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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.
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 30 (18 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
Sense and denotation as algorithm and value
 Lecture Notes in Logic
, 1994
"... In his classic 1892 paper On sense and denotation [12], Frege first contends that in addition to their denotation (reference, Bedeutung), proper names also have a sense (Sinn) “wherein the mode of presentation [of the denotation] is contained. ” Here proper names include common nouns like “the earth ..."
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Cited by 27 (3 self)
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In his classic 1892 paper On sense and denotation [12], Frege first contends that in addition to their denotation (reference, Bedeutung), proper names also have a sense (Sinn) “wherein the mode of presentation [of the denotation] is contained. ” Here proper names include common nouns like “the earth ” or “Odysseus ” and descriptive phrases like “the point of intersection of
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 25 (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.
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 25 (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.
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
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 14 (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
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&q ..."
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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