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]).

Computational complexity, speedable and levelable sets

1977 ◽  
Vol 42 (4) ◽  
pp. 545-563 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Computational Complexity ◽  
Jump Operator ◽  
Information Theoretic ◽  
Recursively Enumerable ◽  
Index Sets ◽  
Close Relationship ◽  
Speed Up ◽  
Theoretic Characterization ◽  
Large Speed

One of the most interesting aspects of the theory of computational complexity is the speed-up phenomenon such as the theorem of Blum [6, p. 326] which asserts the existence of a 0, 1-valued total recursive function with arbitrarily large speed-up. Blum and Marques [10] extended the speed-up definitions from total to partial recursive functions, or equivalently, to recursively enumerable (r.e.) sets, and introduced speedable and levelable sets. They classified the effectively speedable sets as the subcreative sets but remarked that “the characterizations we provided for speedable and levelable sets do not seem to bear a close relationship to any already well-studied class of recursively enumerable sets.” The purpose of this paper is to give an “information theoretic” characterization of speedable and levelable sets in terms of index sets resembling the jump operator. From these characterizations we derive numerous consequences about the degrees and structure of speedable and levelable sets.

Working below a high recursively enumerable degree

1993 ◽  
Vol 58 (3) ◽  
pp. 824-859 ◽  
Author(s):  
Richard A. Shore ◽  
Theodore A. Slaman
Keyword(s):  
Turing Degrees ◽  
Jump Operator ◽  
Recursively Enumerable Set ◽  
Enumerable Degree ◽  
Local Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree ◽  
Turing Reducibility ◽  
Definition Of ◽  
The Relationship

In recent work, Cooper [3, 1990] has extended results of Jockusch and Shore [6, 1984] to show that the Turing jump is definable in the structure given by the Turing degrees and the ordering of Turing reducibility. In his definition of x′ from x, Cooper identifies an order-theoretic property shared by all of the degrees that are recursively enumerable in x and above x. He then shows that x′ is the least upper bound of all the degrees with this property. Thus, the jump of x is identified by comparing the recursively enumerable degrees with other degrees which are not recursively enumerable. Of course, once the jump operator is known to be definable, the relation of jump equivalence x′ = y′ is also known to be a definable relation on x and y. If we consider how much of the global theory of the Turing degrees is sufficient for Cooper's methods, it is immediately clear that his methods can be implemented to show that the jump operator and its weakening to the relation of jump equivalence are definable in any ideal closed under the Turing jump. However, his methods do not localize to , the degrees, or to the recursively enumerable degrees.This paper fits, as do Shore and Slaman [16, 1990] and [17, to appear], within the general project to develop an understanding of the relationship between the local degree-theoretic properties of a recursively enumerable set A and its jump class. For an analysis of the possibility of defining jump equivalence in , consult Shore [15, to appear] who shows that the relation x(3) = y(3) is definable. In this paper, we will restrict our attention to definitions expressed completely in ℛ (Note: All sets and degrees discussed for the remainder of this paper will be recursively enumerable.) Ultimately, one would like to find some degree-theoretic properties definable in terms of the ordering of Turing reducibility and quantifiers over the recursively enumerable degrees that would define the relation of jump equivalence or define one or more of the jump classes Hn = {w∣ wn = 0n+1} or Ln = {w ∣ wn = 0n}. Such a result could very likely then be used as a springboard to other general definability results for the recursively enumerable degrees. It would be especially interesting to know whether every recursively enumerable degree is definable and whether every arithmetical degree-invariant property of the recursively enumerable sets is definable in .

On degrees of unsolvability and complexity properties

1975 ◽  
Vol 40 (4) ◽  
pp. 529-540 ◽  
Author(s):  
Ivan Marques
Keyword(s):  
Turing Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Complete Sets ◽  
Turing Complete ◽  
Degrees Of Unsolvability

In this paper we present two theorems concerning relationships between degrees of unsolvability of recursively enumerable sets and their complexity properties.The first theorem asserts that there are nonspeedable recursively enumerable sets in every recursively enumerable Turing degree. This theorem disproves the conjecture that all Turing complete sets are speedable, which arose from the fact that a rather inclusive subclass of the Turing complete sets, namely, the subcreative sets, consists solely of effectively speedable sets [2]. Furthermore, the natural construction to produce a nonspeedable set seems to lower the degree of the resulting set.The second theorem says that every speedable set has jump strictly above the jump of the recursive sets. This theorem is an expected one in view of the fact that all sets which are known to be speedable jump to the double jump of the recursive sets [4].After this paper was written, R. Soare [8] found a very useful characterization of the speedable sets which greatly facilitated the proofs of the results presented here. In addition his characterization implies that an r.e. degree a contains a speed-able set iff a′ > 0′.

Jump embeddings in the Turing degrees

1991 ◽  
Vol 56 (2) ◽  
pp. 563-591 ◽  
Author(s):  
Peter G. Hinman ◽  
Theodore A. Slaman
Keyword(s):  
Partial Ordering ◽  
Turing Degrees ◽  
First Order ◽  
Upper Semilattice ◽  
Recursively Enumerable ◽  
Locally Countable ◽  
Image Set ◽  
Turing Reducibility ◽  
Image Position ◽  
Order Structures

Since its introduction in [K1-Po], the upper semilattice of Turing degrees has been an object of fascination to practitioners of the recursion-theoretic art. Starting from relatively simple concepts and definitions, it has turned out to be a structure of enormous complexity and richness. This paper is a contribution to the ongoing study of this structure.Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if {P, ≤P) is a partial ordering of cardinality at most ℵ1 which is locally countable—each point has at most countably many predecessors—then there is an embeddingwhere D is the set of all Turing degrees and <T is Turing reducibility. If (P, ≤P) is a countable partial ordering, then the image of the embedding may be taken to be a subset of R, the set of recursively enumerable degrees. Without attempting to make the notion completely precise, we shall call embeddings of the first sort global, in contrast to local embeddings which impose some restrictions on the image set.

A minimal degree not realizing least possible jump

1974 ◽  
Vol 39 (3) ◽  
pp. 571-574 ◽  
Author(s):  
Leonard P. Sasso
Keyword(s):  
Lebesgue Measure ◽  
Minimal Degree ◽  
Partial Ordering ◽  
Recursively Enumerable ◽  
Degree Constructions ◽  
A Priority ◽  
Combinatorial Device ◽  
Priority Argument ◽  
New System

The least possible jump for a degree of unsolvability a is its join a ∪ 0′ with 0′. Friedberg [1] showed that each degree b ≥ 0′ is the jump of a degree a realizing least possible jump (i.e., satisfying the equation a′ = a ∪ 0′). Sacks (cf. Stillwell [8]) showed that most (in the sense of Lebesgue measure) degrees realize least possible jump. Nevertheless, degrees not realizing least possible jump are easily found (e.g., any degree b ≥ 0′) even among the degrees <0′ (cf. Shoenfield [5]) and the recursively enumerable (r.e.) degrees (cf. Sacks [3]).A degree is called minimal if it is minimal in the natural partial ordering of degrees excluding least element 0. The existence of minimal degrees <0” was first shown by Spector [7]; Sacks [3] succeeded in replacing 0” by 0′ using a priority argument. Yates [9] asked whether all minimal degrees <0′ realize least possible jump after showing that some do by exhibiting minimal degrees below each r.e. degree. Cooper [2] subsequently showed that each degree b > 0′ is the jump of a minimal degree which, as corollary to his method of proof, realizes least possible jump. We show with the aid of a simple combinatorial device applied to a minimal degree construction in the manner of Spector [7] that not all minimal degrees realize least possible jump. We have observed in conjunction with S. B. Cooper and R. Epstein that the new combinatorial device may also be applied to minimal degree constructions in the manner of Sacks [3], Shoenfield [6] or [4] in order to construct minimal degrees <0′ not realizing least possible jump. This answers Yates' question in the negative. Yates [10], however, has been able to draw this as an immediate corollary of the weaker result by carrying out the proof in his new system of prioric games.

Countable initial segments of the degrees of unsolvability

1976 ◽  
Vol 41 (2) ◽  
pp. 289-300 ◽  
Author(s):  
A. H. Lachlan ◽  
R. Lebeuf
Keyword(s):  
Initial Segment ◽  
Upper Semilattice ◽  
Order Types ◽  
Degrees Of Unsolvability

In this paper we show that any countable upper semilattice with zero can be embedded as an initial segment of the degrees of unsolvability. This provides a characterization of the order types of the countable initial segments of the degrees since any such initial segment is trivially an initial segment of a countable upper semilattice. Let a segment S of the degrees be a set of degrees such that, if a, b Є S and a < c < b, then c Є S. One may readily observe that our result also characterizes the order types of all countable segments of the degrees of unsolvability.

Skeletal elements of crinoid echinoderms: High-resolution structural and microanalytical results

1983 ◽  
Vol 41 ◽  
pp. 194-195
Author(s):  
D. F. Blake ◽  
L. F. Allard ◽  
D. R. Peacor
Keyword(s):  
High Resolution ◽  
Marine Invertebrates ◽  
Sea Urchins ◽  
Structural Features ◽  
Sea Stars ◽  
High Resolution Tem ◽  
Relationship Of ◽  
The Relationship ◽  
Insight Into

Echinodermata is a phylum of marine invertebrates which has been extant since Cambrian time (c.a. 500 m.y. before the present). Modern examples of echinoderms include sea urchins, sea stars, and sea lilies (crinoids). The endoskeletons of echinoderms are composed of plates or ossicles (Fig. 1) which are with few exceptions, porous, single crystals of high-magnesian calcite. Despite their single crystal nature, fracture surfaces do not exhibit the near-perfect {10.4} cleavage characteristic of inorganic calcite. This paradoxical mix of biogenic and inorganic features has prompted much recent work on echinoderm skeletal crystallography. Furthermore, fossil echinoderm hard parts comprise a volumetrically significant portion of some marine limestones sequences. The ultrastructural and microchemical characterization of modern skeletal material should lend insight into: 1). The nature of the biogenic processes involved, for example, the relationship of Mg heterogeneity to morphological and structural features in modern echinoderm material, and 2). The nature of the diagenetic changes undergone by their ancient, fossilized counterparts. In this study, high resolution TEM (HRTEM), high voltage TEM (HVTEM), and STEM microanalysis are used to characterize tha ultrastructural and microchemical composition of skeletal elements of the modern crinoid Neocrinus blakei.

Microstructural Characterization of Au-Ge-Ni based ohmic contacts to GaAs/AlGaAs modfet device

1987 ◽  
Vol 45 ◽  
pp. 326-327
Author(s):  
A.K. Rai ◽  
A.K. Petford-Long ◽  
A. Ezis ◽  
D.W. Langer
Keyword(s):  
Contact Resistance ◽  
Ohmic Contact ◽  
Ohmic Contacts ◽  
Microstructural Characterization ◽  
Almost Everywhere ◽  
Spacer Layer ◽  
Good Contact ◽  
The Relationship ◽  
Lower Contact

Considerable amount of work has been done in studying the relationship between the contact resistance and the microstructure of the Au-Ge-Ni based ohmic contacts to n-GaAs. It has been found that the lower contact resistivity is due to the presence of Ge rich and Au free regions (good contact area) in contact with GaAs. Thus in order to obtain an ohmic contact with lower contact resistance one should obtain a uniformly alloyed region of good contact areas almost everywhere. This can possibly be accomplished by utilizing various alloying schemes. In this work microstructural characterization, employing TEM techniques, of the sequentially deposited Au-Ge-Ni based ohmic contact to the MODFET device is presented.The substrate used in the present work consists of 1 μm thick buffer layer of GaAs grown on a semi-insulating GaAs substrate followed by a 25 Å spacer layer of undoped AlGaAs.

Characterization of Auditory Disability and Its Relation to the Resilience

2018 ◽  
Vol 5 (2) ◽  
pp. 15
Author(s):  
Bibian Bibeca Bumbila García ◽  
Hernán Andrés Cedeño Cedeño ◽  
Tatiana Moreira Chica ◽  
Yaritza Rossana Parrales Ríos
Keyword(s):  
Social Integration ◽  
Conceptual Analysis ◽  
Disabled People ◽  
Disabled Students ◽  
Hearing Disability ◽  
Technical University ◽  
The Relationship ◽  
Made In

The objective of the work is to establish the characterization of the auditory disability and its relationship with resilience at the Technical University of Manabí. The article shows a conceptual analysis related to the inclusion and social integration of disabled students. Based on the fact that the person with disabilities grows and develops in the same way as that of people without disabilities and what usually happens is that disabled people are rejected and discriminated against based on a prefabricated and erroneous conceptualization of these people. The results associated with the application of the SV-RES test prepared by the researchers are shown (Saavedra & Villalta, 2008b). Characterization of the auditory deficit is made in the students, and the limitations that derive from it are pointed out. We analyze the particularities related to communication with students who have a hearing disability and resilience in this type of student, where some personal highlights that in this sense constitute an example of resilience. Finally, the results related to the study of the relationship between students' hearing disability and the level of resilience dimensions are shown.

The Problem of Divine Hiddenness

2018 ◽  
Author(s):  
Michael C. Rea
Keyword(s):  
Problem Of Evil ◽  
Philosophical Discussion ◽  
Detailed Characterization ◽  
The Past ◽  
Divine Hiddenness ◽  
The Problem Of Evil ◽  
The Relationship

This chapter provides a detailed characterization of the various meanings of the term “divine hiddenness,” carefully and rigorously articulates the version of the problem of divine hiddenness that has dominated contemporary philosophical discussion for the past twenty-five years, and then explains the relationship between that problem and the problem of evil.

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