We call an $\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha $ with $\alpha - a_n < 2^{-n}$ for infinitely many ${n \in \mathbb {N}}$. We also call a set $A\subseteq \mathbb {N}$ regainingly approximable if it is c.e. and the strongly left-computable number $2^{-A}$ is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. K-trivial, we construct such an $\alpha $ such that ${K(\alpha \restriction n)>n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $\mathbf {0}'$-computable low$_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $n\geq 2$, there exists a Polish space $X_n$ such that exactly the high$_{n}$-degrees are required to present the homeomorphism type of $X_n$. Along the way we investigate the computable aspects of Čech homology groups. We also show that no compact Polish space has a least presentation with respect to Turing reducibility.

We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$-c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $-c.a. as shown in [9]. We also show that if cone avoidance is added then there is no level below $\varepsilon _0$ which can be used to characterize the complexity of $A_1$ and $A_2$.

We study for each computably bounded $\Pi ^0_1$ class P the set of degrees of c.e. paths in P. We show, amongst other results, that for every c.e. degree a there is a perfect $\Pi ^0_1$ class where all c.e. members have degree a. We also show that every $\Pi ^0_1$ set of c.e. indices is realized in some perfect $\Pi ^0_1$ class, and classify the sets of c.e. degrees which can be realized in some $\Pi ^0_1$ class as exactly those with a computable representation.

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees $\langle \mathcal {R}_{\mathrm {T}},\leq _{\mathrm {T}}\rangle $ , we do not in general know how to characterize the degrees $\mathbf {d}\in \mathcal {R}_{\mathrm {T}}$ below which L can be bounded. The important characterizations known are of the $L_7$ and $M_3$ lattices, where the lattices are bounded below $\mathbf {d}$ if and only if $\mathbf {d}$ contains sets of “fickleness” $>\omega $ and $\geq \omega ^\omega $ respectively. We work towards finding a lattice that characterizes the levels above $\omega ^2$ , the first non-trivial level after $\omega $ . We introduced a lattice-theoretic property called “ $3$ -directness” to describe lattices that are no “wider” or “taller” than $L_7$ and $M_3$ . We exhaust the 3-direct lattices L, but they turn out to also characterize the $>\omega $ or $\geq \omega ^\omega $ levels, if L is not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some 3-direct lattices, but the removals did not change the levels characterized. This leads us to conjecture that a USL characterizes the same r.e. degrees as the lattice on which the USL is based. We discovered three 3-direct lattices besides $M_3$ that also characterize the $\geq \omega ^\omega $ -levels. Our search for a $>\omega ^2$ -candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four $\geq \omega ^\omega $ -lattices as sublattices.

Abstract prepared by Liling Ko.

E-mail: [email protected]

URL: http://sites.nd.edu/liling-ko/

We solve a longstanding question of Soare by showing that if ${\mathbf d}$ is a non-low $_2$ computably enumerable degree then ${\mathbf d}$ contains a c.e. set with no r-maximal c.e. superset.

We prove a number of elementary facts about computability in partial combinatory algebras (pca’s). We disprove a suggestion made by Kreisel about using Friedberg numberings to construct extensional pca’s. We then discuss separability and elements without total extensions. We relate this to Ershov’s notion of precompleteness, and we show that precomplete numberings are not 1–1 in general.

We show that there is a cuppable c.e. degree, all of whose cupping partners are high. In particular, not all cuppable degrees are ${\operatorname {\mathrm {low}}}_3$-cuppable, or indeed ${\operatorname {\mathrm {low}}}_n$ cuppable for any n, refuting a conjecture by Li. On the other hand, we show that one cannot improve highness to superhighness. We also show that the ${\operatorname {\mathrm {low}}}_2$-cuppable degrees coincide with the array computable-cuppable degrees, giving a full understanding of the latter class.

We examine how degrees of computably enumerable equivalence relations (ceers) under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of $[0,\omega ]$. In fact, for any $n \in [0,\omega ]$, there is a degree d and weakly precomplete ceers ${E_1}, \ldots ,{E_n}$ in d so that any ceer R in d is isomorphic to ${E_i} \oplus D$ for some $i \le n$ and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes.

We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is ${\rm{\Pi }}_3^0$-hard, though the definition is ${\rm{\Sigma }}_4^0$. We close the gap by showing that the index set is ${\rm{\Sigma }}_4^0$-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide.

Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if $\left( {{E_i}} \right)$ is a uniform c.e. sequence of non universal ceers, then $\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$ has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4].

Lastly, we show that there exists an infinite antichain of weakly precomplete ceers.

A set $G \subseteq \omega$ is n-generic for a positive integer n if and only if every ${\rm{\Sigma }}_n^0$ formula of G is decided by a finite initial segment of G in the sense of Cohen forcing. It is shown here that every n-generic set G is properly ${\rm{\Sigma }}_n^0$ in some G-recursive X. As a corollary, we also prove that for every $n > 1$ and every n-generic set G there exists a G-recursive X which is generalized ${\rm{lo}}{{\rm{w}}_n}$ but not generalized ${\rm{lo}}{{\rm{w}}_{n - 1}}$. Thus we confirm two conjectures of Jockusch [4].

We consider two generalizations of the recursion theorem, namely Visser’s ADN theorem and Arslanov’s completeness criterion, and we prove a joint generalization of these theorems.

BON+ is an applicative theory and closely related to the first order parts of the standard systems of explicit mathematics. As such it is also a natural framework for abstract computations. In this article we analyze this aspect of BON+ more closely. First a point is made for introducing a new operation τN, called truncation, to obtain a natural formalization of partial recursive functions in our applicative framework. Then we introduce the operational versions of a series of notions that are all equivalent to semi-decidability in ordinary recursion theory on the natural numbers, and study their mutual relationships over BON+ with τN.

We review the current knowledge concerning strong jump-traceability. We cover the known results relating strong jump-traceability to randomness, and those relating it to degree theory. We also discuss the techniques used in working with strongly jump-traceable sets. We end with a section of open questions.

We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of ${\rm{\Delta }}_2^0$ functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

Let $ \le _c $ be computable the reducibility on computably enumerable equivalence relations (or ceers). We show that for every ceer R with infinitely many equivalence classes, the index sets $\left\{ {i:R_i \le _c R} \right\}$ (with R nonuniversal), $\left\{ {i:R_i \ge _c R} \right\}$, and $\left\{ {i:R_i \equiv _c R} \right\}$ are ${\rm{\Sigma }}_3^0$ complete, whereas in case R has only finitely many equivalence classes, we have that $\left\{ {i:R_i \le _c R} \right\}$ is ${\rm{\Pi }}_2^0$ complete, and $\left\{ {i:R \ge _c R} \right\}$ (with R having at least two distinct equivalence classes) is ${\rm{\Sigma }}_2^0$ complete. Next, solving an open problem from [1], we prove that the index set of the effectively inseparable ceers is ${\rm{\Pi }}_4^0$ complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a ${\rm{\Sigma }}_3^0$ complete preordering relation, a fact that is used to show that the preordering relation $ \le _c $ on ceers is a ${\rm{\Sigma }}_3^0$ complete preordering relation.

The K-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [22, Question 4.2] and later in [25, Problem 5.5.8].

We give a negative answer to this question. In fact, we show the following stronger statement in the c.e. degrees. There exists a K-trivial degree d such that for all degrees a, b which are not K-trivial and a > d, b > d there exists a degree v which is not K-trivial and a > v, b > v. This work sheds light to the question of the definability of the K-trivial degrees in the c.e. degrees.

It is proved that there are two minimal $\Pi_1^0$ classes that are not automorphic.

A d.c.e. (2-computably enumerable) degree d is pseudo-isolated if d itself is non-isolated (in the sense that no computably enumerable (c.e.) degree below d can bound the c.e. degrees below d) and there is a d.c.e. degree \textbf{b} < \textbf{d} bounding all c.e. degrees below d. We prove in this paper that the pseudo-isolated degrees are densely distributed in the c.e. degrees.

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A(a1, …, an) is exponential diophantine, i.e. can be represented in the form

where a1 …, an, x1, …, xm range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B, multiplication, AB, and exponentiation, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns.

Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.