Ershov Hierarchy Research Articles

A classification of low c.e. sets and the Ershov hierarchy

AbstractIn this paper, we prove several results about the Turing jumps of low c.e. sets. We show that only Δ‐levels of the Ershov Hierarchy can properly contain the Turing jumps of c.e. sets and that there exists an arbitrarily large computable ordinal with a normal notation such that the corresponding Δ‐level is proper for the Turing jump of some c.e. set. Next, we generalize the notion of jump traceability to the jump traceability with ‐ and ‐bound for every infinite computable ordinal α. It is known that jump traceability and superlowness coincide on the c.e. sets and we show that for every infinite computable ordinal α, jump traceability with ‐ or ‐bound of a c.e. set A is equivalent to the fact that . Finally, we consider the generalized truth‐table reducibilities and prove that for every (not necessarily the Turing jump of a c.e. set) set A and every limit computable ordinal α, iff .

How to approximate fuzzy sets: mind-changes and the Ershov Hierarchy

Computability theorists have introduced multiple hierarchies to measure the complexity of sets of natural numbers. The Kleene Hierarchy classifies sets according to the first-order complexity of their defining formulas. The Ershov Hierarchy classifies limit computable sets with respect to the number of mistakes that are needed to approximate them. Biacino and Gerla extended the Kleene Hierarchy to the realm of fuzzy sets, whose membership functions range in a complete lattice. In this paper, we combine the Ershov Hierarchy and fuzzy set theory, by introducing and investigating the Fuzzy Ershov Hierarchy.

Rogers Semilattices with Least and Greatest Elements in the Ershov Hierarchy

Rogers Semilattices with Least and Greatest Elements in the Ershov Hierarchy

Computable Linear Orders and the Ershov Hierarchy

Computable Linear Orders and the Ershov Hierarchy

One-element Rogers semilattices in the Ershov hierarchy

One-element Rogers semilattices in the Ershov hierarchy

Normalizing notations in the Ershov hierarchy

AbstractThe Turing degrees of infinite levels of the Ershov hierarchy were studied by Liu and Peng [8]. In this paper, we continue the study of Turing degrees of infinite levels and lift the study of density property to the levels beyond ω2. In doing so, we rely on notations with some nice properties. We introduce the concept of normalizing notations and generate normalizing notations for higher levels. The generalizations of the weak density theorem and the nondensity theorem are proved for higher levels in the Ershov hierarchy. Furthermore, we also investigate the minimal degrees in the infinite levels of the Ershov hierarchy.

CEA Operators and the Ershov Hierarchy

CEA Operators and the Ershov Hierarchy

On Transfinite Levels of the Ershov Hierarchy

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: pengcheng@nankai.edu.cn

CEA-операторы и иерархия Ершова

CEA-операторы и иерархия Ершова

On Universal Pairs in the Ershov Hierarchy

We develop the Ershov theory of C-classes for some finite families of sets in the Ershov hierarchy. We generalize the result by Muchnik on multiple $ m $ -reducibility as follows: There exists an $ m $ -universal pair of disjoint sets for each level of the Ershov hierarchy.

Weak Density and Nondensity among Transfinite Levels of the Ershov Hierarchy

We show that for any ω-r.e. degree d and n-r.e. degree b with d<b, there is an (ω+1)-r.e. degree a strictly between d and b. We also show that there is a maximal incomplete (ω+1)-r.e. degree. As a corollary, Dω is not a Σ1-elementary substructure of Dω+1.

Classifying equivalence relations in the Ershov hierarchy

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. This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the Delta ^0_2 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 on the Sigma ^{-1}_{a}smallsetminus Pi ^{-1}_a equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.

Turing reducibility in the fine hierarchy

In the late 1980s, Selivanov used typed Boolean combinations of arithmetical sets to extend the Ershov hierarchy beyond Δ20 sets. Similar to the Ershov hierarchy, Selivanov's fine hierarchy{Σγ}γ<ε0 proceeds through transfinite levels below ε0 to cover all arithmetical sets. In this paper we use a 0‴ construction to show that the Σ30 Turing degrees are properly contained in the Σωω+2 Turing degrees (to be defined); intuitively, the latter class consists of “non-uniformly Σ30 sets” in the sense that will be clarified in the introduction. The question whether the hierarchy was proper at this level with respect to Turing reducibility remained open for over 20 years.

Weakly precomplete equivalence relations in the Ershov hierarchy

Weakly precomplete equivalence relations in the Ershov hierarchy

Weakly Precomplete Equivalence Relations in the Ershov Hierarchy

We study the computable reducibility ≤c for equivalence relations in the Ershov hierarchy. For an arbitrary notation a for a nonzero computable ordinal, it is stated that there exist a $$ {\varPi}_a^{-1} $$ -universal equivalence relation and a weakly precomplete $$ {\varSigma}_a^{-1} $$ - universal equivalence relation. We prove that for any $$ {\varSigma}_a^{-1} $$ equivalence relation E, there is a weakly precomplete $$ {\varSigma}_a^{-1} $$ equivalence relation F such that E ≤cF. For finite levels $$ {\varSigma}_m^{-1} $$ in the Ershov hierarchy at which m = 4k +1 or m = 4k +2, it is shown that there exist infinitely many ≤c-degrees containing weakly precomplete, proper $$ {\varSigma}_m^{-1} $$ equivalence relations.

Rogers Semilattices for Families of Equivalence Relations in the Ershov Hierarchy

The paper studies Rogers semilattices for families of equivalence relations in the Ershov hierarchy. For an arbitrary notation a of a nonzero computable ordinal, we consider $$\sum\nolimits_a^{- 1} {}$$ -computable numberings of the family of all $$\sum\nolimits_a^{- 1} {}$$ equivalence relations. We show that this family has infinitely many pairwise incomparable Friedberg numberings and infinitely many pairwise incomparable positive undecidable numberings. We prove that the family of all c.e. equivalence relations has infinitely many pairwise incomparable minimal nonpositive numberings. Moreover, we show that there are infinitely many principal ideals without minimal numberings.

Isolated maximal d.r.e. degrees

There are very few results about maximal d.r.e. degrees as the construction is very hard to work with other requirements. In this paper we show that there exists an isolated maximal d.r.e. degree. In fact, we introduce a closely related notion called (m,n)-cupping degree and show that there exists an isolated (2,ω)-cupping degree, and there exists a proper (2,1)-cupping degree. It helps understanding various degree structures in the Ershov Hierarchy.

Rogers semilattices for families of equivalence relations in the Ershov hierarchy

Rogers semilattices for families of equivalence relations in the Ershov hierarchy

Computable Linear Orders and the Ershov Hierarchy

We give the collection of relations on computable linear orders. For any natural number n, the degree spectrum of such relations of some computable linear orders contains exactly all n-computable enumerable degrees. We also study interconnections of these relations among themselves.

Linearisations and the Ershov hierarchy

A partial order is computably well founded if it does not computably embed a copy of ' ‰ ' -, the order type of the negative integers. It is computably scattered if it does not computably embed a copy of ' •, the order type of Q. It is known that, for each of these properties, there are computable partial orders satisfying the property which do not have a computable linear extension with the same property. Rosenstein showed, however, that for both of these properties, every computable partial order satisfying the property has a ' 2 0 linear extension also satisfying the property. Thus, linear extensions of a computable order preserving the properties of computable well foundedness or computable scatteredness can always be found at the ' 2 0 level of the arithmetical hierarchy, but not at the ' 1 0 level. In this paper, we investigate at which level of the Ershov hierarchy such linear extensions can be found. We show that, for both properties, every computable partial order satisfying the property has an ' ‰-c.e. linear extension with the same property. We establish that this is the best possible result within the Ershov hierarchy by constructing, respectively, computably well founded and computably scattered orders which do not have n-c.e. linear extensions which are computably well founded and computably scattered respectively, for any n < ' ‰. In a strengthening of Rosenstein's theorems in another direction, we show that a linear extension preserving each of these properties can be computed using any oracle satisfying an escape property, which includes the class of non-generalised low 2 sets. Finally, we show that the analogue of Rosenstein's theorems do not hold for the property of not computably embedding a copy of , the order type of the integers, by constructing a computable partial ordering which does not embed, but such that every ' 2 0 linear extension of the ordering does admit a computable embedding of.