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1
Albuquerque, Hugo Cardoso. "Operators and strong versions of sentential logics in Abstract Algebraic Logic." Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/394003.
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This dissertation presents the results of our research on some recent devel-opments in Abstract Algebraic Logic (AAL), namely on the Suszko operator, the Leibniz filters, and truth-equational logics. Part I builts and develops an abstract framework which unifies under a common treatment the study of the Leibniz, Suszko, and Frege operators in AAL. Part II generalizes the theory of the strong version of protoalgebraic logics, started in, to arbitrary sentential logics.
The interplay between several Leibniz- and Suszko-related notions led us to consider a general framework based upon the notion of S-operator (inspired by that of "mapping compatible with S-filters" of Czelakowski), which encompasses the Leibniz, Suszko, and Frege operators. In particular, when applied to the Leibniz and Suszko operators, new notions of Leibniz and Suszko S-filters arise as instances of more general concepts inside the abstract framework built. The former generalizes the existing notion of Leibniz filter for protoalgebraic logics to arbitrary logics, while the latter is introduced here for the first time. Sev-eral results, both known and new, follow quite naturally inside this framework, again by instantiating it with the Leibniz and Suszko operators. Among the main new results, we prove a General Correspondence Theorem (Theorem ??), which generalizes Blok and Pigozzi's well-known Correspondence Theorem for protoalgebraic logics, as well as Czelakowski's less known Correspondence The-orem for arbitrary logics. We characterize protoalgebraic logics in terms of the Suszko operator as those logics in which the Suszko operator commutes with inverse images by surjective homomorphisms (Theorem ??). We characterize truth-equational logics in terms of their (Suszko) S-filters (Theorem ??), in terms of their full g-models (Corollary ??), and in terms of the Suszko operator, a characterization which strengthens that of Raftery, as those logics in which the Suszko operator is a structural representation from the set of S-filters to the set of AIg(S)-relative congruences, on arbitrary algebras (Theorem ??). Finally, we prove a new Isomorphism Theorem for protoalgebraic logics (Theorem ??), in the same spirit of the famous one for algebraizable logics and for weakly algebraizable logics.
Endowed with a notion of Leibniz filter applicable to any logic, we are able to generalize the theory of the strong version of a protoalgebraic logic developed by Font and Jansana to arbitrary sentential logics. Given a sentential logic 5, its strong version St is the logic induced by the class of matrices whose truth set is Leibniz filter. We study three definability criteria of Leibniz filters: equational, explicit and logical definability. Under (any of) these assumptions, we prove that the St-filters coincide with Leibniz S-filters on arbitrary algebras. Finally, we apply the general theory developed to a wealth of non-protoalgebraic log-ics covered in the literature. Namely, we consider Positive Modal Logic P,A4,C, Belnap's logic B, the subintuitionistic logics w1C, and Visser's logic VP,C, and Lukasiewicz's infinite-valued logic preserving degrees of truth. We also consider the generalization of the last example mentioned to logics preserving degrees of truth from varieties of integral commutative residuated lattices, and further generalizations to the non-integral case, as well as to the case without multi-plicative constant. We classify all the examples investigated inside the Leibniz and Frege hierarchies. While none of the logics studied is protoalgebraic, all the respective strong versions are truth-equational.Aquesta dissertació presenta els resultats de la nostra recerca sobre alguns temes recents en Lògica Algebraica Abstracta (LAA), concretament, l'operador de Suszko, els filtres de Leibniz, i les lògiques truth-equacionals. La interacció entre vàries nocións relacionades amb els operadors de Leibniz i de Suszko ens va portar a considerar un marc general basat en la noció de S-operador, que abasta els operadors de Leibniz, de Suszko, i de Frege, unificant així aquests tres operadors paradigmàtics de la LAA sota un mateix tractament.
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Engstrom, Ronald W. Retzer Kenneth A. "The effects of logic on achievement in intermediate algebra." Normal, Ill. : Illinois State University, 1988. http://www.mlb.ilstu.edu/articles/dissertations/8818710.PDF.
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Thesis (D.A.)--Illinois State University, 1988.Title from title page screen, viewed Oct. 13, 2004. Dissertation Committee: Kenneth A. Retzer (chair), Lynn H. Brown, John A. Dossey, Lotus D. Hershberger, Albert D. Otto, Walter D. Pierce. Includes bibliographical references (leaves 97-102) and abstract. Also available in print.
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Esteban, María. "Duality Theory and Abstract Algebraic Logic." Doctoral thesis, Universitat de Barcelona, 2013. http://hdl.handle.net/10803/125336.
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In this thesis we present the results of our research on duality theory for non-classical logics under the point of view of Abstract Algebraic Logic (AAL). Firstly, we propose an abstract Spectral-like duality and an abstract Priestley-style duality for every filter distributive finitary congruential logic with theorems. This proposal aims to unify the various dualities for concrete logics that we find in the literature, by showing the abstract template in which all of them fit. Secondly, the dual correspondence of some logical properties is examined. This serves to reveal the connection between our abstract dualities and the concrete dualities related wot concrete logics. We apply those results to get new dualities for suitable expansions of a well-known logic: the implicative fragment of intuitionistic logic. Finally, we develop a new strategy that can be modularly applied to simplify some of the dualities obtained.
The first part of the dissertation is devoted to introduce the preliminaries and the basic notation. In Chapter 1 we fix the mathematical concepts that we assume the reader is familiar with. Of particular interest is the section in which we introduce the basic concepts of AAL, such as "S-filter" or "S-algebra". The notion of "closure operator" plays a fundamental role in AAL, as well as in our dissertation. The notions of filter and ideal associated with a closure operator, and the separation lemmas between them are studied in detail in Chapter 2. Moreover, we briefly review the literature on duality theory for non classical logics in Chapter 3.
In the second part of the dissertation we present an abstract view of the duality theory for non-classical logics. In Chapter 4 we review previous works on this topic, in which our work relies, and we introduce the notions of "referential algebra", "irreducible and optimal S-filter" and "S-semilattice". This lead us to identify a set of necessary conditions that a logic should satisfy in order to develop a Spectra-like/Priestley-style duality for it. These conditions are: "filter distributivity","congruentiality", "finitarity" and "having theorems". Moreover, we carry out a brief digression in which we argue how those notions can also be used to develop an abstract theory of canonical extensions.
The core of the proposed theory consists of the definitions of dual objects and morphisms, for the category of S-algebras and homomorphisms, for any logic S that satisfies the mentioned properties. In Chapter 5 we define a Spectral-like duality and a Priestley-style duality for filter distributive finitary congruential logics with theorems, and we prove the respective duality theorems. Due to the abstraction of our approach, we obtain that the objects of both categories involved in the duality posses algebraic nature. However, through the analysis of the dual correspondence of several well-known logical properties, we can simplify the definitions of the dual categories, provided the logic under consideration satisfies such good logical properties. This analysis is interesting under the point of view of AAL, since our results can be regarded as bridge theorems between logical properties and properties of a Kripke-style semantics. And it is also interesting under the point of view of duality theory, since it confirms the strength of duality theory, that can be developed in a modular way beyond the distributive lattice setting. Moreover, our analysis shows the connection of the general theory proposed with the concrete results that we find in the literature, and lead us to explore the applications of such general theory to obtain new dualities.En esta tesis se presentan los resultados de nuestra investigación acerca de la teoría de la dualidad para lógicas no clásicas desde el punto de vista de la Lógica Algebráica Abstracta (LAA). Un estudio preliminar de las distintas nociones de filtros e ideales lógicos asociados a las álgebras de una lógica cualquiera, y los lemas de separación entre dichas nociones nos lleva a proponer una dualidad abstracta de tipo espectral, y otra de tipo Priestley, para cada lógica congruencial, filtro distributiva, finitaria y con teoremas. Esta propuesta pretende unificar las distintas dualidades de tipo espectral y de tipo Priestley para lógicas no clásicas que encontramos en la literatura, mostrando el esquema abstracto en el que todas ellas encajan e identificando. En segundo lugar es examinada la correspondencia dual de algunas propiedades lógicas, como la propiedad de la conjunción, la propiedad de la disyunción, el teorema de deducción, la propiedad del elemento inconsistente o la propiedad de introducción de la modalidad. Esto sirve, por una parte, para revelar la conexión que existe entre las dualidades abstractas propuestas y las dualidades concretas relacionadas con lógicas no clásicas que habían sido estudiadas previamente, y por otra parte, para obtener nuevas dualidades. Centrándonos en el fragmento implicativo de la lógica intuicionista y en sus expansiones que son filtro distributivas, congruenciales, finitarias y con teoremas, mostramos cómo las dualidades que habían sido estudiadas para algunas de esas lógicas se pueden obtener como casos particulares de la teoría general. Además obtenemos nuevas dualidades para varias de dichas expansiones, algunas de las cuales pueden ser simplificadas dado que las lógicas tienen buenas propiedades. Finalmente, desarrollamos una nueva estrategia que puede ser aplicada de forma modular para simplificar algunas de las dualidades obtenidas. En conclusión, en esta tesis se muestra que la Lógica Algebráica Abstracta provee un marco general teórico apropiado para desarrollar una teoría abstracta de la dualidad para lógicas no clásicas. Dicha teoría uniformiza los diferentes resultados de la literatura, y de ella se deducen nuevos resultados.
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Pretnar, Matija. "Logic and handling of algebraic effects." Thesis, University of Edinburgh, 2010. http://hdl.handle.net/1842/4611.
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In the thesis, we explore reasoning about and handling of algebraic effects. Those are computational effects, which admit a representation by an equational theory. Their examples include exceptions, nondeterminism, interactive input and output, state, and their combinations. In the first part of the thesis, we propose a logic for algebraic effects. We begin by introducing the a-calculus, which is a minimal equational logic with the purpose of exposing distinct features of algebraic effects. Next, we give a powerful logic, which builds on results of the a-calculus. The types and terms of the logic are the ones of Levy’s call-by-push-value framework, while the reasoning rules are the standard ones of a classical multi-sorted first-order logic with predicates, extended with predicate fixed points and two principles that describe the universality of free models of the theory representing the effects at hand. Afterwards, we show the use of the logic in reasoning about properties of effectful programs, and in the translation of Moggi’s computational ¸-calculus, Hennessy-Milner logic, and Moggi’s refinement of Pitts’s evaluation logic. In the second part of the thesis, we introduce handlers of algebraic effects. Those not only provide an algebraic treatment of exception handlers, but generalise them to arbitrary algebraic effects. Each such handler corresponds to a model of the theory representing the effects, while the handling construct is interpreted by the homomorphism induced by the universal property of the free model. We use handlers to describe many previously unrelated concepts from both theory and practice, for example CSS renaming and hiding, stream redirection, timeout, and rollback.
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Townsend, Brian E. "Examining secondary students algebraic reasoning flexibility and strategy use /." Diss., Columbia, Mo. : University of Missouri-Columbia, 2005. http://hdl.handle.net/10355/4131.
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Thesis (Ph. D.)--University of Missouri-Columbia, 2005.The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file viewed on (November 14, 2006) Vita. Includes bibliographical references.
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Lu, Weiyun. "Topics in Many-valued and Quantum Algebraic Logic." Thesis, Université d'Ottawa / University of Ottawa, 2016. http://hdl.handle.net/10393/35173.
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Introduced by C.C. Chang in the 1950s, MV algebras are to many-valued (Łukasiewicz) logics what boolean algebras are to two-valued logic. More recently, effect algebras were introduced by physicists to describe quantum logic. In this thesis, we begin by investigating how these two structures, introduced decades apart for wildly different reasons, are intimately related in a mathematically precise way. We survey some connections between MV/effect algebras and more traditional algebraic structures. Then, we look at the categorical structure of effect algebras in depth, and in particular see how the partiality of their operations cause things to be vastly more complicated than their totally defined classical analogues. In the final chapter, we discuss coordinatization of MV algebras and prove some new theorems and construct some new concrete examples, connecting these structures up (requiring a detour through effect algebras!) to boolean inverse semigroups.
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Phillips, Caitlin. "An algebraic approach to dynamic epistemic logic." Thesis, McGill University, 2010. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=86767.
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In reasoning about multi-agent systems, it is important to look beyond the realm of propositional logic and to reason about the knowledge of agents within the system, as what they know about the environment will affect how they behave. A useful tool for formalizing and analyzing what agents know is epistemic logic, a modal logic developed by philosophers in the early 1960s. Epistemic logic is key to understanding knowledge in multi-agent systems, but insufficient if one wishes to study how the agents' knowledge changes over time. To do this, it is necessary to use a logic that combines dynamic and epistemic modalities, called dynamic epistemic logic. Some formalizations of dynamic epistemic logic use Kripke semantics for the states and actions, while others take a more algebraic approach, and use order-theoretic structures in their semantics. We discuss several of these logics, but focus predominantly on the algebraic framework for dynamic epistemic logic.Past approaches to dynamic epistemic logic have typically been focused on actions whose primary purpose is to communicate information from one agent to another. These actions are unable to alter the valuation of any proposition within the system. In fields such as security and economics, it is easy to imagine situations in which this sort of action would be insufficient. Instead, we expand the framework to include both communication actions and actions that change the state of the system. Furthermore, we propose a new modality which captures both epistemic and propositional changes that result from the agents' actions.
En raisonnement sur les systemes multi-agents, il est important de regarder au-dela du domaine de la logique propositionnelle et de raisonner sur les con- naissances des agents au sein du syst`eme, parce que ce qu'ils savent au sujet de l'environnement influe sur la mani`ere dont ils se comportent. Un outil utile pour l'analyse et la formalisation de ce que les agents savent, est la logique epistemique, une logique modale developpee par les philosophes du debut des annees 1960. La logique epistemique est la cle de la comprehension des connaissances dans les systemes multi-agents, mais elle est insuffisante si l'on veut etudier la facon dont la connaissance des agents evolue a travers le temps. Pour ce faire, il est necessaire de recourir a une logique qui allie des modalites dynamiques et epistemiques, appele la logique epistemique dynamique. Certaines formalisations de la logique epistemique dynamique utilisent la semantique de Kripke pour les etats et les actions, tandis que d'autres prennent une approche algebrique, et utilisent les structures ordonne dans leur semantique. Nous discutons plusieurs de ces logiques, mais nous nous concentrons principalement sur le cadre algebrique pour la logique epistemique dynamique.
Les approches adoptees dans le passe a la logique epistemique dynamique ont generalement ete axe sur les actions dont l'objectif principal est de communiquer des informations d'un agent a un autre. Ces actions sont dans l'impossibilite de modifier l' evaluation de toute proposition au sein du systeme. Dans des domaines tels que la securite et l' economie, il est facile d'imaginer des situations dans lesquelles ce type d'action serait insuffisante. Au lieu de cela, nous etendons le cadre algebrique pour inclure a la fois des actions de communication et des actions qui changent l' etat du systeme. En outre, nous proposons une nouvelle modalite qui permet de capturer a la fois les changements epistemiques et les changements propositionels qui resultent de l'action des agents.
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Silva, Thiago Nascimento da. "Algebraic semantics for Nelson?s logic S." PROGRAMA DE P?S-GRADUA??O EM SISTEMAS E COMPUTA??O, 2018. https://repositorio.ufrn.br/jspui/handle/123456789/24823.
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Al?m da mais conhecida l?gica de Nelson (?3) e da l?gica paraconsistente de Nelson (?4), David Nelson introduziu no artigo de 1959 "Negation and separation of concepts in constructive systems", com motiva??es de aritm?tica e construtividade, a l?gica que ele chamou de "?". Naquele trabalho, a l?gica ? definida por meio de um c?lculo (que carece crucialmente da regra de contra??o) tendo infinitos esquemas de regras, e nenhuma sem?ntica ? fornecida. Neste trabalho n?s tomamos o fragmento proposicional de ?, mostrando que ele ? algebriz?vel (de fato, implicativo) no sentido de Blok & Pigozzi com respeito a uma classe de reticulados residuados involutivos. Assim, fornecemos a primeira sem?ntica para ? (que chamamos de ?-?lgebras), bem como um c?lculo estilo Hilbert finito equivalente ? apresenta??o de Nelson. Fornecemos um algoritmo para construir ?-?lgebras a partir de ?-?lgebras ou reticulados implicativos e demonstramos alguns resultados sobre a classe de ?lgebras que introduzimos. N?s tamb?m comparamos ? com outras l?gicas da fam?lia de Nelson, a saber, ?3 e ?4.
Besides the better-known Nelson logic (?3) and paraconsistent Nelson logic (?4), in Negation and separation of concepts in constructive systems (1959) David Nelson introduced a logic that he called ?, with motivations of arithmetic and constructibility. The logic was defined by means of a calculus (crucially lacking the contraction rule) having infinitely many rule schemata, and no semantics was provided for it. We look in the present dissertation at the propositional fragment of ?, showing that it is algebraizable (in fact, implicative) in the sense of Blok and Pigozzi with respect to a class of involutive residuated lattices. We thus provide the first known algebraic semantics for ?(we call them of ?-algebras) as well as a finite Hilbert-style calculus equivalent to Nelson?s presentation. We provide an algorithm to make ?-algebras from ?-algebras or implicative lattices and we prove some results about the class of algebras which we have introduced. We also compare ? with other logics of the Nelson family, that is, ?3 and ?4.
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PERUZZI, LUISA. "Algebraic approach to paraconsistent weak Kleene logic." Doctoral thesis, Università degli Studi di Cagliari, 2018. http://hdl.handle.net/11584/255936.
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The starting point of this work is the analysis of the logic known as Paraconsistent Weak Kleene (PWK), the 3-valued logic with two designated values defined through the weak Kleene tables. Some philosophical assumptions stand behind the introduction of non-classical logics, which is, basically allowing logics to deal with partial predicates. Despite different non-classical formalisms have found lots more success than PWK logic, this thesis highlights a very surprising connection (which can be further generalized) between such a logic on one side, and the purely algebraic theory of regular varieties, on the other. The latter has been studied in universal algebra since the '60, but had found no application in logic before. In particular, the present work is divided in two different parts, each of which makes use of different machineries and techniques: one is more logically oriented and regards the study of Paraconsistent Weak Kleene logic, under the perspective of Abstract Algebraic Logic, while the other part involves a closer study of the algebraic semantics of the mentioned logic.
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BONZIO, STEFANO. "Algebraic structures from quantum and fuzzy logics." Doctoral thesis, Università degli Studi di Cagliari, 2016. http://hdl.handle.net/11584/266667.
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This thesis concerns the wide research area of logic. In particular, the first
part is devoted to analyze different kinds of relational systems (orthogonal
and residuated), by investigating the properties of the algebras associated
to them. The second part is focused on algebras of logic, in particular, the
relationship between prominent quantum and fuzzy structures with certain
semirings is proved. The last chapter concerns an application of group theory
to some well known mathematical puzzles.
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Moraschini, Tommaso. "Investigations into the role of translations in abstract algebraic logic." Doctoral thesis, Universitat de Barcelona, 2016. http://hdl.handle.net/10803/394028.
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This memoir is divided into two parts, devoted to two topics in (ab-stract) algebraic logic. In the first part we develop a hierarchy in which propositional logics “L” are classified according to the definability conditions enjoyed by the truth sets of the matrix semantics Mod* L. More precisely, we focus on conditions belonging to the conceptual framework of the Leibniz hierarchy, meaning that they can be characterized by means of the order-theoretic behaviour of the Leibniz operator. We study the class of logics such that truth is definable in Mod* L by means of universally quantified equations leaving one variable free. Then we study logics for which truth is implicitly definable in Mod* L and show that the injectivity of the Leibniz operator does not transfer in general from theories to filters over arbitrary algebras. Finally we consider an intermediate condition on the truth sets in Mod* L that corresponds to the order-reflection of the Leibniz operator. We conclude this part of the memoir by taking a computational glimpse to the Leibniz and Frege hierarchies.
In the second part of this memoir we present an algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences. This correspondence provides a general explanation of the correspondence that appears in some well-known trans-lations between logics, e.g., Godel's translation of intuitionistic logic into the gobal modal logic 84 corresponds to the functor that takes an interior algebra to the Heyting algebra of its open elements and Kolmogorov's translation of classical logic into intuitionistic logic corresponds to the functor that takes a Heyting algebra to the Boolean algebra of its regular elements.
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Ross, Brian James. "An algebraic semantics of Prolog control." Thesis, University of Edinburgh, 1992. http://hdl.handle.net/1842/585.
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The coneptual distinction between logic and control is an important tenet of logic programing. In practice, however, logic program languages use control strategies which profoundly affect the computational behavior of programs. For example, sequential Prolog's depth-first-left-first control is an unfair strategy under which nontermination can easily arise if programs are ill-structured. Formal analyses of logic programs therefore require an explicit formalisation of the control scheme. To this ends, this research introduces an algebraic proccess semantics of sequential logic programs written in Milner's calculus of Communicating Systems (CCS). the main contribution of this semantics is that the control component of a logic programming language is conciesly modelled. Goals and clauses of logic programs correspond semantically to sequential AND and OR agents respectively, and these agents are suitably defined to reflect the control strategy used to traverse the AND/OR computation tree for the program. The main difference between this and other process semantics which model concurrency is that the processes used here are sequential. The primary control strategy studied is standard Prolog's left-first-depth-first control. CCS is descriptively robust, however, and a variety of other sequential control schemes are modelled, including breadth-first, predicate freezing, and nondeterministic strategies. The CCS semantics for a particular control scheme is typically defined hierarchically. For example, standard Prolog control is initially defined in basic CCS using two control operators which model goal backtracking and clause sequencing. Using these basic definitions, higher-level bisimilarities are derived, ehich are more closely mappable to Prolog program constructs. By using variuos algebraic properties of the control operators, as well as the stream domain and theory of observational equivalence from CCS, a programming calculus approach to logic program analysis is permitted. Some example applications using the semantics include proving program termination, verifying transformations which use cut, and characterising some control issues of partial evaluation. Since progress algebras have already been used to model concurrency, this thesis suggests that they are an ideal means for unifying the operational semantics of the sequential and concurrent paradigms of logic programming.
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Horáček, Jan [Verfasser]. "Algebraic and Logic Solving Methods for Cryptanalysis / Jan Horáček." Passau : Universität Passau, 2020. http://d-nb.info/1205155228/34.
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Akishev, Galym. "Monadic bounded algebras : a thesis submitted to the Victoria University of Wellington in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics /." ResearchArchive@Victoria e-Thesis, 2009. http://hdl.handle.net/10063/915.
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Favro, Giordano <1985>. "Algebraic structures for the lambda calculus and the propositional logic." Doctoral thesis, Università Ca' Foscari Venezia, 2015. http://hdl.handle.net/10579/8333.
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Nella prima parte della tesi definiamo due famiglie di insiemi, Mn e
Gn, dove n è un indice sui naturali, costituiti da termini muti
detti rispettivamente restricted regular mute e regular mute e definiti induttivamente. Proviamo inoltre che gli
insiemi Mn sono graph easy, ovvero che per ogni termine chiuso t esiste un graph model che eguaglia t a tutti gli elementi di Mn.
Nella seconda parte introduciamo le factor algebras su tipi del primo
ordine. Mostriamo come possano essere usate come controparte
algebrica per le strutture su tipi del primo ordine. Mostriamo che
questa traduzione si estende a formule ed equazioni fra termini e che queste traduzioni
hanno un significato semantico. Utilizzando questi risultati, possiamo studiare la logica del primo ordine tramite tecniche algebriche. Costruiamo quindi
un calcolo algebrico per la logica proposizionale basato sugli assiomi
della varietà generata dalle factor algebras sul tipo della logica
proposizionale. Forniamo inoltre un sistema di riscrittura confluente
e terminante per il calcolo.
Inoltre analizziamo le proprietà algebriche di base delle factor algebras su tipi del primo ordine.
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Pra, Baldi Michele. "An algebraic study of logics of variable inclusion and analytic containment." Doctoral thesis, Università degli studi di Padova, 2018. http://hdl.handle.net/11577/3426841.
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This thesis focuses on a wide family of logics whose common
feature is to admit a syntactic definition based on specific
variable inclusion principles.
This family has been divided into three main components:
logics of left variable inclusion, containment logics, and
the logic of demodalised analytic implication.
We offer a general investigation of such logics within
the framework of modern abstract algebraic logic.
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Sustretov, Dmitry. "Non-algebraic Zariski geometries." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:b67f85d8-6fac-4820-913d-a064d3582412.
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The thesis deals with definability of certain Zariski geometries, introduced by Zilber, in the theory of algebraically closed fields. I axiomatise a class of structures, called 'abstract linear spaces', which are a common reduct of these Zariski geometries. I then describe what an interpretation of an abstract linear space in an algebraically closed field looks like. I give a new proof that the structure "quantum harmonic oscillator", introduced by Zilber and Solanki, is not interpretable in an algebraically closed field. I prove that a similar structure from an unpublished note of Solanki is not definable in an algebraically closed field and explain the non-definability of both structures in terms of geometric interpretation of the group law on a Galois cohomology group H1(k(x), μn). I further consider quantum Zariski geometries introduced by Zilber and give necessary and sufficient conditions that a quantum Zariski geometry be definable in an algebraically closed field. Finally, I take an attempt at extending the results described above to complex-analytic setting. I define what it means for quantum Zariski geometry to have a complex analytic model, an give a necessary and sufficient conditions for a smooth quantum Zariski geometry to have one. I then prove a theorem giving a partial description of an interpretation of an abstract linear space in the structure of compact complex spaces and discuss the difficulties that present themselves when one tries to understand interpretations of abstract linear spaces and quantum Zariski geometries in the compact complex spaces structure.
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Spinks, Matthew (Matthew James) 1970. "Contributions to the theory of pre-BCK-algebras." Monash University, Gippsland School of Computing and Information Technology, 2002. http://arrow.monash.edu.au/hdl/1959.1/7947.
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Dang, Han Hing [Verfasser], and Bernhard [Akademischer Betreuer] Möller. "Algebraic Calculi for Separation Logic / Han Hing Dang. Betreuer: Bernhard Möller." Augsburg : Universität Augsburg, 2015. http://d-nb.info/1077704933/34.
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Vicinansa, Guilherme Scabin. "Algebraic estimators with applications." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/3/3139/tde-21092018-150106/.
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In this work we address the problem of friction compensation in a pneumatic control valve. It is proposed a nonlinear control law that uses algebraic estimators in its structure, in order to adapt the controller to the aging of the valve. For that purpose we estimate parameters related to the valve\'s Karnopp model, necessary to friction compensation, online. The estimators and the controller are validated through simulations.Nessa pesquisa, estudamos o problema de compensação de atrito em válvulas pneumáticas. É proposta uma lei de controle não linear que tem estimadores algébricos em sua estrutura, para adaptar o controlador ao envelhecimento da válvula. Para isso, estimam-se os valores de parâmetros relacionados ao modelo de Karnopp da válvula, necessários à compensação do atrito, de maneira online. Os estimadores e o controlador são validados através de simulações.
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Barker, David Daniel. "Teachers' knowledge of algebraic reasoning its organization for instruction /." Diss., Columbia, Mo. : University of Missouri-Columbia, 2007. http://hdl.handle.net/10355/4858.
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Thesis (Ph. D.)--University of Missouri-Columbia, 2007.The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on November 21, 2007) Vita. Includes bibliographical references.
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Wannenburg, Johann Joubert. "Varieties of De Morgan Monoids." Thesis, University of Pretoria, 2020. http://hdl.handle.net/2263/75178.
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De Morgan monoids are algebraic structures that model certain non-classical logics. The variety DMM of all De Morgan monoids models the relevance logic Rt (so-named because it blocks the derivation of true conclusions from irrelevant premises). The so-called subvarieties and subquasivarieties of DMM model the strengthenings of Rt by new logical axioms, or new inference rules, respectively. Meta-logical problems concerning these stronger systems amount to structural problems about (classes of) De Morgan monoids, and the methods of universal algebra can be exploited to solve them. Until now, this strategy was under-developed in the case of Rt and DMM.
The thesis contributes in several ways to the filling of this gap. First, a new structure theorem for irreducible De Morgan monoids is proved; it leads to representation theorems for the algebras in several interesting subvarieties of DMM. These in turn help us to analyse the lower part of the lattice of all subvarieties of DMM. This lattice has four atoms, i.e., DMM has just four minimal subvarieties. We describe in detail the second layer of this lattice, i.e., the covers of the four atoms. Within certain subvarieties of DMM, our description amounts to an explicit list of all the covers. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids.
Thereafter, we use these insights to identify strengthenings of Rt with certain desirable meta-logical features. In each case, we work with the algebraic counterpart of a meta-logical property. For example, we identify precisely the varieties of De Morgan monoids having the joint embedding property (any two nontrivial members both embed into some third member), and we establish convenient sufficient conditions for epimorphisms to be surjective in a subvariety of DMM. The joint embedding property means that the corresponding logic is determined by a single set of truth tables. Epimorphisms are related to 'implicit definitions'. (For instance, in a ring, the multiplicative inverse of an element is implicitly defined, because it is either uniquely determined or non-existent.) The logical meaning of epimorphism-surjectivity is, roughly speaking, that suitable implicit definitions can be made explicit in the corresponding logical syntax.Thesis (PhD)--University of Pretoria, 2020.
DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)
Mathematics and Applied Mathematics
PhD
Unrestricted
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Solanki, Vinesh. "Zariski structures in noncommutative algebraic geometry and representation theory." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:3fa23b75-9b85-4dc2-9ad6-bdb20d61fe45.
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A suitable subcategory of affine Azumaya algebras is defined and a functor from this category to the category of Zariski structures is constructed. The rudiments of a theory of presheaves of topological structures is developed and applied to construct examples of structures at a generic parameter. The category of equivariant algebras is defined and a first-order theory is associated to each object. For those theories satisfying a certain technical condition, uncountable categoricity and quantifier elimination results are established. Models are shown to be Zariski structures and a functor from the category of equivariant algebras to Zariski structures is constructed. The two functors obtained in the thesis are shown to agree on a nontrivial class of algebras.
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Tiger, Norkvist Axel. "Morphisms of real calculi from a geometric and algebraic perspective." Licentiate thesis, Linköpings universitet, Algebra, geometri och diskret matematik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-175740.
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Noncommutative geometry has over the past four of decades grown into a rich field of study. Novel ideas and concepts are rapidly being developed, and a notable application of the theory outside of pure mathematics is quantum theory. This thesis will focus on a derivation-based approach to noncommutative geometry using the framework of real calculi, which is a rather direct approach to the subject. Due to their direct nature, real calculi are useful when studying classical concepts in Riemannian geometry and how they may be generalized to a noncommutative setting. This thesis aims to shed light on algebraic aspects of real calculi by introducing a concept of morphisms of real calculi, which enables the study of real calculi on a structural level. In particular, real calculi over matrix algebras are discussed both from an algebraic and a geometric perspective.Morphisms are also interpreted geometrically, giving a way to develop a noncommutative theory of embeddings. As an example, the noncommutative torus is minimally embedded into the noncommutative 3-sphere.Ickekommutativ geometri har under de senaste fyra decennierna blivit ett etablerat forskningsområde inom matematiken. Nya idéer och koncept utvecklas i snabb takt, och en viktig fysikalisk tillämpning av teorin är inom kvantteorin. Denna avhandling kommer att fokusera på ett derivationsbaserat tillvägagångssätt inom ickekommutativ geometri där ramverket real calculi används, vilket är ett relativt direkt sätt att studera ämnet på. Eftersom analogin mellan real calculi och klassisk Riemanngeometri är intuitivt klar så är real calculi användbara när man undersöker hur klassiska koncept inom Riemanngeometri kan generaliseras till en ickekommutativ kontext. Denna avhandling ämnar att klargöra vissa algebraiska aspekter av real calculi genom att introducera morfismer för dessa, vilket möjliggör studiet av real calculi på en strukturell nivå. I synnerhet diskuteras real calculi över matrisalgebror från både ett algebraiskt och ett geometriskt perspektiv. Morfismer tolkas även geometriskt, vilket leder till en ickekommutativ teori för inbäddningar. Som ett exempel blir den ickekommutativa torusen minimalt inbäddad i den ickekommutativa 3-sfären.
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Barros, Jose Bernado dos Santos Monteiro Vieira de. "Semantics of non-terminating systems through term rewriting." Thesis, University of Oxford, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.260738.
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Walsh, Alison. "Relations between logic and mathematics in the work of Benjamin and Charles S. Peirce." Thesis, Middlesex University, 1999. http://eprints.mdx.ac.uk/6432/.
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Charles Peirce (1839-1914) was one of the most important logicians of the nineteenth century. This thesis traces the development of his algebraic logic from his early papers, with especial attention paid to the mathematical aspects. There are three main sources to consider. 1) Benjamin Peirce (1809-1880), Charles's father and also a leading American mathematician of his day, was an inspiration. His memoir Linear Associative Algebra (1870) is summarised and for the first time the algebraic structures behind its 169 algebras are analysed in depth. 2) Peirce's early papers on algebraic logic from the late 1860s were largely an attempt to expand and adapt George Boole's calculus, using a part/whole theory of classes and algebraic analogies concerning symbols, operations and equations to produce a method of deducing consequences from premises. 3) One of Peirce's main achievements was his work on the theory of relations, following in the pioneering footsteps of Augustus De Morgan. By linking the theory of relations to his post-Boolean algebraic logic, he solved many of the limitations that beset Boole's calculus. Peirce's seminal paper 'Description of a Notation for the Logic of Relatives' (1870) is analysed in detail, with a new interpretation suggested for his mysterious process of logical differentiation. Charles Peirce's later work up to the mid 1880s is then surveyed, both for its extended algebraic character and for its novel theory of quantification. The contributions of two of his students at the Johns Hopkins University, Oscar Mitchell and Christine Ladd-Franklin are traced, specifically with an analysis of their problem solving methods. The work of Peirce's successor Ernst Schröder is also reviewed, contrasting the differences and similarities between their logics. During the 1890s and later, Charles Peirce turned to a diagrammatic representation and extension of his algebraic logic. The basic concepts of this topological twist are introduced. Although Peirce's work in logic has been studied by previous scholars, this thesis stresses to a new extent the mathematical aspects of his logic - in particular the algebraic background and methods, not only of Peirce but also of several of his contemporaries.
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Bár, Filip. "Infinitesimal models of algebraic theories." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/267026.
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Smooth manifolds have been always understood intuitively as spaces that are infinitesimally linear at each point, and thus infinitesimally affine when forgetting about the base point. The aim of this thesis is to develop a general theory of infinitesimal models of algebraic theories that provides us with a formalisation of these notions, and which is in accordance with the intuition when applied in the context of Synthetic Differential Geometry. This allows us to study well-known geometric structures and concepts from the viewpoint of infinitesimal geometric algebra. Infinitesimal models of algebraic theories generalise the notion of a model by allowing the operations of the theory to be interpreted as partial operations rather than total operations. The structures specifying the domains of definition are the infinitesimal structures. We study and compare two definitions of infinitesimal models: actions of a clone on infinitesimal structures and models of the infinitesimalisation of an algebraic theory in cartesian logic. The last construction can be extended to first-order theories, which allows us to define infinitesimally euclidean and projective spaces, in principle. As regards the category of infinitesimal models of an algebraic theory in a Grothendieck topos we prove that it is regular and locally presentable. Taking a Grothendieck topos as a base we study lifts of colimits along the forgetful functor with a focus on the properties of the category of infinitesimally affine spaces. We conclude with applications to Synthetic Differential Geometry. Firstly, with the help of syntactic categories we show that the formal dual of every smooth ring is an infinitesimally affine space with respect to an infinitesimal structure based on nil-square infinitesimals. This gives us a good supply of infinitesimally affine spaces in every well-adapted model of Synthetic Differential Geometry. In particular, it shows that every smooth manifold is infinitesimally affine and that every smooth map preserves this structure. In the second application we develop some basic theory of smooth loci and formal manifolds in naive Synthetic Differential Geometry using infinitesimal geometric algebra.
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Hassani, Sarah Dossey John A. "Calculus students' knowledge of the composition of functions and the chain rule." Normal, Ill. Illinois State University, 1998. http://wwwlib.umi.com/cr/ilstu/fullcit?p9835906.
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Thesis (D.A.)--Illinois State University, 1998.Title from title page screen, viewed July 3, 2006. Dissertation Committee: John A. Dossey (chair), Roger Day, Michael Marsali, Michael Plantholt. Includes bibliographical references (leaves 196-202) and abstract. Also available in print.
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Brey, Amina. "Multiple representations and cognitive load: words, arrows, and colours when solving algebraic problems." Thesis, Nelson Mandela Metropolitan University, 2013. http://hdl.handle.net/10948/d1020392.
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This study investigates the possible effects that access to selected multiple representations (words, arrows and colours) have in terms of cognitive load and learner achievement when presented with algebraic problems at grade nine level. The presentation of multiple representations (the intervention) was intended to decrease extraneous cognitive load, manage the intrinsic cognitive load (algebraic problems) and optimise germane cognition (schema acquisition and automation). An explanatory sequential mixed-method design was employed with six hundred and seventy three learners in four secondary schools. Quantitative data were generated via pre-, intervention and post-tests/questionnaires, while qualitative data were obtained from open-ended questions in the pre-, intervention, and post-tests/questionnaires, eight learner focus group interviews (n = 32), and four semi-structured, open-ended teacher interviews. Statistically and practically significant improvement in mean test scores from the pre- to intervention test scores in all schools was noted. No statistically and practically significant improvement was noted in further post-tests except for post-test 2 which employed more challenging problems (statistically significant decrease with a small practical effect). Learners expressed their preference for arrows, followed by colours and then words as effective representations. Teacher generated qualitative data suggests that they realise the importance of using multiple representations as an instructional strategy and implicitly understand the notion of cognitive load. The findings, when considered in the light of literature on cognitive load, suggest that a reduction in extraneous cognitive load by using a more effective instructional design (multiple representations) frees working memory capacity which can then be devoted to the intrinsic cognitive load (algebraic problems) and thereby increase germane cognition (schema acquisition and automation).
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Kaplan, Elliot. "Initial Embeddings in the Surreal Number Tree." Ohio University Honors Tutorial College / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ouhonors1429615758.
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Wasman, Deanna G. "An investigation of algebraic reasoning of seventh- and eighth-grade students who have studied from the Connected mathematics project curriculum /." free to MU campus, to others for purchase, 2000. http://wwwlib.umi.com/cr/mo/fullcit?p9988711.
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PUERTO, AUBEL ADRIAN. "Algebraic Structures for the Analysis of Distributability of Elementary Systems and their Processes." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2019. http://hdl.handle.net/10281/241253.
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In questa tesi studio in che modi si possono distribuire i sistemi e i
processi che quei sistemi eseguono. La nozione centrale per raggiungere
l'obiettivo è che, quando un sistema è distribuito, una sua osservazione
"da lontano" richiede uno scambio d'informazioni con le diverse parti
del sistema. Questo approccio si caratterizza per il fatto che la
"sincronizzazione" (o "handshaking") è il modo fondamentale di interazione.
I formalismi impiegati sono presi dalla teoria delle reti di Petri.
I sistemi elementari e i sistemi di condizioni ed eventi in quella teoria
costituiscono le specificazioni di sistemi. Le reti causali e gli insiemi
parzialmente ordinati permettono di modellare processi. In questi modelli,
lo stato dell'arte offre una nozione di sottoprocesso, cui si può
associare una struttura che porta l'informazione su come distribuire il
processo. Formalmente, questa struttura è un reticolo ortomodulare.
Nella tesi mostro che gli elementi minimali non banali di quel reticolo
(sottoprocessi minimali) possono essere ordinati in modo da formare
un'astrazione del processo dato. La natura di questa nozione di
sottoprocesso consente di mostrare che l'astrazione rappresenta le
componenti del processo, cioè le parti che possono operare indipendentemente.
Il comportamento dei sistemi elementari e dei sistemi di condizioni e
eventi è modellato per mezzo di sistemi di transizioni etichettate.
Nella tesi si applica un'interpretazione delle regioni elementari come
proprietà localmente osservabili del sistema, motivata dalla sintesi di
reti elementari. Secondo questa interpretazione, le regioni elementari
offrono una specificazione adeguata dell'infrastruttura su cui si può
distribuire un sistema.
Era già noto che l'insieme delle regioni di un sistema elementare o di
condizioni ed eventi forma un insieme ortomodulare, da cui si può ricavare
un sistema di transizioni etichettate canonico, che contiene tutte le
regioni dell'insieme ortomodulare dato. Stabilire se il sistema canonico
ha più regioni di quelle specificate è un problema aperto.
Il sistema canonico è il più "grande" che si può ottenere dall'insieme
ortomodulare, nel senso che ogni altro sistema conforme alla specificazione
è un suo sottosistema. D'altra parte, non tutti i sottosistemi hanno
la stessa struttura regionale. Definisco una condizione sufficiente per
avere l'isomorfismo. Il risultato si ottiene dotando di un'opportuna
struttura l'insieme degli eventi, o delle etichette, del sistema canonico,
così da riflettere la concorrenza.
Un insieme ortomodulare si dice stabile quando è isomorfo all'insieme delle
regioni del sistema di transizioni canonico derivato. Erano già note
condizioni sotto le quali il primo insieme si immerge nel secondo.
Si congettura che tutti gli insiemi parzialmente ordinati ottenuti come
insiemi di regioni di sistemi elementari (insiemi regionali) sono stabili.
Nella tesi si dà una nuova condizione necessaria perché un insieme ortomodulare
sia regionale, e si mostra che in quel caso l'immersione è forte.
Non tutte le immersioni sono forti, ma tutti gli isomorfismi sono
immersioni forti. Dal risultato segue che l'immersione mappa regioni
minimali su regioni minimali.This work studies systems, and the processes they execute, in the way they can be distributed. To this aim, the central notion is that when a system is distributed, a remote observation requires an exchange of information from the different locations of the system. This approach is characterised by the fact that handshaking is the basic mode of interaction. The chosen formalisms are taken in the framework Petri net theory. Elemen- tary net systems, and condition/event net systems provide specifications for the systems. Causal nets and partially ordered sets allow for modelling processes. With these last formalisations, the state of the art provides a notion of subpro- cesses that can be structured so as to carry information on how a process can be distributed. This structure is formalised as an orthomodular lattice. This work shows that the minimal non trivial elements of this lattice, the minimal subprocesses, can be ordered so as to provide an abstraction of the process. The nature of this notion of subprocess permits to show that this abstraction depicts the localities of the process, parts of the process which can run independently from each other. The behaviour of elementary, and condition/event net systems, is modelled with labelled transition systems. This work adheres to an interpretation of the set of elementary regions, as the one of locally observable properties of the sys- tem, motivated by elementary net synthesis. According to this interpretation, elementary regions represent a suitable specification of the available infrastruc- ture on which to distribute a system. The state of the art shows that the set of regions of an elementary, or condition/event system, forms an orthomodular poset, and a way to retrieve a canonical labelled transition system such that all regions of the orthomodular poset are also regions of it. The question of whether this canonical transition system has more regions than the specified ones is an open problem. The canonical transition system is the largest one can obtain from an orthomodular poset, in the sense that systems complying with the specification, can be found as subsystems of it. However, not all its subsystems display the same regional structure. This work presents a sufficient condition for this to happen. This is achieved by providing a structure to the set of events, or labels, of the canonical system, which reflects concurrency. An orthomodular poset is called stable when it is isomorphic to the set of regions of its canonical transition system. The state of the art shows that when the first poset is of a given class, it embeds in the second. It is conjectured that all posets that arise as the set of elementary regions of an elementary system, regional posets, are stable. This work provides a condition necessary for an orthomodular poset to be regional, and shows that when it holds, the embedding is strong. Not every embedding is strong, but all isomorphisms are, in particular, strong embeddings. This result implies that the embedding maps minimal regions to minimal regions.
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Bergvall, Olof. "Cohomology of the moduli space of curves of genus three with level two structure." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-103062.
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In this thesis we investigate the moduli space M3[2] of curves of genus 3 equipped with a symplectic level 2 structure. In particular, we are interested in the cohomology of this space. We obtain cohomological information by decomposing M3[2] into a disjoint union of two natural subspaces, Q[2] and H3[2], and then making S7- resp. S8-equivariantpoint counts of each of these spaces separately.Målet med denna uppsats är att undersöka modulirummet M3[2] av kurvor av genus 3 med symplektisk nivå 2 struktur. Mer specifikt vill vi hitta informationom kohomologin av detta rum. För att uppnå detta delar vi först upp M[2] i en disjunkt union av två naturliga delrum, Q[2] och H3[2], och räknar därefter punkterna av dessa rum S7- respektive S8-ekvivariant.
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Fernández, Victor Leandro. "Fibrilação de logicas na hierarquia de Leibniz." [s.n.], 2005. http://repositorio.unicamp.br/jspui/handle/REPOSIP/280595.
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Orientador: Marcelo Esteban ConiglioTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias Humanas
Made available in DSpace on 2018-08-04T20:57:48Z (GMT). No. of bitstreams: 1 Fernandez_VictorLeandro_D.pdf: 6531217 bytes, checksum: 2a972c9e9fa860af8f9cc57b3e1bb73d (MD5) Previous issue date: 2005
Resumo: Neste trabalho investigamos com um enfoque abstrato um processo de combinações de lógicas conhecido como Fibrilação de lógicas. Em particular estudamos a transferência, mediante fibrilação, de certas propriedades intrínsecas às lógicas proposicionais. As noções mencionadas são as de protoalgebrizabilidade, equivalencialidade e algebrizabilidade. Ditas noções fazem parte da "Hierarquia de Leibniz" , conceito fundamental da chamada Lógica Algébrica Abstrata. Tal hierarquia classifica as diferentes lógicas segundo o seu grau de algebrizabilidade. Assim, nesta tese estudaremos se, quando duas lógicas possuem alguma dessas propriedades, a fibrilação delas possui também tal característica. Com o objetivo de diferençar os diferentes modos de fibrilação existentes na literatura, analisamos duas maneiras de fibrilar lógicas: Fibrilação categorial (ou C-fibrilação) e Fibrilação no sentido de D. Gabbay (G-fibrilação). Também estudamos uma variante da Gfibrilação de lógicas conhecida como Fusão de lógicas. Assim, damos diferentes condições que devem valer para que a C-fibrilação de uma lógica protoalgébrica seja também protoalgébrica, e procedemos de forma similar com as outras propriedades que constituem a Hierarquia de Leibniz. No caso da G-fibrilação e da fusão de lógicas chegamos a diversos resultados análogos aos anteriores, os quais permitem ter uma visão geral da relação entre Lógica Algébrica Abstrata e as Combinações de lógicas
Abstract: ln this thesis we investigate, with an abstract approach, a process of combinations of logics known as fibring of logics. ln particular we study the transference by fibring of certain properties, intrinsic to propositionallogics: protoalgebricity, equivalenciality and algebraizability. The notions above belong to the "Leibniz Hierarchy", a fundamental concept of the so-called Abstract Algebraic Logic. Such hierarchy classifies the logics according to its algebraizability degree. So, in this thesis we will study whether, given two logics having some of these properties, the fibring of them still has that property. With the aim of distinguishing the different techniques of fibring existing in the literature, we analyze two methods of fibring logics: Categorial Fibring (or C-fibring) and Fibring in D. Gabbay's sense (G-fibring). We also study a variant of G-fibring known as fusion of logics. So, we give different conditions that must hold in order to obtain a protoalgebraic logic by means of C-fibring of protoalgebric logics. We proceed in a similar way with the other properties that constitutes the Leibniz Hierarchy. With respect to G-fibring and fusion, we arrive to similar results which allow us to get an overview of the relation between Abstract AIgebraic Logic and the subject of combinations of logics
Doutorado
Doutor em Filosofia
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Buchele, Suzanne Fox. "Three-dimensional binary space partitioning tree and constructive solid geometry tree construction from algebraic boundary representations /." Digital version accessible at:, 1999. http://wwwlib.umi.com/cr/utexas/main.
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Banks, Christopher Jon. "Spatio-temporal logic for the analysis of biochemical models." Thesis, University of Edinburgh, 2015. http://hdl.handle.net/1842/10512.
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Process algebra, formal specification, and model checking are all well studied techniques in the analysis of concurrent computer systems. More recently these techniques have been applied to the analysis of biochemical systems which, at an abstract level, have similar patterns of behaviour to concurrent processes. Process algebraic models and temporal logic specifications, along with their associated model-checking techniques, have been used to analyse biochemical systems. In this thesis we develop a spatio-temporal logic, the Logic of Behaviour in Context (LBC), for the analysis of biochemical models. That is, we define and study the application of a formal specification language which not only expresses temporal properties of biochemical models, but expresses spatial or contextual properties as well. The logic can be used to express, or specify, the behaviour of a model when it is placed into the context of another model. We also explore the types of properties which can be expressed in LBC, various algorithms for model checking LBC - each an improvement on the last, the implementation of the computational tools to support model checking LBC, and a case study on the analysis of models of post-translational biochemical oscillators using LBC. We show that a number of interesting and useful properties can be expressed in LBC and that it is possible to express highly useful properties of real models in the biochemistry domain, with practical application. Statements in LBC can be thought of as expressing computational experiments which can be performed automatically by means of the model checker. Indeed, many of these computational experiments can be higher-order meaning that one succinct and precise specification in LBC can represent a number of experiments which can be automatically executed by the model checker.
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Berni, Jean Cerqueira. "Some algebraic and logical aspects of C∞-Rings." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-14022019-203839/.
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As pointed out by I. Moerdijk and G. Reyes in [63], C∞-rings have been studied specially for their use in Singularity Theory and in order to construct topos models for Synthetic Differential Geometry. In this work, we follow a complementary trail, deepening our knowledge about them through a more pure bias, making use of Category Theory and accounting them from a logical-categorial viewpoint. We begin by giving a comprehensive systematization of the fundamental facts of the (equational) theory of C∞-rings, widespread here and there in the current literature - mostly without proof - which underly the theory of C∞-rings. Next we develop some topics of what we call a ∞Commutative Algebra, expanding some partial results of [66] and [67]. We make a systematic study of von Neumann-regular C∞-rings (following [2]) and we present some interesting results about them, together with their (functorial) relationship with Boolean spaces. We study some sheaf theoretic notions on C∞-rings, such as ∞(locally)-ringed spaces and the smooth Zariski site. Finally we describe classifying toposes for the (algebraic) theory of ∞ rings, the (coherent) theory of local C∞-rings and the (algebraic) theory of von Neumann regular C∞-rings.Conforme observado por I. Moerdijk e G. Reyes em [63], os anéis C∞ têm sido estudados especialmente tendo em vista suas aplicações em Teoria de Singularidades e para construir toposes que sirvam de modelos para a Geometria Diferencial Sintética. Neste trabalho, seguimos um caminho complementar, aprofundando nosso conhecimento sobre eles por um viés mais puro, fazendo uso da Teoria das Categorias e os analisando a partir de pontos de vista algébrico e lógico-categorial. Iniciamos o trabalho apresentando uma sistematização abrangente dos fatos fundamentais da teoria (equacional) dos anéis C∞, distribuídos aqui e ali na literatura atual - a maioria sem demonstrações - mas que servem de base para a teoria. Na sequência, desenvolvemos alguns tópicos do que denominamos Álgebra Comutativa C∞, expandindo resultados parciais de [66] e [67]. Realizamos um estudo sistemático dos anéis C∞ von Neumann-regulares - na linha do estudo algébrico realizado em [2]- e apresentamos alguns resultados interessantes a seu respeito, juntamente com sua relação (funtorial) com os espaços booleanos. Estudamos algumas noções pertinentes à Teoria de Feixes para anéis ∞, tais como espaços (localmente) ∞anelados e o sítio de Zariski liso. Finalmente, descrevemos toposes classicantes para a teoria (algébrica) dos anéis C∞, a teoria (coerente) dos anéis locais C∞ e a teoria (algébrica) dos anéis C∞ von Neumann regulares.
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Perchy, Yamil Salim. "Opinions, Lies and Knowledge. An Algebraic Approach to Mobility of Information and Processes." Thesis, Université Paris-Saclay (ComUE), 2016. http://www.theses.fr/2016SACLX059/document.
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La notion de système de contraintes (cs – selon l'acronyme anglais) est un concept central aux formalismes de la théorie de la concurrence tels que les algèbres de processus pour la programmation concurrente par contraintes. Les systèmes de contraintes sont souvent représentés par des treillis : ses éléments, appelées contraintes, représentent des informations partiales tandis que l’ordre du treillis correspond à des implications. Récemment, une notion appelée “système de contraintes spatiales à n-agents” a été développée pour représenter l’information dans la programmation concurrente par contraintes où les systèmes sont multi-agents et spatialement distribués.D’un point de vue informatique, un système de contraintes spatiales peut être utilisé pour spécifier l’information partiale contenue dans l'espace d'un certain agent (information locale). D’un point de vue épistémique, un cs spatial peut être utilisé pour représenter l’information qui est considérée vrai pour un certain agent (croyance). Les systèmes de contraintes spatiales, néanmoins, ne fournissent pas de mécanismes pour la spécification de la mobilité de l’information ou des processus d'un espace à un autre. La mobilité de l’information est un aspect fondamental des systèmes concurrents.Dans cette thèse nous avons développé la théorie des systèmes de contraintes spatiales avec des opérateurs pour spécifier le déplacement des informations et processus entre les espaces. Nous étudions les propriétés de cette nouvelle famille de systèmes de contraintes et nous illustrons ses applications.Du point de vue calculatoire, ces nouveaux opérateurs nous apportent de l’extrusion d’informations et/ou des processus, qui est un concept central dans les formalismes pour la communication mobile. Du point de vue épistémique, l’extrusion correspond à une notion que nous avons appelé énonciation ; une information qu’un agent souhaite communiquer à d'autres mais qui peut être inconsistante avec les croyances de l’agent même. Des énonciations peuvent donc être utilisées pour exprimer des notions épistémiques tels que les canulars ou les mensonges qui sont fréquemment utilisés dans les réseaux sociaux.Globalement, les systèmes de contraintes peuvent exprimer des notions épistémiques comme la croyance/énonciation et la connaissance en utilisant respectivement une paire de fonctions espace/extrusion qui représentent l’information locale, et un opérateur spatial dérivé qui représente l’information globale. Par ailleurs, nous montrons qu’en utilisant un type précis de systèmes de contraintes nous pouvons aussi représenter la notion du temps comme une séquence d'instancesThe notion of constraint system (cs) is central to declarative formalisms from concurrency theory such as process calculi for concurrent constraint programming (ccp). Constraint systems are often represented as lattices: their elements, called constraints, represent partial information and their order corresponds to entailment. Recently a notion of n-agent spatial cs was introduced to represent information in concurrent constraint programs for spatially distributed multi-agent systems. From a computational point of view a spatial constraint system can be used to specify partial information holding in a given agent’s space (local information). From an epistemic point of view a spatial cs can be used to specify information that a given agent considers true (beliefs). Spatial constraint systems, however, do not provide a mechanism for specifying the mobility of information/processes from one space to another. Information mobility is a fundamental aspect of concurrent systems.In this thesis we develop the theory of spatial constraint systems with operators to specify information and processes moving between spaces. We investigate the properties of this new family of cs and illustrate their applications. From a computational point of view the new operators provide for process/information extrusion, a central concept in formalisms for mobile communication. From an epistemic point of view extrusion corresponds to what we shall call utterance; information that an agent communicates to others but that may be inconsistent with the agent’s beliefs. Utterances can be used to express instances of epistemic notions such as hoaxes or intentional lies which are common place in social media.On the whole, constraint systems can express the epistemic notions of belief /utterance and knowledge by means of, respectively, a space/extrusion function pair that specifies local information and a derived spatial operator that specifies global information. We shall also show that, by using a specific kind of our constraint systems, we can also encode the notion of time as a sequence of instances
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Petria, Marius. "Generic refinements for behavioral specifications." Thesis, University of Edinburgh, 2011. http://hdl.handle.net/1842/4889.
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This thesis investigates the properties of generic refinements of behavioral specifications. At the base of this investigation stands the view from algebraic specification that abstract data types can be modeled as algebras. A specification of a data type is formed from a syntactic part, i.e. a signature detailing the interface of the data type, and a semantic part, i.e. a class of algebras (called its models) that contains the valid implementations of that data type. Typically, the class of algebras that constitutes the semantics of a specification is defined as the class of algebras that satisfy some given set of axioms. The behavioral aspect of a specification comes from relaxing the requirements imposed by axioms, i.e. by allowing in the semantics of a specification not only the algebras that literally satisfy the given axioms, but also those algebras that appear to behave according to those axioms. Several frameworks have been developed to express the adequate notions of what it means to be a behavioral model of a set of axioms, and our choice as the setting for this thesis will be Bidoit and Hennicker’s Constructor-based Observational Logic, abbreviated COL. Using specifications that rely on the behavioral aspects defined by COL we study the properties of generic refinements between specifications. Refinement is a relation between specifications. The refinement of a target specification by a source specification is given by a function that constructs models of the target specification from the models of the source specification. These functions are called constructions and the source and target specifications that they relate are called the context of the refinement. The theory of refinements between algebraic specifications, with or without the behavioral aspect, has been well studied in the literature. Our analysis starts from those studies and adapts them to COL, which is a relatively new framework, and for which refinement has been studied only briefly. The main part of this thesis is formed by the analysis of generic refinements. Generic refinements are represented by constructions that can be used in various contexts, not just in the context of their definition. These constructions provide the basis for modular refinements, i.e. one can use a locally defined construction in a global context in order to refine just a part of a source specification. The ability to use a refinement outside its original context imposes additional requirements on the construction that represents it. An implementer writing such a construction must not use details of the source models that can be contradicted by potential global context requirements. This means, roughly speaking, that he must use only the information available in the source signature and also any a priori assumption that was made about the contexts of use. We look at the basic case of generic refinements that are reusable in every global context, and then we treat a couple of variations, i.e. generic refinements for which an a priori assumption it is made about the nature of their usage contexts. In each of these cases we follow the same pattern of investigation. First we characterize the constructions that ensure reusability by means of preservation of relations, and then, in most cases, we show that such constructions must be definable in terms of their source signature. Throughout the thesis we use an informal analogy between generic (i.e. polymorphic) functions that appear in second order lambda calculus and the generic refinements that we are studying. This connection will enable us to describe some properties of generic refinements that correspond to the properties of polymorphic functions inferred from their types and named “theorems for free” by Wadler. The definability results, the connection between the assumptions made about the usage contexts and the characterizing relations, and the “theorems for free” for behavioral specifications constitute the main contributions of this thesis.
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Al-Shujary, Ahmed. "Kähler-Poisson Algebras." Licentiate thesis, Linköpings universitet, Matematik och tillämpad matematik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-150620.
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The focus of this thesis is to introduce the concept of Kähler-Poisson algebras as analogues of algebras of smooth functions on Kähler manifolds. We first give here a review of the geometry of Kähler manifolds and Lie-Rinehart algebras. After that we give the definition and basic properties of Kähler-Poisson algebras. It is then shown that the Kähler type condition has consequences that allow for an identification of geometric objects in the algebra which share several properties with their classical counterparts. Furthermore, we introduce a concept of morphism between Kähler-Poisson algebras and show its consequences. Detailed examples are provided in order to illustrate the novel concepts.The series name is corrected in the electronic version of the cover.
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Matsson, Isak. "Algebras in Monoidal Categories." Thesis, Uppsala universitet, Algebra och geometri, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-447430.
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Berg, Sandra. "Global dimension of (higher) Nakayama algebras." Thesis, Uppsala universitet, Algebra och geometri, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-420721.
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Persson, Westin Elin. "Homological properties of some stratified algebras." Licentiate thesis, Uppsala universitet, Algebra och geometri, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-426125.
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Vaso, Laertis. "Cluster Tilting for Representation-Directed Algebras." Licentiate thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-364224.
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Nour, Abir. "Etude de systèmes logiques extensions de la logique intuitionniste." Université Joseph Fourier (Grenoble), 1997. http://www.theses.fr/1997GRE10128.
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Pour modeliser le raisonnement d'un ensemble ordonne t d'agents intelligents, h. Rasiowa a introduit des systemes logiques appeles approximation logics. Dans ces systemes, un ensemble de constantes -bien que difficile a justifier dans les applications- joue un role fondamental. Dans notre travail, nous considerons des systemes logiques appeles l#t#f sans ce type de constantes mais en nous limitant au cas ou t est un ensemble ordonne fini. Nous demontrons un theoreme de deduction faible et nous presentons une liste de proprietes qui seront utilisees par la suite. Nous introduisons aussi une semantique algebrique en utilisant les algebres de heyting avec des operateurs. Pour demontrer la completude du systeme l#t#f par rapport a la semantique algebrique, nous utilisons la methode de h. Rasiowa et r. Sikorski pour la logique du premier ordre. Dans le cas propositionnel, un corollaire nous permet d'affirmer que la question de savoir si une formule du calcul propositionnel est valide ou non est effectivement decidable. Nous etudions aussi certaines relations entre les logiques l#t#f et les logiques intuitionniste et classique. De plus, la methode des tableaux est consideree car elle est connue dans la litterature sur les logiques non classiques. Enfin, nous introduisons une semantique de type kripke. L'ensemble dit des mondes possibles est ici enrichi d'une famille de fonctions indexee par l'ensemble fini t et verifiant certaines conditions. Ce type de semantique nous permet de deduire plusieurs resultats. Nous construisons un modele fini particulier de type kripke qui caracterise le calcul l#t#f propositionnel. Nous introduisons une semantique relationnelle en suivant la methode de e. Orlowska, qui a l'enorme avantage de permettre une interpretation de la logique propositionnelle l#t#f en n'utilisant qu'un type d'objet : les relations. Nous traitons aussi le probleme de la complexite du calcul propositionnel l#t#f.
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Stroiński, Mateusz. "Homological Algebra for Quiver Representations." Thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-354756.
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Restadh, Petter. "The Selfinjective Nakayama Algebras and their Complexity." Thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-354490.
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Matsson, Isak. "The Nichols-Zoeller Theorem for Quasi-Hopf Algebras." Thesis, Uppsala universitet, Algebra och geometri, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-388631.
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Gustavsson, Bim. "Representations of Finite-Dimensional Algebras and Gabriel’s Theorem." Thesis, Uppsala universitet, Algebra och geometri, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-395660.
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Lundkvist, Signe. "Torsion Classes and Support Tilting Modules for Path Algebras." Thesis, Uppsala universitet, Algebra och geometri, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-355364.
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