Marcel Bourgade
email: m.bourgade@pnr-martinique.com

geckos recently introduced to Martinique, and remarks on their current distributions in the Caribbean and their presumed impacts on biodiversity

Keywords: Mabouya, Mabouia, Mabuya, bantu, geckos, Sauria, Africa, Eastern Caribbean, Asia, lizards, non-indigenous, passive anthropochory, intentional introductions, invasive alien species, hemidactyls, Gekko, Lepidodactyls, parthenogenesis, ecological niche, biodiversity

Natural dispersion and introductions of

African hemidactyl geckos called Mabouya in the Caribbean and in South America, and first mentions of the Mourning gecko (Lepidodactylus lugubris Duméril and Bibron, 1836), after that of the Tokay gecko (Gekko gecko Linnaeus, 1758), Asian geckos recently introduced to Martinique, and remarks on their current distributions in the Caribbean and their presumed impacts on biodiversity Marcel Bourgade To cite this version: Marcel Bourgade. Natural dispersion and introductions of African hemidactyl geckos called Mabouya in the Caribbean and in South America, and first mentions of the Mourning gecko (Lepidodactylus lugubris Duméril and Bibron, 1836), after that of the Tokay gecko (Gekko gecko Linnaeus, 1758), Asian geckos recently introduced to Martinique, and remarks on their current distributions in the

The term Mabouya in various Bantu African languages as well as in Taino languages is a name that refers to different things. It is used to name a species of skink found in Martinique called Greater Skink of Martinique : Mabuya mabouya, species extinct today. Mabuya or Mabouia, is also the generic term for naming geckos in general in the Caribbean and in West and South Africa.

The Mabuya then refer to mysterious reptiles with nocturnal habits, capable of vocalizing and emitting sounds, and which, thanks to their tapetum lucidum, have phosphorescent eyes ; they are evocative in the mythologies of African and Caribbean peoples of the "spirits of the night".

African Mabouya, Hemidactyla geckos dispersed naturally from West Africa to America since geological time

The African Mabouya Hemidactylus sp. dispersed across the Atlantic, from geological time estimated between -16 to -6 My well before the very existence of humans. Homo sapiens sapiens having appeared about 80,000 years ago, and the oldest human presence on the American continent being attested more than 35,000 years ago at the site of Pedra Furada on the Atlantic coast of northeastern current Brazil. This fact of a transatlantic biological connection, by crossing the ocean on a plant raft of hemidactyl propagules, which washed up on the emerged lands of the proto-Caribbean, is attested and confirmed by molecular biology data for Brazil. [START_REF] Carranza | Systematics, biogeography, and evolution of Hemidactylus geckos (Reptilia: Gekkonidae) elucidated using mitochondrial DNA sequences Departament de Biologia Animal[END_REF], Saint Lucia (Ilets Maria), and Trinidad, having generated the species H. brasilianus and H. palaichthus, well after the geological separation between Africa and South America, it -105 to -119 My ago [START_REF] Mclaughlin | The breakup history of Gondwana and its impact on pre-Cenozoic floristic provincialism[END_REF].

It must be said that the quality of the limestone of the eggs of hemidactyls as in general those of the Gekkonidae, compared to those of other squamates, prove to be more resistant to exposure to salt water; and that this proven resistance makes it quite possible to cross long distances at sea and therefore the Atlantic on rafts of plants. [START_REF] Carranza | Systematics, biogeography, and evolution of Hemidactylus geckos (Reptilia: Gekkonidae) elucidated using mitochondrial DNA sequences Departament de Biologia Animal[END_REF].

It should be noted that this period considered by Carranza and Arnold, from -16 My to -6 My corresponds globally to the revised dates of the formation of the Isthmus of Panama, leading to the separation of the water masses of the Pacific Ocean with those waters of an Atlantic ocean with new directions of emerging currents. Indeed from -25 My to -10 My, up to -8 My we witness tectonic and volcanic geological phenomena, leading to the formation of the Isthmus of Panama to the gradual closing of the straits which until then gave access to the waters and ocean currents of the Pacific. Isthmus of Panama whose date of complete closure would be much older than previously thought, dating from -10 My instead of -4 to 3 My. (Sepulchre P. et al. 2014).

This separation between water masses of the Pacific and Atlantic oceans will fundamentally modify circulation and marine currents ; the Gulf Stream will configure itself in the dynamics of these profound upheavals naturally impacting the evolution of fauna and flora. This dating of an attested dispersal of hemidactyls for Brazil and globally for the north of the South American continent [START_REF] Carranza | Investigating the origin of transoceanic distributions: mtDNA shows Mabuya lizards (Reptilia, Scincidae) crossed the Atlantic twice[END_REF] is therefore applicable for the eastern Caribbean, and particularly demonstrated for Saint Lucia and Trinidad. It remains to deepen the studies relating to hemidactyls considered as H. mabouia present in the eastern Caribbean, with this complexity of possible hybridizations between these different hemidactyls all originating from the Atlantic coasts of Africa.

The presence of H. palaichthus, on Major Maria islet in Saint Lucia and Trinidad, as in Brazil and in the north of the South American continent, has been attested by David Corke since 1987 [START_REF] Corke | Indies)[END_REF].

Mabouya Hemidactylus palaichthus, on Major Maria islet in Saint Lucia and Trinidad Thus for millions of years the Mabouya Hemidactylus palaichthus has cohabited with the endemic American Caribbean Big Mabouya Thecadactylus rapicauda Houttuyn, 1782.

Thecadactylus rapicauda here specimen on the Sainte-Anne peninsula south-east of Martinique

Comments relating to the chronology of the passive introductions of African Mabouya

The African Mabouya Hemidactylus mabouia, as in the whole of the Caribbean, is present in nature throughout the territory of Martinique, it is observable in arid and dry environments, as in humid environments, for example isolated on the limestone islets of Sainte-Anne, or in the petrifaction savannah and on the Limestone hills of Sainte-Anne, or in the mangroves and in the three types of forest (xerophilous, mesophilic, hygrophilous), where it cohabits with the endemic American Caribbean Big Mabouya Thecadactylus rapicauda Houttuyn, 1782; as it is also very widespread inside homes with different phenotypes. It has been identified in the Eastern Caribbean by several authors, first in Saint Vincent in 1818.

The Hemidactyles observed by ourselves in the wild in Martinique during herpetological surveys as part of the IBR (Rapid Biological Inventory) initiated by the city of Lamentin for studies of the bay of Génipa in cooperation with Cuban biologists, and by elsewhere for the CD2S (Caribbean Center for Sustainable and Solidarity Development) as part of preliminary work in 2013 for the A.B.C. (Atlas of Communal Biodiversity) of SAINTE-ANNE, in humid mangroves, in dry and arid environments of the peninsula of South East Martinique, taking refuge under rocks, presumed to be the same species of African Mabouya Hemidactylus mabouia, present phenotypes and patterns very diverse and varied, compared with those of the Hemidactyls living and breeding in general around and in dwellings.

The massive and certainly homogeneous introduction of hemidactyls from several geographical areas of West and South Africa, an introduction linked to the activity of the so-called "triangular slave trade", i.e. recent passive anthropochoric introductions of hemidactyls accompanying the transatlantic deportation of Africans on slave ships to America to be enslaved there, since 1492 (-500 years), would explain the phenotypic diversity observed today in the Eastern Caribbean of Mabouya considered as H. mabouia; this phenotypic variety is probably the consequence of biological reconnections between hemidactyls introduced in the last 500 years and hemidactyls originating from the same African coasts already introduced before the slavery period; African mabouya Hemidactylus mabouia is indeed present in America and therefore in the Caribbean much earlier than we think.

Three specimens of adult Hemidactylus mabouia and one juvenile, all observed on the Sainte-Anne peninsula

The mere fact that the same word Mabuya or Mabouia, is existing and used in idioms to name the same reality among both African and American peoples before European colonization, is significant for admitting the anthropochoric introduction by Africans of Hemidactylus mabouia in Taïno country long before Christopher Columbus landed in Gwanahani, which he baptized San Salvador. Linguistically, a certain number of other facts should be noted regarding the similarities of African and Amerindian terms for identical names for subjects, animals, plants, objects and utensils before the European presence. This is the case of similarity between African and Amerindian idioms for terms such as: yam, yanm, meaning yam; n'yambi meaning sweet potato; for the term gwanin, designating a species of metal alloy based on gold existing both in West Africa and Central America before colonization by Europeans. The use by peoples of the term Mabouya both in Caribbean America and in Africa, is then part of a series of historical and archaeological facts, of linguistic data, constituting bundles of evidence of an African presence before 1492 in the say Americas. We refer to the work of the Mexican historian Riva PALACIO, "History of America before Christopher Columbus" published in 1900; by Joël ROGERS, in his book "Africa's gift to America"; by Léo WIENER, "Africa and the discovery of America" in three volumes published in 1920, 1922, by the historian John D. BALDWIN in his work "Ancient America" published in 1971; but also to the work of Jean MAZEL, specialist in Mediterranean Civilizations, author of "Presence of the Black World" (1975); without forgetting the work of Ivan VAN SERTIMA "They came before Columbus".

Let us quote Jean MAZEL commenting on the work of WIENER, extract from the chapter "Blacks in the Americas before Christopher Columbus" without having knowledge at the time in 1975 of the archaeological and molecular biology data relating to the site attesting to a human presence coming directly from Oldest Africa in America at Pedra Furada in northeast Brazil:

"It is therefore impossible to deny the presence of blacks in pre-Columbian America. Whether it's Olmec statues, whether it's pottery found everywhere in burials, in the Mayan pyramids. It would seem that from the moment the Aztecs prevailed over the Mayans, from the moment the Aztec civilization took hold, there was a kind of erasure of the old black presence in the Americas. It is generally admitted, at present, that the American continent was originally populated by Indians who came from Asia, through the Bering Strait, at different very ancient times. And it is also now accepted that navigators from the Old World reached the shores of the Americas long before Columbus. It suffices first of all to look at a map to realize that the shortest distance which connects the Eurafrican continent to the Americas is indeed the passage which separates the tip of Cape Verde at the northeastern extremity of Brazil and to the south of the West Indies. We know very well, too, that the regime of trade winds and currents almost necessarily brings to this zone a sufficiently solid boat from the coasts of present-day Senegal. (…) Moreover, it seems inscribed in the geography of the West Indies and of all the islands of this coral fringe, the advanced shield of America towards the east, that they would serve as a bridgehead for the approach and penetration, by the Africans, from the American continent; free approach and free penetration in the oldest times, then, at the time of the slave trade, forced approach and penetration.

In any case, it is via the Antilles that African seeding has always occurred, around the Caribbean Sea in Central America and North America, as in Colombia and even in Peru where we have unearthed a quantity of art objects representing obviously Negro-African human types. » Thus, Central American sites, notably Mexican, just as prehistoric but more recent than the South American site of Pedra Furada in Brazil, lead to the same conclusions: the ancestors of Amerindians from Amazonia and Central America are also African , crossed the Atlantic and inhabited the American coasts from Brazil to Yucatan. The work of Douglas WALLACE then largely questioned the so-called Asian origin of the first Americans or Paleo-Americans; By focusing on the mitochondrial DNA of Native Americans, he succeeded in proving:

-That the settlement of America was made in several migratory waves, the oldest of which dates back, according to the most recent archaeological studies, to 45,000 years ago with the implication of crossings of the Atlantic by "navigators" or "drifts" coming from Africa ; -That successive and massive migrations came from Asia (Behring Strait), but also to a lesser extent from Europe (DNA traces of European or Caucasian having migrated by crossing the North Atlantic).

-That an Australoid origin (Melanesian or African) is therefore observed on human skeletons discovered in several prehistoric sites in South and Central America.

Recent introductions of Asian geckos in America and the Caribbean

The Tokay gecko, Gekko gecko (Linnaeus, 1758), was intentionally introduced in Martinique by a socalled "defender of nature and birds" in the early 1970s during the period when the Washington Convention was drawn up and signed; a certain M. Bon Saint-Côme, although founding president at the time of the SEPANMAR association (approved by the State), then released into the wild on the territory of Saint-Joseph (Bois Neuf) and Lamentin ( Bélème) of the Gecko Tokay that he had brought back from Southeast Asia.

The other two species of Asian Geckos L. lugubris and H. frenatus have been known to have been introduced into the American Caribbean in passive mode and anthropochory; they have had a wide distribution in the greater Caribbean region for more than 70 years, they have only been reported in the insular Caribbean during the last decade. Comparatively, the dynamics of introductions and dispersal of G. gecko on the insular Caribbean are similar to those of L. lugubris and H. frenatus.

Among the three new exotic species, the Mourning gecko, Lepidodactylus lugubris (Duméril and Bibron, 1836), recorded in South America, Pacific coast in Peru since year 1963, has the largest distribution on Curaçao and is probably established since the Longer. The common house gecko, Hemidactylus frenatus (in Duméril, 1836) has a limited distribution and was probably a more recent introduction.

Distribution in the greater Caribbean region of Hemidactylus frenatus

It is therefore highly probable and predictable that Martinique will record the introduction of this third exotic species of Asian gecko.

The Tokay Gecko, Gekko gecko intentionally introduced to the eastern Caribbean first in Martinique

To the responsibility of intentional introduction of M. Bon Saint-Côme in the early 1970s, is added that of a second introduction to the Usine du Galion district in Trinité, on the occasion of hurricane David in 1979, Tokay Geckos have escaped from their enclosure. This second wave of introduction of Gecko Tokay in Martinique is the responsibility of a seasonal employee of the Usine du Galion, a Dutchman who came to Martinique several times during the sugar campaign, who collected said pets, kept aquariums containing different species of reptiles (snakes, lizards, geckos, etc.) which he also traded or trafficked; since he also devoted himself to capturing mabouyas and lizards endemic to the country to transport them and sell them elsewhere.

In red the two sites attested to intentional introductions, release into the wild of G. gecko in 1970 at Saint-Joseph, in 1980 at Trinité (Galion). In orange brown locations of sites where the Tokay Gecko proliferate ; last report at the beginning of June 2023 in the town of Morne Vert.

Distribution of the Gekko gecko in the Caribbean (June 2023). Martinique is the first island in the Caribbean where this Asian species was intentionally introduced in 1970 ; and the populations of G. gekco introduced in Guadeloupe most likely come from Martinique.

Dwarf Gecko, Mourning Lepidodactylus lugubris

The Mourning Gecko (Lepidodactylus lugubris), an Asian dwarf gecko, is one of the most prolific geckonid species in the world ; due to its capacity for parthenogenesis [START_REF] Cuellar | Natural parthenogenesis in the gekkonid lizard Lepidodactylus lugubris[END_REF][START_REF] Borroto-Páez | High abundance of the introduced Mourning Gecko (Lepidodactylus lugubris) in southwestern Grande Terre, Guadeloupe, French West Indies[END_REF]. Like the hemidactyl Mabouya, these geckos produce eggs that are resistant to exposure to salt water and desiccation [START_REF] Bosch | First record from Cuba of the introduced mourning gecko, Lepidodactylus lugubris (Duméril and Bibron, 1836)[END_REF][START_REF] Lorvelec | The status of the introduced Mourning Gecko (Lepidodactylus lugubris) in Guadeloupe (French Antilles) and the high probability of introduction of other species with the same pattern of distribution[END_REF] because their hard-shelled eggs contain the water necessary for their survival. development [START_REF] Andrews | Water vapor permeability of the rigid-shelled gecko egg[END_REF]. Also in addition to the passive anthropochoric introductions of the Mourning Gecko, their presence in the exotic pet trade (EAC), constitutes growing facts of intentional introductions; thus L. lugubris has amplified the opportunities for introductions beyond its natural distribution [START_REF] Lorvelec | The status of the introduced Mourning Gecko (Lepidodactylus lugubris) in Guadeloupe (French Antilles) and the high probability of introduction of other species with the same pattern of distribution[END_REF].

Originally from the coastal areas of Southeast Asia and the Indo-Pacific region [START_REF] Behm | First records of the mourning gecko (Lepidodactylus lugubris Duméril and Bibron, 1836), common house gecko (Hemidactylus frenatus in Duméril, 1836), and Tokay gecko (Gekko gecko Linnaeus, 1758) on Curaçao, Dutch Antilles, and remarks on their Caribbean distributions[END_REF], Mourning geckos have spread to several Pacific islands, North America, and more recently in the Caribbean [START_REF] Nania | Continuous expansion of the geographic range linked to realized niche expansion in the invasive Mourning gecko Lepidodactylus lugubris (Duméril & Bibron, 1836)[END_REF]. The first report of the presence of the species in the Caribbean is from 1976 on Big Corn Island in Nicaragua [START_REF] Henderson | Lepidodactylus lugubris (Reptilia: Gekkonidae). A recent addition to the herpetofauna of Nicaragua[END_REF]).

They have since been reported in Guadeloupe in 2010 [START_REF] Lorvelec | First record of the mourning gecko (Lepidodactylus lugubris) on Guadeloupe, French West Indies[END_REF][START_REF] Gomès | Lepidodactylus lugubris (Mourning Gecko). Distribution[END_REF][START_REF] Borroto-Páez | High abundance of the introduced Mourning Gecko (Lepidodactylus lugubris) in southwestern Grande Terre, Guadeloupe, French West Indies[END_REF] The new Asian dwarf gecko, Mourning Lepidodactylus lugubris, also arboreal but still frequenting the ground, rocks and litter, will come to occupy the ecological niche of the endemic and native lizards of Martinique. Considering its size, it will probably be eaten by larger geckos than it, but also by Anolis.

On the other hand, the Asian dwarf gecko could easily consume Ti Mabouya Sphaérodactyles at night.

But we must not neglect the certain advantage of the Asian dwarf gecko Mourning over other native species whose ecological niches it will occupy to the point that these endemic species disappear there (substitution process); that of its parthenogenetic mode of reproduction giving it an exceptional potential for propagation.

Note that there are now only two species of exotic lizards introduced in Martinique whose mode of reproduction is parthenogenetic:

-The Lepidodactylus lugubris, Asian dwarf gecko probably constituting a threat for the endemic dwarf geckos of Martinique which are the Sphaérodactyles.

-The Gymnophthalmus underwoodi Grant, 1958, originating from the South American continent, very prolific which threatens to replace the endemic species of Saint Lucia and Martinique, the Gymnophthalmus pleii Bocourt, 1879.

  and in other Caribbean countries such as Cuba and Bonaire (Alonso Bosch and Borroto-Páez 2017; Van Buurt and Smulders 2022). The very first record indicating the presence of the species in Martinique dates from December 2020, by an observer member of iNaturalist (https://www.inaturalist.org/observations/66918518) who posted a location in the southeastern part of Martinique on the territory of the city of Sainte-Anne, visibly observed on a ship at anchor near Pointe Marin. A second report of the presence of Lepidodactylus lugubris dated July 2022 observed on the banks of the Lézarde canal in Lamentin Martinique, was recorded by a publication in May 2023, by Alice Armand and Benjamin Ferlay of the Bivouac Naturaliste association on journal: Reptiles & Amphibians (ku.edu) And a third report by the author of this publication, dated the end of December 2022, of the presence of Lepidodactylus lugubris around the Marin marina ; identification of the species confirmed by Dr Ivan INEICH expert in herpetology of the MNHN of Paris. Specimens of Dwarf Gecko, Demi-Deuil Lepidodactylus lugubris observed at Marina du Marin Distribution of Lepidodactylus lugubris in the Caribbean and Central and South America (June 2023). The presence of the Mourning Dwarf Gecko, L. lugubris is now attested in Martinique. This second Asian species present in Martinique was clearly introduced in a passive and anthropochoric mode by propagules most likely coming from Guadeloupe. In red, the first three sites of observation of introductions of the Asian species of Dwarf Mourning Gecko Lepidodactylus lugubris in Martinique. 2020 2020 2022

  Ti Mabouya of the Mornes Limestones of Sainte-Anne Martinique, Dwarf Gecko Sphaérodactylus sp. (b) Ti Mabouya of Saint-Lucia, Dwarf Gecko, Sphaerodactylus microlepis Reinhardt & Lütken 1862 (c) Ti Mabouya of Martinique, Dwarf Gecko, Sphaerodactylus festus Barbour, 1915 (d) Ti Mabouya of Martinique, Dwarf Gecko, Sphaerodactylus vincenti Boulenger, 1891

Presumed impacts on biodiversity of introductions of Asian geckos in Martinique.

For the time being in Martinique only the harmful and destructive impact of biodiversity due to the introduction of the Gekko gecko can be observed and analyzed.

The rule and logic among lizards is that the larger consumes the smaller; and that within a species there is cannibalism, with adults consuming their own offspring. The Tokay Gecko by its larger size comes to Martinique to supplant the endemic Gros Mabouya of the Caribbean, the Thecadactylus rapicauda. Occupying the same ecological niche, the G. gecko becomes, at this size scale, the supreme predator of the herpetofauna in Martinique. Hunter of the night, arboreal, it consumes Anolis, Dactyloa roquet with diurnal habits ; But it also descends from trees to hunt on the ground, thus it attacks all other Mabouyas that are also night owls (Ti Mabouya, Sphaerodactylus festus, ..., Hemidactyls, ...), amphibians, as well as insects within its reach. It even attacks birds, eggs and small chicks in their nests.


Xavier Dolques
email: dolques@lirmm.fr

M Huchard
email: huchard@lirmm.fr

C Nebut
email: nebut@lirmm.fr

From transformation traces to transformation rules: Assisting Model Driven Engineering approach with Formal Concept Analysis

published or not. The documents may come

From transformation traces to transformation rules: Assisting model driven engineering approach with formal concept analysis

Xavier Dolques, Marianne Huchard, Clémentine Nebut

Introduction

Model Driven Engineering (MDE) is a recent paradigm that gives models a predominating role in the software development process. A well known initiative of the Object Management Group (OMG) in this domain is Model Driven Architecture (MDA) [START_REF] Soley | the OMG Staff: Model driven architecture[END_REF]. Rather than developing programs in specific implementation languages, engineers are encouraged to produce and maintain high-level models describing the domain and the specific problem they deal with. Models are written in conformity with metamodels that capture concepts of the modeling language, e.g. classes and attributes for UML class diagrams, or entities and relationships for Entity-Relationship (ER) diagrams. Tools that assist modeling or that semi-automatically transform models in several directions [START_REF] Mens | A taxonomy of model transformation[END_REF], e.g. going from abstract models to code models or translating models from a modeling language to another, accompany the approach. One consequence of this generalized usage of models inside a wide range of tools and contexts is the huge amount of metamodels. Some of them are regrouped in zoos, like the Atlantic Zoo 1 . Successive versions of modeling and meta-modeling languages are another source of diversity. Then, the success of MDE approach strongly depends on the easiness to develop metamodel-to-metamodel transformations, either dedicated to a development step, or to guarantee tool interoperability.

The transformations are developed with general-purpose languages like Java, or with specialized languages, as ATL [START_REF] Jouault | Transforming models with atl[END_REF] or QVT [START_REF] Omg | MOF TM Query / Views / Transformations[END_REF]. Most of them are rather simple, because they mainly associate a pattern in the target model to a pattern in the source model. One case of such transformation changes the metamodel which is used to express concepts of a domain: going from UML to ER, or going from UML to Java are usual transformation tasks for software developers.

Programmers of model transformations must have serious skills in the chosen transformation language and in the involved metamodels, however in practice few programmers have this kind of skills. For example, many programmers cleverly handle UML models, but do not know the underlying metamodel.

In this paper, we propose a method inspired by "Programming By-example" approaches [START_REF] Lieberman | Your Wish is My Command: Giving Users the Power to Instruct their Software[END_REF] to alleviate the writing of transformations. Engineers only need to handle models in their usual (concrete) syntax and to describe main cases of a transformation, namely examples. Such a transformation example includes the source model, the target model and trace links that make explicit how elements from the source model are transformed into elements of the target model. The transformation rules are generated from the transformation traces, using formal concept analysis extended by relations.

We introduce our problem into detail in Section 2, through a running example. Then, in Section 3, part of this example is used to present the extension of Formal Concept Analysis we will use to take into account relations inside models and traces. Section 4 describes the method we use to generate transformation rules from the lattices. Section 5 reports a case study. Related works are presented in Section 6 and we conclude in Section 7.

Problem overview using an example

A classical transformation used during the workshop MTIP '2005 [6] is used to illustrate our proposal. UML models (class diagrams) are converted into Entity-Relationship models. Simplified metamodels (also called abstract syntax) of UML and ER formalisms are presented in Figure 1.

The simplified UML metamodel describes the main concepts, also called metaclasses, that are used in UML structural models (class diagrams): classes which are a subset of types, properties owned by classes, associations connecting classes through properties. Attributes upper and lower indicate how many values a property can have (or how many objects can be connected when property is used in the context of an association). A simple UML model, given in usual concrete syntax in the lhs of Fig. 2, gathers two classes Account and Client, respectively owning properties number and name, and an association owns associating Account and Client through properties owner and ownedAccount. An account is linked to exactly one client while a client can have an unlimited number of accounts. Fig. 1. Simplified metamodels for UML (lhs) and Entity-Relationship (rhs) used in [START_REF] Wimmer | Towards model transformation generation by-example[END_REF].

The metaclasses of the simplified ER metamodel are entities which have attributes, relationships which connect entities through roles and cardinalities that restrict the connection number. The rhs of Fig. 2 presents an ER model composed of two entities Account and Client, with attributes number and name, and connected via the relation possess and roles owner and ownedAccount.

Fig. 2 shows a transformation trace given by a designer who indicated in dashed lines the correspondences between the source and target model.

Another view of the transformation trace is shown in Figure 3, where models are written using abstract syntax. Each element identifier (e.g. Account) is followed by the name of the metaclass (e.g. Class) it belongs to. Going from concrete syntax to abstract syntax is discussed in [START_REF] Wimmer | Towards model transformation generation by-example[END_REF]. This example illustrates a common task in MDE: conversion between similar metamodels. A UML Property can be converted into an Attribute or a Role depending on the context thus even in simple cases, it is not possible to create a transformation rule using only the metaclass of the source elements: Analysing the neighborhood of a property converted into an attribute (for example property number), we find that the property is only connected to a Class through owningClass. In the case of a property converted into a role (for example property ownership) connections are with an association with a Class through type. Inspired by [START_REF] Wimmer | Towards model transformation generation by-example[END_REF]. Some role names are omitted for the sake of clarity.

The found transformation rules can be expressed as follows, in a syntax close to declarative transformation languages.

Rule c10

For all Property p where p is connected to a Class through owningClass and p is not connected to an Association through association create Attribute a As we have seen through this very restricted example, inferring the transformation rules mainly consists in finding common features of source elements and target elements connected by the transformation. Among these common features, the neighborhood of the elements has to be considered. For all these reasons, Formal Concept Analysis can be a relevant approach, if we consider an extension able to include and exploit links in the description of elements.

Relational Concept Analysis

In this section, we briefly recall Formal Concept Analysis [START_REF] Ganter | Formal Concept Analysis[END_REF] and Relational Concept Analysis, the extension we will use in our approach.

Formal Concept Analysis A formal context is denoted by K = (O, A, I), O is an object set, A is an attribute set and I ⊆ O × A. (o, a) ∈ I when a is an attribute of o.
The UML example of Figure 3 is encoded in a formal context where O are the model elements and A is divided in two subsets: A 1 encodes the types and A 2 encodes the types of the neighbors (Tab.1).

(o, a) ∈ (O, A 1 ) is in I if a is the metaclass of o; (o, a) ∈ (O, A 2 ) if a is the metaclass of a neighbor of o.
Table 1. Formal context encoding the UML model of Fig. 3.

type z }| { neighbor type z }| { Class Property Association Class Property Association Account X X Client X X owner X X X ownership X X X owns X X number X X name X X A concept is a pair (X, Y ) with X ⊆ O, Y ⊆ A and X = {o ∈ O|∀y ∈ Y, (o, y) ∈ I} is the extent (covered objects), Y = {a ∈ A|∀x ∈ X, (x, a) ∈ I} is the intent (shared attributes).
The concept lattice associated to Table 1 is given in Figure 4 with a simplified labelling. Each box represents a concept: the name in the upper part, the simplified intent in the middle part and the simplified extent in the lower part. The example of Fig. 4 shows a classification of the model elements depending of their type and their neighbor type. The elements with type Property (concept c2) have in their neighborhood at least an element of type Class. But some of these elements, regrouped in the subconcept c4, also contain in their neighborhood at least an element of type Association. As shown in this example, it is possible with only one incidence relation I to represent several kinds of characteristics of an object set. To have an accurate description, we can use as formal attribute the type of the neighbors of an element, and we could use also the link name which connects the element to its neighbor. Nevertheless, in a single step, we cannot take into account the created concepts. One solution is then to create a context which associates initial objects and new created concepts through a special incidence relation. Applying such an approach characterizes an element by its neighbors, the neighbors of its neighbors, etc. to have a definition of the concept as accurate as possible.

Relational Concept Anaysis Relational Concept Analysis [START_REF] Huchard | Relational concept discovery in structured datasets[END_REF] is one of the extensions of Formal Concept Analysis that considers links between objects in the concept construction. Considering these links leads to take into account concepts created in one step of the process to enhance the object description and create new abstractions at next steps. Connections can be made with other FCA-based proposals to deal with relational descriptions or complex structures including [START_REF] Priss | Classification of meronymy by methods of relational concept analysis[END_REF][START_REF] Prediger | The lattice of concept graphs of a relationally scaled context[END_REF][START_REF] Ganter | Pattern structures and their projections[END_REF][START_REF] Ferré | Arbitrary relations in formal concept analysis and logical information systems[END_REF] to mention just a few.

Relational Concept Analysis (RCA) computes concepts based on a relational context family composed of one or two formal contexts, as well as relational contexts. A relational context describes a relation between objects of two formal contexts (not necessarily different).

A relational context family R is a kind of multicontext [START_REF] Wille | Conceptual structures of multicontexts[END_REF] presented as a pair (K, R). K is a set of formal contexts

K i = (O i , A i , I i ), R is a set of relational contexts R j = (O k , O l , I j ) (O k et O l are the object sets of K k et K l de K). O k is called the source of R j .
With RCA, data are divided into several object sets. In our example, we consider the elements of the models (context K 1 = (O 1 , A 1 , I 1 )) and the metaclasses of the metamodels (context K 2 = (O 2 , A 2 , I 2 )), these two contexts are shown in Figure 5. K 1 objects are described by two kinds of characteristics: their metaclass and their neighbors. The relation which connects an element to its metaclass (resp. neighbor) is represented by a relational context included in

O 1 × O 2 (resp. O 1 × O 1 ).
Each association end of the metamodel (e.g. owningClass) is encoded into a separated relational context (Fig. 7). The attribute set of K 1 is empty. K 2 elements are described by a unique identifier in order to generate a first lattice where each object belongs to a concept different from the others (Fig. 6). The metaclasses are not used as attributes in K 1 to obtain a clearer modeling and an easy evolution. Relations describing metaclasses can be easily added, for example inheritance. For the sake of clarity, inheritance is not encoded and metaclass Type does not appear in K 2 .

New abstractions emerge iterating two steps. The first step is classical concept lattice construction. In the second step, formal contexts are added to relational contexts enhanced by concepts created in previous lattice construction, then lattices are built.

Initialisation step. Lattices are built at this step using FCA. For each formal context K i , a lattice L 0 i is created (in our example, it is shown in Fig. 6).

Step n+1. For each relational context The computed lattice stemming from the context K 1 extended by relational contexts is shown in Figure 8. The final lattice stemming from the context K 2 is still the lattice of Fig. 6 because K 2 is never the source of a relational context.

R j = (O k , O l , I j ), an enhanced re- lational context R s j = (O k , A, I) is created. A is

Generating transformation rules

Our approach includes three steps: first, classification of the elements of source and target models of the transformation; second, classification of the links that show how elements are connected by the transformation; third, transformation of the resulting concepts into transformation rules.

Classification of model elements

Elements of a model (necessarily conform to metamodel) yet belong to an explicit classification driven by their metaclass. But the elements that instantiate a same metaclass may have different meanings and be transformed in different ways: e.g. UML properties can be transformed into ER attribute or roles. To capture these different meanings, we study the neighborhood of elements: an UML Property used as a role is linked to an association, while an UML property used as an attribute has a link owningClass with a class and no link with an association.

The use of RCA allows us to obtain a classification of the elements of a model taking into account this neighborhood. We have seen in the previous section parts of the used modeling. For a given model, a first formal context ModelElement is created which contains all the model elements (it is equivalent to the context K 1 of Section 3). Objects of this context are the model elements and the set of attributes is an empty set. A second formal context called Meta-ModelElement, contains the metamodel elements (it is equivalent to the context K 2 of section 3). A relational context connects ModelElement and MetaMod-elElement: the incidence relation is here the link between an element and its metaclass (its type), this is equivalent to the context metaclass in Figure 7). This leads to classify elements of ModelElement according to their metaclasses. Each relation end R of the metamodel is encoded into a relational context in M odelElement × M odelElement (Tables of Figure 7). Using those relations refines the classification of ModelElement.

The result of this encoding is a concept lattice that describes the elements available in a model. Concept intents will be used to generate part of the transformation rules. Figure 8 presents the concept lattice describing the elements of the UML model. For example, concept c 2 describes Properties which are connected through owningClass links to Classes. More precisely, elements of the simplified extent of c 2 (namely number and name) are not connected to anything else.

Classification of transformation links

Transformation links are given in a transformation example to describe the correspondences between several elements of two models. We consider here 1-1 links (one source element transformed into one target element), or 1-n links (one source element transformed into n target elements). Links are encoded into a formal context MapLinks: objets are the links and there are no attributes. From the MapLinks context, we build the lattice which is used to generate the transformation rules.

For two models A and B involved in the transformation, we create two relational contexts LinkA ⊆ M apLinks × M odelElementA and LinkB ⊆ M apLinks × M odelElementB. An element of MapLinks is connected to a concept of the ModelElement lattice if one element of the concept extent is end of the link. The lattice2 represents a classification of links based on their ends.

Concept interpretation and rule generation

A transformation link is characterized by the concept containing its end in model A and the concept containing its end in model B. The concepts of the lattice built on top of MapLinks regroup links which have common characteristics in their two ends. For example, concept c 10 (lattice of Figure 9) regroups links that connect properties linked to classes and not to associations (concept c 2 , lattice Fig. 8 classifying UML elements), and attributes (concept c 3 of the lattice classifying ER elements). We propose to extract the transformation rules using these characteristics. The description of source elements (linkA values) can be seen as premise of the rule, while the description of target elements (linkB values) can be interpreted as the conclusion. Concept c 10 leads to the Rule 10 of Section 2: the premise is derived from the UML concept c 2 and expresses that the involved source elements are precisely the properties p where p is connected to a Class through owningClass and p is not connected to an Association through association. The conclusion of the rule is derived from the ER concept c 3 , expressing that the target element is an ER attribute.

In other words, we consider a rule as a mapping taking as parameter a source element conform to metamodel A. If this element satisfies given required characteristics, it is transformed into target elements conform to metamodel B and satisfying other required characteristics. All required characteristics are indicated inside the concepts.

For a rule stemming from a concept c of the lattice associated with MapLink, required properties are obtained analysing the concept MEA.c' (for ModelEle-mentA.c' ), which is the most specialized concept of ModelElementA in c intent. There are three categories of characteristics: Mandatory characteristics, described in MEA.c' intent. They are common to all c links.

Authorized characteristics, described in the intents of concepts specializing MEA.c', under the condition that the concept extent includes the end of a link of c. For example, in the case of the concept c 8 of MapLink regrouping links L 6 and L 9 (Figure 3), MEA.c' corresponds to MEA.c 7 ; ownership (end of L 6 ) which belongs to MEA.c 5 and owner (end of L 9 ) which belongs to MEA.c 6 are then authorized.

Forbidden characteristics, described by the intents of the concepts which do not include in their extent the end of a link of c. These characteristics are especially important if they belong to concepts specializing MEA.c'. In our example, if we consider the concept of MapLink regrouping links L 3 et L 4 , MEA.c' corresponds to MEA.c 2 . The rule must not include description coming from MEA.c 6 since no element of its extent is end of L 3 or L 4 .

Case Study

This section provides a proof-of-concept of our technique in the form of a case study. We are interested here in the quality and usability of the obtained rules.

By quality, we mean the rules adequacy with what a developer would have done by hand: ideally, the results obtained with our approach should give similar or identical rules. The usability evaluates if the number of obtained rules does not explode, as could unhappily be expected with lattices.

Preparation

In order to gather experimental data, we organized a session with graduate students around model transformation, during their Model Driven Engineering class. The models and metamodels were written using the Eclipse Modeling Framework (EMF) and the Sample Reflective Ecore Model Editor, the students were familiar with them. We created small models conform to metamodels that students were used to handle. These models were given to the students as source examples and they were asked to create by-hand a transformed model conform to another given metamodel. They were also asked to write the trace links between the two models and the transformation using an imperative language.

These models were given to the students as source examples, as well as the specification of a model transformation. The transformation was specified with written natural language, and also orally explained by the teacher. The target metamodels of the transformations were also familiar to the students. The students were asked to create by-hand the result model. The students were told that their work will be used for a research experiment but they did not know the exact purpose of the experiment. They had all the time they wanted to produce their data, but they worked alone and two nearby students in the classroom were given different models. We then gathered all the data and applied our RCA-technique on them. We have then studied the lattices to extract the rules.

The metamodels used for this case study are simplified versions of L A T E X and HTML, the subjects were asked to make transformations from one to another in both directions. From the obtained data, we gathered the results presented in Table 2. At first, we measured the size of all the handled models. The metamodel size is the number of its EClass. The model size corresponds to the number of its metamodel EClass instances. A trace size is given by its number of links.

We also measured the source metamodel coverage, using the number of EClass for which instances were traced. The final lattice used to generate the rules contains concepts that cannot be used as they do not contain enough information to make a rule. The Rules Space defines the set of concepts that are usable as rules.

During the experiments, a lot of factors could cause error, and we needed to know whether the technique was sensible to errors. We have determined that concepts with the same source but different targets could cause problems, that is what is measured for the detected problems. We then checked if these problems were real and if all problems were detected. By problems, we mean trace link errors or an incoherent target model.

For the last metrics, we applied the rules we obtained on the source example, and we measured the difference between the result obtained and the target model example. As in some cases the rules do not cover all the elements in the source model, we have based our measurement on trace links. As some problems have appeared in the data used to generate the rules, we also made measurement excluding links from the rules that cause problems due to trace errors.

Evaluation

The results on the different models seem satisfying as every rule applied, if it has been generated from sane data, behaves in the expected way. We also see that errors can be treated in most cases: they were all detected on our data and good rules can be obtained from the remaining sane model elements. But we can also see some cases of false positives, that could be more frequent with bigger models.

A problem that could arise when working with RCA is the size of the lattice, and especially the number of possible rules. But, this does not appear to be a problem with the transformation we have here.

Threats to validity -Conclusion validity:

Our results are valid as a proof-of-concept: we conducted our experiments on a little number of models. The subjects knowledge was good enough. -Internal validity: The high number of problems comes from the used tools (Eclipse Model Editor) and the edition of XMI files, this is highly errorprone.

-Construct validity: The correctness of the transformation is measured by the transformation of an element in another element with the good type, but we do not take into account the relations between the transformed elements. -External validity: We are using here little metamodels and examples, as it is needed to make a proof-of-concept, but it makes results hardly generalizable on bigger metamodels. However, metamodels are reasonably different so that the transformations to generate are not obvious.

Related Work

In Model-Driven Engineering domain, the automatic generation of model transformation is a recent and active research topic. Roots and inspiration can be found in the domains of ontology and schema matching [START_REF] Rahm | A survey of approaches to automatic schema matching[END_REF][START_REF] Shvaiko | A survey of schema-based matching approaches[END_REF] and programming by-example or by-demonstration [START_REF] Cypher | Watch What I Do: Programming by Demonstration[END_REF][START_REF] Lieberman | Your Wish is My Command: Giving Users the Power to Instruct their Software[END_REF]. Metamodel alignement-based approaches search for mappings that provide transformation rules. In [START_REF] Roser | An approach to automatically generated model transformations using ontology engineering space[END_REF], metamodels are mapped to a pivot ontology, then an ontology-based reasoning is used to generate a relational-QVT transformation. In [START_REF] Kappel | Lifting metamodels to ontologies: A step to the semantic integration of modeling languages[END_REF], refactoring is applied to metamodels in order to make explicit hidden concepts of metamodels and obtain an ontology where all concepts are reified before mapping. Similarity Flooding [START_REF] Melnik | Similarity flooding: A versatile graph matching algorithm and its application to schema matching[END_REF] propagates similarity values in a labeled graph whose vertices are potential mappings. It is used in several proposals for metamodel alignment such as in [START_REF] Lopes | Schema matching in the context of model driven engineering: From theory to practice[END_REF] and [START_REF] Falleri | Meta-model Matching for Automatic Model Transformation Generation[END_REF].

Another track of research aims at inferring transformation rules from transformation examples (traces). Particle swarm optimization is used in [START_REF] Kessentini | Model Transformation as an Optimization Problem[END_REF] to generate a consistent transformation of a model. The transformation of a model source element is encoded as a particle which has to be placed in the space of possible transformations. Graph transformation rules are semi-automatically derived from mappings given by a user between two models in [START_REF] Balogh | Model transformation by example using inductive logic programming[END_REF]; Analysis is then based on model element neighboring, inductive logics and interaction with an expert. ATL rules are derived from transformation examples written in concrete syntax in [START_REF] Wimmer | Towards model transformation generation by-example[END_REF].

Meta-model alignment approaches are especially suitable to give mappings in the case of rather simple transformations in the context of tool interoperability or version changes. Approaches based on rule inference have a larger application spectrum including complex transformations or transformations where metamodels are quite different. RCA allows us to map connected elements (patterns) rather than isolated elements. Lattices classify results and help navigation among generated rules to choose the relevant ones. Compared to the opaqueness of the optimization approach which just provides a transformation result, inferred rules provide a transformation procedure, and are a clear and easy-to-handle artefact.

Conclusion and future work

In this paper, we propose to generate model transformation rules using examples of transformed models and transformation links between source and target elements. This allows engineers involved in Model Engineering tasks to rapidly have a transformation program even if they are not familiar with transformation languages and metamodels. Using Formal Concept Analysis, rules are classified through a lattice which helps navigation and choice. On simple models, we show that the results are satisfying as the method produces correct rules most of the time and is usable on erroneous data. From these encouraging results we plan to test our approach on bigger and more complex transformations. But to achieve this goal, we need to complete our tools for generating and manipulating rules. By using other scaling operators we expect to enhance the produced rules and detect new patterns for rule premise and conclusion.

Fig. 2 .

 2 Fig.2. UML (lhs) and ER (rhs) models in concrete syntax, and transformation links.

Fig. 3 .

 3 Fig.3. UML model, ER model, in abstract syntax, and transformation links. Inspired by[START_REF] Wimmer | Towards model transformation generation by-example[END_REF]. Some role names are omitted for the sake of clarity.

Fig. 4 .

 4 Fig. 4. Concept lattice associated with Tab.1

  the concept set of the lattice L n l (created at step n). Incidence relation I contains the set of pairs (o, a) s.t. S(R(o), Extent(a)) is true, where S is a scaling operator. The scaling operator we use in the rest of the paper is S ∃ (R(o), Extension(a)), which is true iff ∃x ∈ R(o), x ∈ Extent(a). Other operators could be used, and especially S ∀∃ (R(o), Extent(a)), which is true iff ∀x ∈ R(o), x ∈ Extent(a)∧∃x ∈ R(o), x ∈ Extent(a). For each formal context K i extended with enhanced relations with source O i , the lattice L n+1 i is created.

Fig. 7 .

 7 Fig. 7. Relational contexts. Contexts in the upper part correspond to relations in K1 × K1. Context in the lower part is a relation in K1 × K2.

Table 2 .

 2 Data obtained from the case study.

		d15 d16 d43 d44 d25 d26 d29
	Transformation	latex to html html to latex
	Source MetaModel size	3 3 3 3	7 7	7
	Target MetaModel size	7 7 7 7	3 3	3
	Source Model size	9 12 9 12 13 13 13
	Target Model size	11 14 14 22 12 11 12
	Trace size	9 12 13 21 10 11 10
	Source MetaModel coverage 3 3 3 3	3 4	3
	Rules Space size	5 5 5 4	4 4	3
	Detected problems	2 2 4 3	1 1	0
	Real problems	2 2 2 1	1 1	0
	Number of links	9 12 13 14 9 11 10
	Number of correct links	8 11 11 14 8 11 10
	Number of links from rules	4 10 4 14 8 11 10
	without problems
	Number of correct links from	4 10 4 14 8 11 10
	rules without problems

http://www.emn.fr/x-info/atlanmod/index.php/Atlantic

All the lattices are at this url: http://www.lirmm.fr/˜dolques/publications/data/iccs09