Relativizing, a tree of trees, the jump operator

1979 ◽  
pp. 95-104
Author(s):  
Richard L. Epstein
Keyword(s):  
Jump Operator

Eventually infinite time Turing machine degrees: infinite time decidable reals

2000 ◽  
Vol 65 (3) ◽  
pp. 1193-1203 ◽  
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.

Minimal degrees and the jump operator

1973 ◽  
Vol 38 (2) ◽  
pp. 249-271 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Minimal Degree ◽  
Partial Ordering ◽  
Jump Operator ◽  
Upper Semilattice ◽  
Recursively Enumerable ◽  
Original Number ◽  
Nonzero Degree ◽  
The Relationship ◽  
Degrees Of Unsolvability

The jump a′ of a degree a is defined to be the largest degree recursively enumerable in a in the upper semilattice of degrees of unsolvability. We examine below some of the ways in which the jump operation is related to the partial ordering of the degrees. Fried berg [3] showed that the equation a = x′ is solvable if and only if a ≥ 0′. Sacks [13] showed that we can find a solution of a = x′ which is ≤ 0′ (and in fact is r.e.) if and only if a ≥ 0′ and is r.e. in 0′. Spector [16] constructed a minimal degree and Sacks [13] constructed one ≤ 0′. So far the only result concerning the relationship between minimal degrees and the jump operator is one due to Yates [17] who showed that there is a minimal predecessor for each non-recursive r.e. degree, and hence that there is a minimal degree with jump 0′. In §1, we obtain an analogue of Friedberg's theorem by constructing a minimal degree solution for a = x′ whenever a ≥ 0′. We incorporate Friedberg5s original number-theoretic device with a complicated sequence of approximations to the nest of trees necessary for the construction of a minimal degree. The proof of Theorem 1 is a revision of an earlier, shorter presentation, and incorporates many additions and modifications suggested by R. Epstein. In §2, we show that any hope for a result analogous to that of Sacks on the jumps of r.e. degrees cannot be fulfilled since 0″ is not the jump of any minimal degree below 0′. We use a characterization of the degrees below 0′ with jump 0″ similar to that found for r.e. degrees with jump 0′ by R. W. Robinson [12]. Finally, in §3, we give a proof that every degree a ≤ 0′ with a′ = 0″ has a minimal predecessor. Yates [17] has already shown that every nonzero r.e. degree has a minimal predecessor, but that there is a nonzero degree ≤ 0′ with no minimal predecessor (see [18]; or for the original unrelativized result see [10] or [4]).

scholarly journals Variational characterization of Hp

2019 ◽  
Vol 149 (5) ◽  
pp. 1123-1134 ◽  
Author(s):  
Honghai Liu
Keyword(s):  
Hardy Space ◽  
Approximate Identity ◽  
Jump Operator ◽  
Variational Characterization ◽  
Approximate Identities ◽  
Oscillation Operator

AbstractIn this paper, we obtain the variational characterization of Hardy space Hp for $p\in (((n)/({n+1})),1]$, and get estimates for the oscillation operator and the λ-jump operator associated with approximate identities acting on Hp for $p\in (((n)/({n+1})),1]$. Moreover, we give counterexamples to show that the oscillation and λ-jump associated with some approximate identity cannot be used to characterize Hp for $p\in (((n)/({n+1})),1]$.

DEFINABILITY OF THE JUMP OPERATOR IN THE ENUMERATION DEGREES

2003 ◽  
Vol 03 (02) ◽  
pp. 257-267 ◽  
Author(s):  
I. Sh. KALIMULLIN
Keyword(s):  
Jump Operator ◽  
Enumeration Degrees ◽  
Upper Semilattice

We show that the e-degree 0'e and the map u ↦ u' are definable in the upper semilattice of all e-degrees. The class of total e-degrees ≥0'e is also definable.

scholarly journals Splitting theorems and the jump operator

1998 ◽  
Vol 94 (1-3) ◽  
pp. 45-52 ◽  
Author(s):  
R.G. Downey ◽  
Richard A. Shore
Keyword(s):  
Jump Operator

Jump inversions inside effectively closed sets and applications to randomness

2011 ◽  
Vol 76 (2) ◽  
pp. 491-518 ◽  
Author(s):  
George Barmpalias ◽  
Rod Downey ◽  
Keng Meng Ng
Keyword(s):  
Complete Solution ◽  
Negative Answer ◽  
The Other ◽  
Random Sets ◽  
Jump Operator ◽  
Closed Sets

AbstractWe study inversions of the jump operator on classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0′ sets which are not 2-random. Both of the classes coincide with the degrees above 0′ which are not 0′-dominated. A further application is the complete solution of [24, Problem 3.6.9]: one direction of van Lambalgen's theorem holds for weak 2-randomness, while the other fails.Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [24, Problem 8.2.14]: not all weakly 2-random sets are array computable. In fact, given any oracle X, there is a weakly 2-random which is not array computable relative to X. This contrasts with the fact that all 2-random sets are array computable.

RELATIVIZING CHAITIN'S HALTING PROBABILITY

2005 ◽  
Vol 05 (02) ◽  
pp. 167-192 ◽  
Author(s):  
ROD DOWNEY ◽  
DENIS R. HIRSCHFELDT ◽  
JOSEPH S. MILLER ◽  
ANDRÉ NIES
Keyword(s):  
Computability Theory ◽  
Algorithmic Randomness ◽  
Jump Operator ◽  
Random Real ◽  
Primary Object ◽  
Object Of Study ◽  
Free Machine ◽  
Computable Functions ◽  
Omega Operator

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. 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, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random 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.

scholarly journals Absorption in quantum electrodynamic cavities in terms of a quantum jump operator

2014 ◽  
Vol 90 (3) ◽  
Author(s):  
V. Debierre ◽  
G. Demésy ◽  
T. Durt ◽  
A. Nicolet ◽  
B. Vial ◽  
...  
Keyword(s):  
Quantum Electrodynamic ◽  
Jump Operator ◽  
Quantum Jump

A characterization of jump operators

1988 ◽  
Vol 53 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Howard Becker
Keyword(s):  
Characterization Theorem ◽  
Recursion Theory ◽  
Turing Degrees ◽  
Descriptive Set Theory ◽  
Jump Operator ◽  
General Study ◽  
General Topic ◽  
Descriptive Set ◽  
Increasing Function

The topic of this paper is jump operators, a subject which originated with some questions of Martin and a partial answer to them obtained by Steel [18]. The topic of jump operators is a part of the general study of the structure of the Turing degrees, but it is concerned with an aspect of that structure which is different from the usual concerns of classical recursion theory. Specifically, it is concerned with studying functions on the degrees, such as the Turing jump operator, the hyperjump operator, and the sharp operator.Roughly speaking, a jump operator is a definable ≤T-increasing function on the Turing degrees. The purpose of this paper is to characterize the jump operators, in terms of concepts from descriptive set theory. Again roughly speaking, the main theorem states that all jump operators (other than the identity function) are obtained from pointclasses by the same process by which the hyperjump operator is obtained from the pointclass Π11; that is, if Γ is the pointclass, then the operator maps the real x to the universal Γ(x) subset of ω. This characterization theorem has some corollaries, one of which answers a question of Steel [18]. In §1 we give a brief introduction to this general topic, followed by a brief (and still somewhat imprecise) description of the results contained in this paper.

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