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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 79 (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.
Hypercomputation: computing more than the Turing machine
, 2002
"... In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which al ..."
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Cited by 43 (5 self)
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In this report I provide an introduction to the burgeoning field of hypercomputation – the study of machines that can compute more than Turing machines. I take an extensive survey of many of the key concepts in the field, tying together the disparate ideas and presenting them in a structure which allows comparisons of the many approaches and results. To this I add several new results and draw out some interesting consequences of hypercomputation for several different disciplines. I begin with a succinct introduction to the classical theory of computation and its place amongst some of the negative results of the 20 th Century. I then explain how the ChurchTuring Thesis is commonly misunderstood and present new theses which better describe the possible limits on computability. Following this, I introduce ten different hypermachines (including three of my own) and discuss in some depth the manners in which they attain their power and the physical plausibility of each method. I then compare the powers of the different models using a device from recursion theory. Finally, I examine the implications of hypercomputation to mathematics, physics, computer science and philosophy. Perhaps the most important of these implications is that the negative mathematical results of Gödel, Turing and Chaitin are each dependent upon the nature of physics. This both weakens these results and provides strong links between mathematics and physics. I conclude that hypercomputation is of serious academic interest within many disciplines, opening new possibilities that were previously ignored because of long held misconceptions about the limits of computation.
Accelerated Turing Machines
 Minds and Machines
, 2002
"... Abstract. Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of π contains n consecutive 7s, for any n; solve the Turingmachine halti ..."
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Cited by 35 (2 self)
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Abstract. Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of π contains n consecutive 7s, for any n; solve the Turingmachine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures and the theoretical limits of computability. Contrary to a recent paper by Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle’s Chinese room argument.
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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Cited by 23 (0 self)
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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.
The Length of Infinite Time Turing Machine Computations
 Bulletin of the London Mathematical Society
"... We show that the halting times of infinite time Turing Machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This gives a clarification of the nature of "supertasks" or infinite time computations. The proof furt ..."
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Cited by 23 (8 self)
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We show that the halting times of infinite time Turing Machines (considered as ordinals coded by sets of integers) are themselves all capable of being halting outputs of such machines. This gives a clarification of the nature of "supertasks" or infinite time computations. The proof further yields that the class of sets coded by outputs of halting computations coincides with a level of Godel's constructible hierarchy: namely that of L where is the supremum of halting times. A number of other open questions are thereby answered. 1 Introduction: Infinite Time Turing Machines Hamkins and Lewis in [4] give an account of the construction of these machines (first developed by Kidder and Hamkins in 1989) and develop the basic theory of this notion of computability. The reader should refer to this paper for a clear and full account of their basic properties, from which all the results and definitions of this introduction are taken. But to summarise: an infinite time Turing machine has...
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.
An enhanced theory of infinite time register machines
 CIE 2008. LNCS
, 2008
"... Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times a register content is defined as a lim inf of previous regi ..."
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Cited by 13 (2 self)
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Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times a register content is defined as a lim inf of previous register contents, if that limit is finite; otherwise the register is reset to 0. (A previous weaker version of infinitary register machines, in [6], would halt without a result in case of such an overflow.) The theory of infinite time register machines has similarities to the infinite time Turing machines (ITTMs) of Hamkins and Lewis. Indeed ITRMs can decide all Π11 sets, yet they are strictly weaker than ITTMs.
The basic theory of infinite time register machines
 ARCH. MATH. LOGIC (2010) 49:249–273
, 2010
"... Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operati ..."
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Cited by 11 (7 self)
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Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operations. In this paper, we examine the ITRMs introduced by the third and fourth author (Koepke and Miller in Logic and Theory of Algorithms LNCS, pp. 306–315, 2008), where a register content at a limit time is set to the lim inf of previous register contents if that limit is finite; otherwise the register is reset to 0. The theory of these machines has several similarities to the infinite time Turing machines (ITTMs) of Hamkins and Lewis. The machines can decide all 11 sets, yet are strictly weaker than ITTMs. As in the ITTM situation, we introduce a notion of ITRMclockable ordinals corresponding to the running times of computations. These form a transitive initial segment of the ordinals. Furthermore we prove a Lost Melody theorem: there is a real r such that there is a program P that halts
Ultimate truth vis à vis stable truth
 Journal of Philosophical Logic
, 2003
"... Abstract. We show that the set of ultimately true sentences in Hartry Field’s Revengeimmune solution to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this ..."
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Cited by 10 (6 self)
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Abstract. We show that the set of ultimately true sentences in Hartry Field’s Revengeimmune solution to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second order number theory is needed to establish the semantic values of sentences over the ground model of the standard natural numbers: ¢¡Comprehension Axiom scheme) is insufficient. £¤¦¥¨ § (second order number theory with a ©��
Minimality Arguments for Infinite Time Turing Degrees
"... We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely !, and construct continuum many incomparable such degrees which are minimla over 0. We show that we can apply an argument going back to that of H. Friedman to prove that the set 1d ..."
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Cited by 10 (7 self)
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We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely !, and construct continuum many incomparable such degrees which are minimla over 0. We show that we can apply an argument going back to that of H. Friedman to prove that the set 1degrees of certain \Sigma 1 2 correct KPmodels of the form L oe (oe ! ! L 1 ) have minimal upper bounds. 1 Introduction Obtaining minimality results in degree theory has a long history: the methods go back to those of Spector when he constructed minimal Turing degrees, and to GandySacks, [2], for minimal hyperdegrees. The perfect set construction is the common thread to these proofs. A further feature, which is shared, either directly or indirectly, by such arguments, is the use of a selection principle in order to typically, directly shrink a perfect set T ` ! ! to a T 0 so that a particular function is either continuous onetoone, or constant on the branches of T 0 . For ex...