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A.Lewis, Infinite time turing machines
 Journal of Symbolic Logic
"... Abstract. We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every Π1 1 set, for example, is decidable by such machines, and the semidecidable sets form a portion of the ..."
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Cited by 77 (5 self)
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Abstract. We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every Π1 1 set, for example, is decidable by such machines, and the semidecidable sets form a portion of the ∆1 2 sets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators. In these days of superfast computers whose speed seems to be increasing without bound, the more philosophical among us are perhaps pushed to wonder: what could we compute with an infinitely fast computer? By proposing a natural model for supertasks—computations with infinitely many steps—we provide in this paper a theoretical foundation on which to answer this question. Our model is simple: we simply extend the Turing machine concept into transfinite ordinal time. The resulting machines can perform infinitely many steps of computation, and go on to more computation after that. But mechanically they work just like Turing machines. In particular, they have the usual Turing machine hardware; there is still the same smooth infinite paper tape and the same mechanical head moving back and forth according to a finite algorithm, with finitely many states. What is new is the definition of the behavior of the machine at limit ordinal times. The resulting computability theory leads to a notion of computation on the reals, concepts of decidability and semidecidability for sets of reals as well as individual reals, two kinds of jumpoperator, and a notion of relative computability using oracles which gives a rich degree structure on both the collection of reals and the collection of sets of reals. But much remains unknown; we hope to stir interest in these ideas, which have been a joy for us to think about.
RELATIVISTIC COMPUTERS AND THE TURING Barrier
, 2006
"... We examine the current status of the physical version of the ChurchTuring Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with Deutsch et al [17] that PhCT ..."
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Cited by 22 (10 self)
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We examine the current status of the physical version of the ChurchTuring Thesis (PhCT for short) in view of latest developments in spacetime theory. This also amounts to investigating the status of hypercomputation in view of latest results on spacetime. We agree with Deutsch et al [17] that PhCT is not only a conjecture of mathematics but rather a conjecture of a combination of theoretical physics, mathematics and, in some sense, cosmology. Since the idea of computability is intimately connected with the nature of Time, relevance of spacetime theory seems to be unquestionable. We will see that recent developments in spacetime theory show that temporal developments may exhibit features that traditionally seemed impossible or absurd. We will see that recent results point in the direction that the possibility of artificial systems computing nonTuring computable functions may be consistent with spacetime theory. All these trigger new open questions and new research directions for spacetime theory, cosmology, and computability.
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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Cited by 10 (0 self)
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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.
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,
Comparing causality principles
, 2005
"... The principle of common cause is discussed as a possible fundamental principle of physics. Some revisions of Reichenbach’s formulation of the principle are given, which lead to a version given by Bell. Various similar forms are compared and some equivalence results proved. The further problems of ca ..."
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Cited by 6 (0 self)
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The principle of common cause is discussed as a possible fundamental principle of physics. Some revisions of Reichenbach’s formulation of the principle are given, which lead to a version given by Bell. Various similar forms are compared and some equivalence results proved. The further problems of causality in a quantal system, and indeterministic causal structure, are addressed, with a view to defining a causality principle applicable to quantum gravity. 1
ON BECOMING, COSMIC TIME AND ROTATING UNIVERSES
, 2001
"... In the literature on the compatibility between the time of our experience and the time of physics, the special theory of relativity has enjoyed central stage. By bringing into the discussion the general theory of relativity, I suggest a new analysis of the misunderstood notion of becoming, developed ..."
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Cited by 5 (0 self)
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In the literature on the compatibility between the time of our experience and the time of physics, the special theory of relativity has enjoyed central stage. By bringing into the discussion the general theory of relativity, I suggest a new analysis of the misunderstood notion of becoming, developed from hints in Gödel’s published and unpublished arguments for the ideality of time. I claim that recent endorsements of such arguments, based on Gödel’s own “rotating ” solution to Einstein’s field equation, fail: once understood in the right way, becoming can be shown to be both mindindependent and compatible with spacetime physics. Being a needed tertium quid between views of time traditionally regarded as in conflict, such a new approach to becoming should also help to dissolve a crucial aspect of the centuryold debate between the socalled A and B theories of time.
Logic of spacetime and relativity theory
, 2006
"... 2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4 ..."
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Cited by 5 (3 self)
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2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4
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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Cited by 4 (0 self)
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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
The nature of spacetime singularities
 In 100 Years of Relativity – SpaceTime Structure: Einstein and
, 2005
"... Present knowledge about the nature of spacetime singularities in the context of classical general relativity is surveyed. The status of the BKL picture of cosmological singularities and its relevance to the cosmic censorship hypothesis are discussed. It is shown how insights on cosmic censorship als ..."
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Cited by 3 (0 self)
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Present knowledge about the nature of spacetime singularities in the context of classical general relativity is surveyed. The status of the BKL picture of cosmological singularities and its relevance to the cosmic censorship hypothesis are discussed. It is shown how insights on cosmic censorship also arise in connection with the idea of weak null singularities inside black holes. Other topics covered include matter singularities and critical collapse. Remarks are made on possible future directions in research on spacetime singularities. 1
Can general relativistic computers break the Turing barrier?
"... Abstract. Can general relativistic computers break the Turing barrier? Are there final limits to human knowledge? Limitative results versus human creativity (paradigm shifts). Gödel’s logical results in comparison/combination with Gödel’s relativistic results. Can Hilbert’s programme be carried ..."
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Cited by 3 (2 self)
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Abstract. Can general relativistic computers break the Turing barrier? Are there final limits to human knowledge? Limitative results versus human creativity (paradigm shifts). Gödel’s logical results in comparison/combination with Gödel’s relativistic results. Can Hilbert’s programme be carried through after all? 1 Aims, 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. At that time a general consensus was reached declaring PhCT valid, and indeed in the succeeding decades the PhCT was an extremely useful and valuable maxim in elaborating the foundations of theoretical computer science, logic, foundation of mathematics and related areas. But since PhCT is partly a physical conjecture, we emphasize that this consensus of the 1930’s was based on the physical worldview of the 1930’s. Moreover, many thinkers considered PhCT as being based on