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 PERMUTATIONS OF THE INTEGERS INDUCE ONLY THE TRIVIAL AUTOMORPHISM OF THE TURING DEGREES

2018 ◽  
Vol 24 (2) ◽  
pp. 165-174
Author(s):  
BJØRN KJOS-HANSSEN
Keyword(s):  
Open Problem ◽  
Main Idea ◽  
Computability Theory ◽  
Turing Degrees ◽  
Nontrivial Automorphism

AbstractIs there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers. We show that a permutation of ω cannot induce any nontrivial automorphism of the Turing degrees of members of 2ω, and in fact any permutation that induces the trivial automorphism must be computable.A main idea of the proof is to consider the members of 2ω to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more.

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.

A HIERARCHY OF COMPUTABLY ENUMERABLE DEGREES

2018 ◽  
Vol 24 (1) ◽  
pp. 53-89 ◽  
Author(s):  
ROD DOWNEY ◽  
NOAM GREENBERG
Keyword(s):  
Computability Theory ◽  
Ordinal Notations ◽  
Image Position ◽  
Computably Enumerable

AbstractWe introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

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.

scholarly journals Definable Encodings in the Computably Enumerable Sets

2000 ◽  
Vol 6 (2) ◽  
pp. 185-196 ◽  
Author(s):  
Peter A. Cholak ◽  
Leo A. Harrington
Keyword(s):  
Computability Theory ◽  
Finite Sets ◽  
Warm Up ◽  
Computably Enumerable Sets ◽  
Disjoint Sets ◽  
New Form ◽  
Historical Remarks ◽  
Equivalent Class ◽  
The Ideal ◽  
Computably Enumerable

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is that the guts of the proofs of these theorems uses a new form of definable coding for the computably enumerable sets.We will work in the structure of the computably enumerable sets. The language is just inclusion, ⊆. This structure is called ε.All sets will be computably enumerable non-computable sets and all degrees will be computably enumerable and non-computable, unless otherwise noted. Our notation and definitions are standard and follow Soare [1987]; however we will warm up with some definitions and notation issues so the reader need not consult Soare [1987]. Some historical remarks follow in Section 2.1 and throughout Section 3.We will also consider the quotient structure ε modulo the ideal of finite sets, ε*. ε* is a definable quotient structure of ε since “Χ is finite” is definable in ε; “Χ is finite” iff all subsets of Χ are computable (it takes a little computability theory to show if Χ is infinite then Χ has an infinite non-computable subset). We use A* to denote the equivalent class of A under the ideal of finite sets.

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.

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