Completeness of Pledger's modal logics of one-sorted projective and elliptic planes

Ken Pledger devised a one-sorted approach to the incidence relation of plane geometries, using structures that also support models of propositional modal logic. He introduced a modal system 12g that is valid in one-sorted projective planes, proved that it has finitely many non-equivalent modalities, and identified all possible modality patterns of its extensions. One of these extensions 8f is valid in elliptic planes. These results were presented in his doctoral dissertation. Here we show that 12g and 8f are strongly complete for validity in their intended one-sorted geometrical interpretations, and have the finite model property. The proofs apply standard technology of modal logic (canonical models, filtrations) together with a step-by-step procedure introduced by Yde Venema for constructing two-sorted projective planes

Undecidable First-Order Theories of Affine Geometries

Tarski initiated a logic-based approach to formal geometry that studies first-order structures with a ternary betweenness relation (\beta) and a quaternary equidistance relation (\equiv). Tarski established, inter alia, that the first-order (FO) theory of (R^2,\beta,\equiv) is decidable. Aiello and van Benthem (2002) conjectured that the FO-theory of expansions of (R^2,\beta) with unary predicates is decidable. We refute this conjecture by showing that for all n>1, the FO-theory of monadic expansions of (R^2,\beta) is \Pi^1_1-hard and therefore not even arithmetical. We also define a natural and comprehensive class C of geometric structures (T,\beta), where T is a subset of R^2, and show that for each structure (T,\beta) in C, the FO-theory of the class of monadic expansions of (T,\beta) is undecidable. We then consider classes of expansions of structures (T,\beta) with restricted unary predicates, for example finite predicates, and establish a variety of related undecidability results. In addition to decidability questions, we briefly study the expressivity of universal MSO and weak universal MSO over expansions of (R^n,\beta). While the logics are incomparable in general, over expansions of (R^n,\beta), formulae of weak universal MSO translate into equivalent formulae of universal MSO. This is an extended version of a publication in the proceedings of the 21st EACSL Annual Conferences on Computer Science Logic (CSL 2012).Comment: 21 pages, 3 figure

Comparing theories: the dynamics of changing vocabulary. A case-study in relativity theory

There are several first-order logic (FOL) axiomatizations of special relativity theory in the literature, all looking essentially different but claiming to axiomatize the same physical theory. In this paper, we elaborate a comparison, in the framework of mathematical logic, between these FOL theories for special relativity. For this comparison, we use a version of mathematical definability theory in which new entities can also be defined besides new relations over already available entities. In particular, we build an interpretation of the reference-frame oriented theory SpecRel into the observationally oriented Signalling theory of James Ax. This interpretation provides SpecRel with an operational/experimental semantics. Then we make precise, "quantitative" comparisons between these two theories via using the notion of definitional equivalence. This is an application of logic to the philosophy of science and physics in the spirit of Johan van Benthem's work.Comment: 27 pages, 8 figures. To appear in Springer Book series Trends in Logi

Toward knowledge-based automatic 3D spatial topological modeling from LiDAR point clouds for urban areas

Le traitement d'un très grand nombre de données LiDAR demeure très coûteux et nécessite des approches de modélisation 3D automatisée. De plus, les nuages de points incomplets causés par l'occlusion et la densité ainsi que les incertitudes liées au traitement des données LiDAR compliquent la création automatique de modèles 3D enrichis sémantiquement. Ce travail de recherche vise à développer de nouvelles solutions pour la création automatique de modèles géométriques 3D complets avec des étiquettes sémantiques à partir de nuages de points incomplets. Un cadre intégrant la connaissance des objets à la modélisation 3D est proposé pour améliorer la complétude des modèles géométriques 3D en utilisant un raisonnement qualitatif basé sur les informations sémantiques des objets et de leurs composants, leurs relations géométriques et spatiales. De plus, nous visons à tirer parti de la connaissance qualitative des objets en reconnaissance automatique des objets et à la création de modèles géométriques 3D complets à partir de nuages de points incomplets. Pour atteindre cet objectif, plusieurs solutions sont proposées pour la segmentation automatique, l'identification des relations topologiques entre les composants de l'objet, la reconnaissance des caractéristiques et la création de modèles géométriques 3D complets. (1) Des solutions d'apprentissage automatique ont été proposées pour la segmentation sémantique automatique et la segmentation de type CAO afin de segmenter des objets aux structures complexes. (2) Nous avons proposé un algorithme pour identifier efficacement les relations topologiques entre les composants d'objet extraits des nuages de points afin d'assembler un modèle de Représentation Frontière. (3) L'intégration des connaissances sur les objets et la reconnaissance des caractéristiques a été développée pour inférer automatiquement les étiquettes sémantiques des objets et de leurs composants. Afin de traiter les informations incertitudes, une solution de raisonnement automatique incertain, basée sur des règles représentant la connaissance, a été développée pour reconnaître les composants du bâtiment à partir d'informations incertaines extraites des nuages de points. (4) Une méthode heuristique pour la création de modèles géométriques 3D complets a été conçue en utilisant les connaissances relatives aux bâtiments, les informations géométriques et topologiques des composants du bâtiment et les informations sémantiques obtenues à partir de la reconnaissance des caractéristiques. Enfin, le cadre proposé pour améliorer la modélisation 3D automatique à partir de nuages de points de zones urbaines a été validé par une étude de cas visant à créer un modèle de bâtiment 3D complet. L'expérimentation démontre que l'intégration des connaissances dans les étapes de la modélisation 3D est efficace pour créer un modèle de construction complet à partir de nuages de points incomplets.The processing of a very large set of LiDAR data is very costly and necessitates automatic 3D modeling approaches. In addition, incomplete point clouds caused by occlusion and uneven density and the uncertainties in the processing of LiDAR data make it difficult to automatic creation of semantically enriched 3D models. This research work aims at developing new solutions for the automatic creation of complete 3D geometric models with semantic labels from incomplete point clouds. A framework integrating knowledge about objects in urban scenes into 3D modeling is proposed for improving the completeness of 3D geometric models using qualitative reasoning based on semantic information of objects and their components, their geometric and spatial relations. Moreover, we aim at taking advantage of the qualitative knowledge of objects in automatic feature recognition and further in the creation of complete 3D geometric models from incomplete point clouds. To achieve this goal, several algorithms are proposed for automatic segmentation, the identification of the topological relations between object components, feature recognition and the creation of complete 3D geometric models. (1) Machine learning solutions have been proposed for automatic semantic segmentation and CAD-like segmentation to segment objects with complex structures. (2) We proposed an algorithm to efficiently identify topological relationships between object components extracted from point clouds to assemble a Boundary Representation model. (3) The integration of object knowledge and feature recognition has been developed to automatically obtain semantic labels of objects and their components. In order to deal with uncertain information, a rule-based automatic uncertain reasoning solution was developed to recognize building components from uncertain information extracted from point clouds. (4) A heuristic method for creating complete 3D geometric models was designed using building knowledge, geometric and topological relations of building components, and semantic information obtained from feature recognition. Finally, the proposed framework for improving automatic 3D modeling from point clouds of urban areas has been validated by a case study aimed at creating a complete 3D building model. Experiments demonstrate that the integration of knowledge into the steps of 3D modeling is effective in creating a complete building model from incomplete point clouds

Spheres, cubes and simple

In 1929 Tarski showed how to construct points in a region-based first-order logic for space representation. The resulting system, called the geometry of solids, is a cornerstone for region-based geometry and for the comparison of point-based and region-based geometries. We expand this study of the construction of points in region-based systems using different primitives, namely hyper-cubes and regular simplexes, and show that these primitives lead to equivalent systems in dimension n ≥ 2. The result is achieved by adopting a single set of definitions that works for both these classes of figures. The analysis of our logics shows that Tarski’s choice to take sphere as the geometrical primitive might be intuitively justified but is not optimal from a technical viewpoint

3D Composer: A Software for Micro-composition

The aim of this compositional research project is to find new paradigms of expression and representation of musical information, supported by technology. This may further our understanding of how artistic intention materialises during the production of a musical work. A further aim is to create a software device, which will allow the user to generate, analyse and manipulate abstract musical information within a multi-dimensional environment. The main intent of this software and composition portfolio is to examine the process involved during the development of a compositional tool to verify how transformations applied to the conceptualisation of musical abstraction will affect musical outcome, and demonstrate how this transformational process would be useful in a creative context. This thesis suggests a reflection upon various technological and conceptual aspects within a dynamic multimedia framework. The discussion situates the artistic work of a composer within the technological sphere, and investigates the role of technology and its influences during the creative process. Notions of space are relocated in the scope of a personal compositional direction in order to develop a new framework for musical creation. The author establishes theoretical ramifications and suggests a definition for micro-composition. The main aspect focuses on the ability to establish a direct conceptual link between visual elements and their correlated musical output, ultimately leading to the design of a software called 3D-Composer, a tool for the visualisation of musical information as a means to assist composers to create works within a new methodological and conceptual realm. Of particular importance is the ability to transform musical structures in three-dimensional space, based on the geometric properties of micro-composition. The compositions Six Electroacoustic Studies and Dada 2009 display the use of the software. The formalisation process was derived from a transposition of influences of the early twentieth century avant-garde period, to a contemporary digital studio environment utilising new media and computer technologies for musical expression

Sparse octree algorithms for scalable dense volumetric tracking and mapping

This thesis is concerned with the problem of Simultaneous Localisation and Mapping (SLAM), the task of localising an agent within an unknown environment and at the same time building a representation of it. In particular, we tackle the fundamental scalability limitations of dense volumetric SLAM systems. We do so by proposing a highly efficient hierarchical data-structure based on octrees together with a set of algorithms to support the most compute-intensive operations in typical volumetric reconstruction pipelines. We employ our hierarchical representation in a novel dense pipeline based on occupancy probabilities. Crucially, the complete space representation encoded by the octree enables to demonstrate a fully integrated system in which tracking, mapping and occupancy queries can be performed seamlessly on a single coherent representation. While achieving accuracy either at par or better than the current state-of-the-art, we demonstrate run-time performance of at least an order of magnitude better than currently available hierarchical data-structures. Finally, we introduce a novel multi-scale reconstruction system that exploits our octree hierarchy. By adaptively selecting the appropriate scale to match the effective sensor resolution in both integration and rendering, we demonstrate better reconstruction results and tracking accuracy compared to single-resolution grids. Furthermore, we achieve much higher computational performance by propagating information up and down the tree in a lazy fashion, which allow us to reduce the computational load when updating distant surfaces. We have released our software as an open-source library, named supereight, which is freely available for the benefit of the wider community. One of the main advantages of our library is its flexibility. By carefully providing a set of algorithmic abstractions, supereight enables SLAM practitioners to freely experiment with different map representations with no intervention on the back-end library code and crucially, preserving performance. Our work has been adopted by robotics researchers in both academia and industry.Open Acces

Suitably underspecified: systematic notations and the relations between paper and music

Through building a taxonomy of drawing, and a set of four drawing research studies aimed at generating innovative cross-disciplinary practices, an argument will be developed that systematised drawings such as the music notation are hybrid representational environments, sufficiently different from other inscriptive practices as to merit a separate classification. The taxonomical model will decentralise specific modes of drawing, in favour of a multi-disciplinary view appropriate to the persistence of its subject as a deeply rooted strategic and executive practice, and the four studies will engage the time-factoring of notation systems as transductive environments, setting the conditions for innovative practices both in and outside of the frame of the inscription

Logic and intuition in architectural modelling: philosophy of mathematics for computational design

This dissertation investigates the relationship between the shift in the focus of architectural modelling from object to system and philosophical shifts in the history of mathematics that are relevant to that change. Particularly in the wake of the adoption of digital computation, design model spaces are more complex, multidimensional, arguably more logical, less intuitive spaces to navigate, less accessible to perception and visual comprehension. Such spatial issues were encountered much earlier in mathematics than in architectural modelling, with the growth of analytical geometry, a transition from Classical axiomatic proofs in geometry as the basis of mathematics, to analysis as the underpinning of geometry. Can the computational design modeller learn from the changing modern history, philosophy and psychology of mathematics about the construction and navigation of computational geometrical architectural system model space? The research is conducted through a review of recent architectural project examples and reference to three more detailed architectural modelling case studies. The spatial questions these examples and case studies raise are examined in the context of selected historical writing in the history, philosophy and psychology of mathematics and space. This leads to conclusions about changes in the relationship of architecture and mathematics, and reflections on the opportunities and limitations for architectural system models using computation geometry in the light of this historical survey. This line of questioning was motivated as a response to the experience of constructing digital associative geometry models and encountering the apparent limits of their flexibility as the graph of dependencies grew and the messiness of the digital modelling space increased. The questions were inspired particularly by working on the Narthex model for the Sagrada Família church, which extends to many tens of thousands of relationships and constraints, and which was modelled and repeatedly partially remodelled over a very long period. This experience led to the realisation that the limitations of the model were not necessarily the consequence of poor logical schema definition, but could be inevitable limitations of the geometry as defined, regardless of the means of defining it, the ‘shape’ of the multidimensional space being created. This led to more fundamental questions about the nature of Space, its relationship to geometry and the extent to which the latter can be considered simply as an operational and notational system. This dissertation offers a purely inductive journey, offering evidence through very selective examples in architecture, architectural modelling and in the philosophy of mathematics. The journey starts with some questions about the tendency of the model space to break out and exhibit unpredictable and not always desirable behaviour and the opportunities for geometrical construction to solve these questions is not conclusively answered. Many very productive questions about computational architectural modelling are raised in the process of looking for answers