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24
NonTuring Computers and NonTuring Computability
, 1994
"... possible to perform computational supertasks — that is, an infinite number of computational steps in a finite span of time — in a kind of relativistic spacetime that Earman and Norton (1993) have dubbed a MalamentHogarth spacetime1. ..."
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Cited by 36 (2 self)
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possible to perform computational supertasks — that is, an infinite number of computational steps in a finite span of time — in a kind of relativistic spacetime that Earman and Norton (1993) have dubbed a MalamentHogarth spacetime1.
Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing ..."
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A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard ChurchTuring Thesis.
Deciding arithmetic in Malament–Hogarth spacetimes
, 2001
"... Abstract Presented here are some new results concerning the computational power of socalled SADn computers, a class of Turing machinebased computers that utilise the geometry of MalamentHogarth spacetimes to perform nonTuring computable feats. The main result is that SADn can decide nquantifier ..."
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Abstract Presented here are some new results concerning the computational power of socalled SADn computers, a class of Turing machinebased computers that utilise the geometry of MalamentHogarth spacetimes to perform nonTuring computable feats. The main result is that SADn can decide nquantifier arithmetic but not (n+1)quantifier arithmetic, a result which reveals how neatly SADns map into the Kleene arithmetical hierarchy.
Firstorder logic foundation of relativity theories
 In New Logics for the XXIst Century II, Mathematical Problems from Applied Logics, volume 5 of International Mathematical Series
, 2006
"... Abstract. Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity— including such revolutionary areas as black hole physics, relativistic computers, new cosmology—are presented in this paper. We would like to invite the logician reader to ..."
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Cited by 8 (8 self)
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Abstract. Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity— including such revolutionary areas as black hole physics, relativistic computers, new cosmology—are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Besides general perspective and motivation, we present initial results in this direction.
Predictability, Computability and Spacetime
, 2002
"... thesis is the result of the author’s own work and includes nothing which is the outcome of work done in collaboration. To my Mum and Dad, who succeeded in violating Larkin’s Law. And to my sister Lyn, who recently stopped pulling my hair. Acknowledgements The following have personally helped to shap ..."
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Cited by 7 (0 self)
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thesis is the result of the author’s own work and includes nothing which is the outcome of work done in collaboration. To my Mum and Dad, who succeeded in violating Larkin’s Law. And to my sister Lyn, who recently stopped pulling my hair. Acknowledgements The following have personally helped to shape the ideas in the thesis: Gordon Belot,
Quantum SpeedUp of Computations
 Philosophy of Science
, 2002
"... ChurchTuring Thesis as saying something about the scope and limitations of physical computing machines. Although this was not the intention of Church or Turing, the Physical Church Turing thesis is interesting in its own right. Consider, for example, Wolfram’s formulation: One can expect in fact th ..."
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ChurchTuring Thesis as saying something about the scope and limitations of physical computing machines. Although this was not the intention of Church or Turing, the Physical Church Turing thesis is interesting in its own right. Consider, for example, Wolfram’s formulation: One can expect in fact that universal computers are as powerful in their computational capabilities as any physically realizable system can be, that they can simulate any physical system...Nophysically implementable procedure could then shortcut a computationally irreducible process. (Wolfram 1985) Wolfram’s thesis consists of two parts: (a) Any physical system can be simulated (to any degree of approximation) by a universal Turing machine (b) Complexity bounds on Turing machine simulations have physical significance. For example, suppose that the computation of the minimum energy of some system of n particles takes at least exponentially (in n) many steps. Then the relaxation time of the actual physical system to its minimum energy state will also take exponential time. An even more extreme formulation of (more or less) the same thesis is due to Aharonov (1998): A probabilistic Turing machine can simulate any reasonable physical device in polynomial cost. She calls this The Modern Church Thesis. Aharonov refers here to probabilistic Turing machines that use random numbers in addition to the usual deterministic table of steps. It seems that such machines are capable to perform certain tasks faster than fully deterministic machines. The most famous randomized algorithm of that kind concerns the decision whether a given natural number is prime. A probabilistic algorithm that decides primality in a number of
Abstract SuperTasks, Accelerating Turing Machines and Uncomputability
"... Accelerating Turing machines are abstract devices that have the same computational structure as Turing machines, but can perform supertasks. I argue that performing supertasks alone does not buy more computational power, and that accelerating Turing machines do not solve the halting problem. To sh ..."
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Accelerating Turing machines are abstract devices that have the same computational structure as Turing machines, but can perform supertasks. I argue that performing supertasks alone does not buy more computational power, and that accelerating Turing machines do not solve the halting problem. To show this, I analyze the reasoning that leads to Thomson's paradox, point out that the paradox rests on a conflation of different perspectives of accelerating processes, and conclude that the same conflation underlies the claim that accelerating Turing machines can solve the halting problem.
ZENO MEETS MODERN SCIENCE
, 2005
"... “No one has ever touched Zeno without refuting him”. We will not refute Zeno in this paper. Instead we review some unexpected encounters of Zeno with modern science. The paper begins with a brief biography of Zeno of Elea followed by his famous paradoxes of motion. Reflections on continuity of space ..."
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“No one has ever touched Zeno without refuting him”. We will not refute Zeno in this paper. Instead we review some unexpected encounters of Zeno with modern science. The paper begins with a brief biography of Zeno of Elea followed by his famous paradoxes of motion. Reflections on continuity of space and time lead us to Banach and Tarski and to their celebrated paradox, which is in fact not a paradox at all but a strict mathematical theorem, although very counterintuitive. Quantum mechanics brings another flavour in Zeno paradoxes. Quantum Zeno and antiZeno effects are really paradoxical but now experimental facts. Then we discuss supertasks and bifurcated supertasks. The concept of localisation leads us to Newton and Wigner and to interesting phenomenon of quantum revivals. At last we note that the paradoxical idea of timeless universe, defended by Zeno and Parmenides at ancient times, is still alive in quantum gravity. The list of references that follows is necessarily incomplete but we hope it will assist interested reader to fill in details. PACS numbers: 01.70.+w 1.
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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We discuss the question of how to operationally validate whether or not a “hypercomputer” performs better than the known discrete computational models.