scholarly journals A NOTE ON INITIAL SEGMENTS OF THE ENUMERATION DEGREES

2014 ◽  
Vol 79 (2) ◽  
pp. 633-643 ◽  
Author(s):  
THEODORE A. SLAMAN ◽  
ANDREA SORBI
Keyword(s):  
Partial Order ◽  
Principal Ideal ◽  
Total Degree ◽  
Enumeration Degrees ◽  
Enumeration Degree

AbstractWe show that no nontrivial principal ideal of the enumeration degrees is linearly ordered: in fact, below every nonzero enumeration degree one can embed every countable partial order. The result can be relativized above any total degree: if a,b are enumeration degrees, with a total, and a < b, then in the degree interval (a,b), one can embed every countable partial order.

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.

scholarly journals Degrees of unsolvability of continuous functions

2004 ◽  
Vol 69 (2) ◽  
pp. 555-584 ◽  
Author(s):  
Joseph S. Miller
Keyword(s):  
Continuous Functions ◽  
Turing Degrees ◽  
Total Degree ◽  
Classical Analysis ◽  
Degree Relative ◽  
Enumeration Degrees ◽  
Standard Notion ◽  
Scott Set ◽  
Degrees Of Unsolvability ◽  
Satisfactory Theory

Abstract.We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f ∈ [0,1] computes a non-computable subset of ℕ there is a non-total degree between Turing degrees a <Tb iff b is a PA degree relative to a; ⊆ 2ℕ is a Scott set iff it is the collection of f-computable subsets of ℕ for some f ∈ [0,1] of non-total degree; and there are computably incomparable f, g ∈ [0,1] which compute exactly the same subsets of ℕ. Proofs draw from classical analysis and constructive analysis as well as from computability theory.

Noncappable enumeration degrees below 0e′

1996 ◽  
Vol 61 (4) ◽  
pp. 1347-1363 ◽  
Author(s):  
S. Barry Cooper ◽  
Andrea Sorbi
Keyword(s):  
Enumeration Degrees ◽  
Enumeration Degree

AbstractWe prove that there exists a noncappable enumeration degree strictly below 0e′.

On the jump classes of noncuppable enumeration degrees

2011 ◽  
Vol 76 (1) ◽  
pp. 177-197 ◽  
Author(s):  
Charles M. Harris
Keyword(s):  
Turing Degrees ◽  
Enumeration Degrees ◽  
Enumeration Degree

AbstractWe prove that for every Σ20 enumeration degree b there exists a noncuppable Σ20 degree a > 0e such that and . This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding , that there exist Σ20 noncuppable enumeration degrees at every possible—i.e., above low1—level of the high/low jump hierarchy in the context of .

ON EXTENSIONS OF EMBEDDINGS INTO THE ENUMERATION DEGREES OF THE ${\Sigma_2^0}$-SETS

2005 ◽  
Vol 05 (02) ◽  
pp. 247-298 ◽  
Author(s):  
STEFFEN LEMPP ◽  
THEODORE A. SLAMAN ◽  
ANDREA SORBI
Keyword(s):  
Partial Order ◽  
Enumeration Degrees

We give an algorithm for deciding whether an embedding of a finite partial order [Formula: see text] into the enumeration degrees of the [Formula: see text]-sets can always be extended to an embedding of a finite partial order [Formula: see text].

Interpreting true arithmetic in the -enumeration degrees

2010 ◽  
Vol 75 (2) ◽  
pp. 522-550
Author(s):  
Thomas F. Kent
Keyword(s):  
Standard Model ◽  
Partial Order ◽  
Enumeration Degrees ◽  
First Order ◽  
Standard Models

AbstractWe show that there is a first order sentence φ(x: a, b, l) such that for every computable partial order and -degree u > 0e, there are -enumeration degrees a ≤ u, b, and l such that . Allowing to be a suitably defined standard model of arithmetic gives a parameterized interpretation of true arithmetic in the -enumeration degrees. Finally we show that there is a first order sentence that correctly identifies a subset of the standard models, which gives a parameterless interpretation of true arithmetic in the -enumeration degrees.

Structural properties and Σ20 enumeration degrees

2000 ◽  
Vol 65 (1) ◽  
pp. 285-292 ◽  
Author(s):  
André Nies ◽  
Andrea Sorbi
Keyword(s):  
Structural Properties ◽  
Enumeration Degrees ◽  
Enumeration Degree ◽  
Computably Enumerable

AbstractWe prove that each Σ20 set which is hypersimple relative to ∅′ is noncuppable in the structure of the Σ20 enumeration degrees. This gives a connection between properties of Σ20 sets under inclusion and and the Σ20 enumeration degrees. We also prove that some low non-computably enumerable enumeration degree contains no set which is simple relative to ∅′.

Bounding nonsplitting enumeration degrees

2007 ◽  
Vol 72 (4) ◽  
pp. 1405-1417 ◽  
Author(s):  
Thomas F. Kent ◽  
Andrea Sorbi
Keyword(s):  
Enumeration Degrees ◽  
Enumeration Degree ◽  
Degree Bounds

AbstractWe show that every nonzero enumeration degree bounds a nonsplitting nonzero enumeration degree.

Jumps of quasi-minimal enumeration degrees

1985 ◽  
Vol 50 (3) ◽  
pp. 839-848 ◽  
Author(s):  
Kevin McEvoy
Keyword(s):  
Turing Degree ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Enumeration Degree ◽  
Enumeration Reducibility ◽  
Degree Structure ◽  
Usual Topology ◽  
Definition Of ◽  
Second Category ◽  
Degrees Of Unsolvability

Enumeration reducibility is a reducibility between sets of natural numbers defined as follows: A is enumeration reducible to B if there is some effective operation on enumerations which when given any enumeration of B will produce an enumeration of A. One reason for interest in this reducibility is that it presents us with a natural reducibility between partial functions whose degree structure can be seen to extend the structure of the Turing degrees of unsolvability. In [7] Friedberg and Rogers gave a precise definition of enumeration reducibility, and in [12] Rogers presented a theorem of Medvedev [10] on the existence of what Case [1] was to call quasi-minimal degrees. Myhill [11] also defined this reducibility and proved that the class of quasi-minimal degrees is of second category in the usual topology. As Gutteridge [8] has shown that there are no minimal enumeration degrees (see Cooper [3]), the quasi-minimal degrees are very much of interest in the study of the structure of the enumeration degrees. In this paper we define a jump operator on the enumeration degrees which was introduced by Cooper [4], and show that every complete enumeration degree is the jump of a quasi-minimal degree. We also extend the notion of a high Turing degree to the enumeration degrees and construct a “high” quasi-minimal enumeration degree—a result which contrasts with Cooper's result in [2] that a high Turing degree cannot be minimal. Finally, we use the Sacks' Jump Theorem to characterise the jumps of the co-r.e. enumeration degrees.

Deriving the Partial Order of Documents to Extend Clustering Applications

2019 ◽  
Vol 7 (1) ◽  
pp. 424-430
Author(s):  
A. George Louis Raja ◽  
F. Sagayaraj Francis ◽  
P. Sugumar
Keyword(s):  
Partial Order
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