Degree spectra of intrinsically c.e. relations

2001 ◽  
Vol 66 (2) ◽  
pp. 441-469 ◽  
Author(s):  
Denis R. Hirschfeldt
Keyword(s):  
Computable Structure ◽  
Degree Spectra ◽  
Degree Spectrum

AbstractWe show that for every c.e. degree a > 0 there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. This result can be extended in two directions. First we show that for every uniformly c.e. collection of sets S there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is the set of degrees of elements of S. Then we show that if α ∈ ω ∪ {ω} then for any α-c.e. degree a > 0 there exists an intrinsically α-c.e. relation on the domain of a computable structure whose degree spectrum {0, a}. All of these results also hold for m-degree spectra of relations.

Degree spectra of relations on computable structures in the presence of Δ20 isomorphisms

2002 ◽  
Vol 67 (2) ◽  
pp. 697-720 ◽  
Author(s):  
Denis R. Hirschfeldt
Keyword(s):  
Point Of View ◽  
The Other ◽  
Computable Structure ◽  
Sufficient Condition ◽  
Computable Structures ◽  
Linear Orderings ◽  
Degree Spectra ◽  
Other Hand ◽  
Infinite Degree ◽  
Degree Spectrum

AbstractWe give some new examples of possible degree spectra of invariant relations on Δ20-categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a Δ20-categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [23] to establish the same result for computable relations on computable linear orderings.We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ20-categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode [3] to extend results of Harizanov [14].

Computability of Fraïssé limits

2011 ◽  
Vol 76 (1) ◽  
pp. 66-93 ◽  
Author(s):  
Barbara F. Csima ◽  
Valentina S. Harizanov ◽  
Russell Miller ◽  
Antonio Montalbán
Keyword(s):  
Boolean Algebra ◽  
Necessary Conditions ◽  
Computable Structure ◽  
Finitely Generated ◽  
Degree Spectra ◽  
Atomless Boolean Algebra

AbstractFraïssé studied countable structures through analysis of the age of , i.e., the set of all finitely generated substructures of . We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.

scholarly journals THE COMPLEMENTS OF LOWER CONES OF DEGREES AND THE DEGREE SPECTRA OF STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 997-1006 ◽  
Author(s):  
URI ANDREWS ◽  
MINGZHONG CAI ◽  
ISKANDER SH. KALIMULLIN ◽  
STEFFEN LEMPP ◽  
JOSEPH S. MILLER ◽  
...  
Keyword(s):  
Turing Degrees ◽  
Degree Spectra ◽  
Degree Spectrum ◽  
Image Position ◽  
Countable Structure

AbstractWe study Turing degrees a for which there is a countable structure ${\cal A}$ whose degree spectrum is the collection {x : x ≰ a}. In particular, for degrees a from the interval [0′, 0″], such a structure exists if a′ = 0″, and there are no such structures if a″ > 0‴.

scholarly journals RELATIVE TO ANY NON-HYPERARITHMETIC SET

2013 ◽  
Vol 13 (01) ◽  
pp. 1250007 ◽  
Author(s):  
NOAM GREENBERG ◽  
ANTONIO MONTALBÁN ◽  
THEODORE A. SLAMAN
Keyword(s):  
Linear Ordering ◽  
Degree Spectra ◽  
Degree Spectrum

We prove that there is a structure, indeed a linear ordering, whose degree spectrum is the set of all non-hyperarithmetic degrees. We also show that degree spectra can distinguish measure from category.

scholarly journals TURING DEGREE SPECTRA OF DIFFERENTIALLY CLOSED FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 1-25 ◽  
Author(s):  
DAVID MARKER ◽  
RUSSELL MILLER
Keyword(s):  
Turing Degree ◽  
Turing Degrees ◽  
Closed Field ◽  
Degree Spectra ◽  
Low Degree ◽  
Closed Fields ◽  
Degree Spectrum ◽  
Countable Structure

AbstractThe degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the computable degree 0. Indeed, this is an equivalence, for we also show that if this spectrum contained a low degree, then it would contain the degree 0. From these results we conclude that the spectra of differentially closed fields of characteristic 0 are exactly the jump-preimages of spectra of automorphically nontrivial graphs.

COMPUTABILITY AND UNCOUNTABLE LINEAR ORDERS II: DEGREE SPECTRA

2015 ◽  
Vol 80 (1) ◽  
pp. 145-178
Author(s):  
NOAM GREENBERG ◽  
ASHER M. KACH ◽  
STEFFEN LEMPP ◽  
DANIEL D. TURETSKY
Keyword(s):  
Computability Theory ◽  
Structure Theory ◽  
Computable Structure ◽  
Linear Orders ◽  
Degree Spectra ◽  
Successor Relation ◽  
Computable Structure Theory ◽  
Image Position

AbstractWe study the computable structure theory of linear orders of size $\aleph _1 $ within the framework of admissible computability theory. In particular, we study degree spectra and the successor relation.

Degree Spectra of Structures

2021 ◽  
Author(s):  
I. Sh. Kalimullin ◽  
V. L. Selivanov ◽  
A. N. Frolov
Keyword(s):  
Degree Spectra

scholarly journals Π10 classes and strong degree spectra of relations

2007 ◽  
Vol 72 (3) ◽  
pp. 1003-1018 ◽  
Author(s):  
John Chisholm ◽  
Jennifer Chubb ◽  
Valentina S. Harizanov ◽  
Denis R. Hirschfeldt ◽  
Carl G. Jockusch ◽  
...  
Keyword(s):  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Truth Table ◽  
Linear Ordering ◽  
Computable Structures ◽  
Degree Spectra

AbstractWe study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable subsets of 2ω and Kolmogorov complexity play a major role in the proof.

Relativized Degree Spectra

2007 ◽  
Vol 17 (6) ◽  
pp. 1215-1233 ◽  
Author(s):  
A. A. Soskova
Keyword(s):  
Degree Spectra

scholarly journals Degree Spectra of Structures Relative to Equivalences

2019 ◽  
Vol 58 (2) ◽  
pp. 158-172 ◽  
Author(s):  
P. M. Semukhin ◽  
D. Turetsky ◽  
E. B. Fokina
Keyword(s):  
Degree Spectra
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