Jessie Carbonnel

Marianne Huchard

Clémentine Nebut

Exploring the variability of interconnected product families with relational concept analysis

Keywords: https://en, wikipedia, org/wiki/Category:

Among the various directions that SPLE promotes, extractive adoption of complex product lines is especially valuable, provided that appropriate approaches are made available. Complex variability can be encoded in different ways, including the feature model (FM) formalism extended with multivalued attributes, UML-like cardinalities, and references connecting separate FMs. In this paper, we address the extraction of variability relationships depicting connections between systems from separate families. Because Formal Concept Analysis provides suitable knowledge structures to represent the variability of a given system family, we explore the relevance of Relational Concept Analysis, an FCA extension to take into account relationships between different families, to tackle this issue. We investigate a method to extract variability information from descriptions representing several inter-connected product families. It aims to be used to assist the design of inter-connected FMs, and to provide recommendations during product selection.

Introduction

As families of similar software systems grow in size and complexity, managing them by adopting an approach based on software product line (SPL) engineering becomes more and more relevant. When various software systems of a same family have been individually developed in an undisciplined way, migrating to an SPL may be done through an extractive approach, by analyzing their commonalities and differences, and by identifying the reusable assets and a reference architecture. Complex variability extraction may result in Feature Models (FMs) [KCH + 90, CW07] extended with multivalued attributes, UML-like cardinalities, or references connecting separate FMs [START_REF] Czarnecki | Generative Programming for Embedded Soft.: An Industrial Experience Report[END_REF].

Previous research work has studied variability extraction in the boolean case [LP07, HLE11, RPK11, ACP + 12, AMHS + 14, MZB + 17, SSS17], and in presence of multi-valued attributes and cardinalities [START_REF] Bécan | Synthesis of attributed feature models from product descriptions[END_REF][START_REF] Carbonnel | Towards complex product line variability modelling: Mining relationships from non-boolean descriptions[END_REF]. More specifically, in [START_REF] Carbonnel | Towards complex product line variability modelling: Mining relationships from non-boolean descriptions[END_REF], we show how Formal Concept Analysis (FCA) and its extension to Pattern Structures help in extracting variability relationships from a product family described by multi-valued attributes and cardinalities. It leads to binary implications, groups and mutex involving boolean features as well as attribute values and cardinalities.

In this paper, we address the problem of cross-family variability relationship extraction from descriptions representing several inter-connected product families (see Figure 1). These families may correspond to various concerns or product pieces, and may be connected by different relationships, e.g., use, part-of, compatible-with, provide. We introduce a method based on Relational Concept Analysis (RCA), an FCA extension to take into account these relationships between the product families. This method produces concept lattice families from which are extracted cross-family features and cross-family relationships that can be binary implications, mutex, co-occurrences or groups involving both boolean features and cross-family features. This is part of a general approach we envisage, thus we also discuss how in the future this method can assist the design of FMs with new kinds of references, and the exploration of existing configurations for the sake of recommendation.

Figure 1: Schema of the envisaged approach.

In Section 2, we introduce the foundations of variability extraction with FCA. Addressing cross-family variability extraction with Relational Concept Analysis is developed in Section 3. Then, we discuss how the extracted crossfamily variability can help in inter-connected FMs design and in configuration exploration (Section 4). We develop related work in Section 5, before concluding and drawing perspectives of this work in Section 6.

2 Background: Assisted variability extraction with FCA Variability modeling. Variability modeling appears in the earliest steps for extractive adoption of a software product line from a family of product variants. To elaborate a variability model in the context of a feature-oriented mod-eling approach [KCH + 90], the main existing approaches need to identify (a) representative and discriminating features and (b) relationships between these features (e.g. implications, mutually exclusive features or groups). Then the extracted information is represented through feature models [KCH + 90, CW07, SHTB07], propositional logics [START_REF] Batory | Feature models, grammars, and propositional formulas[END_REF], description logics [START_REF] Bagheri | Modular feature models: Representation and configuration[END_REF], or constraints [SDM + 11]. Relationships between features are usually extracted from the product variant descriptions depicting variants along with their features. For example, these descriptions may take the form of tabular descriptions, such as Product Comparison Matrices that can be found in Wikipedia Formal Concept Analysis (FCA) Formal Concept Analysis (FCA) [START_REF] Ganter | Formal concept analysismathematical foundations[END_REF] can be summarized by the equation: "objects + attributes = concept hierarchy". In other words, based on a set of objects (entities, individuals) described by a set of attributes (properties, characteristics), FCA helps to: (1) group a maximal set of objects sharing a maximal set of attributes into a concept, and

(2) hierarchically organize the set of concepts.

In the simplest form, FCA considers as input a formal context K = (O, A, J), where O is a set of objects, A is a set of attributes and J ⊆ O × A is a binary relationship, where (o, a) ∈ J when "o possesses a". Table 1 represents in a tabular form a formal context KET describing (in a simplified way) Knowledge Engineering (KE) tools (left-hand-side), and a formal context KEC describing basic Knowledge Engineering (KE) components (right-hand-side). The first column of a formal context presents the objects (here in the form of an identifier representing KE tools or components), and the first row gathers the attributes (here the boolean features). KE tools are described by their import/export formats, their distribution mode (commercial or open source) and if they are delivered in SaaS mode (Software as a Service). Basic KE components are dedicated to resource creation (ontology, rules or raw file) or to implement a machine learning algorithm from the categories: rule extraction, decision tree or neural network. The strategy can be supervised or unsupervised. The components can be classified as a symbolic method or as a statistical method.

ket0 × × × × × × ket1 × × × × × × ket2 × × × × × × ket3 × × × × × × ket4 × × × × × × ket5 × × × × × × ket6 × × × × × × ket7 × × × × × × ket8 × × × × × × KEC knowledge
od1 × × × × od2 × × × × rd1 × × × × f1 × × × × re1 × × × × × × × re2 × × × × × × × dt1 × × × × × × × nn1 × × × × × × ×
FCA extracts a set of formal concepts. Each concept C = (E, I) is a maximal group of objects E associated with a maximal group of attributes I these objects share. E = {o ∈ O | ∀a ∈ I, (o, a) ∈ J} is the concept extent and I = {a ∈ A | ∀o ∈ E, (o, a) ∈ J} is the concept intent. For example (using short names), objects E sy = {od1, od2, rd1, f 1, re1, re2} share attributes I sy = {sy, kec}. As these sets cannot be extended, (E sy , I sy ) is a concept.

Extent inclusion (and intent containment) induces a specialization order ≤ CL between concepts: for two concepts C 1 = (E 1 , I 1 ) and C 2 = (E 2 , I 2 ), C 1 ≤ CL C 2 if and only if E 1 ⊆ E 2 (and equivalently I 1 ⊇ I 2 ). For example, if we consider the concept (E rc , I rc ) with E rc = {od1, od2, rd1, f 1} and I rc = {rc, sy, kec}, (E rc , I rc ) ≤ CL (E sy , I sy ). The concept lattice is the set of all concepts C K of the formal context K, provided with the order ≤ CL . Attributeconcepts (resp. Object-concepts) are particular concepts that contain at least an attribute (resp. an object) which is not present in a super-concept (resp. subconcept). An AOC-poset (for Attribute-Object-Concept partially ordered set) is the restriction of the concept lattice to these specific concepts. The AOCposet which classifies the KE tools (resp. components) of the left-hand-side (resp. right-hand-side) of Table 1 is shown in Figure 2 (resp. Figure 3). A concept is shown as a 3-part box containing: (1) the concept identifier, (2) its intent deprived from the attributes inherited from the super-concepts, and (3) its extent deprived from the objects that appear in the sub-concepts. Variability information extraction with FCA Variability information can be extracted from the AOC-poset and used to guide the FM synthesis [START_REF] Carbonnel | Modelling equivalence classes of feature models with concept lattices to assist their extraction from product descriptions[END_REF]. Table 2 explains how variability information can be extracted from AOC-posets and translated into FM constructs. The first row considers the situation in which a concept introducing a feature f 2 is a sub-concept of a concept introducing f 1 (denoted by C f2 < CL C f1 ). In this case, all configurations having f 2 also have f 1 , that leads to the propositional formula. Such an implication can be translated into 3 different FM constructs: a refinement relationship x, an optional relationship y or a requires constraint z.

The second row shows how logical equivalences can be read in AOC-posets. When f 1 and f 2 are introduced in the same concept, this means that features f 1 and f 2 are always present together (f 1 ↔ f 2 ). In a feature diagram, this can be represented: with a mandatory relationship { or with two requires constraints |.

The third row shows how mutual exclusions can be read in an AOC-poset. Two features f 1 and f 2 (respectively introduced in C f1 and C f2 ) are mutually exclusive if the common sub-concepts of C f1 and C f2 (denoted C f1 C f2 ) do not introduce any object.

The fourth row presents the mapping of or-groups ~into AOC-posets. We consider {f 1 , ..f k } the features involved in an or-group, and f 0 the parent-feature of this group. All configurations having one of the features in {f 1 , ..f k } should also have f 0 , and conversely, configurations having the parent-feature f 0 have at least one feature of the or-group. Thus, the union of the extents of concepts introducing features of {f 1 , ..f k } is equal to the extent of the concept introducing f 0 . Moreover, the concept introducing the parent-feature of an or-group is not an object-concept, as at least one feature of {f 1 , ..f k } has to be selected. It is denoted by C f0 ∈ OC, C f0 being the concept introducing the parent-feature, and OC the set of object-concepts of the AOC-poset. Besides, we consider that there always exists a root feature, which appears in the top concept containing all configurations.

The last row presents the mapping of xor-groups into AOC-posets. Xorgroups are like or-groups, but concepts introducing the features of {f 1 , ..f k } do not have a common sub-concept introducing an object. Indeed, features involved in a xor-group and features involved in an exclude cross-tree constraint have a similar behaviour, as they are mutually exclusive in both situations.

Cross-product family variability extraction with RCA

Product line engineering faces inescapable issues related to the growth in size and complexity of software systems and to their evolution [START_REF] Holl | A systematic review and an expert survey on capabilities supporting multi product lines[END_REF]. Decomposing systems and considering separated concerns or gradual construction are solutions that have been investigated, e.g. in multi-product lines [HGR12, [START_REF] Botterweck | Variability and Evolution in Systems of Systems[END_REF][START_REF] Rosenmüller | Modeling dependent software product lines[END_REF]. Extending the variability extraction from a product family to a set of interconnected product families means that we need to consider both intra-family and inter-family (cross-family) variability information.

Relational Concept Analysis Cross-family variability extraction can be assisted by Relational Concept Analysis (RCA) [START_REF] Rouane-Hacene | Relational concept analysis: mining concept lattices from multi-relational data[END_REF] which considers sev-

x f 1 f 2 f2 → f1 Cf 2 ≤CL Cf 1 Cf 2 f2 Cf 1 f1 y f 1 f 2 z f2 → f1 { f1 f2 or f1 f2 f1 ↔ f2 Cf 1 =CL Cf 2 Cf 1,2 f1,f2 | f1 → f2 f2 → f1 } f1 → ¬f2 f1 → ¬f2 or f2 → ¬f1 Ext(Cf 1 Cf 2 ) = ∅ Cf 2 Cf 1 C f2 f1 ~f0 f1 ... fk f0 ↔ (f1 ∨ . . . ∨ fk) ∀fi ∈ {f1, ..fk}, Cf i ≤CL Cf 0 . {Cf 1 , ...Cf k } is the greatest antichain [JRJ94]
verifying:

Ext(Cf 1 ) ∪ ... ∪ Ext(Cf k ) = Ext(Cf 0 ). Cf 0 ∈ OC Cf 0 f 0 Cf j ... Cf i f i f j Cm f0 f1 ... fk f0 ↔ (f1 ⊕. . .⊕fk) ∀fi ∈ {f1, ..fk}, Cf i ≤CL Cf 0 . ∀fi, fj ∈ {f1, ..fk} | fi = fj, Ext(Cf i Cf j ) = ∅. {Ext(Cf 1 ), ..., Ext(Cf k )} is a partition of Ext(Cf 0 ). Cf 0 ∈ OC Cf 0 f 0 Cf j ... Cf i f i f j

Cm

eral object categories (one per product family). While FCA groups objects sharing commonalities in their intrinsic attributes, RCA additionally groups objects sharing commonalities in their similar relations to other objects having themselves commonalities. To that aim, RCA iteratively applies FCA to propagate similarities from objects in one category to objects in (possibly) another category through relations. To follow up our illustrative example, identifying the group of statistical KE components can be propagated to KE tools through a provides relation, leading to identify the group of KE tools dedicated to statistical analyses. The considered dataset in RCA is a Relational Context Family (RCF), which is a set of object-attribute contexts (objects described by intrinsic attributes) and a set of object-object contexts (relations between objects of the different categories). For our example, the RCF is composed of the object-attribute contexts KEC and KET (Table 1), and the object-object relation provides ⊆ KET ×KEC (Table 4). In the general case, an RCF contains n object-attribute contexts To consider the provides relation, information is added to the description of objects of its domain (KET ). A straightforward integration scheme could be to simply add to the object-attribute context KET attributes of the form (provides, kec), and to add a cross when an object of KET provides the corresponding object of KEC. For example, (provides, od1) could be added to the description of ket0 in a new column of KET . This would allow to group KETs that own the same KECs, but we would loose the valuable knowledge given by the KEC concepts. Remember that these concepts indicate what are the shared characteristics of KECs, e.g. dt1 and nn1 are grouped in KEC 8 because both are statistical-based KECs (see Fig. 3). Now if we consider the way provides describes ket7 and ket8 (resp. by (provides, dt1) and (provides, nn1)), these two KETs do not share any KEC, whereas they share the fact that they allow statistical-based learning. To acquire such information, we propagate the knowledge highlighted in the KEC AOC-poset to the KET AOC-poset through relational attributes. These relational attributes are formed using scaling quantifiers inspired by constructors used in descriptions logics [BCM + 03], such as ∃ and ∃∀5 , as illustrated in Table 3. For example, ket7 and ket8 descriptions will be enhanced by a relational attribute ∃provides(KEC 8) to indicate that both ket7 and ket8 have a link to at least one object of the KEC 8 extent.

K i = (O i , A i , J i ) , i
In the RCA process, a scaling quantifier q is associated with each relation r, and systematically applied to form all the relational attributes of the form qr(C) with C a concept formed on the objects of the range of r. These relational attributes are added to the context of the domain of r and a new extended AOCposet can be built. For example, Fig. 4 shows the AOC-poset built for KET extended with the relational attributes formed with the quantifier ∃∀ applied to provides. In this AOC-poset, KET 4 now highlights new shared information about ket7 and ket8 which is the fact that both provide only (∃∀) statisticalbased learning components (the components they provide are all in KEC 8 extent).

RCA can consider complex and possibly cyclic entity-relationship models. A non-trivial modeling phase is central in this perspective, to determine which are the object-attribute contexts (separate product families) and which are the object-object contexts (interconnections between the families), and the adequate scaling quantifiers.

Cross-family features Table 3 shows two examples of the new features that this framework makes available to describe cross-family variability. These features are abstractions of links between an object o (a product configuration of the domain of r in our context) and a concept C (a group of product configurations of the range of r). Existential features (∃r(C)) indicate if a configuration o is linked by r to at least one configuration in the extent of Concept C. In our example, ∃provides(C) is the feature "provides a KEC with the features of C". Feature ∃provides(KEC 8) can be read "provides a statistical learning component". Universal features (∃∀r(C)) indicate if all r links of a configuration o are towards configurations in the extent of Concept C ; for our example, it can be summarized as "provides only KEC with the features of C". As presented previously, ∃∀provides(KEC 8) can be read "provides only statistical learning components". Cross-family variability relationships Variability information can be extracted from the AOC-poset resulting from RCA, using the same mappings as introduced in Table 2. The found relationships also involve cross-family features. Three examples are given below using the AOC-poset of Figure 4.

Example of co-occurrence. By the second row of Table 2, rdf and ∃∀provides(KEC 12) are co-occurrent, since both are introduced in the same KET 5. In other words, since KEC 12 corresponds to symbolic components, that means that tools exporting in rdf only use symbolic methods. 2, KET 1 is below KET 9, thus ∃∀provides(KEC 6) → Saas: the knowledge engineering tools providing components that only deal with ontology definition (KEC 6) are delivered in SaaS mode. This can be translated through several ways in a feature model (see cases x, y and z in Table 2).

Example of implication. By the first row of Table

Example of exclusion. KET 5 and KET 6 do not have a common subconcept introducing a configuration. By the third row of Table 2, ∃∀provides(KEC 6) → ¬pf . We can deduce that none of the KETs providing

provides od1 od2 rd1 f1 re1 re2 dt1 nn1 ket0 × ket1 × × ket2 × × × ket3 × × × ket4 × × × × ket5 × × × × × ket6 × × ket7 × ket8 ×

Exploiting cross-family variability

In this section, we draw tracks of future research to exploit cross-family variability information.

Extended (semi-automated) FM synthesis Intra-family information has been used to assist FM synthesis [CW07, SLB + 11, HLE11, RPK11, ACP + 12, HLE13, AMdSH + 14, SRA + 14, CHN19b]. The methods that use FCA to assist FM synthesis [RPK11, AMdSH + 14, CHN19b] exploit a concept structure to derive automatically an FM or to provide user guidance by reducing their choices during FM construction. For example, the FM of KETs in Fig. 5 (resp. of KECs in Fig. 6) can be extracted from the AOC-poset of Fig. 2 (resp. Fig. 3) by an FM designer, assisted by the rules presented in Sect. 2. Cross-family variability information can in turn be used to assist the synthesis of interconnected FMs, either by writing constraints between two different FMs (as shown in Sect. 3), or to enhance an FM from one family by features coming from knowledge about relations in real configurations. We illustrate this by using information extracted from the AOC-poset of KET (Fig. 4). First, we introduce a reference kec below the FM root, with cardinality 1-many (*). Then the AOC-poset states that in observed real configurations (as shown in KET 4), some KETs propose only supervised learning algorithms (∃∀provides(KEC 9)), thus if a user chooses that option when he designs a KET, he should be guided afterwards to configure only components implementing supervised learning algorithms. The AOCposet also shows (in KET 5) that a group of KETs is dedicated to symbolic approaches only (∃∀provides(KEC 12)) and a subgroup of them is dedicated to ontology definitions only (∃∀provides(KEC 6)). From these observations, the FM designer could decide to introduce a feature supervised ket (sup), a feature symbolic ket (sy) with a sub-feature ontology dedicated (o). These choices are represented as an extension of the FM KET , as shown in Fig. 7 in the grey rectangular box. The introduced features in the KET FM should be used in configuration tools to reduce the choices in the KEC FM when they are selected. It can be tricky for an FM designer to choose to add features to the KET FM depending on the relations that are observed between KET configurations and KEC configurations. This indeed propagates structuring information from the target of provides (KEC) to the source (KET), and in many cases, this could break the separation of concerns. In this case, it should be preferred to use cross-family constraints. Using cross-family constraints to write cross-tree constraints nevertheless requires that the concept carrying the constraint corresponds to a feature. If this is not the case the constraint should be redistributed on the existing children if any exists. . Cross-family features group features of interest from F M 2 into more "abstract features" that will enrich F M 1 with external concerns.

Recommendation Information embodied in AOC-posets can be used to guide the users in their choices when selecting a valid configuration. Concepts represent maximal groups of valid configurations sharing a maximal set of features, that may be intrinsic or cross-family. If we consider that the conjunction of a concept's features forms a query, then the concept's set of configurations can be seen as the result of this query. The concepts kept in an AOC-poset represent the maximal queries that can be formulated by a user, i.e., the maximal feature conjunctions representing each configuration subset. For instance, if a user wants to retrieve all KETs supporting the RDF format, the corresponding maximal query in Figure 4 corresponds to the concept introducing the feature rdf (KET 5), and the result is ket0-3. This can be applied for cross-family features too: it enables to query the KET configurations depending on their relationships to KECs. For instance, if a user wants to retrieve KETs providing only neural network KECs (KEC 5), this query corresponds to the concept KET 13 of Figure 4 introducing ∃∀provides(KEC 5) and it results to the unique configuration ket8. The querying can benefit from the variety of quantifiers and their different semantics [START_REF] Braud | Generalization effect of quantifiers in a classification based on relational concept analysis[END_REF]. If, this time, a user wants to retrieve KETs providing at least one neural network KEC, we should consider the AOC-poset built with the existential quantifier.

The cept formed in an AOC-poset with a quantifier s can be projected in the AOCposet formed with q g , if q g is more general than q s . For instance, ∃ is more general than ∃∀. Choosing the right quantifier can be useful to tune the degree of observed variability and the precision of the recommendation systems.

Concepts' position in the AOC-poset and to each other reveals other useful information about the structured family. Concepts on top of the AOC-poset usually possess less features and more configurations and therefore represent groups of features shared by most of the considered configurations. Dually, concepts at the bottom generally possess more features and less configurations, and represent more specialised features, i.e., the ones shared by few configurations. For instance Figure 3 reveals that there are more symbolic KECs (KEC 12) than statistical ones (KEC 8): this information may be used in recommendation systems to propose, for instance, popular features to a user. It is also noteworthy that the closer two concepts are in the structure, the more similar are their sets of features and configurations. Therefore, navigating from one concept to one of its neighbours represents modifications that can be applied on the corresponding query. This navigation has different properties depending the used conceptual structure (e.g. AOC-poset, concept lattice). Retrieving a concept based on a set of features can be seen as querying information from the relation concept family. Navigating from a concept to another in the structure enables a user to explore the set of available configurations by progressively refining the query. Let us imagine that a user wants to retrieve all the available open source KETs: this query corresponds to KET 7 of Figure 4. If the user wants to refine this query to find a more specific KET, the AOC-poset can be used to recommend the smallest modifications that may be applied: here, it proposes to either explore KETs having features json and saas (i.e., moving to KET 0) or explore KETs supporting rdf format and providing only symbolic KECs (i.e., moving to KET 5). Crossfamily features allow to switch between AOC-posets: here, the user can "jump" to the concept referenced by ∃∀provides(KEC 12) in the KECs' AOC-poset of Figure 3 and then refine the corresponding query. For instance, they can choose to focus on symbolic KECs specialised on ressource creation (KEC 10) and then jump back to the previous KETs' AOC-poset, but this time in the concept introducing ∃∀provides(KEC 10) (i.e., KET 1) with ket0 and ket1.

Related Work

Interconnected variability models Several tracks have been followed to represent modularity, and FMs extensions have been proposed, among which we can notice Feature Models with References (FMR) [START_REF] Czarnecki | Generative Programming for Embedded Soft.: An Industrial Experience Report[END_REF][START_REF] Czarnecki | Staged Configuration Using Feature Models[END_REF] and Modular Feature Models (MFM) [START_REF] Bagheri | Modular feature models: Representation and configuration[END_REF]. These two extensions enable the separation of a single FM into several FMs dedicated to sub-product lines or to separate concerns. In FMRs, a reference is a feature of an FM representing the root of another FM. Cross-FM constraints may be established between features of the two FMs. In MFMs, the authors encourage the definition of FMs modules, and the features in different FM modules are connected through cross-FM implications in module bridges. The approach is developped using the description logic ALCH for describing the FMs and the cross-tree and cross-FM constraints.

Friess et al. [START_REF] Friess | Modelling compositions of modular embedded soft. product lines[END_REF] work on modelling composition rules between different feature models representing independent product lines that can be combined. The authors define what they call a feature configuration, which represents a subset of an FM valid configurations satisfying some constraints. Then, they define composition rules (e.g., uses, parts-of ) between feature configurations of different FMs. Feature configurations permit to finely tune the set of configurations involved in a composition rule. Similarly, Rosenmuller et al. [START_REF] Rosenmüller | Modeling dependent software product lines[END_REF] work on reinforcing constraint expressiveness between different yet connected FMs by relying on product line specialisation [START_REF] Czarnecki | Staged Configuration Using Feature Models[END_REF]. The latter was introduced by Czarnecki et al. [START_REF] Czarnecki | Staged Configuration Using Feature Models[END_REF] as a way to prune the set of valid configurations of an FM by partially configuring this FM. Rosenmuller et al. argue that, when modelling complex product line, relying on domain constraints at the domain level is not sufficient, and that modelling constraints involving product line instances (i.e., valid configurations) is necessary. They use UML class diagrams to model these constraints on configurations, where a class represents an FM, a sub-class represents an FM specialisation and relations are used to define constraints at the configuration level. These "instance models" are used in association with classic domain models such as FMs. Urli et al. [START_REF] Urli | Handling complex configurations in software product lines: a tooled approach[END_REF] propose a domain model regrouping several FMs and relations between these FMs. In this approach, the domain model defines the different abstract concepts of the modelled complex product line, and each FM characterises the variability of one of these concepts. Contrarily to approaches presented before, constraints between FMs involved features and not subsets of configurations. Dhungana et al. [DSB + 11] propose an approach to ease the integration of different kinds of variability models into a unique infrastructure. They present Invar, a framework allowing to manage a repository of different variability models. When a new model is loaded , constraints between the new model and the existing ones need to be specified. These constraints take the form of "if condition then action". Conditions involve the selection or deselection of a feature. Actions may be to include another variability model (similar to model modularisation of [START_REF] Czarnecki | Staged Configuration Using Feature Models[END_REF]), or to select or deselect a feature in another variability model.

Extraction of interconnected variability models Most of the existing methods for FM reverse engineering exclusively focus on boolean FMs [CW07, RPK11, ACP + 12, DDH + 13, HLE13, LLE14]. Becan et al [START_REF] Bécan | Synthesis of attributed feature models from product descriptions[END_REF] were the first to propose a reverse engineering method for more complex FMs taking the form of FMs with attributes. During previous work [START_REF] Carbonnel | Towards complex product line variability modelling: Mining relationships from non-boolean descriptions[END_REF], we introduced the usage of Pattern Structures [START_REF] Ganter | Pattern Struct. and Their Projections[END_REF] to extract complex variability information in the form of logical relationships corresponding to the logical semantics of FMs extended with both multivalued attributes and UML-like cardinalities.

Here we elaborate on RCA, both being possibly combined. To the best of our knowledge, there is still no work about the extraction of "cross-family" variability information.

Conclusion

As software systems grow and the software product line engineering is spreading, collaborative design of huge product lines which combine several concerns will become more and more critical. Complex variability can take various forms, including variability among inter-connected software families. In this paper, we address the aim to extract such cross-family variability from a set of interconnected product configurations. We propose to employ Relational Concept Analysis (RCA), an extension of Formal Concept Analysis to assist this extraction. We introduce cross-family features and cross-family relationships that take the form of binary implications, mutex, co-occurrences or groups that involve both boolean features and cross-family features. Extracting such information has the inherent complexity of related data-mining methods. We expect reasonable complexity for exact extraction of binary implications, mutex, cooccurrences, and probably difficulties, if not unfeasibility, for groups, requiring approximate methods. We then discuss the possibility to use this variability information to assist the design of FMs with new kinds of references, or to explore a set of inter-connected configurations. As future work, we plan to apply the method to inter-connected product descriptions, such as connected PCMs, or by dividing large tabular descriptions, like the one produced for JHipster in [HNA + 17], into separate concerns. Previous work that applied FCA and RCA to Wikipedia's or synthetic PCMs [START_REF] Carbonnel | Variability representation in product lines using concept lattices: Feasibility study with descriptions from wikipedia's product comparison matrices[END_REF] gave us some preliminary results on the feasibility. Exploring the effects of the different RCA scaling quantifiers on these datasets will be very informative to tune the method. We also are interested by the possibilities of composing different inter-connected FMs with the support of our method. Finally, we will study the way this cross-family variability has to be taken into account to guide the users during a product selection/construction.

Figure 2 :

 2 Figure 2: AOC-poset of Knowledge Eng. Tools KET .

Figure 3 :

 3 Figure 3: AOC-poset of Knowledge Eng. Components KEC.

  ∈ {1, ..., n} and m object-object contexts R j = (O k , O l , r j ), j ∈ {1, ..., m}, with r j ⊆ O k × O l is a binary relation such that k, l ∈ {1, ..., n}, O k is the domain of the relation, and O l is the range of the relation.

Table 3 :

 3 Examples of cross-family features Relational attribute Object links vs. concept extent Schema Cross-family Feature r(o) ∩ Extent(C) = ∅ existential ∃r(C) r(o) ⊆ Extent(C) r(o) = ∅ universal ∃∀r(C)

Figure 4 :

 4 Figure4: AOC-poset of Knowledge Engineering Tools KET after integration of information about provides, using ∃∀ quantifier.

Figure 5 :Figure 6

 56 Figure 5: an FM extracted from the AOC-poset of Figure 2.

  different quantifiers are ordered by generalization [BDHB18] thus a con-constraints: pf → co ; co → pf ; sy → rdf ; rdf → sy ; o → saas ; sup → co

Figure 7 :

 7 Figure7: FM that can be extracted from the AOC-poset of Figure4, and cross-family information.

  1 and extracted thanks to API such as OpenCompare 2 , produced by random or manual sampling from generators, such as using the web application generator JHipster 3 [HNA + 17], or by analysing software developed by communities, such as Robocode

4

. These tabular descriptions generally expose characteristics (as columns) and their values (in cells) for the different product variants (as rows). They need to be cleaned, manually, or automatically [BSA + 14] before automated exploitation. From the cleaned tabular descriptions, several methods extract relationships for boolean features, leading to FM synthesis or more simply logical relationships extraction [LP07, HLE11, RPK11, ACP + 12, AMHS + 14, MZB + 17, SSS17, CHN19a]. A few approaches

[START_REF] Bécan | Synthesis of attributed feature models from product descriptions[END_REF][START_REF] Carbonnel | Towards complex product line variability modelling: Mining relationships from non-boolean descriptions[END_REF]

address the problem of extracting more complex variability information to take into account multi-valued attributes and cardinalities. Several of these approaches [LP07, RPK11, AMHS + 14, SSS17, CHN19b, CHN19a] are based on Formal Concept Analysis, which can be seen as a structuring framework for variability analysis and representation, in which some of the other approaches can be embedded, as shown in [AMdSH

+ 14, CHN19a]

.

Table 1 :

 1 Product descriptions of Knowledge Engineering Tools KET (lhs), and Knowledge Engineering Components KEC (rhs).

	KET	knowledge engineering tool (ket)	import export (ie)	rdf	json	proprietay format (pf )	distribution (di)	commercial (co)	open source (os)	saas (sa)

Table 2 :

 2 Mapping between AOC-posets and FMs[START_REF] Carbonnel | Modelling equivalence classes of feature models with concept lattices to assist their extraction from product descriptions[END_REF]. The extent of a context C is denoted by EXT (C)

	FMs	Prop. form.	AOC-posets

Table 4 :

 4 Relation provides between Knowledge Engineering Tools KET (rows) and Knowledge Engineering Components KEC (columns).

		KET_10
	knowledge engineering tool (ket)
		import export (ie)
		distribution (di)
		∃∀ provides( KEC_13)
	KET_7 open source (os)	KET_8 json	KET_9 saas (sa)	KET_6 proprietary format (pf) commercial (co)
	KET_5 rdf ∃∀ provides( KEC_12)	KET_0 ket6	KET_3 ket4 ket5	KET_4 ∃∀ provides( KEC_8) ∃∀ provides( KEC_9) ∃∀ provides( KEC_11)
		KET_1
	KET_2	∃∀ provides( KEC_6)		KET_13	KET_12
	ket2	∃∀ provides( KEC_10)		∃∀ provides( KEC_5)	∃∀ provides( KEC_4)
	ket3	ket0		ket8	ket7
		ket1

∃∀ is used instead of ∀ to avoid assigning to an object o a relational attribute formed on r when r(o) is empty (and would be included in any concept extent).

://github.com/OpenCompare 3 https://www.jhipster.tech/ 4 https://github.com/but4reuse/RobocodeSPL teaching


Lidia Ibáñez

Matthieu Rouleau

Abdelilah Wakkach

Claudine Blin-Wakkach

Gut Microbiome and Bone

Keywords: Osteoclasts, Osteoporosis, Osteoimmunology, T lymphocytes, Gut microbiome

The gut microbiome is now viewed as a tissue that interacts bidirectionally with the gastrointestinal, immune, endocrine, and nervous systems, affecting the cellular responses in numerous organs. Evidence is accumulating of gut microbiome involvement in a growing number of pathophysiological processes, many of which are linked to inflammatory responses. More specifically, data acquired over the last decade point to effects of the gut microbiome on bone mass regulation and on the development of bone diseases (such as osteoporosis) and of inflammatory joint diseases characterized by bone loss. Mice lacking a gut microbiome have bone mass abnormalities that can be reversed by gut recolonization.

Changes in gut microbiome composition have been reported in mice with estrogendeficiency osteoporosis and have also been found in a few studies in humans. Probiotic therapy decreases bone loss in estrogen-deficient animals. The effect of the gut microbiome on bone tissue involves complex mechanisms including modulation of CD4 + T-cell activation, control of osteoclastogenic cytokine production, and modifications in hormone levels. This complexity may contribute to explain the discrepancies across study results depending on age, gender, genetic background, and treatment duration. Further elucidation of the mechanisms involved is needed. However, the available data hold promise that gut microbiome manipulation may prove of interest in the management of bone diseases.

1-Introduction

The 10 14 microorganisms in the gut microbiome (GM) continuously engage in a dynamic dialog with the host cells [START_REF] Marchesi | The normal intestinal microbiota[END_REF]. The GM is now viewed as an organ that contributes to digestive function, transforming complex food components such as fibers and carbohydrates into absorbable nutrients including short-chain fatty acids [START_REF] Topping | Short-chain fatty acids and human colonic function: roles of resistant starch and nonstarch polysaccharides[END_REF]. Commensal bacteria produce factors that modulate the host immune responses, help preserve gut barrier integrity, defend against pathogenic microorganisms, and contribute to immune system development and regulation. These effects are particularly important for the mucosal immune system in the gut, which maintains tolerance toward food allergens and the GM while ensuring protection against pathogenic microorganisms. Thus, the GM regulates the development and function of lymphoid cells; the polarization of gut T cells, most notably Th17 cells; and the production of cytokines [START_REF] Ivanov | Specific microbiota direct the differentiation of IL-17-producing T-helper cells in the mucosa of the small intestine[END_REF][START_REF] Gaboriau-Routhiau | The key role of segmented filamentous bacteria in the coordinated maturation of gut helper T cell responses[END_REF].

The composition of the GM can vary over the life span depending on age, genetic factors, diet, medication intake, and host immune status. The interactions of the GM with the host may be altered by dysbiosis, which is defined as adverse changes in bacterial composition, diversity, and function. When dysbiosis occurs, the GM loses its protective capabilities, the gut barrier is impaired, and the host fails to effectively control the dissemination of GM components into the tissues. The resulting stimulation of the immune system can lead to a variety of diseases. Dysbiosis is associated not only with Crohn's disease, irritable bowel syndrome, and celiac disease; but also with metabolic, cardiovascular, and neurodegenerative diseases, as well as with inflammatory joint disease [START_REF] Hand | Linking the microbiota, chronic disease, and the immune system[END_REF][START_REF] Van De Wiele | How the microbiota shapes rheumatic diseases[END_REF][START_REF] Ni | Gut microbiota and IBD: causation or correlation?[END_REF]. In recent years, the effect of the GM on bone tissue has been evaluated in germ-free animals lacking a GM, animals given antibiotics or probiotics to modify the GM, and humans. Although the study results are somewhat conflicting, they establish the GM as a major regulator of bone mineral density (BMD), chiefly via effects on the immune system.

2-Interactions of bone and immune system --bone resorption

The immune system is central to BMD control under abnormal conditions. The receptor activator of NFκB ligand (RANKL) is the main cytokine involved in osteoclast differentiation. RANKL is produced by mesenchymal cells, osteoblasts, and osteocytes in the bone marrow. During inflammation, activated CD4 + T cells are also a source of RANKL [START_REF] Kong | OPGL is a key regulator of osteoclastogenesis, lymphocyte development and lymph-node organogenesis[END_REF][START_REF] Nakashima | Evidence for osteocyte regulation of bone homeostasis through RANKL expression[END_REF], as well as of other cytokines including interleukin (IL)-17 and tumor necrosis factor alpha (TNF-α), which stimulate osteoclastogenesis [START_REF] Kotake | IL-17 in synovial fluids from patients with rheumatoid arthritis is a potent stimulator of osteoclastogenesis[END_REF][START_REF] Lam | TNF-alpha induces osteoclastogenesis by direct stimulation of macrophages exposed to permissive levels of RANK ligand[END_REF]. Changes in T-cell activation levels therefore affect osteoclast differentiation [START_REF] Kong | Activated T cells regulate bone loss and joint destruction in adjuvant arthritis through osteoprotegerin ligand[END_REF][START_REF] Wakkach | Bone marrow microenvironment controls the in vivo differentiation of murine dendritic cells into osteoclasts[END_REF]. Among helper CD4 + T-cell subsets, only Th17 cells have been found to promote osteoclastogenesis in vitro [START_REF] Sato | Th17 functions as an osteoclastogenic helper T cell subset that links T cell activation and bone destruction[END_REF]. In vivo, Th17 cells are associated with enhanced osteoclast differentiation in mice and humans during inflammatory states [START_REF] Wakkach | Bone marrow microenvironment controls the in vivo differentiation of murine dendritic cells into osteoclasts[END_REF][START_REF] Pöllinger | Th17 cells, not IL-17+ γδ T cells, drive arthritic bone destruction in mice and humans[END_REF][START_REF] Oostlander | T cell-mediated increased osteoclast formation from peripheral blood as a mechanism for Crohn's disease-associated bone loss[END_REF]. Crosstalk between Th17 cells and osteoclasts has also been demonstrated in Crohn's disease in both mice and humans [START_REF] Oostlander | T cell-mediated increased osteoclast formation from peripheral blood as a mechanism for Crohn's disease-associated bone loss[END_REF][START_REF] Ciucci | Bone marrow Th17 TNFα cells induce osteoclast differentiation, and link bone destruction to IBD[END_REF]. Th17 cells activated at sites of bowel inflammation produce high levels of the osteoclastogenic factors RANKL, IL-17, and TNF-α [START_REF] Ciucci | Bone marrow Th17 TNFα cells induce osteoclast differentiation, and link bone destruction to IBD[END_REF]. They migrate to the bone marrow, where they strongly enhance osteoclast differentiation by producing cytokines and upregulating RANKL expression by mesenchymal cells [START_REF] Ciucci | Bone marrow Th17 TNFα cells induce osteoclast differentiation, and link bone destruction to IBD[END_REF]. Th17 cells also upregulate the expression by mesenchymal cells of chemokines (MCP1, MIP1α) that attract the monocytic precursors of osteoclasts, increasing their recruitment to the bone marrow [START_REF] Ciucci | Bone marrow Th17 TNFα cells induce osteoclast differentiation, and link bone destruction to IBD[END_REF]. The peripheral blood of patients with Crohn's disease contains Th17 cells that exhibit the same osteoclastogenic properties and therefore probably contribute to the bone loss often seen in this disease [START_REF] Oostlander | T cell-mediated increased osteoclast formation from peripheral blood as a mechanism for Crohn's disease-associated bone loss[END_REF][START_REF] Ciucci | Bone marrow Th17 TNFα cells induce osteoclast differentiation, and link bone destruction to IBD[END_REF]. Th17 cells activated in the gut are thus crucially involved in inflammatory bone loss destruction.

Interestingly, mice lacking a GM have no Th17 cells in their gut wall. Introducing certain bacterial species into the gut efficiently induces Th17 cell differentiation, demonstrating the importance of the GM in the emergence of these cells [START_REF] Ivanov | Specific microbiota direct the differentiation of IL-17-producing T-helper cells in the mucosa of the small intestine[END_REF][START_REF] Gaboriau-Routhiau | The key role of segmented filamentous bacteria in the coordinated maturation of gut helper T cell responses[END_REF]. Th17 cells are characterized by considerable phenotypic and functional plasticity [START_REF] Peters | The many faces of Th17 cells[END_REF]. They not only protect against pathogenic bacteria [START_REF] Ivanov | Induction of intestinal Th17 cells by segmented filamentous bacteria[END_REF] but also contribute to many chronic inflammatory diseases, some of which have bone loss among their features, such as Crohn's disease [START_REF] Khor | Genetics and pathogenesis of inflammatory bowel disease[END_REF], inflammatory joint disease [START_REF] Wu | Gut-residing segmented filamentous bacteria drive autoimmune arthritis via T helper 17 cells[END_REF], spondyloarthritis [START_REF] Glatigny | Proinflammatory Th17 cells are expanded and induced by dendritic cells in spondylarthritis-prone HLA-B27-transgenic rats[END_REF], and psoriasis [START_REF] Nograles | Th17 cytokines interleukin (IL)-17 and IL-22 modulate distinct inflammatory and keratinocyte-response pathways[END_REF]. These data lend credence to the existence of a link between the GM and bone resorption via the emergence of Th17 cells.

3-Gut microbiome (gm) and bone formation

Several clinical studies reported about two decades ago pointed to an association between excessive bacterial proliferation in the gut and decreased BMD [START_REF] Stotzer | Bone mineral density in patients with small intestinal bacterial overgrowth[END_REF]. Patients with small intestinal bacterial overgrowth syndrome had low BMD values and osteomalacia, and some of them had high levels of the proinflammatory cytokines TNF-α and IL-1, as well as increased osteoclast activation. Another feature of this syndrome is the development of nutritional deficiencies due to nutrient consumption by the gut bacteria. Thus, insufficient intestinal absorption of calcium, phosphate, and vitamin D contribute to the bone manifestations of the syndrome [START_REF] Stotzer | Bone mineral density in patients with small intestinal bacterial overgrowth[END_REF]. Consequently, these studies cannot provide evidence of a direct link between the GM and the bone phenotype.

Convincing evidence of GM involvement in bone tissue development was obtained only a decade later, via studies of germ-free mice, which have no GM. Given the effect of the GM on metabolism, absence of a GM from birth induces a host of physiological and metabolic alterations. The absorption of calories, vitamins, and nutrients is decreased; height and weight gains are delayed, and many organs are abnormally small [START_REF] Wallace | The impact of early life gut colonization on metabolic and obesogenic outcomes: what have animal models shown us?[END_REF]. Another consequence is immaturity of the immune, vascular, endocrine, intestinal, and nervous systems, all of which are involved in regulating bone mass (Figure 1) [START_REF] Clarke | Minireview: Gut microbiota: the neglected endocrine organ[END_REF][START_REF] Kennedy | Kynurenine pathway metabolism and the microbiota-gut-brain axis[END_REF]. Female C57Bl/6J mice lacking a GM from birth have higher BMD values in both trabecular and cortical bone compared to their conventionally reared counterparts [START_REF] Sjögren | The gut microbiota regulates bone mass in mice[END_REF].

These high BMD values reflect decreases in the numbers of osteoclast precursors and mature osteoclasts, which translate into diminished bone resorption. Bone formation is unchanged.

Additional contributors to the reduction in bone resorption include decreases in CD4 + T-cell frequency and in the expression levels of TNF-α and IL-6 in bone marrow [START_REF] Sjögren | The gut microbiota regulates bone mass in mice[END_REF].

Colonization of germ-free mice with GM from conventionally raised mice corrects the BMD and immune parameter values, confirming that the GM controls osteoclast differentiation during skeletal growth [START_REF] Sjögren | The gut microbiota regulates bone mass in mice[END_REF]. Among the receptors involved in recognition of bacteria by the immune system, the nucleotide-binding oligomerization domain proteins NOD1 and NOD2 bind bacterial peptidoglycans and activate the NFκB pathway. In knockout mice for both NOD1 and NOD2, changes in the GM have no effect on BMD or on TNF-α and RANKL expression, demonstrating that NOD1 and NOD2 play a critical role in GM effects on bone [START_REF] Ohlsson | Regulation of bone mass by the gut microbiota is dependent on NOD1 and NOD2 signaling[END_REF].

However, GM effects on bone development are more complex than appears to the casual eye. A study in germ-free juvenile male Balb/c mice reported in 2016 produced contradictory results, with delays in most of the main growth parameters and shorter femurs compared to conventionally raised controls [START_REF] Schwarzer | Lactobacillus plantarum strain maintains growth of infant mice during chronic undernutrition[END_REF]. The discrepancy between the results of the two studies may be ascribable to differences in the genetic background, gender, or age of the mice. The C57Bl/6J and Balb/C strains differ regarding GM diversity, IgA levels (which modulate GM diversity), and the T-cell profile [START_REF] Fransen | BALB/c and C57BL/6 mice differ in polyreactive iga abundance, which impacts the generation of antigen-specific IgA and microbiota diversity[END_REF][START_REF] Stanisavljević | Strain-specific helper T cell profile in the gut-associated lymphoid tissue[END_REF]. Treatment duration is also of the utmost importance. Colonization of germ-free mice by GM from normal mice is followed by increases in both bone resorption and bone formation. The bone resorption enhancement, however, is transient, resolving after 4 weeks. Thus, the GM exerts an acute catabolic effect on bone tissue [START_REF] Yan | Gut microbiota induce IGF-1 and promote bone formation and growth[END_REF]. In the longer term, bone formation predominates, leading to increased skeletal growth and bone mass correction, indicating a long-term anabolic effect of the GM on bone tissue [START_REF] Yan | Gut microbiota induce IGF-1 and promote bone formation and growth[END_REF]. This anabolic effect coincides with an increase in the production of insulin growth factor (IGF)-1. IGF-1 levels in serum and bone marrow are low in germ-free mice and in mice given antibiotics. They increase after gut recolonization or ingestion of a diet high in short-chain fatty acids, which are produced when GM bacteria break down polysaccharides [START_REF] Yan | Gut microbiota induce IGF-1 and promote bone formation and growth[END_REF]. Given the major role for IGF1 in skeletal development [START_REF] Yakar | IGF-1 and bone: New discoveries from mouse models[END_REF], control of IGF1 production is among the mechanisms that may explain the effect of the GM on bone growth.

Another major issue is the influence of gender. Male and female C57Bl/6J mice with significant GM alterations induced by low-dose penicillin treatment started at birth exhibit different phenotypes [START_REF] Cox | Altering the intestinal microbiota during a critical developmental window has lasting metabolic consequences[END_REF]. The females, but not the males, have higher adulthood BMD values compared to untreated animals [START_REF] Cox | Altering the intestinal microbiota during a critical developmental window has lasting metabolic consequences[END_REF]. In addition, the BMD increase seen after gut recolonization of germ-free mice is more marked in the long-term in males than in females [START_REF] Yan | Gut microbiota induce IGF-1 and promote bone formation and growth[END_REF]. Interestingly, the GM in male and female mice is similar before puberty but differs in adulthood, and the differences contribute to increase the susceptibility to certain diseases, such as type I diabetes in females [START_REF] Yurkovetskiy | Gender bias in autoimmunity is influenced by microbiota[END_REF]. Gender-related differences in the GM have also been reported in humans [START_REF] Mueller | Differences in fecal microbiota in different European study populations in relation to age, gender, and country: a cross-sectional study[END_REF][START_REF] Dominianni | Sex, body mass index, and dietary fiber intake influence the human gut microbiome[END_REF]. Thus, two-way interactions between the GM and sex hormones influence the composition of the GM and the response of host tissues, including bone, to the GM [START_REF] Baker | Estrogen-gut microbiome axis: Physiological and clinical implications[END_REF].

4-Gut microbiome and bone destruction

Potential effects of the GM on bone mass during estrogen-deficient osteoporosis have been investigated. At the menopause, the drop in estrogen production leads to a decline in bone mass due to a combination of diminished bone formation and increased bone resorption. In mice with osteoporosis induced by ovariectomy, osteoclast activity is enhanced due to increased production of RANKL and TNF-α in the bone marrow [START_REF] Roggia | Up-regulation of TNF-producing T cells in the bone marrow: A key mechanism by which estrogen deficiency induces bone loss in vivo[END_REF]. The production of these two factors is mediated by CD4 + T cells [START_REF] Roggia | Up-regulation of TNF-producing T cells in the bone marrow: A key mechanism by which estrogen deficiency induces bone loss in vivo[END_REF]. Thus, mice deficient in CD4 + T cells do not lose bone after being ovariectomized [START_REF] Cenci | Estrogen deficiency induces bone loss by enhancing T-cell production of TNF-α[END_REF]. Similarly, compared to premenopausal women and to postmenopausal women without osteoporosis, patients with postmenopausal osteoporosis exhibit RANKL and TNF-α overproduction by peripheral blood CD4 + T cells [START_REF] D'amelio | Estrogen deficiency increases osteoclastogenesis up-regulating T cells activity: a key mechanism in osteoporosis[END_REF].

The central role for the GM in controlling lymphocyte activation [START_REF] Gaboriau-Routhiau | The key role of segmented filamentous bacteria in the coordinated maturation of gut helper T cell responses[END_REF] and its two-way interactions with sex hormones [START_REF] Yurkovetskiy | Gender bias in autoimmunity is influenced by microbiota[END_REF] have prompted studies in mice designed to assess the effect of the GM on the BMD decrease induced by estrogen deprivation. In several studies of ovariectomized mice, a protective effect against bone loss was obtained by the administration of probiotics such as Lactobacillus (L) reuteri, L. paracasei, L. plantarum, Bifidobacterium (B) longum, and mixtures of several species (B. breve, B. longum, B. infantis, L. acidophilus, L. plantarum, L. paracasei, L. bulgaricus, and Streptococcus thermophilus) [START_REF] Parvaneh | Probiotics (Bifidobacterium longum) increase bone mass density and upregulate Sparc and Bmp-2 Genes in rats with bone loss resulting from ovariectomy[END_REF][START_REF] Britton | Probiotic L. reuteri treatment prevents bone loss in a menopausal ovariectomized mouse model[END_REF][START_REF] Ohlsson | Probiotics protect mice from ovariectomyinduced cortical bone loss[END_REF]. This finding was confirmed in a model of osteoporosis induced by the gonadotropin-releasing hormone agonist leuprolide, which suppresses the production of sex hormones, including estrogens, in the long term. The estrogen deficiency led to bone loss in conventionally raised mice but not in germ-free mice [START_REF] Li | Sex steroid deficiency-associated bone loss is microbiota dependent and prevented by probiotics[END_REF]. In conventionally raised mice, estrogen deficiency increased gut wall permeability and the proportion of CD4 + T cells producing RANKL, IL-17, and TNF-α, thereby enhancing osteoclastogenesis (Figure 2) [START_REF] Li | Sex steroid deficiency-associated bone loss is microbiota dependent and prevented by probiotics[END_REF]. Germ-free mice exhibited none of these modifications [START_REF] Li | Sex steroid deficiency-associated bone loss is microbiota dependent and prevented by probiotics[END_REF]. In a small preliminary study of normal controls, patients with osteopenia, and patients with osteoporosis (6 per group), differences in the GM were found across the three groups. This finding requires confirmation in a larger sample size but suggests a role for dysbiosis in human osteoporosis [START_REF] Wang | Diversity analysis of gut microbiota in osteoporosis and osteopenia patients[END_REF]. GM alterations have a major impact in inflammatory joint disease accompanied by bone loss. Mice and rats whose GM is lacking or altered by antibiotics have decreased susceptibility to arthritis and spondyloarthritis [START_REF] Taurog | The germfree state prevents development of gut and joint inflammatory disease in HLA-B27 transgenic rats[END_REF][START_REF] Wu | Gut-residing segmented filamentous bacteria drive autoimmune arthritis via T helper 17 cells[END_REF]. Gut dysbiosis has been reported in patients with rheumatoid arthritis or spondyloarthritis [START_REF] Vaahtovuo | Fecal microbiota in early rheumatoid arthritis[END_REF][START_REF] Zhang | The oral and gut microbiomes are perturbed in rheumatoid arthritis and partly normalized after treatment[END_REF][START_REF] Breban | Faecal microbiota study reveals specific dysbiosis in spondyloarthritis[END_REF]. In addition, in rats with collagen-induced arthritis, the administration of probiotics protects against inflammation and bone loss [START_REF] Amdekar | Lactobacillus casei reduces the inflammatory joint damage associated with collagen-induced arthritis (CIA) by reducing the pro-inflammatory cytokines: Lactobacillus casei: COX-2 inhibitor[END_REF]. Thus, the effect of the GM on bone is not confined to conditions characterized by hormonal changes and can involve a wide variety of mechanisms.

Recent data indicate that osteoporosis and inflammatory joint diseases share a common immune component. The gut barrier is a crucial player in host-GM interactions.

Dysbiosis is associated with gut barrier alterations that promote the dissemination of bacteria and of the factors they produce. The gut barrier is altered in both inflammatory joint disease and estrogen deficiency [START_REF] Li | Sex steroid deficiency-associated bone loss is microbiota dependent and prevented by probiotics[END_REF][START_REF] Kerr | Repeated measurement of intestinal permeability as an assessment of colitis severity in HLA-B27 transgenic rats[END_REF][START_REF] Martínez-González | Intestinal permeability in patients with ankylosing spondylitis and their healthy relatives[END_REF]. In both situations, the alterations are accompanied with enhanced CD4 + T-cell activation and increased production of the proinflammatory and osteoclastogenic cytokines IL-17, TNF-α, IL1 β and RANKL [START_REF] Britton | Probiotic L. reuteri treatment prevents bone loss in a menopausal ovariectomized mouse model[END_REF][START_REF] Ohlsson | Probiotics protect mice from ovariectomyinduced cortical bone loss[END_REF][START_REF] Li | Sex steroid deficiency-associated bone loss is microbiota dependent and prevented by probiotics[END_REF].

By modulating the gut immune response, dysbiosis also alters monocyte and lymphocyte migration to tissues, including the bone marrow. In germ-free mice, the numbers of monocytes and osteoclastic precursors in the bone marrow are diminished but return to normal after gut colonization with GM from control mice [START_REF] Sjögren | The gut microbiota regulates bone mass in mice[END_REF][START_REF] Khosravi | Gut microbiota promote hematopoiesis to control bacterial infection[END_REF]. In addition, changes in the GM translate into alterations in monocyte trafficking [START_REF] Bain | Constant replenishment from circulating monocytes maintains the macrophage pool in the intestine of adult mice[END_REF]. In patients with Crohn's disease, severe bone loss, and dysbiosis, the Th17 cells can migrate to the bone marrow and induce the recruitment of osteoclastic precursors, thereby prompting a massive increase in osteoclastogenesis [START_REF] Ciucci | Bone marrow Th17 TNFα cells induce osteoclast differentiation, and link bone destruction to IBD[END_REF]. Whereas osteoclasts from normal mice induce the generation of regulatory T cells (Tregs), osteoclasts generated during inflammatory states activate CD4 + T cells that produce TNF-α [START_REF] Ibáñez | Inflammatory osteoclasts prime TNFαproducing CD4(+) T cells and express CX3 CR1[END_REF]. This difference is ascribable to the cellular origin of the osteoclasts and reflects differences in osteoclastic precursor recruitment under normal conditions versus during dysbiosis-related inflammatory states.

5-Future directions

A growing number of studies indicate a major role for the GM in regulating bone mass both during growth and during disease states. The GM can act in combination with other factors including the diet, genetic susceptibility, lifestyle, and medications. Another mechanism of action of the GM is a direct effect due to the dissemination of bacteria [START_REF] Burcelin | Metagenome and metabolism: the tissue microbiota hypothesis[END_REF] or of the factors they produce, which then activate the inflammatory responses in various tissues including the bone marrow However, the mechanisms involved are complex, and further investigations are needed to clarify and to control them.

Restoring a balanced GM is now being considered as a therapeutic tool for various diseases. Methods of modifying the GM include dietary changes and supplementation with probiotics such as short-chain fatty acids, oligosaccharides, carbohydrates, and dietary fiber. These supplements are metabolized by certain bacterial strains, whose growth they promote, thereby modifying the composition of the GM. The modified GM stimulates antiinflammatory responses and promotes the intestinal absorption of calcium, thereby increasing BMD, an effect demonstrated in mice given short-chain fatty acids by gavage [START_REF] Yan | Gut microbiota induce IGF-1 and promote bone formation and growth[END_REF]. Oligosaccharides from human milk have shown similar beneficial effects on bone [START_REF] Charbonneau | Sialylated milk oligosaccharides promote microbiota-dependent growth in models of infant undernutrition[END_REF].

Probiotics also have a substantial effect on BMD. Various strains of Lactobacillus and Bifidobacterium have antiinflammatory effects, enhance vitamin D abruption, and diminish osteoclast differentiation, thereby protecting against the bone mass loss induced by ovariectomy in mice [START_REF] Parvaneh | Probiotics (Bifidobacterium longum) increase bone mass density and upregulate Sparc and Bmp-2 Genes in rats with bone loss resulting from ovariectomy[END_REF][START_REF] Britton | Probiotic L. reuteri treatment prevents bone loss in a menopausal ovariectomized mouse model[END_REF][START_REF] Ohlsson | Probiotics protect mice from ovariectomyinduced cortical bone loss[END_REF]. The effects of probiotics in humans are being evaluated in several clinical trials registered on ClinicalTrials.gov, but the results are not yet available.

Another approach is GM transplantation, which has been widely used in mice to demonstrate that the GM was involved in many disease states, including those affecting the bone [START_REF] Sjögren | The gut microbiota regulates bone mass in mice[END_REF][START_REF] Yan | Gut microbiota induce IGF-1 and promote bone formation and growth[END_REF][START_REF] Li | Sex steroid deficiency-associated bone loss is microbiota dependent and prevented by probiotics[END_REF]. In humans, GM transplantation has been used successfully to treat bowel diseases such as colitis due to antibiotic-resistant bacteria [START_REF] Cammarota | Fecal Microbiota Transplantation for the treatment of Clostridium difficile infection: A systematic review[END_REF]. Clinical studies are under way to evaluate the efficacy of GM transplantation in rheumatic diseases. However, no effects on bone have been reported. Interest is growing in GM manipulation as a therapeutic tool, and additional research is therefore needed to elucidate the mechanisms involved and to evaluate the efficacy of this approach in bone diseases. The beneficial effects of GM manipulation in preclinical models suggest promise for the treatment of bone disease.

Figure 1 :

 1 Figure 1: Link between the gut microbiome and bone

Figure 2 :

 2 Figure 2: Gut microbiome alterations and osteoclast differentiation

Acknowledgments

The work conducted by our group is supported by the French National Research Agency (ANR, ANR-16-CE14-0030-01), Biocodex, and the Fondation pour la Recherche Médicale (FRM, ECO20160736019). The figures accompanying this article were created using illustrations from Servier Medical Art (http://smart.servier.com).

Disclosure of interest

CBW has received a research grant from Biocodex.