Corpus ID: 237532177

Structural Highness Notions

@inproceedings{Calvert2021StructuralHN,
  title={Structural Highness Notions},
  author={Wesley Calvert and Johanna N. Y. Franklin and Daniel Turetsky},
  year={2021}
}
We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of backand-forth relations), jump hierarchies, and computing descending sequences in linear orders. 

References

SHOWING 1-10 OF 31 REFERENCES
Degrees that Are Low for Isomorphism
TLDR
It is shown that while there is no clear-cut relationship between this property and other properties related to computational weakness, the low-forisomorphism degrees contain all Cohen 2-generics and are disjoint from the Martin-Lof randoms. Expand
Isomorphism relations on computable structures
TLDR
The notion of FF-reducibility introduced in [9] is used to show completeness of the isomorphism relation on many familiar classes in the context of all equivalence relations on hyperarithmetical subsets of ω. Expand
Lowness for isomorphism, countable ideals, and computable traceability
TLDR
It is shown that within the hyperimmune-free degrees, lowness for isomorphism is entirely independent of computable traceability. Expand
1 1 relations and paths through O
When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [33], [34], we give syntactical conditionsExpand
Degrees of and lowness for isometric isomorphism
TLDR
It is shown that lowness for isomorphism coincides with lownes for isometry of metric spaces, and certain restricted notions of lownedness for isometric is morphism with respect to fixed computable presentations are examined. Expand
Taking the path computably traveled
TLDR
It is proved that lowness for paths in Baire space and lownedess forpaths in Cantor space are equivalent and, furthermore, that these notions are also equivalent to lownes for isomorphism. Expand
LOWNESS AND HIGHNESS PROPERTIES FOR RANDOMNESS NOTIONS
Given two relativizable classes R and P and a real A, we say that A is in Low(R,P) if R P A and that A is in High(R,P) if R A P. In this paper, we survey the current results on highness and lownessExpand
Computable structures and the hyperarithmetical hierarchy
Preface. Computability. The arithmetical hierarchy. Languages and structures. Ordinals. The hyperarithmetical hierarchy. Infinitary formulas. Computable infinitary formulas. The Barwise-KreiselExpand
Computable trees of Scott rank ω1CK, and computable approximation
TLDR
This work shows that there are computable trees of Scott rank ω1CK, and introduces a notion of “rank homogeneity”, in which orbits of tuples can be understood relatively easily in rank homogeneous trees. Expand
A degree-theoretic definition of the ramified analytical hierarchy
Let D be the set of all (Turing) degrees, < the usual partial ordering of D and j the (Turing) jump operator on D. The following relations are shown to be first-order definable in the structure D =Expand
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