The Complexity of Orbits of Computably Enumerable Sets

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, \E, such that the question of membership in this orbit is $\Sigma^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of \E is \wock +1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of \E; for all finite $\alpha \geq 9$, there is a properly $\Delta^0_\alpha$ orbit (from the proof). A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi

Extending Properly n-REA Sets

In [5] Soare and Stob prove that if $A$ is an r.e. set which isn't computable then there is a set of the form $A \oplus W^A_e$ which isn't of r.e. Turing degree. If we define a properly $n+1$-REA set to be an $n+1$-REA set which isn't Turing equivalent to any $n$-REA set (and identify 0-REA sets with the computable sets) this result shows that every properly 1-REA set can be extended to a properly 2-REA set. This result was extended in [1] by Cholak and Hinman who proved that every 2-REA set can be extended to a properly 3-REA set. This leads naturally to the hypothesis that every properly $n$-REA set can be extended to a properly $n+1$-REA set. In this paper, we show this hypothesis is false and that there is a properly $3$-REA set which can't be extended to a properly $4$-REA set

The Complexity of Local Stratification

The class of locally stratified logic programs is shown to be Π11-complete by the construction of a reducibility of the class of infinitely branching nondeterministic finite register machines.nondeterministic finite register machines

Iterated relative recursive enumerability

A result of Soare and Stob asserts that for any non-recursive r.e. set C , there exists a r.e.[ C ] set A such that A ⊕ C is not of r.e. degree. A set Y is called [of] m -REA ( m -REA[ C ] [degree] iff it is [Turing equivalent to] the result of applying m -many iterated ‘hops’ to the empty set (to C ), where a hop is any function of the form X → X ⊕ W e X . The cited result is the special case m =0, n =1 of our Theorem. For m =0,1, and any ( m +1)-REA set C , if C is not of m -REA degree, then for all n there exists a n -r.e.[ C ] set A such that A ⊕ C is not of ( m+n )-REA degree. We conjecture that this holds also for m ≥2.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46068/1/153_2005_Article_BF01278463.pd

The translation theorem

We state and prove the Translation Theorem. Then we apply the Translation Theorem to Soare's Extension Theorem, weakening slightly the hypothesis to yield a theorem we call the Modified Extension Theorem. We use this theorem to reprove several of the known results about orbits in the lattice of recursively enumerable sets. It is hoped that these proofs are easier to understand than the old proofs.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46067/1/153_2005_Article_BF01352931.pd