should be dealt with and what is the impact they have on the acceptability of other arguments. Existing studies are essentially done through the principle-based study of these semantics. Indeed, defining and studying principles drew attention of many scholars in this area.

Consider Equivalence, which is one of the well-known principles, stating that the acceptability degree of an argument should only depend on acceptability degrees of its direct attackers and consider the argumentation graph F ex containing two arguments a and b, and where b is attacked by a self-attacking argument a (i.e., F ex = ({a, b}, {(a, a), (a, b)})). Equivalence implies that a and b should be equally acceptable because a and b are both attacked by a self-attacking argument. However, this is debatable, since the intuition behind a self-attacking argument is that it is inconsistent in one way or another so we would tend to accept b being attacked by a (which is selfattacking) rather than accepting a. Note that, under all semantics returning conflict-free extensions, a self-attacking argument is always rejected, i.e. it does not belong to any extension. Also, regarding the ranking-based and gradual semantics, it was pointed out that it would be natural to attach the worst possible rank to self-attacking arguments [START_REF] Matt | A game-theoretic measure of argument strength for abstract argumentation[END_REF]. Furthermore, two principles were defined to formalise this intuition.

The first one is called Strong Self-Contradiction, and introduced by Matt and Toni [START_REF] Matt | A game-theoretic measure of argument strength for abstract argumentation[END_REF]. It says that the acceptability degree of an argument must be 0 if and only if that argument is self-attacking. The second principle, called Self-Contradiction, was introduced by Bonzon et al. [START_REF] Bonzon | A Comparative Study of Rankingbased Semantics for Abstract Argumentation[END_REF] and states that every self-attacking argument is strictly less acceptable than every non self-attacking argument. Consider the argumentation graph F ex again and note that, under every semantics that satisfies Self-Contradiction, b is strictly more acceptable than a. This example shows that Equivalence and Self-Contradiction are not compatible, i.e. there exists no semantics that satisfies both of them.

To the best of our knowledge, there exists only one semantics (known as M&T) that satisfies Self-Contradiction and Strong Self-Contradiction. That semantics was introduced by Matt and Toni [START_REF] Matt | A game-theoretic measure of argument strength for abstract argumentation[END_REF]. However, this semantics has a limitation that makes it inapplicable in practice. Namely, as noted by Matt and Toni themselves, as the space used to calculate the scores grows exponentially with the number of arguments, even with the optimisation techniques they used it did not scale to more than a dozen of arguments.

The research objective of the present paper is to study the under-explored family of semantics that satisfy Strong Self-Contradiction. Our goals are thus to identify which principles are (in)compatible with Strong Self-Contradiction and to define a semantics, which we call nsa (no self-attacks), that satisfies Strong Self-Contradiction as well as a maximal number of compatible principles. After introducing the formal setting and recalling the existing principles from the literature:

-We prove the incompatibilities between some of the principles, and identify a maximal set of principles that contains (Strong) Self-Contradiction;

-We introduce an iterative algorithm in order to define a new semantics and prove that it always converges. The acceptability of degree of each argument with respect to nsa is then defined as the limit of the corresponding sequence; -We provide a characterisation of nsa, i.e. a declarative (non-iterative) definition and show that the two are equivalent: each semantics satisfying the declarative definition coincides with nsa; -We check which principles are satisfied by nsa and compare it with the h-categorizer semantics [START_REF] Besnard | A logic-based theory of deductive arguments[END_REF] and the M&T semantics in terms of principle satisfaction; -We formally prove that no semantics can satisfy a strict super-set of the set of principles satisfied by nsa; -We experimentally show that nsa is computationally efficient and compare it with the M&T semantics and the h-categorizer semantics. The results confirm the hypothesis that the M&T semantics does not scale.

Formal Setting and Existing Semantics

An argumentation graph (AG) [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] Dung's framework comes equipped with various types of semantics used to evaluate the arguments. These include the extension-based semantics (see [START_REF] Baroni | An introduction to argumentation semantics[END_REF] for an overview), the labelling-based semantics [START_REF] Caminada | On the issue of reinstatement in argumentation[END_REF], the ranking-based semantics (see [START_REF] Bonzon | A Comparative Study of Rankingbased Semantics for Abstract Argumentation[END_REF] for an overview) and the gradual semantics. We refer the reader to [START_REF] Bonzon | Combining extension-based semantics and ranking-based semantics for abstract argumentation[END_REF][START_REF] Amgoud | A replication study of semantics in argumentation[END_REF] for a complete overview of the existing families of semantics in abstract argumentation and the differences between these approaches (e.g., definition, outcome, application). In this article, we focus on gradual semantics which assign to each argument in an argumentation graph a score, called acceptability degree. This degree belongs to the interval [0, 1]. Higher degrees correspond to stronger arguments.

Definition 1 (Gradual semantics).

A gradual semantics is a function S which associates to any argumentation graph

F = (A, R) a function Deg S F : A → [0, 1]. Thus, Deg S F (x)
represents the acceptability degree of x ∈ A.

In the rest of the section we recall two gradual semantics. We first introduce hcategorizer, which is one of the most studied gradual semantics and also satisfies a maximal compatible set of principles from the literature. 5 Then we introduce M&T semantics which is, to the best of our knowledge, the only semantics known in the literature to satisfy Self-Contradiction.

h-categorizer Semantics

The h-categorizer semantics [START_REF] Besnard | A logic-based theory of deductive arguments[END_REF][START_REF] Pu | Argument ranking with categoriser function[END_REF] uses a categorizer function to assign a value to each argument by taking into account the strength of its attackers, which itself takes into account the strength of its attackers, and so on.

Definition 2 (h-categorizer semantics). Let F = (A, R) be an argumentation graph. The h-categorizer semantics is a gradual semantics such that ∀x ∈ A:

Deg h F (x) = 1 1 + y∈Att F (x) Deg h F (y)

M&T Semantics

The gradual semantics introduced by Matt and Toni [START_REF] Matt | A game-theoretic measure of argument strength for abstract argumentation[END_REF] computes the acceptability degree of an argument using a two-person zero-sum strategic game. For an AG F = (A, R) and an argument x ∈ A, the set of strategies for the proponent is the set of all subsets of arguments that contain x: S P (x) = {P | P ⊆ A, x ∈ P } and for the opponent it is the set of all subsets of arguments:

S O = {O | O ⊆ A}. Given two strategies X, Y ⊆ A, the set of attacks from X to Y is defined by Y ←X F = {(x, y) ∈ X ×Y | (x, y) ∈ R}.
From this measurement, Matt and Toni define the notion of degree of acceptability of a set of arguments w.r.t. another one used to compute the reward of a proponent's strategy. Definition 3 (Reward). Let F = (A, R) be an argumentation graph, x ∈ A be an argument, P ∈ S P (x) be a strategy chosen by the proponent and O ∈ S O be a strategy chosen by the opponent. The degree of acceptability of P w.r.t. O is φ(P, O) =

1 2 1 + f (|O ←P F |) -f (|P ←O F |) with f (n) = n n+1
. The reward of P over O, denoted by r F (P, O), is defined by:

r F (P, O) =        0 iff P is not conflict-free 1 iff P is conflict-free and |P ←O F | = 0 φ(P, O) otherwise
Proponent and opponent have the possibility of using a strategy according to some probability distributions, respectively p = (p 1 , p 2 , . . . , p m ) and q = (q 1 , q 2 , . . . , q n ), with m = |S P | and n = |S O |. For each argument x ∈ A, the proponent's expected payoff E(x, p, q) is then given by E(x, p, q) = n j=1 m i=1 p i q j r i,j with r i,j = r F (P i , O j ) where P i (respectively O j ) represents the i th (respectively j th ) strategy of S P (x) (respectively S O ). The proponent can expect to get at least min q E(x, p, q), where the minimum is taken over all the probability distributions q available to the opponent. Hence the proponent can choose a strategy which will guarantee her a reward of max p min q E(x, p, q). The opposite is also true with min q max p E(x, p, q). Definition 4 (M&T semantics). The semantics M&T is a gradual semantics that assigns a score to each argument x ∈ A in F as follows:

Deg MT F (x) = max p min q E(x, p, q) = min q max p E(x, p, q)
3 Principles for Gradual Semantics

Principles have been introduced by [START_REF] Amgoud | Acceptability semantics for weighted argumentation frameworks[END_REF] in order to better understand the behavior of the gradual semantics, choose a semantics for a particular application, guide the search for new semantics, compare semantics with each other, etc. We do not claim that all of these principles are mandatory (we will see later that some of them are incompatible).

In the rest of this section, we introduce the principles. 6The first one, called Anonymity, states that the name of an argument should not impact its acceptability degree.

Principle 1 (Anonymity) A semantics S satisfies Anonymity iff for any two AGs F = (A, R) and

F = (A , R ) for any isomorphism f from F to F , ∀a ∈ A, Deg S F (a) = Deg S F (f (a)).
Independence says that the acceptability degree of an argument should be independent of unconnected arguments.

Principle 2 (Independence) A semantics S satisfies Independence iff, for any two AGs F = (A, R) and

F = (A , R ) such that A ∩ A = ∅, ∀a ∈ A, Deg S F (a) = Deg S F ⊗F (a).
Directionality states that the acceptability of argument x can depend on y only if there is a path from y to x. Neutrality states that an argument with an acceptability degree of 0 should have no impact on the arguments it attacks.

Principle 4 (Neutrality) A semantics S satisfies Neutrality iff, for any AG

F = (A, R) if ∀a, b ∈ A, Att F (b) = Att F (a) ∪ {x} with x ∈ A \Att F (a) and Deg S F (x) = 0 then Deg S F (a) = Deg S F (b).
Equivalence says that if two arguments have the same attackers, or more generally attackers of the same strength, they should have the same acceptability degree.

Principle 5 (Equivalence) A semantics S satisfies Equivalence iff, for any AG

F = (A, R), ∀a, b ∈ A, if there exists a bijective function f from Att F (a) to Att F (b) s.t. ∀x ∈ Att F (a), Deg S F (x) = Deg S F (f (x)) then Deg S F (a) = Deg S F (b).
Maximality states that a non-attacked argument should have the highest acceptability degree.

Principle 6 (Maximality) A semantics S satisfies Maximality iff, for any AG

F = (A, R), ∀a ∈ A, if Att F (a) = ∅ then Deg S F (a) = 1.
Counting states that a non-zero degree attacker should impact the acceptability of the attacked argument.

Principle 7 (Counting) A semantics S satisfies Counting iff for any AG

F = (A, R), ∀a, b ∈ A, if i) Deg S F (a) > 0 and ii) Att F (b) = Att F (a) ∪ {y} with y ∈ A\Att F (a) and Deg S F (y) > 0 then Deg S F (a) > Deg S F (b).
Weakening says that the acceptability of an argument should be strictly lower than 1 if it has at least one attacker with a non-zero acceptability degree.

Principle 8 (Weakening) A semantics S satisfies Weakening iff for any AG

F = (A, R), ∀a ∈ A, if ∃b ∈ Att F (a) s.t. Deg S F (b) > 0, then Deg S F (a) < 1.
Weakening Soundness states that if the acceptability degree of an argument is not maximal, it must be that it is attacked by at least one non-zero degree attacker.

Principle 9 (Weakening Soundness) A semantics S satisfies Weakening Soundness iff, for any AG

F = (A, R), ∀a ∈ A, if Deg S F (a) < 1 then ∃b ∈ Att F (a) such that Deg S F (b) > 0.
Reinforcement states that the acceptability degree increases if the acceptability degrees of attackers decrease.

Principle 10 (Reinforcement) A semantics S satisfies Reinforcement iff for any AG

F = (A, R), ∀a, b ∈ A, if i) Deg S F (a) > 0 or Deg S F (b) > 0, ii) Att F (a)\Att F (b) = {x}, iii) Att F (b)\Att F (a) = {y} and iv) Deg S F (y) > Deg S F (x), then Deg S F (a) > Deg S F (b).
Resilience states that no argument in an argumentation graph can have a acceptability degree of 0. It is certainly not a mandatory principle.

Principle 11 (Resilience) A semantics S satisfies Resilience if for any AG

F = (A, R), ∀a ∈ A, Deg S F (a) > 0.
The last three principles are incompatible with each other. The first principle, called Cardinality Precedence states, roughly speaking, that the greater the number of direct attackers of an argument, the lower its acceptability degree.

Principle 12 (Cardinality Precedence) A semantics S satisfies Cardinality Precedence iff for any AG

F = (A, R), ∀a, b ∈ A, if i) Deg S F (b) > 0, and ii) |{x ∈ Att F (a) s.t. Deg S F (x) > 0}| > |{y ∈ Att F (b) s.t. Deg S F (y) > 0}| then Deg S F (a) < Deg S F (b).
Quality Precedence states, roughly speaking, that the greater the acceptability degree of the strongest attacker of an argument, the lower its acceptability degree.

Principle 13 (Quality Precedence) A semantics S satisfies Quality Precedence if for any AG F

= (A, R), ∀a, b ∈ A, if i) Deg S F (a) > 0 and ii) ∃y ∈ Att F (b) s.t. ∀x ∈ Att F (a), Deg S F (y) > Deg S F (x) then Deg S F (a) > Deg S F (b).
Compensation states that several attacks from arguments with a low acceptability degree may compensate one attack from an argument with high acceptability degree. 7Principle 14 (Compensation) A semantics S satisfies Compensation iff both Cardinality Precedence and Quality Precedence are not satisfied.

In the literature, two principles directly refer to the self-attacking arguments. The first one, called Self-Contradiction, was introduced by [START_REF] Bonzon | A Comparative Study of Rankingbased Semantics for Abstract Argumentation[END_REF] and states that the degree of a self-attacking argument should be strictly lower than the degree of an argument that does not attack itself.

Principle 15 (Self-Contradiction) A semantics S satisfies Self-Contradiction iff, for any AG

F = (A, R) with two arguments a, b ∈ A, if (a, a) ∈ R and (b, b) / ∈ R then Deg S F (b) > Deg S F (a).
The second principle was introduced by Matt and Toni [START_REF] Matt | A game-theoretic measure of argument strength for abstract argumentation[END_REF]. Its original name was "Self-contradiction must be avoided". We rename it for clarity reasons, namely in order to avoid the confusion with the name of Principle 15. This principle states that an argument that attacks itself should have the smallest acceptability degree (i.e. 0).

Principle 16 (Strong Self-Contradiction

) A semantics S satisfies Strong Self-Contra- diction iff, for any AG F = (A, R) with a ∈ A, Deg S F (a) = 0 iff (a, a) ∈ R.

Analysis of Principles and Links Between Them

In this section we analyse the links between principles and identify two maximal mutually compatible sets of principles. Let us first observe that Strong Self-Contradiction implies Self-Contradiction. The next proposition follows directly from the definitions of the respective principles.

Proposition 1. If a gradual semantics S satisfies Strong Self-Contradiction, it satisfies Self-Contradiction.

Proof. Let us suppose that Strong Self-Contradiction is satisfied by S. This means that those and only those arguments that have the minimum score are the self-attacking arguments (∀a ∈ A, Deg S F (a) = 0 iff (a, a) ∈ R). This implies that all arguments that do not attack themselves have an acceptability degree greater than 0. Formally, ∀b ∈ A,

Deg S F (b) > 0 iff (b, b) / ∈ R. Consequently, for two arguments a, b ∈ A, if (a, a) ∈ R and (b, b) / ∈ R then Deg S F (b) > Deg S F (a) = 0.
As discussed in the introduction, the next result shows that Equivalence and Self-Contradiction are incompatible. Proposition 2. There exists no gradual semantics S that satisfies both Equivalence and Self-Contradiction.

Proof. We provide a proof by contradiction. Let us suppose that a gradual semantics S satisfies both Equivalence and Self-Contradiction and consider the argumentation graph F = (A, R) with A = {a, b} and R = {(a, a), (a, b)}. From Self-Contradiction, we have Deg S F (a) < Deg S F (b), while from Equivalence, we have Deg S F (a) = Deg S F (b). Contradiction. Hence, S does not satisfy both Equivalence and Self-Contradiction. Since S was arbitrary, we conclude that there exists no semantics that satisfies both Equivalence and Self-Contradiction.

However, the Equivalence principle is not the only one incompatible with Strong Self-Contradiction. Some other incompatibilities exist mainly because self-attacking arguments are treated differently from other arguments. Indeed, according to Strong Self-Contradiction, self-attacking arguments are directly classified as the worst arguments, whereas the other principles just consider a self-attack as an attack like any other (i.e. an attack between two distinct arguments). Proposition 3. There exists no gradual semantics S that satisfies both Strong Self-Contradiction and Resilience.

Proof. We provide a proof by contradiction. Let us suppose that a gradual semantics S satisfies both Strong Self-Contradiction and Resilience, and consider the argumentation graph F = (A, R) where A = {a} and R = {(a, a)}. From Strong Self-Contradiction, we have Deg S F (a) = 0, while from Resilience, we have Deg S F (a) > 0. Contradiction. Hence, S does not satisfy both Strong Self-Contradiction and Resilience. Since S was arbitrary, there exists no semantics that satisfies both Resilience and Strong Self-Contradiction. Proposition 4. There exists no gradual semantics S that satisfies both Strong Self-Contradiction and Weakening Soundness.

Proof. We provide a proof by contradiction. Let us suppose that a gradual semantics S satisfies both Strong Self-Contradiction and Weakening Soundness, and consider the argumentation graph F = (A, R) where A = {a} and R = {(a, a)}. From Strong Self-Contradiction, we have Deg S F (a) = 0, while from Weakening Soundness, we have Deg S F (a) > 0 because a is the only attacker of a and Deg S F (a) = 0. Contradiction. Hence, S does not satisfy both Strong Self-Contradiction and Weakening Soundness. Since S was arbitrary, there exists no semantics that satisfies both Strong Self-Contradiction and Weakening Soundness. Proposition 5. There exists no gradual semantics S that satisfies both Strong Self-Contradiction and Reinforcement.

Proof. We provide a proof by contradiction. Let us suppose that a gradual semantics S satisfies both Strong Self-Contradiction and Reinforcement, and consider the argumentation graph F = (A, R) represented in Figure 1. Taking these incompatibilities into account, our goal is now to study two maximal mutually compatible sets of principles we are interested in. For this, we need the notion of dominance. A semantics S dominates a semantics S on the set of principles P if the subset of principles from P satisfied by S is a strict superset of the subset of principles from P satisfied by S . In the rest of the discussion, we suppose that P is the set of all principles studied in Section 3. Note that if a semantics S satisfies a maximal for set inclusion set of principles, it is not dominated by any semantics.

A first maximal (for set inclusion) set of principles has been identified by [START_REF] Amgoud | Acceptability semantics for weighted argumentation frameworks[END_REF] and is a direct consequence of their Proposition 1. We define this set of principles as P CREW = {Anonymity, Independence, Directionality, Neutrality, Equivalence, Maximality, Weakening, Counting, Weakening Soundness, Reinforcement, Resilience and Compensation}.

Theorem 1 ([4]

). P CREW is a maximal for set inclusion set of principles.

We can formally show that there is a unique maximal set of principles compatible with Compensation, Resilience, Equivalence and Weakening Soundness.

Theorem 2. Let P be the set of all principles defined in Section 3 (Principles 1-16). Let S be a gradual semantics that satisfies Compensation, Resilience, Equivalence and Weakening Soundness. If S is not dominated w.r.t. P , then S satisfies exactly the principles from P CREW .

Proof. On one hand, we know from the work by [START_REF] Amgoud | Acceptability semantics for weighted argumentation frameworks[END_REF] that h-categorizer satisfies all the principles from P CREW . On the other hand, it is clear from the incompatibility results between the principles that S cannot satisfy Strong Self-Contradiction which is incompatible with Resilience (see Proposition 3), Self-Contradiction which is incompatible with Equivalence (see Proposition 2), Cardinality/Quality Precedence which are both incompatible with Compensation (see [START_REF] Amgoud | Acceptability semantics for weighted argumentation frameworks[END_REF]). Thus, in order not to be dominated by hcategorizer, S must satisfy all the principles from P CREW ; due to the incompatibilities, S cannot satisfy any more principles.

In this paper we choose to explore the space of principles compatible with Strong Self-Contradiction (which is not in P CREW ). One naturally wants to maximise the set of satisfied principles. Can we satisfy Strong Self-Contradiction and all the other principles? The answer is negative (see Propositions 2-6). First, one has to choose between Cardinality Precedence, Quality Precedence and Compensation. In this paper, we explore the possibility of satisfying Compensation. This choice is based on the fact that this principle is satisfied by virtually all semantics, as showed by Amgoud et al. [START_REF] Amgoud | Acceptability semantics for weighted argumentation frameworks[END_REF]. Indeed, Cardinality Precedence and Quality Precedence represent, roughly speaking, drastic or extreme cases and are satisfied only by the semantics specifically designed to satisfy them, like max-based semantics and card-based semantics [START_REF] Amgoud | Acceptability semantics for weighted argumentation frameworks[END_REF] or by semantics having other specificities. For instance, iterative schema [START_REF] Gabbay | Equilibrium states in numerical argumentation networks[END_REF], which satisfies Quality Precedence, is a discrete semantics (it takes only three possible values). This yields another maximal set of principles which includes those two principles. We define this set of principles as P 2S2C = {Anonymity, Independence, Directionality, Maximality, Weakening, Counting, Compensation, Self-Contradiction, Strong Self-Contradiction}. Theorem 3. P 2S2C is a maximal for set inclusion set of principles.

Proof. Firstly, all the principles in P 2S2C are compatible because nsa satisfies all of them (see Proposition 7). Secondly, P 2S2C is maximal because for each remaining principle p ∈ {Equivalence, Weakening Soundness, Neutrality, Reinforcement, Cardinality Precedence, Quality Precedence and Resilience}, there exists (at least) one principle in P 2S2C which is incompatible with p:

-Equivalence and Self-Contradiction are incompatible (see Proposition 2); -Neutrality and Strong Self-Contradiction are incompatible (see Proposition 6); -Reinforcement and Strong Self-Contradiction are incompatible (see Proposition 5); -Weakening Soundness and Strong Self-Contradiction are incompatible (see Proposition 4); -Cardinality Precedence and Compensation are incompatible (see [START_REF] Amgoud | Acceptability semantics for weighted argumentation frameworks[END_REF]); -Quality Precedence and Compensation are incompatible (see [START_REF] Amgoud | Acceptability semantics for weighted argumentation frameworks[END_REF]); -Resilience and Strong Self-Contradiction are incompatible (see Proposition 3);

We now show that there is a unique maximal set of principles compatible with Strong Self-Contradiction and Compensation. This follows from the fact that if a semantics satisfies Strong Self-Contradiction, it cannot satisfy several principles (see Propositions 2-6) but can satisfied all the others (as witnessed by the semantics we introduce in this paper).

Theorem 4. Let P be the set of all principles defined in Section 3 (Principles 1-16). Let S be a gradual semantics that satisfies Strong Self-Contradiction and Compensation. If S is not dominated w.r.t. P , then S satisfies exactly the principles from P 2S2C .

Proof. It is clear that from the incompatibility results between different principles, S cannot satisfy (i) Resilience, Equivalence and Weakening Soundness which are incompatible with Strong Self-Contradiction (or Self-Contradiction), and (ii) Cardinality Precedence and Quality Precedence which are both incompatible with Compensation. The set of remaining principles corresponds exactly to P 2S2C which is a maximal for set inclusion set of principles. However, S cannot satisfy exactly a subset of P 2S2C because, in this case, S will be dominated by a semantics that satisfies the principles of P 2S2C . Consequently, when S satisfies Strong Self-Contradiction and Compensation, the only way to ensure that S is not dominated is when S satisfies exactly the principles from P 2S2C .

To the best of our knowledge, no semantics that satisfy all the principles from P 2S2C has been presented in the literature. In the next section, we define a semantics that satisfies this set of principles.

Before doing that, let us comment on the non satisfaction of some principles. It is tempting to change the principles in order to treat the self-attacks in another way, and consequently make the principles fit some definitions or theorems. We argue that it is better to start by having a full picture of what happens with existing principles. Indeed, the principles should be the most stable part of a theory. We are not against introduction of new principles (or changing the existing ones). This might be part of future work.

No Self-Attack h-categorizer Semantics

In this section, we define a new gradual semantics, called no self-attack h-categorizer (nsa) semantics, inspired by the h-categorizer semantics. The main difference is that we assign 0 degree to the self-attacking arguments.

Definition 5 (nsa). Let F = (A, R) be an AG. We define f F ,i nsa : A → [0, +∞] as follows : for every argument a ∈ A for i ∈ {0, 1, 2, ..},

f F ,i nsa (a) =          0 if (a, a) ∈ R 1 if (a, a) / ∈ R and i = 0 1 1 + b∈Att F (a) f F ,i-1 nsa (b)
if (a, a) / ∈ R and i > 0

(1)

By convention, if Att F (a) = ∅, b∈Att F (a) f F ,i-1 nsa (b) = 0.
Although nsa is inspired by the h-categorizer semantics, the modifications made change the result obtained requiring the verification that nsa also converges to a unique result. Thus, in the next result, we show that for every argumentation graph F = (A, R), for every argument a ∈ A, f F ,i nsa (a) converges as i approaches infinity. Roughly speaking, the goal of the next theorem is to formally check that assigning zero values to self-attacking arguments does not impact the convergence of the scores. Thus, applying nsa to the original argumentation graph F provides the same result as when the h-categorizer semantics is applied on a restricted version of F where the self-attacking arguments are deleted.

Theorem 5. For every argumentation graph

F = (A, R), for every a ∈ A, if (a, a) / ∈ R, we have lim i→∞ f F ,i nsa (a) = Deg h F (a) where F = (A , R ) with A = {x ∈ A| (x, x) / ∈ R} and R = {(x, y) ∈ R | x ∈ A and y ∈ A }.
Proof. Let F = (A, R) be an AG and F = (A , R ) be an AG such that A = {x ∈ A|(x, x) / ∈ R} and R = {(x, y) ∈ R | x ∈ A and y ∈ A }. Without loss of generality, let us denote A = {a 0 , a 1 , . . . , a n }.

Let us recall the iterative version of h-categorizer, that can be used to calculate the scores of arguments [START_REF] Pu | Argument ranking with categoriser function[END_REF]: for every a, for i ∈ N

f F ,i h (a) =      1 if i = 0 1 1 + b∈Att F (a) f F ,i-1 h (b) if i > 0 (2)
We prove by induction on i that for each a ∈ A :

f F ,i nsa (a) = f F ,i h (a)
Base: Let i = 0. From the formal definition of nsa (Definition 5) and equation (2), we have f F ,0 nsa (a) = f F ,0 h (a) = 1. Thus, the inductive base holds.

Step: Let us suppose that the inductive hypothesis is true for every k ∈ {0, 1, . . . i} and let us show that it is true for i + 1. We need to prove :

f F ,i+1 nsa (a) = f F ,i+1 h (a)
From the inductive hypothesis, we know that for each argument a ∈ A , f F ,i nsa (a) = f F ,i h (a). Thus, from equation (1), we have:

f F ,i+1 nsa (a) = 1 1 + b∈Att F (a) f F ,i nsa (b)
From equation ( 2), we have

f F ,i+1 h (a) = 1 1 + b∈Att F (a) f F ,i h (b) Let us note Att F (a) = Att F (a) ∪ {b 0 , . . . , b m } with m ≥ 0 and remark that ∀b ∈ {b 0 , . . . , b m }, we have (b, b) ∈ R. According to equation (1), ∀b ∈ {b 0 , . . . , b m }, f F ,i nsa (b) = 0.
Consequently, as 0 is the neutral element of the addition, we have ∀a ∈ A ,

f F ,i+1 nsa (a) = f F ,i+1 h (a)
. By induction, we conclude that for every i ∈ N and for every a ∈ A

f F ,i nsa (a) = f F ,i h (a)
Since f h converges when i → ∞ and f nsa coincides with f h for every argument of A , we conclude that f nsa converges too. Formally, ∀a ∈ A ,

lim i→∞ f F ,i nsa (a) = lim i→∞ f F ,i h (a) = Deg h F (a)
We can now introduce the formal definition of nsa. We can now show that the acceptability degrees attributed to arguments by nsa satisfy the equation from Definition 5 (naturally, not taking into account the second line of the equation, since it considers the case i = 0). Theorem 6. For any F = (A, R), for any a ∈ A,

Deg nsa F (a) =    0 if (a, a) ∈ R 1 1 + b∈Att F (a) Deg nsa F (b) otherwise Proof. Let F = (A, R
) be an argumentation graph and a ∈ A.

The case where a is a self-attacking argument is trivial.

In the rest of the proof we consider the case where a is not a self-attacking argument. Letting lim i→∞ in the following equality

f i+1 nsa (a) = 1 1 + b∈Att F (a) f i nsa (b)
and using the fact that arithmetical operations and sum are continuous functions, we obtain :

lim i→∞ f i+1 nsa (a) = 1 1 + b∈Att F (a) lim i→∞ f i nsa (b) then Deg nsa F (a) = 1 1 + b∈Att F (a) Deg nsa F (a)
We now show that the equation from Theorem 6 is not only satisfied by nsa, but is also its characterization. More precisely, the next result proves that if an arbitrary semantics D satisfies that equation, it must be that D coincides with nsa.

Theorem 7. Let F = (A, R) be an AG with a ∈ A and D : A → [0, 1] be a function with the following formula:

D(a) =      0 if (a, a) ∈ R 1 1 + b∈Att F (a) D(b) otherwise (3)
then D ≡ Deg nsa F . Proof. Let F = (A, R) be an AG and suppose that D : A → [0, 1] is the function from equation (3).

Let A = {a 1 , .., a n } and let F : [0, 1] n → [0, 1] n be the function such that F (x 1 , .., x n ) = (F 1 (x 1 , .., x n ), ..., F n (x 1 , ..., x n )) where the functions F i are defined by the following equality:

F i (x 1 , . . . , x n ) =        0 if (a i , a i ) ∈ R 1 1 + j:aj ∈Att F (ai) x j otherwise (4)
We also define the partial order ≤ on R n in the following way: if x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) then x ≤ y iff for every i it holds that x i ≤ y i .

Thus, from Equation (3), it follows that

F (D(a 1 ), ..., D(a n )) = (D(a 1 ), ..., D(a n )).
Observe that F is a non-increasing function and that G = F • F is a non-decreasing function, and that :

(f i+1 nsa (a 1 ), ..., f i+1 nsa (a n )) = F ((f i nsa (a 1 ), ..., f i nsa (a n ))) for every i ∈ N. Since (f 0 nsa (a 1 ), ..., f 0 nsa (a n )) ∈ [0, 1] n with f 0 nsa (a i ) = 0 iff (a i , a i ) ∈ R and f 0
nsa (a i ) = 1 otherwise, by the inequalities, we obtain (f 0 nsa (a 1 ), ..., f 0 nsa (a n )) ≥ (D(a 1 ), ..., D(a n ))

From [START_REF] Amgoud | Gradual semantics accounting for varied-strength attacks[END_REF], and since F is non-increasing, we have:

(f 1 nsa (a 1 ), ..., f 1 nsa (a n )) ≤ (D(a 1 ), ..., D(a n )) (6)
From [START_REF] Baroni | An introduction to argumentation semantics[END_REF], and since G = F • F is non-decreasing, we have:

(f 2i nsa (a 1 ), ..., f 2i nsa (a n )) ≥ (D(a 1 ), ..., D(a n )) (7)
and

(f 2i+1 nsa (a 1 ), ..., f 2i+1 nsa (a n )) ≤ (D(a 1 ), ..., D(a n )) (8)
for every i ∈ N. Since all f i converge, from [START_REF] Baroni | Solving semantic problems with odd-length cycles in argumentation[END_REF] an acceptability degree of 0 : Deg nsa F (a 0 ) = Deg nsa F (a 2 ) = 0. The non-attacked arguments or the arguments only attacked by self-attacking arguments have, by definition, the maximum score: Deg nsa F (a 5 ) = 1. Applying the formula from Theorem 6, we obtain the following acceptability degrees for a 1 and a 4 : Deg nsa F (a 1 ) = 0.732 and Deg nsa F (a 4 ) = 0.399. Finally, following the same method, here are the details concerning a 3 :

Deg nsa F (a 3 ) = 1 1 + Deg nsa F (a 1 ) + Deg nsa F (a 2 ) + Deg nsa F (a 4 ) = 1 1 + 0.732 + 0 + 0.399 = 0.477
In order to have an overview of the difference between nsa and the gradual semantics introduced in Section 2, the degrees of acceptability of arguments w.r.t. the h-categorizer semantics and the M&T semantics have also been added in the table of Fig. 3. This comparison clearly shows that nullifying the impact of self-attacking arguments more or less significantly changes the degree of acceptability of other arguments (e.g. a 1 and a 3 ).

Principle-Based Evaluation of Semantics

In this section we evaluate the nsa semantics with respect to principle compliance, and compare the results with two existing semantics, namely M&T and h-categorizer. We first show that nsa satisfies all the principles from P 2S2C , and thus cannot be dominated by any semantics.

Proposition 7. The gradual semantics nsa satisfies all the principles from P 2S2C . The other principles are not satisfied.

In order to axiomatically compare nsa with the two other gradual semantics, let us check for the principles studied in this paper those that are satisfied by M&T and recall those satisfied by the h-categorizer semantics.

Proposition 8. The gradual semantics M&T satisfies Anonymity, Independence, Directionality, Maximality, Weakening, Compensation, Self-Contradiction and Strong Self-Contradiction. The other principles are not satisfied.

Proposition 9 ([3]

). The gradual semantics h-categorizer satisfies all the principles from P CREW . The other principles are not satisfied.

Note that nsa dominates M&T, i.e. it satisfies strictly more principles. Observe that nsa and h-categorizer are incomparable in terms of principles satisfaction. Indeed, nsa represents one choice, i.e. the position to satisfy Strong Self-Contradiction and Compensation. It also satisfies all the compatible principles. h-categorizer represents another choice, namely that to satisfy Compensation, Resilience, Equivalence and Weakening Soundness. Concretely, a semantics satisfying P CREW considers that a self-attacking argument is a path like the other ones. So an argument which attacks itself (and is not attacked by any other argument) can be stronger than an argument which is attacked by several arguments. On the contrary, a semantics which satisfies P 2S2C considers that a self-attacking argument is intrinsically flawed, without even requiring other arguments to defeat it. Note that there exist other maximal sets of compatible principles, for example the one containing Resilience and Self-Contradiction. We leave a detailed study of these maximal sets of compatible principles for future work.

Experimental Results

We now empirically compare nsa with M&T and h-categoriser semantics. We consider a large experimental setting representing three different models used during the ICCMA competition (http://argumentationcompetition.org/) as a way to generate random argumentation graphs: i) the Erdös-Rényi model (ER) which generates graphs by randomly selecting attacks between arguments, ii) the Barabasi-Albert model (BA) which provides networks, called scale-free networks, with a structure in which some nodes have a huge number of links, but in which nearly all nodes are connected to only a few other nodes, and iii) the Watts-Strogatz model (WS) which produces graphs which have small-world network properties, such as high clustering and short average path lengths. The generation of these three types of AGs was done by the AFBench-Gen2 generator [START_REF] Cerutti | Afbenchgen2: A generator for random argumentation frameworks[END_REF]. We generated a total of 2160 AGs evenly distributed between the three models. For each model, the number of arguments varies among Arg = {5, 10, 15, 25, 50, 100, 250, 500} with 90 AGs for each of these values. The parameters used to generate graphs are as follows: for ER, 10 random instances for each (numArg, probAttacks) in Arg × {0.2, 0.3, . . . , 1}; for BA, 9 random instances for each (numArg, probCycles) in Arg×{0, 0.1, . . . , 0.9}; for WS, (numArg, probCycles, β, K) in Arg × {0.25, 0.5, 0.75} × {0, 0.25, 0.5, 0.75, 1} × {k ∈ 2N s.t. 2 ≤ k ≤ |Arg| -1}. We refer the reader to [START_REF] Cerutti | Afbenchgen2: A generator for random argumentation frameworks[END_REF] for the meaning of the parameters.

In order to compare the execution times of the three semantics studied in this paper, we have implemented them in C and ran the program on a cluster of identical computers with dual quad-core processors with 128 GB RAM. 8Figure 4 shows the average execution time obtained by each semantics for the instances classified according to the number of arguments. A first remark is that, unlike the other two semantics, the M&T semantics quickly explodes in time since it systematically reaches the timeout (900 seconds) when the number of arguments is greater than 15. A second remark is that, unsurprisingly, the nsa and h-categorizer semantics have very similar execution times for each of the instances. Figure 5 shows the difference between nsa and h-categorizer semantics more precisely. Moreover, they allow us to quickly compute (with an average smaller than one second) the degree of acceptability of each argument even for large AGs. Only a few very dense instances (i.e. those with a high probability of cycles) require between 1 and 2 seconds when numArg = 500.

Summary

We studied the question of the treatment of self-attacks by gradual semantics following a principle-based approach. We showed links and incompatibilities between principles, defined a new semantics called no self-attack h-categorizer semantics and proved that it dominates the only existing semantics satisfying Self-Contradiction principle. Moreover, we showed that our semantics satisfies a maximal possible amount of principles (i.e. no semantics satisfying Self-Contradiction can satisfy more principles) and is usable in practice as it returns results very quickly (on average less than 1 second) even on large and dense AGs. In addition to the future work already discussed in the paper, we think it would be interesting to extend the approach we used for the h-categorizer semantics to other gradual semantics (if possible). Finally, the work presented in this paper concerns "classic" argumentation graphs but one could naturally ask the same question about AGs containing more information (support relation, weight on arguments and/or attacks, etc.).

  is a directed graph F = (A, R) where A is a finite set of arguments and R a binary relation over A, i.e. R ⊆ A × A. For a, b ∈ A, (a, b) ∈ R means that a attacks b. The notation Att F (a) = {b | (b, a) ∈ R} represents the set of direct attackers of argument a. For two graphs F = (A, R) and F = (A , R ), we denote by F ⊗ F the argumentation graph F = (A ∪ A , R ∪ R ).

Principle 3 (

 3 Directionality) A semantics S satisfies Directionality iff, for any AG F = (A, R) and F = (A, R ) such that a, b ∈ A, R = R∪{(a, b)} it holds that : ∀x ∈ A, if there is no path from b to x, then Deg S F (x) = Deg S F (x).

Fig. 1 :Fig. 2 :

 12 Fig. 1: AG showing that Reinforcement and Strong Self-Contradiction are incompatible.

Definition 6 (

 6 nsa). The no self-attack h-categorizer semantics is a function nsa which associates to any argumentation framework F = (A, R) a function Deg nsa F (a) : A → [0, 1] as follows: Deg nsa F (a) = lim i→∞ f F ,i nsa (a).

Fig. 4 :

 4 Fig. 4: Execution speed for the nsa (in green), the M&T (in blue) and the h-categorizer (in red) semantics. x-axis shows the number of arguments of the instances (Arg = {5, 10, 15, 25, 50, 100, 250, 500}). y-axis shows the execution time in seconds (with a timeout of 900 seconds).

Fig. 5 :

 5 Fig. 5: A zoomed-in version of the graph from Figure 4 to better see the difference between the execution speed for the nsa semantics (in green) and the h-categorizer (in red) semantics.

  and (8) we obtain (Deg nsa F (a 1 ), . . . , Deg nsa F (a n )) ≥ (D(a 1 ), ..., D(a n )) and (Deg nsa F (a 1 ), . . . , Deg nsa F (a n )) ≤ (D(a 1 ), ..., D(a n )) and thus ∀a ∈ A, Deg nsa F(a) = D(a).Below is an example of the nsa semantics applied on an argumentation graph. Let us apply the no self-attack h-categorizer semantics (nsa) on the argumentation graph illustrated in Fig.3. By definition, the self-attacking arguments have On the left, an argumentation graph F and, on the right, the table containing the degrees of acceptability of each argument of F w.r.t. the no self-attack h-categorizer semantics (nsa), the h-categorizer semantics (h) and the semantics M&T (MT).

	Example 1 a0	a1	a2	Deg S F a0	nsa 0	h 0.618	MT 0
				a1	0.732	0.495	0.25
				a2	0	0.618	0
				a3	0.477	0.398	0.167
				a4	0.399	0.401	0.25
	a5	a4	a3	a5	1	1	1
	Fig. 3:

Table 1 :

 1 Principles satisfied by the M&T, h-categorizer and nsa semantics. The shaded cells contain the results already proved in the literature.

	Principles	M&T h-cat nsa
	Anonymity
	Independence
	Directionality
	Neutrality	×	×
	Equivalence	×	×
	Maximality
	Weakening
	Counting	×
	Weakening Soundness	×	×
	Reinforcement	×	×
	Resilience	×	×
	Cardinality Precedence	×	× ×
	Quality Precedence	×	× ×
	Compensation
	Self-Contradiction		×
	Strong Self-Contradiction		×

The final authenticated publication is available online at the following address : https://doi.org/10.1007/978-3-030-89391-0 8

formally: out of the principles from Section 3, no semantics satisfies a strict superset of the principles satisfied by h-categorizer.

We do not include the Proportionality principle since it is only applicable when arguments are attached intrinsic weights.

There are several version of this principle. We use the version that allows to clearly distinguish between the three cases (CP, QP, Compensation). Namely, each semantics satisfies exactly one of the three principles.

The code and benchmarks are available online at https://github.com/jeris90/nsa code.git.

Acknowledgements

Vivien Beuselinck was supported by the ANR-3IA Artificial and Natural Intelligence Toulouse Institute. Srdjan Vesic was supported by Responsible AI Chair a chair in Artificial Intelligence (https://ia-responsable.eu/).