Related papers: On Universality in Real Computation
We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length $\omega$ to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite…
Classical computations can not capture the essence of infinite computations very well. This paper will focus on a class of infinite computations called convergent infinite computations}. A logic for convergent infinite computations is…
We define a generalization of the Turing machine that computes on general sets. Our main theorem states that the class of generalized Turing machine computable functions and the class of Set Recursive functions coincide.
Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification…
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.
For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
Conventional Neural Networks can approximate simple arithmetic operations, but fail to generalize beyond the range of numbers that were seen during training. Neural Arithmetic Units aim to overcome this difficulty, but current arithmetic…
Expanding upon the widely recognized notion of mathematical universality in Turing machines, a concept of thermodynamic universality in Turing machines is introduced. Under the physical Church-Turing thesis, the existence of a…
This paper provides a new and more direct proof of the assertion that a Turing computable function of the natural numbers is primitive recursive if and only if the time complexity of the corresponding Turing machine is bounded by a…
Finite Turing computation has a fundamental symmetry between inputs, outputs, programs, time, and storage space. Standard models of transfinite computational break this symmetry; we consider ways to recover it and study the resulting model…
A remarkable new definition of a self-delimiting universal Turing machine is presented that is easy to program and runs very quickly. This provides a new foundation for algorithmic information theory. This new universal Turing machine is…
We start by an introduction to the basic concepts of computability theory and the introduction of the concept of Turing machine and computation universality. Then se turn to the exploration of trade-offs between different measures of…
Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time. By doing so, they provide a natural model of infinitary computability, a theoretical setting for the analysis of the power and…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
By the sometimes so-called MAIN THEOREM of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered different relaxed notions…
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 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…
The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance)…
The Turing Machine is the paradigmatic case of computing machines, but there are others such as analogical, connectionist, quantum and diverse forms of unconventional computing, each based on a particular intuition of the phenomenon of…