Related papers: A note reviewing Turing's 1936
Although the Turing-machine model of computation is widely used in computer science it is fundamentally inadequate as a foundation for the theory of modern scientific computation. The real-number model is described as an alternative.…
This paper talk about the complexity of computation by Turing Machine. I take attention to the relation of symmetry and order structure of the data, and I think about the limitation of computation time. First, I make general problem named…
We show that for both the unary relation of transcendence and the finitary relation of algebraic independence on a field, the degree spectra of these relations may consist of any single computably enumerable Turing degree, or of those c.e.…
Many people think of analog computing as a historic dead-end in computing. In fact, nothing could be further from the truth as analog computing - together with quantum computing - has the potential to bring computing to new levels with…
Standard expositions of Goedel's 1931 paper on undecidable arithmetical propositions are based on two presumptions in Goedel's 1931 interpretation of his own, formal, reasoning - one each in Theorem VI and in Theorem XI - which do not meet…
Before Alan Turing made his crucial contributions to the theory of computation, he studied the question of whether quantum mechanics could throw light on the nature of free will. This article investigates the roles of quantum mechanics and…
By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous…
We argue that the halting problem for quantum computers which was first raised by Myers, is by no means solved, as has been claimed recently. We explicitly demonstrate the difficulties that arise in a quantum computer when different…
While there is a well-established notion of what a computable ordinal is, the question which functions on the countable ordinals ought to be computable has received less attention so far. We propose a notion of computability on the space of…
The well-known Turing machine is an example of a theoretical digital computer, and it was the logical basis of constructing real electronic computers. In the present paper we propose an alternative, namely, by formalising arithmetic…
In contrast with software-generated randomness (called pseudo-randomness), quantum randomness is provable incomputable, i.e.\ it is not exactly reproducible by any algorithm. We provide experimental evidence of incomputability --- an…
This paper explores and clarifies several issues surrounding Zeno machines and the issue of running a Turing machine for infinite time. Without a minimum hypothetical bound on physical conditions, any magical machine can be created, and…
This essay aims to propose construction theory, a new domain of theoretical research on machine construction, and use it to shed light on a fundamental relationship between living and computational systems. Specifically, we argue that…
This paper reviews the 1994 proof that the spectral gap of Hamiltonian quantum systems capable of universal computation is uncomputable.
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
A proof of quantumness is a protocol through which a classical machine can test whether a purportedly quantum device, with comparable time and memory resources, is performing a computation that is impossible for classical computers.…
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…
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…
In contrast to other constructivist schools, for Brouwer, the notion of "constructive object" is not restricted to be presented as `words' in some finite alphabet of symbols, and choice sequences which are non-predetermined and unfinished…
A century ago, discoveries of a serious kind of logical error made separately by several leading mathematicians led to acceptance of a sharply enhanced standard for rigor within what ultimately became the foundation for Computer Science. By…