A spatial logic is any formal language with geometric interpretation. Research on region-based spatial logics, where variables are set to range over certain subsets of geometric space, have been investigated recently within the qualitative spatial reasoning paradigm in AI. We axiomatised the theory of (ROQ(R 2), conv, ≤) , where ROQ(R 2) is the set of regular open rational polygons of the real plane; conv is the convexity property and ≤ is the inclusion relation. We proved soundness and completeness theorems. We also proved several expressiveness results. Additionally, we provide a historical and philosophical overview of the topic and present contemporary results relating to affine spatial logics.

Traditional recursion theory is the study of computable functions on the natural numbers. This thesis considers recursion theory in first-order arithmetical theories and categories, thus expanding the work of Ritchie and Young, Lambek, Scott, and Hofstra. We give a complete characterisation of the representability of computable functions in arithmetical theories, paying attention to the differences between intuitionistic and classical theories and between theories with and without induction. When considering recursion theory from a category-theoretic perspective, we examine syntactic categories of arithmetical theories. In this setting, we construct a strong parameterised natural numbers object and give necessary and sufficient conditions to construct a Turing category associated to an intuitionistic arithmetical theory with induction.

<p>This thesis is divided into two parts: the Theory of Items, and the Theory of Contexts. The latter is a further elaboration of the former.</p> <p>In the first chapter I argue against the classical doctrine of ontological-referential theory. This classical position may be represented by Russell's and by Quine's position on nonexistent objects.</p> <p>The first position that I propose to reject is the view that in order to say anything true about an object its name or description must have an actual reference. This view is represented by Russell's proposition *14.21: t-:Ψ (rx)(øx) .->. E!(rX)(øx) on which Russell writes: "This proposition shows that if any true statement can be made about (1x)(øx), then (1x)(øx) must exist". (Principia Mathematica)</p> <p>The Theory of Items rejects this view and states that whether a statement about a certain object is true or is false does not depend on the ontic status of that object. Thus, consequently, a true statement about nonexistent objects can be made (without making a distinction between a secondary and a primary occurrence as Russell did).</p> <p>The second position that is to be rejected is the view that nonexistent objects are mere nothings. This is represented by one of Quine's theorems that nonexistent objects are simply empty sets. *197 t- r -(Eβ) (α) (α=β. ≡ ø) ->. (1α)ø = 9¬</p> <p>(Mathematical Logic). For the Theory of Items, nonexistent objects are not nothings, they are somethings for they can be said to have any property whatsoever. Thus if we may have a set that contains existent objects, then we may also have a set that contains nonexistent objects. Nonexistent objects are just as much 'items' as existent ones; this is the reason why I call the theory being proposed here the 'Theory of Items' and not the Theory of Objects. The word 'item' is used instead of 'objects' to indicate the ontic neutrality of the matter that we are talking about.</p> <p>In the second chapter I will present various examples of the classical view and I will try reply to their arguments in the light of the Theory of Items explained previously.</p> <p>In the third chapter I will discuss the Theory of Contexts. I will argue that semantical features (truth and falsity) should be assigned to various statements about various items (existent or nonexistent). I maintain that the assignment of a truth value is very much context-dependent. The characteristics of contexts and various rules that iv. govern them will be discussed. More attention will be given to the fictional items and fictional contexts for no doubt they present some peculiar problems. For example if a fictional item x in a story C1 has a feature that-p, and the same item in a different story C2 has a feature that--~p, then can we validly conclude that the fictional item x is both p and ~p? My argument is based on the analysis of contexts. Only by presenting a satisfactory theory of contexts can that problem (and many other paradoxes) be solved.</p> <p>This thesis is far from being complete. There are some important topics that I do not discuss (due to page and time limitation). For example the problems of: significance and nonsignificance; whether we should take a three value logic (by incorporating significance as the third value) instead of the classical two value system; consistencies; and a possible formal theory for the Theory of Contexts. The last two of these problems are stated very briefly in the Appendix.</p> / Master of Arts (MA)

Orientador: Marcelo Esteban Coniglio / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas / Made available in DSpace on 2018-08-17T19:11:26Z (GMT). No. of bitstreams: 1 Silvestrini_LuizHenriquedaCruz_D.pdf: 1289416 bytes, checksum: aa5f4929e7149a647d28c4fd4df86874 (MD5) Previous issue date: 2011 / Resumo: Mikenberg, da Costa e Chuaqui (1986) introduziram a noção de quase-verdade por meio da noção de estruturas parciais, e para tanto, conceberam os predicados como ternas. O arcabouço conceitual resultante proporcionou o emprego de estruturas parciais na ciência, pois, em geral, não sabemos tudo a respeito de um determinado domínio de conhecimento. Generalizamos a noção de predicados como ternas para fórmulas complexas. A partir desta nova abordagem, obtemos uma definição de quase-verdade via noção de satisfação pragmática de uma fórmula A em uma estrutura parcial E. Introduzimos uma lógica subjacente à nossa nova definição de quase-verdade, a saber, a lógica paraconsistente trivalente LPT1, a qual possui uma axiomática de primeira ordem. Relacionamos a noção de quase-verdade com algumas lógicas paraconsistentes já existentes. Defendemos que a formalização das Sociedades Abertas, introduzidas por Carnielli e Lima-Marques (1999), quando combinada com quantificadores modulados, introduzidos por Grácio (1999), constitui uma alternativa para capturar a componente indutiva presente na atividade científica, e mostramos, a partir disso, que a proposta original de da Costa e colaboradores pode ser explicada em termos da nova noção de sociedades moduladas / Abstract: Newton da Costa and his collaborators have introduced the notion of quasi-truth by means of partial structures, and for this purpose, they conceived the predicates as ordered triples: the set of tuples which satisfies, does not satisfy and can satisfy or not the predicate, respectively (the latter represents lack of information). This approach provides a conceptual framework to analyse the use of (first-order) structures in science in contexts of informational incompleteness. In this Thesis, the notion of predicates as triples is extended recursively to any complex formula of the first-order object language. From this, a new definition of quasi-truth via the notion of pragmatic satisfaction is obtained. We obtain the proof-theoretic counterpart of the logic underlying our new definition of quasi-truth, namely, the three-valued paraconsistent logic LPT1, which is presented axiomatically in a first-order language. We relate the notion of quasi-truth with some existing paraconsistent logics. We defend that the formalization of (open) society semantics when combined with the modulated quantifiers constitutes an alternative to capture the inductive component present in scientific activity, and show, from this, that the original proposal of da Costa and collaborators can be explained in terms of the new concept of modulated societies / Doutorado / Filosofia / Doutor em Filosofia

Thesis (MSc)--Stellenbosch University, 1993. / ENGLISH ABSTRACT: Formal methods aim to apply the rigour of mathematical logic to the problem ofguaranteeing that the behaviour of (critical) software conforms to predetermined requirements. The application of formal methods during program construction centers around a formal specification of the required behaviour of the program. A development attempt is successful if the resulting program can be formally proven to conform to its specification. For any substantial program, this entails a great deal of effort. Thus, some research efforts have been directed at providing mechanical support for the application of formal methods to software development. E.W. Dijkstra's calculus of weakest precondition predicate transformers [39,38] represents one of the first attempts to use program correctness requirements to guide program development in a formal manner. / AFRIKAANSE OPSOMMING: Formele metodes poog om die strengheid van wiskundige logika te gebruik om te waarborg dat die gedrag van (kritiese) programmatuur voldoen aan gegewe vereistes. Die toepassing van formele metodes tydens programontwikkeling sentreer rondom a formele spesifikasie van die verlangde programgedrag. 'n Ontwikkelingspoging is suksesvol as daar formee1 bewys kan word dat die resulterende program aan sy spesifikasie voldoen. Vir enige substansiële program, verteenwoordig dit ‘n aansienlike hoeveelheid werk. Verskeie navorsinspoging is gerig op die daarstelling van meganiese ondersteuning vir die gebruik van formele metodes tydens ontwikkeling van sagteware. E. W. Dijkstra se calculus van swakste voorkondisie (“weakest precondition”) predikaattransformators [39,38] is een van die eerste pogings om vereistes vir programkorrektheid op ‘n formele en konstruktiewe wyse tydens programontwikkeling te gebruik.

This thesis explores the geometric principles underlying many of the known Trichotomy Theorems. The main aims are to unify the field construction in non-linear o-minimal structures and generalizations of Zariski Geometries as well as to pave the road for completely new results in this direction. In the first part of this thesis we introduce a new axiomatic framework in which all the relevant structures can be studied uniformly and show that these axioms are preserved under elementary extensions. A particular focus is placed on the study of a smoothness condition which generalizes the presmoothness condition for Zariski Geometries. We also modify Zilber's notion of universal specializations to obtain a suitable notion of infinitesimals. In addition, families of curves and the combinatorial geometry of one-dimensional structures are studied to prove a weak trichotomy theorem based on very weak one-basedness. It is then shown that under suitable additional conditions groups and group actions can be constructed in canonical ways. This construction is based on a notion of ``geometric calculus'' and can be seen in close analogy with ordinary differentiation. If all conditions are met, a definable distributive action of one one-dimensional type-definable group on another are obtained. The main result of this thesis is that both o-minimal structures and generalizations of Zariski Geometries fit into this geometric framework and that the latter always satisfy the conditions required in the group constructions. We also exhibit known methods that allow us to extract fields from this. In addition to unifying the treatment of o-minimal structures and Zariski Geometries, this also gives a direct proof of the Trichotomy Theorem for "type-definable" Zariski Geometries as used, for example, in Hrushovski's proof of the relative Mordell-Lang conjecture.

This thesis investigates the connections between henselian valuations and absolute Galois groups. There are fundamental links between these: On one hand, the absolute Galois group of a field often encodes information about (henselian) valuations on that field. On the other, in many cases a henselian valuation imposes a certain structure on an absolute Galois group which makes it easier to study. We are particularly interested in the question of when a field admits a non-trivial parameter-free definable henselian valuation. By a result of Prestel and Ziegler, this does not hold for every henselian valued field. However, improving a result by Koenigsmann, we show that there is a non-trivial parameter-free definable valuation on every henselian valued field. This allows us to give a range of conditions under which a henselian field does indeed admit a non-trivial parameter-free definable henselian valuation. Most of these conditions are in fact of a Galois-theoretic nature. Throughout the thesis, we discuss a number of applications of our results. These include fields elementarily characterized by their absolute Galois group, model complete henselian fields and henselian NIP fields of positive characteristic, as well as PAC and hilbertian fields.

<p>The thesis consists of three independent parts.</p><p>Part I: Polynomial amoebas</p><p>We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1.</p><p>Part II: Differential equations in the complex plane</p><p>We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform.</p><p>Part III: Radon transforms and tomography</p><p>This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.</p>

Description logics (DLs) are knowledge representation formalisms with well-understood model-theoretic semantics and computational properties. The DL SROIQ provides the logical underpinning for the semantic web language OWL 2, which is quickly becoming the standard for knowledge representation on the web. A central component of most DL applications is an efficient and scalable reasoner, which provides services such as consistency testing and classification. Despite major advances in DL reasoning algorithms over the last decade, however, ontologies are still encountered in practice that cannot be handled by existing DL reasoners. We present a novel reasoning calculus for the description logic SROIQ which addresses two of the major sources of inefficiency present in the tableau-based reasoning calculi used in state-of-the-art reasoners: unnecessary nondeterminism and unnecessarily large model sizes. Further, we describe a new approach to classification which exploits partial information about the subsumption relation between concept names to reduce both the number of individual subsumption tests performed and the cost of working with large ontologies; our algorithm is applicable to the general problem of deducing a quasi-ordering from a sequence of binary comparisons. We also present techniques for extracting partial information about the subsumption relation from the models generated by constructive DL reasoning methods, such as our hypertableau calculus. Empirical results from a prototypical implementation demonstrate substantial performance improvements compared to existing algorithms and implementations.

In recent years the field of real numbers expanded by a multiplicative subgroup has been studied extensively. In this thesis, the known results will be extended to expansions of the real field. I will consider the structure R consisting of the field of real numbers and an irrational power function. Using Schanuel conditions, I will give a first-order axiomatization of expansions of R by a dense multiplicative subgroup which is a subset of the real algebraic numbers. It will be shown that every definable set in such a structure is a boolean combination of existentially definable sets and that these structures have o-minimal open core. A proof will be given that the Schanuel conditions used in proving these statements hold for co-countably many real numbers. The results mentioned above will also be established for expansions of R by dense multiplicative subgroups which are closed under all power functions definable in R. In this case the results hold under the assumption that the Conjecture on intersection with tori is true. Finally, the structure consisting of R and the discrete multiplicative subgroup 2^{Z} will be analyzed. It will be shown that this structure is not model complete. Further I develop a connection between the theory of Diophantine approximation and this structure.