scholarly journals Cototal enumeration degrees and their applications to effective mathematics

2018 ◽  
Vol 146 (8) ◽  
pp. 3541-3552
Author(s):  
Ethan McCarthy
Keyword(s):  
Enumeration Degrees

Then-rea enumeration degrees are dense

1992 ◽  
Vol 31 (4) ◽  
pp. 277-285 ◽  
Author(s):  
Alistair H. Lachlan ◽  
Richard A. Shore
Keyword(s):  
Enumeration Degrees

Relative Splittings of in the-Enumeration Degrees

2017 ◽  
pp. 44-56
Author(s):  
Marat M. Arslanov ◽  
Andrea Sorbi
Keyword(s):  
Enumeration Degrees

Embedding the diamond in the Σ2 enumeration degrees

1991 ◽  
Vol 56 (1) ◽  
pp. 195-212 ◽  
Author(s):  
Seema Ahmad
Keyword(s):  
Negative Answer ◽  
Diamond Lattice ◽  
Turing Degrees ◽  
Effective Algorithm ◽  
Finite Sets ◽  
Enumeration Degrees ◽  
Lower Case ◽  
One To One ◽  
Fixed Standard

Lachlan [5] has shown that it is not possible to embed the diamond lattice in the r.e. Turing degrees while preserving least and greatest elements; that is, there do not exist incomparable r.e. Turing degrees a and b such that a ∧ b = 0 and a ∨ b = 0′. Cooper [3] has compared the r.e. Turing degrees to the enumeration degrees below 0e′ and has asked if the two structures are elementarily equivalent.In this paper we show that such an embedding is possible in the Σ2enumeration degrees, which implies a negative answer to Cooper's question.Theorem. There are low enumeration degreesaandbsuch thata ∧ b = 0eanda ∨ b = 0e′.Lower case italic letters denote elements of ω while upper case italic letters denote subsets of ω. D, E and F are reserved for finite sets, and K for ′. If D = {x0, x1, …, xn} then the canonical index of D is , and the canonical index of is ∅. Dx denotes the set with canonical index x. {Wi}i∈ω is any fixed standard listing of the r.e. sets, and <·, ·> is any fixed recursive bijection from ω × ω to ω.Intuitively, A is enumeration reducible to B if there is an effective algorithm for producing an enumeration of A from any enumeration of B. There is a natural one-to-one correspondence between all such algorithms and the r.e. sets.

INITIAL SEGMENTS OF THE ENUMERATION DEGREES

2016 ◽  
Vol 81 (1) ◽  
pp. 316-325 ◽  
Author(s):  
HRISTO GANCHEV ◽  
ANDREA SORBI
Keyword(s):  
Local Structure ◽  
Initial Segment ◽  
Nonzero Element ◽  
Enumeration Degrees ◽  
Enumeration Degree ◽  
Image Position

AbstractUsing properties of${\cal K}$-pairs of sets, we show that every nonzero enumeration degreeabounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump asa. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration degree can be capped by degrees of any possible local jump (i.e., any jump that can be realized by enumeration degrees of the local structure); every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degreeacan be capped belowa.

A STRUCTURAL DICHOTOMY IN THE ENUMERATION DEGREES

2020 ◽  
pp. 1-18
Author(s):  
HRISTO A. GANCHEV ◽  
ISKANDER SH. KALIMULLIN ◽  
JOSEPH S. MILLER ◽  
MARIYA I. SOSKOVA
Keyword(s):  
Enumeration Degrees
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