Adriana Peña

Jean Jacques Girardot

Horacio Bozzano
email: horacio.bozzano@presi.unlp.edu.ar

A UNIVERSITY OF SUSTAINABLE DEVELOPMENT AND TRANSFORMATION FOR MERCOSUR. Proposal for its creation within the framework of Territoriesnet, the European Union and the Parliament of MERCOSUR: Case study for Minas, Lavalleja, Uruguay

INTRODUCTION

Is it possible to put together the concepts of sustainable development and transformation with a great variety of activities -regarding different dimensions-that have the objective and goal of developing and transforming our societies and social groups, our territories and places in the local, sub-regional and regional scales in the most sustainable possible way? If this task is possible, could we achieve it while simultaneously maintaining scientific rigor and reaching institutions, organizations and citizens in such a way that they would adopt the outcome as their own? This is the first of a series of publications where we start answering these questions from complementary perspectives -political-institutional and scientific-academic-with common intentions. This paper deals with an initiative for the creation of a Sustainable Development and Transformation University for MERCOSUR in the City of Minas, Uruguay, within the Program of Activities 2010-2015 of the Territoriesnet Global Network, particularly regarding the activities done by the Territorios Posibles Latin American Network; TAG, the scientific node that gave birth to it, as well as the European Network of Territorial Intelligence (ENTI), supported by the Framework Program 6 of the European Union. We also work with the Commission for Regional Sustainable Development [Comisión de Desarrollo Regional Sustentable] of the Parliament of MERCOSUR, with the Uruguay UNDP Articulation of Territorial Networks [Articulación de Redes Territoriales (ART)] Program, with various research groups in Uruguay: UCUDAL, UDELAR, CLAEH and FLACSO, with the Ibero-American General Secretariat [Secretaría General Iberoamericana (SEGIB)] and other initiatives that carry out activities with the same focus on science-society than the current proposal. This writing is divided into two sections. The first one takes living and latent socio-cultural imprints from the local society, together with the political and social decision to transform the state of relative social and economic stagnation of Lavalleja in relation with Uruguay as a whole, especially during the last half century. The second describes matters related to what is understood by development and sustainable development, as well as diverse scopes and meanings of this concept to incorporate them to the debate on the generation of a macro-subject of study and transversal intervention to a large number of scientific disciplines that will be useful and cohesive for the actions of this future institution of higher learning.1

IMPRINTS, STAGNATION AND TRANSFORMATION

The momentum for the creation of this University emerges from the combination of local needs and political-institutional and scientific-academic will. The proposed venue, the City of Minas, has a population of 39,000 and is the capital of the Department of Lavalleja -a territory that encompasses an area one-third the size of Holland, 63,000 inhabitants and many natural resources: landscapes, water, minerals, agriculture and forests, among others. Three pillars justify the choice of this territory and society, in brief: socio-cultural imprints, economic stagnation and demographic exodus, and a decision to transform the current state of affairs.

Socio-cultural imprints. Remarkably valuable imprints and marks deeply rooted in Minas and Lavalleja are the pillar for the rescue of a culture that needs increased public awareness and repositioning -like those shared by previous generations in Lavalleja, Uruguay-on par with the current times. For almost the entire last century Lavalleja has been a cultural referent in Uruguay. Minas was founded in 1783 for economic, military and political reasons. "The settlement of the town of Villa Concepción de las Minas implied an episode in the revitalizing epic of the Spanish colonist, and it is basically of European origin. The first immigrants were Spaniards: from Galicia, Asturias, the Canary Islands and other places that constituted the core of the culturization process. The French who emigrated from the Basque Provinces worked on craftwork; others became builders, smithies, etc. They brought with them their typical customs and created a small world that left its mark in the Uruguayan soil. The Italian were also present and offered their tongue to enhance the local and street languages of the villa, together with their food, religion, etc. By mid 19 th century, other migratory contingents mixed with those of Latin origin: the English, German, Swiss, and before World War II, the Armenian and Lebanese. Today we can observe a diversity of races and families that descend from those migratory groups, whose contributions have been significant to the becoming of Minas history." The Department has been the birthplace of great artists, scientists and nationally, as well as internationally, famous personalities. The Music of Eduardo Fabini (1882-1950), distinguished national classical musician, who travelled all over Europe and the United States with works of his creation, such as "Campo" (Country) and "La isla de los ceibos" (The island of ceibos), among others. Juan José Morosoli, (1899-1957) narrative writer, whose works "Perico" and "Los albañiles de Los Tapes" (The bricklayers of Los Tapes) were translated to many languages. Also Guillermo Cuadri, Capagorry and Espínola Gómez (1921-1955). Many have been the renowned painters: Horacio Espondaburu (1855-1902) is considered the first of Minas' painters. Alfredo Vidal y Fuentes (1863-1926) has published numerous national and international scientific works and held important positions in public health; he was the founder of institutions like the Isla de Flores lazar house. The Ardao brothers, a philosopher and a historian, the Ardao sisters, María Julia, historian, and María Isabel, chemist, who discovered an antibiotic jointly with Clemente Estable, a scientist; their research was deepened at Harvard University in the 40's. Esperanza Viscay de Fuller, educator and founding teacher of the Solar de Artigas School in Paraguay. On the stage, the dancer José Otegui Paul went all over the main theatres in our country, the United States and Europe. He made it to Brussels and in 1954 to Paris, at the Marigny Theatre in the Avenue des Champs-Élysées. Among the local initiatives oriented towards promoting social and environmental development and Culture, we may mention Minas Capital of Culture [Minas Capital Cultural] (1990), the Lolita Rubial (1991) Foundation -creator of the Museum of Comics [Museo de la Historieta]-an institution that brings out national cultural achievements by awarding prizes to renowned people of Uruguay's culture all over the country; the Amigos del Arte Association (1992) and more recently the UNI3 Continuous Adult Education University [Universidad de Adultos de Educación Permanente] (2001). Regarding environmental matters, the Ecological Group Arequita [Grupo Ecológico Arequita (GEA)] (1985) has worked on fruitful activities, mainly in environmental education for the community in the elementary, secondary and post-secondary levels. In Lavalleja there are many intermediate and nongovernmental organizations too. The imprints, understood as invisible marks -practices, actions, symbols, events, myths-from a past in the present, and the marks, as visible traces of the first -museums, works, objects, productsare in Lavalleja a basic sediment, a fertile ground to recover, rescue and contribute to the awakening of the Department, recovering its main strengths but also knowing how to make them part of the future.

Economic stagnation and second demographic exodus: the "segunda redota" (the second defeat). The Redota -a gaucho expression for the word defeat (derrota)-was the name given by the locals to what was also called the "exodus of the Uruguayan people" in 1811, after the battle of Las Piedras against the Spanish Crown. 2 At the time, due to political reasons, between 18 and 20% of the Uruguayan population abandoned the country to march with Artigas; an event that could be considered an escape later became what would be the main imprint in the creation of Uruguay as a nation. A century and a half later, mainly due to economic and labor related issues, "…from the decade of 1960 a second exodus took place, which would transform Uruguay into the nation with the lowest rate of growth in all of Latin America: today, approximately 18% of the country's population lives outside of Uruguay. After being the Switzerland of the Americas, the nation started its path towards the country of the exodus: a country where the youth have lesser and lesser reasons to stay, to educate themselves, to work and to settle and form their families. To a great extent, this situation is experienced and perceived by a large number of Uruguayan people as a second redota, particularly because all remember from their parents and grandparents the golden age when living in Uruguay was different. Expectations have changed. The persistency of this population drain has contributed in half a century to institutionalize for our country a palpable concept in our society, sadly known as the culture to emigrate." (Peña, 2007). Diverse authors point out a trend in Uruguay towards the "exporting of young and prepared minds". 3 In this second exodus or second redota, Lavalleja is sadly one of the leaders among the 19 Departments of Uruguay. For half a century, Lavalleja has neither increased nor decreased its population, an infrequent case for Latin America. Between the census of 1963 and 2006, the Department of Lavalleja has lost approximately 5,000 inhabitants: according to the National Census of Population, Lavalleja had a population of 65,823 in 1963; 65,180 in 1975; 61,466 in 1985; 60,743, in 1996; and 60,925 in 2004. The estimate for 2008 is 61,910 inhabitants. Minas, its capital city, has a population of 37,149. It is the main economic, cultural, industrial, commercial, touristic and administrative center. The main localities in the Department are José Pedro Varela (4,983 inhabitants), with rice-growing, fodder and agricultural machinery manufacturing activities; Solís de Mataojo (2,509 inhabitants) where the main activities are meat processing, fishing and cattle raising; José Batlle y Ordóñez (2,298 inhabitants), Mariscala (1,507 inhabitants), Pirarajá (723 inhabitants), Zapicán (602 inhabitants), Colón (223 inhabitants) and Illescas (133 inhabitants); life in these six towns centers around agricultural and forest production.

2 After the battle of Las Piedras, a clash between the locals and the Spanish government in Montevideo, General Artigas had to retreat to the north. Even though the order was obeyed, it led to a collective decision that took the Uruguayans from every town, villa or hamlet to join the patriot and abandon homes and animals. Women, men, elders and children marched together in greater numbers with what little they had while burning crops and belongings to deny them to the Spaniards. They followed the hero in spite of his refusal, because this delayed his march; they spent 11 months in the north where Artigas showed his character of statesman. This could well be considered the foundational pact of the national sentiment of the Uruguayan people. 3 Adela Pellegrini, Andrea Vigorito y Luis Aguiar, among others.

It is also worth mentioning a number of villages and rural places, mostly devoted to agricultural and/or forest production: Villa Serrana, Villa del Rosario, Aramendia, Estación Solis, Retamosa, Andreoni, Aguas Blancas, Marmarajá, Ombúes de Bentancor, Ortiz, Barrancas, Vejigas, Casupá, Los Tapes, Polanco, El Soldado, Barriga Negra, Campanero, Cerro Pelado, Higueritas, Gaetán, Godoy, Verdún, Ladrillos and Marco de los Reyes. The economic activities in Minas are related, to a great extent, to the name that gave its origin to the City and the Department: the mines. Uruguay's Congressional Legislative Palace was built 103 years ago with marble from Minas' surroundings. There are also world class subterranean sources of mineral water and upwellings, like those exploited and marketed with notable success by the Salus Company at Fuente del Puma since 1898. In the past it used to be sold in pharmacies and in 1912 it obtained international awards in Europe for being one of the purest in the world. The first half of the 20 th century was a time of sustained economic development: we may point out the public railroads [Administración de Ferrocarriles del Estado (AFE)], two flour mills -Ugarte and Monasterolo-and the limestone quarry Diano. Lavalleja's industry diversified through the following decades with vinicultural businesses, sweets and food preserves, and dried red pepper for paprika. The Department's economic boom continued through 50's and the 60's; some large industries and open pit quarries started operating at this time, as well as small and medium-sized businesses. In the mid 70's a slow process of economic stagnation started, which would eventually lead to the closing of the marble, granite, dolomite and gold mines, and factories of importance to the local society, especially those of cardboard manufacturing and flour mills. The Uruguayan railroad decline, high industrial costs and the migration of the workforce from Lavalleja to more dynamic departments were the main factors of this economic stagnation. The last industry that established itself in the department did so 17 years ago: a shoe making factory, with local government support and Brazilian capitals. Currently many of the community leaders and citizens are strongly determined to transform this harsh reality. The needs and expectations of the local societies from Minas and the various localities in the Department are central to changing this state of affairs: there is a will to develop new activities, to transform realities dominated by inertia and population exodus, mainly to Montevideo, Uruguay's seaside resorts and to other countries, especially to Spain, the United States, Australia and Argentina. Minas and Lavalleja represent today a society with a significant cultural development in Uruguay that has been weakened by the exodus of its children and by local inertias; in the last few years, however, a large part of its inhabitants has become convinced that it is necessary to wake up.

Decision to transform the present situation: the four legs. This initiative is based on the combination of wills and decisions of social, politic-institutional, economic and academic-scientific actors, local as well as from other latitudes. Four legs are necessary to drive an initiative of this nature. If one of them teeters or weakens, or if more vertical or less participative styles of work are imposed, the decision may be taken anyway, but its germination, appropriation and projection will be more prone to failure. The conditions for achievement cannot be met just by intentions or decisions. It is then necessary that the initiative goes through process controls, where top-down and bottom-up styles establish a dialog among the four legs that will not only help each participant to become more aware, but will also value his or her knowledge and know-how. One of the most important keys for the success of this enterprise lies there: territorial actors -social, economic, community-each sharing their knowledge and open to co-building with the contributions of politicians and academicians; the institutional actors, activating from their places vectors of decision that truly incorporate popular and scientific knowledge to legitimate transformation in a more solid way; and scientific actors dismounting from their horses and avoiding epistemic obstacles.

Probably the most difficult task will be that of placating and softening our epistemophilic obstacles, alive and latent from ancient times. There is no society without conflicts, contradictions, suspicion and differences; it is necessary to co-operate and complement each other, overcoming inertias and differences, it is part of the path that must be walked. This is a slow process of co-construction of trust between different actors, where we must incorporate win-win mindsets, rescuing the knowledge that provides for the best synergies. Confronting and reproducing conflict permanently, without mediating, co-constructing or agreeing supposes channeling time and energy to other types of development: a development in which conflict is a way of life. In this sort of scenarios, times and energies of politicians, scientists and the public will eventually die down and contribute to the lethargy and even inertia of every project of development and virtuous transformation. For over a year there has been ongoing work, that has not lacked conflicts or obstacles, to develop and strengthen this four legs, to shape the initiative as a virtuous transformation; actions are taken simultaneously in three directions: with local actors, with political-institutional actors, and with academic-scientific actors. Work with the local actors has followed three lines: 1) the creation of a Multi-sectorial Commission for University Development [Comisión Multisectorial de Desarrollo Pro-Universidad] in Minas; 2) the implementation of three survey models -Minas, inland towns and rural places-named: "Minas and Lavalleja: Past, present and future", and 3) a record of projects and ideas for the development and transformation of Minas and Lavalleja. In these three lines participate people of all ages, from universities and other places, in brief, everyone that feels attracted to the project. 4 In the last four years the political-institutional actors that drive the initiative have been opening doors in the most diverse institutional spaces, local, national and international, creating what today has become an institutional map of notable value due to the synergy oriented towards the achievement of the goal. University and scientific actors work, as many, from research, outreach and teaching, connected with the extra-university world -institutions, organizations, citizenship-by places and possible territories more sustainable than the current ones. We would like to point out a great number of scientific and academic work experiences with territorial institutions and actors; among them, three decades linked to the activities of the collective constituted by 200 scientific, institutional and territorial actors that today make up the ENTI in eight European countries, which also performs activities in three other continents: Africa, Asia and North America. It is also worth mentioning the work conducted by the recently constituted TAG (Territory, Actors, Governance) Team, with headquarters at the National University of La Plata, which brings together 72 researchers and people with university education from different institutions and public and territorial actors who have been walking a path parallel to the European one for the last two decades.5

SUSTAINABLE DEVELOPMENT, DEVELOPMENT AND DEVELOPMENTS

Why a Sustainable Development University? In 1987 the Brundtland Report read: "Sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs." Between Stockholm 1972 and Rio 1992 many things have happened, a whole lot more until Copenhagen 2009. Discourses and awareness have notably moved forward, but not enough has been seen in terms of concrete actions or results: the socio-ecologic transition seems to be a valid objective to guide action, if we keep in mind that 100 thousand people die every day out of hunger or sickness, that there are over 212 million unemployed and that if our planet could talk, it would be quite angry with us. The concept of sustainable development acquires value and applicability as we define in the first place the concept of development in nature and scope, so we can then evaluate the feasibility of the components and/or principles of sustainability that will be applied to each particular development we face and wish to implement. That is, in this new University, the meaning of sustainable development, starting from a generic cohesive concept built on five sustainability components -POETA-proposed by Guimaraes (2002) 6 , research and work will be conducted for their application to each development; this aims to building a bridge between real development and virtuous transformation, thus accomplishing its goal. That is why the University will be a Sustainable Development and Transformation one. At the same time, development -together with other numerous words-is being investigated and acted upon from a wide range of disciplines and inter-disciplinary fields of education. We agree with Sergio Boisier, we face "… an increasing conversion of development as a noun into an adjective, this creates not only confusion but also a tautologization of the concept" (Boisier; 2003:1). Today it is very clear that the concept of development is not the one from the 50's and the 60's, we must reduce this tautologization. 7 In a broad sense, from the scientific-academic world, we understand development as an inclusive concept from about 30 disciplines and activities; among them local, endogenous, neighborhood, urban, tourist, regional, territorial, environmental, institutional, strategic, municipal, community, social, cultural, educational, health, rural, agricultural, livestock, forest, mining, commercial, industrial, railroad, road, port, decentralized, capillary, micro-entrepreneurial, entrepreneurial and many other forms of development. And as if that weren't enough, all of them sustainable.

Is it possible to group together these about 30 developments and other ones? Following the shape of Venn diagrams and as concrete projects are included, a swarm of groupings are being built where today it is possible to recognize, broadly speaking and in a preliminary and provisional way, six groups or lines of development. In all cases these are preferentially and non-exclusively, the following groups: 1-local, 2-regional, 3-social, 4-economic, 5-institutional and 6-sectorial. 8 Is there anything shared by knowledge and activities related to sustainable development? Matters, problems and subjects referred to transformations derived from diverse types of development, it seems quite clear that they have different features depending on a large range of subjects of study and inter-and trans-disciplinary crossings. They do, however, share some common characteristics: processes of transformation of heterogeneous and complex realities -in the micro, meso and macro scales-are conceived, designed, built, negotiated, agreed, and implemented -not without conflicts or contradictions-by specific actors in specific places; and even though it may seem obvious, they are not realized by other actors or in other places. 9 Particular combinations of processes, actors and places constitute a mortar: they are the leitmotiv and theoretical and ideological basis for this initiative that intends to co-construct sustainable processes of development and transformation in a democratic way, with social and environmental principles preferably oriented towards the disadvantaged.

6 Guimaraes ( 2002) "…we may propose that sustainability for a community depends of the interconnections among: population (size, composition, density, demographic dynamics); social organization (patterns of production, social stratification, patterns of conflict resolution); environment (physical as well as built environment, environmental processes, natural resources); technology (innovation, technical progress, use of energy); and social aspirations (patterns of consumption, values, culture)" 7 We suggest a text by Sergio Boisier "Desarrollo (local): De qué estamos hablando?" op.cit 8 A version of this "swarm" in the shape of Venn diagrams may be seen at www.territoriosposibles.org. 9 A Research Program for 2010-2012 is being prepared in Territoriesnet for the mapping and systematization of processes, places and actors regarding these four legs -academic-scientific, political-institutional, socio-territorial and economicterritorial-as well as all types of development that can be found in the bibliography and, simultaneously, different types of transformation, considering where it would be operating and/or should be promoted.

Social, economic, employment, educational, local, regional, territorial, environmental and many other problem groups are studied, analyzed and explained as scientific knowledge with the express intention of discovering the real possibilities for change and/or transformation that each subject of research carries in itself, in its potential condition as a target for intervention. This is all about creating scenarios of territorial governance with scientific rigor in the micro, meso and macro scales between the scientific-academic world, government institutions and a diverse range of territorial actors. These concerns exist in our public governmental institutions -local, regional and nationalusually related with the instrumentation of policies and programs in diverse scales that have an impact on vulnerable populations and environments. Also, a growing number of social and local and international community organizations include these as their main concerns. As Costamagna has stated: "There are a large number of challenges, the systemic and integral view of the concept, the links between different levels of government and policies within the same levels, bonds, networks and information, economic resources and municipal autonomies, among others. In this sense, the training of human resources for this challenge and increasing the links between academics, the scientific-technological system and the territory deserve a separate mention, an absolutely necessary encounter for the Development of the countries in the Region in the next years." 10 This initiative intends to be an open and fruitful field of experimentation within the framework of the Territoriesnet International Network, due to the driving force of two of its networks: Territorios Posibles and ENTI, although open to other networks. We hope to little by little create a beacon that will give out the emerging results to a wide collective of researchers with open and powerful communication guidelines, where scientists as well as politicians and the territorial actors involved will be present with their discourses and actions. It will be the combination of the trajectories of the participating groups and lines of research at the university, together with the answer to the needs, interests and expectations of the territorial and institutional actors, what will gradually create the profiles of this institution of higher learning. The current reality indicates that all of Minas and Lavalleja summon and attract tourist, cultural, mining and rural development, among other vocations latent in their society and territory.

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11

• ALBUQUERQUE, Francisco, FERRARO y COSTAMAGNA (2008) Desarrollo Económico Local, descentralización y democracia. UNSAM Edita, Buenos Aires • BARBERO, Jesús M.(2003) De los medios a las mediaciones. Comunicación, cultura y hegemonía. A Bello y G.Gili • BOISIER, Sergio (2001) Desarrollo (local): De qué estamos hablando? En: Transformaciones globales, instituciones y políticas de desarrollo local, Vásquez Barquero y Madoery, Homo Sapiens Ediciones, Rosario. • BOZZANO, Horacio (2009) Territorios posibles: procesos, lugares y actores. Ed.Lumiere, Buenos Aires • DURKHEIM, Émile (1993) Escritos selectos. • KAROL, Jorge (2003) Actores sociales y gestión de la ciudad. Cátedra Planeamiento Urbano, FADU, Universidad Nacional de La Plata, La Plata • LUHMANN, Niklas (1998) Sistemas sociales: Lineamientos para una teoría general. Anthropos, Barcelona. 1°ed • MADOERY, Oscar (2002)

This writing is divided into two sections: the concept of transformation associated to the management of the intended development, and co-constructing transformation with local society: first results (Peña, Girardot and Bozzano:

2010; IT-GO Conference Nantes-Rennes)

The opportunity to create a university center in Minas is neither new nor isolated; a large part of the local society has given its approval to it for the last few years. Even though the idea of a University of Development and Transformation was born as a local political initiative and as an academic initiative from other places, it was quickly shared with the entire society, without distinctions or partisanships.

TAG UNLP, with headquarters in La Plata, is actually formed by Rafaela, Buenos Aires, Córdoba, Rosario, Paraná, Oberá, Mar del Plata, Quilmes and San Juan (Argentina), Minas (Uruguay), Santo Domingo (Dominicana), México D.F., San José (Costa Rica), Bogotá (Colombia), Talca (Chile), Maracaibo (Venezuela), Portoviejo (Ecuador) and other cities and towns, represented by universities, institutions and territorial actors.


Sébastien Ferré

Pierre Allard

Olivier Ridoux

Cubes of Concepts: Multi-Dimensional Exploration of Multi-Valued Contexts

Keywords: formal concept analysis, information systems, data exploration, navigation, multi-valued context, multi-dimensional analysis, OLAP cubes

A number of information systems offer a limited exploration in that users can only navigate from one object to another object, e.g. navigating from folder to folder in file systems, or from page to page on the Web. An advantage of conceptual information systems is to provide navigation from concept to concept, and therefore from set of objects to set of objects. The main contribution of this paper is to push the exploration capability one step further, by providing navigation from set of concepts to set of concepts. Those sets of concepts are structured along a number of dimensions, thus forming a cube of concepts. We describe a number of representations of concepts, such as sets of objects, multisets of values, and aggregated values. We apply our approach to multi-valued contexts, which stand at an intermediate position between many-valued contexts and logical contexts. We explain how users can navigate from one cube of concepts to another. We show that this navigation includes and extends both conceptual navigation and OLAP operations on cubes.

Introduction

Navigation is a convenient way for exploring data, compared to querying, because it provides guiding to users during their search, and supports exploratory search [START_REF] Marchionini | Exploratory search: from finding to understanding[END_REF]. At each step of a navigation path, users are located at a navigation place, and a number of navigation links are suggested to them in order to reach neighbour navigation places. However, most existing systems suffer from a number of limitations, e.g., file systems and the Web. A first limitation is that navigation places are objects (e.g., files and folders, Web pages), and users can therefore only jump from object to object. This makes it difficult to group and compare objects. A second limitation is that the navigation graph is manually drawn. As a consequence, the navigation graph is generally very sparse, and requires from users chance or expertise or both to find their way to the searched objects. For example, if photos are organized in a file system first by date then by topic, it is easy enough to find photos by date, difficult to find photos by topic, and impossible to find them by depicted person. A third limitation is that navigation links may lead to empty results, e.g., an empty folder. This is a frustrating experience for users who have to resort to tedious trial-and-error. A fourth limitation is that the navigation path must be linear. At each step, only one navigation link can be chosen. If users want to explore several navigation links, they have to choose and explore one, then to move backward in the navigation history, and choose a second one, and so on.

Conceptual navigation [START_REF] Godin | Experimental comparison of navigation in a Galois lattice with conventional information retrieval methods[END_REF][START_REF] Carpineto | A lattice conceptual clustering system and its application to browsing retrieval[END_REF][START_REF] Lindig | Concept-based component retrieval[END_REF][START_REF] Ducrou | An intelligent user interface for browsing and search MPEG-7 images using concept lattices[END_REF][START_REF] Ferré | Camelis: a logical information system to organize and browse a collection of documents[END_REF] overcomes the first three limitations by relying on Formal Concept Analysis (FCA) [START_REF] Ganter | Formal Concept Analysis -Mathematical Foundations[END_REF]. Users navigate from concept to concept, and hence from set of objects to set of objects. The navigation graph is the concept lattice, which is automatically derived from data, a formal context, and therefore automatically adapts to changes in data. Finally, it is easy to prevent empty results, by removing the bottom concept from the navigation graph. Regarding the fourth limitation, if the concept lattice is small enough, the line-diagram enables to view all navigation paths at a glance. In practice, however, the concept lattice is much too large to be visualized, and only the current concept and its neighbours are displayed. Existing conceptual navigation systems follow that principle, and only allow for linear navigation paths.

The contribution of this paper is to extend conceptual navigation to nonlinear navigation paths. Concretely, this means that users can choose a set of navigation links at some step, leading to a set of concepts. As several multichoices can be done in sequence, a navigation place becomes a multi-dimensional set of concepts, a cube of concepts. We show that our approach covers and extends the multi-dimensional analysis of OLAP [START_REF] Codd | Providing OLAP (On-line Analytical Processing) to User-Analysts: An IT Mandate[END_REF][START_REF] Vassiliadis | A survey of logical models for OLAP databases[END_REF]. We define cubes of concepts on multi-valued contexts, which we introduce in this paper. They extend manyvalued contexts by allowing several values for the attribute of an object, rather than only one. The advantages of using multi-valued contexts are (1) more general results that directly apply to many-valued contexts and binary contexts, and (2) a data model close to databases [START_REF] Codd | A relational model of data for large shared data banks[END_REF], OLAP [START_REF] Codd | Providing OLAP (On-line Analytical Processing) to User-Analysts: An IT Mandate[END_REF][START_REF] Vassiliadis | A survey of logical models for OLAP databases[END_REF], the Semantic Web [START_REF] Antoniou | A Semantic Web Primer[END_REF], and Logical Information Systems [START_REF] Ferré | An introduction to logical information systems[END_REF].

After introducing useful notations on mappings, multisets, tuples, and FCA in Section 2, Section 3 defines multi-valued contexts, and value domains. Section 4 defines cubes of concepts to be used as navigation places, and cube transformations as navigation links. Those cube transformations are proved safe, i.e. not leading to empty results. Section 5 discusses the representation of cubes of concepts in Abilis. Section 6 compares cubes of concepts and their transformations to OLAP cubes and operators. Section 7 compares our approach to other approaches combining FCA and OLAP.

Preliminaries

A mapping M from a set A to a set B is a set of couples a → b, where a ∈ A and b ∈ B, and such that (a, b

1 ) ∈ M ∧ (a, b 2 ) ∈ M ⇒ b 1 = b 2 .
It is a partial function from A to B that is defined in extension. Its domain is noted dom(M ), and its range is noted ran(M ). A multiset M over a set A is a mapping from A to natural numbers N. It generalizes sets by allowing elements to occur several times in a multiset: a → n ∈ M means that a has multiplicity n in M . For convenience, we define the following notations on a multiset M :

M α = {a → n ∈ M | n ≥ α} is the filtering of multiset M by a minimum multiplicity α, M Y = {a → n ∈ M | a ∈ Y } is the restriction of the domain of M to Y , and #M = Σ{n | a → n ∈ M } is the cardinal of multiset M .
Given a tuple x of dimension n, we denote by x i the i-th component of the tuple. The notation x is a shorthand for x 1 . . . x n . We define the following notations on tuples: x + x ′ = x 1 . . . x n x ′ is the extension of a tuple x by an element x ′ , x-i = x 1 . . . x i-1 x i+1 . . . x n is the restriction of a tuple x by removing the i-th element, x[i ← x ′ ] = x 1 . . . x i-1 x ′ x i+1 . . . x n is the replacement of the i-th element by x ′ , and dom i (X) = {x i | x ∈ X} is the domain of the i-th elements, where X is a set of tuples of same dimension n.

We here recall some basic notations for Formal Concept Analysis (FCA) [START_REF] Ganter | Formal Concept Analysis -Mathematical Foundations[END_REF]. However, we adopt a somewhat novel presentation in that intension is made a particular case of a more general notion: the index. A formal context is a triple K = (O, A, I), where O is the set of objects, A is the set of attributes, and I ⊆ O×A is the incidence relation between objects and attributes. The extension of a set of attributes Y is defined as the set of objects that have all the attributes in Y , i.e., ext(Y ) = {o ∈ O | ∀a ∈ Y : (o, a) ∈ I}. By abuse of notation, we will denote by ext({a}) the expression ext(a) for . The index of a set of objects X is defined as the multiset of attributes of objects in X, where the multiplicity of each attribute is the number of its objects in X, i.e.,

index (X) = {a → n | a ∈ A, n = #(X ∩ ext(a)) = 0}.
The intent of a set of objects X is then defined as the domain of the index of X filtered by the maximum frequency, i.e., int(X) = dom(index (X) #X ). A concept c is a pair (X, Y ) s.t. X = ext(Y ) and Y = int(X): ext(c) = X is called the extent of c, and int(c) = Y is called the intent of c.

Multi-Valued Contexts

In order to better represent real datasets and complex data, a number of extensions or generalizations of FCA have been proposed: many-valued contexts [START_REF] Ganter | Conceptual scaling[END_REF], logical contexts [START_REF] Ferré | A logical generalization of formal concept analysis[END_REF], pattern structures [START_REF] Ganter | Pattern structures and their projections[END_REF], to cite only a few of them. In particular, many-valued contexts are equivalent to tables in relational databases. Logical contexts (and pattern structures) are more general, but miss the distinction between attributes and values, which will be useful in this paper. Conversely, logical contexts enable to describe a photo as depicting both Alice and Bob, which is not possible in a many-valued context where the photo is an object, "depicts" is an attribute, and persons are values: "depicts" is a multi-valued attribute. The term "many-valued" means that an attribute can take one among many values, whereas the term "multi-valued" here means that an attribute can take multiple values at once. Multi-valued attributes do not exist in relational databases because those are normalized into several tables (one table for photos, and one table for the "depicts" relation). As FCA generally handles a single table, the context, it is important to allow for multi-valued attributes. This leads us to the definition of multi-valued contexts as a set of triples (object,attribute,value), like graphs in the RDF data model [START_REF] Antoniou | A Semantic Web Primer[END_REF].

Definition 1 (multi-valued context). A multi-valued context is a quadruple K = (O, A, V, T ), where O is the set of objects, A is the set of (valued) attributes, V is the set of values, and T ⊆ O × A × V is a set of triples. Each triple (o, a, v) means that v is a value taken by the atribute a on object o. This example illustrates the fact that an object can have several values for an attribute (e.g., the depicted persons of photo 4), or no value at all (e.g., the size of photo 6). This example is kept small for illustration purposes, but our approach is designed for datasets with hundreds to thousands of objects, and tens of attributes.

As the set

O × A × V is equivalent to O × A → P(V ),

Value Domains and Attribute-Value Schemas

In practice, there are interdependencies between attributes and values on one hand, and between values on the other hand. Each attribute expects values from a given domain, and the values of a domain can be organized into a generalization ordering (e.g., locations, date intervals). We first define domains of values in order to formalize hierarchies of values, and aspects related to OLAP such as granularity levels, and aggregators.

Definition 2 (value domain).

A value domain is a structure D = (V, ⊑, ⊤, Λ, Γ ), where:

-V is the set of values, -⊑ is a partial ordering (called subsumption) that represents the generalization ordering between values, and ≺ is the corresponding covering relation, -⊤ is a distinguished value, more general than any value, -Λ ⊆ P(V ) is a set of granularity levels, each level λ ⊆ V being defined as a subset of values, -Γ is a set of aggregators over multi-sets of values.

2M, 1234k).

There is a granularity level for each level of precision: units (e.g., 1234567), tens (e.g., 1234.56k), ..., millions (e.g., 1M). Size values represent integer intervals, and are hierarchically organized by inclusion. Applicable aggregators are, in addition to those of dates, the sum and the average.

There are different ways to define the hierarchy of values of a value domain. A first way is to use scale contexts [START_REF] Ganter | Conceptual scaling[END_REF] when the set of values is finite and fixed. A second way is to define logics as in Logical Concept Analysis (LCA, [START_REF] Ferré | An introduction to logical information systems[END_REF]), which works well for infinite domains (e.g., integers, dates, geographic shapes), and allows for the dynamic and automatic insertion of new values in the hierarchy.

Subsumption can be extended from values to granularity levels: for any two levels λ 1 , λ 2 , we have

λ 1 ⊑ λ 2 iff for every v 1 ∈ λ 1 , there exists v 2 ∈ λ 2 s.t. v 1 ⊑ v 2 .
While levels are generally disjoint and totally ordered, they need not be. A classical example is the date domain, with four disjoint levels: days, months, weeks, and years. The level "day" is subsumed by the levels "month" and "week", both of which are in turn subsumed by the level "year", but neither "month" is subsumed by "week" nor the converse. Sometimes, there are no natural levels, and we want to define them from the topology of the hierarchy. Each level corresponds to a certain depth in the hierarchy, and they may overlap. Definition 3 (topological levels). Let D = (V, ⊑, ⊤, Λ, Γ ) be a value domain. The granularity level λ(p) denotes the topological level at depth p, and is recursively defined as follows for every p > 0:

-λ(0) = {⊤}, -λ(p + 1) = {v ∈ V | ∃v ′ ∈ λ(p) : v ≺ v ′ } ∪ {v ∈ λ(p) | ∄v ′ ∈ V : v ′ ≺ v}.

Attribute Contexts and Feature Context

A multi-valued attribute is a binary relationship between objects and values. A formal context can therefore be derived for each attribute, with the help of a value domain for handling subsumption between values. It is called an attribute context, and it plays the same role as a realized scale. 1. The feature context of the multi-valued context Ke on photos (only some levels of values are shown).

person date size photo Alice Bob Charlie 2010 2011 1M 3M 1 × × × 2 × × × 3 × × × 4 × × × × 5 × × × × 6 × × Table
Definition 4 (attribute context). Let K = (O, A, V, T ) be a multi-valued context, a ∈ A be an attribute, and D a value domain for a. The attribute context of a is defined as

K a = (O, V a , I a ), where I a = {(o, v) ∈ O × V a | (o, a, v ′ ) ∈ T, v ′ ⊑ a v}.
In the attribute context K a , ext a (Y ) is the set of objects having all values of Y as values of the attribute a (directly or by subsumption), e.g., the set of photos depicting a set of persons together; index a (X) is the mappings from values in V a to their frequency as the value of the attribute a over the set of objects X, e.g., the distribution of persons that are depicted in a given set of photos. Finally, int a (X) is the set of values in V a occurring in all objects in X through attribute a, e.g., the set of persons depicted by all elements of given set of photos.

The feature context K F is the apposition (see p. 30 in [START_REF] Ganter | Formal Concept Analysis -Mathematical Foundations[END_REF]) of all attribute contexts, distinguishing values by prefixing them with attributes. This is the natural translation of a multi-valued context into a formal context. Definition 5 (feature context). Let K = (O, A, V, T ) be a multi-valued context, and (D a ) a∈A be a collection of value domains. The feature context is defined as K F = (O, F, I F ), where the set of features F , and the incidence relation I F between objects and features is defined by:

-F = {(a, v) | a ∈ A, v ∈ V a }, -I F = {(o, (a, v)) | (o, a, v ′ ) ∈ T, v ′ ⊑ a v}.
In the feature context, the extension ext F (Y ) returns the set of objects having a set of features Y ; and the index index F (X) returns the distribution of features over a set of objects X. Table 1 contains a partial representation of the feature context of K e , showing only one level of values for each attribute (persons, years, and millions of pixels). The FCA operations of the feature context can be related to the FCA operations of attributes contexts as in Lemma 1.

Lemma 1. The following equations holds for every multi-valued context K:

1. ext F ((a, v)) = ext a (v), 2. int F (X) = a∈A {(a, v) | v ∈ int a (X)}, 3. index F (X) = a∈A {(a, v) → n | v → n ∈ index a (X)}.
In conceptual navigation, navigation places are formal concepts. We are here interested in navigating in the concept lattice of the feature context because it contains all the information of a multi-valued context, and its associated value domains. In the following, we assume a multi-valued context K = (O, A, V, T ) together with value domains D a for each a ∈ A, and its derived feature context K F = (O, F, I F ). In this paper, we start from the framework of Logical Information Systems (LIS, [START_REF] Ferré | An introduction to logical information systems[END_REF]) with a restriction to conjunctive queries, whereas disjunction and negation are generally available in LIS. For the sake of simplicity, we will stick in this paper to this restriction, but the following results easily extent to Boolean queries. In LIS, the current concept c is defined by ext(c) = ext F (q), where q ⊆ F is the current query. This query is the result of the successive choices of a single feature (single-choices) among those suggested by the system, starting from the empty set, and hence from the top concept. We now want to extend conceptual navigation to choices of multiple features (multi-choices), in order to escape the linearity limitation, and to provide multi-dimensional analysis. This raises the following questions: What is a natural multi-choice? What is a navigation place after performing a multi-choice? What defines such a navigation place?

Starting from the example multi-valued context K e , suppose a user wants to look at photos by depicted person. He can select in turn each person, successively visiting the concepts 1456, 24, and 35 (for convenience, we name concepts after their extension: 24 is the concept whose extension is {2, 4}). This is tedious and unpractical for comparing the three subsets of photos. Alternately, the user could perform a multi-choice of the three persons, leading him to the set of the three concepts, indexed by the three persons. This results in the mapping {Alice → 1456, Bob → 24, Charlie → 35}. From there, the user wants to focus on photos depicting Alice, and performs a single choice of the feature (person, Alice). The effect of this single-choice applies to each of the three above concepts, leading to the mapping {Alice → 1456, Bob → 4, Charlie → 5}. Bob and Charlie persist as indices because they are depicted with Alice on some photos. Finally, the user performs a second multi-choice of sizes at the level of millions (1M and 3M). The effect of this multi-choice applies to each concept of the last mapping, producing a mapping of mappings or, in a more compact form, a mapping from couples (person, size) to concepts: {(Alice, 1M ) → 15, (Alice, 3M ) → 4, (Bob, 3M ) → 4, (Charlie, 1M ) → 5}. Only concepts whose extension is not empty are retained, so that not all combinations of a person and a size are present.

This example scenario shows that a natural multi-choice is a granularity level of some attribute, which we define as an axis in analogy with graph axes. Definition 6 (axis). An axis x = a/λ combines an attribute a ∈ A and a granularity level λ ∈ Λ a for that attribute. In the following, a(x) denotes the attribute of the axis, λ(x) denotes the level of the axis, and index x (X) is a shorthand for (index a (X)) λ , the a-index over objects X, restricted to the level λ.

The example scenario also shows that navigation places are now mappings from tuples of values to concepts of the feature context. Those mappings are equivalent to n-dimensional arrays, where n is the size of tuples. Because the dimension of those arrays can be any natural number, and in analogy with OLAP, we call them cubes of concepts. Section 7 compares them to OLAP cubes. What determines such a cube is a sequence of single-choices and multiples-choices. The sequence of single-choices amounts to a set of features, the query q, and the sequence of multi-choices amounts to a tuple of axes, the dimension tuple d.

Definition 7 (cube of concepts). Given a query q ⊆ F , and a tuple d of n axes as dimensions, the cube of concepts is defined as a mapping from coordinates (tuples of values) to concepts:

Cube(q, d) = {v → c | v ∈ n i=1 λ(d i ), ext(c) = ext F (q) ∩ n i=1 ext((a(d i ), v i )) = ∅}.
The above example shows that different concepts in a cube of concepts may overlap, and even that a same concept can appear at different coordinates. This comes from multi-valued contexts, where an object can have several values on a same axis (e.g., Alice and Bob as depicted persons). The example also shows that some coordinates, i.e. some combinations of values, may be missing in the cube. The reason is that either no object matches this combination, or an object has no value on some axis (e.g., photo 6 has no size value). Those are key differences with OLAP cubes (Section 7).

It is possible to generalize some definitions from concepts to cubes.

Definition 8. The extension of the cube of concepts is the union of the extensions of the concepts. The intent and index of a cube of concepts can be derived from its extension as usual. ext(C)

= v →c∈C ext(c) int(C) = int F (ext(C)) index (C) = index F (ext(C))
Beware that the extension of a cube is not necessarily a concept extension, because concept extensions are not closed under set union. However, it is important not to close it, so that the index index (ext(C)) properly reflects the contents of the cube, and not the larger ext F (int(C)) that may contain objects not visible in the cube.

In LIS, navigation links are defined by query transformations [START_REF] Ferré | Camelis: a logical information system to organize and browse a collection of documents[END_REF], rather than by the covering relation of the concept lattice. Query transformations are the addition or the removal of a feature, and combinations such as the replacement of a feature. Adding a feature f to a query q, noted q + f , provides downward navigation in the concept lattice. This gives access not only to lower neighbours, but also to concepts deeper in the lattice. All relevant features are suggested as navigation links, and not only those leading to a lower neighbour. Removing a feature f , noted qf , provides upward navigation in the concept lattice. Of course, features can be removed in a different order they were added to the query. Addition and removal can be combined to provide sideward navigation, e.g., shifting from photos of Alice in 2010 to photos of Bob in 2010, and then to photos of Bob in 2011.

In this paper, because navigation places are cubes of concepts, we define navigation links as transformations of cubes of concepts. As a cube of concepts is made of a query q and a tuple of dimensions d, the above query transformations equally apply to cubes of concepts. A second way to transform a cube is to change the dimensions of the cube. Possible transformations are:

d + d ′ : the addition of a dimension d ′ , -di: the removal of a dimension d i , -σ(d): a permutation σ on d to change the ordering of dimensions.

An important property of conceptual navigation is safeness, i.e. to suggest only navigation links that lead to concepts whose extension is not empty. This is important to avoid dead-ends, trial-and-error navigation, and hence frustration for users. With cubes of concepts, navigation is safe if it never leads to empty cubes. The following theorem states the conditions under which a cube transformation is safe.

Theorem 1. Let C = Cube(q, d) be a cube of concepts. The specializing cube transformations are safe under the following conditions:

-Cube(q + f, d) if f ∈ index F (ext(C)), -Cube(q, d + d ′ ), if index d ′ (ext(C)) = ∅.
The generalizing transformations Cube(qf, d) and Cube(q, di), as well as the permutation transformation Cube(q, σ(d)), are necessarily safe.

Proof. We give the proofs for the specializing transformations. The proof for other transformations are trivial.

-Proof for Cube(q + f, d):

f ∈ index F (ext(C)) ⇒ ext F (f ) ∩ ext(C) = ∅ ⇒ ext F (f ) ∩ v →c∈C ext(c) = ∅ (Definition 8) ⇒ ∃v → c ∈ C : ext F (f ) ∩ ext(c) = ∅ (Definition 7) ⇒ ∃v ∈ i λ(d i ) : ext F (f ) ∩ (ext F (q) ∩ i ext((a(d i ), v i ))) = ∅ ⇒ ∃v ∈ i λ(d i ) : ext F (q + f ) ∩ i ext((a(d i ), v i )) = ∅ (because ext(f ) ∩ ext(q) = ext(q ∪ {f }) = ext(q + f )) ⇒ ∃v → c ′ ∈ Cube(q + f, d) ⇒ Cube(q + f, d) = ∅. -Proof for Cube(q, d + d ′ ): index d ′ (ext(C)) = ∅ ⇒ index a(d ′ ) (ext(C)) λ(d ′ ) = ∅ (Definition 6) ⇒ ∃v ′ ∈ λ(d ′ ) : v ′ ∈ index a(d ′ ) (ext(C)) ⇒ ∃v ′ ∈ λ(d ′ ) : (a(d ′ ), v ′ ) ∈ index F (ext(C)) ⇒ ∃v ′ ∈ λ(d ′ ) : ext F ((a(d ′ ), v ′ )) ∩ ext(C) = ∅ (Lemma 1) ⇒ ∃v → c ∈ C : ∃v ′ ∈ λ(d ′ ) : ext((a(d ′ ), v ′ )) ∩ ext(c) = ∅ (Definition 8) ⇒ ∃v ∈ i λ(d i ) : ∃v ′ ∈ λ(d ′ ) : ext((a(d ′ ), v ′ )) ∩ (ext F (q) ∩ i ext((a(d i ), v i ))) = ∅ ⇒ ∃v + v ′ ∈ ( i λ(d i )) × λ(d ′ ) : ext F (q) ∩ (( i ext((a(d i ), v i ))) ∩ ext((a(d ′ ), v ′ ))) = ∅ ⇒ ∃v + v ′ → c ′ ∈ Cube(q, d + d ′ ) ⇒ Cube(q, d + d ′ ) = ∅.
The condition for the addition of a feature to the query is the same as previously known in LIS. It establishes the feature index index F (ext(C)) of a cube C as the set of features that can be added to the query. The condition for the addition of a dimension involves the indexes for each axis, which are included in the feature index (see Lemma 1). Therefore, the extension of conceptual navigation from concepts to cubes of concepts does not entail any increase in the size of suggested navigation links. A suggested dimension is simply an axis that shares values with the feature context. This is consistent with multi-choices being sets of choices.

Representation and Interaction in Abilis

In LIS systems, e.g. Camelis and Abilis1 , the current concept c is represented by the query q that defines it, the extension ext(c) of the concept as a list of objects, and the feature index index (ext(c)) over that extension, which includes the intension int(c) of the current concept. The feature index is organized by attribute, like facets in Faceted Search [START_REF]Dynamic taxonomies and faceted search. The information retrieval series[END_REF], and is displayed as a tree to reflect the generalization ordering between values. The query can be transformed by selecting features in the index. A feature is removed from the query if it belongs to it, otherwise it is added to the query.

Multi-dimensional conceptual navigation with cubes of concepts has been implemented in Abilis, along with rich capabilities to represent cubes of concepts. With a cube of concepts C = Cube(q, d), we still have a query q, an extension ext(C), and an index index F (ext(C)) over that extension. The index is also the support of interaction by suggesting features and axes to be added or removed from the cube. The important difference is that a navigation place is a set of concepts projected at some coordinates instead of a single concept. Each concept defines a unit of knowledge, and has many possible concrete representations. The two obvious representations of a concept are its extension and its intention. Other useful representations are attribute indexes, and also aggregations of attribute indexes. By analogy with OLAP, we call a possible representation of a concept a measure, even if OLAP measures are generally limited to aggregated values.

Definition 9 (measure).

A measure is defined as any function from a concept to a piece of data representing some aspect of that concept. Given a multi-valued context K, the measures for a concept c that we have implemented in Abilis are the extension ext(c) (noted ext), the count #ext(c) (noted count), the index over some axis index x (ext(c)) (noted x), and the aggregated index over some axis γ(index x (ext(c))) (noted γ(x)).

For example, the concept 1456 has the following representations.

measure result ext {1, 4, 5, 6} count 4 person/individual {Alice → 4, Bob → 1, Charlie → 1} date/year {2010 → 1, 2011 → 3} size/million {1M → 2, 3M → 1} sum(size/million) 5M
In Abilis, an extension is represented as a list of objects, which can be displayed only in part if too long. An index over some axis is a multiset of values, and can therefore be represented as a tag cloud, where the font size renders the multiplicity of values. An aggregated index is generally a numerical value, but it could be anything. For example, in a domain where values are geometrical shapes, an aggregated value can be a geometrical shape (e.g., union, centroid, buffer area). If those geometrical shapes are geo-located, they can be rendered on a map.

The definition of a cube of concepts can be refined as a cube of concept measures, which is defined by a tuple of measures in addition to the query and dimensions.

Definition 10 (cube of concept measures). Given a query q ⊆ F , a tuple d of n axes as dimensions, and a tuple of p measures m, the cube of concept measures is defined as

Cube(q, d, m) = {v → (m j (c)) j∈1..p | v → c ∈ Cube(q, d)}.
After describing the possible representations of individual concepts, we need to describe the possible representations of cubes of concepts. In other words, how to represent the multi-dimensional structure of a cube in rich and flexible ways. Abilis provides the following structures: arrays Arrays can be used for all dimensions. They use values as row/column labels, and their cells can contains arbitrary representations of dimensions and measures. There are three kinds of arrays: horizontal arrays, vertical arrays, and two-dimensional arrays (spreadsheets). The later represents two dimensions at a time. bar charts Bar charts can be used to represent the last dimension when the measure is an aggregated numerical value. There are horizontal and vertical bar charts. pie charts Pie charts can be used in the same conditions as bar charts. maps Maps can be used to represent a geographical dimension. The query is at the top left, the feature index is at the bottom left, the selected dimensions and measure are at the top right along with representation choices, and the cube itself, here a two-dimensional array of tag clouds, is at the bottom right. There are many possible ways to navigate to this view from the initial cube Cube(∅, (), ext), because the definition of the query, dimensions, and measures can be interleaved arbitrarily. Here is a possible scenario. Initially, the list of all 208 publications of the context is displayed in a zero-dimensional cube. Then, this list is grouped by LIS team author by choosing the attribute lis author as a dimension. This results in a vertical array of lists of publications indexed by author. Note that a same publication may appear in several lists (in several cells of the array) because a publication may have several authors (lis author is multi-valued). Then, the results are restricted to journal and conference papers by selecting the two features type is "article" and type is "inproceedings". Note that disjunction (and negation) is available in Abilis, like in other LIS systems. Then, by choosing the attribute year as a measure, lists of papers are replaced by tag clouds of years in each cell. From there, it would be possible to apply an aggregator such as average, minimum or maximum. Those tag clouds can be restricted to years since 2000 by adding the feature year = 2k to the query. Finally, in order to get a finer analysis of the distribution of years of publication, the attribute type is selected as an additional dimension, which results in a two-dimensional array of tag clouds. Note how the attribute type is used both as a dimension and in the query, and how the attribute year is used both as a measure and in the query.

OLAP (On-Line Analytical Processing) [START_REF] Codd | Providing OLAP (On-line Analytical Processing) to User-Analysts: An IT Mandate[END_REF][START_REF] Vassiliadis | A survey of logical models for OLAP databases[END_REF] is an approach to the multidimensional exploration of data, and it is part of the domain of business intelligence. OLAP does not add any expressiveness compared to relational databases, and in fact is less expressive, but it makes it much easier and quicker to perform aggregative queries than with SQL. The principle is to let users navigate from view to view, and in this respect, it follows the same goals as conceptual navigation. In this section, we compare our approach to OLAP, and we show that it covers all OLAP representations and operations, and that it allows for more flexibility and expressiveness than OLAP. However, it should be noted that there is a trade-off between expressiveness and efficiency, and that some of the restrictions seen in OLAP are useful to accelerate some computations.

In OLAP, a same structure, called an OLAP cube, is used both for representing data and for representing views. An OLAP cube is a multidimensional database that is defined by n dimensions (e.g., date and place), p measures (e.g., sales), and a mapping from tuples of n values (e.g., December 2011 and Rennes) to aggregated values for the measure (e.g., total sales). Therefore, an OLAP cube is equivalent to a cube of concept measures, where measures are all aggregated indexes. In our approach, data is represented as a multi-valued context, from which many different cubes can be defined by varying dimensions and measures. Moreover, elements of the cubes need not be aggregated values, but can also be sets of objects (extension), and multisets of values (indexes). In fact, the objects from which an OLAP cube may have been defined have been lost because they have been aggregated in the preprocessing stage. On the contrary, multi-valued contexts are object-centered.

The object-centered approach allows for more flexibility and heterogeneity in data, such as multi-valued attributes or missing values. Suppose we have the cube of the sum of the size of photos, by person: Cube(∅, person/individual , sum(size/million)) = {Alice → 5M, Bob → 6M, Charlie → 2M }. This is a valid OLAP cube. Now, suppose we want to aggregate those sizes to get the total size of photos. The result in OLAP would be 13M , whereas the correct result, as given by our approach, is 9M . This is because a photo can have several persons, and this is why OLAP assumes a strict partition between the values of a dimension.

OLAP defines a number of operators on cubes that play the same role as our cube transformations. We translate each of those operators in our approach:

Slice The selection of a sub-cube of dimension n -1 by fixing the value of some dimension. In our approach, this is equivalent to adding a feature to the query, and removing a dimension. A difference is that any feature can be selected, whether it belongs to a dimension or a measure or none. Add dimension The addition of a dimension. In OLAP, this is restricted to predefined dimensions, while in our approach, this can be any attribute. Remove dimension The removal of a dimension. In OLAP, this necessarily entails an aggregation.

Pivot The swap of two dimensions. This is a particular case of permutation in our approach. Drill-down The change of a dimension level for a finer granularity level (e.g., from years to months or weeks). In our approach, this is equivalent to removing a dimension axis a/λ, and adding the axis a/λ ′ , where λ ′ ≺ λ. Abilis provides a direct navigation link for drill-down. Roll-up The converse of drill-down, i.e. the change of a dimension level for a coarser granularity level (e.g., from months to years).

This demonstrates the flexiblity and expressiveness of our approach. All attributes can be used in the query, dimensions, and measures, and a same attribute can play several roles at the same time. An attribute can even be used twice as a dimension, which makes sense when the attribute is multi-valued. For example, the cube Cube(∅, (person/individual , person/individual ), count) displays the number of photos for each couple of persons (and for each person on the diagonal of the array). Finally, our approach allows for drill-down and roll-up on the measures that are based on an axis, because an axis is parameterized by a level. This is particularly useful when the measure is an index, i.e. a multiset of values. For instance, we can display for each person, the multiset of the dates of their photos, and those dates can be displayed at the level of years or months or weeks or days. This has proved useful for discovering functional dependencies and association rules in multi-valued contexts [START_REF] Allard | Discovering functional dependencies and association rules by navigating in a lattice of OLAP views[END_REF].

Related Work

We compare our approach to other approaches combining FCA and OLAP. Stumme [START_REF] Stumme | Conceptual on-line analytical processing[END_REF] describes conceptual OLAP for conceptual information systems. His approach is very similar to OLAP, except for the definition of hierarchies of dimension values, and therefore has the same limitations as OLAP (see Section 6). A many-valued context defines the multi-dimensional space. Each attribute defines a dimension, whose hierarchy of values is the conceptual scale derived from a scale context. Those conceptual scales have no defined levels, apart from the default topological levels as in Definition 3. The measures, called variables, are defined out of the context, as functions from objects of the many-valued context to values. Therefore, dimensions and measures are strongly separated, and only aggregated measures are available. The only structures for representing cubes are the line-diagrams of the conceptual scales, which have to be designed in advance for better presentation. Each line-diagram represents one dimension, and nested line-diagrams are used to represent multi-dimensional cubes. Measure values appear as labels of the concepts of the innermost line-diagrams. Nested line-diagrams are an alternative to nested arrays. They need more space but they better expose complex hierarchies of values.

Penkova and Korobko [START_REF] Penkova | Constructing the integral OLAP-model based on formal concept analysis[END_REF] apply standard FCA on cube schemas, instead of on cubes themselves. They start from a formal context where objects are measures, attributes are dimensions, and the incidence relation is the compatibility relation between measures and dimensions. For instance, in a dataset about the activities of a scientific organization, the dimension "journal name" is compatible with the measure "number of published paper" but not with "number of established conferences". A formal concept is a maximal cube schema: no dimension can be added without removing measures. The concept lattice can be used to guide users on the addition of dimensions (moving downward) or measures (moving upward). This approach is interesting when different kinds of objects are mixed (e.g., established conferences and published papers), and some dimensions only apply to some kinds of objects. In our approach, the feature index offers the same benefits by showing for each attribute whether it applies to all objects in the current cube, or only to a subset. And this comes without the usual limitations with OLAP: assymetry between dimensions and measures, only aggregated values as measures, etc. Applying our approach to the above examples, objects would be the individual published papers and established conferences, and their number would be obtained by choosing count as a measure. Other measures would allow to visualize the objects themselves, or the distribution of authors for papers.

Conclusion

The contribution of this paper is to extend conceptual navigation from singleconcept views to cubes-of-concepts views. This means that, at each navigation step, the view is not limited to a single concept, but to a set of concepts, organized into a multi-dimensional cube. The single-concept view is a special case of a cube of concepts, having dimension 0. An important difference with OLAP is that each cube cell is a concept, which can be represented by a number of measures: a set of objects (the extension), a multiset of values (an attribute index), or an aggregated value. Finally, the cube need not cover the whole dataset, but can focus on a given subset, which is defined by a query.

Our approach generalizes OLAP cubes and navigation between cubes by relaxing a number of constraints. Cubes are derived from an object-centered multivalued context, where no distinction is made between dimensions and measures, and where objects are not aggregated a priori. Objects can have several values for a same attribute. A same attribute can be used in a query, as a dimension, and as a measure at the same time. Drill-down and roll-up equally apply to dimensions and measures.

Our approach retains some constraints from OLAP relative to relational databases in terms of expressiveness. In particular, in a multi-valued context, the entities of a relational database must clearly be separated between objects and values. In another work, we have extended conceptual navigation and LIS to relational data from the Semantic Web [START_REF] Ferré | Conceptual navigation in RDF graphs with SPARQL-like queries[END_REF]. Our goal is now to join the two extensions into one: multi-dimensional and relational conceptual navigation.

Fig. 1 .

 1 Fig. 1. Screenshot of Abilis showing the distribution over years of publications in journals and conferences since 2000 by author and by type of publication.

Figure 1

 1 Figure1is a screenshot of our prototype Abilis. It displays a cube of concept measures showing the distribution over years of publications in journals and conferences since 2000 by author and by type of publication. The query is at the top left, the feature index is at the bottom left, the selected dimensions and measure are at the top right along with representation choices, and the cube itself, here a two-dimensional array of tag clouds, is at the bottom right. There are many possible ways to navigate to this view from the initial cube Cube(∅, (), ext), because the definition of the query, dimensions, and measures can be interleaved arbitrarily. Here is a possible scenario. Initially, the list of all 208 publications of the context is displayed in a zero-dimensional cube. Then, this list is grouped by LIS team author by choosing the attribute lis author as a dimension. This results in a vertical array of lists of publications indexed by author. Note that a same publication may appear in several lists (in several cells of the array) because a publication may have several authors (lis author is multi-valued). Then, the results are restricted to journal and conference papers by selecting the two features type is "article" and type is "inproceedings". Note that disjunction (and negation) is available in Abilis, like in other LIS systems. Then, by choosing the attribute year as a measure, lists of papers are replaced by tag clouds of years in each cell. From there, it would be possible to apply an aggregator such as average, minimum or maximum. Those tag clouds can be restricted to years since 2000 by adding the feature year = 2k to the query. Finally, in order to get a finer analysis of the distribution of years of publication, the attribute type is selected as an additional dimension, which results in a two-dimensional array of tag clouds. Note how the attribute type is used both as a dimension and in the query, and how the attribute year is used both as a measure and in the query.

  a multi-valued context can also be represented like a many-valued context, but allowing for zero, one or several values in each cell. The following table defines a multi-valued context K e of 6 photos described by the persons they depict, the year they were taken, and their size in pixels. It has 19 triples.

	object	person	date	size
	1	Alice	2010-03-19 1.1M
	2	Bob	2010-07-13 3.1M
	3	Charlie	2011-01-30 1.2M
	4	Alice, Bob 2011-04-28 3.2M
	5 Alice, Charlie 2011-08-20 1.3M
	6	Alice	2011-11-11

  The example multi-valued context K e uses three value domains, respectively for persons, dates, and sizes. person Values are either persons (e.g., Alice, Bob, Charlie) or groups (e.g., family, friends). A person is subsumed by every group it belongs to. Persons and groups constitute the two granularity levels: Λ = {individual , group}. The only applicable aggregators are the count (γ(M ) = #M ) and the distinct count (γ(M ) = #dom(M )). date Values are dates at four granularity levels: days (e.g., 2010-03-19), months (e.g., 2010-03), years (e.g., 2010), and calendar weeks (e.g., 2010:42). Date values represent intervals of time, and are hierarchically organized by inclusion. Applicable aggregators are, in addition to the count and distinct count, the earliest date, the latest date, and the median date. size Values are natural numbers at various precisions (e.g., 1M, 1.

Abilis is a Web interface (try it at http://ledenez.insa-rennes.fr/abilis/) on top of Camelis (download it at http://www.irisa.fr/LIS/ferre/camelis/)