scholarly journals On Transfinite Levels of the Ershov Hierarchy

2021 ◽  
Vol 27 (2) ◽  
pp. 220-221
Author(s):  
Cheng Peng
Keyword(s):  
Recursion Theory ◽  
Density Theorem ◽  
Turing Degrees ◽  
Elementary Substructure ◽  
Ordered Structures ◽  
Research Areas ◽  
Ershov Hierarchy ◽  
Sigma 1 ◽  
E Mail

AbstractIn this thesis, we study Turing degrees in the context of classical recursion theory. What we are interested in is the partially ordered structures $\mathcal {D}_{\alpha }$ for ordinals $\alpha <\omega ^2$ and $\mathcal {D}_{a}$ for notations $a\in \mathcal {O}$ with $|a|_{o}\geq \omega ^2$ .The dissertation is motivated by the $\Sigma _{1}$ -elementary substructure problem: Can one structure in the following structures $\mathcal {R}\subsetneqq \mathcal {D}_{2}\subsetneqq \dots \subsetneqq \mathcal {D}_{\omega }\subsetneqq \mathcal {D}_{\omega +1}\subsetneqq \dots \subsetneqq \mathcal {D(\leq \textbf {0}')}$ be a $\Sigma _{1}$ -elementary substructure of another? For finite levels of the Ershov hierarchy, Cai, Shore, and Slaman [Journal of Mathematical Logic, vol. 12 (2012), p. 1250005] showed that $\mathcal {D}_{n}\npreceq _{1}\mathcal {D}_{m}$ for any $n < m$ . We consider the problem for transfinite levels of the Ershov hierarchy and show that $\mathcal {D}_{\omega }\npreceq _{1}\mathcal {D}_{\omega +1}$ . The techniques in Chapters 2 and 3 are motivated by two remarkable theorems, Sacks Density Theorem and the d.r.e. Nondensity Theorem.In Chapter 1, we first briefly review the background of the research areas involved in this thesis, and then review some basic definitions and classical theorems. We also summarize our results in Chapter 2 to Chapter 4. In Chapter 2, we show that for any $\omega $ -r.e. set D and r.e. set B with $D<_{T}B$ , there is an $\omega +1$ -r.e. set A such that $D<_{T}A<_{T}B$ . In Chapter 3, we show that for some notation a with $|a|_{o}=\omega ^{2}$ , there is an incomplete $\omega +1$ -r.e. set A such that there are no a-r.e. sets U with $A<_{T}U<_{T}K$ . In Chapter 4, we generalize above results to higher levels (up to $\varepsilon _{0}$ ). We investigate Lachlan sets and minimal degrees on transfinite levels and show that for any notation a, there exists a $\Delta ^{0}_{2}$ -set A such that A is of minimal degree and $A\not \equiv _T U$ for all a-r.e. sets U.Abstract prepared by Cheng Peng.E-mail: [email protected]

scholarly journals The Structure of d.r.e. Degrees

2021 ◽  
Vol 27 (2) ◽  
pp. 218-219
Author(s):  
Yong Liu
Keyword(s):  
Boolean Algebra ◽  
Recursion Theory ◽  
Density Theorem ◽  
The Other ◽  
Maximal Degree ◽  
Joint Project ◽  
Final Segment ◽  
E Mail

AbstractThis dissertation is highly motivated by d.r.e. Nondensity Theorem, which is interesting in two perspectives. One is that it contrasts Sacks Density Theorem, and hence shows that the structures of r.e. degrees and d.r.e. degrees are different. The other is to investigate what other properties a maximal degree can have.In Chapter 1, we briefly review the backgrounds of Recursion Theory which motivate the topics of this dissertation.In Chapter 2, we introduce the notion of $(m,n)$ -cupping degree. It is closely related to the notion of maximal d.r.e. degree. In fact, a $(2,2)$ -cupping degree is maximal d.r.e. degree. We then prove that there exists an isolated $(2,\omega )$ -cupping degree by combining strategies for maximality and isolation with some efforts.Chapter 3 is part of a joint project with Steffen Lempp, Yiqun Liu, Keng Meng Ng, Cheng Peng, and Guohua Wu. In this chapter, we prove that any finite boolean algebra can be embedded into d.r.e. degrees as a final segment. We examine the proof of d.r.e. Nondensity Theorem and make developments to the technique to make it work for our theorem. The goal of the project is to see what lattice can be embedded into d.r.e. degrees as a final segment, as we observe that the technique has potential be developed further to produce other interesting results.Abstract prepared by Yong Liu.E-mail: [email protected]

The Sacks density theorem and Σ2-bounding

1996 ◽  
Vol 61 (2) ◽  
pp. 450-467 ◽  
Author(s):  
Marcia J. Groszek ◽  
Michael E. Mytilinaios ◽  
Theodore A. Slaman
Keyword(s):  
Recursion Theory ◽  
Density Theorem ◽  
Turing Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Models Of Arithmetic

AbstractThe Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P− + BΣ2. The proof has two components: a lemma that in any model of P− + BΣ2, if B is recursively enumerable and incomplete then IΣ1 holds relative to B and an adaptation of Shore's [9] blocking technique in α-recursion theory to models of arithmetic.

Turing reducibility and Turing degrees

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.

scholarly journals Classifying equivalence relations in the Ershov hierarchy

2020 ◽  
Vol 59 (7-8) ◽  
pp. 835-864
Author(s):  
Nikolay Bazhenov ◽  
Manat Mustafa ◽  
Luca San Mauro ◽  
Andrea Sorbi ◽  
Mars Yamaleev
Keyword(s):  
Equivalence Relations ◽  
Special Focus ◽  
Ershov Hierarchy ◽  
Standard Tool ◽  
Algebraic Properties ◽  
Sigma 1 ◽  
Degree Structure ◽  
Computably Enumerable

Abstract Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $$\leqslant _c$$ ⩽ c . This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the $$\Delta ^0_2$$ Δ 2 0 case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree structure induced by $$\leqslant _c$$ ⩽ c on the $$\Sigma ^{-1}_{a}\smallsetminus \Pi ^{-1}_a$$ Σ a - 1 \ Π a - 1 equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.

An introduction to γ-recursion theory (or what to do in KP – Foundation)

1990 ◽  
Vol 55 (1) ◽  
pp. 194-206 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Set Theory ◽  
Mathematical Theory ◽  
Recursion Theory ◽  
Reverse Mathematics ◽  
Peano Arithmetic ◽  
Turing Degrees ◽  
The Subject ◽  
Admissible Ordinals ◽  
Limited Induction ◽  
Well Foundedness

The program of reverse mathematics has usually been to find which parts of set theory, often used as a base for other mathematics, are actually necessary for some particular mathematical theory. In recent years, Slaman, Groszek, et al, have given the approach a new twist. The priority arguments of recursion theory do not naturally or necessarily lead to a foundation involving any set theory; rather, Peano Arithmetic (PA) in the language of arithmetic suffices. From this point, the appropriate subsystems to consider are fragments of PA with limited induction. A theorem in this area would then have the form that certain induction axioms are independent of, necessary for, or even equivalent to a theorem about the Turing degrees. (See, for examples, [C], [GS], [M], [MS], and [SW].)As go the integers so go the ordinals. One motivation of α-recursion theory (recursion on admissible ordinals) is to generalize classical recursion theory. Since induction in arithmetic is meant to capture the well-foundedness of ω, the corresponding axiom in set theory is foundation. So reverse mathematics, even in the context of a set theory (admissibility), can be changed by the influence of reverse recursion theory. We ask not which set existence axioms, but which foundation axioms, are necessary for the theorems of α-recursion theory.When working in the theory KP – Foundation Schema (hereinafter called KP−), one should really not call it α-recursion theory, which refers implicitly to the full set of axioms KP. Just as the name β-recursion theory refers to what would be α-recursion theory only it includes also inadmissible ordinals, we call the subject of study here γ-recursion theory. This answers a question by Sacks and S. Friedman, “What is γ-recursion theory?”

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.

scholarly journals Intrinsic density, asymptotic computability, and stochasticity

2021 ◽  
Vol 27 (2) ◽  
pp. 220-220
Author(s):  
Justin Miller
Keyword(s):  
Turing Degrees ◽  
Closure Properties ◽  
Computable Set ◽  
Open Questions ◽  
Bernoulli Measure ◽  
Almost All ◽  
E Mail

AbstractThere are many computational problems which are generally “easy” to solve but have certain rare examples which are much more difficult to solve. One approach to studying these problems is to ignore the difficult edge cases. Asymptotic computability is one of the formal tools that uses this approach to study these problems. Asymptotically computable sets can be thought of as almost computable sets, however every set is computationally equivalent to an almost computable set. Intrinsic density was introduced as a way to get around this unsettling fact, and which will be our main focus.Of particular interest for the first half of this dissertation are the intrinsically small sets, the sets of intrinsic density $0$ . While the bulk of the existing work concerning intrinsic density was focused on these sets, there were still many questions left unanswered. The first half of this dissertation answers some of these questions. We proved some useful closure properties for the intrinsically small sets and applied them to prove separations for the intrinsic variants of asymptotic computability. We also completely separated hyperimmunity and intrinsic smallness in the Turing degrees and resolved some open questions regarding the relativization of intrinsic density.For the second half of this dissertation, we turned our attention to the study of intermediate intrinsic density. We developed a calculus using noncomputable coding operations to construct examples of sets with intermediate intrinsic density. For almost all $r\in (0,1)$ , this construction yielded the first known example of a set with intrinsic density r which cannot compute a set random with respect to the r-Bernoulli measure. Motivated by the fact that intrinsic density coincides with the notion of injection stochasticity, we applied these techniques to study the structure of the more well-known notion of MWC-stochasticity.Abstract prepared by Justin Miller.E-mail: [email protected]: https://curate.nd.edu/show/6t053f4938w

Research and Supervision

2017 ◽  
pp. 23-80
Keyword(s):  
Research Process ◽  
Engaged Scholarship ◽  
Doctoral Studies ◽  
Doctoral Level ◽  
Doctoral Research ◽  
Arts And Humanities ◽  
Research Areas ◽  
Research Supervision ◽  
The Social

Doctoral studies are often viewed as a “journey”, with expectations that students will be transformed into independent scholars. Various strategies and approaches are taken to this end, as discussed in Chapter 1. The research performed in doctoral studies is considered by many as the hallmark of such a terminal degree, a type of “rite of passage” for the rest of a scholar's career. This chapter reviews types of research areas, provides a background to mentorship, and engaged scholarship first introduced in research in the social sciences, arts and humanities. The context of research supervision in CS and IT is addressed, and a generic research process is elaborated, with consideration to developing critical thinking in, and imparting knowledge to, a research candidate. Some measures for doctoral research supervision are presented, as well as doctoral level research norms. Frameworks for research supervision are examined, and what is regarded to be successful supervision. Lastly some trends that have emerged in research supervision are reviewed.

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