Finiteness of the universe and computation beyond Turing computability

An anatomy of a quantum adiabatic algorithm that transcends the Turing computability

We give an update on a quantum adiabatic algorithm for the Turing noncomputable Hilbert's tenth problem, and briefly go over some relevant issues and misleading objections to the algorithm.

Transcending the Limits of Turing Computability

Hypercomputation or super-Turing computation is a ``computation'' that transcends the limit imposed by Turing's model of computability. The field still faces some basic questions, technical (can we mathematically and/or physically build a hypercomputer?), cognitive (can hypercomputers realize the AI dream?), philosophical (is thinking more than computing?). The aim of this paper is to address the question: can we mathematically build a hypercomputer? We will discuss the solutions of the Infinite Merchant Problem, a decision problem equivalent to the Halting Problem, based on results obtained i…

Measurability and Computability

The conceptual relation between the measurability of quantum mechanical observables and the computability of numerical functions is re-examined. A new formulation is given for the notion of measurability with finite precision in order to reconcile the conflict alleged by M. A. Nielsen [Phys. Rev. Lett. 79, 2915 (1997)] that the measurability of a certain observable contradicts the Church-Turing thesis. It is argued that any function computable by a quantum algorithm is a recursive function obeying the Church-Turing thesis, whereas any observable can be measured in principle.

Resource dependent undecidability: computability landscape of distinct Turing theories

Can a problem undecidable with classical resources be decidable with quantum ones? The answer expected is no; as both being Turing theories, they should not solve the Halting problem - a problem unsolvable by any Turing machine. Yet, we provide an affirmative answer to the aforesaid question. We come up with a novel logical structure to formulate infinitely many such problems for any pair of distinct Turing theories, including but not limited to the classical and quantum theories. Importantly, a class of other decision problems, such as the Halting one, remains unsolvable in all those theories…

闭合类时曲线的可计算性理论

Computability Theory of Closed Timelike Curves

本文研究配备无界 Deutschian 闭合类时曲线的图灵机的计算能力，证明寻找可数无限维可计算马尔可夫链和量子通道近似不动点的复杂性恰好为 $Δ_2$，并利用量子马尔可夫映射理论确立上界。

Quantum Principles and Mathematical Computability

Taking the view that computation is after all physical, we argue that physics, particularly quantum physics, could help extend the notion of computability. Here, we list the important and unique features of quantum mechanics and then outline a quantum mechanical "algorithm" for one of the insoluble problems of mathematics, the Hilbert's tenth and equivalently the Turing halting problem. The key element of this algorithm is the {\em computability} and {\em measurability} of both the values of physical observables and of the quantum-mechanical probability distributions for these values.