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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.
Computable Isomorphisms, Degree Spectra of Relations, and Scott Families
 Ann. Pure Appl. Logic
, 1998
"... this paper we are interested in those structures in which the basic computations can be performed by Turing machines. ..."
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Cited by 26 (12 self)
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this paper we are interested in those structures in which the basic computations can be performed by Turing machines.
Mass problems and hyperarithmeticity
, 2006
"... A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of we ..."
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Cited by 24 (16 self)
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A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let Pw be the lattice of weak degrees of mass problems associated with nonempty Π 0 1 subsets of the Cantor space. The lattice Pw has been studied in previous publications. The purpose of this paper is to show that Pw partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in Pw which are indexed by the ordinal numbers less than ω CK 1 and which correspond to the hyperarithmetical hierarchy. Namely, for each α < ω CK 1 let hα be the weak degree of 0 (α) , the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p ∗ be the weak degree
Relativizing Chaitin’s halting probability
 J. Math. Log
"... Abstract. As a natural example of a 1random real, Chaitin proposed the halting probability Ω of a universal prefixfree machine. We can relativize this example by considering a universal prefixfree oracle machine U. Let Ω A U be the halting probability of U A; this gives a natural uniform way of p ..."
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Cited by 21 (7 self)
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Abstract. As a natural example of a 1random real, Chaitin proposed the halting probability Ω of a universal prefixfree machine. We can relativize this example by considering a universal prefixfree oracle machine U. Let Ω A U be the halting probability of U A; this gives a natural uniform way of producing an Arandom real for every A ∈ 2 ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, Ω A U can be vastly different for different choices of U. Even for a fixed U, there are oracles A = ∗ B such that Ω A U and Ω B U are 1random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness. 1.
Elementary formal systems, intrinsic complexity, and procrastination
 Information and Computation
, 1997
"... Recently, rich subclasses of elementary formal systems (EFS) have been shown to be identifiable in the limit from only positive data. Examples of these classes are Angluin’s pattern languages, unions of pattern languages by Wright and Shinohara, and classes of languages definable by lengthbounded e ..."
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Cited by 13 (6 self)
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Recently, rich subclasses of elementary formal systems (EFS) have been shown to be identifiable in the limit from only positive data. Examples of these classes are Angluin’s pattern languages, unions of pattern languages by Wright and Shinohara, and classes of languages definable by lengthbounded elementary formal systems studied by Shinohara. The present paper employs two distinct bodies of abstract studies in the inductive inference literature to analyze the learnability of these concrete classes. The first approach, introduced by Freivalds and Smith, uses constructive ordinals to bound the number of mind changes. ω denotes the first limit ordinal. An ordinal mind change bound of ω means that identification can be carried out by a learner that after examining some element(s) of the language announces an upper bound on the number of mind changes it will make before converging; a bound of ω · 2 means that the learner reserves the right to revise this upper bound once; a bound of ω · 3 means the learner reserves the right to revise this upper bound twice, and so on. A bound of ω 2 means that identification can be carried out by a learner that announces an upper bound on the number of times it may revise its conjectured upper bound on the number of mind changes. It is shown in the present paper that the ordinal mind change complexity for identification of languages formed by unions of up to n pattern languages is ω n. It is
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
Defining the Turing Jump
 MATHEMATICAL RESEARCH LETTERS
, 1999
"... The primary notion of effective computability is that provided by Turing machines (or equivalently any of the other common models of computation). We denote the partial function computed by the eth Turing machine in some standard list by # e . When these machines are equipped with an "oracle" for a ..."
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Cited by 10 (6 self)
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The primary notion of effective computability is that provided by Turing machines (or equivalently any of the other common models of computation). We denote the partial function computed by the eth Turing machine in some standard list by # e . When these machines are equipped with an "oracle" for a subset A of the natural numbers #, i.e. an external procedure that answers questions of the form "is n in A", they define the basic notion of relative computability or Turing reducibility (from Turing (1939)). We say that A is computable from (or recursive in) B if there is a Turing machine which, when equipped with an oracle for B, computes (the characteristic function of) A, i.e. for some e, # B e = A. We denote this relation by A # T<F10
Randomness in effective descriptive set theory
 London. Math. Soc
"... Abstract. An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyper ..."
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Cited by 9 (3 self)
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Abstract. An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyperarithmetical, each low for random set is. Finally we study a very strong yet effective randomness notion: Z is strongly random if Z is in no null Π1 1 set of reals. We show that there is a greatest Π1 1 null set, that is, a universal test for this notion. 1.
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 1997
"... A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is ma ..."
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Cited by 8 (3 self)
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A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinaltheoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie "  the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency  technical results in pro...