Thierry Pérez
email: thierry.perez@imbe.fr

César Ruiz

Marcela Villegas-Plazas

Olivier P Thomas

Howard Junca

Specialized microbiome of the cave-dwelling sponge Plakina kanaky (Porifera, Homoscleromorpha)

Keywords: Microbiome, Entotheonella sp, vertical transmission, species-specific community

The recent description of the polychromatic sponge Plakina kanaky revealed original microsymbionts, with some morphotypes recorded for the first time in Homoscleromorpha, and others never before observed in other sponge groups. Illumina 16S amplicon sequencing was used to characterize this microbial community by comparing contents of seven specimens of this Plakinidae with five other sponge species: 1 Homoscleromopha of the Oscarellidae family and 4 Demospongiae. A total of 256,458 sequences of the hypervariable V5-V6 region of the 16S rRNA gene were clustered into 2,829 OTUs at 97% similarity, with Proteobacteria, Poribacteria and Chloroflexi being the most abundant phyla. The Plakina kanaky specific community appeared to be mainly composed by 5 OTUs representing about 10% of the total microbiome. Among these, the filamentous bacterium Candidatus Entotheonella, which was among the dominant morphotypes previously observed in the mesohyl and the larvae of P. kanaky, was detected in all studied specimens. However, other original and dominant morphotypes could not be assigned to a known prokaryotic taxon. This cave dwelling sponge species harbors a distinctive microbiome composition of potential taxonomic and metabolic novelties that may be linked to its ecological success in such extreme environments.

Introduction

Sponges (Phylum Porifera) are well known marine sessile organisms living in close association with symbiotic Bacteria, Archaea and unicellular Eukaryotes. The microbial communities of these holobionts are often complex in terms of diversity and biotic interactions [START_REF] Thomas | Diversity, structure and convergent evolution of the global sponge microbiome[END_REF]. Transmission electron microscopy has allowed the description of a great variety of microbial morphotypes that can be found in the sponge mesohyl, inside specialized cells and even inside their nuclei. In some sponge species, known as bacteriosponges or High Microbial Abundance (HMA) sponges, the microbial part of the holobiont can constitute up to 40% of the total biomass represented by a diverse microbial community [START_REF] Vacelet | Etude en microscopie électronique de l'association entre bactéries et spongiaires du genre Verongia (Dyctioceratida)[END_REF][START_REF] Webster | Marine sponges and their microbial symbionts: love and other relationships[END_REF][START_REF] Hardoim | Phylogenetically and spatially close marine sponges harbour divergent bacterial communities[END_REF][START_REF] Uriz | Endosymbiotic Calcifying Bacteria: A New Cue to the Origin of Calcification in Metazoa?[END_REF]. It is well accepted that sponge-associated communities are therefore a significant source of unique microbial diversity [START_REF] Hentschel | Molecular evidence for a uniform microbial community in sponges from different oceans[END_REF][START_REF] Taylor | Sponge-Associated Microorganisms: Evolution, Ecology, and Biotechnological Potential[END_REF][START_REF] Thomas | Functional genomic signatures of sponge bacteria reveal unique and shared features of symbiosis[END_REF][START_REF] Reveillaud | Host-specificity among abundant and rare taxa in the sponge microbiome[END_REF]. Sponges harbor rather species-specific prokaryotic communities, composed both by generalist and specialist microbes [START_REF] Easson | Phylogenetic signal in the community structure of host-specific microbiomes of tropical marine sponges[END_REF][START_REF] Thomas | Diversity, structure and convergent evolution of the global sponge microbiome[END_REF]. The rate of host specificity depends on the modes of symbiont transmission, which can be horizontal (from the environment), vertical (through reproduction) or a mixture of both (for a review see [START_REF] Thacker | Sponge-Microbe Symbioses[END_REF]. To identify processes of vertical transmission, the symbionts transmitted from parents to offspring can be distinguished using microscopic techniques or traced using molecular tools within the larvae. These associations are often assumed to represent the composite result of co-evolutionary processes, which thus shape the microbiome diversity, influenced by the host specificity and the network of metabolic interactions within the holobiont [START_REF] Thomas | Functional genomic signatures of sponge bacteria reveal unique and shared features of symbiosis[END_REF][START_REF] Pita | The sponge holobiont in a changing ocean: from microbes to ecosystems[END_REF]. Recent advances in next generation sequencing technologies have led to important progress in the study of microorganisms in their natural environments, surpassing some of the limitations of culture-based methodologies, and revealing new insights into sponge symbiotic diversity and functions [START_REF] Schmitt | Assessing the complex sponge microbiota: core, variable and species-specific bacterial communities in marine sponges[END_REF][START_REF] Hentschel | Genomic insights into the marine sponge microbiome[END_REF]. For instance, the retrieval of 16S rRNA gene sequences from metagenomic DNA revealed 52 microbial phyla associated with sponges [START_REF] Webster | The sponge hologenome[END_REF], the most common of which are Proteobacteria, Actinobacteria, Chloroflexi, Bacteroidetes, Acidobacteria, Cyanobacteria, Firmicutes and the candidate phylum Poribacteria. The Archaea lineages Crenarchaeota and Euryachaeota are also well represented [START_REF] Fieseler | Discovery of the novel candidate Phylum "Poribacteria" in marine sponges[END_REF][START_REF] Taylor | Sponge-Associated Microorganisms: Evolution, Ecology, and Biotechnological Potential[END_REF][START_REF] Webster | Deep sequencing reveals exceptional diversity and modes of transmission for bacterial sponge symbionts[END_REF][START_REF] Thomas | Diversity, structure and convergent evolution of the global sponge microbiome[END_REF]). Among Porifera, Homoscleromorpha is the smallest class, currently composed of 124 valid species distributed within two families, Oscarellidae and Plakinidae, and nine genera [START_REF] Van Soest | World Porifera Database[END_REF]. Sponges of this group often inhabit semi-dark or dark habitats such as overhangs or submarine caves in shallow waters. The current knowledge of the symbiotic microorganisms of this group is mostly based on electron microscopy observations, with molecular studies enabling a better assessment of the microbial diversity remaining scarce [START_REF] Gloeckner | Unique and species-specific microbial communities in Oscarella lobularis and other Mediterranean Oscarella species (Porifera: Homoscleromorpha)[END_REF]. The most recent cytological descriptions of various species have revealed dense microbial communities in which the dominant prokaryotic morphotypes are often used as diagnostic traits for the challenging taxonomy of this sponge group. Oscarellidae are mostly LMA sponges, while Plakinidae are generally HMA sponges (Boury-Esnault, [START_REF] Boury-Esnault | Sponges without skeleton: A new Mediterranean genus of Homoscleromopha (Porifera, Demospongiae)[END_REF][START_REF] Muricy | Cytological evidence for cryptic speciation in Mediterranean Oscarella species (Porifera, Homoscleromorpha)[END_REF][START_REF] Muricy | Anatomy, cytology and symbiotic bacteria of four Mediterranean species of Plakina Schulze, 1880 (Demospongiae, Homosclerophorida)[END_REF][START_REF] Gloeckner | Unique and species-specific microbial communities in Oscarella lobularis and other Mediterranean Oscarella species (Porifera: Homoscleromorpha)[END_REF][START_REF] Pérez | Description of the first Caribbean Oscarellidae (Porifera: Homoscleromorpha)[END_REF]. Plakina kanaky [START_REF] Ruiz | Integrative taxonomic description of Plakina kanaky, a new polychromatic sponge species from New Caledonia (Porifera: Homoscleromorpha)[END_REF] is an HMA polychromatic Homoscleromorpha sponge recently described in New Caledonia and harboring an original prokaryotic community, containing, among other morphotypes, a globulous shape never before observed in any Porifera, as well as a filamentous morphology recorded for the first time in Homoscleromorpha [START_REF] Ruiz | Integrative taxonomic description of Plakina kanaky, a new polychromatic sponge species from New Caledonia (Porifera: Homoscleromorpha)[END_REF]. TEM observations indicated a common microbial symbiotic community among the various specimens collected, with only subtle variations suspected to be related to sponge chromotypes. The vertical transmission of the two main morphotypes has also been proposed after their observation inside the coeloblastula larva of P. kanaky (see Ruiz et al. 2015 and Fig 1). This study aims to characterize the prokaryote community associated with P. kanaky , combining the previous morphological description with molecular analyses of the 16S rRNA gene amplicon sequencing in order to: 1) define the patterns of ribotypes detected in P. kanaky individuals compared to other reported sponge microbiomes in the present study; 2) determine if part of the microbial community of P. kanaky is species-specific as a first step towards the identification of the original morphotypes previously observed; and 3) define the putative mode of transmission of the microbial taxa representatives deduced from Operational Taxonomic Units (OTUs) and their reported lifestyles.

[FIG 1]

Material and Methods

Sponge collection and preservation

Seven specimens of P. kanaky were collected in November 2010 and February 2013 by scuba diving in New Caledonia (Tropical Southwestern Pacific Ocean), at Bourail (21°24.303' S, 165°27.811' E), Hienghène (20°36.756' S, 164°56.958' E), and Touho (20°45.503' S, 165°16.964' E), in small dark cavities located between 15 and 25 m depth along the outer slope of the reef (Table 1). These samples were also used for the description of the new species [START_REF] Ruiz | Integrative taxonomic description of Plakina kanaky, a new polychromatic sponge species from New Caledonia (Porifera: Homoscleromorpha)[END_REF]. For molecular analyses, specimens were preserved in ethanol 95%. For comparative and outgroup purposes, we determined the microbial composition using the same approach of four sponge species (one specimen per species): Amphimedon compressa, Aplysina cavernicola, Petrosia ficiformis (Demospongiae) and Oscarella balibaloi (Homoscleromorpha).

[TABLE 1]

Metagenomic DNA extraction, 16S rRNA gene library construction and sequencing

Total DNA extraction from a small (2 cm 3 ) fragment of sponge was performed using a FastDNA TM SPIN KIT (MP Biomedicals). PCR amplicons of the hypervariable V5-V6 region of the 16S rRNA genes were obtained from sponge metagenomic DNA as a template using the primers 807F [5´-GGATTAGATACCCBRGTAGTC-3´] and 1050R [5´-AGYTGDCGACRRCCRTGCA-3´] [START_REF] Bohorquez | In-depth characterization via complementing culture-independent approaches of the microbial community in an acidic hot spring of the Colombian Andes[END_REF]. The PCR cycle conditions and Illumina library preparations were performed as previously described [START_REF] Milici | Low diversity of planktonic bacteria in the tropical ocean[END_REF][START_REF] Villegas-Plazas | Variations in microbial diversity and metabolite profiles of the tropical marine sponge Xestospongia muta with season and depth[END_REF] .

Bioinformatics analysis

All sequence analyses were performed using QIIME 1.9.0 [START_REF] Kuczynski | Using QIIME to analyze 16S rRNA gene sequences from microbial communities[END_REF]) and the Phyloseq R package 1.5.21 [START_REF] Mcmurdie | Phyloseq: An R package for reproducible interactive analysis and graphics of microbiome census data[END_REF]. A set of Illumina reads was initially filtered and split according to the barcodes. Chimeric sequences were identified and excluded from the datasets using Usearch 6.1. A multi-step open-reference OTU picking workflow was performed within the QIIME system. OTUs were picked, assigning the reads to species groups based on 97% sequence similarity. All analyses and multivariate tests were performed using the OTUs relative abundances of the detected microbial community compositions on each sample. Each OTU with a relative abundance higher than 1% was considered as a key microbial community component of a given sample, and among these, the species-specific microbial community composition was defined as the set of OTUs present in all specimens of P. kanaky. Alpha and Beta diversity analyses were performed using QIIME workflow and R software using Phyloseq (McMurdie and Holmes 2013) and Vegan R packages [START_REF] Oksanen | Vegan: community ecology package. R package version 2[END_REF]. All bar plots of abundance, diversity and distribution of all phyla were prepared with the Phyloseq R package. Weighted Unifrac distances were calculated between the microbial communities of the sponge specimens based on OTU presence and relative abundance [START_REF] Lozupone | UniFrac: an effective distance metric for microbial community comparison[END_REF]. The same distance method was used for Beta diversity using the function Adonis using R software. A heatmap based on a Discriminatory Analysis of Partial Least Square Regression PLS-DA analysis was constructed using the online platform MetaboAnalyst [START_REF] Xia | Using MetaboAnalyst 3.0 for Comprehensive Metabolomics Data Analysis[END_REF]. We compared the 50 most important OTUs (Variables Important in Projection, VIP) on the basis of their occurrence in terms of number of reads distributed among all P. kanaky samples.

Results

A total of 256,458 sequences of 250 nucleotides on average were obtained and clustered into 2,829 OTUs at 97% similarity (Table 1). Rarefaction analyses showed that all P. kanaky samples similarly approached an asymptote near 6,000 sequence reads retrieved per amplicon describing more than 96% of the associated microbial communities, according to Good's coverage estimation (Supp Fig. 1). Beta diversity analysis using Permanova test revealed that the microbial community in P. kanaky presents no significant differences among sites (F=0.83; p-value= 0.74) and chromotypes (F=0.78; p-value=0.76). The P. kanaky microbial community is composed of 14 formerly described Bacteria phyla and 1 Archaea lineage. Proteobacteria, Poribacteria and Chloroflexi are the predominant taxa constituting up to 70% of the bacterial community in all samples (Fig. 2A). At class level, 80% of OTUs were correctly assigned, however at the order level we found that 40% of the sequences could not be assigned to a known taxon. Four abundant microbial classes represent between 60 and 80% of the OTUs recorded in the P. kanaky microbiome. They belong to the Deltaproteobacteria (12 -40%), to an unclassified class of Poribacteria (12 -38%), to the SAR202 cluster (7 -22%), and to the Gammaproteobacteria (4 -10%) (Fig. 2A-B). At the order level, only the order Entotheonellales (Deltaproteobacteria) could be identified among the 4 dominant classes recorded in the P. kanaky microbiome. However, other less abundant orders could be identified. They belong to the Acidimicrobiales (Class Acidimicrobia), 2-7% of the OTUs recorded, and to the Caldilineales (Class Anaerolineae) with 0.5-5% of the total microbial community. Archaea also represents less than 2% on average of the total prokaryotic community associated with P. kanaky, and they are represented by two families of the class Thaumarchaeota: Nitrososphaeraceae and Cenarchaeaceae.

[FIG 2]

We performed intraspecific and interspecific comparisons of the microbial communities by generating and including in our datasets the microbiomes of another Homoscleromorpha and of three Demospongiae. This enabled us to compare the ribotypes detected and to characterize the species-specific community associated with P. kanaky specimens. Compared to other sponges analyzed, the largest difference (40%) was found between all P. kanaky specimens and Amphimedon compressa, while the community associated with the Homoscleromorph Oscarella balibaloi appeared as the most similar with less than 20% divergence between both microbial communities. Two Chloroflexi OTUs were found in all sponge samples, one affiliated to the SAR202 cluster and the other one to the class TK17.

The greatest divergences were observed when the abundance was not considered (unweighted). In this case, two groups corresponding to the Homoscleromorpha and Demospongiae were well separated (Fig. 2B). When the presence/absence of OTUs among samples were compared, such unweighted measures suggested that the microbial divergence between Homoscleromorpha samples is about 50%, whereas it reaches 70% with the Demospongiae Amphimedon compressa, whose microbiome is mainly composed by Betaproteobacteria of the EC94 order (89% of the OTUs). Since A. compressa is an HMA sponge species, it represents an exceptional contrasting pattern where phyla proposed to be LMA (Betaproteobacteria) and HMA (SAR202) indicators are coincidentally having a higher relative abundance. Searching for the common features of both Homoscleromorpha, we were able to detect some of the dominant OTUs of P. kanaky in O. balibaloi, one belonging to Poribacteria and two Chloroflexi: the SAR202 cluster and the Caldilineales order. Taken together, these three taxonomic groups, which are shared by both species, represent half of the total microbial community of the Oscarellidae. The species-specific community of P. kanaky appears to be composed of 5 bacterial OTUs recorded in all studied specimens and representing about 10% of the total microbial community. These specific OTUs belong to Acidimicrobiales, an order in the phylum Actinobacteria, to two different SAR202 cluster in the Chloroflexi phylum, to the order Sva0725 in the Acidobacteria phylum, and to the family Endozoicomonadaceae in the Gammaproteobacteria class. Also observed in all P. kanaky samples, the Archaea genus Nitrosopumilus (Cenarchaeaceae) can be considered as a minor part of the species-specific community.

To complete our view of the species-specific community of P. kanaky, we created a heatmap showing the distribution of the 50 VIP OTUs among the various P. kanaky chromotypes. This representation indicates that 6 OTUs are shared by at least 6 P. kanaky specimens. They were assigned to Candidatus solibacter, to the family Methylococcaceae, to SAR202, HTCC2089 (Gammaproteobacteria) and to Candidatus Entotheonella. This candidate genus is actually represented by 6 different OTUs heterogeneously distributed in all P. kanaky specimens (Fig. 3), but because of the genetic differences between Enthotheonella OTUs (divergences greater than 3%), this taxon was not originally detected as a part of the species-specific microbial community. However, even if not a single Entotheonella OTU could be detected in all specimens of P. kanaky, our previous electron microscopy observations of such morphotypes in all specimens and their larvae convinced us to assume this taxon as being part of the species-specific microbial community.

[FIG 3]

Discussion

The availability of high-throughput meta-barcoding approaches allows more accurate descriptions of the microbiomes associated with marine sponges. So far, only a few Homoscleromorpha have been investigated using such methods, although cytological observations of microbial morphotypes are often used as diagnostic traits to distinguish between species [START_REF] Ereskovsky | A new species of the genus Oscarella (Porifera: Homscleromorpha/ Plakinidae) from the North-West Pacific[END_REF][START_REF] Pérez | Oscarella balibaloi, a new sponge species (Homoscleromorpha: Plakinidae) from the Western Mediterranean Sea: cytological description, reproductive cycle and ecology: O. balibaloi: description, reproductive cycle and ecology[END_REF][START_REF] Ruiz | Descriptions of new sponge species and genus, including aspiculate Plakinidae, overturn the Homoscleromorpha classification[END_REF]. Previous work on the HMA Plakortis sp. has highlighted the occurrence of 8 bacterial phyla, among which the Chloroflexi and Acidobacteria that contain many spongespecific clusters [START_REF] Taylor | Sponge-Associated Microorganisms: Evolution, Ecology, and Biotechnological Potential[END_REF]. [START_REF] Sharp | Vertical transmission of diverse microbes in the tropical sponge Corticium sp[END_REF] studied the vertical transmission of some original phylotypes of Proteobacteria, Actinobacteria, Nitrospira and Cloroflexi in Corticium sp.; among these phyla, the closest previously known bacterial relatives were sulfur-oxidizing Gammaproteobacteria, nitrogen-fixing Rhizobiales and Actinobacteria, with the latter phylum being recognized for its ability to produce bioactive compounds [START_REF] Sharp | Vertical transmission of diverse microbes in the tropical sponge Corticium sp[END_REF].

Most of the species-specific community associated with P. kanaky can be classified among Proteobacteria and Poribacteria, but Actinobacteria and Chloroflexi are also recorded in abundance. Proteobacteria is mostly represented by Delta-Gamma-and Alphaproteobacteria. Four orders dominate, Entotheonellales (Deltaproteobacteria with 11 -38 % of relative abundance), an unidentified Poribacteria (12 -38%), a second unidentified order belonging to the class SAR202 (Chloroflexi), and Acidimicrobiales (Actinobacteria with 3 -7 % of relative abundance). Poribacteria have been associated with polyketides and methyl-branched fatty acid production [START_REF] Hochmuth | Linking chemical and microbial diversity in marine sponges: possible role for Poribacteria as producers of methyl-branched fatty acids[END_REF], suggesting also their involvement in the production of secondary metabolites. Three OTUs are related to the Chloroflexi class SAR202 and account for half of the total specific community. SAR202 has also a good number of representatives which have been recorded in soil, freshwater, marine, wastewater, and subsurface environments [START_REF] Morris | Prevalence of the Chloroflexi-Related SAR202 Bacterioplankton Cluster throughout the Mesopelagic Zone and Deep Ocean[END_REF]). While the global bacterial abundance generally decreases with ocean depth, the absolute abundance of SAR202 increases from the subsurface to the meso-and bathypelagic waters [START_REF] Varela | Latitudinal trends of Crenarchaeota and Bacteria in the meso-and bathypelagic water masses of the Eastern North Atlantic[END_REF]. This distribution pattern indicates that SAR202 is well adapted to the environmental conditions of the dark ocean, such as the small dark cavities where P. kanaky dwells. Using MICRO-FISH, [START_REF] Varela | Latitudinal trends of Crenarchaeota and Bacteria in the meso-and bathypelagic water masses of the Eastern North Atlantic[END_REF] found that uptake of L-Asp by SAR202 cells remains stable throughout the water column, suggesting that this cluster may play a key role in lower-surface and deep-ocean biogeochemistry. The order Acidimicrobiales contains mesophilic and moderate thermophilic species, of which all members are obligatory acidophilic and are found in iron-, sulfur-or mineral-sulfide rich environments. Species within this order are capable of ferrous iron and sulfur oxidation and ferric iron reduction [START_REF] Clark | Acidimicrobium ferrooxidans gen. nov., sp. nov.: mixed-culture ferrous iron oxidation with Sulfobacillus species[END_REF][START_REF] Stackebrandt | Proposal for a new hierarchic classification system, Actinobacteria classis nov[END_REF][START_REF] Johnson | Ferrimicrobium acidiphilum gen. nov., sp. nov. and Ferrithrix thermotolerans gen. nov., sp. nov.: heterotrophic, iron-oxidizing, extremely acidophilic actinobacteria[END_REF]. [START_REF] Taylor | Sponge-Associated Microorganisms: Evolution, Ecology, and Biotechnological Potential[END_REF] found several sponge-specific clusters in tropical sponges such as Agelas dilatata, Discodermia dissoluta, Plakortis sp. and Xestospongia muta from the Caribbean Sea, X. testudinaria from Indonesia and Dysidea sp. from the Southern China Sea. Entotheonella sp., first identified in Tetractinellid sponges [START_REF] Schmidt | Identification of the anifungal peptidecontaining symbiont of the marine sponge Theonella swinhoei as a novel delta-Proteobacterim, "Candidatus Entotheonella palauensis[END_REF][START_REF] Schirmer | Metagenomic analysis reveals diverse polyketide synthase gene clusters in microorganisms associated with the marine sponge Discodermia dissoluta[END_REF][START_REF] Brück | Identification of the bacterial symbiont Entotheonella sp. in the mesohyl of the marine sponge Discodermia sp[END_REF], is a bacterial group known for being a unique source of bioactive molecules [START_REF] Bewley | Two classes of metabolites from Theonella swinhoei are localized in distinct populations of bacterial symbionts[END_REF]Jaspars and Challis 2014;[START_REF] Wilson | An environmental bacterial taxon with a large and distinct metabolic repertoire[END_REF]. Widely distributed throughout the ocean, this phylotype is generally considered to be a free-living planktonic bacterium, acquired by sponges from the environment [START_REF] Wilson | An environmental bacterial taxon with a large and distinct metabolic repertoire[END_REF]. However, the cytological description of P. kanaky reported the predominance of a filamentous Entotheonella-like bacteria in all four chromotypes of P. kanaky as well as in their embryos [START_REF] Ruiz | Integrative taxonomic description of Plakina kanaky, a new polychromatic sponge species from New Caledonia (Porifera: Homoscleromorpha)[END_REF]. The present study confirms the occurrence of Entotheonella, and the great novelty of this first record in Homoscleromorph sponges also provides the first clues to its vertical transmission. In the species-specific community of P. kanaky, Entothenonella is represented by 12 different OTUs with 97% similarity. However, we believe that small genetic divergences in the hypervariable V5 -V6 region may explain the assignment of various OTUs to Candidatus Entotheonella sp. Considering the large specialized biosynthetic repertory reported for Entotheonella, its close association to P. kanaky likely confers great properties of adaptation to the whole sponge holobiont. Finally, regarding archaeal groups detected in P. kanaky, they had been reported with steady detection and low abundances in the microbiomes of most sponge species surveyed so far [START_REF] Thomas | Diversity, structure and convergent evolution of the global sponge microbiome[END_REF] with only few reported exceptions were these groups are predominant [START_REF] Turque | Environmental Shaping of Sponge Associated Archaeal Communities[END_REF][START_REF] Jackson | Archaea appear to dominate the microbiome of Inflatella pellicula deep sea sponges[END_REF]Garcia-Bonilla et al. 2019). Among them, the occurring OTUs assigned to Thaumarchaeota are associated with genera playing essential metabolic activities such as ammonia oxidation, which is one of the first reactions carried out to achieve nitrification [START_REF] Steger | Diversity and mode of transmission of ammonia-oxidizing archaea in marine sponges[END_REF]. Members related to the archaeal clade of the Nitrososphaera-Nitrosopumilus cluster [START_REF] Stieglmeier | Nitrososphaera viennensis gen. nov., sp. nov., an aerobic and mesophilic, ammonia-oxidizing archaeon from soil and a member of the archaeal phylum Thaumarchaeota[END_REF]) could be the globulous prokaryotic form described in P. kanaky adults and larvae [START_REF] Ruiz | Integrative taxonomic description of Plakina kanaky, a new polychromatic sponge species from New Caledonia (Porifera: Homoscleromorpha)[END_REF], a morphotype never before observed in sponges. This is a possibility still to be confirmed and requiring further investigation. It must be noted that the use of more specific molecular markers might reveal a much higher diversity of symbiotic Archaea in sponges, particularly in those living in extreme environments, and might thus highlight other putative adaptive capabilities. The microbial community associated to Plakina kanaky has no significant differences among chromotypes and/or sites. It is likely that this community is transferred from parents to offspring, while other OTUs can be acquired from the environment. Both types of acquisition are key processes which may improve the adaptive capabilities of a holobiont living in constraining environments such as underwater caves. The microbial diversity associated with Homoscleromorpha may thus partly explain the ecological success of this poorly known sponge group in dark and oligotrophic environments. This preliminary study may thus serve as a baseline for further research on the adaptative capabilities of Homoscleromorpha.

Figures

Figure 1 .

 1 Figure 1. Coeloblastula of Plakina kanaky with a detailed view of the filamentous morphotype associated with Candidatus Entotheonella sp., found in both the adult and the larvae of P. kanaky.

  Figure 1. Coeloblastula of Plakina kanaky with a detailed view of the filamentous morphotype associated with Candidatus Entotheonella sp., found in both the adult and the larvae of P. kanaky.

Figure 2 .

 2 Figure 2. (A) Occurrence of all phyla inferred from 16S amplicon Illumina reads with an average of 8504 sequences per specimen. (B) Relative class abundance inferred from 16S amplicon Illumina reads with an average of 9820 sequences per specimen and a hierarchical clustering through a weighted UniFrac matrix. (C) Similarity of the samples throughout the Bray Curtis method according to the number of shared OTUs. Plakina kanaky specimen's abbreviations: TO, Touho. BO, Bourail. HE, Hienghène. Bl, bleu. Br, Brick-Red. Ye, Yellow. Gr, Green.

Figure 3 .

 3 Figure 3. Heatmap comparing the 50 most abundant OTUs in Plakina kanaky chromotypes, and dendrogram indicating the intra-and interspecies relationships (top). Heat colours indicate number of reads per taxa, from zero reads (grey) to 3000 reads (yellow). Only the OTUs which are shared by at least 6 chromotypes are marked with black circles and listed in order of appearance (from top to bottom) on the heatmap. Red circles indicate the OTUs assigned to Candidatus Entotheonella. Plakina kanaky specimen's abbreviations: TO, Touho. BO, Bourail. HE, Hienghène. Bl, bleu. Br, Brick-Red. Ye, Yellow. Gr, Green.

Table 1 .

 1 Individuals and collection sites of the studied specimens (11) with the number of reads and OTUs (97 % similarity) per specimen. NC: New Caledonia; FR: France.

	Sponge species	sample code	Chromo type	Collection site	# Reads	# OTUs
	Plakina kanaky	130207-BO_Ye	Yellow (Ye)	Bourail (BO)	17024 852
	P. kanaky	130207BO_Br	Brick-Red (Br)	Bourail (BO)	12913 915
	P. kanaky	101107BO_Bl	Blue (Bl)	Bourail (BO)	5818	376
	P. kanaky	130205TO_Br	Brick-Red (Br)	Touho (TO)	10470 520
	P. kanaky	130205TO_Bl	Blue (Bl)	Touho (TO)	12713 556
	P. kanaky	130205HI_Ye	Yellow (Ye)	Hienghène (HI) 13351 413
	P. kanaky	130205HI_Gr	Green (Gr)	Hienghène (HI) 7201	409
	Amphimedon compressa	ACM.C3	NA	La Martinique	13203 298
	Aplysina cavernicola	FA.M5	NA	Marseille (FR)	38489 919
	Petrosia ficiformis	PF.M5	NA	Marseille (FR)	35031 737
	Oscarella balibaloi	OB.M7	NA	Marseille (FR)

Acknowledgments

We would like to thank EcosNord and Colciencias for financial support though bilateral research cooperation and mobility grants from 2012-2014 and 2015-2017. Funding was provided by the CNRS LIA MARRIO, TOTAL Foundation. The authors are grateful to the Institut de Recherche pour le Développement (IRD) for providing diving facilities in New Caledonia, and to Virginie Baldy (IMBE) for her contribution to the field work. We thank Eric JC Gálvez, Silke Kahl and Iris Plumeier, HZI, Braunschweig, for technical advice on amplicon preparation and sequencing, and to Dr. Erika García-Bonilla, Microbiomas Foundation, and Prof. Andrés Pinzón Velasco, Universidad Nacional de Colombia, for kindly supporting bioinformatic resources and analyses.

Supporting Infomation

Supplementary figure 1. (A) Chao1 indicator. This plot is showing the coverage of sequencing and the richness of all samples. (B) Rank abundance plot of OTUs and sequence accumulation on each specimen observed at 97% similarity in datasets. Plakina kanaky specimen's abbreviations: TO, Touho. BO, Bourail. HE, Hienghène. Bl, bleu. Br, Brick-Red. Ye, Yellow. Gr, Green.


N Alechina

N Bulling

S Demri

B Logan

On the Complexity of Resource-Bounded Logics

We revisit decidability results for resource-bounded logics and use decision problems for vector addition systems with states (VASS) to characterise the complexity of (decidable) model-checking problems. We show that the modelchecking problem for the logic RB±ATL is 2EXPTIME-complete by using recent results on alternating VASS. In addition, we establish that the model-checking problem for RBTL is decidable and has the same complexity as for RBTL * (the extension of RBTL with arbitrary path formulae), namely EXPSPACE-complete, proving a new decidability result as a by-product of the approach. Finally, we establish that the model-checking problem for RB±ATL * is decidable by a reduction to parity games, and show how to synthesise values for resource parameters.

Introduction

Resource-bounded logics [START_REF] Bulling | On the (Un-)Decidability of Model-Checking Resource-Bounded Agents[END_REF][START_REF] Bulling | Expressing properties of resource-bounded systems: The logics RBTL * and RBTL[END_REF][START_REF] Monica | On a logic for coalitional games with priced-resource agents[END_REF][START_REF] Alechina | Decidable model-checking for a resource logic with production of resources[END_REF][START_REF] Alechina | On the boundary of (un)decidability: Decidable model-checking for a fragment of resource agent logic[END_REF][START_REF] Bulling | Model checking resource bounded systems with shared resources via alternating Büchi pushdown systems[END_REF] extend alternating-time temporal logic (ATL) [START_REF] Alur | Alternating-time temporal logic[END_REF] by adding transitions that produce and consume resources to the models. As shown in [START_REF] Alechina | On the boundary of (un)decidability: Decidable model-checking for a fragment of resource agent logic[END_REF], the introduction of implicit counters in the models (i.e. variables interpreted by natural numbers) and the ability to quantify over strategies for a given set of agents can lead to undecidability, or decidability with a very high worst-case upper bound on the complexity of the model checking problem.

The nature of the strategy modalities means that reasoning about resources has similarities to the analysis of runs of vector addition systems with states (a.k.a. VASS) [START_REF] Karp | Parallel program schemata[END_REF], and more specifically to games on VASS, see e.g. [START_REF] Brázdil | Reachability games on extended vector addition systems with states[END_REF]. In this paper, we exploit results on VASS in order to analyse the model-checking problem for resource-bounded logics. Model-checking problems on VASS based on temporal logics and games are not always decidable, or at least quite difficult to solve, but sharp results exist. Temporal logics on VASS often lead to undecidable model-checking problems, see e.g. [START_REF] Esparza | On the decidability of model checking for several µ-calculi and Petri nets[END_REF][START_REF] Esparza | Decidability and complexity of Petri net problems -an introduction[END_REF], and this is more common with branching-time temporal logics such as CTL [START_REF] Esparza | Decidability and complexity of Petri net problems -an introduction[END_REF], or when the atomic formulae can state properties about the counter values [START_REF] Howell | Problems concerning fairness and temporal logic for conflict-free Petri nets[END_REF]. However, there are exceptions. For example, CTL model-checking on one-counter VASS is PSPACE-complete [START_REF] Serre | Parity games played on transition graphs of one-counter processes[END_REF][START_REF] Göller | Branching-time model checking of one-counter processes and timed automata[END_REF] (see also [START_REF] Vester | On the complexity of model-checking branching and alternating-time temporal logics in one-counter systems[END_REF]). Similarly, the control-state repeated reachability problem for VASS is shown to be decidable in [START_REF] Jančar | Decidability of a temporal logic problem for Petri nets[END_REF], and this is generalised to full LTL (for which the atomic formulae are exactly control states) in [START_REF] Habermehl | On the complexity of the linear-time mu-calculus for Petri nets[END_REF], where the model-checking problem for LTL on VASS is shown to be EXPSPACE-complete. Also in [START_REF] Jančar | Decidability of a temporal logic problem for Petri nets[END_REF], a strict fragment of LTL restricted to the "infinitely often" temporal operator GF and atomic formulae stating properties on counter values is shown decidable by a reduction into the reachability problem for VASS.

As far as games for VASS are concerned, the situation is even less encouraging. Two-player games on VASS in which each player can freely update the counter values are undecidable, even with simple winning conditions such as the reachability of a given control state [START_REF] Brázdil | Reachability games on extended vector addition systems with states[END_REF]. However, asymmetric VASS games in which at most one player can freely update the counter values and where the winning conditions are simple, are decidable [START_REF] Raskin | Games for counting abstractions[END_REF]. In addition, the game on asymmetric VASS with reachability of a control state (a slight variant of single-sided VASS in [START_REF] Abdulla | Solving Parity Games on Integer Vectors[END_REF] or alternating VASS in [START_REF] Courtois | Alternating vector addition systems with states[END_REF]) has been shown to be 2EXPTIME-complete [START_REF] Courtois | Alternating vector addition systems with states[END_REF], and decidable with parity conditions [START_REF] Abdulla | Solving Parity Games on Integer Vectors[END_REF][START_REF] Jančar | On reachability-related games on vector addition systems with states[END_REF]. The non-termination problem for symmetric games is 2EXPTIME-complete (the upper bound is from [START_REF] Jurdziński | Fixed-dimensional energy games are in pseudopolynomial time[END_REF] and the lower bound is from [START_REF] Courtois | Alternating vector addition systems with states[END_REF]).

In this paper, we establish formal relationships between model-checking problems for resource-bounded logics and decision problems for alternating VASS (also known as single-sided VASS). We then use these relationships to show new results for the decidability and complexity of model-checking resource-bounded logics. Ours is not the first work in this direction. There are clear similarities between resource values and counter values, and the semantics of resource-bounded logics are inherently gamebased. Previous work has explored the connections with counter machines, either to obtain undecidability, or to obtain lower bounds on complexity, e.g., [START_REF] Bulling | On the (Un-)Decidability of Model-Checking Resource-Bounded Agents[END_REF][START_REF] Alechina | On the boundary of (un)decidability: Decidable model-checking for a fragment of resource agent logic[END_REF].

We give optimal complexity upper bounds and new decidability results, including for resource-bounded logics with enriched path formulae as those in CTL * [START_REF] Emerson | Temporal and modal logic[END_REF]. First, we show that the model-checking problem for RB±ATL is 2EXPTIME-complete (Theorem 1 and Theorem 2), and that RB±ATL restricted to a bounded number of resources is in EXPTIME. The 2EXPTIME lower bound is obtained by a reduction from the state reachability problem for alternating VASS (AVASS) [START_REF] Courtois | Alternating vector addition systems with states[END_REF], whereas the upper bound is established by a reduction to the state reachability and the termination problems for AVASS (both problems are needed). These results are obtained by using formal relationships between strategies in concurrent game structures and proofs in AVASS, and the key observation is that only asymmetric VASS are needed. The formal relationships also allow us to show that the model-checking problem for RB±ATL * (a new logic naturally extending RB±ATL) is decidable by a reduction to the parity game problem for AVASS [START_REF] Abdulla | Solving Parity Games on Integer Vectors[END_REF] (Theorem 4). To the best of our knowledge, the complexity of the parity game problem for AVASS is still open. We also show that resource parameters can be effectively computed in the parameterised version of RB±ATL * (Theorem 5), due to the fact that the Pareto frontier for any parity game on single-sided VASS is computable [START_REF] Abdulla | Solving Parity Games on Integer Vectors[END_REF]Theorem 4]. As far as we know, this is the first time that resource values are synthesised in resource-bounded logics (see also [START_REF] Juhl | Theories of Programming and Formal Methods -Essays Dedicated to Jifeng He on the Occasion of His 70th Birthday[END_REF]). Lastly, we show that the modelchecking problems for RBTL [START_REF] Bulling | Expressing properties of resource-bounded systems: The logics RBTL * and RBTL[END_REF] and its extension RBTL * are EXPSPACE-complete, and that RBTL restricted to a bounded number of resources is in PSPACE.

Alternating VASS Preliminaries

We write N (resp. Z) for the set of natural numbers (resp. integers) and [m, m ] with m, m ∈ Z to denote the set { j ∈ Z : m ≤ j ≤ m }. Given a dimension r ≥ 1 and a ∈ Z, we write a ∈ Z r to denote the vector of dimension r with all components equal to a. For each x ∈ Z r , we write x(1), . . . , x(r) for the entries of x. For each x, y ∈ Z r , x y def ⇔ for every i ∈ [1, r], we have x(i) ≤ y(i). We also write x ≺ y when x y and x y.

A binary tree T, which may contain nodes with one child, is a non-empty subset of {1, 2} * such that, for all n ∈ {1, 2} * and i ∈ {1, 2}, n • i ∈ T implies n ∈ T and, n • 2 ∈ T implies n • 1 ∈ T. The nodes of T are its elements. The root of T is ε, the empty word. All notions such as parent, first child, second child, subtree and leaf, have their standard meanings. The height of T is the length, i.e. the number of nodes, of the longest simple path from the root to a leaf.

An alternating VASS (AVASS) [START_REF] Courtois | Alternating vector addition systems with states[END_REF] is a tuple A = (Q, r, R 1 , R 2 ) such that (1) Q is a finite set of locations (a.k.a. control states) and r ≥ 0 is the number of resource values,

(2) R 1 is a finite subset of Q × Z r × Q (unary rules) and (3) R 2 is a finite subset of Q 3 (fork rules). A derivation skeleton of A is a labelling D : T → (R 1 ∪R 2 ∪{⊥}) such that:
(1) T is a (possibly infinite) binary tree (subset of {1, 2} * with standard conditions), ( 2) if n has one child in T, then

D(n) ∈ R 1 , (3) if n has two children in T, then D(n) ∈ R 2 and (4) if n is a leaf in T, then D(n) =⊥. A derivation of A based on D is a labelling D :
T → Q×Z r such that: (1) if n has one child n in T, D(n) = (q, u, q ) and D(n) = (q, v), then D(n ) = (q , v + u) and ( 2) if n has two children n and n in T, D(n) = (q, q 1 , q 2 ) and D(n) = (q, v), then D(n ) = (q 1 , v) and D(n ) = (q 2 , v). So, fork rules do not update the resources and whence, there is an asymmetry between unary rules and fork rules (this makes a difference with branching VASS, see e.g. [START_REF] Verma | Karp-Miller Trees for a Branching Extension of VASS[END_REF][START_REF] Demri | The covering and boundedness problems for branching vector addition systems[END_REF]). This is a useful feature when dealing with the proponent restriction condition in RB±ATL. A derivation D is admissible whenever D : T → Q × N r , i.e. only natural numbers occur in it. An admissible derivation is also called a proof .

The state reachability problem for AVASS is as follows: given an AVASS A and control states q 0 and q f , is there a finite proof of AVASS whose root is equal to (q 0 , 0) and each leaf belongs to {q f } × N r ? When A has no fork rules, A is essentially a VASS [START_REF] Karp | Parallel program schemata[END_REF] and the above problem is an instance of the coverability problem, which is known to be EXPSPACE-complete [START_REF] Lipton | The reachability problem requires exponential space[END_REF][START_REF] Rackoff | The covering and boundedness problems for vector addition systems[END_REF] (see also [START_REF] Blockelet | Model-checking coverability graphs of vector addition systems[END_REF][START_REF] Demri | On selective unboundedness of VASS[END_REF]). The non-termination problem for AVASS is as follows: given an AVASS A and a control state q 0 , is there a proof whose root is equal to (q 0 , 0) and all the maximal branches are infinite? Proposition 1. [START_REF] Courtois | Alternating vector addition systems with states[END_REF][START_REF] Jurdziński | Fixed-dimensional energy games are in pseudopolynomial time[END_REF] The state reachability and non-termination problems for AVASS are 2EXPTIME-complete.

Decidability of these problems were first established in [START_REF] Raskin | Games for counting abstractions[END_REF] by using monotonicity of the games. The 2EXPTIME upper bound is preserved if we assume that the root is labelled by (q 0 , b) with b ∈ N r encoded with a binary representation (see Lemma 1).

In the sequel, we shall also admit fork rules of any arity α ≥ 1, and therefore, in such slightly extended AVASS the set of fork rules R 2 is a finite subset of β≥2 Q β .

Lemma 1. The following extension of the state reachability and non-termination problems for AVASS remains in 2EXPTIME:

-Fork rules can be α-ary for any α ≥ 1 (but there are only a finite amount of them).

-Reachability is related to a subset Q f ⊆ Q (instead of a singleton set).

-The initial configuration is (q 0 , b) with b ∈ N r instead of the fixed tuple 0.

-The value ω is allowed in b in the initial configuration (q 0 , b), where for all n ∈ Z,

we have ω = n + ω = ω + n.
The rather standard proof consists in using Proposition 1 by simulating a non-binary fork by a linear-size gadget made of unary and binary fork rules and by adding binary fork rules from states in Q f to a new single final state (alternatively, one could add unary rules with effect 0). The third item in Lemma 1 can be handled by adding a new unary rule with effect b whereas the fourth one amounts to ignore the components with the root value ω.

The notions of derivation skeleton, derivation and proof are also extended to general trees T ⊆ (N \ {0}) * . The set of finite words T ⊆ (N \ {0}) * is a (not necessarily binary)

tree iff for all n ∈ (N \ {0}) * and i ∈ (N \ {0}), n • i ∈ T implies n ∈ T and, n • i ∈ T and i > 1 imply n • (i -1) ∈ T. Such AVASS correspond to a single-sided VASS [6,1].
In what follows, by a VASS we mean an alternating VASS without any fork rule and write it as V = (Q, r, R) where R is a finite set of unary rules. Given a VASS V, its transition system TS(V)

def = (W, - →, L) is such that: (1) W def = Q × N r , (2)
L is a truth assigment with elements of Q also understood as propositional variables and L(q) def = {q} × N r and (3) -→ is a binary relation on W such that (q, v) -→ (q , v ) iff there is a unary rule (q, u, q ) in R such that v = v + u where '+' is the component-wise addition on N r . As usual, we also write * -→ to denote the reflexive and transitive closure of -→. Since TS(V) is a Kripke-style structure, it can be used to interpret modal or temporal formulae (e.g., LTL or CTL formulae) where atomic formulae refer to control states. Since alternating-time temporal logics such as ATL or ATL * are strict extensions of CTL or CTL * respectively, complexity hardness results for temporal logics can be lifted to such logics. A known result which will be useful in the sequel is that the model-checking problem for LTL on VASS is EXPSPACE-complete (the atomic formulae/propositions are control states) and it is PSPACE-complete for a fixed number of resources [START_REF] Habermehl | On the complexity of the linear-time mu-calculus for Petri nets[END_REF].

Below, we consider AVASS with a finite set of fork rules included in β≥2 Q β , and where the proofs are trees with nodes labelled by elements in

Q × (N ∪ {ω}) r . Given an AVASS A = (Q, r, R 1 , R 2 ), a colouring col is a map Q → [0, p] for some p ≥ 0.
The parity game problem for AVASS is as follows: given an AVASS A, a control state q 0 , b ∈ (N ∪ {ω}) r and col : Q → [0, p], is there a proof the root of which is equal to (q 0 , b), all the maximal branches are infinite and the maximal colour that appears infinitely often along each branch is even (the colour of each node is induced by col)? Corollary 2] The parity game problem for AVASS is decidable.

Proposition 2. [1,
To be precise, [1, Corollary 2] states the result for single-sided VASS that can be viewed as AVASS such that the set Q of control states is partitioned into

Q = Q 1 Q 2 ,
unary rules start by states in Q 1 , fork rules start by states in Q 2 and there is at most one fork rule starting in the same control state (necessarily, belonging to Q 2 ). The problem for AVASS can be reduced to that for single-sided VASS. It is not difficult to show that the state reachability and non-termination problems for AVASS can be understood as subproblems of the parity game problem and therefore their decidability also follows from [START_REF] Abdulla | Solving Parity Games on Integer Vectors[END_REF]. However, the situation is different in the case of complexity. While the exact complexity of the parity game problem is unknown, the state reachability and non-termination problems for AVASS are shown to be 2EXPTIME-hard in [START_REF] Courtois | Alternating vector addition systems with states[END_REF], the state reachability problem is shown to be in 2EXPTIME in [START_REF] Courtois | Alternating vector addition systems with states[END_REF] and the non-termination problem is proved to be in 2EXPTIME in [START_REF] Jurdziński | Fixed-dimensional energy games are in pseudopolynomial time[END_REF]. It has been shown recently that the parity game problem is in TOWER [START_REF] Jančar | On reachability-related games on vector addition systems with states[END_REF].

This decidability result has been strenghtened in [START_REF] Abdulla | Solving Parity Games on Integer Vectors[END_REF] in the following way. Given A, q 0 and col : Q → [0, p], the set of tuples b ∈ (N ∪ {ω}) r for which there is a positive solution to the parity game problem is upward closed and computable. This means that it can be represented effectively by a Boolean combination of atomic constraints of the form either x i ≥ k where i ∈ [1, r] and k ∈ N or x i = ω. Indeed, since the set is upward closed, by Dickson's Lemma, it has a finite set of minimal elements (with respect to the well-quasi-ordering slightly extended to accomodate the addition of the value ω) that allows one to define easily the symbolic representation in terms of atomic constraints of the form x ≥ k. The Pareto frontier of A, q 0 and col : Q → [0, p] is defined as the set of minimal elements in (N ∪ {ω}) r for which there is a positive solution to the parity game problem. Theorem 4] The Pareto frontier for any parity game on single-sided VASS is computable.

Proposition 3. [1,

The Logic RB±ATL and Variants

We consider the logics RB±ATL and RB±ATL * . The logic RB±ATL was introduced in [START_REF] Alechina | Decidable model-checking for a resource logic with production of resources[END_REF][START_REF] Alechina | Technical report: Model-checking for resource-bounded ATL with production and consumption of resources[END_REF], and extends ATL [START_REF] Alur | Alternating-time temporal logic[END_REF] with resources. RB±ATL * extends RB±ATL to allow path formulae to be any LTL-like formula.

Let PROP be a countably infinite set of atomic propositions. The models for the logics RB±ATL and RB±ATL * are the structures introduced in Definition 1 below. These are concurrent game structures from [START_REF] Alur | Alternating-time temporal logic[END_REF], but enriched with a cost function that specifies how resources are produced or consumed. At some abstract level, a structure is equipped with r counters and the transitions can perform increments and decrements. Definition 1. A resource-bounded concurrent game structure M is a tuple of the form (Agt, S, Act, r, act, cost, δ, L) such that:

-Agt is a non-empty finite set of agents (by default Agt = [1, k] for some k ≥ 1).

-S is a set of states and r ≥ 1 is the number of resources.

-Act is a non-empty set of actions with a distinguished action idle.

act : Agt × S → P(Act) is the action manager function such that for all a and s we have idle ∈ act(a, s). cost : S × Agt × Act → Z r is the (partial) cost function so that cost(s, a, a) is defined exactly when a ∈ act(a, s). Moreover, cost(s, a, idle) = 0. -δ : S × (Agt → Act) → S is the (partial) transition function such that δ is defined on (s, f) whenever for all agents a ∈ Agt, we have f(a) ∈ act(a, s). -L : PROP → P(S) is a truth assignment (the definition can be adapted when finite subsets of PROP are involved).

The map δ is also viewed as a deterministic relation with transitions of the form s (a 1 ,...,a k )

-----→ s where δ(s, f) = s and for all i ∈ [1, k] = Agt, we have f(i) = a i . We say that M is finite whenever S and Act are finite sets and L is restricted to a finite subset of PROP.

For instance, the idle action is considered in [START_REF] Alechina | Decidable model-checking for a resource logic with production of resources[END_REF][START_REF] Alechina | Technical report: Model-checking for resource-bounded ATL with production and consumption of resources[END_REF], where motivations for considering such a distinguished action are given. Given a coalition A ⊆ Agt and a state s, a joint action by A in s is a map f : A → Act such that, for all agents a ∈ A, we have f(a) ∈ act(a, s). The set of joint actions by A in s is denoted D A (s). Given a state s, the set of joint actions by Agt in s is denoted D(s) (instead of D Agt (s)) and the map δ is defined only for such joint actions. We write f g whenever g is a conservative extension of f, i.e. dom(f) ⊆ dom(g) and, f and g agree on dom(f).

Given a joint action f ∈ D A (s), we write out(s, f) to denote {s ∈ S | there is g ∈ D(s) such that f g and s = δ(s, g)}. For instance, out(s, f) is a singleton set when f ∈ D(s) since δ is a map and not a relation. Given a joint action f ∈ D A (s) and a state s, the cost of any transition fired from s following f (restricted to A by definition) is as follows: cost A (s, f) def = a∈A cost(s, a, f(a)). In a sense, the value cost A (s, f) does not depend on the costs related to the agents in (Agt \ A), or equivalently, the cost related to the agents in (Agt \ A) is reduced to zero.

A computation λ is a finite sequence or an ω-sequence of the form s 0

f 0 - → s 1 f 1 - → s 2 . . . such that for all i < |λ| -1, we have s i+1 ∈ δ(s i , f i ).
Here, |λ| denotes the length of λ, each s i is a state and each f i belongs to D(s i ). For instance |s 0

f 0 - → s 1 • • • f n-1 -→ s n | = n + 1 and |s 0 f 0 - → s 1 • • • f n-1 -→ • • • | = ω
for any infinite computation. So, in full generality, in a computation, a transition between two successive states is labelled by a joint action: this is not strictly needed for the forthcoming developments but it provides a more general notion that might be used in other contexts (for instance, if the winning condition of forthcoming strategies depends on the actions of all the agents and not only on those for the agents in A or on the visited states). A strategy F A for the coalition A is a map from the set of finite computations to the set of joint actions by A such that F A (s 0

f 0 - → s 1 • • • f n-1 -→ s n ) ∈ D A (s n ). A computation λ = s 0 f 0 - → s 1 f 1 - → s 2 • • • respects the strategy F A iff for all i < |λ|, we have, s i+1 ∈ out(s i , F A (s 0 f 0 - → s 1 . . . f i-1
-→ s i )). A computation λ that respects F A is maximal whenever it cannot be extended further while respecting the strategy. Note that maximal computations respecting F A are infinite. The set of all maximal computations that respect the strategy F A and that start at the state s is denoted by out(s, F A ). So far, no resource value has been involved in computations. Below, we shall quantify over maximal computations that respect a strategy and therefore for defining a strategy we can restrict ourselves to finite computations that respect it so far.

Given a bound

b ∈ (N ∪ {ω}) r , a computation λ = s 0 f 0 - → s 1 f 1 - → s 2 . . . in out(s, F A ) is b-consistent iff for all i < |λ|, we have 0 ( i-1 j=0 cost A (s j , F A (s 0 f 0 - → s 1 . . . f j-1 -→ s j )) + b). Whenever b(i) = ω,
this can be viewed as a means to disregard what happens on the ith resource (assuming that n + ω = ω for any n ∈ Z). Indeed, b(i) = ω amounts to guarantee from the beginning of the computation that there is an infinite supply of resources on the ith component. Note also that the above condition is slightly different from the one in [START_REF] Alechina | Technical report: Model-checking for resource-bounded ATL with production and consumption of resources[END_REF] but strictly equivalent. We have decided to adopt that notation in order to show more easily the relationships with VASS decision problems. So, for a

computation λ = s 0 f 0 - → s 1 f 1 - → s 2 .
. . and a coalition A, there is an underlying sequence v 0 , v 1 , . . . of resource values so that v 0 def = b and for all i < |λ| -1, we have v

i+1 def = v i + cost A (s i , F A (s 0 f 0 - → s 1 . . . f i-1 -→ s i ))
. The values of the sequence only depend on the agents in A, which is often called the proponent restriction condition.

The set of all the b-consistent (infinite) computations is denoted by out(s, F A , b). A b-strategy F A with respect to s is a strategy such that out(s, F A ) = out(s, F A , b). This definition also slightly differs from the one in [START_REF] Alechina | Technical report: Model-checking for resource-bounded ATL with production and consumption of resources[END_REF] that is not relative to a given state and therefore in [START_REF] Alechina | Technical report: Model-checking for resource-bounded ATL with production and consumption of resources[END_REF] the equality should hold for all the states.

So far, we have provided the main definitions about resource-bounded concurrent game structures and strategies. Let us present now the logic RB±ATL. Given a set of agents Agt = {a 1 , . . . , a k } and r ≥ 1, we write RB±ATL(Agt, r) to denote the resourcebounded logic with k agents and r resources whose models are resource-bounded concurrent game structures with the same parameters. Formulae of RB±ATL(Agt, r) are defined according to the grammar: Since all the maximal computations are infinite, the index i involved for clauses above related to A b or A b U can take any value in N. The presence of the idle action allows the extension of a strategy as soon as a given condition is satisfied along the computations. For instance, M, s | = A b φ is equivalent to the existence of f ∈ D A (s) such that for all g f, we have M, s | = φ with δ(s, g) = s and 0 b+cost A (s, f). Moreover, a strategy modality A b reduces the impact of the function cost in two ways. If the ith component of b is equal to ω, then there are no constraints on the ith resource along the computation. The restriction of cost to opponent agents in (Agt\A) is also reduced to 0 (so without any impact on consistency).

φ ::= p | ¬φ | φ ∧ φ | A b φ | A b φ | A b φUφ,
Obviously, RB±ATL(Agt, r) is a quantitative variant of ATL [START_REF] Alur | Alternating-time temporal logic[END_REF] in which resource values are computed along the computations.

The model-checking problem for RB±ATL is as follows: given k, r ≥ 1 (in unary), a formula φ in RB±ATL([1, k], r), a finite RB±ATL([1, k], r) model M and a state s, is M, s | = φ? The encoding of k and r in unary is unessential since the size of M with an explicit representation of all the transitions is over k + r. Proposition 4. [3, Theorem 1] The model-checking problem for RB±ATL is decidable.

We also consider RB±ATL * , an extension of RB±ATL in which the path formulae can be any LTL-like formula, in particular a temporal operator may no longer be preceded by a cooperation modality. This is a new logic although its definition follows a classical schema for branching-time temporal logics. Given a set of agents Agt = [1, k] and r ≥ 1, we write RB±ATL * (Agt, r) to denote the resource-bounded logic with k agents and r resources whose models are resource-bounded concurrent game structures with the same parameters. The parameterised version of RB±ATL * denoted by ParRB±ATL * admits formulae as RB±ATL * except that the values b ∈ (N ∪ {ω}) r are replaced by tuples of variables within VAR = {x 1 , x 2 , . . .}. Here is a typical formula in ParRB±ATL * :

{1} (x 1 ,x 2 ) Uq f ∧ {2} (x 2 ,x 3 ) Uq f .
Given a parameterised (state or path) formula φ with variables x 1 , . . . , x n and a map v : {x 1 , . . . , x n } → (N ∪ {ω}), we write v(φ) to denote the formula in RB±ATL * obtained from φ by replacing each occurrence of a variable x by v(x). The parameterised model-checking problem for ParRB±ATL * is as follows: given k, r ≥ 1 (in unary), a state formula φ in ParRB±ATL * ([1, k], r), a finite RB±ATL * ([1, k], r) model M and a state s, compute the set of maps v such that M, s | = s v(φ). Here, computing means to be able to characterise the set of maps v with M, s | = s v(φ), by using a symbolic representation with nice computational properties. We can show that we only need Boolean combinations of atomic formulae of the form either x ≥ k where k ∈ N or x = ω.

The Complexity of RB±ATL

The results in this section are obtained by elaborating on correspondences between AVASS decision problems, and the existence of strategies in RB±ATL. We show a 2EX-PTIME-hardness result by a reduction from the state reachability problem for AVASS. This improves the EXPSPACE-hardness result in [START_REF] Alechina | Technical report: Model-checking for resource-bounded ATL with production and consumption of resources[END_REF]. It is also worth noting that in the proof of Theorem 1, the presence of the idle action requires a bit of work.

Theorem 1. The model-checking problem for RB±ATL is 2EXPTIME-hard.

The upper bound is proved by designing a labelling algorithm as done in [START_REF] Alechina | Technical report: Model-checking for resource-bounded ATL with production and consumption of resources[END_REF] but the main difference with [START_REF] Alechina | Technical report: Model-checking for resource-bounded ATL with production and consumption of resources[END_REF] rests on the fact that we explicitly call subroutines that solve decision problems on AVASS. Let M be a finite resource-bounded concurrent game structure, A ⊆ Agt be a coalition, F A be a strategy and s be a state. We construct an AVASS A M,A,s such that the set of computations respecting F A and starting from s corresponds to a derivation skeleton whose root is labelled by a unary rule with first state s. Moreover, if F A is b-strategy w.r.t. s, then the derivation skeleton can be turned into a proof whose root is labelled by (s, b).

Given M = (Agt, S, Act, r, act, cost, δ, L), the AVASS A M,A,s

def = (Q, r, R 1 , R 2 ) is built as follows: Q def = {s} ∪ {(s , f) | s ∈ S, f ∈ D A (s )} ∪ {(g, s ) | s , s ∈ S, g ∈ D(s ), δ(s , g) = s }.
-The set R 1 contains the following rules: (1) for all f ∈ D A (s), (s, cost A (s, f), (s, f)) and ( 2) for all (g, s ) ∈ Q, for all f ∈ D A (s ), ((g, s ), cost A (s , f), (s , f)). -The set R 2 contains the following rules. For all (s , f) ∈ Q, let {(g 1 , s 1 ), . . . , (g α , s α )} = {(g, s ) ∈ Q | s ∈ δ(s , g), g ∈ D(s ), f g}. The set is non-empty thanks to constraints on the idle action. We add the fork rule ((s , f), (g 1 , s 1 ), . . . , (g α , s α )). In order to define unambiguously that rule, we assume an arbitrary ordering on Q.

It is worth noting that s has a special status in Q simply because any proof whose root configuration contains s has no predecessor configuration. Any derivation skeleton from A M,A,s has to alternate the rules in R 1 and the rules in R 2 , by construction. For every (s , f) in Q, there is a unique fork rule starting by (s , f) and the construction applies also in the degenerate cases, i.e. when A = Agt or when A = ∅ (assuming that cost(s , f) = 0 for the unique f ∈ D ∅ (s )). The main property of A M,A,s is stated below.

Lemma 2. There is a b-strategy w.r.t. s in M iff there is a proof in A M,A,s whose root is labelled by (s, b) and every maximal branch is infinite.

Theorem 2. The model-checking problem for RB±ATL is in 2EXPTIME.

An AVASS of the form A S M,A,s is defined as the restriction of A M,A,s in which the opponent coalition has no way to go out of S . The algorithm is given below with the essential property:

GMC(M, ψ) = {s | M, s | = ψ}.
Algorithm 1 An algorithm for RB±ATL model checking. if s S 1 then return ∅ end if 10:

A b ψ 1 Uψ 2 : return {s | A S 1 ∪S 2 M,A,s , (s, b), S 2 is a positive instance of state reachability} with S 1 = GMC(M, ψ 1 ), S 2 = GMC(M, ψ 2 ), S 2 = {(g, s ) ∈ Q | s ∈ S 2 } ∪ {s ∈ Q | s = s, s ∈ S 2 } 11:
end case 12: end procedure Corollary 1. For any fixed r ≥ 1, the model-checking problem for RB±ATL restricted to at most r resources is in EXPTIME. For r ≥ 4, the problem is EXPTIME-hard.

We invoke [START_REF] Jurdziński | Fixed-dimensional energy games are in pseudopolynomial time[END_REF]Theorem 3.4] and [13, Theorem 3.1] since for a bound r, the state reachability and the non-termination problems can be solved in EXPTIME. EXPTIMEhardness (r ≥ 4) is due to [START_REF] Courtois | Alternating vector addition systems with states[END_REF]Proposition 4.2] and to the proof of Theorem 1.

More Path Formulae While Preserving Decidability

In this section, we study the model-checking problem for resource-bounded logics where the path formulae can be any LTL-like formula. In doing so, we also illustrate the versatility of our formalisation, by showing how it can be used to establish complexity results for the model-checking problem for the logics RBTL * and RB±ATL * .

The Logic RBTL * and its Complexity

The models of the logic RBTL * are structures of the form (Q, r, R, L) where (Q, r, R) is a VASS and L is a truth assignment built on elements of Q understood as propositional variables, so that L(q) = {q} (see e.g [10, Section 3]). In order to fit the usual terminology, below, an infinite proof in (Q, r, R) is called a path or run and it can be represented by λ = (q 0 , v 0 ) -→ (q 1 , v 1 ) . . .. We write λ[i, +∞) to denote the run starting from (q i , v i ) taken from λ as a suffix and λ(i) to denote the configuration (q i , v i ).

The state formulae φ and the path formulae Φ of RBTL * are defined by mutual recursion with the grammar (relatively to Q and r)

φ ::= q | ¬φ | (φ ∧ φ) | b Φ Φ ::= φ | ¬Φ | (Φ ∧ Φ) | Φ | (ΦUΦ) | Φ
where q ∈ Q and b ∈ (N∪{ω}) r . Syntactically, every state formula is also a path formula according to this grammar, and this reflects the fact that a path uniquely identifies a control state in which a formula is interpreted: its starting control state. We present the semantics for RBTL * by distinguishing the state formulae from the path formulae. The two satisfaction relations | = s and | = p are defined as follows (clauses for the Boolean connectives are omitted).

M, q | = s q iff q = q M, q | = s b Φ iff there is an infinite run λ starting at (q, b) such that M, λ | = p Φ M, λ | = p φ iff M, q 0 | = s φ for state formulae φ with λ(0) = (q 0 , v 0 ) M, λ | = p Φ iff M, λ[1, +∞) | = p Φ M, λ | = p ΦUΨ iff there is i ≥ 0 such that M, λ[i, +∞) | = p Ψ and for every j ∈ [0, i -1], we have M, λ[ j, +∞) | = p Φ M, λ | = p Φ iff for all i ≥ 0, M, λ[i, +∞) | = p Φ.
The model-checking problem for RBTL * is as follows: given a model M, q and a state formula φ, is it M, q | = s φ? The logic RBTL is the fragment of RBTL * in which any subformula whose outermost connective is in {U, , }, is preceded by some b . The problem for RBTL is already EXPSPACE-hard since the state reachability problem for VASS can be reduced easily to it. The EXPSPACE lower bound for the model-checking problem for RBTL can be matched with the upper bound for RBTL * .

Theorem 3. The model-checking problems for RBTL and RBTL * are EXPSPACE-complete.

We can obtain a improved complexity result if the number of resources is considered fixed.

the automata-based approach (as for temporal logics). This is the most natural way to inherit colours from A to A M,A,s ⊗ A. (I) A M,A,s has a proof the root of which is equal to (s, b), all the maximal branches are infinite and the L n -projection of each infinite branch belongs to the language accepted by A (i.e. to L(A)). (II) A M,A,s ⊗ A has a proof the root of which is equal to ((s, q 0 ), b), all the maximal branches are infinite and the maximal colour that appears infinitely often is even.

Theorem 4. The model-checking problem for RB±ATL * is decidable.

Lemma 3 is essential to establish Theorem 4 since its proof uses the product between an alternating VASS and a deterministic parity automaton recognizing ω-words. This is reminiscent of the proof of [START_REF] Alur | Alternating-time temporal logic[END_REF]Theorem 5.6] about the 2EXPTIME upper bound for the ATL * model-checking problem. Rabin tree automata of the proof of [START_REF] Alur | Alternating-time temporal logic[END_REF]Theorem 5.6] are replaced by deterministic parity automata for encoding the LTL formulae and by alternating VASS (with counters) as outcome of the synchronisation. The proof of Theorem 5 is based on a global model-checking algorithm that is essentially based on Lemma 3 and on [START_REF] Abdulla | Solving Parity Games on Integer Vectors[END_REF]Theorem 4]. Synthesising resource values has been also considered in [START_REF] Juhl | Theories of Programming and Formal Methods -Essays Dedicated to Jifeng He on the Occasion of His 70th Birthday[END_REF].

Concluding Remarks

We have related model-checking problems for resource-bounded logics and decision problems for AVASS. Though such relationships should not come as a complete surprise, we obtained new complexity and decidability results. We prove that the modelchecking problem for RB±ATL introduced in [START_REF] Alechina | Decidable model-checking for a resource logic with production of resources[END_REF][START_REF] Alechina | Technical report: Model-checking for resource-bounded ATL with production and consumption of resources[END_REF] is 2EXPTIME-complete. No complexity upper bound was known so far. We have introduced the logic RB±ATL * that extends RB±ATL, and we have shown that the model-checking problem is decidable. The same hold for the parameterised version ParRB±ATL * , i.e. it is decidable to compute the set of resource bounds for which the given parameterised formula is satisfied. We have also shown that the model-checking problem for RBTL * introduced in [START_REF] Bulling | Expressing properties of resource-bounded systems: The logics RBTL * and RBTL[END_REF] is EXPSPACE-complete. No complexity upper bound for RBTL was known so far as well as the decidability status for RBTL * .We believe that the simple framework we have proposed could be used to obtain further results for new resource-bounded logics.

0 - → s 1

 01 where p ∈ PROP, A ⊆ Agt and b ∈ (N∪{ω}) r . The size of a formula is computed from a DAG representation and the integers are encoded in binary. Note that forthcoming hardness results do not use the conciseness of the DAG representation (with respect to the tree representation).The satisfaction relation | = is defined inductively as follows assuming that M is an RB±ATL(Agt, r) model (we omit the obvious cases for the Boolean connectives):M, s | = p def ⇔ s ∈ L(p)and, M, s | = A b φ def ⇔ there is a b-strategy F A w.r.t. s such that for all s 0 f . . . ∈ out(s, F A ), we have M, s 1 | = φ M, s | = A b φ def ⇔ there is a b-strategy F A w.r.t. s such that for all λ = s 0 f 0 -→ s 1 . . . ∈ out(s, F A ), for all i < |λ|, we have M, s i | = φ M, s | = A b φ 1 Uφ 2 def ⇔ there is a b-strategy F A w.r.t. s such that for all λ = s 0 f 0 -→ s 1 . . . ∈ out(s, F A ), there is some i < |λ| such that M, s i | = φ 2 and for all j ∈ [0, i -1], we have M, s j | = φ 1 .

1 : 1 M

 11 procedure GMC(M, φ) 2: case φ of 3: p: return {s ∈ S | p ∈ L(s)} 4: ¬ψ: return S \ GMC(M, ψ) 5: ψ 1 ∧ ψ 2 : return GMC(M, ψ 1 ) ∩ GMC(M, ψ 2 ) 6: A b ψ: return {s | ∃ f ∈ D A (s), 0 b + cost A (s, f), for all f g ∈ D(s), δ(s, g) ∈ GMC(M, ψ)} 7: A b ψ: S 1 := GMC(M, ψ) 8: if s ∈ S 1 then return {s | A S ,A,s , (s, b) is non-terminating} end if 9:

Lemma 3 .

 3 Let (s, b) ∈ Q × (N ∪ {ω}) r .The statements below are equivalent:

Theorem 5 .

 5 The parameterised model-checking problem for ParRB±ATL * is decidable.

Acknowledgements We would like to thank the anonymous reviewers for their numerous suggestions that helped us improve the quality of the paper.

Corollary 2. For any fixed r ≥ 1, the model-checking problem for RBTL * restricted to at most r resources is in PSPACE.

The PSPACE upper bound is then a consequence of [START_REF] Habermehl | On the complexity of the linear-time mu-calculus for Petri nets[END_REF]Theorem 4.1]. Again, if r is fixed but greater than two, then the model-checking problem for RBTL * restricted to at most r resources is PSPACE-hard since the state reachability problem for VASS of dimension two is PSPACE-complete [START_REF] Blondin | Reachability in two-dimensional vector addition systems with states is PSPACE-complete[END_REF]. When r = 1, the model-checking problem for RBTL * restricted to at most one resource is NP-hard since the state reachability for VASS of dimension one is NP-complete [START_REF] Haase | On the Complexity of Model Checking Counter Automata[END_REF].

Decidability of RB±ATL *

In order to illustrate the reduction from the model-checking problem for RB±ATL * into the parity game problem, we briefly present a notion of synchronisation. Let M = (Agt, S, Act, r, act, cost, δ, L) be a resource-bounded concurrent game structure. Given p 1 , . . . , p n , we write Σ n to denote P({p 1 , . . . , p n }) and L n (s )

be the AVASS defined from M, A and s, and let

) be a deterministic parity automaton over Σ n . The principle of the synchronised product A M,A,s ⊗ A defined below is the following. Any (infinite) branch of a proof of A M,A,s contains control states of the form s, (s , f) or (g, s ) where s is a distinguished state of M, s is any state, f ∈ D A (s ) and g is a joint action in D(s ) with δ(s , g) = s . By construction, (s , f) is preceded by a state of the form either (g, s ) or s (if s = s). So an infinite branch of the form (s 0 , u 0 ) ((s 0 , f 0 ), u 1 ) ((g 1 , s 1 ), u 1 ) ((s 1 , f 1 ), u 2 ) ((g 2 , s 2 ), u 2 ) • • • leads to the ω-word L n (s 0 ) L n (s 1 ) L n (s 2 ) • • • that admits a unique run in A (thanks to determinism). Above, we slightly abuse notation since we identify a branch with its label. Given an infinite branch s 0

The control states of A M,A,s ⊗ A are pairs in Q × Q and the second components are therefore control states in Q as they appear for the unique run on L n (s 0 ) L n (s 1 )

-For each (g, s )

u -→ (s , f) ∈ R 1 , and for each q ∈ Q , R 1 contains the rule ((g, s ), q) u -→ ((s , f), q). So, firing a unary rule from A M,A,s does not change the second component.

-For each ((s , f), (g 1 , s 1 ), . . . , (g α , s α )) ∈ R 2 and for each q ∈ Q , we add in R 2 (((s , f), q), ((g 1 , s 1 ), δ(q, L n (s ))), . . . , ((g α , s α ), δ(q, L n (s )))). Firing a fork rule from A M,A,s changes the second component in a unique way depending on q and L n (s ).

Again, there is a unique fork rule starting by the control state ((s , f), q).

Let us define the colouring col : Q → [0, p] such that for all (q, q ) ∈ Q , we have col ((q, q )) def = col(q ). The synchronised product satisfies the essential property for