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Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
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WORKING WITH STRONG REDUCIBILITIES ABOVE TOTALLY ωC.E. DEGREES
"... Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ω ..."
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Abstract. We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allows us to compute such sets. For example, we prove that a c.e. degree is totally ωc.e. iff every set in it is weak truthtable reducible to a hypersimple, or ranked, set. We also show that a c.e. degree is array computable iff every leftc.e. real of that degree is reducible in a computable Lipschitz way to a random leftc.e. real (an Ωnumber). 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))).
The computable Lipschitz degrees of computably enumerable sets are not dense
 Ann. Pure Appl. Logic
"... Abstract. The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truthtable reducibility [6]. This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the comp ..."
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Abstract. The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truthtable reducibility [6]. This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solovay degrees of strongly c.e. reals are not dense. There are similarities to Barmpalias and Lewis’s proof that the identity bounded Turing degrees of c.e. sets are not dense [2], however the problem for the computable Lipschitz degrees is more complex. 1.
1 Comparing C.E. Sets Based on Their Settling Times
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THE SETTLING TIME REDUCIBILITY ORDERING AND ∆0 2 SETS
"... Abstract. The Settling Time reducibility ordering gives an ordering on computably enumerable sets based on their enumerations. The <st ordering is in fact an ordering on c.e. sets, since it is independent of the particular enumeration chosen. In this paper we show that it is not possible to exten ..."
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Abstract. The Settling Time reducibility ordering gives an ordering on computably enumerable sets based on their enumerations. The <st ordering is in fact an ordering on c.e. sets, since it is independent of the particular enumeration chosen. In this paper we show that it is not possible to extend this ordering in an approximationindependent way to ∆0 2 sets in general, or even to nc.e. sets for any fixed n ≥ 3. 1.