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Due to common misconceptions about the Church-Turing thesis, it has been widely assumed that the Turing machine provides an upper bound on what is computable. This is not so. The new field of hypercomputation studies models of computation…
Computational problems are classified into computable and uncomputable problems. If there exists an effective procedure (algorithm) to compute a problem then the problem is computable otherwise it is uncomputable. Turing machines can…
This paper investigates a variety of unconventional quantum computation devices, including fermionic quantum computers and computers that exploit nonlinear quantum mechanics. It is shown that unconventional quantum computing devices can in…
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…
This paper discusses "computational" systems capable of "computing" functions not computable by predefined Turing machines if the systems are not isolated from their environment. Roughly speaking, these systems can change their finite…
The hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine. Several numerical simulations of a possible hypercomputational algorithm based on…
For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While…
We introduce the notion of universal memcomputing machines (UMMs): a class of brain-inspired general-purpose computing machines based on systems with memory, whereby processing and storing of information occur on the same physical location.…
Turing computability is the standard computability paradigm which captures the computational power of digital computers. To understand whether one can create physically realistic devices which have super-Turing power, one needs to…
Hypercomputation is a relatively new branch of computer science that emerged from the idea that the Church--Turing Thesis, which is supposed to describe what is computable and what is noncomputable, cannot possible be true. Because of its…
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…
We clarify the confusion, misunderstanding and misconception that the physical finiteness of the universe, if the universe is indeed finite, would rule out all hypercomputation, the kind of computation that exceeds the Turing computability,…
This paper serves as a review and discussion of the recent works on memcomputing. In particular, the $\textit{universal memcomputing machine}$ (UMM) and the $\textit{digital memcomputing machine}$ (DMM) are discussed. We review the…
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…
We outline the construction of a molecular system that could, in principle, implement a thermodynamically reversible Universal Turing Machine (UTM). By proposing a concrete-albeit idealised-design and operational protocol, we reveal…
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…
It is common practice to compare the computational power of different models of computation. For example, the recursive functions are strictly more powerful than the primitive recursive functions, because the latter are a proper subset of…
A universal Turing machine is a powerful concept - a single device can compute any function that is computable. A universal spin model, similarly, is a class of physical systems whose low energy behavior simulates that of any spin system.…
With the great success in simulating many intelligent behaviors using computing devices, there has been an ongoing debate whether all conscious activities are computational processes. In this paper, the answer to this question is shown to…
Recent works have independently suggested that Quantum Mechanics might permit for procedures that transcend the power of Turing Machines as well as of `standard' Quantum Computers. These approaches rely on and indicate that Quantum…