Computably Enumerable Sets in the Solovay and the Strong Weak Truth Table Degrees

2005 ◽  
pp. 8-17 ◽  
Author(s):  
George Barmpalias
Keyword(s):  
Truth Table ◽  
Computably Enumerable Sets ◽  
Weak Truth Table Degrees ◽  
Computably Enumerable

Contiguity and distributivity in the enumerable Turing degrees — Corrigendum

2002 ◽  
Vol 67 (4) ◽  
pp. 1579-1580
Author(s):  
Rodney G. Downey ◽  
Steffen Lempp
Keyword(s):  
Truth Table ◽  
Turing Degree ◽  
Turing Degrees ◽  
Computably Enumerable Set ◽  
Enumerable Degree ◽  
Computably Enumerable Sets ◽  
Image Position ◽  
Computably Enumerable Degree ◽  
Computably Enumerable

A computably enumerable Turing degree a is called contiguous iff it contains only a single computably enumerable weak truth table degree (Ladner and Sasso [2]). In [1], the authors proved that a nonzero computably enumerable degree a is contiguous iff it is locally distributive, that is, for all a1, a2, c with a1 ∪a2 = a and c ≤ a, there exist ci, ≤ ai with c1 ∪ c2 = c.To do this we supposed that W was a computably enumerable set and ∪ a computably set with a Turing functional Φ such that ΦW = U. Then we constructed computably enumerable sets A0, A1 and B together with functionals Γ0, Γ1, Γ, and Δ so thatand so as to satisfy all the requirements below.That is, we built a degree-theoretical splitting A0, A1 of W and a set B ≤TW such that if we cannot beat all possible degree-theoretical splittings V0, V1 of B then we were able to witness the fact that U ≤WW (via Λ).After the proof it was observed that the set U of the proof (page 1222, paragraph 4) needed only to be Δ20. It was then claimed that a consequence to the proof was that every contiguous computably enumerable degree was, in fact, strongly contiguous, in the sense that all (not necessarily computably enumerable) sets of the degree had the same weak truth table degree.

The theory of the recursively enumerable weak truth-table degrees is undecidable

1992 ◽  
Vol 57 (3) ◽  
pp. 864-874 ◽  
Author(s):  
Klaus Ambos-Spies ◽  
André Nies ◽  
Richard A. Shore
Keyword(s):  
Partial Order ◽  
Open Problem ◽  
Truth Table ◽  
Recursively Enumerable ◽  
Undecidable Theory ◽  
Weak Truth Table Degrees

AbstractWe show that the partial order of -sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

scholarly journals Definable properties of the computably enumerable sets

1998 ◽  
Vol 94 (1-3) ◽  
pp. 97-125 ◽  
Author(s):  
Leo Harrington ◽  
Robert I. Soare
Keyword(s):  
Computably Enumerable Sets ◽  
Computably Enumerable
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