Dans cette thèse, nous proposons une méthode pour la visualisation d'ensembles chevauchant et de basé sur les diagrammes d'Euler. Les diagrammes d'Euler sont probablement les plus intuitifs pour représenter de manière schématique les ensembles qui partagent des éléments. Cette métaphore visuelle est ainsi un outil puissant en termes de visualisation d'information. Cependant, la génération automatique de ces diagrammes présente encore de nombreux problèmes difficiles. Premièrement, tous les clustering chevauchants ne peuvent pas être dessinées avec les diagrammes d'Euler classiques. Deuxièmement, la plupart des algorithmes existants permettent uniquement de représenter les diagrammes de dimensions modestes. Troisièmement, les besoins des applications réelles requièrent un processus plus fiable et plus rapide.Dans cette thèse, nous décrivons une version étendue des diagrammes d'Euler. Cette extension permet de modéliser l'ensemble des instances de la classe des clustering chevauchants. Nous proposons ensuite un algorithme automatique de génération de cette extension des diagrammes d'Euler. Enfin, nous présentons une implémentation logicielle et des expérimentations de ce nouvel algorithme. / In this thesis, we propose a method for the visualisation of overlapping sets and of fuzzy graph clusterings based on Euler diagrams.Euler diagrams are probably the most intuitive and most used method to depict sets in which elements can be shared. Such a powerful visualisation metaphor could be an invaluable visualisation tool, but the automatic generation of Euler diagrams still presents many challenging problems. First, not all instances can be drawn using standard Euler diagrams. Second, most existing algorithms focus on diagrams of modest dimensions while real-world applications typically features much larger data. Third, the generation process must be reliable and reasonably fast.In this thesis, we describe an extended version of Euler diagrams that can be produced for every input instance. We then propose an automatic procedure for the generation of such diagrams that specifically target large input instances. Finally, we present a software implementation of this method and we describe some output examples generated on real-world data.

An Euler diagram C = {c_1, c_2,..., c_n} is a collection of n simple closed curves (i.e., Jordan curves) that partition the plane into connected subsets, called regions, each of which is enclosed by a unique combination of curves. Typically, Euler diagrams are used to visualize the distribution of discrete characteristics across a sample population; in this case, each curve represents a characteristic and each region represents the sub-population possessing exactly the combination of containing curves' properties. Venn diagrams are a subclass of Euler diagrams in which there are 2^n regions representing all possible combinations of curves (e.g., two partially overlapping circles). In this dissertation, we study the Euler Diagram Generation Problem (EDGP), which involves constructing an Euler diagram with a prescribed set of regions. We describe a graph-theoretic model of an Euler diagram's structure and use this model to develop necessary-and-sufficient existence conditions. We also use the graph-theoretic model to prove that the EDGP is NP-complete. In addition, we study the related Area-Proportional Euler Diagram Generation Problem (w-EDGP), which involves constructing an Euler diagram with a prescribed set of regions, each of which has a prescribed area. We develop algorithms for constructing area-proportional Euler diagrams composed of up to three circles and rectangles, as well as diagrams with an unbounded number of curves and a region of common intersection. Finally, we present implementations of our algorithms that allow the dynamic manipulation and real-time construction of area-proportional Euler diagrams.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This work presents and discusses the extension of traditional Aristotelian syllogistics carried out by John Neville Keynes in the beginning of the twentieth century, through the introduction of a notation for negative terms into logical theory. The primary bibliography used was the fourth edition, dated 1906, of the Keynes‟s textbook on Logic Studies and Exercises in Formal Logic . Keynes has an extensional interpretation of the use of negative terms: they are understood as an extensional complement of the corresponding positive terms relative the universe of discourse; in this sense, his conception of the negation of terms obeys the Principle of Excluded Middle. The extension of traditional syllogistics by the addition of negative terms also leads to an extension of the number logical relations among the categorical propositions, as well as the number of valid immediate inferences. The Square of Oppositions is transformed into an Octagon of Oppositions, to which three new logical relations between the categorical propositions are added, namely, complementarity, sub-complementarity and contra-complementarity; the validity of these new logical relations does not require existential presupposition of any of the involved terms. Regarding immediate inferences, besides the conversion process, three new types of formal processes are obtained: obversion, contraposition (partial and total) and inversion (partial and total). To prove the validity of these formal processes, as well as of any syllogistic inference, Keynes lays out a diagrammatic method based on the well-known Euler method; in Keynes‟s method, however, negative terms are represented. In Keynes‟s version of Euler‟s diagrammatic method validity is understood as preservation of information: a collection of basic diagrams, corresponding to elementary information, is assigned to categorical propositions; and an inference is valid if, and only if, the diagrams assigned to the premises are also assigned to the conclusion. / Este trabalho apresenta e discute a ampliação da silogística tradicional aristotélica, realizada por John Neville Keynes no início do século XX, mediante a introdução de uma notação para termos negativos na teoria lógica. A bibliografia primária utilizada foi a quarta edição, datada de 1906, do manual de Lógica de Keynes Studies and Exercises in Formal Logic. Keynes tem uma interpretação extensional acerca do uso dos termos negativos: eles são entendidos como o complemento extensional do seu correspondente termo positivo em relação ao universo do discurso; neste sentido, a sua concepção da negação terminística obedece ao Princípio do Terceiro Excluído. A ampliação da silogística tradicional pelo acréscimo dos termos negativos leva, também, a uma ampliação do número de relações lógicas entre as proposições categóricas, assim como do número de inferências imediatas válidas. O Quadrado de Oposições é transformado em um Octógono de Oposições, no qual são acrescentadas três novas relações lógicas entre as proposições categóricas, a saber, a complementaridade, a subcomplementaridade, e a contracomplementaridade; a validade destas novas relações lógicas não exige o pressuposto existencial de nenhum dos termos envolvidos. Quanto às inferências imediatas, além do processo de conversão, três novos tipos destes processos formais são obtidos: a obversão, a contraposição (parcial e total) e a inversão (parcial e total). Para provar a validade destes processos formais, assim como de quaisquer inferências na silogística, Keynes apresenta um método diagramático desenvolvido a partir do conhecido método de Euler; no método de Keynes, porém, há a representação de termos negativos. Na versão de Keynes do método diagramático de Euler a validade é entendida como preservação de informação: uma coleção de diagramas básicos, correspondentes às informações elementares, é atribuída às proposições categóricas; e uma inferência é válida se, e somente se, os diagramas atribuídos às premissas também estão atribuídos à conclusão.

Specification plays a vital role in software engineering to facilitate the development of highly dependable software. The importance of specification in software development is to serve, amongst others, as a communication tool for stakeholders in the software project. The specification also adds to the understanding of operations, and describes the properties of a system. Various techniques may be used for specification work. Z is a formal specification language that is based on a strongly-typed fragment of Zermelo-Fraenkel set theory and first-order logic to provide for precise and unambiguous specifications. Z uses mathematical notation to build abstract data, which is necessary for a specification. The role of abstraction is to describe what the system does without prescribing how it should be done. Diagrams, on the other hand, have also been used in various areas, and in software engineering they could be used to add a visual component to software specifications. It is plausible that diagrams may also be used to reason in a semi-formal way about the properties of a specification. Many diagrammatic languages are based on contours and set theory. Examples of these languages are Euler-, Spider-, Venn- and Pierce diagrams. Euler diagrams form the foundation of most diagrams that are based on closed curves. Diagrams, on the other hand, have also been used in various areas, and in software engineering they could be used to add a visual component to software specifications. It is plausible that diagrams may also be used to reason in a semi-formal way about the properties of a specification. Many diagrammatic languages are based on contours and set theory. Examples of these languages are Euler-, Spider-, Venn- and Pierce diagrams. Euler diagrams form the foundation of most diagrams that are based on closed curves. The purpose of this research is to demonstrate the extent to which diagrams can be used to represent a Z specification. A case study is used to transform the specification modelled with Z language into a diagrammatic specification. Euler, spider, Venn and Pierce diagrams are combined for this purpose, to form one diagrammatic notation that is used to transform a Z specification / School of Computing / M. Sc. (Information Systems)

В настоящее время силлогистика, благодаря ее связи с естественным языком, занимает значимую позицию в широком спектре дисциплин: от когнитивных наук до исследований в области искусственного интеллекта. Ряд особенностей восприятия и обработки человеком силлогистических рассуждений часто приводит к ошибкам при решении данного рода задач. Применение интерактивной визуализации символьных контекстов с применением компьютерных моделей снижает вероятность возникновения подобных проблем. Кроме того, повышению эффективности на этапе обучения способствует введение игровых элементов. В силлогистике самыми распространенными вариантами графических представлений являются круговые диаграммы Эйлера и Венна. Первый подход, согласно ряду эмпирических результатов, оказывает более положительный эффект на результаты решения силлогистических задач. В качестве иллюстрации идеи визуальной поддержки освоения базовых принципов силлогистического вывода предложен проект программного решения, предоставляющего возможности интерактивного построения силлогизмов с использованием диаграмм Эйлера в качестве графической репрезентации. / Today syllogistic, due to its connection with natural language, finds broad application in a wide range of research fields: from cognitive sciences to AI. Some specific features of human perception and processing of syllogistic reasoning makes such kind of tasks prone to error. To diminish the impact of these cognitive biases visual interactive computer models are used as a representation of symbolic contexts. Moreover, to increase the effectiveness of learning various game elements could be applied. Euler and Venn diagrams are the most common methods of the syllogistic reasoning visual representation. There are some empirical proofs that Euler diagrams have better effect on the results of the syllogistic tasks solving. To illustrate the idea of a visual support system for syllogistic learning, we have proposed a project of software application that represents the environment for interactive syllogism construction, which relies on Euler diagrams as a visual representation.