Chapter 6 Autostable models and algorithmic dimensions

Гончаров, С. С.

doi:10.1016/s0049-237x(98)80007-4

Cited by 10 publications

(7 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the same paper he introduced the notion of an autostable model, which is equivalent to that of a computably categorical model. Since then, the definition of computable categoricity has been standardized and relativized to arbitrary Turing degrees d, and has been the subject of much study (see, e.g., [4,5]). …”

Section: Preliminariesmentioning

confidence: 99%

Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations

et al. 2015

View full text Add to dashboard Cite

Keywords: index set, structure categorical relative to n-decidable presentations, n-decidable structure categorical relative to m-decidable presentations.We say that a structure is categorical relative to n-decidable presentations (or autostable relative to n-constructivizations) if any two n-decidable copies of the structure are computably isomorphic. For n = 0, we have the classical definition of a computably categorical (autostable) structure. Downey, Kach, Lempp, Lewis, Montalbán, and Turetsky proved that there is no simple syntactic characterization of computable categoricity. More formally, they showed that the index set of computably categorical structures is Π 1 1 -complete. Here we study index sets of n-decidable structures that are categorical relative to m-decidable presentations, for various m, n ∈ ω. If m ≥ n ≥ 0, then the index set is again Π 1 1 -complete, i.e., there is no nice description of the class of n-decidable structures that are categorical relative to m-decidable presentations. In the case m = n − 1 ≥ 0, the index set is Π 0 4 -complete, while if 0 ≤ m ≤ n − 2, the index set is Σ 0 3 -complete. * Supported by Austrian Science Fund FWF (projects V 206 and I 1238).

show abstract

Section: Preliminariesmentioning

confidence: 99%

Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations

et al. 2015

View full text Add to dashboard Cite

show abstract

“…This Handbook also contains other useful survey papers on aspects of effective model theory and algebra and an extensive bibliography. The one most closely related to the theme of this paper is Goncharov [1998]. Another interesting survey is Millar [1999] in The Handbook of Computability Theory (Griffor [1999]).…”

Section: Basic Notionsmentioning

confidence: 99%

Effective Model Theory: The Number of Models and Their Complexity

Khoussainov¹,

Shore²

1999

Models and Computability

View full text Add to dashboard Cite

Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.

show abstract

Section: Introductionmentioning

confidence: 99%

Categoricity Spectra for Rigid Structures

Fokina¹,

Frolov²,

Kalimullin³

2016

Notre Dame J. Formal Logic

View full text Add to dashboard Cite

For a computable structure M , the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of M . If the spectrum has a least degree, this degree is called the degree of categoricity of M . In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity.

show abstract

Chapter 6 Autostable models and algorithmic dimensions

Cited by 10 publications

References 28 publications

Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations

Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations

Effective Model Theory: The Number of Models and Their Complexity

Categoricity Spectra for Rigid Structures

Contact Info

Product

Resources

About