Yusuke Hayashi
email: hayashi@ei.sanken.osaka-u.ac.jp

Jacqueline Bourdeau

Riichiro Mizoguchi

Ontological Modeling Approach to Blending Theories for Instructional and Learning Design

Keywords: Instructional design, Learning design, Ontology, Theory-aware, Design support

This paper proposes a modeling framework for learning and instructional design from the viewpoint of ontological engineering. One of the characteristics of this framework is a theory/paradigm-independent ontology for modeling learning/instruction. This paper discusses how our modeling framework with the theory/paradigm-independent ontology contributes to modeling learning and instruction from a comprehensive viewpoint of various educational theories.

Introduction

Sharing and reuse of information about not only resources but also their structure such as design intention have been brought to public attention as recent development of IT standards in the areas of learning, education, and training. Representative specifications include IEEE LTSC Learning Object Metadata (LOM) standard [START_REF]IEEE LTSC, The Learning Object Metadata standard[END_REF] and IMS Learning Design (LD) specification [START_REF]IMS Learning Design. Version 1.0 Final Specification[END_REF]. These bring about global benefits of exchange format for sharing and reuse of information about learning contents.

However, a problem still remains: how to build "good" design for education using the standards. Of course, considerable achievement has been made in instructional and learning sciences. This being said, even though some educational theories (learning, instructional and instructional design theories) prescribe optimal/desirable methods of learning and instruction, many of the theories are not sufficiently articulated for the use of designers. Such theories allow for diverse interpretation and therefore may be difficult to use in practice.

One of the reasons why these problems come up is that the description of educational theories is made in natural language, using different terminology. As Reigeluth points out, although many theories prescribe the same method for the same situation, these are described in different terminology [START_REF] Reigeluth | Instructional-design: What is it and why is it?[END_REF]. This leads to a diversity of theories that are all open to interpretation. Even for experts, it is sometimes difficult to appropriately use theories while having a clear understanding of the similarities and differences between them. Consequently, we must first establish a common basis for understanding the theories at a conceptual level, along with organized concepts and vocabulary.

The goal of this study is to build an engineering infrastructure that enables designers to select instructional and learning theories and blend them into an instructional/learning design. This paper thus proposes a modeling framework for learning and instructional design from the view point of ontological engineering [START_REF] Mizoguchi | Using Ontological Engineering to Overcome Common AI-ED Problems[END_REF], based on the results of previous research in this respect [START_REF] Bourdeau | Collaborative Ontological Engineering of Instructional Design Knowledge for an ITS[END_REF][2] [START_REF] Psyche | Making Learning Design Standards Work with an Ontology of Educational Theories[END_REF] . In addition, the purpose of this study is not to expose a scientifically valid basis for organizing educational theories nor to reconstruct them on this basis, but rather to find an engineering approximation that allows the building of an engineering infrastructure that enables practitioners to utilize educational theories.

This paper is organized as follows. The next section presents our perspective on modeling of learning and instruction and proposes our resulting modeling framework for them. Section 2 illustrates an example of a model of theoretical knowledge for education based on this modeling framework. Section 3 discusses an application of this framework to design support system for learning scenarios or activities. The final section concludes this paper and states the issues to be examined further.

Ontological modeling framework for instructional/learning design

This study adapts the ontological modeling framework of functional design knowledge in the world of manufacturing industry by Kitamura et al. [START_REF] Kitamura | Deployment of an Ontological Framework of Functional Design Knowledge[END_REF] to model educational theories. The main characteristic of this framework is independent conceptualization of what is achieved and how to achieve the change in the target things. The former is organized as ontologies of functionality, and the latter is as ways of function achievement. The function of artifacts (devices) is modeled as hierarchical structure of component functions linked with ways of function achievement. This model is called "function decomposition tree". Kitamura et al. have confirmed the effectiveness of the framework in: (1) affording a better understanding of the functionality of devices, (2) facilitating the sharing of design rationales of devices, (3) supporting the improvement of functionalities, and so on. Although the domain is different from educational knowledge, we believe that it is applicable to the systematization of theoretical knowledge for instructional/learning design.

For the educational domain, we have developed an ontology of learning, instruction and instructional design (L/I/ID) [START_REF] Bourdeau | Collaborative Ontological Engineering of Instructional Design Knowledge for an ITS[END_REF][2] [START_REF] Psyche | Making Learning Design Standards Work with an Ontology of Educational Theories[END_REF] . This study proposes an ontological modeling framework for education based both on the L/I/ID ontology and Kitamura's framework [START_REF] Hayashi | Ontological Support for a Theory-Eclectic Approach to Instructional and Learning Design[END_REF]. It aims at facilitating [START_REF] Bourdeau | Collaborative Ontological Engineering of Instructional Design Knowledge for an ITS[END_REF] the sharing of a model of instruction/learning and (2) the application of educational theories appropriately to the modeling. IMS Learning Design (LD) specification [START_REF]IMS Learning Design. Version 1.0 Final Specification[END_REF] has recently focused on the sharing aspect, and we believe that our approach will contribute to make IMS LD specification work with educational theories.

Roughly speaking, the L/I/ID ontology is composed of five major concepts: concepts related to Common, Learning, Instructional and Instructional design worlds and Educational event in the learning and the instructional worlds [START_REF] Hayashi | Ontological Support for a Theory-Eclectic Approach to Instructional and Learning Design[END_REF]. This ontology is developed in an effort to model learning and instruction in any paradigm -Behaviorism, Cognitivism and Constructivism -. Ertmer and Newby [START_REF] Ertmer | Behaviorism, cognitivism, constructivism: Comparing critical features from an instructional design perspective[END_REF] assert that although each of the paradigms has many unique features, each describes the same phenomena (learning). In a similar line of thought, one of the notable features of this ontology is the conceptualization of relation between learning and instruction. The meaning of an instruction is defined by the change of learner state as achieved or by the intended result of learning. However, instruction is defined independently of the change of learner state. That is because an instructional action may have different effects on the change depending on the context. Independence is the key to allow for a variety of combinations of instructional actions and effects and to compare strategies provided by educational theories.

The Learning/instruction process is modeled based on our ontological modeling framework for education. A unit of learning/instruction is described as I_L event, which is a sub-class of Educational event. An I_L event is defined as a combination of instructional action and change of learner state caused by a learning action. This definition allows to describe the relation between instruction and learning in a learning/instruction process. The

Way of achievement

Macro -I_L event I_L event decomposition

is-achieved by

Micro -I_L events

(A) (B) (C)

Fig. 1 An example of I_L event decomposition

whole process is modeled as a function decomposition tree. In our framework, this model is a hierarchical structure of I_L event to achieve a certain change of a learner state. Thus, it is called "I_L event decomposition tree". In an I_L event decomposition tree, an upper (macro) I_L event is connected with the lower (micro) ones by way of achievement of change of a learner state (referred to just as "Way" hereafter). For example, consider a situation where an instructor wants a learner to be ready to learn and it is necessary that the learner is motivated into the learning and then that he recalls prior learning. Fig. 1 illustrates this case as I_L event decomposition. An oval node represents an I_L event. Fig. 1 (A) indicates the macro-I_L event, in which "Preparing the learner for learning" is the instructional action and "Being ready to learn" is the change of learner state. A conceivable process to achieve this is to motivate the learner (Fig. 1 (B)) and then to stimulate recall of prior learning (Fig. 1 (C)). The former instructional action brings about the learner state "Motivated" (Fig. 1 (B)), and the latter does "Recalling prior learning" (Fig. 1 (C)). A way is a description of relationship of such a decomposition of the required change into the detailed changes and actions to achieve them. A 'Way' has two sorts of interpretations. One, so-called bottom-up manner, is that the sum of the changes of learner state in micro-I_L event promotes the changes of learner state of the macro-I_L event. This manner, which concentrates on states, is descriptive. It describes which outcome is produced by a sequence of changes of learner state. The other, so-called top-down manner, is that an instructional action of a macro-I_L event is decomposed into detailed/concrete instructional actions of micro-I_L events. This manner, which concentrates on action, is prescriptive. It prescribes which sequence of instructional actions is required for performing the intended instructional action.

Following the top-down interpretation, this study proposes a method to systematize theoretical knowledge. The theories prescribe strategies for planning instructional and learning process according to supposed situations. These strategies may superficially vary from theory to theory but, as discussed in the introduction, some of the differences just come from the difference in terminology that each theory uses. We assume that some essentials of theories characterizing themselves are disclosed if their strategies are resolved conceptually.

In our framework we propose modeling learning/instructional strategies as Way using L/I/ID ontology. A set of Ways derived from a theory characterizes the theory from the prescriptive aspect. In addition, modeling strategies on a common basis, which is L/I/ID ontology, is expected to disclose not only characteristics of each theory but also commonality among theories.

We call Ways derived from theories 'Way-knowledge'. Such Ways can be applied to various instructional/learning designs if the supposed situation is matched. This study aims to support designers blending theories into their own instructional/learning design by providing an engineering infrastructure for accumulating and sharing variety of Way-knowledge derived from theories.

An example of comprehensive model of educational theories

In our ontological modeling framework for education with the L/I/ID ontology, educational theories are modeled as Way-knowledge. In this section, we will discuss how Way-knowledge describes educational theories comprehensively.

Fig. 2 shows an example of an I_L event decomposition tree. This tree represents a decomposition of the I_L event, "Facilitate learning / Knowledge state". An oval node represents I_L event. Its label expresses combination of instructional action and change of learner state in the form of "instructional action/change of learner state". The links between I_L events found below and above a square represent a Way. I_L events joined to a square and located below it represent an AND relationship. This means that, according to a Way, an upper I_L event is achieved when all of the lower I_L events are achieved. If more than one Way leads to an I_L event, it represents OR relationship among Ways, that is to say, there are more than one method to achieve the change of learner state. This relationship provides alternatives to make more detailed instructional/learning design for designers. Such an I_L event decomposition tree with OR relationship is called "General I_L event decomposition tree".

The general I_L event decomposition tree shown in Fig. 2 covers the whole process of instruction/learning -from preinstruction to assessment -. The foundation of this process is the five major learning components by Dick et al [START_REF] Dick | The systematic design of instruction[END_REF]. They are a summary of Gagne's nine events of instruction [START_REF] Gagne | Principles of Instructional Design[END_REF]. These theories are basically considered to be based on cognitivism approach but to be somewhat eclectic in view of paradigms -behaviorism, cognitivism and constructivism - [START_REF] Snelbecker | Is instructional theory alive and well ?[END_REF]. This model is intended to be a comprehensive model of instructional/learning theories by giving a shape to Dick's components based on various theories in different paradigms. Now, we take a close look at the decomposition below the I_L event, "Exercise the learner / Absorbed (Fig. 2 (B))", because this decomposition includes both cognitivism and constructivism approaches with OR relationship of Ways. Ways included in this decomposition are based on cognitive approach -Gagne's nine events of instruction [START_REF] Gagne | Principles of Instructional Design[END_REF] and Merrill's component display theory [START_REF] Merrill | Component display theory[END_REF] -and constructivist approach -Collins's cognitive apprenticeship [START_REF] Collins | Cognitive apprenticeship: Teaching the crafts of reading, writing and mathematics[END_REF] and Jonassen's model for designing constructivist learning environments [START_REF] Jonassen | Designing constructivist learning environment[END_REF] -.

The point of selecting principles from multiple theories is the change of learner state. According to Carey [START_REF] Carey | Library skills, information skills, and information literacy: Implications for teaching and learning[END_REF] and Ertmer and Newby [START_REF] Ertmer | Behaviorism, cognitivism, constructivism: Comparing critical features from an instructional design perspective[END_REF], aspects of constructivist approach can be compatible with aspects of prescriptive approach for specified types of learners and learning outcomes [START_REF] Dick | The systematic design of instruction[END_REF]. From this viewpoint, Dick relates constructivist strategy to the five major learning components. In the similar line of these thought, we proposes a modeling method for organizing any theories in terms of what and how their strategies intend to achieve. This study sets up the working hypothesis that the idea of states in the learning process is common while the assumed mechanism of developing knowledge is different for each paradigm 1 . With this consideration, we have organized learner states intended to be shared among different educational theories. "Being ready to learn (Fig. 2 (A))" and "Self-aware (Fig. 2 (D))" are examples of such learning states. Making attempts to organize common concepts and to interpret (supposed) intention of theories as the change of learner state will afford a better understanding and utilization of educational theories.

Let us examine blending of these approaches in more detail. The I_L event "Exercise the learner / Absorbed (Fig. 2 (B))" is decomposed into two sub-I_L events: "Eliciting performance / Absorbed (Fig. 2 (C))" and "Providing informative feedback / Self-aware (Fig. 2 (D))". This decomposition is derived from Gagne's events. To put it briefly in terms of what to be achieved, the former objective is to develop what is learned through practice and the latter one is to become self-aware in order to make the practice more efficient.

These I_L events can be decomposed by both of cognitivist and constructivist approaches. For example, there are two Ways to decompose "Provide informative feedback / Self-aware". Cognitivist approach (Fig. 2 (10)) gives relatively explanatory feedback, e.g. giving correct answer or help. On the other hand, constructivist approach (Fig. 2 [START_REF] Jonassen | Designing constructivist learning environment[END_REF]) gives relatively inquiring feedback, e.g. providing hint or assisting learners to articulate. Constructivist approach also has other characteristics such as authenticity and self-construction of knowledge, though this is not shown in Fig. 2 because of presenting a contrast to explanatory methods. These characteristics can be described in a more detailed model but are too complicated to explain in detail here.

The key point of the blending of these approaches is, as discussed in the previous section, to take particular note of the change of learner state. These theories indeed have differences in the method. However, at the same time, the objective (intended change of learner state) can be described as common in becoming self-aware in this example. From this viewpoint, we set both the cognitivist and constructivist approaches as a Way to decompose "Providing informative feedback / Self-aware (Fig. 2 (D))". In the same manner as this, other cognitivist and constructivist methods are also organized as Way knowledge shown in Fig. 2 (1) -( 13).

Toward a theory-aware instructional/learning design support system

One of the remarkable applications of our ontological modeling framework for education is theory-aware design support system for learning contents [START_REF] Hayashi | Ontological Support for a Theory-Eclectic Approach to Instructional and Learning Design[END_REF]. By "Theory-aware" [START_REF] Mizoguchi | Using Ontological Engineering to Overcome Common AI-ED Problems[END_REF], we mean the capability of information systems which can support the activities of users based on the understanding of relevant theories.

Nkambou et al. [START_REF] Nkambou | CREAM-Tools : An Authoring Environment for Knowledge Engineering in Intelligent Tutoring Systems[END_REF] discuss the benefits of accessing theories and required functionalities of theory-aware ITS authoring environment. The benefits are: 1) make decisions (macro, micro) after reflection and reasoning, 2) communicate about or explain their design decisions, 3) check consistency among design decisions, intra-theory and inter-theories, 4) produce scrutable learning environments, 5) have heuristic knowledge grounded in theoretical knowledge. The required functionalities are that authors can: 1) ask the system what theories apply best to this or that learning situation/goal, 2) ask the system to show examples, 3) ask the system for advice on whether this element of a theory can be combined to an element from another theory, what is the risk associated to doing so, any preferable other solution, etc.

Referring to their argument, this study focuses on the following two points as the requirement for realization of theory-awareness in an instructional/learning design support system: The system can (1) help designers to build theoretically valid model of learning/instruction, and (2) explain the help with the theoretical justification to designers. We now discuss how our ontological modeling framework for education contributes to these two advantages.

As discussed in the previous sections, learning/instructional strategies included in educational theories are described as Way-knowledge in our modeling framework. Way-knowledge is a relationship between macro-I_L event and some micro ones with desired change of learner state (goal) and necessary condition for the change (situation). In addition, Way-knowledge is described with not each theory's own terminology but our L/I/ID ontology. This ontology is designed to accept terminological difference among educational theories and paradigms on a conceptual level as mentioned before. Based on this ontology, Way-knowledge is not only human-readable but also machine-readable. This enables systems to expound educational theories described as Way-knowledge and to interpret models of learning/instruction built by learning/ instructional designers founded on educational theories. These will be quite helpful to learning/instructional designers for in-depth understanding of theories and for communicating about their design decisions and products.

Another characteristic of our modeling framework is that a theory is described as a set of Way-knowledge. In other words, a theory is split into pieces of strategy and each strategy is described as a Way-knowledge. For example, in fig. 2, Way-knowledge (1), ( 2), ( 4), and ( 6) is derived from Gagne and Briggs's theory, and ( 8) and ( 12) derives from Merrill's component display theory. Such modeling of theories helps learning/instructional designers to make a model of learning/instruction from various viewpoints such as those that follow. One viewpoint is that a support system can pick and choose applicable strategies within a theory according to the grain size of process. If a designer wants to decompose complete learning/instructional process by him/herself, Way knowledge (1) is recommended. On the other hand, if he/she wants to do more detailed process, Way knowledge (4) or ( 6) are recommended.

The other is that a support system can also suggest whether an element of a process derived from a theory can be exchanged by an element derived from another theory. Way-knowledge [START_REF]IMS Learning Design. Version 1.0 Final Specification[END_REF] and ( 11) illustrated below the I_L event "Provide informative feedback /Self-aware (Fig. 2 (D))" is a good example to illustrate the possibility of exchange. Such guidance becomes possible with inter-theory comparison in terms of strategies included in theories or derived from them.

Conclusion

We have discussed a modeling framework for learning/instructional design based on ontological engineering. The characteristics of this framework include: 1) a theory/ paradigm-independent ontology for modeling learning/instruction, 2) compatibility between prescriptive and descriptive models derived from educational theories, 3) theory-awareness brought out by an ontological modeling framework.

In this paper, we have concentrated on how the theory/paradigm-independent ontology contributes to modeling learning/instruction from a comprehensive viewpoint of various educational theories. Our modeling framework based on this ontology will be helpful to blend of educational theories as discussed in section 2, and to enhance the quality of support by information systems for learning/instructional design as discussed in section 3.

The following issues to be examined further still remain: 1) investigating the theory/paradigm-independence of the L/I/ID ontology through organizing much more educational theories in our modeling framework, 2) considering the relation between learning objects and abstract design of learning contents described as an I_L event decomposition tree, 3) implementation of a theory-aware design support system for learning contents, and 4) compliance with standards for semantic web and e-learning.

Note that we are not saying all the learning theories share the same learning states.

Acknowledgments

We would like to thank to Dr. Daniele Allard for her helpful comments.


Josianne Basque
email: jbasque@teluq.uquebec.ca

Kim Chi Dao
email: kdao@teluq.uquebec.ca

Participating Virtually in a Scientific Conference: A Collaborative E-Learning Scenario for Authentic Learning in Higher Education

This paper presents a collaborative e-learning scenario inspired by socio-constructivist and situated learning theories which encourage authentic learning. Developed for a graduate distance education course, this scenario requires students to participate virtually in an asynchronous scientific conference. This paper presents the learning scenario, the technological environment developed to implement this scenario, as well as the results of a preliminary investigation which examines students' perceptions and satisfaction towards the scenario and the environment.

Introduction

Over the last few years, online learning has become increasingly popular, not only for distance education universities, but for campus-based universities as well. At the same time, the education sector, at all levels, has been undergoing a shift of paradigm towards socioconstructivist and situated approaches to learning [START_REF] Brown | Situated Cognition And The Culture Of Learning[END_REF][START_REF] Hung | Constructivism and E-Learning: Balancing Between the Individual and Social Levels of Cognition[END_REF][START_REF] Jonnaert | Compétences et socioconstructivisme: Un cadre théorique[END_REF]Jonnaert & Vander Borght, 1999;[START_REF] Lafortune | Accompagnement socioconstructiviste. Pour s'approprier une réforme en éducation[END_REF]Kirshner & Whitson, 1997;[START_REF] Lave | Situated Learning: Legitimate Peripheral Participation[END_REF][START_REF] Orey | The Impact of Situated Cognition: Instructional Design Paradigms in Transition[END_REF]Wilson & Madsen Myers, 1999).

Instructional designers are looking for ways to implement socioconstructivist and situated learning principles in the design of online courses. Computer-supported collaborative learning (CSCL) is one main approach used [START_REF] Dillenbourg | Collaborative Learning: Cognitive and Computational Approaches[END_REF][START_REF] Henri | Apprentissage collaboratif à distance: Pour comprendre et concevoir les environnements d'apprentissage virtuels[END_REF][START_REF] Roberts | Computer-supported collaborative learning in higher education[END_REF]. Authentic learning is another mainstream approach suggested by many authors. For example, [START_REF] Duffy | Constructivism: New Implications for Instructional Technology?[END_REF] propose that students should use tools to perform activities which are similar to those found in their future professional fields. [START_REF] Savery | Problem-Based Learning: An Instructional Model and its Constructivist Framework[END_REF] highlight the importance of creating situations which let the students practice the competencies required by the professional environments in which they will eventually be working. For [START_REF] Reeves | Authentic Activities and Online Learning[END_REF], authentic learning has ten main characteristics: (1) activities matching real-world tasks, (2) ill-defined activities, (3) complex tasks engaging students over a sustained period of time, (4) opportunity to examine the task from different perspectives, using a variety of resources, (5) opportunity to collaborate, (6) opportunity to reflect, (7) interdisciplinary perspectives enabling students to play diverse roles, (8) activities integrated with assessment, (9) creation of polished products and (10) allowance for competing solutions and diversity of outcomes.

In this paper, we present a collaborative and authentic e-learning scenario designed for graduate students. This scenario invites them to participate in an asynchronous virtual scientific conference (VSC1 ). A scientific conference is an event that these future high level researchers and professionals are likely to attend during and after their graduate studies. This scenario is thus compatible with the Legitimate Peripheral Participation Theory proposed by [START_REF] Lave | Situated Learning: Legitimate Peripheral Participation[END_REF], who assumes that "the mastery of knowledge and skill require newcomers to move toward full participation in the sociocultural practices of a community" (p. 29).

To our knowledge, few collaborative online activities reported in the literature use the scientific conference analogy to structure the interactions amongst students. Fjuk & Sorensen (1997) describe what they call "Pedagogical On-line Seminars", which essentially consist in virtual forums moderated by a professor or an expert in a given domain. [START_REF] Clemson | Using Virtual Poster Sessions in a Distance Education Course[END_REF] reports an online course which includes a "virtual poster session", which is a typical activity of a scientific conference; however, it is conducted in a synchronous mode and implies essentially the sharing of files and chat discussions. In our scenario, all the learners' interactions occur in an asynchronous mode and it includes three main events: a virtual poster session, a symposium and a plenary session. This paper describes first the instructional scenario of the VSC and the technological environment developed to implement this scenario within the context of an online course. Then, the results of a preliminary study assessing the students' perceptions and their level of satisfaction regarding both the scenario and the environment of the virtual conference are presented.

The Learning Scenario

The VSC learning scenario was designed and tested in a graduate course, which is offered entirely at a distance at the Tele-universite of Quebec, a French-Canadian distance education university. Up to now, 68 students from five different cohorts have successfully completed this 135-hour course entitled Information Technology and Cognitive Development.

The entire course is structured around the metaphor of a scientific conference. The "Conference program" includes four (learning) activities (see Figure 1): Preparing for the conference; Participating in a poster session; Attending a symposium; and Participating in the plenary session.

Figure 1. The Conference Program

In the first activity, students get acquainted with the conference environment and program and complete the conference "registration" process by presenting themselves to other participants. They also use a questionnaire to activate their prior knowledge related to the topic of the conference and continue to build on this knowledge by reading some introductory documents on the subject. Finally, with other participants, they use the forum tool to discuss how those first readings have already changed their prior knowledge in the domain.

During the second activity, students participate in a virtual poster session where they produce a poster (with PowerPoint) which summarizes the results of two published research papers related to the conference topic. Students can find the papers on their own or they may select them from a list provided in the VSC environment. Each student must then write a comment or formulate a question regarding one of the posters. Finally, the poster's author must reply to these comments and questions.

In the third activity, students participate in a virtual symposium about the effects of ICT on learning and cognitive development. The papers presented at this symposium are, in fact, a collection of published papers reporting different points of view from experts in the field. Thus, students "attend" this symposium and write a text in which they critique or defend one of the issues presented. Each participant must then comment on one participant's text, who, in turn, replies to the comments he or she receives.

In the fourth and final activity, learners participate in a plenary session that takes place within one of the forums of the course. The goal of this activity is to reflect upon and discuss the conclusions they have gathered and the knowledge they have acquired as a result of the interactions which occurred within the forums, during the poster session and finally, during the symposium.

All of the work produced throughout the conference (posters, debates, discussions in the forums, comments and replies) is used for summative assessment of learning. Throughout the course, students are assisted by a tutor whose main tasks consist of moderating the forums, responding to questions submitted by email or posted in the forums and evaluating students' productions.

The Virtual Environment of the Scientific Conference

Unlike certain collaborative learning platforms [START_REF] Faerber | Une métaphore spatiale et des outils intégrés pour des apprentissages coopératifs à distance: ACOLAD[END_REF], the VSC environment is not structured according to a spatial metaphor. It could be described as a functional metaphor, which, according to some authors, is sufficient to induce a sense of immersion (Daele et al., 2000;[START_REF] Jensen | Time, Place, and Identity in Project Work on the Net[END_REF]. [START_REF] Harrison | Re-Place-ing Space: The Roles of Place and Space in Collaborative Systems[END_REF] argue that the critical property of computer-supported collaborative work system "is not rooted in the properties of space at all. Instead, it is rooted in sets of mutually-held, and mutually available, cultural understandings about behaviour and action. In contrast to 'space', we call this a sense of 'place'. Our principle is: 'Space is the opportunity; place is the understood reality'." (p. 67). The two virtual zones created, although called the Poster Session Room (Salle des posters) and the Symposium Room (Salle des conferences), have interfaces that are free of icons which would represent actual, physical rooms. Indeed, these two "rooms" are displayed as electronic tables which include various posting spaces.

Basically, both rooms are organized similarly and they offer the same functionalities. The Poster Session Room is illustrated on Figure 2. Both rooms allow users to post and display their papers and posters, and they also allow users to discuss these documents. Users select an interface button to indicate the action they wish to perform; for example, they can post a file, view a posted document, ask a question or make a comment, reply. In order to ensure that all participants have the opportunity to receive and respond to a question or comment, a "first come, first serve" principle was implemented in the system. The buttons displayed in these interfaces are dynamic, meaning that they appear gradually, according to how the interactions unfold. This permits us to control the actions that each participant can do at one point in the scenario. For example, as long as a posting area remains empty, a single button appears to all participants: the "Speak" button (Intervenir). Once a participant has manifested his intention to "speak" in one of the posting spaces in a room, his name appears in the "Contributors" column (Intervenants) and the "Posting'' button (Déposer) appears in the column "Possible actions" of this specific participant only. Later, once the "contributor" participant has posted his or her work, the ''View'' button (Voir) appears automatically in the same column, inviting all participants to view the contributor's posting. When a participant posts a comment or question concerning the poster, his or her name appears in the "Authors of questions'' column. Finally, when the contributor responds, a check mark appears in the ''Answered Question'' column. By selecting the ''View'' button, a single window then appears where all participants can view (1) the contributor's poster or symposium paper, (2) the comment or question formulated by another participant, and (3) the contributor's reply.

Figure 2. The Poster Session Room

Once a participant posts a file, it can be modified within a certain period fixed by the professor (in this case, six days). When this delay has expired, the system prevents users from performing any type of modification. This feature ensures that students who prepare a response to another's work do not encounter a different document at the moment they are about to post their response.

Data Collection

In order to assess students' satisfaction and perceptions towards the learning scenario and the VSC environment, two sources of data were used: 1. Questionnaire. At the end of the course, students are asked to volunteer their opinions by filling out a questionnaire anonymously. This questionnaire is designed to assess the course in general and its overall structure. It also includes a section which specifically addresses the VSC (15 questions). Up to now, 22 students have filled out and returned this questionnaire. 2. Spontaneous comments regarding the VSC, posted by students in the forums.

Results from a descriptive preliminary analysis are presented below.

Preliminary Results

A preliminary analysis of the data collected provides a portrait of the students' perceptions regarding (1) the similarities between the VSC and other types of conferences, (2) the technological aspects of the VSC, and (3) the effect the VSC had, if any, on their motivation and learning.

Perceived Similarities between the VSC and Other Types of Conferences

More than a half of the 22 learners who filled out the questionnaire indicated they had previously attended a ''live'' scientific conference (N=13). Eight respondents said they had previously attended synchronous virtual conferences, and 9 that they had participated in another asynchronous virtual conference aside from this one. These last results are startling as the virtual conference analogy is not widely used in e-learning. We hypothesize that students who offered an affirmative answer associated this strategy to various online collaborative activities, such as discussions in forums or chat, for example.

Thirtheen respondents feel that the dynamic of the discussions within the VSC is ''very" or "moderately" similar to those occurring at live conferences (see Table 1). As one participant indicated, ''It's not exactly the same as attending a live conference; however, discussions through the question and answer approach provide a close replica of these events.'' Ten respondents indicated that they encountered the same dynamics at the VSC as those typically found in synchronous virtual conferences.

Statements

Very Moderately A little Not at all Don't know TOTAL

The VSC reproduces the dynamic of the discussions in live conferences. The VSC reproduces the dynamic of the discussions in synchronous virtual conferences.

6 4 3 0 6 19
Table 1. Results for the dimension "Perceived Similarities between the VSC and other types of conferences"

In their comments, respondents mentioned certain advantages of the VSC compared to live conferences. For example, they noted that the information is available at all times, that participants can attend according to their own schedule, and that users can research a topic before getting involved in a discussion. ''In a live conference, it is difficult to gather all of the information. However, the asynchronous conference offers the possibility of consulting reference work and relevant documents'', indicates a student. Others add that ''The conference formula allowed me to attend the conference at my own pace'' and that ''This approach provides us with the opportunity to research certain topics before responding." However, some learners mentioned that the limits imposed on the number of comments that could be made on each production in the VSC differ significantly from the reality of live conferences. Another person noted that participants must ''get into the game". In fact, the suspension of disbelief seems to be an essential condition to engage in authentic online learning environments [START_REF] Herrington | Patterns of engagement in authentic online learning environments[END_REF] and virtual worlds [START_REF] Hand | Other faces of virtual reality[END_REF].

The asynchronous conference paradigm becomes problematic for students who require a course extension (for personal, professional or health reasons, for example). Thus, students who complete the course beyond the regular 15-week course time frame are alienated from the group, its activities, and can no longer participate in the discussions related to the documents (posters and papers), thereby missing out on the emulation that ensues from the VSC scenario.

Evaluating the Technological Aspect of the VSC

Users had mixed reactions as to whether the rooms were user-friendly (see Table 2). To the statement '"the organization and operation of the Poster Session Room are user-friendly'', thirteen learners indicate that they ''agree'' or that they ''strongly agree'', although many indicated they had difficulty understanding how to use the Poster Session Room, which was the first one used in this course. Seven students indicated that they had technical problems using the VSC rooms. Two said they had difficulty posting their work. Another one claims that the documents posted suddenly disappeared. Some learners dislike being prevented from modifying their files after it has been in a room for a certain period of time.

Statements

Strongly agree Agree Disagree Strongly disagree TOTAL

The organization and operation of the Poster Session Room are user-friendly. Table 2. Results for the dimension "Technological aspects of the VSC"

Many of these issues were brought up during the first semester the course was offered, which resulted in a number of modifications to the operation and aspect of the rooms. Furthermore, a concise user's guide to help learners navigate within the rooms was created and the tutor provides, if needed, some technical support to the participants, especially during the first activity held in the Poster Session Room.

The Impact of VSC on Motivation and Learning

In general, students have a positive attitude towards the VSC, which is considered a motivating and stimulating tool (see Table 3). Almost all respondents (N=20) feel that the conference analogy is stimulating, and all (N=21) agree that posting their work and viewing others' work is a stimulating factor that favors learning. Most of them (N=17) feel that the peer discussions are very motivating, if not essential: ''In my case, it favored motivation, overachievement, collaboration and sharing expertise''; ''Fantastic idea! We must act as experts, return to our assignments and reading, take a stand in front of the rest of the group…'' One indicated that one advantage of the CSV is that it ''allows learners to compare their knowledge with that of their peers''. However, two respondents would have would have liked to see asynchronous activities supplemented with synchronous discussions All respondents enjoyed the virtual poster session (N=21) and most of them the symposium (N=19). Furthermore, the majority (N=19) indicated that they would recommend the use of a VSC in other e-learning courses.

As to how students perceive the learning contribution of the VSC, all respondents confirm that they have learnt from participating in the VSC, whether from the virtual poster session (N=21) or during the symposium (N=18). ''A wonderful synthesis!'', claims one of the respondents. When asked whether they have learnt from the course overall, all respondents indicate they have expanded their knowledge "a lot" (N=19) or "moderately" (N=2). Furthermore, the estimated mean percentage of reaching their own learning goals is 92% (SD = 2,11).. Finally, 82% of the respondents would recommend the course to other students (N=18).

Statements

Strongly agree Agree Disagree Strongly disagree TOTAL

The scientific conference metaphor is stimulating for learning.

Conclusion

For learners enrolled in graduate studies, the VSC scenario integrates many characteristics of authentic learning [START_REF] Basque | L'apprentissage « situé » dans les cours en ligne : Le cas du colloque scientifique virtuel (CSV)[END_REF]. It is an "ideal formula to favor discussions in a distance learning course", as one student commented. Compared to other types of conferences, it is even seen as an advantageous avenue and it is considered a stimulating and motivating learning tool. Thus it seems that most learners were satisfied with this experience and that they met their own learning objectives. However, we should be cautious with this conclusion, as our sample size was small.

Based upon the results of this preliminary study, a list of recommendations and improvements has been compiled to enhance the VSC:

-Investigate the ergonomics of the room's interface; -Reconsider the necessity of setting a time limit in order to prevent users from modifying a posted file while considering the discussion process; -Review the scenario to allow for multiple comments on a single production; -Prepare a more detailed description of the role of the tutor in the VSC and develop room management tools for him or her; -Investigate possible adaptations of the VSC in other learning contexts.

Work on two of those issues has already begun. First, we elaborated the tutor's guide. This manual includes a description of the various tasks tutors must perform to ensure that the conference runs smoothly, tips to help students during the conference, help tools to assist individual students and the entire group, as well as various instructions to deal with difficult or delicate situations. Second, the VSC was recently integrated in another course, at the undergraduate level. In one of the learning activities, students from different countries have to elaborate a case study on biodiversity: each of them present the plan of their case study in the form of a poster in the Poster Session Room and post their paper (reporting the case study) in the Symposium Room. In this case, students are the main "speakers" in the symposium, as opposed to the initial design of this activity in the VSC. Moreover, each participant must comment at least two papers or others comments in each of the two rooms of the VSC, instead of only one in the initial application of the VSC.

Table 3 .

 3 Results for the dimension "Impact of the VSC on motivation and learning"

	10	10	1	0	21

In French, the acronym CSV stands for Colloque scientifique virtuel.


Valéry Psyché
email: vpsyche@licef.teluq.uquebec.ca

Riichiro Mizoguchi

Jacqueline Bourdeau
email: bourdeau@licef.teluq.uquebec.ca

Ontology Development at the Conceptual Level for Theory-Aware ITS Authoring Systems

Keywords: Ontology, Ontological Engineering, Intelligent Tutoring Systems, Authoring Environments, Instructional Science, Learning Scenario

This paper presents ontology engineering at the conceptual level, based on case analysis where an author (a human or software agent), assisted by an ITS authoring system, needs to select an instructional strategy (method) in order to develop a learning scenario or environment. A detailed description of the development process at the conceptual level is provided, followed by the presentation of its results. Outstanding issues raised therein will be shared for further reflection.

Introduction

This paper presents ontology engineering (OE) at the conceptual level, based on case analysis where an author (a human or software agent), assisted by an ITS authoring system, needs to select an instructional strategy in order to develop a learning scenario or environment. In this case the author benefits from having access to the theories on which such strategies rely. A poster will illustrate our development process at the conceptual level, which follows MizLab's view of OE [START_REF] Mizoguchi | A Step Towards Ontological Engineering[END_REF]. Results of this process are presented in this paper, including: 1) a use case; 2) an ontology; and 3) models of the learning scenario. Furthermore, key questions raised during this process are shared for further reflection.

Ontological engineering at the conceptual level

In our conceptual development process, we closely followed OE methodological guidelines provided by MizLab [START_REF] Mizoguchi | A Step Towards Ontological Engineering[END_REF], although ad hoc elements developed within the process were also necessary. The conceptual level we speak of corresponds to Mizoguchi's "level 1" [START_REF] Mizoguchi | A Step Towards Ontological Engineering[END_REF]. The level 1 methodology should be seen as the articulation of a world of interest, and consists in: the structuring of a collection of terms; the elicitation of concepts; the constitution of a so-called "is-a" hierarchy among these; the establishment of brief concept definitions; and the establishment of simple relations to prevent unexpected interpretations of concepts. This conceptual development was conducted using Hozo, an Ontology environment developed at Mizlab, composed of an ontology editor and an ontology and models server.

Use-cases

In the first step, we built use cases by anticipating a situation where an author needs an instructional strategy to design a learning scenario or environment. In order to achieve his goal, the author would like to query an ITS authoring environment about the instructional theories on which the available strategies are based. Theses queries depend on learning conditions previously identified. To expose the dependency between theory and strategy, we rely on work found in a book edited by Reigeluth [START_REF] Reigeluth | Instructional theories in action: lessons illustrating, selected theories and models[END_REF] in which alternative lessons for the same content are presented according to various theories and authors.

An iterative modelling process

As mentioned, our guidelines for this stage were those suggested by Mizoguchi [START_REF] Mizoguchi | A Step Towards Ontological Engineering[END_REF]. He proposed that one should: 1) thoroughly investigate the "world" one wants to represent, and even aspects which appear self-evident; 2) uncover hidden assumptions; 3) formulate and formalize fundamental concepts; 4) formulate and formalize relations among concepts. Following these guidelines enabled us to uncover concepts that are implicit to various instructional theories. Then, by defining a common vocabulary for other concepts that were not in the theories, ontological commitment was taken in order to share an understanding of the world of interest among team members. Concepts were then organized under hierarchies by first identifying "is-a", and then "part-of" relations. The "Is-a" hierarchy organizes concepts from the abstract to the concrete. The "part-of" hierarchy allows the elaboration of more semantically complex relations that are sometimes necessary to express a concept correctly. After constituting these hierarchies, simple connectivity between concepts was established with the "participate-in" relation. A good explanation of these relations can be found in [START_REF] Kozaki | Development of an environment for building ontologies which is based on a fundamental consideration of relationship and role[END_REF]. Again following Mizoguchi, this stage was an iterative one, which stopped once the concepts were stabilized. Only then did we start creating models (or instances). We should also mention that, with regard to the present work, level 1 was specified without any axiomatic constraints.

Presentation of the results

As a result of this OE process, we have built: 1) use cases to illustrate an instructional scenario for teaching a concept in optics to secondary school learners; 2) a core ontology; and 3) three models. The use cases are based on: the Gagné-Briggs Theory of Instruction, the Merrill Component Display Theory and the Collins Theory of Inquiry Teaching. In each use case, the author is presented with learning situation elements that correspond to one of the three theories. (Types of elements include: prerequisites of the lesson, learning content, the teaching strategy, teaching material, the type of assessment, and the activities involved in achieving the lesson objective and assessing learners.) For example, activities suggested to the author are based on either the Gagne instructional events, the Merrill performance/content matrix, or the Collins instructional techniques. The ontology developed at the modelling stage, which we call the core ontology, consists of a representation of a partial domain of (secondary school) optics and three kinds of learning scenarios (see figure 1). Three models that rely on the use cases and the core ontology were built. These models focus on the teaching/learning interaction, according to each instructional theory. As examples, we will present six of the nine Gagné events of instruction, one of the sixteen elements of the Merrill performance/content matrix and six of the ten Collins techniques of instruction.

Issues for further reflection

A number of issues were raised which would deserve further deep thinking and discussion for further work, among which are the following: 1) Are we allowed to harmonize terminology belonging to different theories or even different paradigms? 2) Is it risky to integrate elements from different theories into a single scenario? 3) Should a learning scenario be inextricably linked to an instructional theory? That is, can we use the hierarchy of Gagne's intellectual skills to represent knowledge associated with an intellectual task, no matter the theory on which the strategy is based? 4) Should we consider using test queries (via the ITS) to help validate the formalized ontology (at level 2) -and how can we evaluate feasibility at the functional level (level 3)? 6) Concretely, what could ontological engineering bring to existing ITS authoring systems [START_REF] Murray | Authoring intelligent tutoring systems: an analysis of the state of the art[END_REF]?

Conclusion

We have presented the iterative process applied by team members and the results obtained at level 1. Further work will include: 1) Developing more ontologies according to theories from each paradigm; 2) Interfacing these ontologies with an existing authoring system, and experimenting with a complete scenario; 3) Developing functionalities for authoring tasks based on "2)"; and 4) Conducting an evaluation process.

Figure 1 .

 1 Figure 1. Core Ontology in tree hierarchy window (left) and graphical representation window (right)


Jia Long Shyu
email: b03401072@ntu.edu.tw

Physical Interpretation of the Unitary Representations of the Poincaré Group

The quantum field theory describes microscopic physical phenomena with experimentally determined parameters. In this study, an attempt of identifying the physical nature with mathematical objects is made in pursuit of interpreting the parameters. For this purpose, the unitary representations of the Poincaré group are studied. Inspired by the structure of the unitary representations, the phenomenon of self-loops is defined, and it gives an equation for the masses of elementary particles. The masses of leptons and W, Z particles are calculated using this equation, and the results are interpreted as a relationship between the self-loop amplitudes and the masses of elementary particles.

I. INTRODUCTION

The Standard Model is the most successful model describing the microscopic physical phenomena. However, with the need for more than ten parameters to be determined by experiments, doubts are that the Standard Model cannot be the ultimate theory of physics, and a more fundamental theory underlies.

In pure mathematics, mathematical structures possess well-defined behavior. This characteristic resembles the physical nature -that it is ruled by laws of nature. A conjecture arises thereafter: the entities of our physical world are mathematical objects, and one can find the complete set of physical phenomena in the corresponding mathematical objects.

We cannot verify this conjecture in the experimental level, therefore it is not a conjecture that can be answered in the realm of physics. This conjecture hence serves only as a motivation for one to develop a physical theory from mathematical structures. Based on these restrictions, the only way to qualify a mathematical structure as the entity of physics is the identification of all physical phenomena.

Wigner [1] studied the unitary representations of the Poincaré group and obtained a correspondence between particle states and the representations. We shall develop the above proposal on this structure.

In Section II, we introduce the unitary irreducible representation of the Poincaré group, and impose the Feynman rules from the quantum field theory to the states within the representations. An essential phenomenon of self-loops arises in this model. The self-loops are viewed as the clocks of elementary particles. With this interpretation, one can derive an equation for the masses of elementary particles. This equation has undetermined parameters, and several choices are made to explore its validity. The masses of leptons, Z, and W particles are then calculated and compared to the experimental values. In Section III, we discuss the physical meaning of these results, the possible limitations, and future work.

The Poincaré group is defined as the symmetry group of the Minkowski spacetime [START_REF] Tung | Group Theory in Physics[END_REF], including the homogeneous and the inhomogeneous transformation. Its unitary irreducible representation is characterized by two parameters: the rest mass M and the spin λ. Wigner identified these representations as particle states. For states from representations of any (M, λ), if λ = 1/2, we call the state a spinor. For λ = 1, the state is called a vector. These names are derived from their behavior under the Poincaré group.

One may identify elementary particles in the quantum field theory as particle states in this representation by imposing different gauge transformations to these states. Furthermore, one can define the Feynman diagrams and impose Feynman rules to these diagrams. This makes defining transition amplitudes between states possible, and identifies the above construction with quantum field theory in the amplitude level.

B. The Self-Loops

In the above construction, if we allow state transition by Feynman diagrams, then the only processes for a spinor to remain as itself are the self-loops. To investigate the physical meaning of the self-loops, consider the thought experiments in Einstein's discussion in the special relativity [START_REF] Einstein | [END_REF]. For the measurer, a clock is necessary to obtain measurements of space and time coordinates. However, suppose that only one electron is present, then there are no clocks, and the only apparatus left are the self-loops. This picture suggests that the self-loop is related to the time evolution.

On the other hand, in relativistic quantum mechanics, the wave equation for particles describes the wave function of particles at rest to be in proportion to e -imt . This factor suggests that a frequency is associated with particles at rest, and is in proportion to the mass of the particle. Again, in the above discussion, the only possible attribute of the particle at rest to be related to the frequency is the self-loop.

With the above two considerations combined, we conjecture that the self-loops are the evolution of time, and the frequency of the self-loops is in proportion to the mass of the particle. Particles with different masses possess clocks with different ticking frequencies.

C. The M Equation

The self-loop amplitude of an elementary particle is related to the frequency of its intrinsic clock. If we identify the time evolution using this clock to the formula from relativistic quantum mechanics, we have

2πΓ ∝ m, ( 1
)
where Γ is the rate of the self-loops and m is the particle mass. The formula for the infinitesimal decay rate is [START_REF] Peskin | An Introduction to Quantum Field Theory[END_REF]

dΓ = 1 2m a Π f d 3 p f (2π) 3 1 2E f |M| 2 (2π) 4 δ (4) (p A -Σp f ), (2)
where M is the amplitude of the self-loop, m a is the mass of the incoming particle, E f are the energies of the outgoing particles, and p A , p f are the four-momentum of the incoming and outgoing momentum, respectively. Inspired by the above formulas, we propose: 4π|M| = m 2 .

(3)

Note that we only conjecture that the self-loop amplitude is in proportion to the mass squared, and the factor is not determined. The factor 4π is chosen to fit the calculation results with experimental data. This is an equation of m, because M depends on the particle mass. Therefore, we can solve this equation to obtain masses of elementary particles. This equation is called the M equation because the letter m is present on both sides of the equation. This equation is applied to calculate masses of leptons, Z, and W particles. In the calculation of leptonic masses, since e -, µ -, and τ -are equivalent in terms of Feynman diagrams, we propose a factor of genus g to reflect the fact that there are three generations of leptons, which gives

g × 4π|M| = m 2 . ( 4
)
The details of calculations are discussed in the appendix, including the specific choices of loops and genus for each particle. The results are compared to experimental values from the Particle Data Group [START_REF] Tanabashi | [END_REF], shown in Table I.

There is a factor two in the ratio between calculated leptonic masses and the masses W and Z particles. One possible explanation is that under unitary transformations, the spinors rotate with half of the speed of the vectors. However, when viewed from external observers, the half rotation is counted as one cycle without noticing the internal 180-degree flip of spinors. Hence, the fermion mass should be 2m.

The neutrino masses are also calculated. The results are smaller than experimental estimations as discussed in the appendix.

III. DISCUSSION

A. The Significance of the M Equation

The M equation gives the masses of Z 0 , W ± , e -, µ -. However, there exists arbitrariness such as the choice of the coefficient 4π, the leptonic genus, the loops, and the argument for the one-half solution of the leptonic masses. Since the M equation is proposed from theoretical considerations, and involves the unobservable self-loops, we cannot justify it as the correct equation for the masses of the elementary particles by studying the self-loops in experiments. Nor can we claim the validity of any specific choice of parameters. One must note that in general, even if we can eliminate all the arbitrariness from this equation, it could still be a numerical accident for the calculated masses to match experimental data. Strictly speaking, we can only claim the discovery of an equation which is satisfied by the masses of elementary particles, and relates the self-loop amplitudes to the masses of the particle.

B. Remarks on the Renormalization

In the renormalization of quantum field theories, bare parameters of Lagrangians are assumed to be infinite and absorb the divergences present in loop diagrams. In the calculation of elementary particle's masses, the assumption that they are in proportion to the maximal momentum gives a mass that is truly infinite as Λ goes to infinity. Still, the compatibility of this theory with the renormalization procedure and the running parameters requires further study.

C. Future Work

The calculation with M equation employs parameters without radiative corrections. Corrections with running parameters are necessary to verify the validity of the results. The choice of loops also needs further investigation. In the calculation of masses of W ± and Z 0 , we adopt choices of loops that lack physical interpretation. For the amplitudes of µ -, we discard the W loop as the major decay channel. In the case of τ -, both Z 0 and W -loops are dropped and a result with error up to 12% is obtained. This reflects our ignorance of the essence of lepton generations, or an inappropriate choice for the coefficients of self-loops. A more thorough analysis of the influence of decays on the coefficients of loops should be carried out.

We can also perform the calculation to the quarks. Among the gluon, γ, W and Z self-loops for quarks, the relatively large gluon coupling constant causes the quark to have divergent mass. This might reflect the fact that there are no free quarks, and when quarks are bound together, the gluon loops disappear due to asymptotic freedom, and are turned on only when the constituents are torn apart.

Despite the results of calculation of masses of elementary particles, to be qualified as the mathematical structure of physical reality, this theory must include all the other fundamental physical phenomena. There are still other parameters undetermined theoretically: the coupling constants, the quark masses, hardron masses, the Cabibbo angle, mixing of neutrinos, and Higgsmechanism-related parameters.

Appendix A: Calculation of Masses of Z and W

We apply the Feynman rules [START_REF] Griffiths | Introduction to Elementary Particles[END_REF][START_REF] Romao | [END_REF] in quantum field theories to calculate the self-loop amplitudes. The amplitudes are simplified using form [8], and the formulas are input in python for integration.

As mentioned above, there are arbitrarinesses when applying the M equation. For instance, the coefficient of the equation and the choice of self-loops for each particle. We adopt several choices of the parameters to explore the validity of the M equation. For the calculation of masses of the vector bosons, we choose the W loop and the Higgs loop for the Z particle, and the Z loop for the W particle.

We demonstrate the calculation with the amplitude of the W loop M ZW of Z particle:

M ZW = ε µ * d 4 k (2π) 4 V λσµ Π σβ W (k)Π λα W (p -k)V βαν ε ν , (
A1) where Π σβ W (k) is the W particle propagator, and V λσµ is the vertex factor for three-point interaction of Z 0 and W ± , and p is the four-momentum of the Z particle at rest. The integration domain of k is understood to be the R 4 .

We can use the Feynman parametrization to combine the two denominators of propagators, and the integral becomes

d 4 k (2π) 4 1 0 dx V λσµ V βαν g σβ -k σ k β m 2 W g λα -(p-k) λ (p-k) α m 2 W [(k -xp) 2 -∆] 2 , (A2) where ∆ = (x 2 -x)m 2 Z + m 2 W .
Next, change the integration variable from k to l = k -xp, contract the vertices, apply the Wick rotation l 0 → il 0 to get

d 4 l (2π) 4 1 0 dx Φ(x, l) (l 2 + ∆) 2 , (A3)
where Φ(x, l) is the numerator of Equation A2 with k replaced by l + xp. At this step, we assume that there is a maximal selfclock frequency Λ, and the Z clock should be in proportion to that clock, m Z = Λ/Λ Z . The form of the 1/Λ Z factor is defined for future convenience. With this assumption, by changing the integration variable from l to l = k/m Z , dividing m W by m Z and p by m Z , we obtain

|l| 2 ≤Λ 2 Z d 4 l (2π) 4 1 0 dx m 2 Z Φ(x, l) l 2 + ∆ 2 , (A4)
where Φ(x, l) = Φ(x, l)/m 2 Z and ∆ = ∆/m 2 Z . Note that the m 2 Z will be cancelled by the right-hand side of the M equation. The same manipulations are performed on all the self-loops under consideration. After summing up all the self-loop amplitudes under considertaion, we are ended up with a equation of Λ Z . Solving the equation with the Weinberg angle cosθ W = 0.8768 and g e = 0.3028, we obtain Λ Z = 2.3751, Λ W = 2.6966. This choice of parameters possesses one special feature. Suppose that we include the condition of m W /m Z = cosθ W from the Higgs mechanism, and solve cosθ W and the M equation for m Z , m W simultaneously, we can get Λ Z = 2.3455, Λ W = 2.6602, and cosθ W = 0.8817.

We can use another choice of parameter and obtain similar results. We change the coefficient of the M equation to be 8π. Use all possible loops and bubbles except for the fermionic ones, and perform the same procedures as above, we have Λ Z = 4.7547 and Λ W = 2.6812. Note that if we divide Λ Z by 2, we have Λ Z = 2.3773 and cosθ W = 0.8860. electron has three self-loops: the γ loop, W -loop and Z loop. µ -decays easily in W -channel, while Z and W -channel are forbidden for τ -. The approximation of zero neutrino mass is made in the propagator of neutrinos, and we use m Z = 2.3455 and cosθ W = 0.8768 for the mass parameters of Z and W loops.

To calculate the mass of the electron, use the γ loop, W -loop, Z loop and genus 1, define m e = Λ/Λ e , apply the Feynman parameter and Wick rotation, we can obtain Λ e . The muon has the γ and W -loop, with genus 3/2. Only the γ loop is considered for the case of tauon, with genus 2. The results are normalized by m Z and given in Table I. Note that we divide the Λ values for all leptons by 2 to obtain the results in the table, as explained in Section II C.

The neutrinos all undergo W + and Z loops. If we assume that the neutrinos are left-handed, the amplitudes are zero. We also calculated the case assuming neutrinos to be Dirac spinors, and the M equation does not give a solution below Λ ν = 10 12 , with genus 1, 2, 3, or 4. This suggests that the neutrinos may have a mass smaller than the current estimated values [START_REF] Tanabashi | [END_REF].

TABLE I

 I

		. M Equation Mass Calculations
	Particle	Mass	Mass	Error
		(normalized by mz) (calculated)
	Z 0	1	1
	W -	8.815 × 10 -1	8.817 × 10 -1 0.023%
	e -	5.604 × 10 -6	5.762 × 10 -6 2.817%
	µ -	1.159 × 10 -3	1.165 × 10 -3 0.529%
	τ -	1.949 × 10 -2	2.167 × 10 -2 11.2%

Appendix B: Calculation of Masses of Leptons

There are twelve elementary leptons, which are e -, µ -, τ -, ν e , ν µ , ν τ , and their anti-particles, with three generations of leptons ordered by increasing masses. It is assumed that leptons of different generations rotates with different genus. For electrons, muons and tauons, the genus is chosen to be 1, 3/2 and 2, respectively. When choosing the self-loops, we require that the particles cannot decay through the chosen channels. Therefore, the Appendix C: Remarks on the Top Quarks

The top quark is the heaviest elementary particle observed. To investigate whether the mass of the top quark, m t , is the maximal frequency of self-clocks, we divide m t by the mass of Z 0 , which gives 1.898 using m t = 173.1 GeV/c 2 . If we assume that Λ Z = Λ/m Z = 1.898, and calculate the masses of electrons as above, we obtain m e /m Z = 5.514 × 10 -6 , which is of -1.6% error compared to the experimental values, without the factor two mentioned in Section II C.

There is a choice of self-loop parameters related to this observation. Use 8π as the coefficient of the M equation. Adopt all bubbles and loops of the Z particle except for the W loop, and all the loops of the W particle. Together with m W /m Z = cosθ W , the M equation gives Λ Z = 1.873, Λ W = 2.142, and cosθ W = 0.8745. Using the above results with cosθ W = 0.8768, we have m e = 5.790 × 10 -6 , with 1.387% error. Note that for this choice, the genus of electrons is chosen to be 1/2. Again, the m e in this choice does not require the factortwo remedy, and the drop of W loops may be a result of the Z-particle loss during the process. However, we do not have reasonable results for masses of µ -and τ -. This may imply an unclear understanding of the generations of elementary particles.


Alain Gély
email: alain.gely@loria.fr

Miguel Couceiro
email: miguel.couceiro@loria.fr

Amedeo Napoli
email: amedeo.napoli@loria.fr

Steps Towards Achieving Distributivity in Formal Concept Analysis

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Context and Motivations

Formal Concept Analysis (FCA) has proved to be an eective tool in data analysis and knowledge discovery in several application domains [START_REF] Carpineto | Concept Data Analysis: Theory and Applications[END_REF][START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF]. Concept lattices provide a valuable support for several tasks, such as classication, information retrieval and pattern recognition. Besides lattices, trees and their extensions [START_REF] Bertrand | Multilevel clustering models and interval convexities[END_REF][START_REF] Bertrand | Pyramids and weak hierarchies in the ordinal model for clustering[END_REF][START_REF] Diatta | Galois weak hierarchies: Theoretical and computational issues[END_REF] are used in biology, notably, in phylogeny, for modeling inter-species liations. In this domain, one of the main problems is to nd evolution trees for representing existing species from accessible DNA fragments. When several trees are leading to the same inter-species liations, the preferred ones are the most parsimonious, i.e. the number of modications such as mutations for example, is minimal for the considered species. However, several possible parsimonious trees may exist. Such a situation arises with inverse or parallel mutations, e.g., when a gene goes back to a previous state or the same mutation appears for two non-linked species. This asks for a generic representation of such a family of trees.

Bandelt [START_REF] Bandelt | Median algebras[END_REF][START_REF] Bandelt | Median-joining networks for inferring intraspecic phylogenies[END_REF] proposes the notion of median graph to overcome this issue, since he noticed that a median graph is capable of encoding all parsimonious trees. A median graph is a connected graph such that for any three vertices a, b, c, there is exactly one vertex x which lies on a shortest path between each pair of vertices in ta, b, cu. Alternatively, median graphs can be thought of as a generalization distributive lattices [START_REF] Birkho | A ternary operation in distributive lattices[END_REF][START_REF] Grätzer | General Lattice Theory[END_REF]. However, the extraction of such structures directly from data remained unaddressed.

Uta Priss [START_REF] Priss | Concept lattices and median networks[END_REF][START_REF] Priss | Representing median networks with concept lattices[END_REF] made a rst attempt to use algorithmic machinery of FCA and the links between distributive lattices and median graphs, to analyze phylogenetic trees. However, not every concept lattice is distributive, and thus FCA alone does not necessarily outputs median graphs. In [START_REF] Priss | Representing median networks with concept lattices[END_REF] Uta Priss sketches an algorithm to convert any lattice into a median graph. The key step is to transform any lattice into a distributive lattice.

In this article, we propose an algorithm supporting such a transformation that minimizes the changes introduced to the original lattice. Using the context of an initial concept lattice as input, the algorithm outputs the context of a distributive lattice, without necessarily building the lattice. Our approach relies on Birkho 's representation of distributive lattices [START_REF] Birkho | On the combination of subalgebras[END_REF][START_REF] Birkho | Lattice theory[END_REF]. Moreover, we illustrate our approach with a generic example that reveals the diculties of transforming of a concept lattice into a distributive lattice. We do not settle this issue entirely, but we propose major steps and an approach towards its solution.

The paper is organized as follows. In Sections 2 and 3 we recall the basic background and notation as well as some key results on distributive lattices. The transformation algorithm is presented in Section 4 and we discuss the strengths and limitations of our approach in Section 5.

Denitions and Notations

In this section we recall basic notions and notation needed throughout the paper.

We will mainly adopt the formalism of [START_REF] Ganter | Formal Concept Analysis: Mathematical Foundations[END_REF], and we refer the reader to [START_REF] Caspard | Ensembles Ordonnés Finis: Concepts, Résultats et Usages[END_REF][START_REF] Davey | Introduction to Lattices and Order[END_REF] for further background.

Partially Ordered Sets, Lattices and Homomorphisms

A partially ordered set (or poset for short) is a pair pP, ¤q where P is a set and ¤ is a partial order on P , that is, a reexive, antisymmetric and transitive binary relation on P . A poset pP I , ¤ I q is a subposet of pP, ¤q if P I P and ¤ I ¤. For a subset X P , let Ó X ty P : y ¤ x for some x Xu and Ò X ty P : x ¤ y for some x Xu. If X : txu, we use the notation Ó x and Ò x instead of Ó txu and Ò txu, respectively. In this paper, we will only consider nite posets pP, ¤q and, when there is no danger of ambiguity, we will refer to them by their underlying universes P .

A set X P is a (poset ) ideal (resp. lter) if X Ó X (resp. X Ò Xq. If X Ó x (resp. X Ò x) for some x P , then X is said to be a principal ideal (resp. lter) of P . For x, y P , the greatest element of Ó x Ó y (resp. least element Ò x Ò y) if it exists, is called the inmum (resp. supremum) of x and y. A lattice is a poset pL, ¤q such that the inmum and the supremum of any pair x, y L exist, and they are denoted respectively by x y and x y. A subset X L is a sublattice of L if for every x, y X we have that x y, x y X. As for posets, we will only consider nite lattices pL, ¤q and we will refer to them by their underlying universes L.

In this nite setting, posets and lattices can be represented and clearly visualized by their Hasse-diagrams [START_REF] Davey | Introduction to Lattices and Order[END_REF]. Also, the notions of inmum and supremum naturally extend from pairs to any subset of elements of a given lattice L. In this way, the notions ofand -irreducible elements (that constitute the building blocks of lattices) can be dened as follows. For x L, let x ¦ pÒ xztxuq and x ¦ pÓ xztxuq. Then x L is said to be a -irreducible element of L if x $ x ¦ . Dually, x is said to be a -irreducible element of L if x $ x ¦ . We will denote the set of -irreducible elements and -irreducible elements of L by M pLq and JpLq, respectively. Observe that both M pLq and JpLq are posets when ordered by ¤.

We now recall the notions of poset and lattice homomorphisms.

Let pP, ¤q and pP I , ¤ I q be two posets. A mapping f : P Ñ P I is said to be a (poset ) homomorphism if x ¤ y implies f pxq ¤ I f pyq. In addition, if f : P Ñ P I is injective (one-to-one), then it is called a (poset ) embedding. If it is a bijection and an embedding such that, for every x I , y I P I , x I ¤ I y I implies f ¡1 px I q ¤ f ¡1 py I q, then it is called a (poset ) isomorphism.

In the case of lattices, the notions of homomorphism, embedding and isomorphism become more stringent. Let pL, , q and pL I , I , I q be two lattices. A mapping f : L Ñ L I is said to be a (lattice ) homomorphism if f px yq f pxq I f pyq and f px yq f pxq I f pyq. In addition, if f : L Ñ L I is injective, then it is called a (lattice ) embedding. If it is a bijection and it is an embedding such that f ¡1 is also an embedding, then it is called a (lattice ) isomorphism.

When it is clear from the context, we will drop (poset) and (lattice) and simply refer to homomorphism, embedding and isomorphism.

It is noteworthy that the image f pLq of a homomorphism f : L Ñ L I is a sublattice of L I , and that two isomorphic lattices have the same Hasse diagram. In particular, two lattices L and L I are isomorphic if and only if both (1) JpLq and JpL I q, and (2) M pLq and M pL I q are isomorphic. In the case of distributive lattices, Birkho [START_REF] Birkho | Lattice theory[END_REF] showed that (1) suce to guarantee that L and L I are isomorphic (pJ, ¤q and pM, ¤q are isomorphic). The latter result is key ingredient in Birkho's representation of distributive lattices that we will discuss in Section 3, and that we will use in Section 4 to devise an algorithm to modify any nite lattice into an optimal distributive lattice containing it.

Formal Concept Analysis

Reduced Contexts, Concepts and Concept Lattices. We denote by C pO, A, Iq a formal context where O is a set of objects, A a set of attributes and I an incidence relation between objects and attributes. In phylogenetic data, objects are usually species, attributes are mutations, and po, aq I or oIa when mutation a is spotted in species o.

Denition 1 (Galois connections). For a set X O, Y A we dene:

X I ty A | xIy for all x Xu Y I tx O | xIy for all y Y u
Then a formal concept is a pair pX, Y q, where X O, Y A and X I Y and Y I X. X is the extent and Y is the intent of the concept. The set of all formal concepts ordered by inclusion of the extents dually the intents denoted by ¤ generates the concept lattice of the context C pO, A, Iq.

For o O, γo po P , o I q denotes the concept introducing object o. For a A, µa pa I , a P q denotes the concept introducing attribute a.

A claried context is a context such that x I y I implies x y for any element of O and any element of A. Moreover, a claried context is reduced i it contains:

no vertex x O such that x I X I with X O, x X no vertex x A such that x I X I with X A, x X
The reduced context is also called a standard context. ¢ if j ¤ m where ¤ is the partial ordering in the concept lattice,

Ó if j Ó m, Ò if j Ò m, Ø if j Ó m and j Ò m,
otherwise an empty cell.

1 ¢ Ø Ø 2 Ø ¢ Ø 3 Ø Ø ¢ N5 a b c 1 ¢ Ó Ø 2 Ø ¢ ¢ 3 Ò Ø ¢ Distributive lattice a b c d 1 ¢ Ø ¢ ¢ 2 Ø ¢ ¢ ¢ 3 ¢ ¢ Ø 4 ¢ Ø ¢
Fig. 1. Three lattices and their reduced contexts with arrow relations.

Distributive Lattices and Their Representation

A lattice is distributive if and are distributive one with respect to the over. Formally, a lattice L is distributive if for every x, y, z L, we have that one (or, equivalently, both) of the following identities holds: piq x py zq px yq px zq, piiq x py zq px yq px zq.

Distributive lattices appear naturally in any classication task or as computation and semantic models; see, e.g., [START_REF] Caspard | Ensembles Ordonnés Finis: Concepts, Résultats et Usages[END_REF][START_REF] Davey | Introduction to Lattices and Order[END_REF][START_REF] Hopcroft | Introduction to Automata Theory, Languages and Computability[END_REF][START_REF] Mattern | Virtual time and global states of distributed systems[END_REF]. This is partially due to the fact that any distributive lattice can be thought of as a sublattice of a power-set lattice, i.e., the set PpXq of subsets of a given set X. This result is a corollary to Birkho 's representation of distributive lattices that we will further discuss in Subsection 3.2.

Characterization of Distributive Lattice

The distributive property of lattices has been equivalently described in several ways. We recall a few useful characterizations that we will use in the following sections of the paper. The rst characterization establishes a correspondence between distributive lattices and median algebras. Indeed, a median algebra is a structure pM, mq where M is a nonempty set and m : M 3 Ñ M is an operation, called median operation, that satises the following conditions mpa, a, bq a and mpmpa, b, cq, d, eq mpa, mpb, c, dq, mpb, c, eqq, for every a, b, c, d, e M . It is not dicult to see that if L is distributive, then mpa, b, cq pa bq pb cq pc aq is a median operation. The connection to median graphs was established by Avann [START_REF] Avann | Median algebras[END_REF] who showed that every median graph is the Hasse diagram of a median algebra (thought of as a semilattice). For further background see, e.g., [START_REF] Bandelt | Median algebras[END_REF].

The second characterization describes distributive lattices in terms of two forbbiden structures, namely, M 3 and N 5 (see Fig. 1) that are, up to isomorphism, the smallest non distributive lattices. The third characterization is given in terms of formal contexts and it is also illustrated in Fig. 1: neither M 3 nor N 5 are distributive since for M 3 , there are two double arrows by row and column; for N 5 , there is one double arrow by row and column, but additional simple arrows.

Distributive Lattices and Ideal Lattices

Let pP, ¤q be a poset and consider the set OpP q of ideals of P , i.e., OpP q t ¤ xX Ó x | X P u.

It is well-known that for every poset P , the set OpP q ordered by inclusion is a distributive lattice, called ideal lattice of P , and that two posets P and P I are iso- morphic if and only if OpP q and OpP I q are isomorphic as lattices. Furthermore, the poset of -irreducible elements of OpP q is JpOpP qq tÓ x | x P u and it is (order) isomorphic to P .

Dually, we saw in Subsection 2.1 that for any lattice L the set JpLq ofirreducible elements of L is a poset ordered by inclusion, and that if two lattices L and L I are isomorphic, then JpLq and JpL I q are also isomorphic (as posets).

Moreover, for any lattice L the set OpJpLqq of ordered ideals of JpLq is a distributive lattice that contains an isomorphic copy of L as a subposet. In particular, if L is isomorphic to OpJpLqq, then L must be distributive. The representation theorem of Birkho [START_REF] Birkho | On the combination of subalgebras[END_REF] states that the converse is true.

Birkho's Representation Theorem 1 Let L be a (nite) distributive lattice and JpLq. Then the mapping x ÑÓ x JpLq is a (lattice) isomorphism from L to OpJpLqq.

As immediate consequences we have that every (nite) distributive lattice can be thought of as a sublattice of a powerset lattice or, equivalently, as a lattice of ideals of a poset. Figure 2 illustrates the latter assertion: on the left is a poset P , and on the right is the lattice of ideals of P . For an arbitrary lattice L, not necessarily distributive, there may be several lattices such that their poset of -irreducible elements are isomorphic but only one of them is a distributive lattice [START_REF] Bordalo | The lattice of strict completions of a poset[END_REF][START_REF] Nation | The lattice of completions of an ordered set[END_REF]. Our goal is to make use of the previous results to present an algorithmic approach that receives a lattice L as input, and outputs an optimal distributive lattice L d such that pJpLq, ¤q is isomorphic topJ pL d q, ¤ d q. Here, by optimal it should be understood with the least number of modications (notably, insertions).

Proposal for Building a Distributive Lattice

From any lattice L, we want to obtain a distributive one L d . Moreover, we want L d to be similar to L. In this work, L d is considered as similar to L if the posets of -irreducible elements of L d and of L are isomorphic. In this case, L can be embedded in L d (L d is a -completion of L).

With this denition of similar (which can dually be applied to -irreducible elements), we can use Birkho Representation Theorem to compute L d or its reduced context.

The main idea of algorithm 1 is to compute the context of L d from the reduced context of L as input. Our approach relies on the underlying poset pJ, ¤q which is used to compute M d .

Property 1. Algorithm 1 outputs the reduced context of the ideal lattice of JpLq.

Proof. By construction, there is only one double-arrow by row and by column, and no other arrows. It follows that C d is the context of a distributive lattice As discussed in section 2.1, this lattice is the ideal poset of pJpLq, ¤q. It follows that pJpLq, ¤q and pJpL d q, ¤ d q are isomorphic.

Algorithm 1: Construction of context of distributive lattice.

Data: Reduced context CpJ, M, Iq Step 1. Computation of m 1 using Jz Ò 1. The algorithm computes therepresentation of m 1 , the -irreducible element such that 1 Ø m 1 . At the end of this step of the loop, C d has a unique column which correspond to m 1 .

Result: Reduced context C d pJ, M d , I d q of pOpJq, , , q M d Ð r I d Ð r foreach j J do Òj Ð r foreach i J do if j 1 i 1 then Òj ÐÒj i M d Ð M d mj // add a -irreducible element mj such that j Ø mj X Ð Jz Òj //
m 1 1 2 ¢ 3 ¢
Step 2. Computation of m 2 using Jz Ò 2. The algorithm computes therepresentation of m 2 , the -irreducible element such that 2 Ø m 2 . At the end of this step of the loop, C d has two columns which correspond to m 1 and to a newly computed element m 2 .

m 1 m 2 1 ¢ 2 ¢ 3 ¢
Step 3. Computation of m 3 using Jz Ò 3. The algorithm computes therepresentation of m 3 , the -irreducible element such that 3 Ø m 3 . At the end of this step of the loop, C d has three columns which correspond to m 1 , m 2 and to a newly computed element m 3 .

m 1 m 2 m 3 1 ¢ ¢ 2 ¢ ¢ 3 ¢
The whole context C d for L d is now computed. By construction, the only arrow relations are double arrows between j and m j Below, L d is drawn with black circles for concepts which were present in L and white circles for new concepts.

m 1 m 2 m 3 1 Ø ¢ ¢ 2 ¢ Ø ¢ 3 ¢ Ø 5

Discussion and Conclusion

Motivated by the work of Priss [START_REF] Priss | Representing median networks with concept lattices[END_REF] on the use of FCA on phylogenetic problems, we have proposed an algorithmic approach to compute the reduced context of a distributive lattice L d from the reduced context of any lattice L, that ensures an order embedding from L into L d that preserves . So, L d can be considerated not too far from L and thus suitable for applications in phylogeny. In the remainder of this nal section, we discuss some features of this algorithm.

First, we discuss an interpretation of the behavior of the algorithm for phylogenetic data. The algorithm computes -irreducible elements of L d without any consideration for -irreducible elements of L but, as discussed in Subsection 3.2, this is not a problem. Now, for real data, two cases may appear:

1. µm j L: in this case, we can use the initial label m of the object (this label may represent a particular gene mutation);

2. µm j L: this case suggests a gene mutation that is not spotted in the data, but that is necessary to provide a parsimonious tree.

Similarly, it is possible that m M pLq but m M pL d q but in any case, µm exists in L and L d . This is the case when a mutation m is regarded as the inmum of other mutations.

Second, we propose an algorithm to build the context of a distributive lattice from any context. However, it is only a partial solution to the problem considered in [START_REF] Priss | Representing median networks with concept lattices[END_REF]:

an algorithm for converting a concept lattice [into a median graph] consists of omitting the bottom node and then checking every principal lter for distributivity and turning it into a distributive lattice if it is not already one.

In the following, we discuss the whole process presented in [START_REF] Priss | Representing median networks with concept lattices[END_REF]. We have proposed an algorithmic approach to turning it into a distributive lattice if it is not already one. However, there is still some work to do as the suggestion in [START_REF] Priss | Representing median networks with concept lattices[END_REF] does provide suitable solutions. This is illustrated by the example given in Indeed, if we were to follow the steps suggested by Priss [START_REF] Priss | Representing median networks with concept lattices[END_REF] on this example, the procedure would not provide a correct solution (i.e., a distributive lattice for principal lters). Consider a local approach on Ò 1 and Ò 2. The rst step is to compute the reduced context of Ò 1 (since the example is symmetric for 1 and 2, we only give details for 1). The reduced context C 1 of L 1 Ò 1 can be built from CpJ, M, Iq by rst observing that 1 I ta, b, du, which entails the following context:

a b d 1 ¢ ¢ ¢ 2 ¢ 3 ¢ ¢ 4 5 ¢ 6
and that reduces to:

a b d 2 ¢ 3 ¢ ¢ 5 ¢ Algorithm 1 then returns the context C 1 d of a distributive lattice; similarly, Algorithm 1 returns context C 2 d of L 2 Ò2. C 1 d m 2 m 3 m 5 2 ¢ ¢ 3 ¢ ¢ 5 ¢ C 2 d m 1 m 4 m 6 1 ¢ ¢ 4 ¢ ¢ 6 ¢
Moreover, in the whole lattice, every m M 1 is greater than 1 and every m M 2 is greater than 2. Hence we obtain the left context for the whole lattice and the reduced context on the right:

m 2 m 3 m 5 m 1 m 4 m 6 2 ¢ ¢ ¢ ¢ ¢ 3 ¢ ¢ 5 ¢ 1 ¢ ¢ ¢ ¢ ¢ 4 ¢ ¢ 6 ¢ m 2 m 5 m 1 m 6 2 ¢ ¢ ¢ 3 ¢ ¢ 5 ¢ 1 ¢ ¢ ¢ 4 ¢ ¢ 6 ¢
The resulting lattice is presented in Figure 4.a; not every principal lter is distributive. The problem comes from the fact that the modied parts of the lattice belong to intersection of Ò1 and Ò2. The new added elements in a lter may belong to other lters, and this may break the consistency achieved in the other lters. Now we applied this procedure in parallel for Ò1 and Ò2, and someone could argue that it should be iterated lter by lter until a xed point is reached.

Nevertheless, an optimal solution cannot be found through the general procedure suggested by Priss [START_REF] Priss | Representing median networks with concept lattices[END_REF], since all lters must be considered simultaneously. In the present case, there exists an optimal solution with only one new concept. This solution is given in We are currently working on these two variations of the problem. The objective is to establish an operational bridge between FCA (concept lattices) and distributive lattices to allow the use of FCA algorithms in phylogeny.

Fig. 1

 1 Fig. 1 shows three examples of reduced contexts with arrow relations C pJ, M, I, Ó , Ò q and the corresponding concept lattices. The two rst lattices on the left are respectively named M 3 and N 5 and they are the smallest nondistributive lattices. The third lattice on the right is a distributive lattice.

Fig. 2 .

 2 Fig. 2. Illustration of Birkho's Representation Theorem.

Figure 3 . 6 ¢Fig. 3 .

 363 Figure 3.

Figure 4 Fig. 4 .Problem 1 (Problem 2 (

 4412 Fig. 4. paq Solution obtained after a local approach and optimal solution.

  Theorem 1. A lattice L is distributive if and only if one (or, equivalently, all) of the following conditions hold: 1. px yq py zq pz xq px yq py zq pz xq; 2. L does not contain neither N 5 nor M 3 as sublattice; 3. the reduced context of L with arrow relations contains exactly one doublearrow Ø in each row and in each column, and no other arrows.

  elements of poset J which are not greater than j foreach x X do I d Ð I d px, mjq To illustrate the algorithm, we use N 5 context as input. At the beginning of the algorithm, the context C d has |J| rows but zero columns. Each step of the external loop computes m j , a new -irreducible element of C d such that j Ø

m j .