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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 Tu ..."
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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.
General relativistic hypercomputing and foundation of mathematics
"... Abstract. Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task, ..."
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Abstract. Looking at very recent developments in spacetime theory, we can wonder whether these results exhibit features of hypercomputation that traditionally seemed impossible or absurd. Namely, we describe a physical device in relativistic spacetime which can compute a nonTuring computable task, e.g. which can decide the halting problem of Turing machines or decide whether ZF set theory is consistent (more precisely, can decide the theorems of ZF). Starting from this, we will discuss the impact of recent breakthrough results of relativity theory, black hole physics and cosmology to well established foundational issues of computability theory as well as to logic. We find that the unexpected, revolutionary results in the mentioned branches of science force us to reconsider the status of the physical Church Thesis and to consider it as being seriously challenged. We will outline the consequences of all this for the foundation of mathematics (e.g. to Hilbert’s programme). Observational, empirical evidence will be quoted to show that the statements above do not require any assumption of some physical universe outside of our own one: in our specific physical universe there seem to exist regions of spacetime supporting potential nonTuring computations. Additionally, new “engineering ” ideas will be outlined for solving the socalled blueshift problem of GRcomputing. Connections with related talks at the Physics and Computation meeting, e.g. those of Jerome DurandLose, Mark Hogarth and Martin Ziegler, will be indicated. 1
On the Possibilities of Hypercomputing Supertasks
 FORTHCOMING IN MINDS AND MACHINES
, 2011
"... This paper investigates the view that digital hypercomputing is a good reason for rejection or reinterpretation of the ChurchTuring thesis. After suggestion that such reinterpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses prop ..."
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This paper investigates the view that digital hypercomputing is a good reason for rejection or reinterpretation of the ChurchTuring thesis. After suggestion that such reinterpretation is historically problematic and often involves attack on a straw man (the ‘maximality thesis’), it discusses proposals for digital hypercomputing with “Zenomachines”, i.e. computing machines that compute an infinite number of computing steps in finite time, thus performing supertasks. It argues that effective computing with Zenomachines falls into a dilemma: either they are specified such that they do not have output states, or they are specified such that they do have output states, but involve contradiction. Repairs though noneffective methods or special rules for semidecidable problems are sought, but not found. The paper concludes that hypercomputing supertasks are impossible in the actual world and thus no reason for rejection of the ChurchTuring thesis in its traditional interpretation.
ACCELERATING MACHINES
, 2006
"... This paper presents an overview of accelerating machines. We begin by exploring the history of the accelerating machine model and the potential power that it provides. We look at some of the problems that could be solved with an accelerating machine, and review some of the possible implementation me ..."
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This paper presents an overview of accelerating machines. We begin by exploring the history of the accelerating machine model and the potential power that it provides. We look at some of the problems that could be solved with an accelerating machine, and review some of the possible implementation methods that have been presented. Finally, we expose the limitations of accelerating machines and conclude by posing some problems for further research.
UNIVERSITY OF PITTSBURGH FACULTY OF ARTS AND SCIENCES
, 2003
"... This dissertation was presented by ..."
12345efghi UNIVERSITY OF WALES SWANSEA REPORT SERIES
"... Newtonian mechanics and infinitely parallel computation by ..."
Abstract Embedding infinitely parallel computation in Newtonian kinematics
"... First, we reflect on computing sets and functions using measurements from experiments with a class of physical systems. We call this experimental computation. We outline a programme to analyse theoretically experimental computation in which a central problem is: Given a physical theory T, explore an ..."
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First, we reflect on computing sets and functions using measurements from experiments with a class of physical systems. We call this experimental computation. We outline a programme to analyse theoretically experimental computation in which a central problem is: Given a physical theory T, explore and classify the computational models that can be embedded in, and abstracted from, the physical systems specified by the physical theory T. We consider the embedding of arbitrary sets, functions, programs, and computers into designs for systems that can be specified in subtheories or fragments T of Newtonian kinematics in order to explore some of the physical assumptions of T that allows its systems to qualify as hypercomputers, i.e. physical models that compute sets and functions that cannot be computed in classical computability theory. In designing systems we work strictly within the chosen theory T and do not concern ourself with whether or not T is valid of the world today. We are interested in exploring the subtheory from a computational point of view and especially in restrictions on the assumptions of T that allow us to return from hypercomputation to classical computation. Secondly, we give a construction of an infinitely parallel machine that can decide all the arithmetical sets of natural numbers. We embed this hypercomputer as system in 3dimensions obeying the laws of a fragment of Newtonian kinematics. In particular, the example shows that communication allowable in Newtonian kinematics is especially powerful. We conclude with further reflections and open problems.
Abstract Analog computation beyond the Turing limit
"... The main purpose of this paper is quite uncontroversial. First, we recall some models of analog computations (including these allowed to perform Turing uncomputable tasks). Second, we support the suggestions that such hypercomputable capabilities of such systems can be explained by the use of infini ..."
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The main purpose of this paper is quite uncontroversial. First, we recall some models of analog computations (including these allowed to perform Turing uncomputable tasks). Second, we support the suggestions that such hypercomputable capabilities of such systems can be explained by the use of infinite limits. Additionally, the inner restrictions of analog models of computations are indicated.
New Physics and Hypercomputation
"... Abstract. Does new physics give us a chance for designing computers, at least in principle, which could compute beyond the Turing barrier? By the latter we mean computers which could compute some functions which are not Turing computable. Part of what we call “new physics ” is the surge of results i ..."
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Abstract. Does new physics give us a chance for designing computers, at least in principle, which could compute beyond the Turing barrier? By the latter we mean computers which could compute some functions which are not Turing computable. Part of what we call “new physics ” is the surge of results in black hole physics in the last 15 years, which certainly changed our perspective on certain things. The two main directions in this line seem to be quantum computers and relativistic, i.e. spacetimetheorybased, ones. We will concentrate on the relativistic case. Is there a remote possibility that relativity can give some feedback to its “founding grandmother”, namely, to logic? 1 General perspective The Physical ChurchTuring Thesis, PhCT, is the conjecture that whatever physical computing device (in the broader sense) or physical thought experiment will be designed by any future civilization, it will always be simulatable by a Turing machine. The PhCT was formulated and generally accepted in the 1930’s.