Related papers: Incomputability after Alan Turing
In the first of this pair of papers, it was proven that that no physical computer can correctly carry out all computational tasks that can be posed to it. The generality of this result follows from its use of a novel definition of…
Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms.
There is much to learn from what Turing hastily dismissed as Lady Lovelace s objection. Digital computers can indeed surprise us. Just like a piece of art, algorithms can be designed in such a way as to lead us to question our understanding…
The Turing machine, as it was presented by Turing himself, models the calculations done by a person. This means that we can compute whatever any Turing machine can compute, and therefore we are Turing complete. The question addressed here…
Computing is an indispensable component of nearly all technologies and is ubiquitous for vast segments of society. It is also essential to discoveries and innovations in most disciplines. However, while past grand challenges in science have…
We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of…
In this first of two papers, strong limits on the accuracy of physical computation are established. First it is proven that there cannot be a physical computer C to which one can pose any and all computational tasks concerning the physical…
Quantum computing exposes the brilliance of quantum mechanics through computer science and, as such, gives oneself a marvelous and exhilarating journey to go through. This article leads along that journey with a historical and current…
We initiate a formal study of logical inferences in context of the measure problem in cosmology or what we call cosmic logic. We describe a simple computational model of cosmic logic suitable for analysis of, for example, discretized…
Computational functionalism posits that consciousness is a computation. Here we show, perhaps surprisingly, that it cannot be a Turing computation. Rather, computational functionalism implies that consciousness is a novel type of…
I survey recent progress on a classic and challenging problem in social choice: the fair division of indivisible items. I discuss how a computational perspective has provided interesting insights into and understanding of how to divide…
Optimization problems are a staple of today's scientific and technical landscape. However, at present, solvers of such problems are almost exclusively run on digital hardware. Using Turing machines as a mathematical model for any type of…
Quantum computers take advantage of interfering quantum alternatives in order to handle problems that might be too time consuming with algorithms based on classical logic. Developing quantum computers requires new ways of thinking beyond…
This text presents the research field of natural/unconventional computing as it appears in the book COMPUTING NATURE. The articles discussed consist a selection of works from the Symposium on Natural Computing at AISB-IACAP (British Society…
The origin of the uncertainty inherent in quantum measurements has been discussed since quantum theory's inception, but to date the source of the indeterminacy of measurements performed at an angle with respect to a quantum state's…
A mathematical concept is identified and analyzed that is implicit in the 2012 paper Turing Incomputable Computation, presented at the Alan Turing Centenary Conference (Turing-100, Manchester). The concept, called dynamic level sets, is…
We discuss historical attempts to formulate a physical hypothesis from which Turing's thesis may be derived, and also discuss some related attempts to establish the computability of mathematical models in physics. We show that these…
The laptops, cell phones, and internet applications commonplace in our daily lives are all rooted in the idea of zeros and ones - in bits. This foundational element originated from the combination of mathematics and Claude Shannon's Theory…
The Church-Turing Thesis confuses numerical computations with symbolic computations. In particular, any model of computability in which equality is not definable, such as the lambda-models underpinning higher-order programming languages, is…
Since its beginnings in the 1940s, automated reasoning by computers has become a tool of ever growing importance in scientific research. So far, the rules underlying automated reasoning have mainly been formulated by humans, in the form of…