Maximality and collapse in the hierarchy of α-c.a. degrees

Computability ◽  
2021 ◽  
pp. 1-34
Author(s):  
Katherine Arthur ◽  
Rod Downey ◽  
Noam Greenberg
Keyword(s):  
Turing Degrees ◽  
Princeton University ◽  
Related Notion ◽  
Computably Enumerable

In (A Hierarchy of Turing Degrees: A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability (2020), Annals of Mathematics Studies, Princeton University Press), Downey and Greenberg define a transfinite hierarchy of low 2 c.e. degrees – the totally α-c.a. degrees, for appropriately small ordinals α. This new hierarchy is of particular interest because it has already given rise to several natural definability results, and provides a new definable antichain in the c.e. degrees. Several levels of this hierarchy contain maximal degrees. We discuss how maximality interacts with upper cones, and the related notion of hierarchy collapse in upper cones. For example, we show that there is a totally ω-c.a. degree above which there is no maximal totally ω-c.a. degree.

Nonlowness is independent from fickleness

Computability ◽  
2021 ◽  
pp. 1-18
Author(s):  
Liling Ko
Keyword(s):  
Lattice Structure ◽  
Turing Degrees ◽  
Direct Construction ◽  
Computable Set ◽  
Princeton University ◽  
Computably Enumerable

It was recently shown that the computably enumerable (c.e.) degrees that embed the critical triple and the M 5 lattice structure are exactly those that are sufficiently fickle. Therefore the embeddability strength of a c.e. degree has much to do with the degree’s fickleness. Nonlowness is another common measure of degree strength, with nonlow degrees expected to compute more degrees than low ones. We ask if nonlowness and fickleness are independent measures of strength. Downey and Greenberg (A Hierarchy of Turing Degrees: A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability (AMS-206) (2020) Princeton University Press) claimed this to be true without proof, so we present one here. We prove the claim by building low and nonlow c.e. sets with arbitrary fickle degrees. Our construction is uniform so the degrees built turn out to be uniformly fickle. We base our proof on our direct construction of a nonlow array computable set. Such sets were always known to exist, but also never constructed directly in any publication we know.

A hierarchy for the plus cupping Turing degrees

2003 ◽  
Vol 68 (3) ◽  
pp. 972-988 ◽  
Author(s):  
Yong Wang ◽  
Angsheng Li
Keyword(s):  
Turing Degrees ◽  
Computably Enumerable

AbstractWe say that a computably enumerable (c. e.) degree a is plus-cupping, if for every c.e. degree x with 0 < x ≤ a, there is a c. e. degree y ≠ 0′ such that x ∨ y = 0′. We say that a is n-plus-cupping, if for every c. e. degree x, if 0 < x ≤ a, then there is a lown c. e. degree I such that x ∨ I = 0′. Let PC and PCn be the set of all plus-cupping, and n-plus-cupping c. e. degrees respectively. Then PC1 ⊆ PC2 ⊆ PC3 = PC. In this paper we show that PC1 ⊂ PC2, so giving a nontrivial hierarchy for the plus cupping degrees. The theorem also extends the result of Li, Wu and Zhang [14] showing that LC1 ⊂ LC2, as well as extending the Harrington plus-cupping theorem [8].

scholarly journals Computably enumerable Turing degrees and the meet property

2015 ◽  
Vol 144 (4) ◽  
pp. 1735-1744 ◽  
Author(s):  
Benedict Durrant ◽  
Andy Lewis-Pye ◽  
Keng Meng Ng ◽  
James Riley
Keyword(s):  
Turing Degrees ◽  
Computably Enumerable

Embedding finite lattices into the ideals of computably enumerable turing degrees

2001 ◽  
Vol 66 (4) ◽  
pp. 1791-1802
Author(s):  
William C. Calhoun ◽  
Manuel Lerman
Keyword(s):  
Turing Degrees ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Lattices ◽  
Necessary And Sufficient ◽  
Lattice Of Ideals ◽  
Computably Enumerable

Abstract.We show that the lattice L20 is not embeddable into the lattice of ideals of computably enumerable Turing degrees (ℐ), We define a structure called a pseudolattice that generalizes the notion of a lattice, and show that there is a Π2 necessary and sufficient condition for embedding a finite pseudolattice into ℐ.

scholarly journals Randomness, lowness and degrees

2008 ◽  
Vol 73 (2) ◽  
pp. 559-577 ◽  
Author(s):  
George Barmpalias ◽  
Andrew E. M. Lewis ◽  
Mariya Soskova
Keyword(s):  
Random Number ◽  
Turing Degrees ◽  
Computably Enumerable

AbstractWe say that A ≤LRB if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever ∝ is not GL2 the LR degree of ∝ bounds degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees.

Decomposition and infima in the computably enumerable degrees

2003 ◽  
Vol 68 (2) ◽  
pp. 551-579 ◽  
Author(s):  
Rodney G. Downey ◽  
Geoffrey L. Laforte ◽  
Richard A. Shore
Keyword(s):  
Turing Degrees ◽  
Computably Enumerable

AbstractGiven two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c = (a ∪ c) ∩ (b ∪ c), a ∪ c ∣ b ∪ c, and c < a ∪ b.

scholarly journals EXACT PAIRS FOR THE IDEAL OF THEK-TRIVIAL SEQUENCES IN THE TURING DEGREES

2014 ◽  
Vol 79 (3) ◽  
pp. 676-692 ◽  
Author(s):  
GEORGE BARMPALIAS ◽  
ROD G. DOWNEY
Keyword(s):  
Negative Answer ◽  
Turing Degrees ◽  
Halting Problem ◽  
The Ideal ◽  
Computably Enumerable

AbstractTheK-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [22, Question 4.2] and later in [25, Problem 5.5.8].We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists aK-trivial degreedsuch that for all degreesa, bwhich are notK-trivial anda > d, b > dthere exists a degreevwhich is notK-trivial anda > v, b > v. This work sheds light to the question of the definability of theK-trivial degrees in the c.e. degrees.

scholarly journals Hierarchy of Computably Enumerable Degrees II

10.53733/133 ◽  
2021 ◽  
Vol 52 ◽  
pp. 175-231
Author(s):  
Rod Downey ◽  
Noam Greenberg ◽  
Ellen Hammatt
Keyword(s):  
Computability Theory ◽  
Turing Degrees ◽  
Computably Enumerable

A transfinite hierarchy of Turing degrees of c.e.\ sets has been used to calibrate the dynamics of families of constructions in computability theory, and yields natural definability results. We review the main results of the area, and discuss splittings of c.e.\ degrees, and finding maximal degrees in upper cones.

scholarly journals Cone Avoidance of Some Turing Degrees

2017 ◽  
Vol 9 (4) ◽  
pp. 69
Author(s):  
Patrizio Cintioli
Keyword(s):  
Turing Degrees ◽  
Computably Enumerable

We study the cone avoidance and the upper cone avoidance of two substructures of m-introimmune Turing degrees. We show that the substructure of the m-introimmune Turing degrees satisfies the cone avoidance property, and that the substructure of the computably enumerable m-introimmune Turing degrees satisfies the upper cone avoidance property.

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