O Fovet

M Ndom

A Crave

A Pannard

Catherine Grimaldi

Influence of dams on river water-quality signatures at event and seasonal scales: the Sélune River (France) case study

Keywords: Dams, river-water concentrations, hydrochemical signatures, anions, nutrients, suspended sediments

Dams and reservoirs are known to disturb river-water composition, among other impacts, with potential implications for downstream river ecosystems and water uses. Existing studies have emphasized the variable influence of dams on water composition according to the element, its speciation (dissolved vs. particulate), reservoir properties (residence times), reservoir functions (e.g. hydropower, irrigation), and management (water releases). A now common approach to analyzing hydrological, geochemical, and biological controls on element export from unregulated rivers is to study hydrochemical signatures such as concentration-flow relationships. We investigated a case study to analyze hydrochemical signatures of a regulated river (Sélune River, western France) upstream and downstream of a chain of two hydropower dams, assuming that the dams disturbed the river's signatures, and that those disturbances would provide information about processes occurring in the reservoirs. Both seasonal and event-scale signatures were analyzed over two contrasting hydrological years and a range of storm events. The dams induced a chemostatic downstream response to storm events whenever elements were diluted or concentrated upstream. Dams did not disturb the seasonality of major anions but did modify silica and phosphorus concentration-flow relationships, especially during low flow. Such changes in dynamics of river-water composition may affect downstream biological communities. This study presents an initial state of the hydrochemical signature of the downstream river, before the removal of the two dams.

Introduction

Dams are a critical water infrastructure with social, economic, and environmental impacts that challenge the concept of sustainability [START_REF] Moran | Sustainable hydropower in the 21st century[END_REF][START_REF] Vedachalam | Small is beautiful? State of the dams and management implications for the future[END_REF]. In terms of environmental impacts, dams strongly alter river hydrological, sediment, hydrochemical, and biological connectivity and continuity [START_REF] Bednarek | Undamming Rivers: A Review of the Ecological Impacts of Dam Removal[END_REF][START_REF] Kelly | Influence of reservoirs on solute transport: a regional-scale approach[END_REF][START_REF] Liro | Dam reservoir backwater as a field-scale laboratory of human-induced changes in river biogeomorphology: a review focused on gravel-bed rivers[END_REF][START_REF] Pestana | The impact of hydroelectric dams on mercury dynamics in South America: A review[END_REF][START_REF] Vörösmarty | Anthropogenic sediment retention: major global impact from registred river impoudments[END_REF]. In terms of water chemistry, dams and the reservoirs they form are known to disturb river-water composition [START_REF] Aristi | Flow regulation by dams affects ecosystem metabolism in Mediterranean rivers[END_REF][START_REF] Mantel | Ecological impacts of small dams on South African rivers Part 1: Drivers of change -water quantity and quality[END_REF][START_REF] Palmer | Downstream effects of impoundments on the water chemistry of the buffalo river (Eastern Cape)[END_REF][START_REF] Pozo | Effects of the Cernadilla-Valparaiso reservoir system on the River Tera[END_REF][START_REF] Van Cappellen | Rivers in the Anthropocene: Global scale modifications of riverine nutrient fluxes by damming[END_REF] in the following ways:

1) Dams modify the flow regime in downstream reaches according to their management, and water chemical composition varies with flow [START_REF] Carling | Evaluating natural and anthropogenic trace element inputs along an alpine to urban gradient in the Provo River, Utah, USA[END_REF][START_REF] Chittoor Viswanathan | Does river restoration affect diurnal and seasonal changes to surface water quality? A study along the Thur River, Switzerland[END_REF][START_REF] Graf | Damage Control: Restoring the Physical Integrity of America's Rivers[END_REF]. 2) Dams modify the chemical weathering rates which are controlled by residence times in watersheds [START_REF] Brink | Weathering rates and origin of inorganic carbon as influenced by river regulation in the boreal sub-arctic region of Sweden[END_REF][START_REF] Gao | Impacts of river impoundment on the riverine water chemistry composition and their response to chemical weathering rate[END_REF]. 3) Reservoirs are physical barriers known to trap suspended materials [START_REF] Carling | Evaluating natural and anthropogenic trace element inputs along an alpine to urban gradient in the Provo River, Utah, USA[END_REF][START_REF] Eiriksdottir | The impact of damming on riverine fluxes to the ocean: A case study from Eastern Iceland[END_REF] and likely to concentrate some solutes by evaporation [START_REF] Kelly | Influence of reservoirs on solute transport: a regional-scale approach[END_REF], but they are also biogeochemically reactive [START_REF] Garnier | Biogeochemical mass-balances (C, N, P, Si) in three large reservoirs of the Seine basin (France)[END_REF][START_REF] Maavara | Worldwide retention of nutrient silicon by river damming: From sparse data set to global estimate[END_REF] and thus likely to affect nutrient cycles. 4) Reservoirs are also likely to modify exchanges between groundwater and the river by changing local piezometric gradients [START_REF] Murgulet | Impact of hydrological alterations on river-groundwater exchange and water quality in a semi-arid area: Nueces River, Texas[END_REF][START_REF] Taleb | The role of the hyporheic zone in the nitrogen dynamics of a semi-arid gravel bed stream located downstream of a heavily polluted reservoir (Tafna Wadi, Algeria)[END_REF] Studies that investigated the influence of dams on water nutrients  nitrogen (N), phosphorus (P), carbon (C), and silica (Si) concentrations (see review in Table 1) found that the influence varied according to the element, its speciation, reservoir properties, dam function and management. The downstream response partly depends on the biogeochemical and ecological functioning of lakes, controlled by their temperature and vertical mixing driven by wind and heat exchanges with the atmosphere [START_REF] Spigel | Mixing processes relevant to phytoplankton dynamics in lakes[END_REF]. Seasonal stratification often leads to late summer hypoxic deep waters, while sinking organic carbon originating from surface phytoplankton accumulates in the hypolimnion [START_REF] Dodds | Freshwater ecology: concepts and environmental applications of limnology[END_REF], leading to sediment anoxic after a few millimeters. If so the trapping ability of sediment decreases and benthic fluxes increase [START_REF] Wetzel | Limnology: lake and river ecosystems[END_REF] except if redox-insensitive P-binding systems, such as aluminium hydroxide, are present [START_REF] Hupfer | Oxygen Controls the Phosphorus Release from Lake Sedimentsa Long-Lasting Paradigm in Limnology[END_REF]. Anoxic deepwater in lakes remobilizes phosphates, providing nutrients for phytoplankton by upwelling of deep water and internal waves [START_REF] Macintyre | Boundary mixing and nutrient fluxes in Mono Lake, California[END_REF]. The mixing regime in dams thus regulate the water quality downstream [START_REF] Winton | Reviews and syntheses: Dams, water quality and tropical reservoir stratification[END_REF].

The typical approach for identifying the influence of dams on water chemistry is thus to compare concentrations or fluxes upstream and downstream of dams for periods ranging from one year to several decades. In contrast, export regime of dissolved or suspended element in unregulated rivers and streams are widely studied by analyzing water quality signatures such as concentration-flow (C-Q) relationships [START_REF] Bieroza | The concentration-discharge slope as a tool for water quality management[END_REF][START_REF] Bowes | Phosphorus-discharge hysteresis during storm events along a river catchment: the River Swale, UK[END_REF][START_REF] Minaudo | Seasonal and eventbased concentration-discharge relationships to identify catchment controls on nutrient export regimes[END_REF][START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF][START_REF] Musolff | Emergent archetype patterns of coupled hydrologic and biogeochemical responses in catchments[END_REF]. Looking at the relative change in river-water concentrations as a function of flow allows for normalizing watershed behavior for several elements to compare either event behaviors [START_REF] Bieroza | Seasonal variation in phosphorus concentration-discharge hysteresis inferred from high-frequency in situ monitoring[END_REF][START_REF] Bowes | Phosphorus-discharge hysteresis during storm events along a river catchment: the River Swale, UK[END_REF][START_REF] Butturini | Diversity and temporal sequences of forms of DOC and NO3-discharge responses in an intermittent stream: Predictable or random succession[END_REF][START_REF] Evans | Causes of concentration/discharge hysteresis and its potential as a tool for analysis of episode hydrochemistry[END_REF] or watersheds [START_REF] Godsey | Concentration-discharge relationships reflect chemostatic characteristics of US catchments[END_REF][START_REF] Musolff | Catchment controls on solute export[END_REF][START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF][START_REF] Rose | Concentration-discharge relationships describe solute and sediment mobilization, reaction, and transport at event and longer timescales[END_REF]. These relationships provide indications on elements source location, abundance or biogeochemical processes leading to their retention or transformation [START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF].

Since dams can have major hydrological and biogeochemical impacts on downstream river reaches, one can assume the following:

1) Dams disturb hydrochemical signatures of rivers; thus, anyone seeking to use these signatures to infer watershed processes should carefully consider the role of any dams present. 2) Modification of these signatures by dams can provide information about the processes occurring in the reservoir-river system.

Therefore, we investigated a case study to analyze hydrochemical signatures of a regulated river (Sélune River, western France) upstream and downstream of a chain of two hydropower dams. The choice of the site was also motivated by the perspective of the two dams removal by 2021. We selected the following signatures: the ratio of coefficients of variation (CV) of C and Q [START_REF] Thompson | Relative dominance of hydrologic versus biogeochemical factors on solute export across impact gradients[END_REF], and the slope of the C-Q regression in the log domain [START_REF] Godsey | Concentration-discharge relationships reflect chemostatic characteristics of US catchments[END_REF]. We analyzed both seasonal and event-scale signatures over two hydrological years and for a range of events.

Materials and methods

Study site

The Sélune River drains a 1083 km 2 watershed before flowing into Mont-Saint-Michel Bay (Fig. 1), in western France. The river is 91 km long and regulated by two hydroelectric dams, Vezin and La-Rochequi-Boit (36 and 16 m high, respectively). Climate is temperate and oceanic, with annual mean (± 1 standard deviation) precipitation as rainfall of 751 mm ± 224 mm for the study period 2015-2018. Precipitation peaks in January (109 ± 51 mm) and is lowest in July (48 ± 24 mm). Because of seasonal variations in Potential Evapotranspiration, stream flow shows seasonal fluctuations: high flow period occurring in Winter and low flow period at the end of Summer (SI Fig. S1).

The bedrock is composed of granite in the north and south of the watershed, and of Brioverian schist in the middle. Because of the low permeability of the substratum, groundwater bodies are particularly shallow, leading to hydromorphic soils in bottomlands, while hillslopes have mainly well-drained Cambisols (USS Working Group WRB, 2006). Most slopes are moderate and less than 3% but can reach 13% in the river gorge and in valleys of some tributaries. The watershed is dominated (94%) by agriculture with mixed crop-livestock farming, leading to dominant grass and arable land-uses.

The general reservoirs characteristics are given in Table 2. The French government plans to remove these two dams on the Sélune River in 2021. This operation is associated with a 16-year multidisciplinary scientific program to evaluate dam removal as a tool for restoring aquatic ecosystems.

In this context, a monitoring strategy has been developed to record water, suspended, and dissolved fluxes before, during, and after dam removal. This study focuses on two years of pre-removal observations.

Data acquisition

On the river, two stations (Fig. 1), one upstream and one downstream of both dams, were instrumented in 2015 with sensors for continuous monitoring (every 6 min to 1 h) of river water level, temperature, electrical conductivity, and turbidity. Stream discharge was determined from water level using a rating curve established for each station specifically by the Regional Directorate for Environment (DREAL).

The upstream station, Virey, is located 4.6 km upstream of the lotic-lentic transition zone, corresponding to a drainage area of 629 km 2 . This upstream station is equipped with sensors for monitoring dissolved oxygen and pH. The downstream station, Signy, is located 4 km downstream of the downstream dam (la Roche-Qui-Boit), corresponding to a drainage area of 777 km 2 . The downstream station is equipped with a fluorescence probe (AlgaeTorch) to monitor chlorophyll a concentration continuously. In this study, we also used the dissolved oxygen recorded just downstream the dam of La-Roche-Qui-Boit.

River water was sampled every week at the upstream and downstream stations, and some storm events were also sampled at both stations using automatic samplers (ISCO). At each sampling, 500 mL of water was filtered at 0.45 µm, and then the suspended sediments (SS) collected were dried and weighed to determine their concentration. Another unfiltered sub-sample was used to determine TP colorimetrically by reaction with ammonium molybdate after mineralization with potassium peroxydisulfate. A third subsample was filtered at 0.45 µm to determine major anion (chloride (Cl), NO3, sulfate (SO4)) concentrations by ionic chromatography, and PO4 and dissolved Si by colorimetry based on reaction with ammonium molybdate (in acid conditions for Si). For ca. half of the samples, a final sub-sample was filtered to determine C and N content of SS using an elemental analyzer (Thermo Finnigan CN FLASH EA 1112).

Vertical profiles of temperature, dissolved oxygen and fluorescence were measured monthly in 2015 and 2016 in the lakes using a multiparametric probe (Idronaut, OceanSeven 316Plus). Concentrations of NO3 and PO4 were measured as described above in 2L of water sampled with a Niskin bottle at 3 depths (1 m, euphotic zone, 1 m above the sediment).

Signatures selected

Signatures were analyzed separately for base-flow concentrations (obtained from regular weekly sampling) and storm-flow concentrations (obtained from automatic samplers). High and low base flows regimes were identified by dividing the hydrographs at each station by their respective median daily flow (SI Fig. S1) recorded over the period, following [START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF]. Among the storm events sampled, we selected only those sampled at both stations (SI Fig. S5).

The ratio of coefficients of variation of C and Q (CVC:CVQ) was proposed by [START_REF] Thompson | Relative dominance of hydrologic versus biogeochemical factors on solute export across impact gradients[END_REF] as a characteristic of export regime. Ratios lower than 0.5 define chemostatic export regimes, indicating that element load variability is controlled by flow variability, while ratios greater than 0.5 qualify chemodynamic regimes. A chemostatic regime does not imply invariant concentration [START_REF] Thompson | Relative dominance of hydrologic versus biogeochemical factors on solute export across impact gradients[END_REF]. The non-linearity in C-Q relationships is usually accounted using a power law relationship [START_REF] Godsey | Concentration-discharge relationships reflect chemostatic characteristics of US catchments[END_REF][START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF][START_REF] Musolff | Catchment controls on solute export[END_REF]. This corresponds to a linear regression in log domain (Eq. 1).

log(𝐶) = 𝑎 + 𝑏 log (𝑄) (1)

Despite the large dispersion often observed in C-Q plots, this regression provides a simple metric to categorize C-Q in archetypes using the sign of exponent b [START_REF] Musolff | Emergent archetype patterns of coupled hydrologic and biogeochemical responses in catchments[END_REF]. We fitted the linear regressions using the least square method. Non-significant regression (p-value>0.005) were classified as flat pattern, suggesting a uniform distribution of element, whereas significant regressions suggest more heterogeneous source in the catchment. Dilution patterns (b<0) are associated to a control by element production or abundance. Enrichment patterns (b>0) are rather controlled by transport capacity or flowpaths extension [START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF]. Biogeochemical reactions can enhance or attenuate these characteristics [START_REF] Minaudo | Seasonal and eventbased concentration-discharge relationships to identify catchment controls on nutrient export regimes[END_REF]. The CV ratio and b sign are not independent [START_REF] Musolff | Catchment controls on solute export[END_REF] as strong enrichment or dilution cannot co-occur with very low CV ratios but flat pattern can be associated to high variance in concentrations leading to scattered C-Q plot, and CV ratios lower than 0.5 can coexist with weak dilution patterns (slightly negative b values).

For event-scale, the dispersion in the C-Q relationships can also be related to hysteresis effects. Direction of hysteresis loops were also compared between the two stations and described as clockwise, anticlockwise or absent.

Results

Reservoirs stratification

Reservoirs undergo summer thermal stratification from mid-April to mid-October (Fig. 2). Phytoplankton developed in the epilimnion of both reservoirs with summer concentrations fluctuating between 15 and 70 g chla L -1 (data not shown). Stratification was more stable in Vezins dam (twice deeper) than in Roche-qui-Boit (Fig. S9). The epilimnion in Vezins fluctuated between 4 and 5 m depth, with a thick metalimnion, and a hypolimnion at 11 to 13 m depth (Fig. 2). The turbine intakes (generally located at 6m depth) thus drawed water from the top of the metalimnion in that reservoir. Roche-qui-Boit was mainly stratified in July and August, with an epilimnion fluctuating between 5 and 7 m depth and a thermal gradient until the bottom of the lake, making the transition metalimnion to hypolimnion difficult to characterize. In both eutrophic reservoirs, bottom waters showed concentrations in oxygen below 1 mg L -1 in July and August, leading potentially to anaerobic conditions at the sediment-water interface. The hypolimnetic oxygen depletion coincided with a 52% P enrichment in Vezins and a 120% P enrichment in Roche-qui-Boit, compared with the epilimnion. It also coincided with a small vertical decrease in NO3 concentrations, but always remaining higher than 4 mg N-NO3 L -1 in the hypolimnion (Table 2). A strong anoxia would have reduced these concentrations close to zero. Changes in nutrients concentrations associated with stratification thus remained limited here, all the more the turbine intakes never reached hypolimnetic anoxic waters.

Seasonal variation in river base-flow concentrations

Base-flow concentrations lay in the range of those previously observed in western France (Figs. S2-S3).

The C-Q relationships were consistent between years at each station for a given flow regime (Fig. 3).

CV ratios distinguished SS and Phosphorus (TP and SRP) with highly chemodynamic exports, and the dissolved elements with lower relative variance in their concentration (Table 3). The dams did not affect the export regimes excepted for Si during low flow that is observed to be more chemodynamic at the downstream station.

Poorly significant regressions were usually associated with low C-Q slopes (median |b|<0.1) whereas significant regressions were associated to absolute slopes higher than 0.1 and high Pearson correlation coefficients: average |rP| of 0.6 (Table 3).

Enrichment patterns (b>0) were associated to SS at the upstream station during high flow and to Phosphorus concentrations during low flow. Dilution patterns were observed for Cl for all stations and flow regimes, and for NO3 only during low flow and only significant at the upstream station. SO4-Q plots remained flat at both stations and whatever the flow regime. Dissolved Si showed an enrichment pattern at the downstream station reflecting the very low concentration occurring at low flow (Fig. S4) while Si-Q plots were flat at the upstream station. These C-Q patterns are consistent with the covariations observed in the time domain (Fig S4).

Storm-flow responses

As expected, storm-flow responses of each element's concentration varied among storms (SI Table S1, Fig S6 -S8). Nevertheless, clear differences appeared between two groups: P and SS vs. anions and dissolved Si (Fig. 4).

CV ratios associated to storm-flow indicated that usually Phosphorus and SS concentrations at both stations were associated to the most chemodynamic storm export regimes (Table 4). Storm export regimes associated to dissolved anions and Silica are more chemostatic, and even more at the downstream station (SI Table S1).

For storm flow, poorly significant regressions were also usually associated with low C-Q slopes. Significant regressions were associated to high |rP| (Table S1): in average 0.85 (upstream) and 0.66 (downstream).

At the upstream station storms TP, PO4 and SS were frequently associated to a strong enrichment patterns according to C-Q slopes >0.1 while Cl, NO3, SO4 and Si were usually associated to a dilution pattern (Table 4 andS1). On the contrary, at the downstream station, all storm C-Q were flat with unsignificant regressions or absolute slope close to zero.

For most events, the C-Q relationships exhibited hysteretic loops described by their direction (Table 4, and SI 2). For SS, TP, and Cl concentrations, most events exhibited a clockwise loop at the event scale at both stations, with both directions for PO4 at the downstream stations. NO3 and SO4 concentrations shifted from counter-clockwise loops upstream to clockwise loops downstream. Si exhibited both directions at the upstream station but mostly counter-clockwise loops at the downstream station.

Discussion

Dams influence seasonal variations of TP and major anions less than the flow regime

Highly chemodynamic export regimes have been reported for sediments and phosphorus concentrations in previous studies too [START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF][START_REF] Thompson | Relative dominance of hydrologic versus biogeochemical factors on solute export across impact gradients[END_REF]. [START_REF] Thompson | Relative dominance of hydrologic versus biogeochemical factors on solute export across impact gradients[END_REF] suggested it can be related to heterogeneously distributed sources activation, threshold-driven transport mechanism (as bed load transport) and high reactivity of elements such as Phosphorus via sorption/desorption processes. The lower chemodynamism found for dissolved Si and anions was also expected from these previous studies and suggests that the dissolved element exported loads are mainly controlled by flow variability. It is usually related to the existence of large legacy storages, indeed emphasized in such agricultural contexts for NO3 (e.g. [START_REF] Howden | Nitrate pollution in intensively farmed regions: What are the prospects for sustaining high-quality groundwater[END_REF][START_REF] Musolff | Groundwater head controls nitrate export from an agricultural lowland catchment[END_REF] and Cl [START_REF] Aubert | Solute transport dynamics in small, shallow groundwater-dominated agricultural catchments: insights from a high-frequency, multisolute 10 yr-long monitoring study[END_REF][START_REF] Hrachowitz | Transit time distributions, legacy contamination and variability in biogeochemical 1/fα scaling: how are hydrological response dynamics linked to water quality at the catchment scale?[END_REF].

At the upstream station, patterns observed for SS (flat in low flow, enrichment in high flow) are consistent with results from [START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF], and suggest an increased mobilization of sediment sources (soils or river bed) with transport capacity. The flatter SS-Q pattern at the downstream station reveals a disruption in sediment continuity along the river due to storage of particles in the reservoirs [START_REF] Carling | Evaluating natural and anthropogenic trace element inputs along an alpine to urban gradient in the Provo River, Utah, USA[END_REF][START_REF] Pozo | Effects of the Cernadilla-Valparaiso reservoir system on the River Tera[END_REF]. The difference of TP and PO4-Q pattern with enrichment limited to low flow could be explained by the relation between the phosphorus sorption capacity of the sediments and their size, e.g. [START_REF] Landers | Hysteresis in suspended sediment to turbidity relations due to changing particle size distributions[END_REF]. At low flow, mobilized SS would be finer and thus would have higher P content whereas during high flow the mobilization of bigger particles, with lower P content, would increase. Dams store sediments in the reservoirs but finest sediments (size < 10 µm), with negligible sedimentation rate and associated to high phosphorus content, cross the reservoirs.

In dams, seasonal stratification drives P release from sediment and oxygen depletion in hypolimnion [START_REF] Hupfer | Oxygen Controls the Phosphorus Release from Lake Sedimentsa Long-Lasting Paradigm in Limnology[END_REF]. As turbine intakes occurred here at the top of the metalimnion, trapping efficiency of the dams increased and nutrients fluxes from hypolimnetic waters was limited. Consequently, for most major anions, the chemical composition of downstream water was little impacted by summer stratification. Cl is assumed non-reactive, its dilution pattern suggests a limited abundance of Cl in the catchment when water flow increase, as found by others [START_REF] Godsey | Concentration-discharge relationships reflect chemostatic characteristics of US catchments[END_REF][START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF]. Previous authors found a dominance of NO3 enrichment pattern attributed to biological retention during low flows [START_REF] Minaudo | Seasonal and eventbased concentration-discharge relationships to identify catchment controls on nutrient export regimes[END_REF][START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF] in contrast to the dilution pattern observed here. Despite its reactivity, NO3 behaved conservatively suggesting a moderate riverine biological effect on NO3 even at low flow. At the downstream station, the flatter pattern of tNO3 may express a moderate increase of biological N retention in the reservoirs. The seasonal variations of dissolved anions as a function of flow regime are not modified by the reservoirs (Fig. 3; Fig. S4).

Retention and biogeochemical reactions affect Si and PO4 signatures mainly during low flow

Chemostatic export regimes for Si observed at the upstream station are consistent with other findings [START_REF] Godsey | Concentration-discharge relationships reflect chemostatic characteristics of US catchments[END_REF][START_REF] Moatar | Elemental properties, hydrology, and biology interact to shape concentration-discharge curves for carbon, nutrients, sediment, and major ions[END_REF] and can be attributed to higher weathering production rates when flow increases [START_REF] Thompson | Relative dominance of hydrologic versus biogeochemical factors on solute export across impact gradients[END_REF]. At the downstream station, higher chemodynamism and enrichment pattern indicate an increased biological retention or uptake of Si. Low flows co-occur with summer-autumn period, where light and temperature conditions are more favorable to phytoplankton growth in reservoirs, and particularly diatoms (Fig. 2,S9). The hypothesis of biological uptake during low flow was also supported by measured variations in temperature, dissolved oxygen concentration, and chlorophyll a concentration (Fig. 5).

The low flow enrichment pattern in PO4, flatter at the downstream station than at the upstream station, may also reflect an increased biological retention in the reservoirs through absorption by phytoplankton cells then incorporated in zooplankton and by sinking processes of both cells and feacal pellets [START_REF] Lyche | Mesocosm tracer studies. 1. Zooplankton as sources and sinks in the pelagic phosphorus cycle of a mesotrophic lake[END_REF]. At the bottom of lakes, P retention is counterbalanced by P release, associated with deeper microbial decomposition of organic matter and chemical dissolving process of sorbed P [START_REF] Hupfer | Oxygen Controls the Phosphorus Release from Lake Sedimentsa Long-Lasting Paradigm in Limnology[END_REF]. The importance of a minimum hypolimnetic nitrate concentration here limits the accumulation of phosphates in the hypolimnion [START_REF] Schauser | Effects of nitrate on phosphorus release: comparison of two Berlin lakes[END_REF]. The biogeochemical reactivity of the dams remained limited here, excepting for the retention capacity, and downstream transfer of remobilized solutes is unlikely.

Damping storm flow by mixing and retention

During storm events, concentration dynamics were consistently damped from the upstream to the downstream station whenever the upstream station had higher or lower solute concentrations.

Unsurprisingly, most SS, TP, and PO4 were highly exported during storm events associated with much higher concentrations than those during base flow [START_REF] Jordan | Characterising phosphorus transfers in rural catchments using a continuous bank-side analyser[END_REF]. Highly chemodynamic regime and enrichment patterns reflect the processes of sediment mobilization from catchment soils by surface flowpaths and from river bed by transport capacity which are both increased during storms. Dominance of clockwise C-Q loops in such enrichment patterns suggest a depletion of particle sources leading to higher concentrations during rising limb than during the falling limb of the storm hydrograph as reported in a sub-basin of the Selune river [START_REF] Dupas | Distinct export dynamics for dissolved and particulate phosphorus reveal independent transport mechanisms in an arable headwater catchment[END_REF][START_REF] Vongvixay | Contrasting suspended sediment export in two small agricultural catchments: Cross-influence of hydrological behaviour and landscape degradation or stream bank management[END_REF] and elsewhere [START_REF] Mcdowell | The effect of antecedent moisture conditions on sediment and phosphorus loss during overland flow: Mahantango Creek catchment, Pennsylvania, USA[END_REF][START_REF] Seeger | Catchment soil moisture and rainfall characteristics as determinant factors for discharge/suspended sediment hysteretic loops in a small headwater catchment in the Spanish pyrenees[END_REF]. The drastic decrease in SS, TP, and PO4 storm concentrations at the downstream station resulted from their storage in the reservoirs by deposition due to the decrease in flow velocity. Some of the SS and P measured at the downstream station may have come from the river bed itself. Indeed, the last reservoir and the downstream station are separated by 4 km of river, along which some fine sediments could remobilize during storm events.

The event dilution observed for Cl, SO4, NO3, and Si at the upstream station reflects a limitation of these elements abundance despite the increase of water flow. The storm dilutions were not observed at the downstream station, due to mixing with the water volume of the reservoirs. The dominant hysteresis loop direction changed between upstream and downstream stations for biologically reactive solutes (Si, NO3, SO4, PO4) but not for conservative Cl (Table 4) suggesting that mixing is also affected by retention and release processes in the reservoirs water and sediments. Despite a management rule that aimed to mimic upstream flow variations at the outlet of the dams (Fig. S1), the dams completely filtered chemical signatures during storm events.

Conclusion

We used hydrochemical signatures usually applied to unregulated rivers to provide information about biological and hydrological controls on chemical elements exported by rivers, to analyze how dams influence these signatures. The analysis of the Sélune River case study confirmed the first hypothesis that dams modify these signatures at both seasonal and event scales. At the event scale, dams promote chemostaticity of the downstream river for all elements, whenever elements are diluted or concentrated during storm events upstream. For base flow, major anions and TP usually exhibited the same differences in their C-Q relationships between low and high flows at both stations, even though mean concentrations could differ between the two stations. This result suggests that the dams do not influence the processes controlling seasonal fluctuations of these elements. In contrast, C-Q relationships for Si and PO4 differed between the stations, with larger differences during low flow. This alteration of river hydrochemical signatures reflects the biogeochemical processes that occur in the reservoirs themselves. Despite dam management designed to deliver an outflow similar to the inflow, our results show that the water released downstream of the dam has a different hydrochemical signature than natural river water, which may have potential effects on downstream biological communities.

Comparison of C-Q relationships instead of absolute concentrations appears valuable to help understand hydrological or biogeochemical processes altered by dams, supporting our second hypothesis. This highlights that the presence of reservoirs should be carefully considered when using such signatures to infer hydrological and biogeochemical processes in watersheds.

Figure 2 .

 2 Figure 2. Profiles of temperature (left) and oxygen concentration (right), showing summer stratification in Vezins (top) and Roche-qui-boit dams (bottom), depending on dates. The black curves show summer profiles. The star shows the depth of the turbine intake to within 20 cm.

Figure 4 .

 4 Figure 4. Concentration-flow relationships during storm event 1 at the upstream (blue) and downstream (red) station for each element. White circles identify the first sample of the event. Flow is expressed in m 3 s -1 , while concentrations are expressed in mg.L -1 .

Figure 5 .

 5 Figure 5. Continuous records of i) chlorophyll a concentration (black dots) compared to punctual measurements associated with phytoplankton and biofilm samples (red dots), ii) water temperature (blue dots) and iii) dissolved oxygen concentration (gray dots) at the downstream station during the 2017 lowflow period.

Table 1 .

 1 Summary of case studies that investigated the influence of dams on water chemical concentrations.

	Location	Parameters	Response	Source
		Si, NO3, PO4, TP,	High retention
	Tera River (Spain)	NH4	Hypolimnetic release	Pozo et al. (1997)
			3 km to 10 km
	Buffalo River (South	PO4,NO3	Shorter during low flow	Palmer and Okeeffe
	Africa)			(1990)
	Tafna River (Algeria)	organic matter, NH4	Retention through deposition Release High flow dilution/ low flow enrichment	Taleb, Belaidi, and Gagneur (2004)
	130 agricultural watersheds (Iowa, USA)	Si NO3	No retention in reservoirs dominated by cyanobacteria Slight decrease when reservoir residence times were sufficient	Downing, Cherrier, and Fulweiler (2016)
	Provo River (Utah, USA)	trace-element isotopic ratios	Lower temporal variability, due to settling of particle-attached elements and homogenization	Carling et al. (2015)
	Glacial river (Iceland)	annual elemental Retention of most particulate fluxes and release of most dissolved fluxes	Eiriksdottir et al. (2017)
	Yangtze River (China)	major ions	Increase in all major ion concentrations following establishment of the reservoir	Wang et al. (2018)
	Rio Grande and Colorado basins (USA)	P, NO3	Retention Decrease or increase depending on watershed	Kelly (2001)
	20 reservoirs across the	Si	Retention of the global reactive Si load	Maavara et al.,
	world			(2014)

Table 2 .

 2 Morphological and chemical characteristics of the two dams (year of build is given in parenthesis).

	Parameters	Dam of Vezins (1932)	Dam of Roche-qui-Boit (1919)
	Surface area (km 2 )	1.7	0.29
	Maximum volume (m 3 )	18 000 000	90 000
	Maximum depth (m)	31	16
	Mean depth (m)	12	3.1
	Mean residence time (days)	14 days (from Nov. to Apr.)	1 day (from Nov. to Apr.)
		47 days (from Jun. to Sept.)	4 days (from Jun. to Sept.)
	Summer concentrations
	(mean  standard deviation):
	Phosphates at surface (g P L -1 )	7.4  2.7	7.4  3.0
	Phosphates at bottom (g P L -1 )	11.3  3.6	16.5  7.3
	Nitrates at surface (mg N L -1 )	4.9  2.3	5.3  1.3
	Nitrates at bottom (mg N L -1 )	4.0  2.0	4.6  2.0

Table 3 .

 3 Descriptors of the concentration (C) -flow (Q) relationship (U)pstream and (D)ownstream: the ratio of coefficients of variation (CV) of C and Q; the slope (b) of the adjusted regressions between C and Q (𝑙𝑜𝑔(𝐶) = 𝑎 + 𝑏 𝑙𝑜𝑔(𝑄)); associated p-value of the regressions (bold slopes values are associated to significant regression) and Pearson correlation coefficient rP between log transformed C and Q.TP = total phosphorus, SS = suspended sediments.

	Descriptor	Location	Flow	Cl	NO3	SO4	Si	TP	PO4	SS
		U	low	0,34	0,31	0,48	0,21	0,82	1,21	1,17
	CVC:CVQ ratio		high	0,22	0,21	0,16	0,13	1,19	0,61	0,76
		D	low high	0,38 0,32	0,38 0,35	0,45 0,28	0,62 0,26	1,61 1,18	1,97 1,41	2,52 1,67
	slope of	U	low high	-0,25 -0,24	-0,17 0,04	0,07 0,02	-0,07 0,00	0,49 0,39	1,04 -0,10	0,43 0,56
	log(C)-log(Q)	D	low high	-0,15 -0,22	-0,14 0,09	0,18 -0,09	0,33 0,17	1,09 0,39	0,88 0,64	0,57 1,06
	p-value	U D	low high low high	0,0000 0,0000 0,0470 0,0100	0,0060 0,5400 0,0800 0,4030	0,4370 0,5940 0,0580 0,2430	0,1360 0,9000 0,0150 0,0120	0,0130 0,1040 0,0013 0,2890	0,0010 0,6440 0,0190 0,0780	0,0870 0,0068 0,3080 0,0620
	rP.	U D	low high low high	-0,80 -0,89 -0,40 -0,65	-0,57 0,13 -0,35 0,24	0,17 0,12 0,38 -0,33	-0,34 0,03 0,48 0,63	0,53 0,36 0,60 0,29	0,65 -0,11 0,47 0,46	0,38 0,57 0,22 0,53

Table 4 .

 4 Mean and standard deviations (sd) of descriptors of the concentration (C) -flow (Q) relationship for eight storm events (U)pstream and (D)ownstream: the ratio of coefficients of variation (CV) of C and Q; the slope (b) of the significant adjusted regressions (p-value ≤0.05) between C and Q (𝑙𝑜𝑔(𝐶) = 𝑎 + 𝑏 𝑙𝑜𝑔(𝑄)). Number of events exhibiting a clockwise (Cw) or counter-clockwise (CCw) direction when concentration-flow relationships show an identifiable loop for each parameter at (U)pstream or (D)ownstream stations. Event values of the descriptors are provided in SI (TableS1 & S2). TP = total phosphorus, SS = suspended sediments.

	Descr.	station		Cl	NO3	SO4	Si	TP	PO4	SS
		U	mean	0,33	0,55	0,43	0,46	1,77	1,00	3,13
	CVc:CVQ		sd	0,13	0,33	0,15	0,22	0,57	0,40	1,47
		D	mean	0,21	0,09	0,05	0,21	1,26	1,91	2,18
			sd	0,33	0,09	0,03	0,22	0,98	1,40	1,22
		U	mean	-0,35	-0,59	-0,13	-0,42	1,47	1,18	1,16
	slope of significant C-Q		sd	0,07	0,23	0,40	0,09	0,44	-	2,08
	regressions	D	mean	-	-0,03	-0,01	-	-	-	-
			sd	-	-	0,05	-	-	-	-
	Cw events	U D	number number	2 4	1 3	0 3	2 1	3 3	5 2	4 4
	CCw events	U D	number number	1 0	2 1	3 1	2 2	1 0	0 2	0 0

The Agence de l'Eau Seine Normandie funded the equipment and the analyses as well as the salary of Mamadou Ndom since 2016. Chemical analyses have been conducted by Béatrice Trinkler, INRA, UMR SAS,

Data availability statement

Upstream flow data and downstream dissolved oxygen data were provided by EDF. The other data, from the database of the Sélune Research Program, are available from the authors upon request.


Andreas Herzig
email: andreas.herzig@irit.fr

Emiliano Lorini
email: emiliano.lorini@irit.fr

Elise Perrotin
email: elise.perrotin@irit.fr

Fabián Romero

Franc ¸ois Schwarzentruber
email: francois.schwarzentruber@ens-rennes.fr

A Logic of Explicit and Implicit Distributed Belief

We present a new logic of explicit and implicit distributed belief with a formal semantics exploiting the notion of belief base. A coalition's distributed belief of explicit type corresponds to a piece of information contained in the collective belief base of the coalition, which is obtained by pooling together the individual belief bases of its members. A coalition's distributed belief of implicit type corresponds to a piece of information that is derivable from the collective belief base of the coalition. We study axiomatic aspects of our logic as well as complexity of model checking. As distributed belief can be inconsistent (contrary to distributed knowledge), we also study a consistency-preserving variant of distributed belief inspired by the literature on belief merging.

Introduction

Epistemic logic (in the broad sense) is the variant of modal logic at the intersection between philosophy [START_REF] Hintikka | Knowledge and Belief[END_REF], artificial intelligence (AI) [START_REF] Fagin | Reasoning about Knowledge[END_REF][START_REF] Meyer | Epistemic logic for AI and computer science[END_REF] and economics [START_REF] Lismont | On the logic of common belief and common knowledge[END_REF] that is devoted to the formal representation of epistemic attitudes of agents including belief and knowledge. It has been shown to have important applications in AI ranging from security protocols [START_REF] Halpern | A knowledge-based analysis of the blockchain protocol[END_REF][START_REF] Burrows | A logic of authentication[END_REF] and blockchain protocol [START_REF] Halpern | A knowledge-based analysis of the blockchain protocol[END_REF][START_REF] Marinkovic | A temporal epistemic logic with a non-rigid set of agents for analyzing the blockchain protocol[END_REF] to epistemic planning [START_REF] Bolander | Epistemic planning for single-and multi-agent systems[END_REF] and agent communication protocols [START_REF] Herzig | How to share knowledge by gossiping[END_REF].

Epistemic logic supports reasoning not only about propositional epistemic attitudes but also about higher-order epistemic attitudes, where the order of an agent's epistemic attitude is defined inductively as follows: (i) an agent's epistemic attitude has order 1 if and only if its content is a propositional formula that does not mention epistemic attitudes of others; (ii) an agent's epistemic attitude has order k with k > 1 if and only it is an epistemic attitude about an agent's epistemic attitude of order k -1 (possibly the same agent).

Epistemic logic languages are traditionally interpreted in Kripke's possible worlds semantics. The type of structures used in these semantics are the so-called multi-agent Kripke models, namely, multirelational structures equipped with valuation functions for the interpretation of atomic formulas. Binary relations in a multi-agent Kripke model are called epistemic accessibility relations.

Different types of collective attitudes have been defined in the epistemic logic framework and the mathematical properties of their corresponding extensions have been investigated, including axiomatizability and computational complexity. This includes the notions of shared knowledge and belief [START_REF] French | On the succinctness of some modal logics[END_REF], common knowledge and belief [START_REF] Vanderschraaf | Common knowledge[END_REF][START_REF] Halpern | A guide to completeness and complexity for modal logics of knowledge and belief[END_REF], distributed knowledge and belief [START_REF] Halpern | A guide to completeness and complexity for modal logics of knowledge and belief[END_REF][START_REF] Wáng | Public announcement logic with distributed knowledge: expressivity, completeness and complexity[END_REF][START_REF] Ågotnes | Resolving distributed knowledge[END_REF][START_REF] Roelofsen | Distributed knowledge[END_REF][START_REF] Van Der Hoek | Group knowledge is not always distributed (neither is it always implicit)[END_REF] and collective acceptance [START_REF] Lorini | The logic of acceptance: Grounding institutions on agents' attitudes[END_REF].

Distributed epistemic attitudes are the results of pooling together or aggregating the agents' individual epistemic attitudes. For instance, for a coalition G of agents to distributively believe that ϕ, ϕ must be included in the set of facts that are "aggregatedly" believed by the agents in G. This notion of aggregation is rendered in possible worlds semantics by intersecting the individual epistemic accessibility relations of the coalition's members. However this could be considered too strong as that intersection can easily be empty thereby making distributed belief inconsistent.

The notion of belief aggregation was studied in parallel in the area of belief merging [START_REF] Konieczny | Merging information under constraints: A logical framework[END_REF][START_REF] Konieczny | On the difference between merging knowledge bases and combining them[END_REF][START_REF] Pigozzi | Belief merging and judgment aggregation[END_REF]. Nonetheless, there are some fundamental differences between the notion of aggregation studied in epistemic logic, via the notions of distributed knowledge and belief, and the notion of aggregation studied in belief merging. First of all, belief merging focuses exclusively on aggregation of propositional beliefs, while epistemic logic allows us to represent aggregation of higherorder epistemic attitudes. Secondly, the belief merging approach is essentially syntactical, as the aggregation operation is defined on individual belief bases, while the epistemic logic approach is essentially semantical, as aggregation is made at the level of the agents' epistemic accessibility relations.

The aim of this paper is to reconcile the two approaches by presenting a new epistemic logic, with a formal semantics exploiting the notion of belief base like in belief merging, that distinguishes explicit and implicit distributed belief. A coalition's distributed belief of explicit type corresponds to a piece of information contained in the collective belief base of the coalition, which is obtained by pooling together the individual belief bases of its members. A coalition's distributed belief of implicit type corresponds to a piece of information that is derivable from the collective belief base of the coalition. The belief semantics for epistemic logic that we use was introduced in [START_REF] Lorini | In praise of belief bases: Doing epistemic logic without possible worlds[END_REF] (see also [START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF][START_REF] Lorini | Decision procedures for epistemic logic exploiting belief bases[END_REF][START_REF] Lorini | Exploiting belief bases for building rich epistemic structures[END_REF]). In this paper, we show that it is wellsuited to model aggregation of agents' higher-order individual beliefs. The latter concept is relevant for some AI applications such as recommendation systems, e-democracy, artificial trading agents or autonomous vehicles. For example, in an online recommendation system, it would be interesting to merge not only the agents' individual beliefs about the (good and bad) qualities of a given product, but also the agents' meta-evaluations, namely, what they believe about the others' beliefs about the product qualities. 6 Similarly, in online pools, it could be useful to compute what the agents think about a certain political issue (e.g., efficacy of immigration policies in Europe, increase of the price of gasoline for contrasting climate change, etc.), as well as what they think the other agents think about the same issue. On top of helping with compactness, making a distinction between implicit and explicit belief allows us to separate the aggregation process (at the level of the explicit beliefs) from the deduction process (from explicit to implicit belief), which we believe to be interesting from a conceptual point of view.

The paper is organized as follows. In Section 2, we present the language of our logic of individual and distributed beliefs. Section 3 is devoted to illustrating a semantics for this language which exploits the notion of belief base. In Section 4, we present an axiomatics for the logic, while in Section 5 we study complexity of its model checking problem. Section 6 introduces an introspective variant of the logic in which coalitions are assumed to have introspection over their distributed beliefs. As distributed belief can be inconsistent (contrary to distributed knowledge), in Section 7, we study a consistencypreserving variant of distributed belief inspired by the literature on belief merging. In Section 8, we conclude.

A language for distributed doxastic attitudes

This section presents a language for representing agents' individual beliefs and coalitions' distributed beliefs of both explicit and implicit type. It extends the language of individual explicit and implicit belief presented in [START_REF] Lorini | In praise of belief bases: Doing epistemic logic without possible worlds[END_REF] with distributed belief. Assume a countably infinite set of atomic propositions Atm = {p, q, . . .} and a finite set of agents Agt = {1, . . . , n}. We define the language in two steps.

We first define the language L0(Atm, Agt) by the following grammar in Backus-Naur Form (BNF):

α ::= p | ¬α | α1 ∧ α2 | iα,
where p ranges over Atm and i ranges over Agt. L0(Atm, Agt) is the language for representing distributed explicit beliefs of coalitions. The formula iα is read "agent i explicitly believes that α".

The language L1(Atm, Agt) extends the language L0(Atm, Agt) by modal operators of distributed implicit belief and is defined by the following grammar:

ϕ ::= α | ¬ϕ | ϕ1 ∧ ϕ2 | 2Gϕ,
where α ranges over L0(Atm, Agt) and G ranges over 2 Agt * = 2 Agt \ {∅}. For notational convenience we write L0 instead of L0(Atm, Agt) and L1 instead of L1(Atm, Agt), when the context is unambiguous.

The other Boolean constructions , ⊥, ∨, → and ↔ are defined from α, ¬ and ∧ in the standard way.

The formula 2Gϕ is read "coalition G has the distributed implicit belief that ϕ". We define the dual operator 3G as follows:

3Gϕ def = ¬2G¬ϕ.
3Gϕ has to be read "ϕ is compatible (or consistent) with coalition G's explicit beliefs".

For notational convenience, for every i ∈ Agt, we simply write 2iϕ instead of 2 {i} ϕ to represent agent i's implicit belief that ϕ.

Belief base semantics

Following [START_REF] Lorini | In praise of belief bases: Doing epistemic logic without possible worlds[END_REF], in this section, we present a formal semantics for the language L1 exploiting belief bases. Unlike the standard Kripke semantics for epistemic logic in which the notions of possible world (or state) and epistemic alternative are given as primitive, in this semantics they are are defined from the primitive concept of belief base.

Definition 1 (State)

A state is a tuple B = (B1, . . . , Bn, V ) where:

• for every i ∈ Agt, Bi ⊆ L0 is agent i's belief base, • V ⊆ Atm is the actual state.
The set of all states is denoted by S.

The sublanguage L0(Atm, Agt) is interpreted with respect to states, as follows.

Definition 2 (Satisfaction relation) Let B = (B1, . . . , Bn, V ) ∈ S. Then:

B |= p ⇐⇒ p ∈ V , B |= ¬α ⇐⇒ B |= α, B |= α1 ∧ α2, ⇐⇒ B |= α1 and B |= α2, B |= iα ⇐⇒ α ∈ Bi.
Observe in particular the set-theoretic interpretation of the distributed explicit belief operator: agent i explicitly believes that α if and only if α is included in her belief base.

It is also worth considering belief correct states, according to which every fact that an agent explicitly believes has to be true.

Definition 3 (Belief correct state)

The state B = (B1, . . . , Bn, V ) is said to be belief correct if and only if, for every i ∈ Agt and for every α ∈ Bi, if α ∈ Bi then B |= α. The set of all belief correct states is denoted by SBC .

The following definition introduces the notion of "pooling", a simple form of belief base aggregation which consists in taking the union of the belief bases of the agents in the coalition.

Definition 4 (Pooling) Let B = (B1, . . . , Bn, V ) ∈ S and let G ∈ 2 Agt * . Then Pool G(B) = i∈G Bi.
A multi-agent belief model (MAB) is defined to be a state supplemented with a set of states, called context. The latter includes all states that are compatible with the common ground [START_REF] Stalnaker | Common ground[END_REF], i.e., the body of information that the agents commonly believe to be the case. BRGB means that B is a doxastic alternative for coalition G at B. The idea of the previous definition is that B is a doxastic alternative for coalition G at B if and only if, B satisfies all facts that are in the database resulting from pooling together the belief bases of the agents in the coalition.

The collective doxastic relation computed from the collective belief base is equal to the intersection of the individual doxastic relations computed from the individual belief bases:

Proposition 1 Let G ∈ 2 Agt * . Then RG = i∈G R {i} .
The following definition extends Definition 2 to the full language L1. Its formulas are interpreted with respect to MABs. (We omit Boolean cases, as they are defined in the usual way.) Definition 7 (Satisfaction relation (cont.)) Let (B, Cxt) be a MAB. Then:

(B, Cxt) |= α ⇐⇒ B |= α, (B, Cxt) |= 2Gϕ ⇐⇒ for all B ∈ Cxt : if BRGB then (B , Cxt) |= ϕ.
We consider the subclass of MABs that guarantee correctness of the agents' beliefs. Saying that (B, Cxt) satisfies BC is the same thing as saying that B ∈ Cxt and, for every i ∈ Agt, the relation Ri ∩ (Cxt × Cxt) is reflexive. The condition B ∈ Cxt in Definition 8 is necessary to make the agents' implicit beliefs correct, i.e., to make the formula 2iϕ → ϕ valid. For example, for B = ({p}, {p → q}, {p, q}) we have that both (B, {B}) and (B, {B, (∅, ∅, {p, q})}) satisfy BC .

As the following proposition highlights, belief correctness for MABs is completely characterized by the fact that the actual world is included in the agents' common ground and that the agents' explicit beliefs are correct in the sense of Definition 3. Let ϕ ∈ L1. We say that ϕ is valid for the class MAB (resp. MABBC ) if and only if, for every (B, Cxt) ∈ MAB (resp. (B, Cxt) ∈ MABBC ) we have (B, Cxt) |= ϕ. We say that ϕ is satisfiable for the class MAB (resp. MABBC ) if and only if ¬ϕ is not valid for the class MAB (resp. MABBC ).

Before concluding this section let us illustrate the belief base semantics for distributed explicit and implicit belief in a concrete example. Example Suppose Agt is a community of autonomous vehicles that have to get from a point A to a point B. There are two routes from A to B, r0 and r1. Route r0 is the shortest one so that each vehicle prefers to take it rather than to take route r1, if it does not believe that the others will also take it. Indeed, if all vehicles take the same route, then traffic will be congested and every vehicle will waste a considerable amount of time. The propositional variable takei,r x denotes that agent i takes route x, for x ∈ {0, 1}. Let us assume that each vehicle has the following pieces of information in its belief base:

• if a vehicle does not believe explicitly that the other vehicles will take route r0, then it will take route r0, • and if all vehicles take route r0 (resp. r1), then the traffic in r0 (resp. r1) will be congested.

Moreover, let us assume that each vehicle:

• explicitly believes that it does not believe explicitly that the other vehicles will take route r0.

In formal terms, for every i ∈ Agt we assume:

Bi = {β, γ, δi}
where

β def = i∈Agt,x∈{0,1}
¬ itake Agt\{i},r 0 → takei,r 0 ,

γ def = x∈{0,1} (takeAgt,r x → congested rx ), δi def = ¬ itake Agt\{i},r 0 ,
and, for every G ∈ 2 Agt * and for every x ∈ {0, 1}:

takeG,r x def = i∈G takei,r x .
Note that the explicit belief that β is an agent's explicit belief of order 2 since it mentions the agents' explicit beliefs of order 1. It is a routine exercise to verify that, for every B = (B1, . . . , Bn, V ) ∈ S where V is an arbitrary valuation, we have:

(B, S) |= i∈Agt ¬2icongested r 0 ∧ 2Agt congested r 0 .
This means that, while none of the vehicles has the implicit belief that route r0 will be congested, the community of vehicles has the distributed implicit belief that route r0 will be congested.

Axiomatics

We define the logic LDA D n axiomatized by the following principles:

Propositional calculus (P C) 2iϕ ∧ 2i(ϕ → ψ) → 2iψ (K2 i ) iα → 2iα (Int i ,2 i ) ϕ 2iϕ (N ec 2 i ) 2Gϕ → 2 G ϕ if G ⊆ G (M on2 G )
where Agt = {1, . . . , n}, i ∈ Agt and G, G ⊆ Agt.

LDA stands for "Logic of Doxastic Attitudes", an acronym introduced in [START_REF] Lorini | In praise of belief bases: Doing epistemic logic without possible worlds[END_REF] (see also [START_REF] Lorini | Rethinking epistemic logic with belief bases[END_REF]). For the axioms we take inspiration from that same paper as well as [START_REF] Fagin | Reasoning about Knowledge[END_REF]. We also define LDA D,T n by adding to the above the following axiom:

2Agt ϕ → ϕ (T2 Agt )
Soundness of LDA D n relative to MAB and of LDA D,T n relative to MABBC is straightforward. To prove completeness of the first item, we first show completeness relative to another class of models, namely notional doxastic models with distributed beliefs (NDMDs). The class of notional doxastic models (NDMs) was introduced for the first time in [START_REF] Lorini | In praise of belief bases: Doing epistemic logic without possible worlds[END_REF]. 7To do this we start from completeness relative to a weaker class of models called quasi-NDMDs, and then we show how to apply several transformations to a quasi-NDMD satisfying a given formula ϕ in order to turn it into a NDMD satisfying ϕ. Completeness of the second item follows by remarking that a reflexivity property is preserved throughout all of these transformations. For the sake of brevity we will omit finer details of the proofs. Definition 9 (NDMD) A notional doxastic model with distributed beliefs is a tuple M = (W, B, R, V ) where W is a set of worlds, B : Agt ×W -→ 2 L 0 is a doxastic function, R : 2 Agt ×W -→ 2 W is a notional function and V : Atm -→ 2 W is a valuation function, and, following the semantics:

(M, w) |= p ⇐⇒ w ∈ V (p), (M, w) |= ¬ϕ ⇐⇒ (M, w) |= ϕ, (M, w) |= ϕ ∧ ψ ⇐⇒ (M, w) |= ϕ and (M, w) |= ψ, (M, w) |= iα ⇐⇒ α ∈ B(i, w), (M, w) |= 2Gα ⇐⇒ ∀v ∈ R(G, w) : (M, v) |= ϕ.
M satisfies the following conditions:

R({i}, w) = α∈B(i,w) α M (NDMD1) R(G, w) = i∈G R({i}, w) (NDMD2)
where α M is the truth set of α in M . If w ∈ R(Agt, w) for all w ∈ W then we say that M is reflexive. The next step is to recover (NDMD2). For this we first use a filtration argument so that we can work with finite quasi-NDMDs.

Definition 10 (Quasi-NDMD) A quasi-NDMD is a model M = (W, B, R, V )
Lemma 1 If a formula ϕ is satisfiable in a (reflexive) quasi-NDMD then it is satisfiable in a finite (reflexive) quasi-NDMD. SKETCH OF PROOF. If M = (W, B, R, V ) is a quasi-NDMD and Σ is a finite set of formulas closed under subformulas, define MΣ = (WΣ, BΣ, RΣ, VΣ) such that:

• WΣ = {[w]Σ : w ∈ W }, where [w]Σ = {v ∈ W : ∀ϕ ∈ Σ, (M, w) |= ϕ iff (M, v) |= ϕ}; • for all i ∈ Agt and [w]Σ ∈ WΣ, BΣ(i, [w]Σ) = Σ ∩ w ∈[w] Σ B(i, w ); • for all G ⊆ Agt and [w]Σ ∈ WΣ, RΣ(G, [w]Σ) = {[v]Σ : ∃w ∈ [w]Σ, ∃v ∈ [v]Σ, v ∈ R(G, w )}; • for all p ∈ Atm, VΣ(p) = {[w]Σ : (M, w) |= p} if p ∈ Atm(Σ) and V (p) = ∅ otherwise.
Then MΣ is a finite quasi-NDMD such that for any w ∈ W and any formula ψ ∈ Σ, (M, w) |= ψ iff (MΣ, [w]Σ) |= ψ. By taking Σ = sub(ϕ), where sub(ϕ) is the set of subformulas of ϕ, we obtain the desired model. Moreover if M is reflexive then MΣ is also reflexive.

Lemma 2 For any quasi-NDMD

M = (W, B, R, V ) there is an equivalent model M = (W, B, R , V ) such that for all w ∈ W and G, G ⊆ Agt, if G ⊆ G then R (G , w) ⊆ R (G, w).
Our meaning of equivalent is: for any formula ϕ and w ∈ W , (M, w) |= ϕ iff (M , w) |= ϕ, and if M is reflexive then M is also reflexive. We call models satisfying this property of R downwards-closed.

If M = (W, B, R, V ) is a finite downwards-closed quasi-NDMD where W = {w1, . . . , wn}, define the expansion M of M as follows: M = (W , B , R , V ) where

• W = i n Wi, where for all i n, Wi = {zσ : σ ∈ S1,i × • • • × Sn,i}, and for all i, k n, S k,i = {G ⊆ Agt : wi ∈ R(G, w k ) and ∀G ⊆ Agt}, if G G then wi ∈ R(G , w k ); • for all i n, w ∈ Wi and j ∈ Agt, B (j, w) = B(j, wi); • for all i, j n, w ∈ Wi, v = zσ ∈ Wj and G ⊆ Agt, v ∈ R (G, w) iff G ⊆ σ[i]; • for all p ∈ Atm, i n and w ∈ Wi, w ∈ V (p) iff wi ∈ V (p).
Lemma 3 M is a finite quasi-NDMD verifying (NDMD2). Moreover for any formula ϕ, i n and w ∈ Wi, (M , w) |= ϕ iff (M, wi) |= ϕ, and if M is reflexive then M is also reflexive.

Proposition 4 If ϕ is satisfiable in a (reflexive) quasi-NDMD then there exists a finite (reflexive) quasi-NDMD M = (W, B, R, V ) satisfying ϕ and verifying (NDMD2). SKETCH OF PROOF. If ϕ is satisfiable in a (reflexive) quasi-NDMD then ϕ is also satisfiable in a finite (reflexive) quasi-NDMD, and in a finite (reflexive) downwards-closed quasi-NDMD. By expanding this last model in the manner described above we get a model satisfying ϕ which verifies the desired properties.

We can now recover (NDMD1) by expanding belief bases in order to contract α∈B(i,w) α M for a given i and w in a given model M . Let's take stock. We have presented a new semantics for epistemic logic exploiting the concept of belief base and shown that it offers a natural framework for modelling the notion of distributed belief and, more generally, belief aggregation in a multi-agent setting. Unlike the belief merging approach that only deals with aggregation of propositional beliefs, our approach takes aggregation of higher-order beliefs into consideration. We have studied four notions of distributed belief, one based on belief pooling (i.e., 2G), its variant for potentially introspective coalitions (i.e., 2 pi G ), another one based on belief combination which guarantees collective belief consistency (i.e., 2 dc G ), and its variant for potentially introspective coalitions (i.e., 2 pi,dc

G

). The logical framework we presented is static, as the agents' belief bases do not change. Future work will be devoted to bringing into our framework different types of belief change operations including belief base expansion, contraction and revision. The extended framework will allow us to study the interplay between belief change and belief aggregation. For instance, following [START_REF] Roelofsen | Distributed knowledge[END_REF][START_REF] Van Der Hoek | Group knowledge is not always distributed (neither is it always implicit)[END_REF], we plan to investigate whether it should be possible for the members of a coalition to come to individually believe that a certain fact ϕ is true, through information exchange, when they distributively believe that ϕ.

Definition 5 (

 5 Multi-agent belief model) A multi-agent belief model (MAB) is a pair (B, Cxt), where B ∈ S and Cxt ⊆ S. The class of MABs is denoted by MAB.Note that in this definition we do not require B ∈ Cxt. A MAB (B, Cxt) such that Cxt = S is said to be complete, since S is conceivable as the complete (or universal) context which contains all possible states. The following definition introduces the concept of doxastic alternative. Definition 6 (Doxastic alternatives) Let G ∈ 2 Agt * . Then RG is the binary relation on the set of states S such that, for all B = (B1, . . . , Bn, V ), B = (B 1 , . . . , B n , V ) ∈ S: BRGB if and only if ∀α ∈ Pool G(B) : B |= α.

Definition 8 (

 8 Belief correct MAB) The MAB (B, Cxt) is belief correct (BC ) if and only if B ∈ Cxt and, for every i ∈ Agt and for every B ∈ Cxt, B RiB . The class of MABs satisfying BC is denoted by MABBC .

Proposition 2 A

 2 MAB (B, Cxt) satisfies BC if and only if B ∈ Cxt and Cxt ⊆ SBC .

2 ) 3

 23 following the same definition as above except conditions (NDMD1) and (NDMD2) are replaced by the following:Proposition If a given formula ϕ is consistent in LDA D n (resp. LDA D,T n ) then it is satisfiable in a quasi-NDMD (resp. a reflexive quasi-NDMD). PROOF. If ϕ is consistent in LDA D n then it is satisfiable in the canonical model M C = (W C , B C , R C , V C), where:• W C is the set of maximal consistent sets for LDA D n ; • for all w ∈ W C , i ∈ Agt, and α ∈ L0, α ∈ B C (i, w) iff iα ∈ w; • for all w, v ∈ W C and G ⊆ Agt, v ∈ R(G, w) iff for all ϕ such that 2Gϕ ∈ w, ϕ ∈ v; • for all w ∈ W C and p ∈ Atm, w ∈ V C (p) iff p ∈ w. M C is a quasi-NDMD. If ϕ is consistent in LDA D,Tn then we can define in the same manner a canonical model M C T in which worlds are the maximal consistent sets for LDA D,T n . M CT is a reflexive quasi-NDMD.

Proposition 5

 5 If ϕ is satisfiable in a finite (reflexive) quasi-NDMD M = (W, B, R, V ) verifying (NDMD2), then ϕ is satisfiable in a finite (reflexive) NDMD.SKETCH OF PROOF. Let T (M ) = w∈W i∈AgtAtm(B(i, w)) be the (finite) terminology of M and let f : Agt × W -→ Atm \ (T (M ) ∪ Atm(ϕ) be an injective function (such a function exists because Atm is infinite). Define M = (W, B , R, V ) such that for all i ∈ Agt and w ∈ W , B (i, w) = B(i, w) ∪ {f (i, w)} andV (p) =      V (p) if p ∈ T (M ) ∪ Atm(ϕ) R({i}, w) if p = f (i, w) ∅ otherwise.

In the case of conformity-a well-known phenomenon studied in social psychology[START_REF] Asch | Studies of independence and conformity: a minority of one against a unanimous majority[END_REF][START_REF] Festinger | Informal social communication[END_REF]-, the agents' beliefs about the product qualities are generally aligned with (and supported by) their higher-order beliefs about the others' beliefs about the product qualities. On the contrary, in the case of anticonformity[START_REF] Willis | Two dimensions of conformity-nonconformity[END_REF], the agents' 1-order beliefs and their higher-order beliefs about the others' beliefs are generally misaligned.

The term 'notional' is borrowed from[START_REF] Dennett | The Intentional Stance[END_REF] (see, also,[START_REF] Konolige | A deduction model of belief[END_REF]). According to Dennett, an agent's notional world is a world at which all the agent's explicit beliefs are true.

ACKNOWLEDGEMENTS

Support from the ANR-3IA Artificial and Natural Intelligence Toulouse Institute and from the ANR project CoPains ("Cognitive Planning in Persuasive Multimodal Communication", grant number ANR-18-CE33-0012) is gratefully acknowledged.

We can show by induction on the structure of ψ that for all ψ ∈ sub(ϕ) and for all w ∈ W , (M, w) |= ψ iff (M , w) |= ψ. In particular (M , w) |= ϕ. Moreover M is a finite NDMD, and if M is reflexive then M is still reflexive.

Proposition 6 If ϕ is satisfiable in a finite NDMD (resp. a finite reflexive NDMD) then ϕ is satisfiable in MAB (resp. MABBC ). SKETCH OF PROOF. If Agt = {1, . . . , n}, let M = (W, B, R, V ) be a finite NDMD satisfying ϕ. Define for each w ∈ W , B w = (B w 1 , . . . , B w n , V w ) where V w = {p ∈ Atm : w ∈ V (p)} and for all i ∈ Agt, B w i = B(i, w). Define Cxt = {B w : w ∈ W }. Then for any formula ψ and any w ∈ W , (M, w) |= ψ iff (B w , Cxt) |= ψ, and in particular if (M, w) |= ϕ for some w ∈ W then (B w , Cxt) |= ϕ. Moreover if M is reflexive then (T2 Agt ) is valid in M and therefore in Cxt, hence (B w , Cxt) satisfies BC .

Corollary 1 If a formula ϕ is consistent in LDA D n (resp. LDA D,T n ) then it is satisfiable in MAB (resp. MABBC ).

Model checking

The following is a compact formulation of the model checking problem for the language L1.

Model checking Given: ϕ ∈ L1 and a finite B ∈ S.

Question: Do we have (B, S) |= ϕ?

Belief correct model checking Given: ϕ ∈ L1 and a finite B ∈ SBC .

Question: Do we have (B, SBC ) |= ϕ?

where the state B = (B1, . . . , Bn, V ) is said to be finite if V and every Bi are finite. Note that, thanks to Proposition 2, the MAB (B, SBC ) in the belief correct variant of model checking belongs to the model class MABBC , as expected.

In [START_REF] Lorini | Exploiting belief bases for building rich epistemic structures[END_REF], it is proved that the previous two problems are PSPACEhard, when considering the fragment of L1 with only singleton coalition operators of type 2 {i} . We are going to prove that the two problems are in PSPACE.

Figure 1 shows a generic algorithm for model checking of a formula ϕ in a given finite state B with an abstract function rel. With the logic defined in this section, the call rel(B, B , G) checks that BRGB . Since α in checking that B |= α (see Definition 6) does not contain any implicit belief operator, it is reducible to the propositional problem by stating any explicit belief as a fresh proposition, hence, checking BRGB can be performed in polynomial time and space.

The following theorem establishes PSPACE membership of the model checking problem.

Theorem 1 If rel is evaluated in polynomial space, then the generic model checking algorithm of Figure 1 runs in polynomial space. SKETCH OF PROOF. The depth of nested calls in mc(B, ϕ) is bounded by the size of ϕ. The local memory used by the recursive call is polynomial in the size of the initial B and the size of ϕ. Note that the loop "for all B . . . " can be performed by enumerating the B containing correct subformulas of formulas in the initial B and in the initial formula ϕ. There is an exponential number of such B but storing the current B requires only a polynomial amount of space.

For belief correct model checking we just check each time that the states B under consideration are correct: this can be done in polynomial time, hence in polynomial space. Note that the algorithm for belief correct model checking is the same, except that we check that each state considered during the execution is belief correct (Definition 3).

Corollary 2 Both model checking and belief correct model checking are in PSPACE.

The algorithm of Figure 1 and Theorem 1 are important for the other PSPACE-membership results given in the rest of the paper.

Introspective variant

The following definition introduces a variant of the notion of doxastic alternative for potentially introspective (pi) coalitions.

Definition 11 (Doxastic alternatives for pi coalitions) Let G ∈ 2 Agt * . Then, R pi G is the binary relation on the set S such that, for all B = (B1, . . . , Bn, V ), B = (B 1 , . . . , B n , V ) ∈ S:

BR pi

G B means that B is a doxastic alternative for the potentially introspective coalition G at B. According to the previous definition, if coalition G is potentially introspectively then her set of doxastic alternatives at B (i.e., R pi G (B)) includes all and only those states that satisfy the coalition's explicit beliefs, and that are for coalition G subjectively equivalent to B. Note that R pi G ⊆ RG since item (i) is exactly the definition of the relation RG.

As the following proposition indicates, if a coalition is potentially introspective, then its implicit beliefs are closed under positive and negative introspection. PROOF. We first prove transitivity. Suppose

The following proposition generalizes Proposition 1 to the introspective case.

By means of the new relation R pi G , we can define a variant of the language L1(Atm, Agt), denoted by L2(Atm, Agt), in which every implicit belief operator 2G is replaced by a potentially introspective variant of it of the form 2 pi G , where 2 pi G ϕ has to be read "if coalition G was potentially introspective, it would implicitly believe that ϕ" (or "coalition G implicitly believes that ϕ, under the assumption that it is potentially introspectively"). The language L2(Atm, Agt) is defined by the following grammar:

where α ranges over L0(Atm, Agt) and G ranges over 2 Agt * . Like the operator 2G, the operator 2 pi G is interpreted relative to a MAB (B, Cxt), as follows:

Model checking and belief correct model checking for L2 are the same as the model checking problems defined in Section 5 except that the input formula ϕ is now in L2.

Theorem 2 Model checking and belief correct model checking for L2 is in PSPACE.

SKETCH OF PROOF. To satisfy the hypothesis of Theorem 1, we need to check that BR pi G B can be checked in polynomial space. Note that formulas α are in L0 in Point (i) in Definition 11. So checking that B |= α can be performed in polynomial time, since it is close to model checking of a propositional formula. Therefore BR pi G B can be checked in polynomial time, and thus in polynomial space.

Consistency

The pooling operation we defined in Section 3 does not necessarily guarantee consistency of the collective belief base resulting from the aggregation of the individual belief bases. Indeed, two agents in a coalition may have contradictory explicit beliefs so that, by pooling together their belief bases, we obtain an inconsistent belief base for the collective. In this section we study a different belief aggregation operation, called belief base combination, that warrants belief base consistency. Before defining it, we need some preliminary notions.

We denote by MCS G(B) the set of maximally consistent subsets of coalition G's belief base.

Definition 12 Let G ∈ 2 Agt * and B ∈ S. Then, X ∈ MCS G(B) if and only if:

• ||X|| S = ∅, and • there is no X ⊆ Pool G(B) such that X ⊂ X and ||X || S = ∅, where, for every Y ⊆ L0, ||Y || S = {B ∈ S : ∀α ∈ Y, B |= α}.

It is worth noting that the definition of MCS G(B) can be formulated in terms of propositional consistency. To see this, let L PROP be the propositional language built from the set of atomic formulas Atm + = Atm ∪ {pi,α : i ∈ Agt and α ∈ L0} and let tr be the following translation from L0 to L PROP :

For each X ⊆ L0, we define tr (X) = {tr (α) : α ∈ X}. Moreover, we say that X is propositionally consistent if and only if ⊥ ∈ Cn tr (X) , where Cn is the classical deductive closure operator over the propositional language L PROP . The following holds.

Proposition 9 Let X ⊆ L0. Then, ||X|| S = ∅ if and only if tr (X) is propositionally consistent.

PROOF. Let W be the set of all valuations for the propositional language L PROP . There exists a bijection f : W -→ S such that (f (w), S) |= α iff w |= tr (α), for all w ∈ W and α ∈ L0. Propositional consistency of tr (X) means that we can find a valuation w ∈ W where all formulas in tr X hold. Thus, there exists B ∈ S such that f (w) = B and (B , S) |= α for all α ∈ X. The left-to-right direction can be proved in an analogous way.

By the previous proposition, the set MCS G(B) can be defined in an equivalent way by replacing ||X|| S = ∅ by ⊥ ∈ Cn tr (X) in the first item and ||X || S = ∅ by ⊥ ∈ Cn tr (X ) in the second item of Definition 12.

The following definition introduces the notion of belief base combination.

Definition 13 (Combining) Let B = (B1, . . . , Bn, V ) ∈ S. Then,

CombG(B) is the result of combining the individual belief bases in B of coalition G's members. This notion of belief base combination is inspired by [START_REF] Baral | Combining multiple knowledge bases[END_REF][START_REF] Baral | Combining knowledge bases consisting of first-order analysis[END_REF]. It is slightly different from belief merging in the sense of [START_REF] Konieczny | Merging information under constraints: A logical framework[END_REF] since it takes the intersection of all maximally consistent subsets of coalition G's belief base, while in belief merging the intersection of a selection of the MCSs is taken. This parallels the distinction between full meet contraction and partial meet contraction in the belief revision literature [START_REF] Alchourrón | On the logic of theory change: Partial meet contraction and revision functions[END_REF]. 8 We leave for future work the analysis of a notion of distributed belief more in line with such a notion of belief merging using a selection function on the set MCS G(B).

Doxastically consistent (dc) coalitions are coalitions whose doxastic accessibility relations are computed after having combined the individual belief bases of its members. Definition 14 (Doxastic alternatives for dc coalitions) Let G ∈ 2 Agt * . Then, R dc G is the binary relation on the set S such that, for all B = (B1, . . . , Bn, V ), B = (B 1 , . . . , B n , V ) ∈ S: 8 For more details about the differences between belief base combination and belief base merging, see [START_REF] Konieczny | On the difference between merging knowledge bases and combining them[END_REF].

Accessibility relations of type R dc G can be used to define a variant of the language L1(Atm, Agt), denoted by L3(Atm, Agt), for doxastically consistent coalitions. It is defined by the following grammar:

where α ranges over L0(Atm, Agt) and G ranges over 2 Agt * . The operator 2 dc G ϕ has the following semantics:

As the following proposition highlights, in a complete model, a doxastically consistent coalition cannot implicitly believe a contradiction. SKETCH OF PROOF. To satisfy the hypothesis of Theorem 1, we only need to show that CombG(B) can be computed in polynomial space in B. To compute CombG(B), we consider all subsets X ⊆ Pool G(B) by lexicographic order. For each such X, we check whether it is consistent and any superset of it is not. Checking propositional consistency of a subset of L0-formulas is in NP, thus in PSPACE. So CombG(B) can be computed in polynomial space.

In order to deal with potentially introspective coalitions, the notions of MCS and belief base combination need to be redefined, in order to warrant belief consistency. We denote by MCS pi G (B) the set of maximally consistent subsets of a potentially introspective (pi) coalition G's belief base.

Definition 15 Let G ∈ 2 Agt * and B ∈ S. Then, X ∈ MCS pi G (B) if and only if:

Belief base combination for pi coalitions goes as follows.

Definition 16 (Combining for pi coalitions) Let B = (B1, . . . , Bn, V ) ∈ S. Then,

X.

Comb pi G (B) is the result of combining the individual belief bases in B of the pi coalition G's members.

The following definition introduces the notion of doxastic accessibility relation for coalitions which are, at the same time, potentially introspective (pi) and doxastically consistent (dc). Accessibility relations of type R pi,dc G can be used to define a fourth variant of the language L1(Atm, Agt), denoted by L4(Atm, Agt), for potentially introspective, doxastically consistent coalitions. It is defined by the following grammar:

where α ranges over L0(Atm, Agt) and G ranges over 2 Agt * . The operator 2 pi,dc G ϕ has the following semantic interpretation:

The following proposition is the counterpart of Proposition 10 for pi coalitions: in a complete model, a potentially introspective, doxastically consistent coalition cannot implicitly believe a contradiction. SKETCH OF PROOF. Again, to satisfy the hypothesis of Theorem 1, we only need to show that Comb pi G (B) can be computed in polynomial space in B. In the spirit of the proof of Theorem 3, it is sufficient to show that checking X ∈ MCS pi G (B) (Definition 15) can be turned into an algorithm in polynomial space. Let us just explain how to check that ||X|| ∼ G (B,S) = ∅. For that, we enumerate in polynomial space all B with Pool G(B) = Pool G(B ) and accepts if formulas in X are true in B (that can be checked in polynomial time since formulas in X are in L0).