Enumerable Degrees Research Articles

Non-cuppable enumeration degrees via finite injury

We exhibit finite injury constructions of high Σ02 enumeration degree incomparable with all intermediate Δ02 enumeration degrees, as also of both an upwards properly Σ02 high and low non-cuppable Σ02 enumeration degree.We also outline how to apply the same methods to prove that, for every Σ02 enumeration degree b there exists non-cuppable degree such that b' ≤a' and a ≤ b, thus showing that there exist non-cuppable Σ02 enumeration degrees at every possible level of the high/low jump hierarchy.

A semilattice generated by superlow computably enumerable degrees

We prove that a partially ordered set of all computably enumerable (c. e.) degrees that are the least upper bounds of two superlow c. e. degrees is an upper semilattice not elementary equivalent to the semilattice of all c. e. degrees.

Splitting and nonsplitting in the [formula omitted] enumeration degrees

This paper continues the project, initiated in (Arslanov, Cooper and Kalimullin 2003) [3], of describing general conditions under which relative splittings are derivable in the local structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting. The main results below include a proof that any high total e -degree below 0 e ′ is splittable over any low e -degree below it, a non-cupping result in the high enumeration degrees which occurs at a low level of the Ershov hierarchy, and a 0̸ ‴ -priority construction of a Π 1 0 e -degree unsplittable over a 3-c.e. e -degree below it.

Decomposability of low 2-computably enumerable degrees and turing jumps in the Ershov hierarchy

In this paper we prove the following theorem: For every notation of a constructive ordinal there exists a low 2-computably enumerable degree that is not splittable into two lower 2-computably enumerable degrees whose jumps belong to the corresponding Δ-level of the Ershov hierarchy.

Embedding countable partial orderings in the enumeration degrees and the -enumeration degrees

In this article, we examine the complexity of three degree structures in terms of the quantity of partial orderings embeddable in them. We prove that every countable partial ordering can be embedded in any interval of enumeration degrees with upper bound, a degree which contains a member with a good approximation. Next we prove that a similar result is true of the ω-enumeration degrees. Finally we consider the structure of the ∑02ω-enumeration degree modulo iterated jump and show that there as well one can embed densely every countable partial ordering.

Embedding distributive lattices in the Formula enumeration degrees

We prove that every countable distributive lattice is embeddable in the Σ02 enumeration degrees via a 0—1 preserving monomorphism. Moreover, we prove that every countable distributive lattice is embeddable below arbitrary Δ02 degree via a 0 preserving monomorphism.

Diamond embeddings into the enumeration degrees

We show that the diamond lattice can be embedded into the Σ02 enumeration degrees preserving 0 and 1, with atoms one high and Π01, and the other one low.

A superhigh diamond in the c.e. tt-degrees

The notion of superhigh computably enumerable (c.e.) degrees was first introduced by (Mohrherr in Z Math Logik Grundlag Math 32: 5---12, 1986) where she proved the existence of incomplete superhigh c.e. degrees, and high, but not superhigh, c.e. degrees. Recent research shows that the notion of superhighness is closely related to algorithmic randomness and effective measure theory. Jockusch and Mohrherr proved in (Proc Amer Math Soc 94:123---128, 1985) that the diamond lattice can be embedded into the c.e. tt-degrees preserving 0 and 1 and that the two atoms can be low. In this paper, we prove that the two atoms in such embeddings can also be superhigh.

Presentations of the successor relation of computable linear ordering

We prove that a nontrivial degree spectrum of the successor relation of either strongly η-like or non-η-like computable linear orderings is closed upwards in the class of all computably enumerable degrees. We also show that the degree spectrum contains 0 if and only if either it is trivial or it contains all computably enumerable degrees.

Interpreting true arithmetic in the -enumeration degrees

AbstractWe show that there is a first order sentence φ(x: a, b, l) such that for every computable partial order and -degree u > 0e, there are -enumeration degrees a ≤ u, b, and l such that . Allowing to be a suitably defined standard model of arithmetic gives a parameterized interpretation of true arithmetic in the -enumeration degrees. Finally we show that there is a first order sentence that correctly identifies a subset of the standard models, which gives a parameterless interpretation of true arithmetic in the -enumeration degrees.

Isolated 2-computably enumerable Q-degrees

In this work we prove that for every pair of computably enumerable degrees a<Q b there exists a properly 2-computably enumerable degree d such that a <Q d <Q b, a isolates d from below, and b isolates d from above. Two corollaries follow from this result. First, there exists a 2-computably enumerable degree which is Q-incomparable with any nontrivial (different from 0 and 0′) computably enumerable degree. Second, every nontrivial computably enumerable degree isolates some 2-computably enumerable degree from below and some 2-computably enumerable degree from above.

Strong noncuppability in low computably enumerable degrees

We prove the existence of noncomputable low computably enumerable degrees b < a such that b is strongly noncuppable to a in the class R.

The limitations of cupping in the local structure of the enumeration degrees

We prove that a sequence of sets containing representatives of cupping partners for every nonzero \({\Delta^0_2}\) enumeration degree cannot have a \({\Delta^0_2}\) enumeration. We also prove that no subclass of the \({\Sigma^0_2}\) enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to \({\mathbf{0}_e'}\) by a single incomplete \({\Sigma^0_2}\) enumeration degree.

Joining to high degrees via noncuppables

Cholak, Groszek and Slaman proved in J Symb Log 66:881–901, 2001 that there is a nonzero computably enumerable (c.e.) degree cupping every low c.e. degree to a low c.e. degree. In the same paper, they pointed out that every nonzero c.e. degree can cup a low2 c.e. degree to a nonlow2 degree. In Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) improved the latter result by showing that every nonzero c.e. degree c is cuppable to a high c.e. degree by a low2 c.e. degree b. It is natural to ask in which subclass of low2 c.e. degrees can b in Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) be located. Wu proved in Math Log Quart 50:189–201, 2004 that b can be cappable. We prove in this paper that b in Jockusch, Li and Yang’s result can be noncuppable, improving both Jockusch, Li and Yang, and Wu’s results.

Splitting in 2-computably enumerable degrees with avoiding cones

In this paper we show that for any pair of properly 2-c. e. degrees 0 < d < a such that there are no c. e. degrees between d and a, the degree a is splittable in the class of 2-c. e. degrees avoiding the upper cone of d. We also study the possibility to characterize such an isolation in terms of splitting.

The structure of the [formula omitted]-degrees contained within a single [formula omitted]-degree

For any enumeration degree a let D a s be the set of s -degrees contained in a . We answer an open question of Watson by showing that if a is a nontrivial Σ 2 0 -enumeration degree, then D a s has no least element. We also show that every countable partial order embeds into D a s . Finally, we construct Σ 2 0 -sets A and B such that B ≤ e A but for every X ≡ e B , X ≰ s A .

The jump operator on the [formula omitted]-enumeration degrees

The jump operator on the ω -enumeration degrees was introduced in [I.N. Soskov, The ω -enumeration degrees, J. Logic Computat. 17 (2007) 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure D ω ′ of the ω -enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of D ω ′ and of the enumeration degrees are isomorphic. In the second part of the paper we study the jumps of the ω -enumeration degrees below 0 ω ′ . We define the ideal of the almost zero degrees and obtain a natural characterization of the class H of the ω -enumeration degrees below 0 ω ′ which are high n for some n and of the class L of the ω -enumeration degrees below 0 ω ′ which are low n for some n .

Enumeration Degrees and Enumerability of Familes

We study the enumerability of families relative to the enumeration degrees. It is shown that if a family of finite sets is e-reducible to every non-zero e-degree, then the family is computably enumerable (c.e). On the another hand, we will find a non-c.e. family which is e-reducible to all non-zero e-degree. This allows to construct a model, whose (extended) degree spectrum coincides with the non-zero e-degrees.

A non-splitting theorem in the enumeration degrees

We complete a study of the splitting/non-splitting properties of the enumeration degrees below 0 e ′ by proving an analog of Harrington’s non-splitting theorem for the Σ 2 0 enumeration degrees. We show how non-splitting techniques known from the study of the c.e. Turing degrees can be adapted to the enumeration degrees.

Prime models of computably enumerable degree

AbstractWe examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b′ = 0′), This extends Csima's result that every CAD theory has a low prime model. We also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d &lt; c and c not low2 (c″ &gt; 0″), then for any CAD theory T, there exists a c.e. degree b with d &lt; b &lt; c such that T has a prime model of degree b, where b can be chosen so that b′ is any degree c.e. in c with d′ ≤ b′. As a corollary, we show that for any degree c with 0 &lt; c &lt; 0′, every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models.