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.

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.

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.

scholarly journals The $\Pi _3$-theory of the computably enumerable Turing degrees is undecidable

1998 ◽  
Vol 350 (7) ◽  
pp. 2719-2736 ◽  
Author(s):  
Steffen Lempp ◽  
André Nies ◽  
Theodore A. Slaman
Keyword(s):  
Turing Degrees ◽  
Computably Enumerable

scholarly journals On the existence of a strong minimal pair

2015 ◽  
Vol 15 (01) ◽  
pp. 1550003 ◽  
Author(s):  
George Barmpalias ◽  
Mingzhong Cai ◽  
Steffen Lempp ◽  
Theodore A. Slaman
Keyword(s):  
Minimal Pair ◽  
Turing Degrees ◽  
Computably Enumerable

We show that there is a strong minimal pair in the computably enumerable (c.e.) Turing degrees, i.e. a pair of nonzero c.e. degrees a and b such that a∩b = 0 and for any nonzero c.e. degree x ≤ a, b ∪ x ≥ a.

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