The Length of Infinite Time Turing Machine Computations

Welch, Philip D.

doi:10.1112/s0024609399006657

Cited by 43 publications

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“…Let us just remark that the proof is a straightforward generalization of the arguments in [3] to the present context.…”

Section: ⊓ ⊔mentioning

confidence: 73%

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Cardinal-Recognizing Infinite Time Turing Machines

Habič

2013

Lecture Notes in Computer Science

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Abstract. We introduce a model of infinitary computation which enhances the infinite time Turing machine model slightly but in a natural way by giving the machines the capability of detecting cardinal stages of computation. The computational strength with respect to ITTMs is determined to be precisely that of the strong halting problem and the nature of the new characteristic ordinals (clockable, writable, etc.) is explored.Keywords: infinite computation, infinite time Turing machine, length of computation Various notions of infinitary computability have now been studied for several decades. The concept that we can somehow utilize infinity to accommodate our computations is at the same time both appealing and dangerous; appealing, since we are often in a position where we could answer some question if only we could look at the output of some algorithm after an infinite amount of steps, and dangerous, since this sort of greediness must inevitably lead to disappointment when we suddenly reach the limits of our model. At that point we must decide whether to push on and strengthen our model in some way or to abandon it in favour of some (apparently) alternative model. But if we do not wish to abandon our original idea, how to strengthen it in such a way that it remains both interesting and intuitive?The behaviour of infinite time Turing machines (ITTMs), first introduced in [1], has by now been extensively explored and various characteristics have been determined. While we are far from reaching full understanding of the model, we nevertheless already feel the urge to generalize further. Perhaps the most direct generalization are the ordinal Turing machines of [2], where both the machine tape and running time are allowed to range into the transfinite. But perhaps this modification seems too strong; with it all constructible sets of ordinals are computable. We would be satisfied with the minimal nontrivial expansion of the ITTM model, i.e. something which computes the appropriate halting problem but no more. Of course, we can easily achieve this goal within the ITTM framework by considering oracle computations, but this somehow doesn't seem satisfactory. We intend to give what we feel is a more natural solution to this problem but which falls short of the 'omnipotence' of ordinal Turing machines.

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“…Let us just remark that the proof is a straightforward generalization of the arguments in [3] to the present context.…”

Section: ⊓ ⊔mentioning

confidence: 73%

“…γ C = sup{α; α is CRITTM-clockable} Welch shows in [3] that γ = λ. We intend to prove a somewhat similar statement in the context of CRITTMs.…”

Section: ⊓ ⊔mentioning

confidence: 99%

Cardinal-Recognizing Infinite Time Turing Machines

Habič

2013

Lecture Notes in Computer Science

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“…To get any further in eventual computability-in particular to prove the other direction of Theorem 1.2-we need the following important results of Welch [12].…”

Section: The Height Of O ++mentioning

confidence: 99%

Infinite time extensions of Kleene’s $${\mathcal{O}}$$

Klev

2009

Arch. Math. Logic

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Using infinite time Turing machines we define two successive extensions of Kleene's O and characterize both their height and their complexity. Specifically, we first prove that the one extension-which we will call O + -has height equal to the supremum of the writable ordinals, and that the other extension-which we will call O ++ -has height equal to the supremum of the eventually writable ordinals. Next we prove that O + is Turing computably isomorphic to the halting problem of infinite time Turing computability, and that O ++ is Turing computably isomorphic to the halting problem of eventual computability. Keywords Infinite time Turing machines · Ordinal notation systemsMathematics Subject Classification (2000) 03D10 · 03D70 · 03F15 IntrodutionOne natural motivation for work on the theory of infinite time Turing machines is the question of how notions and objects from classical computability theory carry over into infinite time. An instance of that question is the motivation for the present paper: we will here study two infinite time analogues of Kleene's O. Our exposition will presuppose some familiarity with these machines and their theory, comparable to what can be got from reading, for instance, Hamkins and Lewis' papers [2] and [3]; material on Kleene's O can be found in Sacks' book [8].To distinguish the classical notion of computability from infinite time notions, we will call the former Turing computability; so, for example, instead of saying that A and A. M. Klev (B)

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“…The growing body of literature on infinite time Turing machines includes [HL00], [Wel00b], [Wel00a], [HS01], [L01], [HL02], [Ham02], [HW03], [DHS05], [Ham05], [Wel05], [Koe05].…”

Section: Introductionmentioning

confidence: 99%

Infinite Time Computable Model Theory

Hamkins

Miller

Seabold

et al. 2008

New Computational Paradigms

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Abstract. We introduce infinite time computable model theory, the computable model theory arising with infinite time Turing machines, which provide infinitary notions of computability for structures built on the reals R. Much of the finite time theory generalizes to the infinite time context, but several fundamental questions, including the infinite time computable analogue of the Completeness Theorem, turn out to be independent of zfc.

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The Length of Infinite Time Turing Machine Computations

Cited by 43 publications

References 0 publications

Cardinal-Recognizing Infinite Time Turing Machines

Cardinal-Recognizing Infinite Time Turing Machines

Infinite time extensions of Kleene’s $${\mathcal{O}}$$

Infinite Time Computable Model Theory

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