Zeno machines and hypercomputation

This paper reviews the Church-Turing Thesis (or rather, theses) with reference to their origin and application and considers some models of "hypercomputation", concentrating on perhaps the most straight-forward option: Zeno machines (Turing machines with accelerating clock). The halting problem is briefly discussed in a general context and the suggestion that it is an inevitable companion of any reasonable computational model is emphasised. It is hinted that claims to have "broken the Turing barrier" could be toned down and that the important and well-founded role of Turing computability in the mathematical sciences stands unchallenged.Comment: 11 pages. First submitted in December 2004, substantially revised in July and in November 2005. To appear in Theoretical Computer Scienc

Matrix theory of gravitation

A new classical theory of gravitation within the framework of general relativity is presented. It is based on a matrix formulation of four-dimensional Riemann-spaces and uses no artificial fields or adjustable parameters. The geometrical stress-energy tensor is derived from a matrix-trace Lagrangian, which is not equivalent to the curvature scalar R. To enable a direct comparison with the Einstein-theory a tetrad formalism is utilized, which shows similarities to teleparallel gravitation theories, but uses complex tetrads. Matrix theory might solve a 27-year-old, fundamental problem of those theories (sec. 4.1). For the standard test cases (PPN scheme, Schwarzschild-solution) no differences to the Einstein-theory are found. However, the matrix theory exhibits novel, interesting vacuum solutions.Comment: 24 page

Nature as a Network of Morphological Infocomputational Processes for Cognitive Agents

This paper presents a view of nature as a network of infocomputational agents organized in a dynamical hierarchy of levels. It provides a framework for unification of currently disparate understandings of natural, formal, technical, behavioral and social phenomena based on information as a structure, differences in one system that cause the differences in another system, and computation as its dynamics, i.e. physical process of morphological change in the informational structure. We address some of the frequent misunderstandings regarding the natural/morphological computational models and their relationships to physical systems, especially cognitive systems such as living beings. Natural morphological infocomputation as a conceptual framework necessitates generalization of models of computation beyond the traditional Turing machine model presenting symbol manipulation, and requires agent-based concurrent resource-sensitive models of computation in order to be able to cover the whole range of phenomena from physics to cognition. The central role of agency, particularly material vs. cognitive agency is highlighted

Notes on the Mathematical Foundations of Analogue Computation

Digital computing has its mathematical foundations in (classical) recursion theory and constructive mathematics. The implicit, working, assumption of those who practice the noble art of analog computing may well be that the mathematical foundations of their subject is as sound as the foundations of the real analysis. That, in turn, implies a reliance on the soundness of set theory plus the axiom of choice. This is, surely, seriously disturbing from a computation point of view. Therefore, in this paper, I seek to locate a foundation for analog computing in exhibiting some tentative dualities with results that are analogous to those that are standard in computability theory. The main question, from the point of view of economics, is whether the Phillips Machine, as an analog computer, has universal computing properties. The conjectured answer is in the negative.

Quantum walks: a comprehensive review

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.Comment: Paper accepted for publication in Quantum Information Processing Journa