Because of the importance of undecidability and the concern with the high complexity of automated reasoning, a few interval-based temporal description logics (ITDLs) have been designed. Moreover, most existing ITDLs are not able to specify the lengths of intervals. In other words, they are not metric. On the other hand, some domains (e.g., medicine) are inherently interval-based, and require a metric logic in order to formalize defined processes and to check process consistency. Hence, a metric interval-based temporal description logic is required. In this thesis, we introduce such a logic (MITDL) along with two algorithms for the satisfiability checking of its formulas. We first introduce an interval-based temporal logic, called IMPNL, inspired by Metric Propositional Neighbourhood Logic. We also present a sound, com- plete and terminating tableau-based algorithm for checking the satisfiability of IMPNL formulas. Afterwards, we combine a restricted version of IMPNL (IMPNL without a negation operator) with the ALC description logic to form a MITDL. We propose two tableau-based algorithms for checking the satisfia- bility of MITDL formulas. We show and prove they are sound, complete and terminate. These algorithms have PSpace and 2NExp-Time complexities. As a proof of concept, we use IMPNL and MITDL to model some clinical practice guidelines (CPG) and check their consistency. We compare MITDL with several languages commonly used for modeling CPGs. / Thesis / Doctor of Science (PhD)

Acumen is a language and tool for modeling and simulating cyber-physical systems. It allows the user to conduct simulations using a technique called rigorous simulation that produces results with explicit error bounds, expressed as intervals. This feature can be useful when designing and testing systems where the reliability of results or taking uncertainty into account is important. Unfortunately, analyzing these simulation results can be difficult, as Acumen supports only two ways of presenting them: raw data tables and 2D-plots. These views of the data make certain kinds of analysis cumbersome, such as understanding correlations between variables. This is especially true when the model in question is large. This project proposes a new way of visualising rigorous simulation results in Acumen. The goal of this project is to create a method for visualising intervallic values in 3D, and implement it in Acumen. To achieve that, every span of values is represented as a series of overlapping objects. This family of objects, which constitutes an under-approximation of the true simulation result, is then wrapped inside a semi-translucent box that is a conservative over-approximation of the simulation result. The resulting implementation makes for a combination of mathematical correctness (rigour), and mediation of intervals in question. It enables the user to explore the results of his rigorous simulations as conveniently as with the existing, non-rigorous simulation methods, using the 3D visualisation to simplify the study of real-life problems. To our knowledge, no existing software features visualisation of interval-based simulation results, nor is there any convention for doing this. Some ways in which the proposed solution could be improved are suggested at the end of this report

Cette thèse contribue au développement de formalismes efficaces pour représenter l’information incertaine. Les formalismes existants tels que la théorie des probabilités ou la théorie des possibilités sont parmi les cadres les plus connus et utilisés pour représenter ce type d’information. Différentes extensions (e.g. théorie des probabilités imprécises, théorie des possibilités à intervalles) ont été proposées pour traiter des informations incomplètes ou des connaissances mal-connues, ainsi que pour raisonner avec les connaissances d’un groupe d’experts. Les contributions de cette thèse sont divisées en deux parties. Dans la première partie, nous développons le conditionnement dans le cadre des possibilités à intervalles et dans le cadre des possibilités ensemblistes. Conditionner dans le cadre standard diffère que l’on considère l’échelle possibiliste qualitative ou quantitative. Notre travail traite les deux définitions du conditionnement possibiliste. Ce qui nous amène à étudier une nouvelle extension de la logique possibiliste, définie comme logique possibiliste ensembliste, et son opérateur de conditionnement dans le cadre possibiliste qualitatif. Ces résultats, plus spécialement en termes de complexité, nous amène à étudier les transformations, plus précisément des transformations du cadre probabiliste vers le cadre possibiliste. En effet, nous analysons des propriétés les tâches de raisonnement comme la marginalisation et le conditionnement. Nous nous attaquons aussi aux transformations des probabilités imprécises vers les possibilités avec un intérêt particulier pour l’inférence MAP. / This thesis contributes to the development of efficient formalisms to handle uncertain information. Existing formalisms such as probability theory or possibility theory are among the most known and used settings to represent such information. Extensions and generalizations (e.g. imprecise probability theory, interval-based possibilistic theory) have been provided to handle uncertainty such as incomplete and ill-known knowledge and reasoning with the knowledge of a group of experts. We are particularly interested in reasoning tasks within these theories such as conditioning. The contributions of this thesis are divided in two parts. In the first part, we tackle conditioning in interval-based possibilistic framework and set-valued possibilistic framework. The purpose is to develop a conditioning machinery for interval-based possibilistic logic. Conditioning in a standard possibilistic setting differs whether we consider a qualitative or quantitative scale. Our works deal with both definitions of possibilistic conditioning. This leads us to investigate a new extension of possibilisticlogic, defined as set-valued possibilistic logic, and its conditioning machinery in the qualitative possibilistic setting. These results, especially in terms of complexity, lead us to study transformations, more precisely from probability to possibility theories. The second part of our contributions deals with probability-possibility transformation procedures. Indeed, we analyze properties of reasoning tasks such as conditioning and marginalization. We also tackle transformations from imprecise probability theory to possibility theory with a particular interest in MAP inference.

Electric power systems continue to increase in complexity because of the deployment of market mechanisms, the integration of renewable generation and distributed energy resources (DER) (e.g., wind and solar), the penetration of electric vehicles and other price sensitive loads. These revolutionary changes and the consequent increase in uncertainty and dynamicity call for significant modifications to power system operation models including unit commitment (UC), economic load dispatch (ELD) and optimal power flow (OPF). Planning and operation of these ???smart??? electric grids are expected to be impacted significantly, because of the intermittent nature of various supply and demand resources that have penetrated into the system with the recent advances. The main focus of this thesis is on the application of the Affine Arithmetic (AA) method to power system operational problems. The AA method is a very efficient and accurate tool to incorporate uncertainties, as it takes into account all the information amongst dependent variables, by considering their correlations, and hence provides less conservative bounds compared to the Interval Arithmetic (IA) method. Moreover, the AA method does not require assumptions to approximate the probability distribution function (pdf) of random variables. In order to take advantage of the AA method in power flow analysis problems, first a novel formulation of the power flow problem within an optimization framework that includes complementarity constraints is proposed. The power flow problem is formulated as a mixed complementarity problem (MCP), which can take advantage of robust and efficient state-of-the-art nonlinear programming (NLP) and complementarity problems solvers. Based on the proposed MCP formulation, it is formally demonstrated that the Newton-Raphson (NR) solution of the power flow problem is essentially a step of the traditional General Reduced Gradient (GRG) algorithm. The solution of the proposed MCP model is compared with the commonly used NR method using a variety of small-, medium-, and large-sized systems in order to examine the flexibility and robustness of this approach. The MCP-based approach is then used in a power flow problem under uncertainties, in order to obtain the operational ranges for the variables based on the AA method considering active and reactive power demand uncertainties. The proposed approach does not rely on the pdf of the uncertain variables and is therefore shown to be more efficient than the traditional solution methodologies, such as Monte Carlo Simulation (MCS). Also, because of the characteristics of the MCP-based method, the resulting bounds take into consideration the limits of real and reactive power generation. The thesis furthermore proposes a novel AA-based method to solve the OPF problem with uncertain generation sources and hence determine the operating margins of the thermal generators in systems under these conditions. In the AA-based OPF problem, all the state and control variables are treated in affine form, comprising a center value and the corresponding noise magnitudes, to represent forecast, model error, and other sources of uncertainty without the need to assume a pdf. The AA-based approach is benchmarked against the MCS-based intervals, and is shown to obtain bounds close to the ones obtained using the MCS method, although they are slightly more conservative. Furthermore, the proposed algorithm to solve the AA-based OPF problem is shown to be efficient as it does not need the pdf approximations of the random variables and does not rely on iterations to converge to a solution. The applicability of the suggested approach is tested on a large real European power system.

L’optimisation globale fiable est dédiée à la recherche d’un minimum global en présence d’erreurs d’arrondis. Les seules approches fournissant une preuve numérique d’optimalité sont des méthodes d’intervalles qui partitionnent l’espace de recherche et éliminent les sous-espaces qui ne peuvent contenir de solution optimale. Ces méthodes exhaustives, appelées branch and bound par intervalles, sont étudiées depuis les années 60 et ont récemment intégré des techniques de réfutation et de contraction, issues des communautés d’analyse par intervalles et de programmation par contraintes. Il est d’une importance cruciale de calculer i) un encadrement précis de la fonction objectif et des contraintes sur un sous-domaine ; ii) une bonne approximation (un majorant) du minimum global. Les solveurs de pointe sont généralement des méthodes intégratives : ils invoquent sur chaque sous-domaine des algorithmes d’optimisation locale afin d’obtenir une bonne approximation du minimum global. Dans ce document, nous nous intéressons à un cadre coopératif combinant des méthodes d’intervalles et des algorithmes évolutionnaires. Ces derniers sont des algorithmes stochastiques faisant évoluer une population de solutions candidates (individus) dans l’espace de recherche de manière itérative, dans l’espoir de converger vers des solutions satisfaisantes. Les algorithmes évolutionnaires, dotés de mécanismes permettant de s’échapper des minima locaux, sont particulièrement adaptés à la résolution de problèmes difficiles pour lesquels les méthodes traditionnelles peinent à converger. Au sein de notre solveur coopératif Charibde, l’algorithme évolutionnaire et l’algorithme sur intervalles exécutés en parallèle échangent bornes, solutions et espace de recherche par passage de messages. Une stratégie couplant une heuristique d’exploration géométrique et un opérateur de réduction de domaine empêche la convergence prématurée de la population vers des minima locaux et évite à l’algorithme évolutionnaire d’explorer des sous-espaces sous-optimaux ou non réalisables. Une comparaison de Charibde avec des solveurs de pointe (GlobSol, IBBA, Ibex) sur une base de problèmes difficiles montre un gain de temps d’un ordre de grandeur. De nouveaux résultats optimaux sont fournis pour cinq problèmes multimodaux pour lesquels peu de solutions, même approchées, sont connues dans la littérature. Nous proposons une application aéronautique dans laquelle la résolution de conflits est modélisée par un problème d’optimisation sous contraintes universellement quantifiées, et résolue par des techniques d’intervalles spécifiques. Enfin, nous certifions l’optimalité de la meilleure solution connue pour le cluster de Lennard-Jones à cinq atomes, un problème ouvert en dynamique moléculaire. / Reliable global optimization is dedicated to finding a global minimum in the presence of rounding errors. The only approaches for achieving a numerical proof of optimality in global optimization are interval-based methods that interleave branching of the search-space and pruning of the subdomains that cannot contain an optimal solution. The exhaustive interval branch and bound methods have been widely studied since the 1960s and have benefitted from the development of refutation methods and filtering algorithms, stemming from the interval analysis and interval constraint programming communities. It is of the utmost importance: i) to compute sharp enclosures of the objective function and the constraints on a given subdomain; ii) to find a good approximation (an upper bound) of the global minimum. State-of-the-art solvers are generally integrative methods, that is they embed local optimization algorithms to compute a good upper bound of the global minimum over each subspace. In this document, we propose a cooperative framework in which interval methods cooperate with evolutionary algorithms. The latter are stochastic algorithms in which a population of individuals (candidate solutions) iteratively evolves in the search-space to reach satisfactory solutions. Evolutionary algorithms, endowed with operators that help individuals escape from local minima, are particularly suited for difficult problems on which traditional methods struggle to converge. Within our cooperative solver Charibde, the evolutionary algorithm and the intervalbased algorithm run in parallel and exchange bounds, solutions and search-space via message passing. A strategy combining a geometric exploration heuristic and a domain reduction operator prevents premature convergence toward local minima and prevents the evolutionary algorithm from exploring suboptimal or unfeasible subspaces. A comparison of Charibde with state-of-the-art solvers based on interval analysis (GlobSol, IBBA, Ibex) on a benchmark of difficult problems shows that Charibde converges faster by an order of magnitude. New optimality results are provided for five multimodal problems, for which few solutions were available in the literature. We present an aeronautical application in which conflict solving between aircraft is modeled by an universally quantified constrained optimization problem, and solved by specific interval contractors. Finally, we certify the optimality of the putative solution to the Lennard-Jones cluster problem for five atoms, an open problem in molecular dynamics.

The complexity in multiscale systems design is significantly greater than in conventional systems because in addition to interactions between components, couplings between physical phenomena and scales are also important. This complexity amplifies two design challenges: a) complexity of coupled simulation models prohibits design space exploration, and b) unavailability of complete simulation models that capture all the interactions. Hence, the challenge in design of multiscale systems lies in managing this complexity and utilizing the available simulation models and information in an efficient manner to support effective decision-making. In order to address this challenge, our primary hypothesis is that the information and computational resources can be utilized in an efficient manner by designing design-processes (meta-design) along with the products. The primary hypothesis is embodied in this dissertation as a framework for integrated design of products and design processes. The framework consists of three components 1) a Robust Multiscale Design Exploration Method (RMS-DEM), 2) information-economics based metrics and methods for simplification of complex design processes and refinement of simulation models, and 3) an information modeling strategy for implementation of the theoretical framework into a computational environment. The framework is validated using the validation-square approach that consists of theoretical and empirical validation. Empirical validation of the framework is carried out using various examples including: pressure vessel design, datacenter cooling system design, linear cellular alloy design, and multifunctional energetic structural materials design. The contributions from this dissertation are categorized in three research domains: a) multiscale design methodology, b) materials design, and c) computer-based support for collaborative, simulation-based multiscale design. In the domain of design methodology, new methods and metrics are developed for integrating the design of products and design processes. The methods and metrics are applied in the field of materials design to develop design-processes and specifications for Multifunctional Energetic Structural Materials. In the domain of computer-based support for design, an information modeling strategy is developed to provide computational support for meta-design. Although the framework is developed in the context of multiscale systems it is equally applicable to design of any other complex system.

Ce mémoire porte sur le fruit de mon travail accompli lors de ma maîtrise en composition instrumentale de l’Université de Montréal. J’y présente mon approche poétique envers lacomposition où des expériences imaginées de paysages et de déroulements sont évoqués par des éléments musicaux. J’y discute également des méthodes que j’utilise pour parvenir aux agencements intervalliques afin de créer ces expériences. J’y traiterai aussi de ma nouvelle approche intervallique vis-à-vis la modalité à travers les modes multicycliques, la majorité de mon travail ayant été le développement de ce système. J’y aborderai ensuite les explorations hors mode dans des stades moins avancés, mais qui m’ont amené des solutions pour des procédés intervalliques moins réalisables dans un contexte strictement modal, à travers des traitements de cellules, des jeux d’intervalles purs et des sonorités provenant des idiomes folkloriques Européen de l’est, précisément des Balkans. Enfin, j’analyse également la forme de chaque œuvre composée lors de ma maîtrise en approfondissant davantage sur l’imaginaire extramusical qui détermine la forme ainsi que le caractère de chaque groupe thématique, y compris le choix du mode s’il y a lieu. / This memoir elaborates on the outcome of my work during my master's degree in the instrumental composition program of Université de Montréal. It begins with a presentation of my poetic approach towards composition, where experiences of landscapes and happenings within them are illustrated through musical devices. What follows is a discussion of all methods I apply to harness the power of intervals to create these experiences. A significant chapter will be devoted to a rather intervallic approach towards modality in the form of multicyclical modes, as most of my work of this master's degree focuses on development of their use. The next chapter focus on all my explorations outside the modal structures, which are at earlier stages of development but have already given me solutions to procedures that are less applicable in a strictly modal context. That includes the treatment of cells, working with just pure intervals and integration of colours derived from the folk traditions of Eastern Europe, more specifically the Balkans. Finally, there will be a chapter that analyzes every individual work included here while digging deeper into the extra-musical imagination behind every work, determining its form and the character of all the sections including the mode of choice if there is one.