acronyme -...

Projet TRECOLOCOCO

DOCUMENT SCIENTIFIQUE

Important Ce document, hors annexes, ne doit pas dépasser 40 pages, corps de texte en

police de taille 11. Ce point constitue un critère de recevabilité de la proposition de projet. Les propositions de projets ne satisfaisant pas aux critères de

recevabilité ne seront pas évaluées.

Nom et prénom du coordinateur / coordinator’s name

Santocanale Luigi

Acronyme / Acronym TRECOLOCOCO

Titre de la proposition de projet

Interactions des Treillis :Combinatoire, Logique, Connaissances, Concurrence

Proposal title Interactions of Lattices:Combinatorics, Logic, Knowledge, Concurrency

Comité d’évaluation / Evaluation committee

SIMI 2

Projet multidisciplinaire / multidisciplinary proposal

X OUI NONSi oui, indiquer un comité secondaire :

Type de recherche / Type of research

X Recherche Fondamentale / Basic Research Recherche Industrielle / Industrial Research Développement Expérimental / Experimental Development

Coopération internationale / International cooperation

OUI X NON

Aide totale demandée / Grant requested

534 658,80 €Durée de la proposition de projet / Proposal duration

48 mois

ANR-GUI-AAP-04 – Doc Scientifique 2011 1/59

Projet TRECOLOCOCO


1. RÉSUME DE LA PROPOSITION DE PROJET / PROPOSAL ABSTRACT.....................92. CONTEXTE, POSITIONNEMENT ET OBJECTIFS DE LA PROPOSITION / CONTEXT,

POSITIONING AND OBJECTIVES OF THE PROPOSAL.....................................102.1. Contexte de la proposition de projet / Context of the proposal .....................102.2. État de l'art et position de la proposition de projet / state of the art and

positionning of the proposal......................................................................102.3. Objectifs et caractère ambitieux et/ou novateur de la proposition de projet /

Objectives, originality and/or novelty of the proposal...............................123. PROGRAMME SCIENTIFIQUE ET TECHNIQUE, ORGANISATION DE LA PROPOSITION

DE PROJET / SCIENTIFIC AND TECHNICAL PROGRAMME, PROPOSAL ORGANISATION.................................................................................12

3.1. Programme scientifique, structuration de la proposition de projet/ Scientific programme, proposal structure................................................................13

3.2. Description des travaux par tâche / Description by task................................153.2.1 Tâche 1 / Task 1 163.2.2 Tâche 2 / Task 2 173.2.3 Tâche 3 / Task 3 193.2.4 Tâche 4 / Task 4 Lattices: logical aspects 233.2.5 Tâche 6 / Task 6 : On the model theory of Heyting algebras 253.2.6 Tâche 6 / Task 6 The variety question for classes of lattices 28

3.3. Calendrier des tâches, livrables et jalons / Tasks schedule, deliverables and milestones................................................................................................30

4. STRATÉGIE DE VALORISATION, DE PROTECTION ET D’EXPLOITATION DES RÉSULTATS / DISSEMINATION AND EXPLOITATION OF RESULTS, INTELLECTUAL PROPERTY........................................................................................32

5. DESCRIPTION DU PARTENARIAT / CONSORTIUM DESCRIPTION .......................345.1. Description, adéquation et complémentarité des partenaires / Partners

description and relevance, complementarity............................................345.2. Qualification du coordinateur de la proposition de projet/ Qualification of the

proposal coordinator.................................................................................355.3. Qualification, rôle et implication des participants / Qualification and

contribution of each partner.....................................................................356. JUSTIFICATION SCIENTIFIQUE DES MOYENS DEMANDÉS / SCIENTIFIC JUSTIFICATION

OF REQUESTED RESSOURCES...............................................................376.1. Partenaire 1 / Partner 1 : LIF .........................................................................376.2. Partenaire 2 / Partner 2 : L3I..........................................................................39

7. ANNEXES / ANNEXES...........................................................................407.1. Références bibliographiques / References.....................................................407.2. Biographies / CV, resume..............................................................................487.3. Implication des personnes dans d’autres contrats / Staff involvment in other

contracts...................................................................................................56


Projet TRECOLOCOCO



Projet TRECOLOCOCO


Avant de soumettre ce document :- Supprimer toutes les instructions en rouge (par exemple en faisant Format

Styles Menu contextuel du style « Instructions » Sélectionner toutes les occurrences suppr.)

- Mettre la table des matières à jour (bouton droit sur la table des matières mettre à jour les champs Mettre à jour toute la table).

- Donner toutes les références bibliographiques en annexe 7.1.

- Ce document, hors annexes, ne doit pas dépasser 40 pages, corps de texte en police de taille 11. Ce point constitue un critère de recevabilité de la proposition de projet.

1. RÉSUME DE LA PROPOSITION DE PROJET / PROPOSAL ABSTRACTRecopier le résumé utilisé dans le document administratif et financier (dit document de soumission).

Traditional research fields at the intersection of theoretical computer science and mathematics – logic, programming language semantics, concurrency theory – rely on the same algebraic and order-theoretic structure, that of a lattice. More recently this structure has attracted computer scientists working in the domain of databases, knowledge representation, and classification – to the point of becoming their main working tool. In mathematics, undergoing researches at the border of geometry, algebra, and combinatorics consider ordered sets and make use of their lattice properties – such as for the theory of finite Coxeter groups and the theory of Hopf algebras .

Despite the many interactions of lattices with various areas of computer science and mathematics, few research groups have been able to link theoretic research on lattices with these areas of interactions.

Our team, composed by experts from different areas, share a common interest for lattices and a strong will to use ideas and tools from lattice theory to deepen the insights on these areas.

A first aim is to strengthen this link between theoretic research and areas of interactions. On the one hand, tackling the traditional and new problems from a lattice theoretic point of view is a novel approach that we wish to exploit. Conversely, the interactions with these areas call for a deepening and renewal of even classical lattice-theoretical knowledge.

A second aim – more of a social nature but not less important – is toconsolidate our collaborative project -- whose birth was somewhat fortuitous – and to grow up to constitute a solid scientific community.

Our researches shall be organised along two axes.


Projet TRECOLOCOCO


A first axis, finite lattices, is at the core of recent researches in computer science (database theory, knowledge representation, classification, but also programming language semantics and concurrency theory) and combinatorics. We shall focus on some lattice theoretic concepts and problems that, arising transversely in all these contexts, appear to be more fundamental. Among them, the OD-graph of a finite lattice and the canonical direct basis of a closure system. Typically, these have to be computed from a Galois representation of a lattice; it is this computation that provides new challenges, both of a theoretic and of an algorithmic nature.

Our study of finite lattices shall be accompanied by the implementation of a library for handling them. This development task shall interact with our theoretic research by collecting ideas and rising new questions. It will also be an important way to animate a dialogue between mathematicians and computer scientists.

A second axis shall focus on infinite lattices that are of interest to other areas of computer science and mathematics, logic and universal algebra. We shall attack some classical (and apparently difficult) problems from lattice theory. These problems are not standing alone, as any attempt to solve them shall interact with and/or depend on other tasks of our project, our researches on finite lattices, but also those of a more logical nature. A main question here is at issue: do the fundamental concepts that work for finite lattices (namely the OD-graph and the canonical direct basis) lift to infinite lattices, in any reasonable way? Depending on the possible answers to this question, we shall develop duality theories and correspondence theories for non-distributive lattices. We shall generalize in this way a wide number of tools and results from classical, intuitionistic, and modal logics, to logics not obeying the distribution law of conjunction and disjunction.

2. CONTEXTE, POSITIONNEMENT ET OBJECTIFS DE LA PROPOSITION / CONTEXT, POSITIONING AND OBJECTIVES OF THE PROPOSAL

A titre indicatif : de 1 à 10 pages pour ce chapitre.Présentation générale du problème qu’il est proposé de traiter dans la proposition de projet et du cadre de travail (recherche fondamentale, industrielle ou développement expérimental).

2.1. CONTEXTE DE LA PROPOSITION DE PROJET / CONTEXT OF THE PROPOSAL Décrire le contnexte et les enjeux éventuels (industriels, économiques, sociaux, réglementaires, médicaux, environnementaux, …) dans lequel (lesquels) se situe la proposition de projet. Présenter une analyse de la pertinence et de la portée par rapport aux besoins.

Lattice theory has recently become quite an important trend within the scientific community studying databases. The reasons for this have been sketched before: namely, a lattice can encode and at the same time organise a state of knowledge. Some algorithmic issues of databases coincide with algorithmic


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issues from lattice theory. Yet the lattice theoretic questions that such a community is confronted to are restricted to such domain of applications. Our experience with French researchers in that area is that too often the word lattice has become a synonym of concept lattice (or Galois lattice) and that problems of pure database-theoretical nature are claimed to be lattice-theoretic.

We do not wish to negate the importance of these problems or the breath of freshness and vitality that database theory has given to lattice theory. Yet we do not wish to identify the two areas, which eventually will lead to flatten down the richness of lattice theory. Instead, we aim at approaching (and advertising for) fundamental problems. Here we should explain in which sense a problem is fundamental, the measure being the following. Possibly, a problem exhibits its fundamental nature exactly when recurring in different areas, applicative or not.

2.2. ÉTAT DE L'ART ET POSITION DE LA PROPOSITION DE PROJET / STATE OF THE ART AND POSITIONNING OF THE PROPOSAL

Présenter un état de l’art national et international en dressant l’état des connaissances sur le sujet, y inclure les éventuelles contributions des partenaires. Positionner la proposition de projet par rapport à ces connaissances.Indiquer si la proposition de projet s’inscrit dans la continuité de projet(s) antérieur(s) déjà financé(s) par l’ANR. Dans ce cas, présenter brièvement les résultats acquis.Faire apparaître d’éventuels résultats préliminaires. Inclure les références bibliographiques nécessaires en annexe 7.1.

A primary goal of the project is to consolidate a community of French researchers, mathematicians and/or computer scientists, who develop research within the scope of lattice theory. The project leader got opportunities to meet some of them (Caspard, Wehrung) at international conferences (AAA, OAL), and observed at the same time a lack of coordination in this research area at the national level. He organised therefore the workshop “Treilis Marseillais” (held at CIRM, Marseilles, during April 2007, within the scope of the ANR JC SOAPDC) where he met most of the other participants to the project. Since then, a number of other researchers joined the community, either because lattices are a keyword in their daily research, or because lattices offer a framework of interactions between logic and combinatorics, close to their personal research skills. The community is presently formed by Karell Bertet (L3I, La Rochelle), François Brucker (LIF, Marseille), Nathalie Caspard (LACL, Paris), Luck Darnière (LAREMA, Angers), Sévérine Fratani (LIF, Marseille), Rémi Morin (LIF, Marseille), Frédéric Olive (LIF, Marseille), Maurice Pouzet (ICJ, Lyon), Luigi Santocanale (LIF, Marseille), Friedrich Wehrung (LMNO, Caen).


Projet TRECOLOCOCO


Let us comment on the need to consolidate this community. Most of the mentioned scientists above do not share a common pat; for example, none of them had the same PhD thesis advisor. Similarly, and possibly as a consequence of what we just observed, the team suffers from a geographic dispersion on the French territory: apart from Marseilles, there is a bijection between participants to the project and laboratories they belong to. The choice of a second “partenaire” reflects more an administrative need to ease handling of resources than a thematic subdivision. While this kind of observation could be considered as a weakness of the project -- and indeed we are aware of the difficulties and obstacles that might arise – it also witnesses the novelty coming with the TRECOLOCOCO project. For example, the fact that each scientist had a distinct PhD director witnesses that there is no existing school for which our group could share some strong feeling of membership; in some sense, we are in an opposite situation of the T. H. Kuhn ``normal science''. On the other hand, the project leader strongly wishes to emphasize the need of laying the bases for such a normal scientific context (again in the sense of Kuhn); as a matter of fact, some of our team members (other than the project leader) are among the strongest and most recognized international scientists in lattice theory. In this respect, let us observe that the project proposes to develop and invest into some scientific areas – lattice theory, algebraic logic, universal algebra – that are considerably under-represented in France.

We presented a first proposal of our project to the INSMI institute of the CNRS, within the PEPS (Projets Exploratoires Pluridisciplinaires) call for proposals. The help received (3000 euros) made it possible for us to organise two meetings (Angers, July 1-2 2010 and Marseilles, November 17-19 2010 ), where researchers got opportunities to present their research, strengthen their interactions capabilities, focus on common goals. The project favoured the collaboration between some of the members, who payed respective visits (Darnière-Santocanale, Santocanale-Wehrung).

As a result of this, some important advances were made on the problem of characterising the equational theory of Associahedra and Permutohedra (see task X).

The PEPS TRECOLOCOCO project made it possible for us to test the working flow of a possible ANR TRECOLOCOCO project. For example, the members agreed that three annual meetings of the project had to be organised, and not more than three, due to the teaching activities in which the members of the project are involved. On the other hand, most of the members agreed on the need of increasing the frequency of mutual visits, as these have shown to be the most productive part in our research. With respect to the original PEPS proposition, we decided not to emphasize the Concurrency part, as the members most implied in this theme had more difficulties – for different and understandable reasons – to get involved into the project.


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2.3. OBJECTIFS ET CARACTÈRE AMBITIEUX ET/OU NOVATEUR DE LA PROPOSITION DE PROJET / OBJECTIVES, ORIGINALITY AND/OR NOVELTY OF THE PROPOSAL

Décrire les objectifs de la proposition de projet et détailler les verrous scientifiques et techniques à lever par la réalisation du projet. Insister sur le caractère ambitieux et/ou novateur de la proposition.Présenter les résultats escomptés et décrire l’(les) éventuel(s) produit(s) final(aux) développé(s).Pour les projets multidisciplinaires, préciser dans quelle mesure l’interaction entre les deux disciplines présente un aspect ambitieux et/ou novateur.

As fundamental research, it is difficult to immediately justify our project with immediate applications. Also, we already mentioned the novelty the project constitute in France, as many fields that we are willing to engage into are under-represented or non-existing in this country. Let us notice, however, that the algorithmic considerations and the libraries we shall develop for handling lattices can have some important impact for applications in the area of databases. We indeed expect this to be the case.

The main importance of the project is the recognition of two facts: on the one side the fact that the lattice structure is increasingly needed in other fields of mathematics and computer science. On the other side, we often remark that researchers dealing with lattices lack the technical skills and various experiences that present in our group. We are therefore strongly convinced that the our team has a potential that need to be amplified.

3. PROGRAMME SCIENTIFIQUE ET TECHNIQUE, ORGANISATION DE LA PROPOSITION DE PROJET / SCIENTIFIC AND TECHNICAL PROGRAMME, PROPOSAL ORGANISATION

A titre indicatif : de 5 à 12 pages pour ce chapitre, en fonction du nombre de tâches.Les tâches représentent les grandes phases du projet. Elles sont en nombre limité. La première tâche correspond à une tâche de coordination.

3.1. PROGRAMME SCIENTIFIQUE, STRUCTURATION DE LA PROPOSITION DE PROJET/ SCIENTIFIC PROGRAMME, PROPOSAL STRUCTURE

Présenter le programme scientifique et justifier la décomposition en tâches du programme de travail en cohérence avec les objectifs poursuivis. Utiliser un diagramme pour présenter les liens entre les différentes tâches (organigramme technique).Pour les projets multidisciplinaires, montrer l'articulation entre les disciplines scientifiques.


Projet TRECOLOCOCO


A lattice can be defined as an ordered set with the property that every finite subset of elements has a least upper bound (a sup) as well as a greatest lower bound (an inf). An elementary result of algebra states that a lattice can also be defined in algebraic terms, that is, using terms over a signature and a list of axioms stating the identity between terms. By its own nature, the lattice structure lies at the intersection of different kind of mathematics, combinatorics, and algebra. As infs and sups model conjunctions and disjunctions, logic is another branch of mathematics that traditionally enjoys close ties with lattices. The intrinsic richness of this mathematical structure has turned it into an object of study in many areas of computer science : database and knowledge representation theory, concurrency theory, verification of computer systems, programming language semantics.

Two main representation theorems constitute the ground of the theory of finite lattices. The first of the two states that every finite lattice is isomorphic to a Galois lattice. Let us recall what this means and the construction we are referring to. Given a table, that is, a relation T ⊆ O × A – where O and T are considered as a set of objects and, respectively, of attributes – the elements of the Galois lattice are the subset of O of the form TY, for Y ⊆ A; here TY = { x ∈ O | xTy for all y ∈ Y }. The ordering of the Galois lattice is given by subset inclusion. The second representation theorem states that every finite lattice is isomorphic to a lattice of closed sets, where the closure operator can be defined by an implicational basis. Here the keywords are numerous: Moore family, closure operator, closed sets, and, among them, the most interesting are canonical direct bases of implications, and the OD-graph of a lattice. An implicational basis on a set P is a collection of pairs Y → y , where Y ⊆ P and y ∈ P . A subset X ⊆ P is then said to be closed if Y ⊆ X et Y → y implies y ∈ X. The collection of closed subsets happens to be closed under intersection, a condition sufficient to ensure that closed subsets form a lattice when ordered by subset inclusion. The second representation theorem is actually quite strong, as every finite lattice can be obtained in this way, through a particular implicational basis known as the canonical direct one.

Even if these two representation theorems and constructions are part of the daily working practice of both mathematicians and computer scientists, they rarely have been under focus and properly studied. We believe that the available knowledge about them is far from being satisfactory. For example, we can ask the following question: what is the cost of computing the canonical direct basis from a table? Of course, the question has a trivial answer, as it is possible to first construct the entire lattice from the table, and then construct the canonical basis from the lattice. However such a way of proceeding would be algorithmically inefficient, as it possibly requires an exponential space for storing the lattice.

The problem of relating the two representations of a lattice, and the computational questions we mentioned above, are among the issues that we want to tackle within the TRECOLOCOCO project. This problem was presented first as it appears, implicitly or explicitly, among a list of other problems/issues/goals that we are proposing to tackle – and possibly solve – that


Projet TRECOLOCOCO


arise from various areas of mathematics and computer science; let us give a brief survey about those.

In knowledge representation and data mining, a lattice describes and organises a complex state of knowledge. One of its representations, either through a table or an implicational basis, is to be considered as a concise description of such a state. We wish, in principle, to dispose of a concise description which is both intuitive for the user and adequate for answering queries in an efficient way. These two objectives are often in contradiction; they can be achieved if we dispose of both the two representations of the lattice and an algorithmically effective way of translating between them.The problem of relating the two representations of a lattice arises again in combinatorics and logic. A noteworthy case where this occurs is the study of the weak Bruhat orders – which are actually lattices – of finite Coxeter groups. These lattices are well understood through their Galois representation. As logicians we propose to characterise the equational theory of these lattices, a task that can better be achieved knowing the representation through the direct canonical basis. Thus, once more, we have to construct the direct canonical basis given the Galois representation.

We also propose to investigate these two representation theorems – that form the core of finite lattice theory – for infinite lattices. Indeed, infinite analogues of these representation theorems work for distributive lattices only. This is the (limited) scope of the Stone-Priestley type of dualities. We wish to understand whether these limits are intrinsic, or whether a general duality theory for all lattices can be developed following the example of the known dualities for finite lattices. A byproduct of this work shall be the development of semantics for substructural logics, which could reasonably pretend, in view of their coherence and usefulness, the role of Kripke semantics of intuitionistic and modal logics. The problem of relating the two representations arises again, in a wider setting and in a more fundamental way. Indeed we are led to confront two quite different ways of approaching logic: one of them puts the emphasis on symmetry, as the Galois representation of a lattice does; the other one, typical of a semantic approach to logic, aims at destroying the symmetry, in an analogous way to the representation of a lattice through the canonical direct basis.

The following diagram abstract away how we structured the project, Two groups of task might be recognized, depending on their proximity to the scientific discipline. Goal 2 of Task 3, the development of a library for handling lattices, has a primary role on gluing our team and amalgamating the ongoing works on different tasks.


Projet TRECOLOCOCO


3.2. DESCRIPTION DES TRAVAUX PAR TÂCHE / DESCRIPTION BY TASK

Pour la tâche de coordination (tâche 1), préciser les aspects organisationnels de la proposition de projet et les modalités de coordination.Pour les tâches suivantes, décrire : - les objectifs et éventuels indicateurs de succès.- le responsable et les partenaires impliqués.- le programme détaillé des travaux.- les livrables.- les contributions des partenaires (le « qui fait quoi »).- la description des méthodes et des choix techniques et de la manière dont

les solutions seront apportées.- les risques et les solutions de repli envisagées.

3.2.1 TÂCHE 1 / TASK 1

Coordination.Responsible for the task: Luigi Santocanale.


Projet TRECOLOCOCO


We shall organize three meetings of the project per year. We already mentioned we do not wish to organize more than three meetings, due the teaching duties of the members of the project.

The coordinator of the project shall also try to encourage mutual visits among the members of the project that happen to live in different cities and work on the same task. This is, in his opinion, the most fruitful way to achieve scientific results: being for some time far away from the usual Some visits have already been payed (Caspard-Wehrung, Darnière-Santocanale, Pouzet-Santocanale, Wehrung-Santocanale) and collaborations are alrady undergoing (Olive-Santocanale, Pouzet-Santocanale, Wehrung-Santocanale). Among the two-side collaborations that we wish to encourage more let us mention: Bertet-Brucker, Caspard-Wehrung, Darnière-Wehrung.

The coordinator shall be responsible to offer a computer platform to help and document the state and development of the project. The platform will include:

1. an usual project web homepage, with general information on the project,2. a list of upcoming events of the project, and events related to the project,3. a bibliographical database, subdivided into a general section for the

literature of interest to the project, and another section reserved to the scientific production of the project,

4. an email distribution list for easing communication among the members of the project,

5. an svn respository, to help collaboration among members, in particular redaction of scientific papers, and implementation issues,

6. a demo-page of the library developed by us (see task 3).

Finally, it will be important to promote our researches within France and abroad. To this goal we shall be organisers, during these four years, some of the French meetings of the series “treillis marsaillais, rochalais, montpeillerains, ...”.

We also wish to organise a main conference on the subject, such for example OAL or TACL.

An important point in coordination will be to promote the interactions among the participants that might be said computer scientists and those who have a mathematical background. As a matter of fact, even the following tasks might be subdivided in those who more closely pertain computer science and those pertaining mathematics. We believe then that the development and use of the library and automatic tools (see task 3) is therefore central to glue our project members in one team. On the one hand, the computer scientists shall develop automatic tools for the daily research of mathematicians. The daily work of mathematicians in our group entails many verifications, which difficult to do by hand, could be easily automatized. Thus, in one direction, the computer scientists shall listen to some specific problems of mathematicians. In the opposite direction, such a kind of interaction shall help the mathematicians to the specific sensibility and language of computer scientists. As a by-product, mathematicians will be more receptive to the problems in lattice theory that


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arise from computers nowadays. Also, this will give an opportunity to amply test the code we shall develop.

3.2.2 TÂCHE 2 / TASK 2Finite lattices, lattices from combinatorics and concurrencyResponsible for the task : Luigi Santocanale.Participants: Karell Bertet, Nathalie Caspard, Maurice Pouzet, Luigi Santocanale, Fred Wehrung

Experienced mathematicians have often met, during their scientific career, the weak Bruhat order on permutations. Many of us have a particular interest for this order [Cas00, CBM04, San07b, SaWe10, OS09] and some of us might be considered expert in this area. This shared interest was also main factor in finding out about our scientific proximity. Our common approach to the study of the weak Bruhat order relies on the fact that this order is a lattice; consequently, it puts in foreground the perspective of lattice theory and its specific tools. Among these tools, let us mention once the canonical direct basis of a closure system and the OD-graph, that have intensively been studied by our group [MC97, GW03, BM10, San09a]. Our interest for the order on permutations extend to other collections of objects from combinatorics that also enjoy the lattice structure. Among them, the collections of binary trees having a fixed number of leaves, known as Tamari lattices [Tam62,HT72, CB04]. A weak Bruhat order can also be defined with respect to an arbitrary finite Coxeter group [BB05, CBM04]; it turns out to be a lattice [Bjö84]. Binary trees relative to an arbitrary finite Coxeter group [Rea06] can be defined, and also turn out to form a lattice.

Lattice theory can, in our opinion, bring an alternative and innovating perspective to these traditional fields from combinatorics (and algebraic combinatorics). This approach – to combinatorics using lattice theory – constitutes therefore a promising path which requires to be fully explored. Our team, with her competences in lattice theory, wishes to do that.

Goal 1. Determine the canonical basis (or OD-graph) of the lattices associated to the finite Coxeter groups of type Bn and Dn; determine the canonical basis of the lattices of trees [Rea06] that can be constructed from these groups.

This goal constitutes a first step in exploring the equational theory of these lattices. Indeed, many equational properties of a lattice naturally translate into properties of its canonical basis (see Task 4, which focuses on this correspondence).

We have already approached this problem and identified a main difficulty that has to be solved. The existing knowledge on the OD-graph makes it easy to compute the OD-graph of a lattice L if there exists a surjective homomorphism from a lattice M onto L, and the OD-graph is known. This is, for example, the case of the OD-graph of the Tamari lattices which can be easily deduced from the OD-graphs lattices of permutations. The situation is less understood when L is a sublattice of M and we dispose of the OD-graph of M. This is the case of the


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lattices of type Bn, which are sublattices of lattices of type A2n (lattices of permutations on 2n+1 elements). Therefore, our investigation of the lattices Bn

necessitate a previous study of the way an OD-graph is transformed along lattice monomorphisms. Although some tools to approach this problem exist in the literature, such as distances and norms valued in a lattice [Grä03, Sem05], we do not dispose yet of a complete theoretical account of this problem, capable for example to suggest an algorithm to compute the OD-graph of a sublattice.

Goal 2. Recovering a Coxeter group structure from the lattice structure.

This very ambitious goal was mentioned to us by Claude Barbut, Henry Crapo,1

and Nathalie Caspard [BCC07].

The weak Bruhat order on a finite Coxeter group is the transitive closure of its Cayley graph; we recall that it is a lattice. The following question arises: can we find a list of conditions characterizing – up to isomorphism – lattices arising from a finite Coxeter group? This goal is similar to the previous one, which aims, roughly speaking, at characterizing the equational theory of lattices of some family of Coxeter groups. Yet, this question goes in another direction, as the conditions occurring in this list need not to be expressed in equational logic, they need not be identities. In principle these conditions could live in some stronger logic, such as second order logic. Logicians and universal algebraists may appreciate Goal 2 in a lesser extent; yet the kind of list we are looking for might be much more useful for the standard mathematicians, working either in group theory or in combinatorics.

Goal 3. Combinatorics and concurrency, event structures. Event structures were introduced by Nielsen, Plotkin, and Winskel [NPW81,NW94] as a model for distributed systems. There are usually presented as a set, the set of events, equipped with an order and a binary irreflexive relation, the conflict relation. These structures are example of a duality: they are associated to some special lattices, in the same way as posets are associated to distributive lattices. Long standing questions concern the existence of finite nice labellings for event structures. Here, a labelling of the events is said to be nice if two distinct events with the the same label either are in temporal causality relation or they are not the initial occurences of actions in conflict. The most striking one was asked in [ABCR94]: is there a function f such that every event structure with no more that n elements, pairwise concurrent or in a minimal conflict, has a nice labelling with at most f(n) colours? The best known result at this moment [PS09], is that the function f, if any, grows faster than every polynomial. The proof makes use of known results on lexicographical power of graphs and fractional coloring. The problem can be formulated in terms of colouring of the Hasse diagram of a chopped lattice. It is our hope that expertises from different sides, notably from order and lattices could shed new light on these question. Our team has already worked on this problem [San08,PS09] and shall also try to understand how the tools developed for the

1 This colleagues has recently retired, this being the only reason not to include them as members of the project.


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nice labellings might be applied for other interesting open problems concerning event structures and concurrency theory [BDR99,Thi02]. Goal 4. From a concept lattice to the canonical basis, and back. Correspondence Galois lattices and OD-graphs.

The following problem is, in our opinion, a central one in the theory of finite lattices: is it possible to establish some sort of dictionary between the two representation theorems that are the foundations for the theory of finite lattices? The problem might be exemplified in many ways, in a pure lattice-theoretical setting as well as in a more applicative setting. It might be asked, for example, whether it is possible to build the OD-graph (and the canonical direct basis) of a lattice, starting from its representation as a concept lattice, i.e. the canonical table T between sup-irreducible elements and inf-irreducible elements. The question has a trivial answer, since it is enough to build the lattice in its entirety, and compute then the OD-graph. Yet such an answer would not be satisfactory, at least from an algorithmic point of view, because the construction of the lattice takes in general exponential spaces compared to the original size of the representations. We modify this question as follows: It is possible to build the OD-graph of a lattice from its Galois representation, in an algorithmically effective way?

An answer to this question requires to considerably deepen our understanding -- both algebraic and combinatorial -- of finite lattices. A first enquiry among experts in lattice theory (e.g. R. Freese) revealed that few results are known which could help to answer the question. The importance of a solution to this problem can well be exemplified in the domain of the knowledge representation and data mining. Although the ontologies described by objects-attributes tables have a rather intuitive philosophical interpretation, the algorithmic exploitation of such ontology is not necessarily easy. The representation of the same ontology by a system of rule seems to be adapted for this purpose; whence the need for disposing of computing the canonical direct basis.

As this problem is rather natural from a mathematical point of view and very few results are available, it might be guessed to be difficult one. In approaching it, we shall try first to solve it for restricted classes of finite lattices, for example the bounded lattices and the semi-distributive ones. For these lattices more structure is available and some tools – we are thinking mainly of Nation duality [FJN95, II.5] – are already available that might be used to approach the problem.

3.2.3 TÂCHE 3 / TASK 3

Finite lattices: knowledge representation, classification, combinatorics , algorithms.

Responsible for task 3 : Karell Bertet.Participants: K. Bertet, F. Brucker, F. Olive, L. Santocanale.


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Lattices for knowledge representation

Formal Concept Analysis [GW99] is an approach to knowledge modelling capable to gather in the last years an increasingly large community, see for example the conferences ICFAC, CLA, etc. Its foundation strongly relies on lattice theory and, particularly, on the first representation theorem. FCA's increasing importance has many reasons. On one side, new scientific areas have recently began to incorporate computer technologies, on a large scale, through data-bases -- whence a large production of data and the need of handling them. On the other side, the increasing power of computers permits to automatize tasks that might have, in worst case, exponential time-space costs. Among them, the typical problems from FCA, related to the representation of a lattice that could have exponential size with respect to the size of the original data.

In data mining, the problem of classification in naturally related to the notion of concept from FCA. The non supervised classification consists in grouping objects that have close attributes while taking apart those having distant attributes; the supervised classification groups objects having a same label (called class), while it distinguishes those having different labels. The notion of concept – as we said, a maximal set of objects having the same attributes – is then repeatedly used within in applications for classifying data, in a supervised way or not.

It is only in recent time that the second representation theorem has become the object of interest in this community. For relational data-bases, the OD-graph of a lattice and the canonical direct basis are known under the name of minimal functional dependencies. For example, the problem of finding minimal keys, with consequences on optimisation of queries, arises in this context. In data mining, dependences between attributes have been classically represented by associative rules – which can be exact or approximate. The combinatorial explosion of the number of rules, inherent to the generation of the associative rules, and the growing volume of data to handle has made necessary the use of concise representations named bases.

There can be many equivalent bases – where two bases are equivalent if they give rise to the same closure operator and to the same Moore family. Our team has deeply contributed to understand them ([BN04] and [BM10]), by focusing on the canonical direct basis. This is the only system of rules which is direct, meaning that the closure of an arbitrary set can be computed by just one application of the rules to the set, and which at the same time is minimal among the direct systems. In [BM10] it is argued about the equality between the canonical direct basis and five other bases known in the literature. The first one, the canonical direct basis, introduced in 2004 [BN04], was originally defined in an algorithmic way, by repeated transformations of equivalent systems of rules – the transformation corresponding to the computation of resolvent clauses followed by a treatment of non redundancy. The basis associated to the dependency relation is defined in terms of a classical tool from lattice theory, the dependency relation between join-irreducible elements [Mon90]. This latter characterization makes it immediate to relate it to the OD-graph of a lattice introduced in [Nat90]. The canonical iteration-free basis was introduced in 1994


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in [Wil94] and defined by means of the notion of free subset – which easily makes it possible to establish the minimality of the basis. The weak implicational basis, introduced in 1995 [RW95], was defined from minimal transversals of a family, called “co-points”. In this way we established a connection with knowledge spaces, I.e. families of sets containing the empty set and closed under-represented unions (or co-Moore families). Finally, the left minimal basis, defined by rules whose premises have minimal cardinality, made it possible to show the strict relationship with minimal functional dependencies in relational data-bases [Mai83] and proper implications in data mining [TB02]. Proper implications, defined by rules with minimal generator as premises, are extended to approximative generic basis in [TB02] and [GBMS06], thus the equivalence between the exact rules of these generic base and the canonical direct basis.

In many practical data analysis, the goal is either to embed the data into a particular model (a hierarchical structure for instance) or to exhibit the global structure (the whole lattice) in order to interpret the data [BB07]. To address those problems one have to design/use particular kind of lattice, relevant for given field (like dichotomic lattices for image classification or crown-free lattices for phylogenetic use). Those specific models have strong properties (polynomial number of cluster, decomposition scheme, ..) that give powerful tools for analysing real data sets.

The goals we shall tackle within this task are three. The first one is devoted to the study of general properties of lattices which can be used in knowledge representation (non direct bases and binary-lattices). A second focuses on two special models useful in data analysis (dichotomic lattice and crown free lattices). The third will lead to the development of a library which will implement the algorithm of the two.

Goal 1. General lattice theory: non direct bases, binary part of lattices.

Our work has been focusing till now on the canonical direct basis. This basis has been proposed only recently, it is less well-known and exploited. Yet the connection with the OD-graph still need to be completely clarified, see for example [MC97]. The relationships to other bases that are not directed still have to be studied. Let us mention, among them, the canonical basis or Sterm's basis, a reference in fundamental contexts as well as for applications. This is actually the same basis as the minimal basis of conjunctive rules, defined in 1986 [GD86].

When the lattices are constructed from data described by a dissimilarity measure, we have shown that only its binary part carries the information [BB08]: there is a bijection between some valued lattices and dissimilarity measures. Nevertheless, this bijection need to be refined in order to fully characterize these lattices without any reference to a valuation. Such a bijection will lead to a better understanding of the relationships between the two most used data representation in knowledge representation: data matrices and dissimilarity measures.

Goal 2. Two relevant classes of lattices: dichotomic lattices, crown-free lattices


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Coming from the field of data mining techniques, most of the literature on the subject using Galois lattices relies on selection-based strategies, which consists of selecting/choosing the concepts of a lattice which encode the most relevant information from the huge amount of available data [MEP05]. Generally, the classification step is then processed by a classical classifier such as the k-nearest neighbours rule or the Bayesian classifier. Opposed to these selection-based strategies are navigation-based approaches which perform the classification stage by navigating through the complete lattice (similar to the navigation in a classification tree), without applying any selection operation. Our approach, named Navigala [VBO10], proposes an original navigation-based approach for supervised classification, applied in the context of noisy symbol recognition.When defined from binary attributes obtained after a discretization pre-processing step in disjoint intervals, this method constructs specific lattices that we called “dichotomic”. More generally, dichotomic lattices can be recognized as they are co-atomistic with a certain complementation property. Structural links between decision trees and dichotomic lattices are a direct consequence of the complementation property: we prove both that every decision tree is included in the dichotomic lattice and that the dichotomic lattice is the merger of all the decision trees [BVG09]. Co-atomisticity gives raise to a local discretization that improve classification results [GBV09].

We have shown in [BG09, BG10] that crown-free lattices [Riv74] can be seen as analogous of trees within the class of lattices: they are in bijection with strongly chordal graphs and admit an elimination scheme that is similar to the leaves elimination scheme for usual trees in graph theory. In order to be used for real data sets, we have to develop practical tools for manipulating them (using their strong decomposition properties, for instance), producing them (efficient algorithm for constructing them from a real data set described by a data matrix; efficient algorithm have been already designed when the data are described by a dissimilarity measure) and graphically representing them (a promising way of doing it can be to generalize the notion of dendrogram used for hierarchical structures).

Goal 3. Algorithmic aspects of lattice theory, development of a library. The recent use of lattices in applicative contexts of computer science has led to recognize the importance of linking fundamental research and applications within lattice theory. Lattice generation tools are mainly developed in the framework of FCA, where a lattice is obtained as a concept lattice of a context. These tools focuses on context edition, graph edition and graph layout for lattices in different format. An interesting overview of FCA softwares is proposed in [Pri08]. However, these softwares do not integrate tools for exploring the lattice exploiting some finer results from lattice theory - in particular they do not rely on the canonical direct basis and the OD-Graph of a lattice. In this spirit, we aim at identifying the set of algorithmic problems on lattices that might occur in an applicative setting. Results from fundamental research shall be used to develop a set of algorithms, hopefully effective and optimal for handling lattices. These algorithms shall be implemented under the form of a library.


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While a first prototype of this library is already developed – using java and python programming languages -- it is still far away from being usable. The project will give us many opportunities in this respect. We plan to make an intensive testing of the library through our community. Also, the library will be core of some automated tools, whose implementation shall be part of the project, having as objective to accompany, support, and complement the more fundamental research of other tasks. This library will include all the algorithms developed in this task :

1. general lattices decomposing scheme using direct and non-direct bases,

2. approximation algorithm for computing dichotomic lattices and crown free lattices,

3. manipulation tools for exploring the lattice structure,4. graphical representation of general and particular lattices

(dichotomic and crown free ones).

3.2.4 TÂCHE 4 / TASK 4 LATTICES: LOGICAL ASPECTSResponsible for the task: Luigi Santocanale.Participants : Luck Darnière, Luigi Santocanale, Friedrich Wehrung.

Within this task, a first aim is to confront, compare, and join our respective skills in domains which, traditionally considered as distinct areas of logic and mathematics, have important overlaps. These domains include modal logic, intuitionistic logic, substructural logics, lattice theory, and universal algebra. This first exchange will provide us a deeper understanding of these fields and the skill to transfer mathematical tools from a domain to the other – a skill which should ensure serendipity.

A traditional approach in the study of modal and intuitionistic propositional logic is the use of algebra: these logics are considered as algebraic systems. Consequently, studying the models of these systems – Boolean algebras with operators, Heyting algebras, residuated lattices – becomes an important and unavoidable part in our understanding of a logic. This approach, of a semantic nature,2 often reveals its power when providing answers to problems which we would rather qualify of a syntactic nature, when logic is about formal systems, and (decision) procedures. A large theory concerning these algebraic models is major and unavoidable part of modern mathematics. A fundamental result of the theory is Stone's theorem [Sto36]: every Boolean algebra can be represented as the set of clopen subsets of a topological space; and this representation theorem generalizes to a duality of categories. Logicians have generalized this result to some of the other algebras we mentioned: Priestley [Pri70] did it for distributive lattices, and Esakia [Esa74] for Heyting algebras. These theorems laid the foundations for other ideas and results. They allowed to define, in a coherent way, the notion of Kripke model of a logic and to consider restricted classes of models. It became possible to relate tautologies of a logic to first order formulae defining restricted classes of models, what is known now as correspondence theory. Finally, the question of canonicity of the tautologies was tackled: the

2 This approach is certainly more popular abroad than in France. We see here a strong argument to propose this project: the French research has to catch up a delay within this domain.


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results obtained by modal logicians [Sah75, GHV04] answering these questions can be considered as uniform answers to the completeness and decidability problems for many modal logics.

Paradoxically, the scope of these theorems us restricted to distributive lattices and extend to algebraic models of logics satisfying the distributivity law between conjunction and disjunction. The non-distributive lattices do not fit this framework; this is for us quite annoying, as non-distributive lattices have a special role within modern logics -- they are algebraic models of the additive fragment of the linear logic. Presently, no recognized notion of Kripke model for additive linear logic nor a duality theorem for lattices is available. Some ideas in this direction have been proposed [Geh06] and one of them [Gol06] is very close to our own proposal [San09a]. However, at the present stare of art, no proposal has been retained by the scientific community for its value and usefulness. It is a few years researchers have been trying to develop duality theories for lattices. A remark that might be made to these researches if that they have been oriented more from algebraic logic than lattice theory. The skills in both fields of our team shall better balance our research and be a guarantee of success.

In the project, we would like to follow a path which in some way simulates the path already followed in traditional modal logic. We propose next some specific goals.

Goal 1. Duality: from modal logic to lattice theory.

We wish to generalize the duality for finite lattices developed in [San09a] to the class of all lattices, comprising the infinite ones. The following considerations suggest the feasibility of this goal. We recently worked on monotone modal logic ML [SaVe09]. In this logic, the possibility operator of modal logic does not satisfy the distributivity law with respect to disjunction. We can consider monotone modal logic ML(S4), the analogous of the well- known modal logic S4: here non-distributive possibility modal operator is a closure operator. It is very easy to code lattice theory into ML(S4) and, consequently and in principle, to use knowledges and tools developed by modal logicians within lattice theory. The situation is similar to the standard coding of intuitionistic logic into modal logic S4. An easy but important remark is that that a Kripke model for ML(S4) is exactly a direct basis in the sense of [BM10] and an OD-graph as in [San09a].

Such a coding of lattice theory into monotone modal logic – thus the possibility of using ideas and tools well-known by modal logicians within a lattice theoretic framework -- shall be our main tool to achieve the goal of generalizing the duality to all lattices.

Goal 2. Correspondence theory and canonicity for lattices.

A remark in [San09a] is that many identities (equations between terms) in the signature of lattices have first order correspondents in OD-graphs. This means that, for such an identity, a finite lattice satisfies the identity and only if its OD-graph is a model of the correspondent, a first order formula. Not all identities behave that nicely: for example, it is not known whether the Arguesian identity


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(a classical strengthening of the modular identity) has such a correspondent on the OD-graph.

This situation reminds of an important chapter of modal logic: several equations are true in one modal algebra if and only if their dual, a Kripke model, is a model of a first order formula, called the correspondent. The following question arises naturally, for modal logic as for lattices: what is the class of equations which possesses a correspondent in first order logic? It is a difficult question which we can approach by approximating the ideal answer: can we identify classes of equations which admit correspondents, in some reasonable extension of first order logic? In modal logic, an already rather complete theory, at the same time effervescent of developments, answers this type of questions [Sah75].

This theory is strictly intertwined with another theme, canonicity [GHV04] of modal equations. Canonicity allows to prove completeness of modal axiomatisations. A modal equation is said to be canonical if it can be lifted from a modal algebra to its Stone completion. Any modal equation in the so called Sahlqvist fragment is canonical. The Stone completion of a Boolean algebra has been generalized to arbitrary lattices, and is known today as the canonical compact completion [GH01]. The question about canonicity therefore can be asked for lattice identities as well. Known results [GH01] indicate that, if we restrict ourselves to the language of the pure lattice theory (without additional operators) very few equations belong to the Sahlqvist fragment. Researchers are now investigating extensions of the Sahlqvist fragment [CP09, Suz08] which preserve canonicity. These works appear more interesting and promising from a pure lattice theoretic perspective. On the other hand, the question of canonicity of equations, as originally posed by modal logicians, has many generalizations, depending on the lattice completion under consideration. For example, the following problem was raised independently by Friedrich Wehrung: does any lattice in a fixed variety embed into an algebraic spatial lattice in the same variety? This was proved to hold for modular lattices in Herrmann, Pickering, and Roddy [HPR94]. In [SaWe09], we solve this question in the negative, by finding a join-semidistributive lattice that cannot be embedded, within its variety, into any algebraic and spatial lattice. We shall focus on this problem, restricted to particular varieties, during the project, and notice its similarity with the original canonicity problem. This question is also a source of completeness and decidability results for some equational axiomatisations. We exploited it in the case of the theory of n-distributive lattices [SaWe09].

3.2.5 TÂCHE 6 / TASK 6 : ON THE MODEL THEORY OF HEYTING ALGEBRASResponsible for the task : Luck Darnière.Participants: Luck Darnière, Maurice Pouzet, Luigi Santocanale, Friedrich Wehrung

The model theory of Heyting algebras is the arena for one among the most interesting disputes in contemporary logic. We recall that a topos [Joh02a, Joh02b] is a sort of universe of sets whose internal logic is intuitionistic; and that the set of truth values of a topos has the structure of a Heyting algebra. A natural question arises: is every Heyting algebra the set of truth values of a


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topos? This is, according to Peter Johnstone, the most important open problem in topos theory. It is also a very attractive question for us, given its proximity, at least in spirit, with a famous problem in lattice theory: is every distributive algebraic lattice the congruence lattice of some lattice? This problem was solved, in the negative, only recently by Fred Wehrung [Weh07]. A positive answer the topos-theoretical question requires to interpret higher order intuitionistic logic in its propositional fragment. When trying to disprove this, A. Pitts noticed that the first order fragment of intuitionistic logic can well be interpreted in the propositional fragment [Pit92]. It is not clear whether the question is still open, and a sort of scientific quarrel is undergoing. A Georgian logician, Dimitri Pataraia, claimed then the answer was indeed positive [Pat03, Pat07]; yet the scientific community is still waiting for a written document proving this result. P. Johnstone [Joh09] ended up presenting his own partial reformulation of the results announced by Pataraia.

The discussion on the most important open problem of topos theory shows that the research on Heyting algebras is active today more than ever. We do not wish, within this project, to attack this fundamental problem of topos theory - it would require large and specific skills in this domain. Yet the project will give us the occasion to understand closer the difficulties raised by this question. There are many other questions raised by Pitts' works that are in more direct relation with our research. In [GZ02] the authors reinterpret Pitts' result as stating the existence of a model completion of the theory of Heyting algebras.

The reader might wish to consult the usual literature on model completion [Rob77]. The existence of such a model completion is implied by Pitts' result together with the fact that Heyting algebras have the amalgamation property. It implies that any Heyting algebra can be embedded into a “rather special” Heyting algebra, in which any equation of the form p(x) = q(x), where p, q are polynomials of the theory of Heyting algebras, possesses a solution if it is possess a solution in some extension. In particular the class of “rather special” Heyting algebras is elementary, in the sense that it can be axiomatized by first order sentences. A similar result holds for the theory of Boolean algebras: this theory also possesses a model completion, a well-known fact since the origins of model theory. For Boolean algebras, the meaning of the words “rather special” is perfectly clear: these are the atomless Boolean algebras. Yet, atomless Heyting algebras are not the "rather special" ones. The existence of a recursive set of axioms for this model completion was proved in [GZ02], but no intuitive axiomatization, finite or not, is known up today.

We specify next some concrete problems and goals.

Goal 1. Characterize the models of the model completion of the theory of Heyting algebras.

Order-theoretic properties of these models (i.e. of the model completion of the theory of Heyting algebras) are already known [GZ02], for example such a model does not have join-prime nor meet-prime elements. This generalizes the fact that model-complete Boolean algebras have no atoms. Can we find a complete finite list of these properties, as for Boolean algebras? This question


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might be rephrased by asking whether the model completion of Heyting algebras has a finite axiomatisation.

Two axioms – suggested by geometrical and topological considerations – were recently introduced by us [DJ10], which made it possible to prove a model completion result for theories of certain varieties of locally finite Heyting algebras. However, the general case still resists. A possible obstacle to achieve our goal is the absence of concrete models of this theory: we have only general abstract recipes to build them. It is thus a question in itself to discover “concrete” examples of these models.

Let us emphasize that the approaches proposed up to now to study this problem come from very different backgrounds: we already mentioned the work by of A. Pitts [Pit92], who privileged a rather syntactic approach, whereas S. Ghilardi and M. Zawadowski [GZ02] were more oriented towards universal algebra, algebraic logic, and category theory. On the other hand, the techniques of [DJ10] come from model theory applied to algebra, combined with a few real geometry. Our team groups together for the first time scientists coming from all these backgrounds (for example L. Santocanale has a specific knowledge of the works S. Ghilardi [GS03] whereas L. Darnière is a co-author of [DJ10]). The project shall be a unique opportunity to produce new advances in this domain.

Goal 2. Solvability of equations with parameters in a finitely generated free Heyting algebra.

The existence of a model completion uniformly answers the question of the solvability of an equation of the form p(x) = q(x) in extensions given Heyting algebras. Yet, we can still try to characterize the equations which have solutions in a given algebra, without passing to extensions. Obviously this question depends on the chosen algebra, and we shall pay a particular attention to free Heyting algebras; this is because of their status among all Heyting algebras, i.e. their universal nature using a categorical language. For example, we shall test the following conjecture, whose origin comes from fixed point theory: any system of polynomial equations of the form x = p(x) admits a least and a greatest solution in a finitely generated free Heyting algebra. Another related question is to know whether it is sufficient for an equation p(x) = q(x) to possess a solution in a free finitely generated Heyting algebra, to have a solution in the profinite completion of this algebra. The underlying idea is that this completion is at the same time its profinite completion and the metric completion for a certain ultra-metric distance, as showed in [DJ09]. We are interested in this problem as a similar situation is met in many other areas of mathematics, in particular in the study of discrete valued rings. In this context Hensel's lemma, holding in the completion, was the trigger for the most important advances in model theory and in arithmetic. Roughly speaking, this lemma states that if a system of polynomial equations has an approximate solution (under certain conditions of smoothness) then it possesses an exact solution. A similar statement, if it were valid for a freely finitely generated Heyting algebra or in its completion, would open in this domain completely new perspectives. Our experience has shown that the study of this problem requires skills in domains other than model theory, in particular in combinatorics. Kripke


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models, to which modal logicians of our team are used to, could have here, on the border between these two disciplines, an important role to play.

Goal 3. Metric spaces over Heyting algebras. Many results on metric spaces have flourished in the last few years in which the Urysohn space plays a pivotal role, see e.g. [KPT05]. Many of them start with a subset V of the non-negative reals and study the class of metric spaces whose distance values are in V, looking for example for the existence of an Urysohn space. In this respect, the study of ultrametric spaces is almost satisfactory, see [DLPS07, DLPS08, VThe10]. We propose to study metric spaces over Heyting algebras. Fixing a specific Heyting algebra, say H, one may look at the category of metric spaces over H, morphisms being non-expansive mappings. As shown in [JMP86, KP98] this frame is general enough to include posets, graphs, transition systems as well as metric spaces. There are representation results; e.g. the fact that every metric space isometrically embeds into a power of H. But this does not say much. For H finite, even for V finite, deep tools of Ramsey theory are needed to study the category of metric spaces over H [VTS10]. For a better understanding of finite metric spaces we intend to clarify the link between countable homogeneous metric spaces and projective limits of finite metric spaces and particularly profinite Heyting algebras.

3.2.6 TÂCHE 6 / TASK 6 THE VARIETY QUESTION FOR CLASSES OF LATTICESResponsible for the task : Friedrich Wehrung.Participants: Nathalie Caspard, Luigi Santocanale, Friedrich Wehrung

Among the hardest currently open problems in lattice theory, there is a collection of problems, all of which can be loosely formulated in the following fashion: we are given a class C of (finite or infinite) lattices, and we are asking whether C is closed under homomorphic images. Most of the time, the class C is already closed under reduced products (or, in the finite case, finite products) and substructures, so we are asking (in the infinite case) whether C is a variety, and thus refer to the problem above as the variety question for the class C. Furthermore, we are asking, in case the answer to the above question is affirmative, to find an explicit equational basis for that variety.

The most well-known open problem among those is probably the following, originating in Haiman [Haim87, Haim91].

Problem 1. Let C be the class of all linear lattices, that is, the class of all lattices of permuting equivalence relations. Is C a variety? Equivalently, is C closed under homomorphic images?

Most variants of this problem, formulated for related classes of modular lattices, are open as well (and probably very difficult). For example, the variety problem is open for sublattices of complemented modular lattices (lattices of subgroups of abelian groups, normal subgroups of groups, subspaces of vector spaces, and so on). Setting Problem 1 as a main goal of the project would sound quite ambitious, on the other hand it is certainly a good place where to aim---especially in view of Littlewood's quote that ``we should always try to solve


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difficult problems; we may not be able to solve them, but we are almost sure to find something interesting''.

Analogues of Problem 1 have been solved for large classes of non-modular lattices. For example, in Semenova and Wehrung [SeWe04a], the variety problem is solved positively for lattices of order-convex subsets of posets. This result is extended to posets of given finite length in Semenova and Wehrung [SeWe03], and also to products of chains in Semenova and Wehrung [SeWe04b]. In each case, it turns out that the variety is finitely based, and a finite basis of equations is constructed explicitly. It can be observed that in all those cases, no direct proof of the closure of the class under homomorphic images is known.

The lattices considered in the works by Semenova and Wehrung are far from modular---indeed, they are join-semidistributive.

The Tamari lattice, or associahedron, on n letters, is defined in [Tam62] as the set of all bracketings on n letters, endowed with a certain ordering (which turns out to be a lattice). For n greater than or equal to 4, the associahedron on n letters is never modular. Say that a finite lattice is sub-Tamari if it can be embedded, as a lattice, into some Tamari lattice.

We shall now record another open question, first formulated in Grätzer [Grä71].

Problem 2. When is a finite lattice sub-Tamari?

Although Problem 2 looks basically ill-formulated (it is rather an answer than a question), it can be easily reformulated into a mathematical question: for example, is it algorithmically decidable whether a finite lattice is sub-Tamari?At the time Problem 2 was stated, a lattice-theoretical tool that is nowadays common knowledge had not come into existence yet, namely the one of a bounded lattice, originating from McKenzie [McK72]. In particular, a finite lattice is bounded if the join-dependency relation on its join-irreducibles has no cycle, and dually. The five-element non-modular lattice is bounded, but the five-element modular non-distributive lattice is not. It was observed in Urquhart [Ur78] that every Tamari lattice is bounded. As every sublattice of a finite bounded lattice is bounded, it follows that every sub-Tamari lattice is bounded. The evidence for the converse was sufficient for Geyer to ask in [Gey94] the following question:

Problem 3. Is every finite bounded lattice sub-Tamari?

Of course, a positive answer to Problem 3 would imply a positive answer to Problem 2. However, Santocanale and Wehrung prove in [SaWe10] that there are finite bounded lattices that are not sub-Tamari. A full solution to Problem 1 is still missing, although such a solution would be a reasonable goal of the project. The following particular instance of Problem 1 is a possible formulation of the variety question for sub-Tamari lattices, and it is also open.

Problem 4. Is the class of sub-Tamari lattices closed under homomorphic images?


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Among the surprises brought by [SaWe10] is the result that the permutohedron on four letters is not sub-Tamari. So, while every permutohedron is bounded (Caspard [Cas00]) and every associahedron is a retract of a permutohedron, not every permutohedron can be embedded into some associahedron! This makes appear as plausible the conjecture that every finite bounded lattice can be embedded into some permutohedron; however, this is also disproved in [SaWe10]. These results open up a whole new field of research, both in lattice theory and its connection with outside research topics, in particular trying to extend those results to combinatorial lattices such as weak Bruhat orders on finite Coxeter groups. It is unlikely that the intended duration of the project will be sufficient to go much beyond scratching the surface of such a program.

Even if the answer to Problem 4 (or its analogue for permutohedra) is negative, the following problem is still of interest.

Problem 5. Determine the equational theory of all associahedra and of all permutohedra.

Among the lattices mentioned in Task 2, Goal 1, those of type An are the lattices of permutations on n+1 elements. The combinatorial structure of these lattices, in particular their OD-graphs, is now well understood. For these lattices we are ready to tackle a question, more fundamental from a logical perspective: does the variety generated by those lattices have a decidable word problem? Namely, is there an algorithm that makes it possible to recognize the lattice-theoretical equations that hold in all permutohedra from the others? Until recently, we believed that the answer was positive, simply because we did not know whether every finite bounded lattice embeds into some permutohedron; and there is no nontrivial equation satisfied by all finite bounded lattices. Now that we found, in [SaWe10], a finite bounded lattice that cannot be embedded into any permutohedron, the evidence turns the other direction. Note that we also found, in [SaWe10], an infinite collection of non-trivial identities that hold in all the Tamari lattices. Thus, at the present, the goal of characterizing all the equations that hold in every associahedron (resp., every permutohedron) is far from being achieved; yet these recent discoveries open new wide paths, towards solving this problem.

The problems above can appeal to both specialists of finite lattices and infinite lattices: for example, tackling the finite statements could take great benefits from implementations such as the ones projected in Task 3. Conversely, having precise, theoretical problems in mind could substantially contribute bringing structure and speeding up the progress for Task 3.

3.3. CALENDRIER DES TÂCHES, LIVRABLES ET JALONS / TASKS SCHEDULE, DELIVERABLES AND MILESTONES

Présenter sous forme graphique un échéancier des différentes tâches et leurs dépendances (diagramme de Gantt par exemple).


Projet TRECOLOCOCO


Présenter un tableau synthétique de l'ensemble des livrables de la proposition de projet (numéro de tâche, date, intitulé, responsable).Préciser de façon synthétique les jalons scientifiques et/ou techniques, les principaux points de rendez-vous, les points bloquants ou aléas qui risquent de remettre en cause l'aboutissement du projet ainsi que les réunions prévues.

With exception of task 1, coordination, we refrain to impose a precise order and calendar for the other tasks. Our experience suggests that successful l research in mathematics and theoretical computer science is too often propelled by some kind of enthusiasm which is intrinsically difficult to redirect at our will. Forcing a precise order of events would, most probably, slow down undergoing research on other goals that is “riding the wave”. We limit ourself to remark some logical dependencies among tasks and their goals, which most often are inter-dependencies.

Task 4 (Logical aspects) has a natural dependency on Task 5 (Heyting algebras). On the one hand, Goal 2 of Task 5, which requires to master Kripke models of intuitionistic logic to obtain results on Heyting algebras, shall provide a way to deepen our working knowledge on classical duality theories, that we want to export to lattice theory (Task 4, Goal 1). On the other hand, we believe opportune to tackle Task 5 at the beginning of the project, as this shall solidify the collaborations between Darnière and Wehrung, and Darnière and Santocanale, thus laying down a solid ground for the following works.We already mentioned (Paragraph 3.1) that Goal 3 of Task 2 (on computing the canonical basis from a table) is fundamental for many other tasks and goals, where it is a matter of computing the canonical basis (Task 2 Goal 1, Task 3 Goal 4, Task 4 Goal 2, Task 6 Problem 5).

Other tasks might considered to have a mutual dependency. For example, Task 6 settles a list of problems that we want to attack. These problems might be considered as a concrete ground of work and experimentations within lattice theory; they are also meant to suggest the right paths and ideas that have to be explored in relation to Task 4. Here we wish to develop a clean theory, accessible to a community of logicians, of what we do on a concrete ground within Task 6. On the opposite direction, the theory we wish to develop in Task 4 might have a strong impact on Task 6, by clarifying, for example, what are the most effective tools and most promising paths to solve the problems. Goal 1 of Task 2 can be considered to be complementary of Task 6 Problem 5. This problem (axiomatisation of a class of lattices) can be raised in a more generality, the class considered could be the one of all lattices arising from some finite Coxeter group. The knowledge on their canonical basis of is a pre-requisite to such a work.Goal 2 of Task 3 deserves a particular discussion. On one side, the order development of such a development goal might be made more precise. The successful achievement of this goal depends on obtaining the resources we are asking for to realize the goal, a research engineer. We also mentioned that – by means of the development of a library and of some applications to verify properties of lattices – we want to animate the interactions among the participants of the project that have different backgrounds. As a matter of fact,


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most tasks (namely Task 2, Task 3, Task 4, and Task 6) might strongly benefit by interacting with this goal.

We come now to the coordination task and propose a calendar. Let us recall that we preview to group together the participants to the project three times per year.

A consolidation phase shall take place during the first year. During this period each responsible shall present to the other participants – at the meetings of the project – the state of art for the task. A pedagogical effort shall be emphasized, for example by giving such a presentation in the form of a mini-course with some written course notes.

At the second year of the project a second phase shall begin, lasting two years. This phase shall be devoted to promote scientific production, with the goal of increasing our usual productivity and enhancing collaborations. A possible measure of achievement for this phase is to count the number of scientific productions co-signed by members of the project (a graph, which should approximate a complete graph or a graph such that participants to the same Task form a clique).

A last phase (starting at the last year of the project) shall emphasize promotion of the project and of the results. This, in particular, will include organisation of a main scientific event, such as a main international conference in the scientific area of the project. Let us recall that, as part of our demand, we are asking to dispose for an important number of personnel working in research (48 month over the 4 years). This is conform to our main goal of constituting a scientific community. In case part of this personnel shall be a doctoral student, then an important deliverable ant the end of the project shall be a doctoral thesis, which, at the same time will constitute an important criterium of success of the project.

Let us come to detail a calendar from Goal 3 from Task 2 : end of 1st year : implementation of the core of the library: class lattice,

and first algorithms end of 2nd year: expanded library including new algorithms created by us end of 3rd year: prototype including previous deliverable and a first

interface for handling, producing, approximating lattices.

As a first core of the library is already available, the presence of a research ingeneer will be required from the second year.

4. STRATÉGIE DE VALORISATION, DE PROTECTION ET D’EXPLOITATION DES RÉSULTATS / DISSEMINATION AND EXPLOITATION OF RESULTS, INTELLECTUAL PROPERTY

A titre indicatif : 0,5 à 2 pages pour ce chapitre.Parmi les points suivants, présenter la (les) stratégie(s) de valorisation des résultats attendus :


Projet TRECOLOCOCO


- la communication scientifique.- la communication auprès du grand public (un budget spécifique peut être

prévu),- les retombées scientifiques, techniques, industrielles, économiques, …- la place du projet dans la stratégie industrielle des entreprises partenaires

du projet.- autres retombées (brevet, normalisation, information des pouvoirs

publics, ...).- les échéances et la nature des retombées technico- économiques attendues.- l’incidence éventuelle sur l’emploi, la création d’activités nouvelles, …

Présenter les grandes lignes des modes de protection et d’exploitation des résultats.Pour les projets partenariaux organismes de recherche/entreprises, les partenaires devront conclure, sous l’égide du coordinateur du projet, un accord de consortium dans un délai de un an si le projet est retenu pour financement. Pour les projets académiques, l’accord de consortium n’est pas obligatoire mais conseillé.

The scientific results of the project shall be made public using the traditional scheme of our disciplines: they shall be presented at conferences and published consequently, on conference proceedings (for Computer Science) and selected journals (Mathematics and Computer Science).

Moreover, two kind of actions will be taken to the goal of promoting our results.

Firstly, we shall take the charge of organising some recurrent meetings that within larger scientific communities. On the national level, we shall organise during these four years at least two workshops of the series “Treillis”, see treillis.org. At the international level, we shall organise a main conference, such as OAL (Order Algebra and Logic), or TACL (Topology, Algebra, and Categories in Logic – we are already organising this conference for 2011), the workshop on General Algebra ....

In principle we would also like to organise some kind of doctoral school, on the national level. The fact that we do not consider our community yet well established in France – this being a main reason for submitting this project – might be problematic in this respect; a possible solution would be to bring in France some well established international school, such as the Summer School on General Algebra and Ordered Sets.

The implementation goal (Task 2, Goal 3) deserves a special discussion. We do not wish to deposit brevets; instead, we make the source code of our library available open source. Thus the need of making it accessible to a large


Projet TRECOLOCOCO


public will enforce to producing documentation for this code (manuals) and also tie the code to a standard public licence (GPL).

5. DESCRIPTION DU PARTENARIAT / CONSORTIUM DESCRIPTION A titre indicatif : de 2 à 5 pages pour ce chapitre, en fonction du nombre de partenaires.

5.1. DESCRIPTION, ADÉQUATION ET COMPLÉMENTARITÉ DES PARTENAIRES / PARTNERS DESCRIPTION AND RELEVANCE, COMPLEMENTARITY

(Maximum 0,5 page par partenaire)Décrire brièvement chaque partenaire et fournir ici les éléments permettant d’apprécier la qualification des partenaires dans le projet (le « pourquoi qui fait quoi »). Il peut s’agir de réalisations passées, d’indicateurs (publications, brevets), de l’intérêt du partenaire pour le projet, …

Montrer en quoi la constitution de ce consortium donne une synergie par rapport à la simple somme des contributions individuelles. (1 page maximum).

From a formal perspective, the project puts together two partners, the LIF in South of France, and the L3I in North-West of France. If many researchers participating to the project are affiliated to the LIF, the rest of them are affiliated to distinct laboratories located in the North-West of France. The choice of a second main partner thus reflects the need (and will) of easing handling of the project, in particular with respect to administration. Karell Bertet, affiliated to the L3I, has kindly agreed to take on herself the responsibility of coordinating the scientists in the North-West; she is also agreed to coordinate scientific task 3. The project thus comprises members from the following laboratories: LIF, L3I, LAREMA, LMNO, LACL, ICJ. We mentioned LIF and L3I as formal partners, following the suggestion not to trespass the number of 4 partners. We interpreted this suggestion as a demand from the ANR to ease administrative handling of the project and the relationship with the contractors. We could decide to mention LIF as the only partner, but we believed important to have a collaborator on the administrative side from our team. Thus the choice of collecting the researchers into 2 groups: South-Est (LIF, ICJ) and North-West (L3I, LAREMA, LMNO, LACL). This is mainly a geographical criterion which also ends up balancing the subdivision. We followed this criterion considering the constraints arising when organising the project meetings. Yet, from a scientific perspective, we could have divided the members of the project into other kind of groups, almost always having non-empty intersections. For instance, a possible thematic subdivision could have grouped researchers working on finite lattices and combinatorics (tasks 2,3: LIF, ICJ, L3I, LACL) and researchers working on infinite ones (tasks 4,5,6: LIF, LAREMA, LMNO, ICJ).

We already mentioned the complementarity of our team: very few of us had a common scientific path. The skills present in our team are thus naturally wide in scope; the scientific excellence of some of us ensures an important depth as well.


Projet TRECOLOCOCO


Let us mention that many of us have already had a way to test their scientific proximity through collaborations. More in general the team already met in several occasions (workshop Treillis Marseillais, conference TACL 2007, workshops for the PEPS project TRECOLOCOCO) and had a way test the respective affinities. The enthusiasm by which the members welcomed the suggestion of an ANR proposal and the many contributions gathered during writing of the project witness the cooperating strength of our team.

5.2. QUALIFICATION DU COORDINATEUR DE LA PROPOSITION DE PROJET/ QUALIFICATION OF THE PROPOSAL COORDINATOR

(0,5 page maximum)Fournir les éléments permettant de juger la capacité du coordinateur à coordonner le projet.

Luigi Santocanale has a strong publication record in the areas concerned by the project. The value of his research has been recently recognized at the national (and local) level when he was granted a PES (Prime d'Excellence Sciéntifique) for the period from 01/09/2010 to 30/08/2014. He coordinated scientific projects in many occasions. When he had no permanent position yet, he was granted by then European Community a Marie Curie Individual Fellowship with the goal to develop the project “Circular proofs and Automata” in LaBRI (Bordeaux), under the supervision of Pr. André Arnold. Since he got a permanent position, firstly as “Maître de conférences” and then as a Professor at Université de Provence, he was coordinator of three projects:

1. SOAPDC, Structures d'Ordre et Application au Calcul Distribué et Concurrent, ANR project jeunes chercheurs no.

2. MFL, Modal Fixpoint Logics, Egide/Van Gogh project , in collaboration with the Intitute of Logic Language and Computation of Amsterdam, whose partnaire coordinator was Yde Venema,

3. TRECOLOCOCO, a PEPS (Projet Exploratoire Pluridisciplinaire) project funded last year by the INSMI, with the goal to put in movement this yeas ANR project.

He was also participant to many other projects (CHOCO, TAGADA, GEOCAL).His implication into promotion of research might be appreciated also considering the number of scientific events he organized: 3 workshops at CIRM (Centre International de Rencontres Mathématiques) of which one was part of the project SOAPDC and the other of the project TRECOLOCOCO, 1 workshop in Amsterdam as part of the MFL project, and the organisation of the workshop Fixed Points in Computer Science 2010, as a satellite workshop to CSL and MFCS 2010. He is now organizing the conference Topology, Algebra, and Categories in Logic, which shall take place in Marseilles from 26 to 20 july 2011.

5.3. QUALIFICATION, RÔLE ET IMPLICATION DES PARTICIPANTS / QUALIFICATION AND CONTRIBUTION OF EACH PARTNER

(2 pages maximum)


Projet TRECOLOCOCO


Pour chaque partenaire, remplir le tableau ci-dessous qui précisera la qualification, les activités principales et les compétences propres de chaque participant :

Partenaire / partner Nom / Name

Prénom / First name

Emploi actuel / Position

Discipline / Field of research

Personne.mois* / Person.month

Rôle/Responsabilité dans la proposition de projet/ Contribution to

the proposal

4 lignes max

Exemple LATIFI Fatima Professeur Caractérisation des facteurs de transcription recombinants en système in vitro …

Coordinateur/responsable (affiliated to Partner 1)

Santocanale

Luigi Professeur Computer Science

38,4 Coordinator.

Responsible for Task 2.

Participating to tasks 2,4,5,6.

Willling to cooperate on Task 3, time permitting.

Autres membres

Partner 1 Brucker François Professeur Computer Science

24 Participant to Task 3.

Study of lattices arising from knowledge representation and classification.

Partner 1 Fratani Séverine Maître de Conférences

Computer Science

9,6 Participant to Tasks 2 and 3.

Partner 1 Olive Fréderic Maître de Conférences

Computer Science

24 Participant to Tasks 2 and 3.

Enumeration problems related to queries.

Partner 1 Pouzet Maurice Professeur (emeritus)

Mathematics

24 Participants to Tasks 2 and 5.

Event structures, Heyting algebras.

Partner 1 Morin Remi Professeur Computer Science

9,6 Participant to task 2.

Event structures.

Partner 2 : L3I

Coordinator Bertet Karell Maître de Conférences

Computer Science

24 Responsible for Task 3 and, in particular for Goal 3 of this task.

Partner 2 Caspard Nathalie Maître de Conférences

Computer Science

28,8 Participant to Tasks 2 and 6.

Tamari lattices, lattices of permutations, lattices of finite Coxeter groups.

Partner 2 Darnière Luck Maître de Conférences

Mathematics

24 Responsiblle for task 5.

Participants to tasks 4,5,6.

Heyting algebras.

Partner 2 Wehrung Friedrich Research Mathemat 28,8 Responsible for Task 6.


Projet TRECOLOCOCO


Director ics Participants to tasks 2,4,5,6.

Lattice theory, algebraic logic, universal algebra.

* à renseigner par rapport à la durée totale du projet

Pour les personnes impliquées à plus de 25% de leur temps dans le projet, placer une biographie en annexe 7.2. Pour les personnes impliquées dans d’autres projets, remplir le tableau en annexe 7.3.

6. JUSTIFICATION SCIENTIFIQUE DES MOYENS DEMANDÉS / SCIENTIFIC JUSTIFICATION OF REQUESTED RESSOURCES

Présenter ici la justification scientifique et technique des moyens demandés par partenaire tel que rempli en ligne sur le site de soumission.Chaque partenaire justifiera les moyens qu’il demande en distinguant les différents postes de dépenses.(Maximum 2 pages par partenaire)

6.1. PARTENAIRE 1 / PARTNER 1 : LIF

1. Équipement / Equipment

None.

2. Personnel / StaffJustifier le personnel à financer par l’ANR sur le projet.

Two post-doctoral students (12 months each). These two students shall be supervised by François Brucker and Luigi Santocanale (and, possibly, co-supervised by other members of the project). Taking into consideration that Brucker and Santocanale are both professors at University (Ecole Centrale and Université de Provence, respectively) and the many responsibilities of this kind of job, these two students shall strongly increase the productivity of the LIF partner. Also, these post-doctoral students shall be a mean to achieve our goal of constituting a scientific community,

One research engineer (12 months). The research engineer shall be a mean to achieve Goal 3, Task 3. He shall be directed by François Brucker and, possibly, co-supervised by other members of the project. We preview the possibility of using part of this amount to hire Master students to work on implementations issues of the project, within some “stage” at the level of Master 2.


Projet TRECOLOCOCO


Estimated cost of staff: 140410 euros

3. Prestation de service externe / SubcontractingNone.

4. Missions / Travel

Missions due to the annual TRECOLOCOCO meetings: we shall organise 2 of them yearly. Our past experience with the PEPS Trecolococo project has suggested a budget of 2000 euros for meeting. Thus we need 16000 euros.

Mutual visits: we want to incite mutual visits among members of the project. We preview 86 days of visits and 2000 euros for related travels. Considering the cost per day being of 80 euros, the sum per year is of 8880 euros.Thus we need 35800 euros.

Visits from some colleague exterior to the project: we want to able to host some colleague from outside the project (mostly some colleague from abroad) for short periods. We preview 45 days of such visits and 1200 euros for related travels. Considering the cost per day being of 80 euros, the sum per year is of 4800 euros.Thus we need 19200 euros.

Travels to conferences: we wish to be able to present our reserches at national and international conferences. We preview 9 conferences per year. We consider the total cost of attending a conference 1300 euros. We need 11700 euros per year.Thus we need 46800 euros.

Organisation of a conference: we wish to organise a conference at the end of the 4th year, and to this goal to pay 10 invited speakers. We count the cost of an invited speaker 1000 euros.Thus we need 10000 euros.

Total on the 4 years: 124800 euros.

5. Dépenses justifiées sur une procédure de facturation interne / Costs justified by internal invoicies

None.

6. Autres dépenses de fonctionnement / Other expenses

Organisation of national meetings “Treillis” and of a doctoral school.To this goal we estimate a need of 5000 euros.


Projet TRECOLOCOCO


Small material: we plan to buy 6 personal computers, whose unitary cost is estimated at 3000 euros. Thus we need 18000 euros.

Total of other expenses: 23000 euros.

6.2. PARTENAIRE 2 / PARTNER 2 : L3I

7. Personnel / Staff

Two post-doctoral students (12 months each). These two students shall be supervised by Karell Bertet and Friedrich Wehrung.

For Karell Bertet, some remark similar to those for Brucker and Santocanale is valild: she is MdC, and considering the many responsibilities she's involved into, this student shall strongly increase the productivity of the L2I partner; moreover, this post-doctoral student shall be a mean to achieve our goal of constituting a scientific community,As far as Friedrich Wehrung is concerned, the coordinator strongly insisted on him directing a post-doctoral student, simply to give the opportunity to a young researcher to develop research in collaboration with this scientist.

One research engineer (12 months). The research engineer shall be a mean to achieve Goal 3, Task 3. He shall be directed by Karell Bertet and, possibly, co-supervised by other member of the project.

Estimated cost of staff: 140410 euros

8. Missions / Travel

Missions due to the annual TRECOLOCOCO meetings: we shall organise 1 of them yearly. Our past experience with the PEPS Trecolococo project has suggested a budget of 2000 euros for meeting. Thus we need 8000 euros.

Mutual visits: we want to incite mutual visits among members of the project. We preview 75 days of visits and 2200 euros for related travels. Considering the cost per day being of 80 euros, the sum per year is of 8200 euros.Thus we need 32800 euros.

Visits from some colleague exterior to the project: we want to able to host some colleague from outside the project (mostly some colleague from abroad) for short periods. We preview 33 days of such visits and 2200 euros for related travels. Considering the cost per day being of 80 euros, the sum per year is of 4400 euros.Thus we need 17760 euros.

Travels to conferences: we wish to be able to present our researches at national and international conferences. We preview 5 conferences per year. We consider


Projet TRECOLOCOCO


the total cost of attending a conference 1300 euros. We need 6500 euros per year.Thus we need 26000 euros.

Total cost of travels on the 4 years: 84560 euros.

9. Autres dépenses de fonctionnement / Other expenses

Small material: we plan to buy 4 personal computers, whose unitary cost is estimated at 3000 euros. Thus we need 12000 euros.

7. ANNEXES / ANNEXESLes annexes ne sont pas comptabilisées dans la limite des 40 pages à respecter.

7.1. RÉFÉRENCES BIBLIOGRAPHIQUES / REFERENCES

Inclure la liste des références bibliographiques utilisées dans la partie « Etat de l’art ».

[AM06] Marcelo Aguiar and Swapneel Mahajan. Coxeter groups and Hopf algebras, volume 23 of Fields Institute Monographs. American Mathematical Society, Providence, RI, 2006. With a foreword by Nantel Bergeron.

[ABCR94] M. R. Assous, V. Bouchitté, C. Charretton, B. Rozoy, Finite labelling problem in event structures, Theor. Comput. Sci. 123 (1) (1994) 9—19.

[BB05] Anders Björner and Francesco Brenti. Combinatorics of Coxeter groups, volume 231 of Graduate Texts in Mathematics. Springer, New York, 2005.

[BB07] F. Brucker and J.-P. Barthélemy. Eléments de classification. Hermes, Londres, 2007.

[BB08] J.-P. Barthélemy and F. Brucker. Binary clustering. Discrete Applied Mathematics, 156 :1237-- 1250, 2008.

[BCC07] Claude Barbut, Nathalie Caspard, and Henry Crapo. Les arrangements de lignes des groupes de coxeter. Exposé à l'occasion de la recontre Treillis Marsaillais, CIRM, Marseille, April 2007.

[BDR99] E. Badouel, P. Darondeau, J.-C. Raoult, Context-free event domains are recognizable, Inf. Comput. 149 (2) (1999) 134–172.

[BDK97] Felipe Bracho, Manfred Droste, and Dietrich Kuske. Representation of computations in concurrent automata by dependence orders. Theor. Comput. Sci., 174(1-2) :67--96, 1997.


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[BG09] F. Brucker, A. Gély, 2009, "Parsimonious cluster Systems", Advances in Data Analysis and Classification, 3, 189-204.

[BG10] F. Brucker, A. Gély, 2010,"Crown free Lattices and their Related Graphs", Order, accepted.

[Bjö84] Anders Björner. Orderings of Coxeter groups. In Combinatorics and algebra (Boulder, Colo., 1983), volume 34 of Contemp. Math., pages 175--195. Amer. Math. Soc., Providence, RI, 1984.

[BM70a] Marc Barbut and Bernard Monjardet. Ordre et classification : algèbre et combinatoire. Tome I. Librairie Hachette, Paris, 1970. Méthodes Mathématiques des Sciences de l'Homme, Collection Hachette Université.

[BM70b] Marc Barbut and Bernard Monjardet. Ordre et classification : algèbre et combinatoire. Tome II. Librairie Hachette, Paris, 1970. Méthodes Mathématiques des Sciences de l'Homme, Collection Hachette Université.

[BM10] K. Bertet and B. Monjardet. The multiple facets of the canonical direct unit implicational basis. Theoretical Computer Science, 2010. À paraître, available at http://halshs. archives-ouvertes.fr/halshs-00308798.

[BN04] Karell Bertet and Mirabelle Nebut. Effcient algorithms on the Moore family associated to an implicational system. Discrete Math. Theor. Comput. Sci., 6(2) :315--338 (electronic), 2004.

[Bru08] F. Brucker. Classifications en classes empiétantes, modèles et algorithmes. Habilitation à diriger des recherches, Université Paul Verlaine de Metz, 2008.

[Cas00] Nathalie Caspard. The lattice of permutations is bounded. Internat. J. Algebra Comput., 10(4) :481--489, 2000.

[CB04] Nathalie Caspard and Claude Barbut. Tamari lattices are bounded : a new proof. Technical Report TR-2004-03, LACL, Université Paris XII, 2004.

[CBM04] Nathalie Caspard, Claude Barbut, and Michel Morvan. Cayley lattices of finite Coxeter groups are bounded. Adv. in Appl. Math., 33(1) :71--94, 2004.

[CLM07] Nathalie Caspard, Bruno Leclerc, and Bernard Monjardet. Ensembles ordonnés finis : concepts, résultats et usages Mathématiques & Applications (Berlin) [Mathematics & Applications, volume 60]. Springer, Berlin, 2007.

[CP09] Willem Conradie and Alessandra Palmigiano. Expanding sahlqvist correspondence for lattice- based logics : the inductive fragment. talk at the conference TACL09, 2009.

[DF99] Jean-Paul Doignon and Jean-Claude Falmagne. Knowledge spaces. Springer-Verlag, Berlin, 1999.


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[DJ09] Luck Darnière and Markus Junker. Codimension and pseudometric in co-Heyting algebras, 2009. To appear in Algebra Universalis.

[DJ10] Luck Darnière and Markus Junker. Model-completion of varieties of co-heyting algebras. Prépublications mathématiques d'Angers 298, january 2010. http ://math.univ-angers.fr.

[DLPS07] C. Delhomme, C. Laflamme, M. Pouzet, N. Sauer. Divisibility of countable metric spaces, European J. of Combinatorics, 28 (2007) 1746-1769.

[DLPS08] C. Delhomme, C.Laflamme, M.Pouzet, N. Sauer. Indivisible ultrametric spaces, Topology and its Applications, 155, (2008)1462-1478. Special Issue: Workshop on the Urysohn space.

[DP02] B. A. Davey and H. A. Priestley. Introduction to lattices and order. Cambridge University Press, New York, second edition, 2002.

[DR95] V. Diekert and G. Rozenberg. The Book of Traces. World Scientific, Singapore, 1995.

[Esa74] L. L. Esakia. Topological Kripke models. Dokl. Akad. Nauk SSSR, 214 :298--301, 1974.

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[GBMS06] G. Gasmi, S. Ben-Yahia, E. Mephu-Nguifo, and Y. Slimani. IGB : une nouvelle base générique informative des règles d'association. I3, Sciences du Traitement de l'Information, 2006.

[GD86] J.L. Guigues and V. Duquenne. Familles minimales d'implications informatives résultant d'un tableau de données binaires. Mathématiques et Sciences Humaines, 95 :5--18, 1986.

[GW99] Bernhard Ganter and Rudolf Wille. Formal concept analysis. Springer-Verlag, Berlin, 1999. Mathematical foundations, Translated from the 1996 German original by Cornelia Franzke.

[Geh06] Mai Gehrke. Generalized Kripke frames. Studia Logica, 84(2) :241--275, 2006.

[GH01] Mai Gehrke and John Harding. Bounded lattice expansions. J. Algebra, 238(1) :345--371, 2001.

[Gey94] W. Geyer. On Tamari lattices, Discrete Math. 133 (1994), 99—122.

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[GHV04] Robert Goldblatt, Ian Hodkinson, and Yde Venema. Erd®s graphs resolve Fine's canonicity problem. Bull. Symbolic Logic, 10(2) :186--208, 2004.

[Gol06] Robert Goldblatt. A Kripke-Joyal semantics for noncommutative logic in quantales. In Advances in modal logic. Vol. 6, pages 209--225. Coll. Publ., London, 2006.

[GZ02] Silvio Ghilardi and Marek Zawadowski. Sheaves, games, and model completions, volume 14 of Trends in Logic, Studia Logica Library. Kluwer Academic Publishers, Dordrecht, 2002. A categorical approach to nonclassical propositional logics.

[GV06] Valentin Goranko and Dimiter Vakarelov. Elementary canonical formulae : extending Sahlqvist's theorem. Ann. Pure Appl. Logic, 141(1-2) :180--217, 2006.

[Grä71] George Grätzer, Lattice Theory. First Concepts and Distributive Lattices, W. H. Freeman and Co., San Francisco, Calif., 1971. xv+212 p.

[Grä03] George Grätzer. General lattice theory. Birkhäuser Verlag, Basel, 2003. With appendices by B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. T. Schmidt, S. E. Schmidt, F. Wehrung and R. Wille,

[GS62] George Grätzer and E. T. Schmidt. On congruence lattices of lattices. Acta Math. Acad. Sci. Hungar., 13 :179--185, 1962.

[GW03] George Grätzer and Friedrich Wehrung. On the number of join-irreducibles in a congruence representation of a finite distributive lattice. Algebra Universalis, 49(2) : 165--178, 2003. Dedicated to the memory of Gian-Carlo Rota.

[GD86] J. L. Guigues and V. Duquenne. Familles minimales d'implications informatives résultant d'un tableau de données binaires. Math. Sci. Humaines, (95) :5--18, 83, 1986.

[Haim87] Mark D. Haiman. Arguesian lattices which are not linear, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 121—123.

[Haim91] Mark D. Haiman. Arguesian lattices which are not type-1, Algebra Universalis 28 (1991), 128—137.

[Ham09] Tarek Hamrouni. Mining concise representations of frequent patterns through conjunctive and disjunctive search spaces. PhD thesis, University of Tunis El Manar, 2009.

[Har92] Gerd Hartung. A topological representation of lattices. Algebra Universalis, 29(2) :273--299, 1992.


Projet TRECOLOCOCO


[HD97] C. Hartonas and J. M. Dunn. Stone duality for lattices. Algebra Universalis, 37(3) :391--401, 1997.

[HPR94] C. Herrmann, D. Pickering, and M. Roddy. A geometric description of modular lattices, Algebra Universalis 31 (1994), 365—396.

[HM00] Jean-François Husson and Rémi Morin. On recognizable stable trace languages. In Jerzy Tiuryn, editor, FoSSaCS, volume 1784 of Lecture Notes in Computer Science, pages 177--191. Springer, 2000.

[HT72] Samuel Huang and Dov Tamari. Problems of associativity : A simple proof for the lattice property of systems ordered by a semi-associative law. J. Combinatorial Theory Ser. A, 13 :7--13, 1972.

[JM09] Peter Jipsen and M. Andrew Moshier. Topological duality and lattice expansions part i : A topological construction of canonical extensions. talk given at TACL 2009., July 2009.

[JMP86] E. Jawhari, D. Misane, M. Pouzet, Retracts: graphs and ordered sets from the metric point of view, in Combinatorics and ordered sets, I.RIVAL Ed., Contemporary Math. , Vol 57 , (1986), p.175-226, AMS Pub.

[Joh02a] Peter T. Johnstone. Sketches of an elephant : a topos theory compendium. Vol. 1, volume 43 of Oxford Logic Guides. The Clarendon Press Oxford University Press, New York, 2002. 26

[Joh02b] Peter T. Johnstone. Sketches of an elephant : a topos theory compendium. Vol. 2, volume 44 of Oxford Logic Guides. The Clarendon Press Oxford University Press, Oxford, 2002.

[Joh09] Peter T. Johnstone. High order cylindric Heyting algebras. Exposé à l'occasion de la conférence CT2009, Le Cap, July 2009.

[KB10] E. MephuNguifo K. Bertet, T. Hamrouni. A new algorithm computing minimal generators. En cours de rédaction, 2010.

[KM02] Dietrich Kuske and Rémi Morin. Pomsets for local trace languages. Journal of Automata, Languages and Combinatorics, 7(2) :187--224, 2002.

[KP98] M. Kabil, M.Pouzet, Injective envelope of graphs and transition systems, Discrete Math., 192(1998), 145--186

[KPT05] A.S. Kechris, V.G. Pestov and S. Todorcevic, Fraissé limits, Ramsey theory, and topological dynamics of automorphism groups. Geom. Funct. Anal. 15:106–189, 2005.

[LR02] Jean-Louis Loday and María O. Ronco. Order structure on the algebra of permutations and of planar binary trees. J. Algebraic Combin., 15(3) :253--270, 2002.


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[Mai83] David Maier. The theory of relational databases. Computer Software Engineering Series. Computer Science Press, Rockville, MD, 1983.

[McK72] Ralph McKenzie. Equational bases and nonmodular lattice varieties.Trans. Amer. Math. Soc., 174 :1--43, 1972.

[Mon90] B. Monjardet. Arrowian characterizations of latticial federation consensus functions. Math. Social Sci., 20(1) :51--71, 1990.

[MC97] Bernard Monjardet and Nathalie Caspard. On a dependence relation in finite lattices. Discrete Math., 165/166 :497--505, 1997. Graphs and combinatorics (Marseille, 1995).

[Mor05] Rémi Morin. Concurrent automata vs. asynchronous systems. In Joanna Jedrzejowicz and Andrzej Szepietowski, editors, MFCS, volume 3618 of Lecture Notes in Computer Science, pages 686--698. Springer, 2005.

[Nat90] J. B. Nation. An approach to lattice varieties of finite height. Algebra Universalis, 27(4) :521--543, 1990.

[NPW81] Mogens Nielsen, Gordon D. Plotkin, and Glynn Winskel. Petri nets, event structures and domains, part i. Theor. Comput. Sci., 13 :85--108, 1981.

[NW95] M. Nielsen, G. Winskel, Models for concurrency, in: Handbook of logic in computer science, Vol. 4 of Handbook. Log. Comput. Sci., Oxford Univ. Press, New York, 1995, pp. 1--148.

[OS09] Frédéric Olive and Luigi Santocanale. Lattices of biconvex sets. Notes privés, 2009.

[Pat03] Dimitri Pataraia. The high order cylindric algebras and topoi. Exposé à l'occasion de la conférence TANCL03, Tblissi, July 2003.

[Pat07] Dimitri Pataraia. Interpretation of higher order propositional quantification in some linear orders. Exposé à l'occasion de la conférence TANCL07, Oxford, August 2007.

[Pet73] C. A. Petri. Concepts of net theory. In MFCS, pages 137--146, 1973.

[Pig09] Adrian Pigors. Categories of generalized frames. talk given at TACL 2009., July 2009.

[Pit92] Andrew M. Pitts. On an interpretation of second-order quantification in first-order intuitionistic propositional logic. J. Symbolic Logic, 57(1) :33--52, 1992.

[Pri70] H. A. Priestley. Representation of distributive lattices by means of ordered stone spaces. Bull. London Math. Soc., 2 :186--190, 1970.


Projet TRECOLOCOCO


[Pri08] U. Priss. Fca software interoperability. In Sixth International Conference on Concept Lattices and their Applications (CLA’08), pages 193–205, Olomouc, Czech Republic, October 21-23 2008.

[PS09] Maurice Pouzet and Luigi Santocanale. On a conjecture by Assous et al. En redaction, August 2009.

[Rea06] Nathan Reading. Cambrian lattices. Adv. Math., 205(2) :313--353, 2006.

[Riv74] Ivan Rival. Lattices with doubly irreducible elements. Canadian Mathematical Bulletin, 17(1) :91-- 95, 1974.

[Rob77] A. Robinson. Complete theories. North-Holland Publishing Co., Amsterdam, second edition, 1977.

[RT91] Brigitte Rozoy and P. S. Thiagarajan. Event structures and trace monoids.Theor. Comput. Sci., 91(2) :285--313, 1991.

[RW95] A. Rush and R. Wille. Knowledge spaces and formal concept analysis. In H.H. Bock and W. Polasek, editors, Data Analysis and Information Systems, pages 427--436, Berlin, 1995. Springer Verlag. 27

[Sah75] Henrik Sahlqvist. Completeness and correspondence in the first and second order semantics for modal logic. In Proceedings of the Third Scandinavian Logic Symposium (Univ. Uppsala, Uppsala, 1973), pages 110--143. Stud. Logic Found. Math., Vol. 82, Amsterdam, 1975. North-Holland.

[San07a] Luigi Santocanale. A nice labelling for tree-like event structures of degree 3. In Luís Caires and Vasco Thudichum Vasconcelos, editors, CONCUR 2007, volume 4703 ofLecture Notes in Computer Science, pages 151--165. Springer, 2007. Proceedings of the 18th International Conference on Concurrency Theory, Lisbon, Portugal, September 3-8, 2007.

[San07b] Luigi Santocanale. On the join dependency relation in multinomial lattices. Order, 24(3) :155--179, 2007.

[San08] Luigi Santocanale. A nice labelling for tree-like event structures of degree 3. To appear in a special issue of the journal Information and Computation dedicated to conference CONCUR 2007, May 2008.

[San09a] Luigi Santocanale. A duality for finite lattices. Preprint available at http://hal. archives-ouvertes.fr/hal-00432113, 2009.

[San09b] Luigi Santocanale. A duality for finite lattices. talk given at TACL 2009., July 2009.

[SaVe09] Luigi Santocanale and Yde Venema. Notes on monotone modal logic. Notes privés, 2009.


Projet TRECOLOCOCO


[SaWe09] Luigi Santocanale and Friedrich Wehrung. Varieties of lattices with geometric descriptions. Notes privées, 2010.

[SaWe10] Luigi Santocanale and Friedrich Wehrung. Vegetable gardens. Notes privées, 2010.

[Sem05] M. V. Semenova. On lattices that are embeddable into lattices of suborders. Algebra and Logic, 44(4) :270--285, 2005.

[SeWe03] M. V. Semenova and F. Wehrung. Sublattices of lattices of order-convex sets, II. Posets of finite length, Internat. J. Algebra Comput. 13, no. 5 (2003), 543—564.

[SeWe04a] M. V. Semenova and F. Wehrung. Sublattices of lattices of order-convex sets, I. The main representation theorem, J. Algebra 277, no. 2 (2004), 825—860.

[SeWe04b] M. V. Semenova and F. Wehrung. Sublattices of lattices of order-convex sets, III. The case of totally ordered sets, Internat. J. Algebra Comput. 14, no. 3 (2004), 357—387.

[Sto36] M. H. Stone. The theory of representations for Boolean algebras. Trans. Amer. Math. Soc., 40(1) : 37--111, 1936.

[Suz08] Tomoyuki Suzuki. Canonicity results of substructural and lattice-based logics. accepted by The Review of Symbolic Logic, 2008.

[Tam62] D. Tamari. The algebra of bracketings and their enumeration, Nieuw Arch. Wisk. (3) 10 (1962), 131--146.

[TB02] R. Taouil and Y. Bastide. Computing proper implications. In 9th International Conference on Conceptual Structures, Stanford, USA, 2002.

[Thi02] P. S. Thiagarajan. Regular event structures and finite petri nets : A conjecture. In Wilfried Brauer, Hartmut Ehrig, Juhani Karhumäki, and Arto Salomaa, editors, Formal and Natural Computing, volume 2300 of Lecture Notes in Computer Science, pages 244--256. Springer, 2002.

[Ur78] A. Urquhart, A topological representation theory for lattices, Algebra Universalis 8 (1978), 45--58.

[VBO10] M. Visani, K. Bertet and J-M. Ogier. Navigala: an Original Symbol Classifier Based on Navigation through a Galois Lattice. International Journal on Pattern Recognition and Artificial Intelligence (IJPRAI), accepté en Novembre 2010, à paraître en 2011.

[VTS10] Nguyen Van Thé, Lionel and Sauer, Norbert. "The Urysohn sphere is oscillation stable" GAFA (Geometric and Functional Analysis) (to appear)


Projet TRECOLOCOCO


[VThe10] Lionel Nguyen Van Thé, Structural Ramsey theory of metric spaces and topological dynamics of isometry groups, Memoirs of the Amer. Math. Soc., 968 (206), 155 pages, 2010. arXiv

[Weh07] Friedrich Wehrung. A solution of Dilworth's congruence lattice problem. Adv. Math., 216(2): 610--625, 2007.

[Wil94] M. Wild. A theory of finite closure spaces based on implications. Advances in Mathematics, 108(1): 118-139, 1994.

7.2. BIOGRAPHIES / CV, RESUME

(1 page maximum par personne)Pour chacune des personnes dont l’implication dans le projet est supérieure à 25% de son temps sur la totalité du projet (c'est-à-dire une moyenne de 3 personnes.mois par année de projet), une biographie d’une page maximum est souhaitée. Elle comportera :- Nom, prénom, âge, cursus, situation actuelle- Autres expériences professionnelles- Liste des cinq publications (ou brevets) les plus significatives des cinq

dernières années, nombre de publications dans les revues internationales ou actes de congrès à comité de lecture.

- Prix, distinctions


Projet TRECOLOCOCO


Karell Bertet

Age : 38 ans,Situation actuelle : Maître de ConférenceEtablissement : Laboratoire L3I, Université de La Rochelle

Cursus : 1998 : Doctorat d'Algorithmique - Sur quelques aspects structurels et

algorithmiques des treillis - LIAFA (ex-LITP), Université Paris 7 1995 : DEA d'Informatique fondamentale - Université Paris 7 1994 : Maitrise d'Informatique - Université Bordeaux I - Mention : Bien. 1993 : Licence d'Informatique - Université Bordeaux I - Mention : A.

Bien. 1992 : DUT d'Informatique - Université Bordeaux I.

Expérience professionnelle 2009 : Congé pour Recherches ou Conversions Thématiques depuis Septembre 1999 : Maître de conférence à l'université de La

Rochelle. Janvier 1999 : Qualification CNU section 27. Septembre 1998 - Août 1999 : Attachée Temporaire à l'Enseignement

et à la Recherche (50%), Université de Paris 7. Octobre 1995 - Août 1998 : Monitrice, Université Paris 7. Octobre 1995 - Août 1998 : Allocataire de recherche au LIAFA (ex-

LITP), Université Paris 7.

5 publications les plus significatives : [1] M. Visani, K. Bertet and J-M. Ogier. Navigala: an Original Symbol Classifier Based on Navigation through a Galois Lattice. International Journal on Pattern Recognition and Artificial Intelligence (IJPRAI), accepté en Novembre 2010, à paraître en 2011.[2] K. Bertet and B. Monjardet. The multiple facets of the canonical direct basis. Theoretical Computer Science. 411(22-24) : 2155-2166, ISNN 0304-3975, May 2010[3] K. Bertet, M. Visani, N. Girard. Treillis dichotomiques et arbres de décision. Traitement du Signal, numéro spécial, volume 26, numéro 5, p 407-416, 2009.[4] K. Bertet and M. Nebut. Efficient algorithms on the family associated to an implicationnal system. Discrete Mathematical and Theoretical Computer Science. 6:315-338, 2004.[5] K. Bertet and N. Caspard. Doubling convex sets in lattices: characterisations and recognition algorithms. Order, 19(2):181--207, Septembre 2002

Nombre total de publications dans revues ou actes de congrès à comité de lecture (parues où à paraître): 5 revues internationales + 3 revues nationales + 5 lncs + 19 conférences


Projet TRECOLOCOCO


Francois Brucker

Age : 35 ansSituation actuelle : professeur, Ecole Centrale de Marseille.

Formation, cursus : HDR Informatique, 2008.Doctorat mention mathématiques et informatique, EHESS (Ecoles des

Hautes Etudes en Sciences Sociales), mention très honorable et les félicitations du Jury, 2001

DEA MIASH (mathématiques, informatique et applications aux sciences de l'homme) cohabilité par Paris IV, l'EHESS, Paris V, Paris I et l'ENST Bretagne, mention très-bien.

Diplôme d'ingénieur de l'École Nationale Supérieure des Télécommunications de Bretagne (ENST Bretagne), 1998.

Responsabilités actuelles : Professeur des Universités, Ecole Centrale de Marseille,Membre du comité de rédaction de la revue Mathématiques et sciences

humaines

Rattachement scientifique: LIF

Expériences professionnelles passées : 2007 - 2009 : Maître de conférences à l'université Paul Verlaine de Metz, 2002 - 2007 : enseignant chercheur à Télécom Bretagne, 2001 : ATER à l'université Louis Pasteur de Strasbourg.

Principales Publications : (2006 - 2010) :

[1] F. Brucker, A. Gély, 2010, "Crown free Lattices and their Related Graphs", Order, accepted.[2] F. Brucker, A. Gély, 2009, "Parsimonious cluster Systems", Advances in Data Analysis and Classification, 3, 189-204.[3] J.-P. Barthélemy, F. Brucker, 2008, ``Binary Clustering'', Journal of Discrete Applied Mathematics, 156, 1237-1250.[4] F. Brucker et J.-P. Barthélemy, 2007, Éléments de Classification, Hermes, Londres, 438p.[5] F. Brucker, 2006, ``Sub-dominant Theory in Numerical Taxonomy'', Discrete Applied Mathematics, 154, 1085--1099.

Nombre de publications revue internationales et congrès avec comité de lectures : 11


Projet TRECOLOCOCO


Nathalie Caspard

Née le 22 août 1972.Situation actuelle : MCF en informatique, Université Paris XII, affiliée au LACL

Cursus: De 1990 à 1994 : DEUG, Licence et Maîtrise MASS (Paris 1 et Paris 7). 1995 : DEA MIASH (Mathématiques, Informatique et Applications aux

Sciences de l'Homme), Paris 1-Paris 5-EHESS, sous la direction de Bernard Monjardet .

Juillet 1995-août 1998 : Allocataire de recherche en informatique à Paris 1. Décembre 1998 : Doctorat en informatique à Paris 1. Titre de la thèse : "Etude structurelle et algorithmique de classes de treillis obtenus par duplications". Jury composé de Bernard Monjardet (directeur), Michel Habib, Daniel Krob, Jens Gustedt, Maurice Pouzet, Jean-Pierre Barthélemy.

1998-2000 : ATER en informatique à l'université Paris 1.. Depuis Septembre 2000 : MCF en informatique au LACL (ESIAG, UPEC).

Co-responsable du diplôme L3 MIAGE.

5 publications les plus significatives:

[1] N. Caspard, C. Le Conte de Poly-Barbut et M. Morvan, "Cayley lattices of finite Coxeter groups are bounded", Advances in Applied Mathematics, 33(1), 71-94, 2004.

[2] N. Caspard, B. Leclerc et B. Monjardet, "Les ensembles ordonnés finis : concepts, résultats et usages", Springer, 2007, 340 pages. Lien : http://www.springer.com/math/algebra/book/978-3-540-73755-1

[3] N. Caspard, B. Leclerc et B. Monjardet, "Finite ordered sets : concepts, results and uses", à paraître chez Cambridge University Press, 2011.

[4] G. Bordalo, N. Caspard et B. Monjardet, "Going down in (semi)lattices of Moore families and convex geometries", Czechoslovak Mathematical Journal, 59(1), 249-271, 2009.

[5] N. Caspard, "Des chaînes et des antichaînes dans les ensembles ordonnés finis", Mathematics and Social Science 190(2), 19-40, 2010.

Nombre total de publications dans revues ou actes de congrès à comité de lecture (parues où à paraître): 12 articles parus ou acceptés dans des revues à comité de lecture, 6 contributions à des actes de conférences ou congrès à comité de lecture. Je compte 2 ouvrages (le même, rédigé d'abord en français, puis en anglais).


Projet TRECOLOCOCO


Luck Darnière

Né le 10 avril 1969 à Abidjan, Côte d'Ivoire.

Cursus:

Doctorat en Mathématiques à l'université de Rennes en janvier 1998. Maître de Conférences en Mathématiques à l'université d'Angers depuis

septembre 1998.

Je compte 4 articles parus ou acceptés dans des revues à comité de lecture. Je compte 5 contributions à des actes de conférences ou congrès à comité de lecture. Je compte 3 preprint.

Publications récentes :

[1] Luck Darnière and Markus Junker. On Bellissima's construction of the finitely generated free Heyting algebras, and beyond. Submitted to Archive for Mathematical Logic.

[2] Luck Darnière and Markus Junker. Model-completion of varieties of co-heyting algebras. 298, january 2010. http://math.univ-angers.fr.

[3] Luck Darnière and Markus Junker. Codimension and pseudometric in co-Heyting algebras, 2009. To appear in Algebra Universalis.

[4] Luck Darnière. Model-completion of scaled lattices. arXiv math 0606792, June 2006. Prépublications mathématiques d'Angers


Projet TRECOLOCOCO


Frederic Olive

Né le 16 juillet 1964 à Paris.

Cursus : Depuis Septembre 1998 : MCF en informatique au LIF (université de

Provence). 1997/98 : ATER en informatique au Labri (Bordeaux 1). 1996/97 : ATER en mathématiques à l’université de Cergy-Pontoise. Juillet 1996 : thèse de doctorat de l’université de Caen - mention très hon-

orable et félicitations du jury. Titre : "Caractérisation des problèmes NP : robustesse et normalisation ". Jury : Patrick Cégielski, Georg Gottlob, Etienne Grandjean (directeur), Serge Grigorieff, Jean-Jacques Hébrard, James F. Lynch, Jean-Eric Pin, Denis Richard.

1992/96 : allocataire de recherche en informatique à l’université de Caen. 1992/95 : moniteur en mathématiques à l’université Paris 7.

5 articles parus ou acceptés dans des revues à comité de lecture, 3 contributions à des actes de conférences ou congrès à comité de lecture.

5 publications les plus significatives :

[1] G. Bagan, A. Durand, E. Grandjean, F. Olive. Computing the j-th solution of a first-order query, RAIRO, vol. 42(1), pp. 147-164, 2008.

[2] A. Durand, F. Olive. First-order queries over one unary function, LNCS, Proc. Annual Conference of the EACSL, vol. 4207, pp. 334-348, 2006

[3] E. Grandjean, F. Olive, Graph properties checkable in linear time in the number of vertices, Journal of Computer and System Sciences, 68, pp. 546-597, 2004

[4] B. Courcelle, F. Olive, Une axiomatisation au premier ordre des arrangements de pseudodroites, Annales de l’Institut Fourier, T. 9, fascicule 3, pp. 883-903, 1999.

[5] E. Grandjean, F. Olive, Monadic Logical Definability of nondeterministic linear time, Journal of Computational Complexity, vol. 7, pp. 54-98, 1998.


Projet TRECOLOCOCO


Maurice Pouzet

Nom/prenom : POUZET MauriceDate et lieu de naissance : 19 Octobre 1945 à Brive la Gaillarde (Corrèze).Divorcé, 4 enfants, 5 petits-enfants.

Cursus :

Etudes Doctorat de 3ième cycle: “Sur certaines algèbres préordonnées par divisibilité”; Université Claude-Bernard, 13 Avril 1970. Doctorat d’état: “Sur la théorie des relations”, Université Claude-Bernard, 23 Janvier 1978, Jury: J.Braconnier, G.Choquet, E.Corominas, R.Fraissé, E.Specker.

Carrière : Assistant Université Claude-Bernard Lyon 1, 1er Octobre 1969; Maître-Assistant, 1er Octobre 1971. Professeur de 2ième classe: 1er Février 1983. Professeur de 1ère classe, 1er Janvier 1987. Retraité le 18 Octobre 2008.

Situation actuelle : Professeur émérite Université Claude-Bernard, 19 Octobre 2008–Adjunct-Professor, The University of Calgary, July 1, 2008–. Membre de l’ICJ (Institut Camille Jordan) de l’Université Claude-Bernard.

Mon activité porte sur les Mathématiques Discrètes et leur interaction avec des sciences appliqués comme l’Informatique. Les recherches que je mène portent plus précisément sur l’ordre et son intervention dans d’autres domaines comme la combinatoire des structures finies (et, plus récemment, la modélisation des calculs, l’analyse ordinale des donnés). Le bilan consiste en 112 articles publiés, 19 directions ou co-direction de thèses soutenues (dont 2 thèses d’ état), 2 en cours et 3 habilitations; 7 rencontres internationales organisés, seul ou en collaboration depuis 1982; coédition des actes de 3 rencontres; membre du comit ́ de rédaction d’une revue.

CINQ PUBLICATIONS SIGNIFICATIVES

[1] C.DELHOMME, C.LAFLAMME, M.POUZET, N.SAUER. Divisibility of countable metric spaces, European J. of Combinatorics, 28 (2007) 17461769.

[2] M.POUZET, When is the orbit algebra of a group an integral domain ? Proof of a conjecture of P.J.Cameron. RAIRO-Theor. Inf. Appl. 42(2008)83-103.

[3] M.COUCEIRO, M.POUZET, On a quasi-ordering on Boolean functions. Theoret. Comput. Sci. 396 (2008), 71-87.

[4] J.DAMMAK, G.LOPEZ, M.POUZET, M.SIKADDOUR, Hypomorphy of graphs up tocomplementation, J. Combin. Theory Ser.B 99 (2009) 84-96.

[5] Y.BOUDABBOUS, M.POUZET, The morphology of infinite tournaments. Application to the growth of their profile. European J. of combinatorics. 31 (2010) 419-676


Projet TRECOLOCOCO



Projet TRECOLOCOCO


Luigi Santocanale

Né le 13 novembre 1967 à Milan, Italie.Situation actuelle : Professeur en Informatique à l'université de Provence Affiliation : LIF

Cursus: Doctorat en Mathématiques à l'université du Québec à Montréal en Avril

2000. Titre de la thèse : Sur les mu-treillis libres. Directeur de thèse : André Joyal, Jury : A. Blass, G. Winskel, C. Reutenauer, L. Bélair, A. Joyal.

Maître de Conférences en Informatique à l'université de Provence depuis septembre 2003.

HDR, université de Provence en decembre 2008. Titre : Structures algébriques et d'ordre en logique et concurrence. Jury :

Professeur en Informatique à l'université de Provence depuis septembre 2009.

Publications les plus significatives (2006-2010):

[1] L. Santocanale, Y. Venema, Completeness for flat modal fixpoint logics, Annals of Pure and Applied Logic 162 (1) (2010) 55-82.[2] L. Santocanale, A nice labelling for tree-like event structures of degree 3, Information and Computation 208 (3) (2010) 652-665, a special issue dedicated to conference CONCUR 2007.[3] W. Belkhir, L. Santocanale, The variable hierarchy for the games μ-calculus, Annals of Pure and Applied Logic 161 (5) (2010) 690-707. [4] L. Santocanale, Derived semidistributive lattices, Algebra Universalis 63 (2) (2010) 101-130.[5] J. Robin B. Cockett and Luigi Santocanale. On the word problem for ΣΠ-categories, and the properties of two-way communication. In Erich Grädel and Reinhard Kahle, editors, CSL 2009, volume 5771 of LNCS, pages 194--208. Springer, 2009.

Je compte 12 articles parus dans des revues internationales à comité de lecture, 15 contributions à des actes de conférences internationales à comité de lecture.

Prix et distinctions: Prime d'excellence sciéntifique du 01/09/2010 au 30/08/2014. Responsable des projets SOAPDC (Programme ANR JC 2005), Modal

Fixpoint Logics (Van Gogh 2008), Trecolococo (PEPS 2010). Participation, en 2010, aux jurys de thèse de Pierre Clairambault (Paris),

Andrea Montoli (Milan), Gaelle Fontaine (Amsterdam). Direction de la thèse de Walid Bellhkir (2005-2008).


Projet TRECOLOCOCO


Friedrich Wehrung

Né le 8 juin 1961.

Cursus : Elève à l'E.N.S. Ulm de 1981 à 1985. Agrégation de mathématiques en 1983. DEA de mathématiques pures en 1983, sous la direction de J. L. Krivine

(titre ` Forcing; mesure et catégorie dans le modèle de Levy-Solovay''). Assistant Normalien (=AMN) à l'Université de Caen de 1985 à 1987. Service d'enseignement, niveaux allant du DEUG à l'Agrégation. Doctorat de l'Université de Caen en 1987, titre ``Non absoluité de

l'injection élémentaire associée à un ultrafiltre complet'', jury composé de P. Dehornoy (directeur), M. Foreman, J. L. Krivine, A. Louveau, et J. Stern.

Teaching assistant à Ohio State University de Juillet 1987 à Juillet 1988. Chargé de recherche au CNRS de novembre 1988 à juillet 2005.

Promotion CR 1 en novembre 1992. Promotion DR 2 en juillet 2005. Habilitation, titre ``Aspects algébriques et ensemblistes des monoïdes

positivement ordonnés'', présentée le 21 septembre 1992, jury composé de P. Dehornoy, M. Foreman, J. L. Krivine, M. Pouzet, J. P. Ressayre, M. P. Schützenberger, R.M. Shortt.

DR2 (CNRS) depuis juillet 2005.

5 publications les plus significatives:

[1] (avec K. R. Goodearl), The Complete Dimension Theory of Partially Ordered Systems with Equivalence and Orthogonality, Memoirs of the American Mathematical Society, Vol. 176, no.~831 (July 2005), viii+117~p.

[2] Von Neumann coordinatization is not first-order, Journal of Mathematical Logic 6, no. 1 (2006), 1—24.

[3] (with K.R. Goodearl and E. Pardo), Semilattices of groups and inductive limits of Cuntz algebras, Journal für die Reine und Angewandte Mathematik 588 (2005), 1—25.

[4] A solution to Dilworth's Congruence Lattice Problem, Advances in Mathematics 216, no. 2 (2007), 610—625.

[5] Large semilattices of breadth three, Fundamenta Mathematicae 210, no. 1 (2010), 1--21.

Nombre total de publications dans revues ou actes de congrès à comité de lecture (parues où à paraître): 77.

7.3. IMPLICATION DES PERSONNES DANS D’AUTRES CONTRATS / STAFF INVOLVMENT IN OTHER CONTRACTS

(Un tableau par partenaire)


Projet TRECOLOCOCO


Mentionner, pour chacune des personnes, leur implication dans d’autres projets en cours (Contrats publics et privés, soit au sein de programmes de l’ANR, soit auprès d’organismes, de fondations, à l’Union Européenne, etc…) que ce soit comme coordinateur ou comme partenaire. Pour chacun, donner le nom de l’appel à projets, le titre du projet et le nom du coordinateur.

Part.Nom de la personne

participant au projet / name

Personne. Mois /

PM

Intitulé de l’appel à projets, source de

financement, montant attribué / Project name,

financing institution, grant allocated

Titre du projet :

Project title

Nom du coordinateur /

coordinator name

Date début &

Date fin / Start and end dates

N° 1 Santocanale 7,2 Programme Blanc 2007

Curry-Howard

and Concurrency Theory (CHOCO)

Thomas Ehrhard

From 01/11/07

to 30/04/11

N°1 Morin 7,2 Programme Blanc 2007

Curry-Howard

and Concurrency Theory (CHOCO)

Thomas Ehrhard

From 01/11/07

to 30/04/11

N°1 Olive 24 Programme Blanc 2007

Algorithms and

Complexity for

Answer Enumerati

on (ENUM)

Arnaud Durand

From 01/10/07

to 30/09/11

N°1 Fratani 24 Programme jeunes

chercheuses et

jeunes chercheurs

2009

Etude et Conceptio

n de Systèmes

avec Perturbatio

ns

(ECSPER)

Pierre-Alain Reynier

From 01/10/09

to 30/09/13

Members of partner 2, L3I are not, at the present time, members of any other ANR project.


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