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15
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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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.
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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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.
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
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.
Quantum Hypercomputation—Hype or Computation?
, 2007
"... A recent attempt to compute a (recursion–theoretic) non–computable function using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion–theoretic) notion of computability. ..."
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A recent attempt to compute a (recursion–theoretic) non–computable function using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion–theoretic) notion of computability. A speculation is then offered as to where the putative power of quantum computers may come from.
THE ALEPHZERO OR ZERO DICHOTOMY (New and extended version with new arguments)
, 804
"... Abstract. This paper proves the existence of a dichotomy which being formally derived from the topological successiveness of ω ∗order leads to the same absurdity of Zeno’s Dichotomy II. It also derives a contradictory result from the first Zeno’s Dichotomy. ..."
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Abstract. This paper proves the existence of a dichotomy which being formally derived from the topological successiveness of ω ∗order leads to the same absurdity of Zeno’s Dichotomy II. It also derives a contradictory result from the first Zeno’s Dichotomy.
Physical Hypercomputation and the Church–Turing
"... Abstract. 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 c ..."
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Abstract. 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. Key words: Church–Turing thesis, effective computation, Gandy’s thesis, physical hypercomputation, supertasks A hypercomputer is a physical or an abstract system that computes functions that cannot be computed by a universal Turing machine. Turing (1939) was perhaps the first to introduce hypercomputers. He called them omachines, for machines with oracles. Other examples of hypercomputers are described in Copeland and Sylvan (1999). In what follows we describe a possible physical device that computes a nonTuring computable function. This physical hypercomputer is a version of the devices introduced by Pitowsky (1990) and Hogarth (1992, 1994). We consider
Quantum gravity computers: On the
, 2008
"... theory of computation with indefinite causal structure ..."
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Effective computation by humans and machines
, 2002
"... There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. ..."
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There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing argument for CTT by analyzing the processes carried out by a human computer. I then contend that if the Gandy–Sieg account is correct, then the notion of effective computability has changed after 1936. Today computer scientists view effective computability in terms of finite machine computation. My contention is supported by the current formulations of CTT, which always refer to machine computation, and by the current argumentation for CTT, which is different from the main arguments advanced by Turing and Church. I finally turn to discuss Robin Gandy’s characterization of machine computation. I suggest that there is an ambiguity regarding the types of machines Gandy was postulating. I offer three interpretations, which differ in their scope and limitations, and conclude that none provides the basis for claiming that Gandy characterized finite machine computation.