Dissertations / Theses on the topic 'Set Theory'
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Dieterly, Andrea K. "Set Theory." Bowling Green State University / OhioLINK, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1304689030.
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Corella, Francisco. "Mechanizing set theory." Thesis, University of Cambridge, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.334076.
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Atmai, Rachid. "Contributions to Descriptive Set Theory." Thesis, University of North Texas, 2015. https://digital.library.unt.edu/ark:/67531/metadc804953/.
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In this dissertation we study closure properties of pointclasses, scales on sets of reals and the models L[T2n], which are very natural canonical inner models of ZFC. We first characterize projective-like hierarchies by their associated ordinals. This solves a conjecture of Steel and a conjecture of Kechris, Solovay, and Steel. The solution to the first conjecture allows us in particular to reprove a strong partition property result on the ordinal of a Steel pointclass and derive a new boundedness principle which could be useful in the study of the cardinal structure of L(R). We then develop new methods which produce lightface scales on certain sets of reals. The methods are inspired by Jackson’s proof of the Kechris-Martin theorem. We then generalize the Kechris-Martin Theorem to all the Π12n+1 pointclasses using Jackson’s theory of descriptions. This in turns allows us to characterize the sets of reals of a certain initial segment of the models L[T2n]. We then use this characterization and the generalization of Kechris-Martin theorem to show that the L[T2n] are unique. This generalizes previous work of Hjorth. We then characterize the L[T2n] in term of inner models theory, showing that they actually are constructible models over direct limit of mice with Woodin cardinals, a counterpart to Steel’s result that the L[T2n+1] are extender models, and finally show that the generalized contiuum hypothesis holds in these models, solving a conjecture of Woodin.
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Dance, Cody. "Contributions to Descriptive Set Theory." Thesis, University of North Texas, 2016. https://digital.library.unt.edu/ark:/67531/metadc955115/.
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Assume AD+V=L(R). In the first chapter, let W^1_1 denote the club measure on \omega_1. We analyze the embedding j_{W^1_1}\restr HOD from the point of view of inner model theory. We use our analysis to answer a question of Jackson-Ketchersid about codes for ordinals less than \omega_\omega. In the second chapter, we provide an indiscernibles analysis for models of the form L[T_n,x]. We use our analysis to provide new proofs of the strong partition property on \delta^1_{2n+1}
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Longo, Cristiano. "Set theory for knowledge representation." Doctoral thesis, Università di Catania, 2012. http://hdl.handle.net/10761/1031.
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The decision problem in set theory has been intensively
investigated in the last decades, and decision procedures
or proofs of undecidability have been provided for several
quantified and unquantified fragments of set theory.
In this thesis we study the decision problem for three novel
quantified fragments of set theory, which allow the explicit manipulation
of ordered pairs.
We present a decision procedure for each
language of this family, and prove that all of these procedures are optimal (in the sense that they run
in nondeterministic polynomial-time) when restricted to formulae with quantifier
nesting bounded by a constant.
The expressive power of
languages of this family is then measured in terms of set-theoretical
constructs they allow to express. In addition, these languages can
be profitably employed in knowledge representation,
since they allow to express a large amount description logic
constructs.
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Corson, Samuel M. "Applications of Descriptive Set Theory in Homotopy Theory." BYU ScholarsArchive, 2010. https://scholarsarchive.byu.edu/etd/2401.
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This thesis presents new theorems in homotopy theory, in particular it generalizes a theorem of Saharon Shelah. We employ a technique used by Janusz Pawlikowski to show that certain Peano continua have a least nontrivial homotopy group that is finitely presented or of cardinality continuum. We also use this technique to give some relative consistency results.
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Popham, S. J. "Some studies in 'finitary' set theory." Thesis, University of Bristol, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.372022.
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Fernandes, Arias A. "The exceptional set in Nevanlinna theory." Thesis, Imperial College London, 1985. http://hdl.handle.net/10044/1/37689.
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Barton, Neil. "Executing Gödel's programme in set theory." Thesis, Birkbeck (University of London), 2017. http://bbktheses.da.ulcc.ac.uk/201/.
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The study of set theory (a mathematical theory of infinite collections) has garnered a great deal of philosophical interest since its development. There are several reasons for this, not least because it has a deep foundational role in mathematics; any mathematical statement (with the possible exception of a few controversial examples) can be rendered in set-theoretic terms. However, the fruitfulness of set theory has been tempered by two difficult yet intriguing philosophical problems: (1.) the susceptibility of naive formulations of sets to contradiction, and (2.) the inability of widely accepted set-theoretic axiomatisations to settle many natural questions. Both difficulties have lead scholars to question whether there is a single, maximal Universe of sets in which all set-theoretic statements are determinately true or false (often denoted by ‘V ’). This thesis illuminates this discussion by showing just what is possible on the ‘one Universe’ view. In particular, we show that there are deep relationships between responses to (1.) and the possible tools that can be used in resolving (2.). We argue that an interpretation of extensions of V is desirable for addressing (2.) in a fruitful manner. We then provide critical appraisal of extant philosophical views concerning (1.) and (2.), before motivating a strong mathematical system (known as‘Morse-Kelley’ class theory or ‘MK’). Finally we use MK to provide a coding of discourse involving extensions of V , and argue that it is philosophically virtuous. In more detail, our strategy is as follows: Chapter I (‘Introduction’) outlines some reasons to be interested in set theory from both a philosophical and mathematical perspective. In particular, we describe the current widely accepted conception of set (the ‘Iterative Conception’) on which sets are formed successively in stages, and remark that set-theoretic questions can be resolved on the basis of two dimensions: (i) how ‘high’ V is (i.e. how far we go in forming stages), and (ii) how ‘wide’ V is (i.e. what sets are formed at successor stages). We also provide a very coarse-grained characterisation of the set-theoretic paradoxes and remark that extensions of universes in both height and width are relevant for our understanding of (1.) and (2.). We then present the different motivations for holding either a ‘one Universe’ or ‘many universes’ view of the subject matter of set theory, and argue that there is a stalemate in the dialectic. Instead we advocate filling out each view in its own terms, and adopt the ‘one Universe’ view for the thesis. Chapter II (‘G¨odel’s Programme’) then explains the Universist project for formulating and justifying new axioms concerning V . We argue that extensions of V are relevant to both aspects of G¨odel’s Programme for resolving independence. We also identify a ‘Hilbertian Challenge’ to explain how we should interpret extensions of V , given that we wish to use discourse that makes apparent reference to such nonexistent objects. Chapter III (‘Problematic Principles’) then lends some mathematical precision to the coarse-grained outline of Chapter I, examining mathematical discourse that seems to require talk of extensions of V . Chapter IV (‘Climbing above V ?’), examines some possible interpretations of height extensions of V . We argue that several such accounts are philosophically problematic. However, we point out that these difficulties highlight two constraints on resolution of the Hilbertian Challenge: (i) a Foundational Constraint that we do not appeal to entities not representable using sets from V , and (ii) an Ontological Constraint to interpret extensions of V in such a way that they are clearly different from ordinary sets. 5 Chapter V (‘Broadening V ’s Horizons?’), considers interpretations of width extensions. Again, we argue that many of the extant methods for interpreting this kind of extension face difficulties. Again, however, we point out that a constraint is highlighted; a Methodological Constraint to interpret extensions of V in a manner that makes sense of our naive thinking concerning extensions, and links this thought to truth in V . We also note that there is an apparent tension between the three constraints. Chapter VI (‘A Theory of Classes’) changes tack, and provides a positive characterisation of apparently problematic ‘proper classes’ through the use of plural quantification. It is argued that such a characterisation of proper class discourse performs well with respect to the three constraints, and motivates the use of a relatively strong class theory (namely MK). Chapter VII (‘V -logic and Resolution’) then puts MK to work in interpreting extensions of V . We first expand our logical resources to a system called V -logic, and show how discourse concerning extensions can be thereby represented. We then show how to code the required amount of V -logic usingMK. Finally, we argue that such an interpretation performs well with respect to the three constraints. Chapter VIII (‘Conclusions’) reviews the thesis and makes some points regarding the exact dialectical situation. We argue that there are many different philosophical lessons that one might take from the thesis, and are clear that we do not commit ourselves to any one such conclusion. We finally provide some open questions and indicate directions for future research, remarking that the thesis opens the way for new and exciting philosophical and mathematical discussion.
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Ahmed, Shehzad. "Progressive Ideals in Combinatorial Set Theory." Ohio University / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1554379497651916.
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Kieftenbeld, Vincent. "Three Topics in Descriptive Set Theory." Thesis, University of North Texas, 2010. https://digital.library.unt.edu/ark:/67531/metadc28441/.
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This dissertation deals with three topics in descriptive set theory. First, the order topology is a natural topology on ordinals. In Chapter 2, a complete classification of order topologies on ordinals up to Borel isomorphism is given, answering a question of Benedikt Löwe. Second, a map between separable metrizable spaces X and Y preserves complete metrizability if Y is completely metrizable whenever X is; the map is resolvable if the image of every open (closed) set in X is resolvable in Y. In Chapter 3, it is proven that resolvable maps preserve complete metrizability, generalizing results of Sierpiński, Vaintein, and Ostrovsky. Third, an equivalence relation on a Polish space has the Laczkovich-Komjáth property if the following holds: for every sequence of analytic sets such that the limit superior along any infinite set of indices meets uncountably many equivalence classes, there is an infinite subsequence such that the intersection of these sets contains a perfect set of pairwise inequivalent elements. In Chapter 4, it is shown that every coanalytic equivalence relation has the Laczkovich-Komjáth property, extending a theorem of Balcerzak and Głąb.
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Ziegler, Albert. "Large sets in constructive set theory." Thesis, University of Leeds, 2014. http://etheses.whiterose.ac.uk/8370/.
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This thesis presents an investigation into large sets and large set axioms in the context of the constructive set theory CZF. We determine the structure of large sets by classifying their von Neumann stages and use a new modified cumulative hierarchy to characterise their arrangement in the set theoretic universe. We prove that large set axioms have good metamathematical properties, including absoluteness for the common relative model constructions of CZF and a preservation of the witness existence properties CZF enjoys. Furthermore, we use realizability to establish new results about the relative consistency of a plurality of inaccessibles versus the existence of just one inaccessible. Developing a constructive theory of clubs, we present a characterisation theorem for Mahlo sets connecting classical and constructive approaches to Mahloness and determine the amount of induction contained in the assertion of a Mahlo set. We then present a characterisation theorem for 2-strong sets which proves them to be equivalent to a logically simpler concept. We also investigate several topics connected to elementary embeddings of the set theoretic universe into a transitive class model of CZF, where considering different equivalent classical formulations results in a rich and interconnected spectrum of measurability for the constructive case. We pay particular attention to the question of cofinality of elementary embeddings, achieving both very strong cofinality properties in the case of Reinhardt embeddings and constructing models of the failure of cofinality in the case of ordinary measurable embeddings, some of which require only surprisingly low conditions. We close with an investigation of constructive principles incompatible with elementary embeddings.
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Pearsall, Sam Alfred. "The Cantor set." CSUSB ScholarWorks, 1999. https://scholarworks.lib.csusb.edu/etd-project/1528.
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Schlee, Glen A. (Glen Alan). "On the Development of Descriptive Set Theory." Thesis, University of North Texas, 1988. https://digital.library.unt.edu/ark:/67531/metadc500836/.
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In the thesis, the author traces the historical development of descriptive set theory from the work of H. Lebesgue to the introduction of projective descriptive set theory. Proofs of most of the major results are given. Topics covered include Corel lattices, universal sets, the operation A, analytic sets, coanalytic sets, and the continuum hypothesis The appendix contains a translation of the famous letters exchanged between R. Baire, E. Borel, J. Hadamard and H. Lebesgue concerning Zermelo's axiom of choice.
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Tsiknis, George Konstantinos. "Applications of a natural deduction set theory." Thesis, University of British Columbia, 1991. http://hdl.handle.net/2429/32181.
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The goal of this thesis is to demonstrate the versatility and suitability of a logic and set theory NaDSet for providing logical foundations to disparate areas of mathematics and computer science. Category theory has been chosen as the area of mathematics, while programming language semantics and semantics for the lambda calculus are the areas of computer science. In each of the three areas NaDSet provides a logical foundation using exactly the same "logistic" method: Basic concepts are defined as terms of the logic and then the logic is used to derive all theorems; no assumptions in the form of additional axioms or rules of deduction are needed. The thesis demonstrates the ease and directness with which this can be done for the three areas, suggesting that in other, as yet unexplored areas, NaDSet may prove to be equally useful.Science, Faculty of
Computer Science, Department of
Graduate
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Mao, Hongwei. "Estimating labour productivity using fuzzy set theory." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape8/PQDD_0019/MQ47065.pdf.
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Kusalik, Timothy. "The continuum hypothesis in algebraic set theory." Thesis, McGill University, 2009. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=32368.
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In "Sheaf Theory and the Continuum Hypothesis", Lawvere and Tierney proved the consistency of the negation of the continuum hypothesis with the theory of Set-like toposes. In this thesis, I generalize the Lawvere-Tierney result in two directions. Lawvere and Tierney's result relies upon the law of excluded middle and the axiom of choice, and I provide a formulation and proof of the consistency of the negation of the continuum hypothesis which abandons this assumption. Moreover, I generalize the work that's been done on the continuum hypothesis and its consistency from the context of topos theory presented in the Lawvere-Tierney proof to the context of algebraic set theory.Dans "Sheaf Theory and the Continuum Hypothesis", Lawvere et Tierney ont démontré la compatibilité de la négation de l'hypothèse du continu avec la théorie des topos qui ressemblent au Set. Dans cette thèse, j'universalise le résultat de Lawvere-Tierney dans deux directions. Le résultat de Lawvere-Tierney compte sur le principe du tiers exclu et l'axiome du choix, et je fournis une formulation et une démonstration de la consistance de la négation de l'hypothèse du continu qui abandonne cette assomption. Aussi, j'universalise tous ces résultats sur l'hypothèse du continu et sa consistance de la contexte de la théorie du topos à la contexte de la théorie algébrique des ensembles.
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Sharma, Jonathan. "STASE: set theory-influenced architecture space exploration." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/52330.
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The first of NASA's high-level strategic goals is to extend and sustain human activities across the solar system. As the United States moves into the post-Shuttle era, meeting this goal is more challenging than ever. There are several desired outcomes for this goal, including development of an integrated architecture and capabilities for safe crewed and cargo missions beyond low Earth orbit. NASA's Flexible Path for the future human exploration of space provides the guidelines to achieve this outcome.
Designing space system architectures to satisfy the Flexible Path starts early in design, when a downselection process works to reduce the broad spectrum of feasible system architectures into a more refined set that contains a handful of alternatives that are to be considered and studied further in the detailed design phases. This downselection process is supported by what is referred to as architecture space exploration (ASE). ASE is a systems engineering process which generates the design knowledge necessary to enable informed decision-making.
The broad spectrum of potential system architectures can be impractical to evaluate. As the system architecture becomes more complex in its structure and decomposition, its space encounters a factorial growth in the number of alternatives to be considered. This effect is known in the literature as combinatorial explosion. For the Flexible Path, the development of new space system architectures can occur over the period of a decade or more. During this time, a variety of changes can occur which lead to new requirements that necessitate the development of new technologies, or changes in budget and schedule. Developing comprehensive and quantitative design knowledge early during design helps to address these challenges.
Current methods focus on a small number of system architecture alternatives. From these alternatives, a series of 'one off' -type of trade studies are performed to refine and generate more design knowledge. These small-scale studies are unable to adequately capture the broad spectrum of possible architectures and typically use qualitative knowledge. The focus of this research is to develop a systems engineering method for system-level ASE during pre-phase A design that is rapid, exhaustive, flexible, traceable, and quantitative.
Review of literature found a gap in currents methods that were able to achieve this research objective. This led to the development of the Set Theory-Influenced Architecture Space Exploration (STASE) methodology. The downselection process is modeled as a decision-making process with STASE serving as a supporting systems engineering method. STASE is comprised of two main phases: system decomposition and system synthesis.
During system decomposition, the problem is broken down into three system spaces. The architecture space consists of the categorical parameters and decisions that uniquely define an architecture, such as the physical and functional aspects. The design space contains the design parameters that uniquely define individual point designs for a given architecture. The objective space holds the objectives that are used in comparing alternatives.
The application of set theory across the system spaces enables an alternative form of representing system alternatives. This novel application of set theory allows the STASE method to mitigate the problem of combinatorial explosion. The fundamental definitions and theorems of set theory are used to form the mathematical basis for the STASE method.
A series of hypotheses were formed to develop STASE in a scientific way. These hypotheses are confirmed by experiments using a proof of concept over a subset of the Flexible Path. The STASE method results are compared against baseline results found using the traditional process of representing individual architectures as the system alternatives. The comparisons highlight many advantages of the STASE method. The greatest advantage is that STASE comprehensively explores the architecture space more rapidly than the baseline. This is because the set theory-influenced representation of alternatives has a summation growth with system complexity in the architecture space. The resultant option subsets provide additional design knowledge that enables new ways of visualizing results and comparing alternatives during early design. The option subsets can also account for changes in some requirements and constraints so that new analysis of system alternatives is not required.
An example decision-making process was performed for the proof of concept. This notional example starts from the entire architecture space with the goal of minimizing the total cost and the number of launches. Several decisions are made for different architecture parameters using the developed data visualization and manipulation techniques until a complete architecture was determined. The example serves as a use-case example that walks through the implementation of the STASE method, the techniques for analyzing the results, and the steps towards making meaningful architecture decisions.
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Incurvati, Luca. "Set theory : its justification, logic and extent." Thesis, University of Cambridge, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.608586.
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Fendrich, Samuel. "From axiomatization to generalizatrion of set theory." Thesis, London School of Economics and Political Science (University of London), 1987. http://etheses.lse.ac.uk/3272/.
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The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within the mathematical approach, in the light of the significance of Cohen's Independence results.
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Lee, Cary. "Descriptive set theory of reduced abelianp-groups /." The Ohio State University, 1995. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487864485228735.
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Mecay, Stefan Terence. "Maximum-Sized Matroids with no Minors Isomorphic to U2,5, F7, F7¯, OR P7." Thesis, University of North Texas, 2000. https://digital.library.unt.edu/ark:/67531/metadc2514/.
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Let M be the class of simple matroids which do not contain the 5-point line U2,5 , the Fano plane F7 , the non-Fano plane F7- , or the matroid P7 , as minors. Let h(n) be the maximum number of points in a rank-n matroid in M. We show that h(2)=4, h(3)=7, and h(n)=n(n+1)/2 for n>3, and we also find all the maximum-sized matroids for each rank.
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Ki, Haseo Kechris A. S. Kechris A. S. "Topics in descriptive set theory related to number theory and analysis /." Diss., Pasadena, Calif. : California Institute of Technology, 1995. http://resolver.caltech.edu/CaltechETD:etd-10112007-111738.
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Gebser, Martin. "Proof theory and algorithms for answer set programming." Phd thesis, Universität Potsdam, 2011. http://opus.kobv.de/ubp/volltexte/2011/5542/.
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Answer Set Programming (ASP) is an emerging paradigm for declarative programming, in which a computational problem is specified by a logic program such that particular models, called answer sets, match solutions. ASP faces a growing range of applications, demanding for high-performance tools able to solve complex problems.
ASP integrates ideas from a variety of neighboring fields. In particular, automated techniques to search for answer sets are inspired by Boolean Satisfiability (SAT) solving approaches. While the latter have firm proof-theoretic foundations, ASP lacks formal frameworks for characterizing and comparing solving methods. Furthermore, sophisticated search patterns of modern SAT solvers, successfully applied in areas like, e.g., model checking and verification, are not yet established in ASP solving.
We address these deficiencies by, for one, providing proof-theoretic frameworks that allow for characterizing, comparing, and analyzing approaches to answer set computation. For another, we devise modern ASP solving algorithms that integrate and extend state-of-the-art techniques for Boolean constraint solving. We thus contribute to the understanding of existing ASP solving approaches and their interconnections as well as to their enhancement by incorporating sophisticated search patterns.
The central idea of our approach is to identify atomic as well as composite constituents of a propositional logic program with Boolean variables. This enables us to describe fundamental inference steps, and to selectively combine them in proof-theoretic characterizations of various ASP solving methods. In particular, we show that different concepts of case analyses applied by existing ASP solvers implicate mutual exponential separations regarding their best-case complexities. We also develop a generic proof-theoretic framework amenable to language extensions, and we point out that exponential separations can likewise be obtained due to case analyses on them.
We further exploit fundamental inference steps to derive Boolean constraints characterizing answer sets. They enable the conception of ASP solving algorithms including search patterns of modern SAT solvers, while also allowing for direct technology transfers between the areas of ASP and SAT solving. Beyond the search for one answer set of a logic program, we address the enumeration of answer sets and their projections to a subvocabulary, respectively. The algorithms we develop enable repetition-free enumeration in polynomial space without being intrusive, i.e., they do not necessitate any modifications of computations before an answer set is found.
Our approach to ASP solving is implemented in clasp, a state-of-the-art Boolean constraint solver that has successfully participated in recent solver competitions. Although we do here not address the implementation techniques of clasp or all of its features, we present the principles of its success in the context of ASP solving.Antwortmengenprogrammierung (engl. Answer Set Programming; ASP) ist ein Paradigma zum deklarativen Problemlösen, wobei Problemstellungen durch logische Programme beschrieben werden, sodass bestimmte Modelle, Antwortmengen genannt, zu Lösungen korrespondieren. Die zunehmenden praktischen Anwendungen von ASP verlangen nach performanten Werkzeugen zum Lösen komplexer Problemstellungen. ASP integriert diverse Konzepte aus verwandten Bereichen. Insbesondere sind automatisierte Techniken für die Suche nach Antwortmengen durch Verfahren zum Lösen des aussagenlogischen Erfüllbarkeitsproblems (engl. Boolean Satisfiability; SAT) inspiriert. Letztere beruhen auf soliden beweistheoretischen Grundlagen, wohingegen es für ASP kaum formale Systeme gibt, um Lösungsmethoden einheitlich zu beschreiben und miteinander zu vergleichen. Weiterhin basiert der Erfolg moderner Verfahren zum Lösen von SAT entscheidend auf fortgeschrittenen Suchtechniken, die in gängigen Methoden zur Antwortmengenberechnung nicht etabliert sind. Diese Arbeit entwickelt beweistheoretische Grundlagen und fortgeschrittene Suchtechniken im Kontext der Antwortmengenberechnung. Unsere formalen Beweissysteme ermöglichen die Charakterisierung, den Vergleich und die Analyse vorhandener Lösungsmethoden für ASP. Außerdem entwerfen wir moderne Verfahren zum Lösen von ASP, die fortgeschrittene Suchtechniken aus dem SAT-Bereich integrieren und erweitern. Damit trägt diese Arbeit sowohl zum tieferen Verständnis von Lösungsmethoden für ASP und ihrer Beziehungen untereinander als auch zu ihrer Verbesserung durch die Erschließung fortgeschrittener Suchtechniken bei. Die zentrale Idee unseres Ansatzes besteht darin, Atome und komposite Konstrukte innerhalb von logischen Programmen gleichermaßen mit aussagenlogischen Variablen zu assoziieren. Dies ermöglicht die Isolierung fundamentaler Inferenzschritte, die wir in formalen Charakterisierungen von Lösungsmethoden für ASP selektiv miteinander kombinieren können. Darauf aufbauend zeigen wir, dass unterschiedliche Einschränkungen von Fallunterscheidungen zwangsläufig zu exponentiellen Effizienzunterschieden zwischen den charakterisierten Methoden führen. Wir generalisieren unseren beweistheoretischen Ansatz auf logische Programme mit erweiterten Sprachkonstrukten und weisen analytisch nach, dass das Treffen bzw. Unterlassen von Fallunterscheidungen auf solchen Konstrukten ebenfalls exponentielle Effizienzunterschiede bedingen kann. Die zuvor beschriebenen fundamentalen Inferenzschritte nutzen wir zur Extraktion inhärenter Bedingungen, denen Antwortmengen genügen müssen. Damit schaffen wir eine Grundlage für den Entwurf moderner Lösungsmethoden für ASP, die fortgeschrittene, ursprünglich für SAT konzipierte, Suchtechniken mit einschließen und darüber hinaus einen transparenten Technologietransfer zwischen Verfahren zum Lösen von ASP und SAT erlauben. Neben der Suche nach einer Antwortmenge behandeln wir ihre Aufzählung, sowohl für gesamte Antwortmengen als auch für Projektionen auf ein Subvokabular. Hierfür entwickeln wir neuartige Methoden, die wiederholungsfreies Aufzählen in polynomiellem Platz ermöglichen, ohne die Suche zu beeinflussen und ggf. zu behindern, bevor Antwortmengen berechnet wurden.
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Rittberg, Colin Jakob. "Methods, goals and metaphysics in contemporary set theory." Thesis, University of Hertfordshire, 2016. http://hdl.handle.net/2299/17218.
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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.
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Liu, Yongwen. "Cloud services selection based on rough set theory." Thesis, Troyes, 2016. http://www.theses.fr/2016TROY0018/document.
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Avec le développement du cloud computing, de nouveaux services voient le jour et il devient primordial que les utilisateurs aient les outils nécessaires pour choisir parmi ses services. La théorie des ensembles approximatifs représente un bon outil de traitement de données incertaines. Elle peut exploiter les connaissances cachées ou appliquer des règles sur des ensembles de données. Le but principal de cette thèse est d'utiliser la théorie des ensembles approximatifs pour aider les utilisateurs de cloud computing à prendre des décisions. Dans ce travail, nous avons, d'une part, proposé un cadre utilisant la théorie des ensembles approximatifs pour la sélection de services cloud et nous avons donné un exemple en utilisant les ensembles approximatifs dans la sélection de services cloud pour illustrer la pratique et analyser la faisabilité de cette approche. Deuxièmement, l'approche proposée de sélection des services cloud permet d’évaluer l’importance des paramètres en fonction des préférences de l'utilisateur à l'aide de la théorie des ensembles approximatifs. Enfin, nous avons effectué des validations par simulation de l’algorithme proposé sur des données à large échelle pour vérifier la faisabilité de notre approche en pratique. Les résultats de notre travail peuvent aider les utilisateurs de services cloud à prendre la bonne décision et aider également les fournisseurs de services cloud pour cibler les améliorations à apporter aux services qu’ils proposent dans le cadre du cloud computingWith the development of the cloud computing technique, users enjoy various benefits that high technology services bring. However, there are more and more cloud service programs emerging. So it is important for users to choose the right cloud service. For cloud service providers, it is also important to improve the cloud services they provide, in order to get more customers and expand the scale of their cloud services.Rough set theory is a good data processing tool to deal with uncertain information. It can mine the hidden knowledge or rules on data sets. The main purpose of this thesis is to apply rough set theory to help cloud users make decision about cloud services. In this work, firstly, a framework using the rough set theory in cloud service selection is proposed, and we give an example using rough set in cloud services selection to illustrate and analyze the feasibility of our approach. Secondly, the proposed cloud services selection approach has been used to evaluate parameters importance based on the users’ preferences. Finally, we perform experiments on large scale dataset to verity the feasibility of our proposal.The performance results can help cloud service users to make the right decision and help cloud service providers to target the improvement about their cloud services
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Romanovski, Iakov. "Connections between descriptive set theory and HF-logic." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/MQ37160.pdf.
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Raghfar, Hossein. "Application of fuzzy set theory to poverty analysis." Thesis, University of Essex, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.343582.
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Artemenko, A. "The main set of foreign exchange regulation theory." Thesis, Sumy State University, 2019. https://essuir.sumdu.edu.ua/handle/123456789/77008.
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The paper is devoted to the study of the evolution of foreign exchange regulation
formation according to political and economic changes in the world. The common and
different features of "foreign exchange regulation" and "foreign exchange control"
have been indicated. The main stages of foreign exchange regulation process have been
identified and characterized.
The question of regulating the foreign exchange market as a necessary platform
for producing cross-border movement of currency values is topical in today's global
world.
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Saboo, Jai Vardhan. "An investment analysis model using fuzzy set theory." Thesis, Virginia Polytechnic Institute and State University, 1989. http://hdl.handle.net/10919/50087.
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Traditional methods for evaluating investments in state-of-the-art technology are sometimes found lacking in providing equitable recommendations for project selection. The major cause for this is the inability of these methods to handle adequately uncertainty and imprecision, and account for every aspect of the project, economic and non-economic, tangible and intangible. Fuzzy set theory provides an alternative to probability theory for handling uncertainty, while at the same time being able to handle imprecision. It also provides a means of closing the gap between the human thought process and the computer, by enabling the establishment of linguistic quantifiers to describe intangible attributes. Fuzzy set theory has been used successfully in other fields for aiding the decision-making process.
The intention of this research has been the application of fuzzy set theory to aid investment decision making. The research has led to the development of a structured model, based on theoretical algorithms developed by Buckley and others. The model looks at a project from three different standpoints- economic, operational, and strategic. It provides recommendations by means of five different values for the project desirability, and results of two sensitivity analyses. The model is tested on a hypothetical case study. The end result is a model that can be used as a basis for promising future development of investment analysis models.Master of Science
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Conley, Clinton Taylor. "Some applications of combinatorics in descriptive set theory." Diss., Restricted to subscribing institutions, 2009. http://proquest.umi.com/pqdweb?did=1876263421&sid=1&Fmt=2&clientId=1564&RQT=309&VName=PQD.
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Chen, Ray-Ming. "Independence and conservativity results for intuitionistic set theory." Thesis, University of Leeds, 2010. http://etheses.whiterose.ac.uk/1439/.
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There are two main parts to this thesis. The first part will deal with some independence results. In 1979, Lifschitz in [13] introduced a realizability interpretation for Heyting's arithmetic, HA, that could differentiate between Church's thesis with uniqueness condition, CT0!, and the general form of Church's thesis, CT0. The objective here is to extend Lifschitz' realizability to intuitionistic Zermelo-Fraenkel set theory with two sorts, IZFN. In addition to separating Church's thesis with uniqueness condition from its general form in intuitionistic set theory, I also obtain several interesting corollaries. The interpretation repudiates a weak form of countable choice, ACN2, asserting that every countable family of inhabited subsets of {0,1} has a choice function. The second part will be concerned with Constructive Zermelo-Fraenkel Set Theory and other intuitionistic set theories augmented by various principles, notably choice principles. It will be shown that the addition of these (choice) principles does not change the stock of provable arithmetical theorems. This type of conservativity result has its roots in a theorem of Goodman[9] who showed that Heyting arithmetic in all nite types augmented by the axiom of choice for all levels is conservative over HA. The technique I employ here to obtain such results for intuitionistic set theories, however, owes a lot to a paper by Beeson published in 1979. In [2] he showed how to construe Goodman's Theorem as the composition of two interpretations, namely relativized realizability and forcing. In this thesis, I adopt the same approach and employ it to a plethora of intuitionistic set theories.
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朱鏡江 and Kan-Kong Chu. "Exceptional set problems on some additive equations." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1994. http://hub.hku.hk/bib/B31212220.
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Chu, Kan-Kong. "Exceptional set problems on some additive equations /." [Hong Kong] : University of Hong Kong, 1994. http://sunzi.lib.hku.hk/hkuto/record.jsp?B14763898.
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Buquicchio, Luke J. "Variational Open Set Recognition." Digital WPI, 2020. https://digitalcommons.wpi.edu/etd-theses/1377.
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In traditional classification problems, all classes in the test set are assumed to also occur in the training set, also referred to as the closed-set assumption. However, in practice, new classes may occur in the test set, which reduces the performance of machine learning models trained under the closed-set assumption. Machine learning models should be able to accurately classify instances of classes known during training while concurrently recognizing instances of previously unseen classes (also called the open set assumption). This open set assumption is motivated by real world applications of classifiers wherein its improbable that sufficient data can be collected a priori on all possible classes to reliably train for them. For example, motivated by the DARPA WASH project at WPI, a disease classifier trained on data collected prior to the outbreak of COVID-19 might erroneously diagnose patients with the flu rather than the novel coronavirus. State-of-the-art open set methods based on the Extreme Value Theory (EVT) fail to adequately model class distributions with unequal variances. We propose the Variational Open-Set Recognition (VOSR) model that leverages all class-belongingness probabilities to reject unknown instances. To realize the VOSR model, we design a novel Multi-Modal Variational Autoencoder (MMVAE) that learns well-separated Gaussian Mixture distributions with equal variances in its latent representation. During training, VOSR maps instances of known classes to high-probability regions of class-specific components. By enforcing a large distance between these latent components during training, VOSR then assumes unknown data lies in the low-probability space between components and uses a multivariate form of Extreme Value Theory to reject unknown instances. Our VOSR framework outperforms state-of-the-art open set classification methods with a 15% F1 score increase on a variety of benchmark datasets.
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Edvardsson, Karin. "How to set ratiohnal environmental goals : theory and applications." Licentiate thesis, KTH, Philosophy and History of Technology, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3875.
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Environmental goals are commonly set to guide work towards ecological sustainability. The aim of this thesis is to develop a precise terminology for the description of goals in terms of properties that are important in their practical use as decision-guides and to illustrate how it can be used in evaluations of environmental policy.
Essay I (written together with Sven Ove Hansson) identifies a set of rationality criteria for individual goals and discusses them in relation to the typical function of goals. For a goal to perform its typical function, i.e., to guide and induce action, it must be precise, evaluable, approachable (attainable), and motivating.
Essay II argues that for a goal system to be rational it must not only satisfy the criteria identified in Essay I but should also be coherent. The coherence of a goal system is made up of the relations that hold among the goals, most notably relations of support and conflict, but possibly also relations of operationalization. A major part of the essay consists in a conceptual analysis of the three relations.
Essay III contains an investigation into the rationality of five Swedish environmental objectives through an application of the rationality criteria identified in Essays I-II. The paper draws the conclusion that the objectives are not sufficiently rational according to the suggested criteria. It also briefly points at some of the difficulties that are associated with the use of goals in environmental policy and managemen
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Ma, Ka Leung. "In solving the dominating set problem : group theory approach." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape10/PQDD_0005/NQ40311.pdf.
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Edvardsson, Karin. "How to Set Rational Environmental Goals : theory and applications." Licentiate thesis, Stockholm, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3875.
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Stammers, Diana. "Set theory in the perception of atonal pitch relations." Thesis, University of Cambridge, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.296742.
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Bhattacharyya, Kakali. "Classification of rock masses based on fuzzy set theory." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2003. http://hub.hku.hk/bib/B29490352.
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Espíndola, Christian. "Achieving completeness: from constructive set theory to large cardinals." Doctoral thesis, Stockholms universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-130537.
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This thesis is an exploration of several completeness phenomena, both in the constructive and the classical settings. After some introductory chapters in the first part of the thesis where we outline the background used later on, the constructive part contains a categorical formulation of several constructive completeness theorems available in the literature, but presented here in an unified framework. We develop them within a constructive reverse mathematical viewpoint, highlighting the metatheory used in each case and the strength of the corresponding completeness theorems. The classical part of the thesis focuses on infinitary intuitionistic propositional and predicate logic. We consider a propositional axiomatic system with a special distributivity rule that is enough to prove a completeness theorem, and we introduce weakly compact cardinals as the adequate metatheoretical assumption for this development. Finally, we return to the categorical formulation focusing this time on infinitary first-order intuitionistic logic. We propose a first-order system with a special rule, transfinite transitivity, that embodies both distributivity as well as a form of dependent choice, and study the extent to which completeness theorems can be established. We prove completeness using a weakly compact cardinal, and, like in the constructive part, we study disjunction-free fragments as well. The assumption of weak compactness is shown to be essential for the completeness theorems to hold.
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Reeder, Patrick F. "Internal Set Theory and Euler's Introductio in Analysin Infinitorum." The Ohio State University, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=osu1366149288.
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St, John Gavin. "On formally undecidable propositions of Zermelo-Fraenkel set theory." Youngstown State University / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1369657108.
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Abbas, Mujahid. "Soft Set Theory: Generalizations, Fixed Point Theorems, and Applications." Doctoral thesis, Universitat Politècnica de València, 2015. http://hdl.handle.net/10251/48470.
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Mathematical models have extensively been used in problems related to
engineering, computer sciences, economics, social, natural and medical sciences
etc. It has become very common to use mathematical tools to solve,
study the behavior and different aspects of a system and its different subsystems.
Because of various uncertainties arising in real world situations,
methods of classical mathematics may not be successfully applied to solve
them. Thus, new mathematical theories such as probability theory and fuzzy
set theory have been introduced by mathematicians and computer scientists
to handle the problems associated with the uncertainties of a model. But
there are certain deficiencies pertaining to the parametrization in fuzzy set
theory. Soft set theory aims to provide enough tools in the form of parameters
to deal with the uncertainty in a data and to represent it in a useful
way. The distinguishing attribute of soft set theory is that unlike probability
theory and fuzzy set theory, it does not uphold a precise quantity. This
attribute has facilitated applications in decision making, demand analysis,
forecasting, information sciences, mathematics and other disciplines.
In this thesis we will discuss several algebraic and topological properties
of soft sets and fuzzy soft sets. Since soft sets can be considered as setvalued
maps, the study of fixed point theory for multivalued maps on soft
topological spaces and on other related structures will be also explored.
The contributions of the study carried out in this thesis can be summarized
as follows:
i) Revisit of basic operations in soft set theory and proving some new
results based on these modifications which would certainly set a new
dimension to explore this theory further and would help to extend its
limits further in different directions. Our findings can be applied to
develop and modify the existing literature on soft topological spaces
ii) Defining some new classes of mappings and then proving the existence
and uniqueness of such mappings which can be viewed as a positive
contribution towards an advancement of metric fixed point theory
iii) Initiative of soft fixed point theory in framework of soft metric spaces
and proving the results lying at the intersection of soft set theory and
fixed point theory which would help in establishing a bridge between
these two flourishing areas of research.
iv) This study is also a starting point for the future research in the area of
fuzzy soft fixed point theory.Abbas, M. (2014). Soft Set Theory: Generalizations, Fixed Point Theorems, and Applications [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/48470
TESIS
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Anna, Slivková. "Partial closure operators and applications in ordered set theory." Phd thesis, Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, 2018. https://www.cris.uns.ac.rs/record.jsf?recordId=107201&source=NDLTD&language=en.
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In this thesis we generalize the well-known connections between closure operators, closure systems and complete lattices. We introduce a special kind of a partial closure operator, named sharp partial closure operator, and show that each sharp partial closure operator uniquely corresponds to a partial closure system. We further introduce a special kind of a partial clo-sure system, called principal partial closure system, and then prove the representation theorem for ordered sets with respect to the introduced partial closure operators and partial closure systems.Further, motivated by a well-known connection between matroids and geometric lattices, given that the notion of matroids can be naturally generalized to partial matroids (by dening them with respect to a partial closure operator instead of with respect to a closure operator), we dene geometric poset, and show that there is a same kind of connection between partial matroids and geometric posets as there is between matroids and geometric lattices. Furthermore, we then dene semimod-ular poset, and show that it is indeed a generalization of semi-modular lattices, and that there is a same kind of connection between semimodular and geometric posets as there is betweensemimodular and geometric lattices.Finally, we note that the dened notions can be applied to im-plicational systems, that have many applications in real world,particularly in big data analysis.U ovoj tezi uopštavamo dobro poznate veze između operatora zatvaranja, sistema zatvaranja i potpunih mreža. Uvodimo posebnu vrstu parcijalnog operatora zatvaranja, koji nazivamo oštar parcijalni operator zatvaranja, i pokazujemo da svaki oštar parcijalni operator zatvaranja jedinstveno korespondira parcijalnom sistemu zatvaranja. Dalje uvodimo posebnu vrstu parcijalnog sistema zatvaranja, nazvan glavni parcijalni sistem zatvaranja, a zatim dokazujemo teoremu reprezentacije za posete u odnosu na uvedene parcijalne operatore zatvaranja i parcijalne sisteme zatvaranja. Dalje, s obzirom na dobro poznatu vezu između matroida i geometrijskih mreža, a budući da se pojam matroida može na prirodan nacin uopštiti na parcijalne matroide (definišući ih preko parcijalnih operatora zatvaranja umesto preko operatora zatvaranja), definišemo geometrijske uređene skupove i pokazujemo da su povezani sa parcijalnim matroidima na isti način kao što su povezani i matroidi i geometrijske mreže. Osim toga, definišemo polumodularne uređene skupove i pokazujemo da su oni zaista uopštenje polumodularnih mreža i da ista veza postoji između polumodularnih i geometrijskih poseta kao što imamo između polumodularnih i geometrijskih mreža. Konačno, konstatujemo da definisani pojmovi mogu biti primenjeni na implikacione sisteme, koji imaju veliku primenu u realnom svetu, posebno u analizi velikih podataka.
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Berndal, Oskar. "Logical properties of morphisms between models of set theory." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-210839.
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The developments of algebraic set theory have given its models of settheory good closure properties under certain algebraic operations on thecategories which constitute the models. However, there does not yet seemto exist an established notion of morphism between such models. In thispaper, we develop a suggestion for such a notion by drawing on inspirationfrom the logical properties of the morphisms naturally arising from forcingin material set theory.
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Murali, V. "A study of universal algebras in fuzzy set theory." Thesis, Rhodes University, 1988. http://hdl.handle.net/10962/d1001983.
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This thesis attempts a synthesis of two important and fast developing branches of mathematics, namely universal algebra and fuzzy set theory. Given an abstract algebra [X,F] where X is a non-empty set and F is a set of finitary operations on X, a fuzzy algebra [I×,F] is constructed by extending operations on X to that on I×, the set of fuzzy subsets of X (I denotes the unit interval), using Zadeh's extension principle. Homomorphisms between fuzzy algebras are defined and discussed. Fuzzy subalgebras of an algebra are defined to be elements of a fuzzy algebra which respect the extended algebra operations under inclusion of fuzzy subsets. The family of fuzzy subalgebras of an algebra is an algebraic closure system in I×. Thus the set of fuzzy subalgebras is a complete lattice. A fuzzy equivalence relation on a set is defined and a partition of such a relation into a class of fuzzy subsets is derived. Using these ideas, fuzzy functions between sets, fuzzy congruence relations, and fuzzy homomorphisms are defined. The kernels of fuzzy homomorphisms are proved to be fuzzy congruence relations, paving the way for the fuzzy isomorphism theorem. Finally, we sketch some ideas on free fuzzy subalgebras and polynomial algebras. In a nutshell, we can say that this thesis treats the central ideas of universal algebras, namely subalgebras, homomorphisms, equivalence and congruence relations, isomorphism theorems and free algebra in the fuzzy set theory setting
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McKenzie, Zachiri Jason. "Automorphisms of models of set theory and extensions of NFU." Thesis, University of Cambridge, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.648220.
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Lambie-Hanson, Christopher. "Covering Matrices, Squares, Scales, and Stationary Reflection." Research Showcase @ CMU, 2014. http://repository.cmu.edu/dissertations/368.
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In this thesis, we present a number of results in set theory, particularly in the areas of forcing, large cardinals, and combinatorial set theory. Chapter 2 concerns covering matrices, combinatorial structures introduced by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. In the course of this proof and subsequent work with Sharon, Viale isolated two reflection principles, CP and S, which can hold of covering matrices. We investigate covering matrices for which CP and S fail and prove some results about the connections between such covering matrices and various square principles. In Chapter 3, motivated by the results of Chapter 2, we introduce a number of square principles intermediate between the classical and (+). We provide a detailed picture of the implications and independence results which exist between these principles when is regular. In Chapter 4, we address three questions raised by Cummings and Foreman regarding a model of Gitik and Sharon. We first analyze the PCF-theoretic structure of the Gitik-Sharon model, determining the extent of good and bad scales. We then classify the bad points of the bad scales existing in both the Gitik-Sharon model and various other models containing bad scales. Finally, we investigate the ideal of subsets of singular cardinals of countable cofinality carrying good scales. In Chapter 5, we prove that, assuming large cardinals, it is consistent that there are many singular cardinals such that every stationary subset of + reflects but there are stationary subsets of + that do not reflect at ordinals of arbitrarily high cofinality. This answers a question raised by Todd Eisworth and is joint work with James Cummings. In Chapter 6, we extend a result of Gitik, Kanovei, and Koepke regarding intermediate models of Prikry-generic forcing extensions to Radin generic forcing extensions. Specifically, we characterize intermediate models of forcing extensions by Radin forcing at a large cardinal using measure sequences of length less than. In the final brief chapter, we prove some results about iterations of w1-Cohen forcing with w1-support, answering a question of Justin Moore.
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Gutekunst, Todd M. "Subsets of finite groups exhibiting additive regularity." Access to citation, abstract and download form provided by ProQuest Information and Learning Company; downloadable PDF file, 128 p, 2008. http://proquest.umi.com/pqdweb?did=1605136271&sid=5&Fmt=2&clientId=8331&RQT=309&VName=PQD.
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