Contiguity and Distributivity in the Enumerable Turing Degrees

Contiguity and Distributivity in The Enumerable Turing Degrees - Corrigendum

Embedding of N5 and the contiguous degrees

Downey and Lempp 8] have shown that the contiguous computably enumerable (c.e.) degrees, i.e. the c.e. Turing degrees containing only one c.e. weak truth-table degree, can be characterized by a local distributivity property. Here we extend their result by showing that a c.e. degree a is noncontiguous if and only if there is an embedding of the nonmodular 5-element lattice N5 into the c.e. degre...

Universal computably enumerable sets and initial segment prefix-free complexity

We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set A with non-trivial prefixfree initial segment complexity, there exists a Turing complete computably enumerable set B with complexity strictly less than the complexity of A. On the other hand it is known that s...

Extensions of Embeddings below Computably Enumerable Degrees

Toward establishing the decidability of the two quantifier theory of the ∆ 2 Turing degrees with join, we study extensions of embeddings of upper-semi-lattices into the initial segments of Turing degrees determined by computably enumerable sets, in particular the degree of the halting set 0. We obtain a good deal of sufficient and necessary conditions.

The Π3-theory of the Computably Enumerable Turing Degrees Is Undecidable

We show the undecidability of the Π3-theory of the partial order of computably enumerable Turing degrees.