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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 75 (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.
Turing computations on ordinals
 Bulletin of Symbolic Logic 11 (2005
, 2005
"... 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. We show that a set of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of Gödel ..."
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Cited by 12 (4 self)
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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. We show that a set of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of Gödel’s constructible universe L. This characterization can be used to prove the generalized continuum hypothesis in L. 1 Introduction. A standard Turing computation may be visualized as a timelike sequence of elementary readwritemove operations carried out by one or more “heads ” on “tapes”. The sequence of actions is determined by the initial tape contents and by a finite Turing program. The specific choice of alphabet, operations
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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Cited by 10 (3 self)
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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 4 (0 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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Cited by 2 (1 self)
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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
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
unknown title
, 2010
"... (will be inserted by the editor) Abstract geometrical computation 3: black holes for classical and analog computating ..."
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(will be inserted by the editor) Abstract geometrical computation 3: black holes for classical and analog computating