scholarly journals Uniform upper bounds on ideals of turing degrees

1978 ◽  
Vol 43 (3) ◽  
pp. 601-612 ◽  
Author(s):  
Harold T. Hodes
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Turing Degrees ◽  
Hierarchy Theory ◽  
Countable Ordinal ◽  
Limit Ordinal ◽  
Theoretic Motivation ◽  
Increasing Function

Given I, a reasonable countable set of Turing degrees, can we find some sort of canonical strict upper bound on I? If I = {a ∣ a ≤ b}, the upper bound on I which springs to mind is b′. But what if I is closed under jump? This question arises naturally out of the question which motivates a large part of hierarchy theory: Is there a canonical increasing function from a countable ordinal, preferably a large one, into D, the set of Turing degrees? If d is to be such a function, it is natural to require that d(α + 1) = d(α)′; but how should d(λ) depend on d ↾ λ, where λ is a limit ordinal?For any I ⊆ D, let MI, = ⋃I. Towards making the above questions precise, we introduce ideals of Turing degrees.Definition 1. I ⊆ D is an ideal iff I is closed under jump and join, and I is downward-closed, i.e., if a ≤ b & b ϵ I then a ϵ I.The following definition reflects the hierarchy-theoretic motivation for this paper.Definition 2. For I ⊆ D and A ⊆ ω, I is an A-hierarchy ideal iff for some countable ordinal α, MI = Lα[A]∩ ωω.All hierarchy ideals are ideals, but not conversely.Early in the game Spector knocked out the best sort of canonicity for upper bounds on ideals, proving that no set of degrees closed under jump has a least upper bound.

scholarly journals More about uniform upper bounds on ideals of turing degrees

1983 ◽  
Vol 48 (2) ◽  
pp. 441-457
Author(s):  
Harold T. Hodes
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Turing Degrees ◽  
Trial And Error ◽  
Minimal Member

AbstractLet I be a countable jump ideal in = 〈The Turing degrees, ≤〉. The central theorem of this paper is:a is a uniform upper bound on I iff a computes the join of an I-exact pair whose double jump a(1) computes.We may replace “the join of an I-exact pair” in the above theorem by “a weak uniform upper bound on I”.We also answer two minimality questions: the class of uniform upper bounds on I never has a minimal member; if ⋃I = Lα[A] ⋂ ωω for α admissible or a limit of admissibles, the same holds for nice uniform upper bounds.The central technique used in proving these theorems consists in this: by trial and error construct a generic sequence approximating the desired object; simultaneously settle definitely on finite pieces of that object; make sure that the guessing settles down to the object determined by the limit of these finite pieces.

Upper bounds on locally countable admissible initial segments of a Turing degree hierarchy

1981 ◽  
Vol 46 (4) ◽  
pp. 753-760 ◽  
Author(s):  
Harold T. Hodes
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Turing Degree ◽  
Turing Degrees ◽  
Locally Countable

AbstractWhere AR is the set of arithmetic Turing degrees, 0(ω) is the least member of {a(2) ∣ a is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's . This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on results of Jensen and is detailed in [3] and [4]. In § 1 we review the basic definitions from [3] which are needed to state the general results.

Minimal upper bounds for ascending sequences of α-recursively enumerable degrees

1976 ◽  
Vol 41 (1) ◽  
pp. 250-260
Author(s):  
C. T. Chong
Keyword(s):  
Upper Bound ◽  
Recursive Function ◽  
Upper Bounds ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
Admissible Ordinals

Let a be an admissible ordinal and let ∧ ≤ α be a limit ordinal. A sequence of a-r.e. degrees is said to be ascending, simultaneous and of length ∧ if (i) there is an α-recursive function t: α × ∧ → α such that, for all ϒ < ∧, Aϒ = {t(σ, ϒ)∣ σ < α} is of degree aϒ; (ii) if ϒ < ⊤ < ∧, then aϒ ≤αaτ and (iii) for all ϒ < ∧, there is a ⊤ > ϒ with aϒ, >αaϒ. Lerman [4] showed that such an exists for every ∧ ≤ α. An upper bound a of is an α-r.e. degree in which every element of is α-recursive. a is minimal if there is no α-r.e. degree b <αa which is also an upper bound of . Sacks [6] proved that every ascending sequence of simultaneously ω-r.e. degrees of length ω cannot have 0ω′, the complete ω-r.e. degree, as a minimal upper bound. In contrast, Cooper [2] showed that there exists an ascending sequence of simultaneously ω-r.e. degrees of length to having a minimal upper bound which is an ω-r.e. degree. In this paper we investigate the behavior of ascending sequences of simultaneously α-r.e. degrees for admissible ordinals α > ω. Call α Σ∞-admissibIe if it is Σn-nadmissible for all n. Let Φ(∧) say: No ascending sequence of simultaneously α-r.e. degrees of length ∧ can have 0α′, the complete α-r.e. degree, as a minimal upper bound. Our main result in this paper is:Let α be either a constructible cardinal with σ2ci(α) < α or Σ∞-admissible. Then σ2cf(α) is the least ordinal ν for which every ∧ ≤ α of cofinality ν (over Lα) can satisfy Φ(∧).

Finite level Borel games and a problem concerning the jump hierarchy

1984 ◽  
Vol 49 (4) ◽  
pp. 1301-1318
Author(s):  
Harold T. Hodes
Keyword(s):  
Upper Bound ◽  
Turing Degrees ◽  
Limit Ordinal ◽  
Locally Countable ◽  
Central Result

The jump hierarchy of Turing degrees assigns to each ξ < (ℵ1)L the degree 0(ξ); we presuppose familiarity with its definition and with the basic terminology of [5]. Let λ be a limit ordinal, λ < (ℵ1)L. The central result of [5] concerns the relation between 0(λ) and exact pairs on Iλ = {0(ξ) ∣ ξ < λ}. In [6] this question is raised: Where a is an upper bound on Iλ, how far apart are a and 0(λ)? It is there shown that if λ is locally countable and admissible, they may be very far apart: 0(λ) = the least member of {a(Ind(λ))∣, a is an upper bound on Iλ}; this is rather pathological, for Ind(λ) may be larger than λ. If λ is locally countable but neither admissible nor a limit of admissibles, we are essentially in the case of λ < ; by results of Sacks [12] and Enderton and Putnam [2], 0(λ) = the least member of {a(2) ∣ a is an upper bound on Iλ}. If λ is not locally countable, Ind(λ) is neither admissible nor a limit of admissibles, so we are again in a case like that of λ < . But what if λ is locally countable and nonadmissible, but is a limit of admissibles? For the rest of this paper let λ be such an ordinal. The central result of this paper answers this question for some such λ.

Least upper bounds for minimal pairs of α-R.E. α-degrees

1974 ◽  
Vol 39 (1) ◽  
pp. 49-56 ◽  
Author(s):  
Manuel Lerman
Keyword(s):  
Characteristic Function ◽  
Upper Bound ◽  
Recursion Theory ◽  
Upper Bounds ◽  
Minimal Pair ◽  
Upper Semilattice ◽  
Minimal Pairs ◽  
Limit Ordinal ◽  
Recursion Theorem ◽  
Admissible Ordinals

The application of priority arguments to study the structure of the upper semilattice of α-r.e. α-degrees for all admissible ordinals α was first done successfully by Sacks and Simpson [5] who proved that there exist incomparable α-r.e. α-degrees. Lerman and Sacks [3] studied the existence of minimal pairs of α-r.e. α-degrees, and proved their existence for all admissible ordinals α which are not refractory. We continue the study of the α-r.e. α-degrees, and prove that no minimal pair of α-r.e. α-degrees can have as least upper bound the complete α-r.e. α-degree.The above-mentioned theorem was first proven for α = ω by Lachlan [1]. Our proof for α = ω differs from Lachlan's in that we eliminate the use of the recursion theorem. The proofs are similar, however, and a knowledge of Lachlan's proof will be of considerable aid in reading this paper.We assume that the reader is familiar with the basic notions or α-recursion theory, which can be found in [2] or [5].Throughout the paper a will be an arbitrary admissible ordinal. We identify a set A ⊆ α with its characteristic function, A(x) = 1 if x ∈ A, and A(x) = 0 if x ∉ A.If A ⊆ α and B ⊆ α, then A ⊕ B will denote the set defined byA ⊕ B(x) = A(y) if x = λ + 2z, λ is a limit ordinal, z < ω and y = λ + z,= B(y) if x = λ + 2z + 1, λ is a limit ordinal, z < ω, and y = λ + z.

Upper bounds on the energy dissipation in turbulent precession

1996 ◽  
Vol 321 ◽  
pp. 335-370 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Experimental Data ◽  
Energy Dissipation ◽  
Viscous Dissipation ◽  
Upper Bound ◽  
Dissipation Rate ◽  
Oblate Spheroid ◽  
Upper Bounds ◽  
Precession Rate ◽  
Long Cylinder

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed

Models of arithmetic and upper bounds for arithmetic sets

1994 ◽  
Vol 59 (3) ◽  
pp. 977-983 ◽  
Author(s):  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Models Of Arithmetic

AbstractWe settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

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.

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