On relative enumerability of Turing degrees

Shamil Ishmukhametov

doi:10.1007/s001530050139

THE n-r.e. DEGREES: UNDECIDABILITY AND Σ1 SUBSTRUCTURES

Journal of Mathematical Logic ◽

10.1142/s0219061312500055 ◽

2012 ◽

Vol 12 (01) ◽

pp. 1250005 ◽

Cited By ~ 1

Author(s):

MINGZHONG CAI ◽

RICHARD A. SHORE ◽

THEODORE A. SLAMAN

Keyword(s):

Turing Degrees ◽

First Order ◽

Global Properties

We study the global properties of [Formula: see text], the Turing degrees of the n-r.e. sets. In Theorem 1.5, we show that the first order of [Formula: see text] is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, [Formula: see text] is not a Σ1-substructure of [Formula: see text].

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PERMUTATIONS OF THE INTEGERS INDUCE ONLY THE TRIVIAL AUTOMORPHISM OF THE TURING DEGREES

Bulletin of Symbolic Logic ◽

10.1017/bsl.2018.15 ◽

2018 ◽

Vol 24 (2) ◽

pp. 165-174

Author(s):

BJØRN KJOS-HANSSEN

Keyword(s):

Open Problem ◽

Main Idea ◽

Computability Theory ◽

Turing Degrees ◽

Nontrivial Automorphism

AbstractIs there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers. We show that a permutation of ω cannot induce any nontrivial automorphism of the Turing degrees of members of 2ω, and in fact any permutation that induces the trivial automorphism must be computable.A main idea of the proof is to consider the members of 2ω to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more.

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GLOBAL PROPERTIES OF THE TURING DEGREES AND THE TURING JUMP

Computational Prospects of Infinity - Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore ◽

10.1142/9789812794055_0002 ◽

2008 ◽

pp. 83-101 ◽

Cited By ~ 8

Author(s):

Theodore A. Slaman

Keyword(s):

Turing Degrees ◽

Global Properties

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Eventually infinite time Turing machine degrees: infinite time decidable reals

Journal of Symbolic Logic ◽

10.2307/2586695 ◽

2000 ◽

Vol 65 (3) ◽

pp. 1193-1203 ◽

Cited By ~ 21

Author(s):

P.D. Welch

Keyword(s):

Turing Machine ◽

Initial Segment ◽

Upper Bounds ◽

Turing Degrees ◽

Turing Machines ◽

Jump Operator ◽

Infinite Time

AbstractWe characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

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Turing reducibility and Turing degrees

Routledge Encyclopedia of Philosophy ◽

10.4324/9780415249126-y006-1 ◽

2018 ◽

Author(s):

Harold Hodes

Keyword(s):

Computational Complexity ◽

Recursion Theory ◽

Turing Degrees ◽

Natural Numbers ◽

Mathematical Objects ◽

Classical Setting ◽

Turing Reducibility

A reducibility is a relation of comparative computational complexity (which can be made precise in various non-equivalent ways) between mathematical objects of appropriate sorts. Much of recursion theory concerns such relations, initially between sets of natural numbers (in so-called classical recursion theory), but later between sets of other sorts (in so-called generalized recursion theory). This article considers only the classical setting. Also Turing first defined such a relation, now called Turing- (or just T-) reducibility; probably most logicians regard it as the most important such relation. Turing- (or T-) degrees are the units of computational complexity when comparative complexity is taken to be T-reducibility.

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