Zeno-machines And The Metaphysics Of Time

This paper aims to explore the nature of Zeno-machines by examining their conceptual coherence, from the perspective of contemporary theories on the passage of time. More specifically, it will analyse the following questions: Are Zeno-machines and supertasks coherent if we adopt the eternalist theory of time? What conclusions can be drawn from choosing the eternalist thesis, or the presentist thesis, when examining Zeno-machines? To this end, an overview of the opposing theories of time is provided, alongside the usual objections to Zeno-machines and their theoretical foundations from Zeno's dichotomy paradox.17216116

ON THE FOUNDATIONS OF COMPUTABILITY THEORY

The principal motivation for this work is the observation that there are significant deficiencies in the foundations of conventional computability theory. This thesis examines the problems with conventional computability theory, including its failure to address discrepancies between theory and practice in computer science, semantic confusion in terminology, and limitations in the scope of conventional computing models. In light of these difficulties, fundamental notions are re-examined and revised definitions of key concepts such as “computer,” “computable,” and “computing power” are provided. A detailed analysis is conducted to determine desirable semantics and scope of applicability of foundational notions. The credibility of the revised definitions is ascertained by demonstrating by their ability to address identified problems with conventional definitions. Their practical utility is established through application to examples. Other related issues, including hidden complexity in computations, subtleties related to encodings, and the cardinalities of sets involved in computing, are examined. A resource-based meta-model for characterizing computing model properties is introduced. The proposed definitions are presented as a starting point for an alternate foundation for computability theory. However, formulation of the particular concepts under discussion is not the sole purpose of the thesis. The underlying objective of this research is to open discourse on alternate foundations of computability theory and to inspire re-examination of fundamental notions

Master index

Pla general, del mural ceràmic que decora una de les parets del vestíbul de la Facultat de Química de la UB. El mural representa diversos símbols relacionats amb la química

Zeno-machines and the metaphysics of time

This paper aims to explore the nature of Zeno-machines by examining their conceptual coherence, from the perspective of contemporary theories on the passage of time. More specifically, it will analyse the following questions: Are Zeno-machines and supertasks coherent if we adopt the eternalist theory of time? What conclusions can be drawn from choosing the eternalist thesis, or the presentist thesis, when examining Zeno-machines? To this end, an overview of the opposing theories of time is provided, alongside the usual objections to Zeno-machines and their theoretical foundations from Zeno’s dichotomy paradox.Keywords: Zeno-machines, philosophy of time, supertasks, Zeno’s paradox

Scale-invariant cellular automata and self-similar Petri nets

Two novel computing models based on an infinite tessellation of space-time are introduced. They consist of recursively coupled primitive building blocks. The first model is a scale-invariant generalization of cellular automata, whereas the second one utilizes self-similar Petri nets. Both models are capable of hypercomputations and can, for instance, "solve" the halting problem for Turing machines. These two models are closely related, as they exhibit a step-by-step equivalence for finite computations. On the other hand, they differ greatly for computations that involve an infinite number of building blocks: the first one shows indeterministic behavior whereas the second one halts. Both models are capable of challenging our understanding of computability, causality, and space-time.Comment: 35 pages, 5 figure

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Counterpossibles in Science: The Case of Relative Computability

I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as 'If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable,' which is true, and 'If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable,' which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What’s more, I argue that these counterpossibles don’t just appear in the periphery of relative computability theory but instead they play an ineliminable role in the development of the theory. Finally, I present and discuss a model theory for these counterfactuals that is a straightforward extension of the familiar comparative similarity models

Computable Diagonalizations and Turing's Cardinality Paradox

A. N. Turing's 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing's Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing's objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing machine, computability, computable sequences, and Turing's effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing's Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing's discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing's two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers