We consider a number of temporal logics, some interval-based and some instant-based, and the choices that have to be made if we need to construct a computational framework for such a logic. We consider the axiomatisation of the accessibility relations of the underlying temporal structures when we are using a modal language as well as the formulation of axioms for distinguishing concepts like actions, events, processes and so on for systems using first-order languages. Finally, we briefly discuss the fields of application of temporal logics and list a number of fields that looks promising for further research. / Computer Science & Information Systems / M.Sc.(Computer Science)

<p>This thesis consists of four papers in domain theory and a summary. The first two papers deal with the problem of defining effectivity for continuous cpos. The third and fourth paper present the new notion of an admissible domain representation, where a domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to X via a continuous function from E to D. </p><p>In Paper I we define a cartesian closed category of effective bifinite domains. We also investigate the method of inducing effectivity onto continuous cpos via projection pairs, resulting in a cartesian closed category of projections of effective bifinite domains. </p><p>In Paper II we introduce the notion of an almost algebraic basis for a continuous cpo, showing that there is a natural cartesian closed category of effective consistently complete continuous cpos with almost algebraic bases. We also generalise the notion of a complete set, used in Paper I to define the bifinite domains, and investigate what closure results that can be obtained. </p><p>In Paper III we consider admissible domain representations of topological spaces. We present a characterisation theorem of exactly when a topological space has a λ-admissible and κ-based domain representation. We also show that there is a natural cartesian closed category of countably based and countably admissible domain representations. </p><p>In Paper IV we consider admissible domain representations of convergence spaces, where a convergence space is a set X together with a convergence relation between nets on X and elements of X. We study in particular the new notion of weak κ-convergence spaces, which roughly means that the convergence relation satisfies a generalisation of the Kuratowski limit space axioms to cardinality κ. We show that the category of weak κ-convergence spaces is cartesian closed. We also show that the category of weak κ-convergence spaces that have a dense, λ-admissible, κ-continuous and α-based consistently complete domain representation is cartesian closed when α ≤ λ ≥ κ. As natural corollaries we obtain corresponding results for the associated category of weak convergence spaces.</p>

This thesis consists of four papers in domain theory and a summary. The first two papers deal with the problem of defining effectivity for continuous cpos. The third and fourth paper present the new notion of an admissible domain representation, where a domain representation D of a space X is λ-admissible if, in principle, all other λ-based domain representations E of X can be reduced to X via a continuous function from E to D. In Paper I we define a cartesian closed category of effective bifinite domains. We also investigate the method of inducing effectivity onto continuous cpos via projection pairs, resulting in a cartesian closed category of projections of effective bifinite domains. In Paper II we introduce the notion of an almost algebraic basis for a continuous cpo, showing that there is a natural cartesian closed category of effective consistently complete continuous cpos with almost algebraic bases. We also generalise the notion of a complete set, used in Paper I to define the bifinite domains, and investigate what closure results that can be obtained. In Paper III we consider admissible domain representations of topological spaces. We present a characterisation theorem of exactly when a topological space has a λ-admissible and κ-based domain representation. We also show that there is a natural cartesian closed category of countably based and countably admissible domain representations. In Paper IV we consider admissible domain representations of convergence spaces, where a convergence space is a set X together with a convergence relation between nets on X and elements of X. We study in particular the new notion of weak κ-convergence spaces, which roughly means that the convergence relation satisfies a generalisation of the Kuratowski limit space axioms to cardinality κ. We show that the category of weak κ-convergence spaces is cartesian closed. We also show that the category of weak κ-convergence spaces that have a dense, λ-admissible, κ-continuous and α-based consistently complete domain representation is cartesian closed when α ≤ λ ≥ κ. As natural corollaries we obtain corresponding results for the associated category of weak convergence spaces.

The specification of enterprise information systems using formal specification languages enables the formal verification of these systems. Reasoning about the properties of a formal specification is a tedious task that can be facilitated much through the use of an automated reasoner. However, set theory is a corner stone of many formal specification languages and poses demanding challenges to automated reasoners. To this end a number of heuristics has been developed to aid the Otter theorem prover in finding short proofs for set-theoretic problems. This dissertation investigates the applicability of these heuristics to next generation theorem provers. / Computing / M.Sc. (Computer Science)

We consider a number of temporal logics, some interval-based and some instant-based, and the choices that have to be made if we need to construct a computational framework for such a logic. We consider the axiomatisation of the accessibility relations of the underlying temporal structures when we are using a modal language as well as the formulation of axioms for distinguishing concepts like actions, events, processes and so on for systems using first-order languages. Finally, we briefly discuss the fields of application of temporal logics and list a number of fields that looks promising for further research. / Computer Science and Information Systems / M.Sc.(Computer Science)

Orientador: Walter Alexandre Carnielli / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias Humanas / Made available in DSpace on 2018-08-12T22:46:52Z (GMT). No. of bitstreams: 1 Freire_RodrigodeAlvarenga_D.pdf: 761227 bytes, checksum: 3b1a0de92aa93b50f2bfc602bf6173bc (MD5) Previous issue date: 2009 / Resumo: Esta Tese tem como objetivo elucidar, ao menos parcialmente, a questão do significado da Teoria de Modelos para uma reflexão sobre o conhecimento matemático no século XX. Para isso, vamos buscar, primeiramente, alcançar uma compreensão da própria reflexão sobre o conhecimento matemático, que será denominada de Fundamentos do Pensamento Matemático no século XX, e da própria relevância fundacional. Em seguida, analisaremos, dentro do contexto fundacional estabelecido, o papel da Teoria de Modelos e da sua interação com a Álgebra, em geral, e, finalmente, empreenderemos um estudo de caso específico. Nesse estudo de caso mostraremos que a Teoria de Galois pode ser vista como um conteúdo lógico, e buscaremos compreender o significado fundacional desse enquadramento modelo-teórico para uma parte da Álgebra clássica. / Abstract: The aim of the present Thesis is to bring some light to the question about the status and relevance of Model Theory to a reflection about the mathematical knowledge in the twentieth century. To pursue this target, we will, first of all, try to reach a comprehension of the reflection about the mathematical knowledge, itself, what will be designated as Foundations of Mathematical Thought in the twentieth century, and of the foundational relevance, itself. In the sequel, we will provide an analysis, of the role of Model Theory and its interaction with Algebra, in general, within the established foundational setting and, finally, we will discuss a specific study case. In this study case we will show that Galois Theory can be seen as a logical content, and we will try to understand the foundational meaning of this model-theoretic framework for some part of classical Algebra. / Doutorado / Logica / Doutor em Filosofia

Orientador: Marcelo Esteban Coniglio / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias Humanas / Made available in DSpace on 2018-08-11T21:55:18Z (GMT). No. of bitstreams: 1 Reis_TeofilodeSouza_M.pdf: 733611 bytes, checksum: 0e64d330d9e71079eddd94de91f141c2 (MD5) Previous issue date: 2008 / Resumo: Neste trabalho apresentamos um novo formalismo de decomposição de Lógicas, as Coberturas por Traduções Possíveis, ou simplesmente CTPs. As CTPs constituem uma versão formal das Semânticas de Traduções Possíveis, introduzidas por W. Carnielli em 1990. Mostramos como a adoção de um conceito mais geral de morfismo de assinaturas proposicionais (usando multifunções no lugar de funções) nos permite definir uma categoria Sig?, na qual os conectivos, ao serem traduzidos de uma assinatura para outra, gozam de grande flexibilidade. A partir de Sig?, contruímos a categoria Log? de lógicas tarskianas e morfismos (os quais são funções obtidas a partir de um morfismo de assinaturas, isto é, de uma multifunção). Estudamos algumas características de Sig? e Log?, afim de verificar que estas categorias podem de fato acomodar as construções que pretendemos apresentar. Mostramos como definir em Log? o conjunto de traduções possíveis de uma fórmula, e a partir disto definimos a noção de CTP para uma lógica L. Por fim, exibimos um exemplo concreto de utilização desta nova ferramenta, e discutimos brevemente as possíveis abordagens para uma continuação deste trabalho. / Abstract: We present a general study of a new formalism of decomposition of logics, the Possible- Translations Coverings (in short PTC 's) which constitute a formal version of Possible-Translations Semantics, introduced by W. Carnielli in 1990. We show how the adoption of a more general notion of propositional signatures morphism allows us to define a category Sig?, in which the connectives, when translated from a signature to another one, enjoy of great flexibility. Essentially, Sig? -morphisms will be multifunctions instead of functions. From Sig? we construct the category Log? of tarskian logics and morphisms between them (these .are functions obtained from signature morphisms, that is, from multifunctions) . We show how to define in Log? the set of possible translations of a given formula, and we define the notion of a PTC for a logic L. We analyze some properties of PTC 's and give concrete examples of the above mentioned constructions. We conclude with a discussion of the approaches to be used in a possible continuation of these investigations. / Mestrado / Mestre em Filosofia

In classical propositional logic (CPL) logical reasoning is formalised as logical entailment and can be computed by means of tableau and resolution proof procedures. Unfortunately CPL is not expressive enough and using first order logic (FOL) does not solve the problem either since proof procedures for these logics are not decidable. Modal propositional logics (MPL) on the other hand are both decidable and more expressive than CPL. It therefore seems reasonable to apply tableau and resolution proof systems to MPL in order to compute logical entailment in MPL. Although some of the principles in CPL are present in MPL, there are complexities in MPL that are not present in CPL. Tableau and resolution proof systems which address these issues and others will be surveyed here. In particular the work of Abadi & Manna (1986), Chan (1987), del Cerro & Herzig (1988), Fitting (1983, 1990) and Gore (1995) will be reviewed. / Computing / M. Sc. (Computer Science)

The study constituted an investigation for concept images and mathematical reasoning of Grade 11 learners on the concepts of reflection, translation and stretch of functions. The aim was to gain awareness of any conceptions that learners have about these transformations. The researcher’s experience in high school and university mathematics teaching had laid a basis to establish the research problem. The subjects of the study were 96 Grade 11 mathematics learners from three conveniently sampled South African high schools. The non-return of consent forms by some learners and absenteeism during the days of writing by other learners, resulted in the subsequent reduction of the amount of respondents below the anticipated 100. The preliminary investigation, which had 30 learners, was successful in validating instruments and projecting how the main results would be like. A mixed method exploratory design was employed for the study, for it was to give in-depth results after combining two data collection methods; a written diagnostic test and recorded follow-up interviews. All the 96 participants wrote the test and 14 of them were interviewed. It was found that learners’ reasoning was more based on their concept images than on formal definitions. The most interesting were verbal concept images, some of which were very accurate, others incomplete and yet others exhibited misconceptions. There were a lot of inconsistencies in the students’ constructed definitions and incompetency in using graphical and symbolical representations of reflection, translation and stretch of functions. For example, some learners were misled by negative sign on a horizontal translation to the right to think that it was a horizontal translation to the left. Others mistook stretch for enlargement both verbally and contextually. The research recommends that teachers should use more than one method when teaching transformations of functions, e.g., practically-oriented and process-oriented instructions, with practical examples, to improve the images of the concepts that learners develop. Within their methodologies, teachers should make concerted effort to be aware of the diversity of ways in which their learners think of the actions and processes of reflecting, translating and stretching, the terms they use to describe them, and how they compare the original objects to images after transformations. They should build upon incomplete definitions, misconceptions and other inconsistencies to facilitate development of accurate conceptions more schematically connected to the empirical world. There is also a need for accurate assessments of successes and shortcomings that learners display in the quest to define and master mathematical concepts but taking cognisance of their limitations of language proficiency in English, which is not their first language. Teachers need to draw a clear line between the properties of stretch and enlargement, and emphasize the need to include the invariant line in the definition of stretch. To remove confusion around the effect of “–” sign, more practice and spiral testing of this knowledge could be done to constantly remind learners of that property. Lastly, teachers should find out how to use smartphones, i-phones, i-pods, tablets and other technological devices for teaching and learning, and utilize them fully to their own and the learners’ advantage in learning these and other concepts and skills / Mathematics Education / D.Phil. (Mathematics, Science and Technology Education)

In classical propositional logic (CPL) logical reasoning is formalised as logical entailment and can be computed by means of tableau and resolution proof procedures. Unfortunately CPL is not expressive enough and using first order logic (FOL) does not solve the problem either since proof procedures for these logics are not decidable. Modal propositional logics (MPL) on the other hand are both decidable and more expressive than CPL. It therefore seems reasonable to apply tableau and resolution proof systems to MPL in order to compute logical entailment in MPL. Although some of the principles in CPL are present in MPL, there are complexities in MPL that are not present in CPL. Tableau and resolution proof systems which address these issues and others will be surveyed here. In particular the work of Abadi & Manna (1986), Chan (1987), del Cerro & Herzig (1988), Fitting (1983, 1990) and Gore (1995) will be reviewed. / Computing / M. Sc. (Computer Science)