Geoffrey LaForte

Randomness and reducibility

We study reducibilities that act as measures of relative randomness on reals, concentrating particularly on their behavior on the computably enumerable reals. One such reducibility, called domination or Solovay reducibility, has already proved to be a powerful tool in the study of randomness of effectively presented reals. Motivated by certain shortcomings of Solovay reducibility, we introduce two new measures of relative randomness and investigate their properties and the relationships between them and Solovay reducibility.

On Schnorr and computable randomness, martingales, and machines

We examine the randomness and triviality of reals using notions arising from martingales and prefix-free machines. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Presentations of computably enumerable reals

We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefix-free set of binary strings.

Why Go¨del's theorem cannot refute computationalism

Abstract Godels theorem is consistent with the computationalist hypothesis. Roger Penrose, however, claims to prove that Godels theorem implies that human thought cannot be mechanized. We review his arguments and show how they are flawed. Penroses arguments depend crucially on ambiguities between precise and imprecise senses of key terms. We show that these ambiguities cause the Godel/Turing diagonalization argument to lead from apparently intuitive claims about human abilities to paradoxical or highly idiosyncratic conclusions, and conclude that any similar argument will also fail in the same ways.

Computably enumerable sets and quasi-reducibility

We consider the computably enumerable sets under the relation of Q-reducibility. We first give several results comparing the upper semilattice of c.e. Q-degrees, 〈RQ, ⩽Q〉, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of 〈RQ, ⩽Q〉 is undecidable.

The Isolated D. R. E. Degrees are Dense in the R. E. Degrees

In the present paper we prove that the isolated differences of r. e. degrees are dense in the r. e. degrees. Mathematics Subject Classification: 03D25.

Completing pseudojump operators

Abstract We investigate operators which take a set X to a set relatively computably enumerable in and above X by studying which such sets X can be so mapped into the Turing degree of K . We introduce notions of nontriviality for such operators, and use these to study which additional properties can be required of sets which can be completed to the jump by given operators of this kind.

Undecidability of the structure of the Solovay degrees of c.e. reals

We show that the elementary theory of the structure of the Solovay degrees of computably enumerable reals is undecidable.

Isolation in the CEA hierarchy

Abstract.Examining various kinds of isolation phenomena in the Turing degrees, I show that there are, for every n>0, (n+1)-c.e. sets isolated in the n-CEA degrees by n-c.e. sets below them. For n≥1 such phenomena arise below any computably enumerable degree, and conjecture that this result holds densely in the c.e. degrees as well. Surprisingly, such isolation pairs also exist where the top set has high degree and the isolating set is low, although the complete situation for jump classes remains unknown.

Pseudojumps and Π10 Classes

For a pseudojump VX and a Π10 class P, we consider properties of the set {VX:X ∈ P}.We show that if P is Medvedev complete or if P has positive measure, and O′ ≤T C, then there exists X ∈ P with VX ≡T C. We examine the consequences when VX is Turing incomparable with VY for X ≠ Y in P and when WeX=WeY for all X, Y ∈ P. Finally, we give a characterization of the jump in terms of Π10 classes.