Turing's Fallacies

This paper reveals two fallacies in Turing's undecidability proof of first-order logic (FOL), namely, (i) an 'extensional fallacy': from the fact that a sentence is an instance of a provable FOL formula, it is inferred that a meaningful sentence is proven, and (ii) a 'fallacy of substitution': from the fact that a sentence is an instance of a provable FOL formula, it is inferred that a true sentence is proven. The first fallacy erroneously suggests that Turing's proof of the non-existence of a circle-free machine that decides whether an arbitrary machine is circular proves a significant proposition. The second fallacy suggests that FOL is undecidable

Arguement in the humanities: A knowledge based approach

In this thesis I have a threefold purpose. I will attempt: (a) to present a generic design for a tool - the Argument Support Program - which can be of use in supporting the reasoning of archaeologists (and others especially, but not exclusively, in the humanities); (b) I will present a model of argumentation and debate as the theoretical orientation within which the model is developed; and, (c) I will suggest that this approach is a natural development of several strands of research within the artificial intelligence community. A tripartite model of argument is presented in terms of arguers, the argument structure produced and the argument domain or field. This model subsumes reasoning, interpretation and argument exchange or debate. It is maintained, further, that while this model is generally applicable, specific domains have particular styles of argument. The notion of argument style is discussed in terms of the types of reasoning used. The related concept of relevance in argument is discussed in terms of the specific tokens of these types which may be used in a particular argument. It is argued that archaeology is characterized, at least in part, by the use of argument by analogy and argument from theoretical principles or models. A design for a generic program - the Argument Support Program (ASP) - based on the theoretical principles is delineated. Details of the partial implementation of the model as a constrained debater in the domain of archaeology (ASP for archaeology or ASParch) are presented. Example runs which illustrate how the characterizing features of archaeology are dealt with are also presented as are examples of the various domain and system knowledge bases needed. The application of ASPs to other domains and areas such as literary criticism, legal reasoning and Darwinian theory is discussed. In the final chapter, the achievements and inadequacies of this research are summarized, possible reasons are presented for the inadequacies in the resulting system and future directions discussed

Computability with polynomial differential equations

Tese dout., Matemática, Inst. Superior Técnico, Univ. Técnica de Lisboa, 2007Nesta dissertação iremos analisar um modelo de computação analógica, baseado em equações diferenciais polinomiais. Começa-se por estudar algumas propriedades das equações diferenciais polinomiais, em particular a sua equivalência a outro modelo baseado em circuitos analógicos (GPAC), introduzido por C. Shannon em 1941, e que é uma idealização de um dispositivo físico, o Analisador Diferencial. Seguidamente, estuda-se o poder computacional do modelo. Mais concretamente, mostra-se que ele pode simular máquinas de Turing, de uma forma robusta a erros, pelo que este modelo é capaz de efectuar computações de Tipo-1. Esta simulação é feita em tempo contínuo. Mais, mostramos que utilizando um enquadramento apropriado, o modelo é equivalente à Análise Computável, isto é, à computação de Tipo-2. Finalmente, estudam-se algumas limitações computacionais referentes aos problemas de valor inicial (PVIs) definidos por equações diferenciais ordinárias. Em particular: (i) mostra-se que mesmo que o PVI seja definido por uma função analítica e que a mesma, assim como as condições iniciais, sejam computáveis, o respectivo intervalo maximal de existência da solução não é necessariamente computável; (ii) estabelecem-se limites para o grau de não-computabilidade, mostrando-se que o intervalo maximal é, em condições muito gerais, recursivamente enumerável; (iii) mostra-se que o problema de decidir se o intervalo maximal é ou não limitado é indecídivel, mesmo que se considerem apenas PVIs polinomiais

Can morality be computed? An exploration of whether machines can be moral agents

A Research Report submitted to the Faculty of Humanities, University of the Witwatersrand, Johannesburg, in partial fulfilment of the requirements for the degree of Master of Arts, Applied Ethics for Professionals 31 May, 2015, JohannesburgTechnology is an integral part of our daily lives and continues to advance rapidly, impacting our physical and social worlds. We increasingly rely on advanced machines to act in more autonomous and sophisticated ways. What would happen if artificial forms of intelligence developed to the point where machines behaved more like us and we started treating them as people? Ethics should anticipate and account for such possibilities so that science does not move faster than our moral understanding. My thesis states that when we are able to feel gratitude or resentment towards the actions of artificially intelligent machines we can be said to see them as morally responsible agents. I argue that standard ethics frames morality developmentally – only when we reach adulthood are we deemed able to enter into the type of relationships where we can hold one another morally responsible for our actions. I apply a more abstract notion of moral development to future versions of technology and couple this with a definition of morality as a relational or social construct. This allows me to argue that machines could develop to a point in the future where we react to them morally as we would to humans. Questions on whether we ought to react in this way are muted as relationally we quite simply would be unable to feel otherwise. Objections from definitions of moral agency based on innate qualities, specifically those associated with the concept of intelligence, are dispelled in favour of a relational definition

Sets, Logic, Computation: An Open Introduction to Metalogic

An introductory textbook on metalogic. It covers naive set theory, first-order logic, sequent calculus and natural deduction, the completeness, compactness, and Löwenheim-Skolem theorems, Turing machines, and the undecidability of the halting problem and of first-order logic. The audience is undergraduate students with some background in formal logic

Modelling a Distributed Data Acquisition System

This thesis discusses the formal modelling and verification of certain non-real-time aspects of correctness of a mission-critical distributed software system known as the ALICE Data Point Service (ADAPOS). The domain of this distributed system is data acquisition from a particle detector control system in experimental high energy particle physics research. ADAPOS is part of the upgrade effort of A Large Ion Collider Experiment (ALICE) at the European Organisation for Nuclear Research (CERN), near Geneva in France/Switzerland, for the third run of the Large Hadron Collider (LHC). ADAPOS is based on the publicly available ALICE Data Point Processing (ADAPRO) C++14 framework and works within the free and open source GNU/Linux ecosystem. The model checker Spin was chosen for modelling and verifying ADAPOS. The model focuses on the general specification of ADAPOS. It includes ADAPOS processes, a load generator process, and rudimentary interpretations for the network protocols used between the processes. For experimenting with different interpretations of the underlying network protocols and also for coping with the state space explosion problem, eight variants of the model were developed and studied. Nine Linear Temporal Logic (LTL) properties were defined for all those variants. Large numbers of states were covered during model checking even though the model turned out to have a reachable state space too large to fully exhaust. No counter-examples were found to safety properties. A significant amount of evidence hinting that ADAPOS seems to be safe, was obtained. Liveness properties and implementation-level verification among other possible research directions remain open

Digital Humanism

This open access book deals with cultural and philosophical aspects of artificial intelligence (AI) and pleads for a “digital humanism”. This term is beginning to be en vogue everywhere. Due to a growing discontentment with the way digitalization is being used in the world, particularly formulated by former heroes of Internet, social media and search engine companies, philosophical as well as industrial thought leaders begin to plead for a humane use of digital tools. Yet the term “digital humanism” is a particular terminology that lacks a sound conceptual and philosophical basis and needs clarification still – and this gap is exactly filled by this book. It propagates a vision of society in which digitization is used to strengthen human self-determination, autonomy and dignity and whose time has come to be propagated throughout the world. The advantage of this book is that it is philosophically sound and yet written in a way that will make it accessible for everybody interested in the subject. Every chapters begins with a film scene illustrating a precise philosophical problem with AI and how we look at it – making the book not only readable, but even entertaining. And after having read the book the reader will have a clear vision of what it means to live in a world where digitization and AI are central technologies for a better and more humane civilization

The Significance of Evidence-based Reasoning for Mathematics, Mathematics Education, Philosophy and the Natural Sciences

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines

Digital Humanism

This open access book deals with cultural and philosophical aspects of artificial intelligence (AI) and pleads for a “digital humanism”. This term is beginning to be en vogue everywhere. Due to a growing discontentment with the way digitalization is being used in the world, particularly formulated by former heroes of Internet, social media and search engine companies, philosophical as well as industrial thought leaders begin to plead for a humane use of digital tools. Yet the term “digital humanism” is a particular terminology that lacks a sound conceptual and philosophical basis and needs clarification still – and this gap is exactly filled by this book. It propagates a vision of society in which digitization is used to strengthen human self-determination, autonomy and dignity and whose time has come to be propagated throughout the world. The advantage of this book is that it is philosophically sound and yet written in a way that will make it accessible for everybody interested in the subject. Every chapters begins with a film scene illustrating a precise philosophical problem with AI and how we look at it – making the book not only readable, but even entertaining. And after having read the book the reader will have a clear vision of what it means to live in a world where digitization and AI are central technologies for a better and more humane civilization