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13
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
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Computability and Incomputability
"... The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal cr ..."
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The conventional wisdom presented in most computability books and historical papers is that there were several researchers in the early 1930’s working on various precise definitions and demonstrations of a function specified by a finite procedure and that they should all share approximately equal credit. This is incorrect. It was Turing alone who achieved the characterization, in the opinion of Gödel. We also explore Turing’s oracle machine and its analogous properties in analysis. Keywords: Turing amachine, computability, ChurchTuring Thesis, Kurt Gödel, Alan Turing, Turing omachine, computable approximations,
Soare, Bounding homogeneous models
"... A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a ddecidable homogeneous model A, i.e., the elementary diagram D e (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithm ..."
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A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a ddecidable homogeneous model A, i.e., the elementary diagram D e (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogeneous model of T has a PA degree. 1
The settlingtime reducibility ordering
 Journal of Symbolic Logic
"... Abstract. To each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω, there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B>st A) ..."
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Abstract. To each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω, there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B>st A) if for every computable function f, for all but finitely many x, mB(x)> f(mA(x)). This settlingtime ordering, which is a natural extension to an ordering of the idea of domination, was first introduced by Nabutovsky and Weinberger in [3] and Soare [6]. They desired a sequence of sets descending in this relationship to give results in differential geometry. In this paper we examine properties of the <st ordering. We show that it is not invariant under computable isomorphism, that any countable partial ordering embeds into it, that there are maximal and minimal sets, and that two c.e. sets need not have an inf or sup in the ordering. We also examine a related ordering, the strong settlingtime ordering where we require for all computable f and g, for almost all x, mB(x)> f(mA(g(x))).
Effective packing dimension of Π 0 1classes
, 2007
"... We construct a Π0 1class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relatio ..."
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We construct a Π0 1class X that has classical packing dimension 0 and effective packing dimension 1. This implies that, unlike in the case of effective Hausdorff dimension, there is no natural correspondence principle (as defined by Lutz) for effective packing dimension. We also examine the relationship between upper box dimension and packing A major theme of computability theory is the effectivization of classical mathematics. To do this one takes an existing (i.e. classical) mathematical notion and develops a new computabilitytheoretic analogue of that notion. Afterwards, one tries to determine the similarities and differences between the
1 Comparing C.E. Sets Based on Their Settling Times
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Continuity of capping in EbT
, 2007
"... A set A ⊆ ω is called computably enumerable (c.e., for short), if there is an algorithm to enumerate the elements of it. For sets A, B ⊆ ω, we say that A is bounded Turing reducible to B if there is a Turing functional, Φ say, with a computable bound of oracle query bits such that A is computed by Φ ..."
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A set A ⊆ ω is called computably enumerable (c.e., for short), if there is an algorithm to enumerate the elements of it. For sets A, B ⊆ ω, we say that A is bounded Turing reducible to B if there is a Turing functional, Φ say, with a computable bound of oracle query bits such that A is computed by Φ equipped with an oracle B, written A ≤bT B. Let EbT be the structure of the c.e. bTdegrees, the c.e. degrees under the bounded Turing reductions. In this paper we study the continuity properties in EbT. We show that for any c.e. bTdegree b � = 0, 0 ′ , there is a c.e. bTdegree a> b such that for any c.e. bTdegree x, b ∧ x = 0 if and only if a ∧ x = 0. This is the first continuity property of the c.e. bTdegrees. 1