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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 83 (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.
R.: Register computations on ordinals
 Arch. Math. Log
, 2008
"... We generalise ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. O rdina l registe r machine s are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read o the truth predicate satises a ..."
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Cited by 20 (10 self)
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We generalise ordinary register machines on natural numbers to machines whose registers contain arbitrary ordinals. O rdina l registe r machine s are able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read o the truth predicate satises a natural theory SO. SO is the theory of the sets of ordinals in a model of the Z ermeloF raenkel axioms ZFC. This allows the following characterisat ion of computable sets: a set of ordinals is ordinal register computable if and only if i t i s an element of Gödel ' s construct ible universe L. 1 Introduction. There are many equivalent machine models for dening the class of intuitively computable sets. We model computations on ordinals on the unlimited register machines (URM) presented in [ 2]. An URM has registers R0; R1; which can hold natural numbers, i. e., elements of the set! = f 0; 1; g. A register program consists of commands to increase or to reset a register. The pro
P ̸= NP∩coNP for infinite time turing machines
 Journal of Logic and Computation
, 2005
"... Abstract. Extending results of Schindler [Sch] and Hamkins and Welch [HW03], we establish in the context of infinite time Turing machines that P is properly contained in NP ∩coNP. Furthermore, NP ∩coNP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P ..."
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Abstract. Extending results of Schindler [Sch] and Hamkins and Welch [HW03], we establish in the context of infinite time Turing machines that P is properly contained in NP ∩coNP. Furthermore, NP ∩coNP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P + = NP + ∩coNP + = NP ∩coNP, though P ++ is properly contained in NP ++ ∩coNP ++. Within any contiguous block of infinite clockable ordinals, we show that Pα ̸ = NPα ∩coNPα, but if β begins a gap in the clockable ordinals, then Pβ = NPβ ∩coNPβ. Finally, we establish that P f ̸ = NP f ∩coNP f for most functions f: R → ord, although we provide examples where P f = NP f ∩coNP f and P f ̸ = NP f.
Infinite time computable model theory
"... 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 se ..."
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Cited by 5 (1 self)
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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. 1.
Computing a model of set theory
 in: [CoLöTo05
, 2005
"... Abstract. We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. The generalized Turing machine is able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ..."
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Abstract. We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. The generalized Turing machine is able to compute a recursive bounded truth predicate on the ordinals. The class of sets of ordinals which can be read off the truth predicate satisfies a natural theory SO. SO is the theory of the sets of ordinals in a model of the ZermeloFraenkel axioms ZFC. Hence a set of ordinals is ordinal computable from ordinal parameters if and only if it is an element of Gödel’s constructible universe L. 1
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 Proceedings of the Fourteenth Conference on Uncertainty in Arti cial Intelligence
, 1998
"... Rights to individual papers and abstracts reside with authors. Conference Foreword ..."
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Rights to individual papers and abstracts reside with authors. Conference Foreword
Supertask Computation
, 2002
"... Abstract. Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms. 1 Supertasks What would you compute with an infinitely ..."
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Abstract. Infinite time Turing machines extend the classical Turing machine concept to transfinite ordinal time, thereby providing a natural model of infinitary computability that sheds light on the power and limitations of supertask algorithms. 1 Supertasks What would you compute with an infinitely fast computer? What could you compute? To make sense of these questions, one would want to understand the algorithms that the machines would carry out, computational tasks involving infinitely many steps of computation. Such tasks, known as supertasks, have been studied since antiquity from a variety of viewpoints. Zeno of Elea (ca. 450 B.C.) was perhaps the first to grapple with supertasks, in his famous paradox that it is impossible to go from here
Infinite Time Register Machines, Enhanced ⋆
"... Abstract. 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 pre ..."
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Abstract. 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 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 Π 1 1 sets, yet they are strictly weaker than ITTMs.