Computable categoricity and the Ershov hierarchy

Khoussainov, Bakhadyr; Stephan, Frank; Yue, Yang

doi:10.1016/j.apal.2008.06.010

Cited by 9 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We give definitions of F a -and G a -categoricities for computable structures, as in [11]. We say that a function F : ω → ω is in a class G a if one of the following conditions is satisfied:…”

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

Bazhenov

2015

Algebra Logic

View full text Add to dashboard Cite

Computable categoricity in the Ershov hierarchy is studied. We consider F a -categorical and G a -categorical structures. These were introduced by B. Khoussainov, F. Stephan, and Y. Yang for a, which is a notation for a constructive ordinal. A generalization of the branching theorem is proved for F a -categorical structures. As a consequence we obtain a description of F a -categorical structures for classes of Boolean algebras and Abelian p-groups. Furthermore, it is shown that the branching theorem cannot be generalized to G a -categorical structures.Research on computable categoricity of structures was initiated by B. L. van der Waerden who studied the question whether there exists an effective algorithm for constructing an isomorphism between two different algebraic closures of a field built in different ways. Subsequently, starting with 1950s, computable categoricity of structures was explored by A. I. Mal'tsev, A. Fröhlich, J. Shepherdson, S. S. Goncharov, J. Remmel, C. Ash, J. Knight, and many other authors.In investigating computably categorical structures, one of the most important problems is to gain an insight into computable categoricity relative to classes of known hierarchies of sets. In particular, of special interest is the Ershov hierarchy introduced in [1-3]. There are quite many works dealing with the Ershov hierarchy. (Among these, it is worth mentioning resent papers [4-10] that focus primarily on numberings in the Ershov hierarchy.) Research on computable categoricity relative to levels of the Ershov hierarchy was started in [11]. There, for an arbitrary a, which is *

show abstract

Section: Preliminariesmentioning

confidence: 99%

“…(Among these, it is worth mentioning resent papers [4-10] that focus primarily on numberings in the Ershov hierarchy.) Research on computable categoricity relative to levels of the Ershov hierarchy was started in [11]. There, for an arbitrary a, which is *…”

mentioning

confidence: 99%

The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

Bazhenov

2015

Algebra Logic

View full text Add to dashboard Cite

show abstract

“…In computable model theory equivalence relations have also been a subject of study, e.g., [2,5,17], etc. In these papers equivalence relations of rather low complexity were studied computable, Σ 0 1 , Π 0 1 , having degree in the Ershov hierarchy .…”

Section: Introductionmentioning

confidence: 99%

On Σ¹₁ equivalence relations over the natural numbers

Fokina

Friedman

2011

Mathematical Logic Qtrly

View full text Add to dashboard Cite

We study the structure of Σ 1 1 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly Σ 1 1 i.e., Σ 1 1 but not Δ 1 1 equivalence classes. We also show the existence of incomparable Σ 1 1 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ 1 1 equivalence relations (under both reducibilities) and show that existence of infinitely many properly Σ 1 1 equivalence classes that are complete as Σ 1 1 sets (under the corresponding reducibility on sets) is necessary but not sufficient for a relation to be complete in the context of Σ 1 1 equivalence relations.

show abstract

“…Things started to change only recently. The transfinite levels of Ershov hierarchy appeared naturally in many areas, for example, in recursion theory and recursive model theory, see Downey and Gale [13] and Khoussainov, Stephan and Yang [17], and in inductive inference, see Ambainis, Freivalds and Smith [2], Carlucci, Case and Jain [5] and Freivalds and Smith [12]. Yet we still feel that Ershov's theorems did not achieve the popularity which they deserved.…”

Section: Introductionmentioning

confidence: 99%

Turing Degrees and the Ershov Hierarchy

Stephan¹,

Yue²,

Yu³

2009

Proceedings of the 10th Asian Logic Conference

Self Cite

View full text Add to dashboard Cite

Abstract. An n-r.e. set can be defined as the symmetric difference of n recursively enumerable sets. The classes of these sets form a natural hierarchy which became a well-studied topic in recursion theory. In a series of ground-breaking papers, Ershov generalized this hierarchy to transfinite levels based on Kleene's notations of ordinals and this work lead to a fruitful study of these sets and their many-one and Turing degrees. The Ershov hierarchy is a natural measure of complexity of the sets below the halting problem. In this paper, we survey the early work by Ershov and others on this hierarchy and present the most fundamental results. We also provide some pointers to concurrent work in the field.

show abstract

Computable categoricity and the Ershov hierarchy

Cited by 9 publications

References 11 publications

The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

On Σ¹₁ equivalence relations over the natural numbers

Turing Degrees and the Ershov Hierarchy

Contact Info

Product

Resources

About

Computable categoricity and the Ershov hierarchy

Cited by 9 publications

References 11 publications

The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

The Branching Theorem and Computable Categoricity in the Ershov Hierarchy

On Σ11 equivalence relations over the natural numbers

Turing Degrees and the Ershov Hierarchy

Contact Info

Product

Resources

About

On Σ¹₁ equivalence relations over the natural numbers