A real X is defined to be relatively c.e. if there is a real Y such that X is c.e.(Y) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X \not\leq_T Y}$$\end{document}. A real X is relatively simple and above if there is a real Y (...) nonempty \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} class contains a member which is not relatively c.e. and that every 1-generic real is relatively simple and above. (shrink)

In the paper we introduce and study regular enumerations for arbitrary recursive ordinals. Several applications of the technique are presented.

We give several new characterizations of the continuous enumeration degrees. The main one proves that an enumeration degree is continuous if and only if it is not half of a nontrivial relativized $\mathcal {K}$ -pair. This leads to a structural dichotomy in the enumeration degrees.

The jump operator on the ω-enumeration degrees was introduced in [I.N. Soskov, The ω-enumeration degrees, J. Logic Computat. 17 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure of the ω-enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of and of the enumeration degrees are isomorphic. In the second part of the paper we (...) study the jumps of the ω-enumeration degrees below . We define the ideal of the almost zero degrees and obtain a natural characterization of the class H of the ω-enumeration degrees below which are high n for some n and of the class L of the ω-enumeration degrees below which are low n for some n. (shrink)

Enumeration reducibility is a notion of relative computability between sets of natural numbers where only positive information about the sets is used or produced. Extending e‐reducibility to partial functions characterises relative computability between partial functions. We define a polynomial time enumeration reducibility that retains the character of enumeration reducibility and show that it is equivalent to conjunctive non‐deterministic polynomial time reducibility. We define the polynomial time e‐degrees as the equivalence classes under this reducibility and investigate their structure on the recursive (...) sets, showing in particular that the pe‐degrees of the computable sets are dense and do not form a lattice, but that minimal pairs exist. We define a jump operator and use it to produce a characterisation of the polynomial hierarchy. (shrink)

We give several new characterizations of the continuous enumeration degrees. The main one proves that an enumeration degree is continuous if and only if it is not half a nontrivial relativized K-pair. This leads to a structural dichotomy in the enumeration degrees.

Recall that B is PA relative to A if B computes a member of every nonempty $\Pi ^0_1(A)$ class. This two-place relation is invariant under Turing equivalence and so can be thought of as a binary relation on Turing degrees. Miller and Soskova [23] introduced the notion of a $\Pi ^0_1$ class relative to an enumeration oracle A, which they called a $\Pi ^0_1{\left \langle {A}\right \rangle }$ class. We study the induced extension of the relation B is PA relative (...) to A to enumeration oracles and hence enumeration degrees. We isolate several classes of enumeration degrees based on their behavior with respect to this relation: the PA bounded degrees, the degrees that have a universal class, the low for PA degrees, and the ${\left \langle {\text {self}\kern1pt}\right \rangle }$ -PA degrees. We study the relationship between these classes and other known classes of enumeration degrees. We also investigate a group of classes of enumeration degrees that were introduced by Kalimullin and Puzarenko [14] based on properties that are commonly studied in descriptive set theory. As part of this investigation, we give characterizations of three of their classes in terms of a special sub-collection of relativized $\Pi ^0_1$ classes—the separating classes. These three can then be seen to be direct analogues of three of our classes. We completely determine the relative position of all classes in question. (shrink)

Ash, C.J., Generalizations of enumeration reducibility using recursive infinitary propositional sentences, Annals of Pure and Applied Logic 58 173–184. We consider the relation between sets A and B that for every set S if A is Σ0α in S then B is Σ0β in S. We show that this is equivalent to the condition that B is definable from A in a particular way involving recursive infinitary propositional sentences. When α = β = 1, this condition is that B is (...) enumeration reducible to A. We establish further generalizations involving infinitely many sets and ordinals. (shrink)

These are the lecture notes from a 10-hour course that the author gave at the University of Notre Dame in September 2010. The objective of the course was to introduce some basic concepts in computable structure theory and develop the background needed to understand the author’s research on back-and-forth relations.

We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees non-total. Several fundamental results are proved: a (...) continuous function with non-total degree has no least degree representation, settling a question asked by Pour-El and Lempp; every non-computable f $\epsilon \mathcal{C}[0, 1]$ computes a non-computable subset of $\mathbb{N}$ ; there is a non-total degree between Turing degrees $a _\eqslantless_{\tau}$ b iff b is a PA degree relative to a; $\mathcal{S} \subseteq 2^{\mathbb{N}}$ is a Scott set iff it is the collection of f-computable subsets of $\mathbb{N}$ for some f $\epsilon \mathcal{C}[O, 1]$ of non-total degree; and there are computably incomparable f, g $\epsilon \mathcal{C}[0, 1]$ which compute exactly the same subsets of $\mathbb{N}$ . Proofs draw from classical analysis and constructive analysis as well as from computability theory. (shrink)

One of the main problems in effective model theory is to find an appropriate information complexity measure of the algebraic structures in the sense of computability. Unlike the commonly used degrees of structures, the structure degree measure is total. We introduce and study the jump operation for structure degrees. We prove that it has all natural jump properties (including jump inversion theorem, theorem of Ash), which show that our definition is relevant. We study the relation between the structure degree jump (...) (in the sense of Soskov) and the jump degrees of a structure (in the sense of Jockusch) and give necessary and sufficient conditions for their existence in the terms of structure degrees. We show some properties, distinguishing the structure degrees from the enumeration degrees. (shrink)