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The nature of uncertainty cannot be generically defined as it is domain and context specific. With that being the case, there have been several proposed models, all of which have their own associated benefits and shortcomings. From these models, it was decided that an R-fuzzy approach would provide for the most ideal foundation from which to enhance and expand upon. An R-fuzzy set can be seen as a relatively new model, one which itself is an extension to fuzzy set theory. It makes use of a lower and upper approximation bounding from rough set theory, which allows for the membership function of an R-fuzzy set to be that of a rough set. An R-fuzzy approach provides the means for one to encapsulate uncertain fuzzy membership values, based on a given abstract concept. If using the voting method, any fuzzy membership value contained within the lower approximation can be treated as an absolute truth. The fuzzy membership values which are contained within the upper approximation, may be the result of a singleton, or the vast majority, but absolutely not all. This thesis has brought about the creation of a significance measure, based on a variation of Bayes' theorem. One which enables the quantification of any contained fuzzy membership value within an R-fuzzy set. Such is the pairing of the significance measure and an R-fuzzy set, an intermediary bridge linking to that of a generalised type-2 fuzzy set can be achieved. Simply by inferencing from the returned degrees of significance, one is able to ascertain the true significance of any uncertain fuzzy membership value, relative to other encapsulated uncertain values. As an extension to this enhancement, the thesis has also brought about the novel introduction of grey analysis. By utilising the absolute degree of grey incidence, it provides one with the means to measure and quantify the metric spaces between sequences, generated based on the returned degrees of significance for any given R-fuzzy set. As it will be shown, this framework is ideally suited to domains where perceptions are being modelled, which may also contain several varying clusters of cohorts based on any number of correlations. These clusters can then be compared and contrasted to allow for a more detailed understanding of the abstractions being modelled.

This thesis confronts Penelope Maddy's Second Philosophical study of set theory with a philosophical analysis of a part of contemporary set-theoretic practice in order to argue for three features we should demand of our philosophical programmes to study mathematics. In chapter 1, I argue that the identification of such features is a pressing philosophical issue. Chapter 2 presents those parts of the discursive reality the set theorists are currently in which are relevant to my philosophical investigation of set-theoretic practice. In chapter 3, I present Maddy's Second Philosophical programme and her analysis of set-theoretic practice. In chapters 4 and 5, I philosophically investigate contemporary set-theoretic practice. I show that some set theorists are having a debate about the metaphysical status of their discipline{ the pluralism/non-pluralism debate{ and argue that the metaphysical views of some set theorists stand in a reciprocal relationship with the way they practice set theory. As I will show in chapter 6, these two stories are disharmonious with Maddy's Second Philosophical account of set theory. I will use this disharmony to argue for three features that our philosophical programmes to study mathematics should have: they should provide an anthropology of mathematical goals; they should account for the fact that mathematical practices can be metaphysically laden; they should provide us with the means to study contemporary mathematical practices.

We can classify several quite different calculi for automated reasoning in first-order logic as instantiation-based methods (IMs). Broadly speaking, unlike in traditional calculi such as resolution where the first-order satisfiability problem is tackled by deriving logical conclusions, IMs attempt to reduce the first-order satisfiability problem to propositional satisfiability by intelligently instantiating clauses. The Inst-Gen-Eq method is an instantiation-based calculus which is complete for first-order clause logic modulo equality. Its distinctive feature is that it combines first-order reasoning with efficient ground satisfiability checking, which is delegated in a modular way to any state-of-the-art ground solver for satisfiability modulo theories (SMT). The first-order reasoning modulo equality employs a superposition-style calculus which generates the instances needed by the ground solver to refine a model of a ground abstraction or to witness unsatisfiability. The thesis addresses the main issue in the Inst-Gen-Eq method, namely efficient extraction of instances, while providing powerful redundancy elimination techniques. To that end we introduce a novel labelled unit superposition calculus with sets, AND/OR trees and ordered binary decision diagrams (OBDDs) as labels. The different label structures permit redundancy elimination each to a different extent. We prove completeness of redundancy elimination from labels and further integrate simplification inferences based on term rewriting. All presented approaches, in particular the three labelled calculi are implemented in the iProver-Eq system and evaluated on standard benchmark problems.

Text in Afrikaans, abstract in Afrikaans and English / Die gebruik van simbole maak wiskunde eenvoudiger en kragtiger, maar ook moeiliker verstaanbaar. Laasgenoemde kan voorkom word as slegs eenvoudige en noodsaaklike simbole gebruik word, met die verduidelikings en motiverings in woorde. Die krag van simbole le veral in die feit dat simbole as substitute vir konsepte kan dien. Omdat die krag van simbole hierin le, skuil daar 'n groot gevaar in die gebruik van simbole. Wanneer simbole los is van sinvolle verstandsvoorstellings, is daar geen krag in simbole nie. Dit is die geval met die huidige benadering in skoolalgebra. Voordat voldoende verstandsvoorstellings opgebou is, word daar op die manipulasie van simbole gekonsentreer. Die algebraiese historiese-kenteoretiese perspektief maak algebra meer betekenisvol vir leerders. Hiervolgens moet die leerlinge die geleentheid gegun word om oplossings in prosavorm te skryf en self hul eie wiskundige simbole vir idees spontaan in te voer. Hulle moet self die voordeel van algebraiese simbole beleef. / The use of symbols in algebra both simplifies and strengthens the subject, but it also increases its level of complexity.This problem can be prevented if only simple and essential symbols are used and if the explanations are fully verbalised. The power of symbols stems from their potential to be used as substitutes for concepts. As this constitutes the crux of mathematical symbolic representation, it also presents a danger in that the symbols may not be comprehended. If symbols are not related to mental representations, the symbols are meaningless. This is the case in the present approach to algebra. Before sufficient mental representations are built, there is a concentration on the manipulation of symbols. The algebraic historical epistemological perspective makes algebra more meaningful for learners. Learners should be granted the opportunities to write their solutions in prose and to develop their own symbols for concepts. / Mathematics Education / M. Sc. (Wiskunde-Onderwys)

Text in Afrikaans, abstract in Afrikaans and English / Die gebruik van simbole maak wiskunde eenvoudiger en kragtiger, maar ook moeiliker verstaanbaar. Laasgenoemde kan voorkom word as slegs eenvoudige en noodsaaklike simbole gebruik word, met die verduidelikings en motiverings in woorde. Die krag van simbole le veral in die feit dat simbole as substitute vir konsepte kan dien. Omdat die krag van simbole hierin le, skuil daar 'n groot gevaar in die gebruik van simbole. Wanneer simbole los is van sinvolle verstandsvoorstellings, is daar geen krag in simbole nie. Dit is die geval met die huidige benadering in skoolalgebra. Voordat voldoende verstandsvoorstellings opgebou is, word daar op die manipulasie van simbole gekonsentreer. Die algebraiese historiese-kenteoretiese perspektief maak algebra meer betekenisvol vir leerders. Hiervolgens moet die leerlinge die geleentheid gegun word om oplossings in prosavorm te skryf en self hul eie wiskundige simbole vir idees spontaan in te voer. Hulle moet self die voordeel van algebraiese simbole beleef. / The use of symbols in algebra both simplifies and strengthens the subject, but it also increases its level of complexity.This problem can be prevented if only simple and essential symbols are used and if the explanations are fully verbalised. The power of symbols stems from their potential to be used as substitutes for concepts. As this constitutes the crux of mathematical symbolic representation, it also presents a danger in that the symbols may not be comprehended. If symbols are not related to mental representations, the symbols are meaningless. This is the case in the present approach to algebra. Before sufficient mental representations are built, there is a concentration on the manipulation of symbols. The algebraic historical epistemological perspective makes algebra more meaningful for learners. Learners should be granted the opportunities to write their solutions in prose and to develop their own symbols for concepts. / Mathematics Education / M. Sc. (Wiskunde-Onderwys)

Les systèmes robotiques et autonomes ne cessent d’évoluer et deviennent de plus en plus impliqués dans les missions à coût considérable et/ou dans les milieux humains. Par conséquent, les simulations et campagnes de tests ne sont plus adaptées à la problématique de sûreté et fiabilité des systèmes robotiques et autonomes compte tenu (i) du caractère sérieux des défaillances éventuelles dans les contextes susmentionnés (un dommage à un robot très coûteux ou plus dramatiquement une atteinte aux vies humaines) et (ii) de la nature non exhaustive de ces techniques (les tests et simulations peuvent toujours passer à côté d’un scénario d’exécution catastrophique.Les méthodes formelles, quant à elles, peinent à s’imposer dans le domaine de la robotique autonome, notamment au niveau fonctionnel des robots, i.e. les composants logiciels interagissant directement avec les capteurs et les actionneurs. Elle est due à plusieurs facteurs. D’abord, les composants fonctionnels reflètent un degré de complexité conséquent, ce qui mène souvent à une explosion combinatoire de l’espace d’états atteignables (comme l’exploration se veut exhaustive). En outre, les composants fonctionnels sont décrits à travers des languages et frameworks informels (ROS, GenoM, etc.). Leurs spécifications doivent alors être traduites en des modèles formels avant de pouvoir y appliquer les méthodes formelles associées. Ceci est souvent pénible, lent, exposé à des erreurs, et non automatique, ce qui implique un investissement dans le temps aux limites de la rentabilité. Nous proposons, dans cette thèse, de connecter GenoM3, un framework de développement et déploiement de composants fonctionnels robotiques, à des langages formels et leurs outils de vérification respectifs. Cette connexion se veut automatique: nous développons des templates en mesure de traduire n’importe quelle spécification de GenoM3 en langages formels. Ceci passe par une formalisation de GenoM3: une sémantique formelle opérationnelle est donnée au langage. Une traduction à partir de cette sémantique est réalisée vers des langages formels et prouvée correcte par bisimulation. Nous comparons de différents langages cibles, formalismes et techniques et tirerons les conclusions de cette comparaison. La modélisation se veut aussi, et surtout, efficace. Un modèle correct n’est pas forcément utile. En effet, le passage à l’échelle est particulièrement important.Cette thèse porte donc sur l'applicabilité des méthodes formelles aux compo-sants fonctionnels des systèmes robotiques et autonomes. Le but est d'aller vers des robots autonomes plus sûrs avec un comportement plus connu et prévisible. Cela passe par la mise en place d'un mécanisme de génération automatique de modèles formels à partir de modules fonctionnels de sys-tèmes robotiques et autonomes. Les langages et outils cibles sont Fiacre/TINA et UPPAAL (model checking), UPPAAL-SMC (statistical model checking), BIP/RTD-Finder (SAT solving), et BIP/Engine (enforcement de propriétés en ligne). Les modèles générés sont exploités pour vérifier des propriétés quali-tatives ou temps-réel, souvent critiques pour les systèmes robotiques et auto-nomes considérés. Parmi ces propriétés, on peut citer, à titre d'exemple, l'ordonnançabilité des tâches périodiques, la réactivité des tâches spora-diques, l'absence d’interblocages, la vivacité conditionnée (un évènement tou-jours finit par suivre un autre), la vivacité conditionnée bornée (un évène-ment toujours suit un autre dans un intervalle de temps borné), l'accessibilité (des états “indésirables” ne sont jamais atteints), etc.La thèse propose éga-lement une analyse du feedback expérimental afin de guider les ingénieurs à exploiter ces méthodes et techniques de vérification efficacement sur les mo-dèles automatiquement générés. / The goal of this thesis is to add to the efforts toward the long-sought objective of secure and safe robots with predictable and a priori known behavior. For the reasons given above, formal methods are used to model and verify crucial properties, with a focus on the functional level of robotic systems. The approach relies on automatic generation of formal models targeting several frameworks. For this, we give operational semantics to a robotic framework, then several mathematically proven translations are derived from such semantics. These translations are then automatized so any robotic functional layer specification can be translated automatically and promptly to various frameworks/languages. Thus, we provide a mathematically correct mapping from functional components to verifiable models. The obtained models are used to formulate and verify crucial properties (see examples above) on real-world complex robotic and autonomous systems. This thesis provides also a valuable feedback on the applicability of formal frameworks on real-world, complex systems and experience-based guidelines on the efficient use of formal-model automatic generators. In this context, efficiency relates to, for instance, how to use the different model checking tools optimally depending on the properties to verify, what to do when the models do not scale with model checking (e.g. the advantages and drawbacks of statistical model checking and runtime verification and when to use the former or the latter depending on the type of properties and the order of magnitude of timing constraints).

This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour.

Automated deduction procedures for modal logics, and related decidable fragments of first-order logic, are used in many real-world applications. A popular way of obtaining decision procedures for these logics is to base them on semantic tableau calculi. We focus on calculi that use unification, instead of the more widely employed approach of generating ground instantiations over the course of a derivation. The most common type of tableaux with unification are so-called free-variable tableaux, where variables are treated as global to the entire tableau. A long-standing open problem for procedures based on free-variable tableaux is how to ensure fairness, in the sense that "equivalent" applications of the closure rule are prevented from being done over and over again. Some solutions such as using depth-first iterative deepening are known, but those are unnecessary in theory, and not very efficient in practice. This is a main reason why there are hardly any decision procedures for modal logics based on free-variable tableaux. In this thesis, we review existing work on incorporating unification into first-order and modal tableau procedures, show how the closure fairness problem arises, and discuss existing solutions to it. For the first-order case, we outline a calculus which addresses the closure fairness problem. As opposed to free-variable tableaux, closure fairness is much easier to achieve in disconnection tableaux and similar clausal calculi. We therefore focus on using clausal first-order tableau calculi for decidable classes, in particular the two-variable fragment. Using the so-called unrestricted blocking mechanism for enforcing termination, we present the first ground tableau decision procedure for this fragment. Even for such a ground calculus, guaranteeing that depth-first terminations terminate is highly non-trivial. We parametrise our procedure by a so-called lookahead amount, and prove that this parameter is crucial for determining whether depth-first derivations terminate or not. Extending these ideas to tableaux with unification, we specify a preliminary disconnection tableau procedure which uses a non-grounding version of the unrestricted blocking rule.

We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0.