André Nies

Automatic structures: richness and limitations

This paper studies the existence of automatic presentations for various algebraic structures. The automatic Boolean algebras are characterised, and it is proven that the free Abelian group of infinite rank and many Fraisse limits do not have automatic presentations. In particular, the countably infinite random graph and the universal partial order do not have automatic presentations. Furthermore, no infinite integral domain is automatic. The second topic of the paper is the isomorphism problem. We prove that the complexity of the isomorphism problem for the class of all automatic structures is /spl Sigma//sub 1//sup 1/-complete.

Interpretability and Definability in the Recursively Enumerable Degrees

We investigate definability in R, the recursively enumerable Turing degrees, using codings of standard models of arithmetic (SMA’s) as a tool. First we show that an SMA can be interpreted in R without parameters. Building on this, we prove that the recursively enumerable T–degrees satisfy a weak form of the biinterpretability conjecture which implies that all jump classes Lown and Highn−1 (n ≥ 2) are definable in R without parameters and, more generally, that all relations on R that are definable in arithmetic and invariant under the double jump are actually definable in R. This partially answers Soare’s Question 3.7 [35, XVI].

Computable Models of Theories with Few Models

In this paper we investigate computable models of א1-categorical theories and Ehrenfeucht theories. For instance, we give an example of an א1categorical but not א0-categorical theory T such that all the countable models of T except its prime model have computable presentations. We also show that there exists an א1-categorical but not א0-categorical theory T such that all the countable models of T except the saturated model, have computable presentations.

RELATIVIZING CHAITIN'S HALTING PROBABILITY

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that and are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness.

Lowness for the Class of Schnorr Random Reals

We answer a question of Ambos-Spies and Kucera in the affirmative. They asked whether, when a real is low for Schnorr randomness, it is already low for Schnorr tests.

Randomness, Computability, and Density

We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1975] and studied by Calude, Hertling, Khoussainov, and Wang [Theoret. Comput. Sci., 255 (2001), pp. 125--149], Calude [Theoret. Comput. Sci., 271 (2002), pp. 3--14], Kucera and Slaman [SIAM J. Comput., 31 (2002), pp. 199--211], and Downey, Hirschfeldt, and LaForte [Mathematical Foundations of Computer Science 2001, Springer-Verlag, Berlin, 2001, pp. 316--327], among others. This measure is called domination or Solovay reducibility and is defined by saying that

Randomness and differentiability

\alpha

Automatic Structures: Richness and Limitations

dominates

Demuth randomness and computational complexity

\beta

From Automatic Structures to Borel Structures

if there are a constant c and a partial computable function