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.

COMPUTABILITY AND UNCOUNTABLE LINEAR ORDERS I: COMPUTABLE CATEGORICITY

2015 ◽  
Vol 80 (1) ◽  
pp. 116-144
Author(s):  
NOAM GREENBERG ◽  
ASHER M. KACH ◽  
STEFFEN LEMPP ◽  
DANIEL D. TURETSKY
Keyword(s):  
Computability Theory ◽  
Structure Theory ◽  
Computable Structure ◽  
Linear Orders ◽  
Computable Categoricity ◽  
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 characterize which of these linear orders are computably categorical.

COMPUTABLE POLISH GROUP ACTIONS

2018 ◽  
Vol 83 (2) ◽  
pp. 443-460
Author(s):  
ALEXANDER MELNIKOV ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Set Theory ◽  
Group Actions ◽  
General Setting ◽  
Structure Theory ◽  
Computable Structure ◽  
Descriptive Set Theory ◽  
Sufficient Condition ◽  
Computable Structure Theory ◽  
Image Position ◽  
Descriptive Set

AbstractUsing methods from computable analysis, we establish a new connection between two seemingly distant areas of logic: computable structure theory and invariant descriptive set theory. We extend several fundamental results of computable structure theory to the more general setting of topological group actions. As we will see, the usual action of ${S_\infty }$ on the space of structures in a given language is effective in a certain algorithmic sense that we need, and ${S_\infty }$ itself carries a natural computability structure (to be defined). Among other results, we give a sufficient condition for an orbit under effective ${\cal G}$-action of a computable Polish ${\cal G}$ to split into infinitely many disjoint effective orbits. Our results are not only more general than the respective results in computable structure theory, but they also tend to have proofs different from (and sometimes simpler than) the previously known proofs of the respective prototype results.

scholarly journals Enumerations in computable structure theory

2005 ◽  
Vol 136 (3) ◽  
pp. 219-246 ◽  
Author(s):  
Sergey Goncharov ◽  
Valentina Harizanov ◽  
Julia Knight ◽  
Charles McCoy ◽  
Russell Miller ◽  
...  
Keyword(s):  
Structure Theory ◽  
Computable Structure ◽  
Computable Structure Theory

scholarly journals Analytic computable structure theory and $L^p$ spaces

2019 ◽  
Vol 244 (3) ◽  
pp. 255-285 ◽  
Author(s):  
Joe Clanin ◽  
Timothy H. McNicholl ◽  
Don M. Stull
Keyword(s):  
Structure Theory ◽  
Computable Structure ◽  
Computable Structure Theory

scholarly journals Degrees of and lowness for isometric isomorphism

2020 ◽  
Vol 12 ◽  
Author(s):  
Johanna N.Y. Franklin ◽  
Timothy H. McNicholl
Keyword(s):  
Metric Spaces ◽  
Structure Theory ◽  
Computable Structure ◽  
Lebesgue Spaces ◽  
Isometric Isomorphism ◽  
Metric Structures ◽  
Computable Structure Theory ◽  
Computable Presentations

We contribute to the program of extending computable structure theory to the realm of metric structures by investigating lowness for isometric isomorphism of metric structures. We show that lowness for isomorphism coincides with lowness for isometric isomorphism and with lowness for isometry of metric spaces. We also examine certain restricted notions of lowness for isometric isomorphism with respect to fixed computable presentations, and, in this vein, we obtain classifications of the degrees that are low for isometric isomorphism with respect to the standard copies of certain Lebesgue spaces.

Computable embeddability for algebraic structures

2021 ◽  
Author(s):  
Nikolay Bazhenov ◽  
Manat Mustafa
Keyword(s):  
Computable Function ◽  
Structure Theory ◽  
Distributive Lattices ◽  
Computable Structure ◽  
Algebraic Structures ◽  
Standard Tool ◽  
Computable Structure Theory ◽  
Complete Preorder ◽  
Primitive Recursive ◽  
Natural Way

In computability theory, the standard tool to classify preorders is provided by the computable reducibility. If [Formula: see text] and [Formula: see text] are preorders with domain [Formula: see text], then [Formula: see text] is computably reducible to [Formula: see text] if and only if there is a computable function [Formula: see text] such that for all [Formula: see text] and [Formula: see text], [Formula: see text] [Formula: see text][Formula: see text]. We study the complexity of preorders which arise in a natural way in computable structure theory. We prove that the relation of computable isomorphic embeddability among computable torsion abelian groups is a [Formula: see text] complete preorder. A similar result is obtained for computable distributive lattices. We show that the relation of primitive recursive embeddability among punctual structures (in the setting of Kalimullin et al.) is a [Formula: see text] complete preorder.

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