259 research outputs found
Real Hypercomputation and Continuity
By the sometimes so-called 'Main Theorem' of Recursive Analysis, every
computable real function is necessarily continuous. We wonder whether and which
kinds of HYPERcomputation allow for the effective evaluation of also
discontinuous f:R->R. More precisely the present work considers the following
three super-Turing notions of real function computability:
* relativized computation; specifically given oracle access to the Halting
Problem 0' or its jump 0'';
* encoding real input x and/or output y=f(x) in weaker ways also related to
the Arithmetic Hierarchy;
* non-deterministic computation.
It turns out that any f:R->R computable in the first or second sense is still
necessarily continuous whereas the third type of hypercomputation does provide
the required power to evaluate for instance the discontinuous sign function.Comment: previous version (extended abstract) has appeared in pp.562-571 of
"Proc. 1st Conference on Computability in Europe" (CiE'05), Springer LNCS
vol.352
Effective Physical Processes and Active Information in Quantum Computing
The recent debate on hypercomputation has arisen new questions both on the
computational abilities of quantum systems and the Church-Turing Thesis role in
Physics. We propose here the idea of "effective physical process" as the
essentially physical notion of computation. By using the Bohm and Hiley active
information concept we analyze the differences between the standard form
(quantum gates) and the non-standard one (adiabatic and morphogenetic) of
Quantum Computing, and we point out how its Super-Turing potentialities derive
from an incomputable information source in accordance with Bell's constraints.
On condition that we give up the formal concept of "universality", the
possibility to realize quantum oracles is reachable. In this way computation is
led back to the logic of physical world.Comment: 10 pages; Added references for sections 2 and
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
Some Thoughts on Hypercomputation
Hypercomputation is a relatively new branch of computer science that emerged
from the idea that the Church--Turing Thesis, which is supposed to describe
what is computable and what is noncomputable, cannot possible be true. Because
of its apparent validity, the Church--Turing Thesis has been used to
investigate the possible limits of intelligence of any imaginable life form,
and, consequently, the limits of information processing, since living beings
are, among others, information processors. However, in the light of
hypercomputation, which seems to be feasibly in our universe, one cannot impose
arbitrary limits to what intelligence can achieve unless there are specific
physical laws that prohibit the realization of something. In addition,
hypercomputation allows us to ponder about aspects of communication between
intelligent beings that have not been considered befor
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