A note on the differences of computably enumerable reals

We show that given any non-computable left-c.e. real α there exists a left-c.e. real β such that α≠β+γ for all left-c.e. reals and all right-c.e. reals γ. The proof is non-uniform, the dichotomy being whether the given real α is Martin-Loef random or not. It follows that given any universal machine U, there is another universal machine V such that the halting probability of U is not a translation of the halting probability of V by a left-c.e. real. We do not know if there is a uniform proof of this fact

Arithmetic complexity via effective names for random sequences

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals

Kolmogorov complexity and computably enumerable sets

We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefix-free complexity. A computably enumerable set $A$ is $K$-trivial if and only if the family of sets with complexity bounded by the complexity of $A$ is uniformly computable from the halting problem

Computability Theory

Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science

New definitions in the theory of Type 1 computable topological spaces

Motivated by the two remarks, that the study of computability based on the use of numberings -- Type 1 computability -- does not have to be restricted to countable sets equipped with onto numberings, and that computable topologies have been in part developed with the implicit hypothesis that the considered spaces should be computably separable, we propose new definitions for Type 1 computable topological spaces. We define computable topological spaces without making reference to a basis. Following Spreen, we show that the use of a formal inclusion relation should be systematized, and that it cannot be avoided if we want computable topological spaces to generalize computable metric spaces. We also compare different notions of effective bases. The first one, introduced by Nogina, is based on an effective version of the statement "a set $O$ is open if for any point in $O$, there is a basic set containing that point and contained in $O$''. The second one, associated to Lacombe, is based on an effective version of "a set $O$ is open if it can be written as a union of basic open sets''. We show that neither of these notions of basis is completely satisfactory: Nogina bases do not permit to define computable topologies unless we restrict our attention to countable sets, and the conditions associated to Lacombe bases are too restrictive, and they do not apply to metric spaces unless we add effective separability hypotheses. We define a new notion of basis, based on an effective version of the Nogina statement, but adding to it several classically empty conditions, expressed in terms of formal inclusion relations. Finally, we obtain a new version of the theorem of Moschovakis which states that the Lacombe and Nogina approaches coincide on countable recursive Polish spaces, but which applies to sets equipped with non-onto numberings, and with effective separability as a sole hypothesis.Comment: 50 pages, 2 figure