19,580 research outputs found
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
Non-classical computing: feasible versus infeasible
Physics sets certain limits on what is and is not computable. These limits are very far from having been reached by current technologies. Whilst proposals for hypercomputation are almost certainly infeasible, there are a number of non classical approaches that do hold considerable promise. There are a range of possible architectures that could be implemented on silicon that are distinctly different from the von Neumann model. Beyond this, quantum simulators, which are the quantum equivalent of analogue computers, may be constructable in the near future
A quantum-information-theoretic complement to a general-relativistic implementation of a beyond-Turing computer
There exists a growing literature on the so-called physical Church-Turing
thesis in a relativistic spacetime setting. The physical Church-Turing thesis
is the conjecture that no computing device that is physically realizable (even
in principle) can exceed the computational barriers of a Turing machine. By
suggesting a concrete implementation of a beyond-Turing computer in a spacetime
setting, Istv\'an N\'emeti and Gyula D\'avid (2006) have shown how an
appreciation of the physical Church-Turing thesis necessitates the confluence
of mathematical, computational, physical, and indeed cosmological ideas. In
this essay, I will honour Istv\'an's seventieth birthday, as well as his
longstanding interest in, and his seminal contributions to, this field going
back to as early as 1987 by modestly proposing how the concrete implementation
in N\'emeti and D\'avid (2006) might be complemented by a
quantum-information-theoretic communication protocol between the computing
device and the logician who sets the beyond-Turing computer a task such as
determining the consistency of Zermelo-Fraenkel set theory. This suggests that
even the foundations of quantum theory and, ultimately, quantum gravity may
play an important role in determining the validity of the physical
Church-Turing thesis.Comment: 27 pages, 5 figures. Forthcoming in Synthese. Matches published
versio
Quantum Hypercomputation - Hype or Computation?
A recent attempt to compute a (recursion--theoretic) non--computable function using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion--theoretic) notion of computability. A speculation is then offered as to where the putative power of quantum computers may come from
Fibers and global geometry of functions
Since the seminal work of Ambrosetti and Prodi, the study of global folds was
enriched by geometric concepts and extensions accomodating new examples. We
present the advantages of considering fibers, a construction dating to Berger
and Podolak's view of the original theorem. A description of folds in terms of
properties of fibers gives new perspective to the usual hypotheses in the
subject. The text is intended as a guide, outlining arguments and stating
results which will be detailed elsewhere
Infinity and the Sublime
In this paper we intend to connect two different strands of research
concerning the origin of what I shall loosely call "formal" ideas: firstly, the
relation between logic and rhetoric - the theme of the 2006 Cambridge
conference to which this paper was a contribution -, and secondly, the impact
of religious convictions on the formation of certain twentieth century
mathematical concepts, as brought to the attention recently by the work of L.
Graham and J.-M. Kantor. In fact, we shall show that the latter question is a
special case of the former, and that investigation of the larger question adds
to our understanding of the smaller one. Our approach will be primarily
historical.Comment: 29 pages and 3 figure
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