647 research outputs found
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
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
Synthetic Philosophy of Mathematics and Natural Sciences Conceptual analyses from a Grothendieckian Perspective
ISBN-13: 978-0692593974. Giuseppe Longo. Synthetic Philosophy of Mathematics and Natural Sciences, Conceptual analyses from a Grothendieckian Perspective, Reflections on âSynthetic Philosophy of Contemporary Mathematicsâ by F. Zalamea, Urbanomic (UK) and Sequence Press (USA), 2012. Invited Paper, in Speculations: Journal of Speculative Realism, Published: 12/12/2015, followed by an answer by F. Zalamea.International audienceZalameaâs book is as original as it is belated. It is indeed surprising, if we give it a momentâs thought, just how greatly behind schedule philosophical reflection on contemporary mathematics lags, especially considering the momentous changes that took place in the second half of the twentieth century. Zalamea compares this situation with that of the philosophy of physics: he mentions DâEspagnatâs work on quantum mechanics, but we could add several others who, in the last few decades, have elaborated an extremely timely philosophy of contemporary physics (see for example Bitbol 2000; Bitbol et al. 2009). As was the case in biology, philosophy â since Kantâs crucial observations in the Critique of Judgment, at least â has often ârun aheadâ of life sciences, exploring and opening up a space for reflections that are not derived from or integrated with its contemporary scientific practice. Some of these reflections are still very much auspicious today. And indeed, some philosophers today are saying something truly new about biology..
On the performance and programming of reversible molecular computers
If the 20th century was known for the computational revolution, what will the 21st be known for? Perhaps the recent strides in the nascent fields of molecular programming and biological computation will help bring about the âComing Era of Nanotechnologyâ promised in Drexlerâs âEngines of Creationâ. Though there is still far to go, there is much reason for optimism. This thesis examines the underlying principles needed to realise the computational aspects of such âenginesâ in a performant way. Its main body focusses on the ways in which thermodynamics constrains the operation and design of such systems, and it ends with the proposal of a model of computation appropriate for exploiting these constraints.
These thermodynamic constraints are approached from three different directions. The first considers the maximum possible aggregate performance of a system of computers of given volume, V, with a given supply of free energy. From this perspective, reversible computing is imperative in order to circumvent the Landauer limit. A result of Frank is refined and strengthened, showing that the adiabatic regime reversible computer performance is the best possible for any computerâquantum or classical. This therefore shows a universal scaling law governing the performance of compact computers of ~V^(5/6), compared to ~V^(2/3) for conventional computers. For the case of molecular computers, it is shown how to attain this bound. The second direction extends this performance analysis to the case where individual computational particles or sub-units can interact with one another. The third extends it to interactions with shared, non-computational parts of the system. It is found that accommodating these interactions in molecular computers imposes a performance penalty that undermines the earlier scaling result. Nonetheless, scaling superior to that of irreversible computers can be preserved, and appropriate mitigations and considerations are discussed. These analyses are framed in a context of molecular computation, but where possible more general computational systems are considered.
The proposed model, the Ś-calculus, is appropriate for programming reversible molecular computers taking into account these constraints. A variety of examples and mathematical analyses accompany it. Moreover, abstract sketches of potential molecular implementations are provided. Developing these into viable schemes suitable for experimental validation will be a focus of future work
- âŠ