Three forms of physical measurement and their computability

info:eu-repo/semantics/publishedVersio

Quantum Algorithm for Hilbert's Tenth Problem

We explore in the framework of Quantum Computation the notion of {\em Computability}, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle--that is, if certain hamiltonian and its ground state can be physically constructed according to the proposal--quantum computability would surpass classical computability as delimited by the Church-Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles

Non-Turing computations via Malament-Hogarth space-times

We investigate the Church-Kalm\'ar-Kreisel-Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church-Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church-Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.) We also look at various ``obstacles'' to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the ``design'' of our future computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.Comment: Final, published version: 25 pages, LaTex with two eps-figures, journal reference adde

The Computability-Theoretic Content of Emergence

In dealing with emergent phenomena, a common task is to identify useful descriptions of them in terms of the underlying atomic processes, and to extract enough computational content from these descriptions to enable predictions to be made. Generally, the underlying atomic processes are quite well understood, and (with important exceptions) captured by mathematics from which it is relatively easy to extract algorithmic con- tent. A widespread view is that the difficulty in describing transitions from algorithmic activity to the emergence associated with chaotic situations is a simple case of complexity outstripping computational resources and human ingenuity. Or, on the other hand, that phenomena transcending the standard Turing model of computation, if they exist, must necessarily lie outside the domain of classical computability theory. In this article we suggest that much of the current confusion arises from conceptual gaps and the lack of a suitably fundamental model within which to situate emergence. We examine the potential for placing emer- gent relations in a familiar context based on Turing's 1939 model for interactive computation over structures described in terms of reals. The explanatory power of this model is explored, formalising informal descrip- tions in terms of mathematical definability and invariance, and relating a range of basic scientific puzzles to results and intractable problems in computability theory

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