45 research outputs found

    The Logic of Turing Progressions

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    [eng] This dissertation is devoted to developing modal logical tools that can be used in the field of proof theory and ordinal analysis. More precisely, we focus on the relation between strictly positive modal logics and both Turing progressions and ordinal notation systems. With respect to the former one, we introduce the system TSC that is tailored to generate exactly all relations that hold between different Turing progressions given a particular set of natural consistency notions. We also present an arithmetical interpretation for this modal system, named the Formalized Turing progressions interpretation. The logic is proven to be arithmetically sound and complete with respect to this interpretation. After exploring the arithmetical semantics of TSC, we investigate the relational semantics of this system. For this purpose, we make use of the universal model of the closed fragment of Go¨del-Lo¨b’s Polymodal Logic (GLP), namely Ignatiev’s universal frame. By slightly modifying the relations defined in this model, we obtain a new frame which is proven to be a universal model for TSC. Moreover, we show how the domain of this frame can be reduced to sequences with finite support while keeping the completeness of the system. As for ordinal notations systems, we present the logic BC (for Bracket Calculus). Unlike other provability logics, BC is based on a purely modal signature that gives rise to an ordinal notation system instead of modalities indexed by some ordinal given a priori. Moreover, since the order between these notations can be established in terms of derivability within the calculus, the inferences in this system can be carried out without using any external property of ordinals. The presented logic is proven to be equivalent to Reflection Calculus (RCΓ0 ), that is, to the strictly positive fragment of GLPΓ0 .[spa] El objetivo de esta tesis es desarrollar herramientas de lógica modal que puedan ser utilizadas en el campo de la teoría de la demostración y el análisis ordinal. Más precisamente, nos centramos en la relación entre las lógicas modales estrictamente positivas y las progresiones de Turing, y entre dichas lógicas y los sistemas de notación ordinal que surgen de ellas. Con respecto a la primera parte, hemos introducido el sistema TSC, diseñado para generar exactamente todas las relaciones válidas entre las diferentes progresiones de Turing, dado un conjunto particular de nociones de consistencia naturales. También presentamos una interpretación aritmética para este sistema modal, denominada interpretación de las Progresiones de Turing formalizadas. Demostramos que la lógica es aritméticamente correcta y completa con respecto a esta interpretación. Tras de estudiar la semántica aritmética de TSC, investigamos la semántica relacional de este sistema. Para este propósito, hacemos uso del modelo universal para el fragmento cerrado de Gödel-Löb’s Polymodal Logic (GLP), a saber, el marco universal de Ignatiev. Modificando ligeramente las relaciones definidas en este modelo, obtenemos un nuevo marco. Demostramos que éste es un modelo universal para TSC. Asimismo, mostramos cómo el dominio de este marco puede reducirse a secuencias con soporte finito manteniendo la completud del sistema. Respecto a los sistemas de notación ordinal, presentamos la lógica BC (por Bracket Calculus). A diferencia de otras lógicas de la demostrabilidad, BC se basa en un lenguaje puramente modal que da lugar a un sistema de notación ordinal, en lugar de estar construido mediante modalidades indexadas por algún ordinal dado a priori. Además, ya que el orden entre estas notaciones puede establecerse en términos de derivabilidad dentro del cálculo, las inferencias en este sistema pueden llevarse a cabo sin usar ninguna propiedad externa de los ordinales. Demostramos que la lógica presentada es equivalente al Reflection Calculus (RCΓ0 ), es decir, al fragmento estrictamente positivo de GLPΓ0

    A self-contained provability calculus for {Γ\Gamma}_{\mbox0}

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    Reverse mathematics of first-order theories with finitely many models

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    We examine the reverse-mathematical strength of several theorems in classical and effective model theory concerning first-order theories and their number of models. We prove that, among these, most are equivalent to one of the familiar systems RCA(0), WKL0, or ACA(0). We are led to a purely model-theoretic statement that implies WKL0 but refutes ACA(0) over RCA(0)

    Engineering Self-Adaptive Collective Processes for Cyber-Physical Ecosystems

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    The pervasiveness of computing and networking is creating significant opportunities for building valuable socio-technical systems. However, the scale, density, heterogeneity, interdependence, and QoS constraints of many target systems pose severe operational and engineering challenges. Beyond individual smart devices, cyber-physical collectives can provide services or solve complex problems by leveraging a “system effect” while coordinating and adapting to context or environment change. Understanding and building systems exhibiting collective intelligence and autonomic capabilities represent a prominent research goal, partly covered, e.g., by the field of collective adaptive systems. Therefore, drawing inspiration from and building on the long-time research activity on coordination, multi-agent systems, autonomic/self-* systems, spatial computing, and especially on the recent aggregate computing paradigm, this thesis investigates concepts, methods, and tools for the engineering of possibly large-scale, heterogeneous ensembles of situated components that should be able to operate, adapt and self-organise in a decentralised fashion. The primary contribution of this thesis consists of four main parts. First, we define and implement an aggregate programming language (ScaFi), internal to the mainstream Scala programming language, for describing collective adaptive behaviour, based on field calculi. Second, we conceive of a “dynamic collective computation” abstraction, also called aggregate process, formalised by an extension to the field calculus, and implemented in ScaFi. Third, we characterise and provide a proof-of-concept implementation of a middleware for aggregate computing that enables the development of aggregate systems according to multiple architectural styles. Fourth, we apply and evaluate aggregate computing techniques to edge computing scenarios, and characterise a design pattern, called Self-organising Coordination Regions (SCR), that supports adjustable, decentralised decision-making and activity in dynamic environments.Con lo sviluppo di informatica e intelligenza artificiale, la diffusione pervasiva di device computazionali e la crescente interconnessione tra elementi fisici e digitali, emergono innumerevoli opportunità per la costruzione di sistemi socio-tecnici di nuova generazione. Tuttavia, l'ingegneria di tali sistemi presenta notevoli sfide, data la loro complessità—si pensi ai livelli, scale, eterogeneità, e interdipendenze coinvolti. Oltre a dispositivi smart individuali, collettivi cyber-fisici possono fornire servizi o risolvere problemi complessi con un “effetto sistema” che emerge dalla coordinazione e l'adattamento di componenti fra loro, l'ambiente e il contesto. Comprendere e costruire sistemi in grado di esibire intelligenza collettiva e capacità autonomiche è un importante problema di ricerca studiato, ad esempio, nel campo dei sistemi collettivi adattativi. Perciò, traendo ispirazione e partendo dall'attività di ricerca su coordinazione, sistemi multiagente e self-*, modelli di computazione spazio-temporali e, specialmente, sul recente paradigma di programmazione aggregata, questa tesi tratta concetti, metodi, e strumenti per l'ingegneria di ensemble di elementi situati eterogenei che devono essere in grado di lavorare, adattarsi, e auto-organizzarsi in modo decentralizzato. Il contributo di questa tesi consiste in quattro parti principali. In primo luogo, viene definito e implementato un linguaggio di programmazione aggregata (ScaFi), interno al linguaggio Scala, per descrivere comportamenti collettivi e adattativi secondo l'approccio dei campi computazionali. In secondo luogo, si propone e caratterizza l'astrazione di processo aggregato per rappresentare computazioni collettive dinamiche concorrenti, formalizzata come estensione al field calculus e implementata in ScaFi. Inoltre, si analizza e implementa un prototipo di middleware per sistemi aggregati, in grado di supportare più stili architetturali. Infine, si applicano e valutano tecniche di programmazione aggregata in scenari di edge computing, e si propone un pattern, Self-Organising Coordination Regions, per supportare, in modo decentralizzato, attività decisionali e di regolazione in ambienti dinamici
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