4,716 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
Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?
Classical interpretations of Goedel's formal reasoning imply that the truth
of some arithmetical propositions of any formal mathematical language, under
any interpretation, is essentially unverifiable. However, a language of
general, scientific, discourse cannot allow its mathematical propositions to be
interpreted ambiguously. Such a language must, therefore, define mathematical
truth verifiably. We consider a constructive interpretation of classical,
Tarskian, truth, and of Goedel's reasoning, under which any formal system of
Peano Arithmetic is verifiably complete. We show how some paradoxical concepts
of Quantum mechanics can be expressed, and interpreted, naturally under a
constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version
is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht
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
Computable functions, quantum measurements, and quantum dynamics
We construct quantum mechanical observables and unitary operators which, if
implemented in physical systems as measurements and dynamical evolutions, would
contradict the Church-Turing thesis which lies at the foundation of computer
science. We conclude that either the Church-Turing thesis needs revision, or
that only restricted classes of observables may be realized, in principle, as
measurements, and that only restricted classes of unitary operators may be
realized, in principle, as dynamics.Comment: 4 pages, REVTE
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