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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 91 (6 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.
NonTuring computations via MalamentHogarth spacetimes
 Int. J. Theoretical Phys
, 2002
"... We investigate the Church–Kalmár–Kreisel–Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church–Turingtype Theses (not only ..."
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Cited by 73 (8 self)
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We investigate the Church–Kalmár–Kreisel–Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church–Turingtype Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church– Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.) We also look at various “obstacles ” to computing a nonrecursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the “design ” of our future computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.
Even Turing Machines Can Compute Uncomputable Functions
 Unconventional Models of Computation
, 1998
"... Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability. 2 1. Introduction Neither Turing nor Post ..."
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Cited by 18 (3 self)
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Accelerated Turing machines are Turing machines that perform tasks commonly regarded as impossible, such as computing the halting function. The existence of these notional machines has obvious implications concerning the theoretical limits of computability. 2 1. Introduction Neither Turing nor Post, in their descriptions of the devices we now call Turing machines, made much mention of time (Turing 1936, Post 1936). 1 They listed the primitive operations that their devices perform  read a square of the tape, write a single symbol on a square of the tape (first deleting any symbol already present), move one square to the right, and so forth  but they made no mention of the duration of each primitive operation. The crucial concept is that of whether or not the machine halts after a finite number of operations. Temporal considerations are not relevant to the functioning of the devices as described, nor  so we are clearly supposed to believe  to the soundness of the proofs that Turi...
The many forms of hypercomputation
 Applied Mathematics and Computation
, 2006
"... This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. ..."
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Cited by 17 (0 self)
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This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess.
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.
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.
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 8 (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,
An explicit solution to Post’s Problem over the reals
, 2008
"... In the BSS model of real number computations we prove a concrete and explicit semidecidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post’s Problem over the reals significantly differs from its classical, discrete ..."
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Cited by 8 (3 self)
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In the BSS model of real number computations we prove a concrete and explicit semidecidable language to be undecidable yet not reducible from (and thus strictly easier than) the real Halting Language. This solution to Post’s Problem over the reals significantly differs from its classical, discrete variant where advanced diagonalization techniques are only known to yield the existence of such intermediate Turing degrees. Then we strengthen the above result and show as well the existence of an uncountable number of incomparable semidecidable Turing degrees below the real Halting Problem in the BSS model. Again, our proof will give concrete such problems representing these different degrees. Finally we show the corresponding result for the linear BSS model, that is over (R, +, −,<)rather than (R, +, −, ×, ÷,<).
The extent of computation in MalamentHogarth spacetimes
, 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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Cited by 7 (0 self)
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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.