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Physical Hypercomputation and the Church–Turing Thesis
, 2003
"... We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a ..."
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Cited by 14 (1 self)
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We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy’s thesis.
Set Theory and Physics
 FOUNDATIONS OF PHYSICS, VOL. 25, NO. 11
, 1995
"... Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of soli ..."
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Cited by 8 (7 self)
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Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid threedimensional objects, (iii) in the theory of effective computability (ChurchTurhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.
Abstract geometrical computation: Turingcomputing ability and undecidability
, 2004
"... In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this ideali ..."
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Cited by 3 (1 self)
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In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating Euclidean lines on (continuous) spacetime diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are continuous and not discrete (i.e. R instead of Z). In this article, the model is restricted to Q in order to remain inside Turingcomputation theory. We prove that our model can carry out any Turingcomputation through twocounter automata simulation and provide some undecidability results.
How to acknowledge hypercomputation?
, 2007
"... We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models. ..."
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Cited by 1 (0 self)
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We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models.
Supertask Computation
, 2002
"... Abstract. Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms. 1 Supertasks What would you compute with an infinitely ..."
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Abstract. Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms. 1 Supertasks What would you compute with an infinitely fast computer? What could you compute? To make sense of these questions, one would want to understand the algorithms that the machines would carry out, computational tasks involving infinitely many steps of computation. Such tasks, known as supertasks, have been studied since antiquity from a variety of viewpoints. Zeno of Elea (ca. 450 B.C.) was perhaps the first to grapple with supertasks, in his famous paradox that it is impossible to go from here
and
, 712
"... We discuss the question of how to operationally validate whether or not a “hypercomputer ” performs better than the known discrete computational models. 1 ..."
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We discuss the question of how to operationally validate whether or not a “hypercomputer ” performs better than the known discrete computational models. 1
The extent of computation in MalamentHogarth
, 2008
"... We analyse the extent of possible computations following Hogarth [7] in MalamentHogarth (MH) spacetimes, and Etesi and Németi [3] in the special subclass containing rotating Kerr black holes. [7] had shown that any arithmetic statement could be resolved in a suitable MH spacetime. [3] had shown tha ..."
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We analyse the extent of possible computations following Hogarth [7] in MalamentHogarth (MH) spacetimes, and Etesi and Németi [3] in the special subclass containing rotating Kerr black holes. [7] had shown that any arithmetic statement could be resolved in a suitable MH spacetime. [3] had shown that some ∀ ∃ relations on natural numbers which are neither universal nor couniversal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. The purpose of this note is to address this question, and further show that MH spacetimes can compute far beyond the arithmetic: effectively Borel statements (so hyperarithmetic in second order number theory, or the structure of analysis) can likewise be resolved: Theorem A. If H is any hyperarithmetic predicate on integers, then there is an MH spacetime in which any query?n ∈ H? can be computed. In one sense this is best possible, as there is an upper bound to computational ability in any spacetime which is thus a universal constant of the spacetime M. Theorem C. Assuming the (modest and standard) requirement that spacetime manifolds be paracompact and Hausdorff, for any MH spacetime M there will be a countable ordinal upper bound, w(M), on the complexity of questions in the Borel hierarchy resolvable in it. 1