Enumerable Degrees Research Articles

Strong minimal pairs in the enumeration degrees

We prove that there are strong minimal pairs in the enumeration degrees and that the degrees of the left and right sides of strong minimal pairs include Σ20 degrees, although it is unknown if there is a strong minimal pair in the Σ20 enumeration degrees. We define a stronger type of minimal pair we call a strong super minimal pair, and show that there are none of these in the enumeration degrees, answering a question of Lempp et al. [6]. We leave open the question of the existence of a super minimal pair in the enumeration degrees.

Extensions of two constructions of Ahmad

In her 1990 thesis, Ahmad showed that there is a so-called “Ahmad pair”, i.e., there are incomparable Σ 2 0 -enumeration degrees a 0 and a 1 such that every enumeration degree x < a 0 is ⩽ a 1 . At the same time, she also showed that there is no “symmetric Ahmad pair”, i.e., there are no incomparable Σ 2 0 -enumeration degrees a 0 and a 1 such that every enumeration degree x 0 < a 0 is ⩽ a 1 and such that every enumeration degree x 1 < a 1 is ⩽ a 0 . In this paper, we first present a direct proof of Ahmad’s second result. We then show that her first result cannot be extended to an “Ahmad triple”, i.e., there are no Σ 2 0 -enumeration degrees a 0 , a 1 and a 2 such that both ( a 0 , a 1 ) and ( a 1 , a 2 ) are an Ahmad pair. On the other hand, there is a “weak Ahmad triple”, i.e., there are pairwise incomparable Σ 2 0 -enumeration degrees a 0 , a 1 and a 2 such that every enumeration degree x < a 0 is also ⩽ a 1 or ⩽ a 2 ; however neither ( a 0 , a 1 ) nor ( a 0 , a 2 ) is an Ahmad pair.

ON CUPPING AND AHMAD PAIRS

Abstract Working toward showing the decidability of the $\forall \exists $ -theory of the ${\Sigma ^0_2}$ -enumeration degrees, we prove that no so-called Ahmad pair of ${\Sigma ^0_2}$ -enumeration degrees can join to ${\mathbf 0}_e'$ .

Computability Theory

Over the last decade computability theory has seen many new and fascinating developments that have linked the subject much closer to other mathematical disciplines inside and outside of logic. This includes, for instance, work on enumeration degrees that has revealed deep and surprising relations to general topology, the work on algorithmic randomness that is closely tied to symbolic dynamics and geometric measure theory. Inside logic there are connections to model theory, set theory, effective descriptive set theory, computable analysis and reverse mathematics. In some of these cases the bridges to seemingly distant mathematical fields have yielded completely new proofs or even solutions of open problems in the respective fields. Thus, over the last decade, computability theory has formed vibrant and beneficial interactions with other mathematical fields. The goal of this workshop was to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work.

PA RELATIVE TO AN ENUMERATION ORACLE

AbstractRecall 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.

Enumeration Degrees of the Bounded Sets

The term ``total enumeration degree'' is related to the fact that the e-degree is total if and only if it contains a graph of some total function. In a number of works by the author and a group of mathematicians from the University of Wisconsin-Madison, the so-called ``graph-cototal enumeration degrees'' were considered, i.e. e-degrees containing the complement of the graph of some total function $f(x)$. In this article, the next step is taken -- the enumeration degrees of sets bounded from above or below by a graph of a total function are considered. More precisely, the set A is bounded from above if $A=\\{\\langle x,y\\rangle:y f(x)\\}$ for some total function $f(x)$. The article presents a number of results showing the existence of nontotal enumeration degrees containing bounded sets, and the constructed e-degrees are quasi-minimal. An important result is the one stating that bounded sets have the Friedberg property related to the jump inversion.

Novikov groups are right-orderable

Novikov groups were introduced as examples of finitely presented groups with unsolvable conjugacy problem. It was Bokut who showed that each Novikov group has a standard basis and thus a solvable word problem. Further, he showed that for every recursively enumerable degree of unsolvability d there is a Novikov group whose conjugacy problem is of degree d. In the present work, we show that Novikov groups are also right-orderable, thus exhibiting the first known examples of finitely presented right-orderable groups with solvable word problem and unsolvable conjugacy problem.

Point Degree Spectra of Represented Spaces

Abstract We introduce the point degree spectrum of a represented space as a substructure of the Medvedev degrees, which integrates the notion of Turing degrees, enumeration degrees, continuous degrees and so on. The notion of point degree spectrum creates a connection among various areas of mathematics, including computability theory, descriptive set theory, infinite-dimensional topology and Banach space theory. Through this new connection, for instance, we construct a family of continuum many infinite-dimensional Cantor manifolds with property C whose Borel structures at an arbitrary finite rank are mutually nonisomorphic. This resolves a long-standing question by Jayne and strengthens various theorems in infinite-dimensional topology such as Pol’s solution to Alexandrov’s old problem.

Nonlowness is independent from fickleness

It was recently shown that the computably enumerable (c.e.) degrees that embed the critical triple and the M 5 lattice structure are exactly those that are sufficiently fickle. Therefore the embeddability strength of a c.e. degree has much to do with the degree’s fickleness. Nonlowness is another common measure of degree strength, with nonlow degrees expected to compute more degrees than low ones. We ask if nonlowness and fickleness are independent measures of strength. Downey and Greenberg (A Hierarchy of Turing Degrees: A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability (AMS-206) (2020) Princeton University Press) claimed this to be true without proof, so we present one here. We prove the claim by building low and nonlow c.e. sets with arbitrary fickle degrees. Our construction is uniform so the degrees built turn out to be uniformly fickle. We base our proof on our direct construction of a nonlow array computable set. Such sets were always known to exist, but also never constructed directly in any publication we know.

Hierarchy of Computably Enumerable Degrees II

A transfinite hierarchy of Turing degrees of c.e.\ sets has been used to calibrate the dynamics of families of constructions in computability theory, and yields natural definability results. We review the main results of the area, and discuss splittings of c.e.\ degrees, and finding maximal degrees in upper cones.

ON SUPERSETS OF NON-LOW SETS

AbstractWe 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.

Degrees of sets having no subsets of higher m- and t t-degree

We consider sets without subsets of higher m- and t t-degree, that we call m-introimmune and t t-introimmune sets respectively. We study how they are distributed in partially ordered degree structures. We show that: each computably enumerable weak truth-table degree contains m-introimmune Π 1 0 -sets; each hyperimmune degree contains bi-m-introimmune sets. Finally, from known results we establish that each degree a with a ′ ⩾ 0 ″ covers a degree containing t t-introimmune sets.

Maximality and collapse in the hierarchy of α-c.a. degrees

In (A Hierarchy of Turing Degrees: A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability (2020), Annals of Mathematics Studies, Princeton University Press), Downey and Greenberg define a transfinite hierarchy of low 2 c.e. degrees – the totally α-c.a. degrees, for appropriately small ordinals α. This new hierarchy is of particular interest because it has already given rise to several natural definability results, and provides a new definable antichain in the c.e. degrees. Several levels of this hierarchy contain maximal degrees. We discuss how maximality interacts with upper cones, and the related notion of hierarchy collapse in upper cones. For example, we show that there is a totally ω-c.a. degree above which there is no maximal totally ω-c.a. degree.

Analog reducibility

Abstract In this paper we introduce and characterize two ‘analog reducibility’ notions for $[0,1]$-valued oracles on $\omega $ obtained by applying the syntactic characterizations of Turing and enumeration reducibility in terms of (positive) relatively $\varSigma _1$ and $\varPi _1$ formulas to formulas in continuous logic (Ben Yaacov, Berenstein, Henson and Usvyatsov, 2008, Model Theory for Metric Structures, vol. 2 of London Mathematical Society Lecture Note Series, pp. 315–427. Cambridge University Press.). The resulting analog and analog enumeration degree structures, $\mathscr {D}_a$ and $\mathscr {D}_{ae}$, naturally extend $\mathscr {D}_T$ and $\mathscr {D}_e$ in a compatible way. To show that these extensions are proper we prove that a sufficiently generic total $[0,1]$-valued oracle does not ‘analog enumerate’ any non-c.e. discrete set and that a sufficiently generic positive $[0,1]$-valued oracle neither ‘analog enumerates’ a non-c.e. discrete set nor ‘analog computes’ a non-trivial total $[0,1]$-valued oracle. We also provide a characterization of the continuous degrees among $\mathscr {D}_{ae}$ as precisely $\mathscr {D}_e \cap \mathscr {D}_a$. Finally we characterize a generalization of r.i.c.e. relations to metric structures via $\varSigma _1$ formulas in the ‘hereditarily compact superstructure’, which was the original motivation for the concepts in this paper.

The automorphism group and definability of the jump operator in the $$\omega $$-enumeration degrees

In the present paper, we show the first-order definability of the jump operator in the upper semi-lattice of the $$\omega $$ -enumeration degrees. As a consequence, we derive the isomorphicity of the automorphism groups of the enumeration and the $$\omega $$ -enumeration degrees.

Fragments of the theory of the enumeration degrees

We prove that every finite distributive lattice can be strongly embedded into the enumeration degrees as an interval, i.e., that there is an interval [a0,a1] of enumeration degrees isomorphic to the lattice, and any enumeration degree b≤a1 lies in this interval or below a0. As corollaries, we conclude that the ∃∀∃-theory of De is undecidable, while the extension of embeddings problem (a subproblem of the ∀∃-theory) is decidable.

Isolation from Side in 2-Computably Enumerable Degrees

In this work we consider isolation from side in different degree structures, in particular, in the 2-computably enumerable wtt-degrees and in low Turing degrees. Intuitively, a 2-computably enumerable degree is isolated from side if all computably enumerable degrees from its lower cone are bounded from above by some computably enumerable degree which is incomparable with the given one. It is proved that any properly 2-computably enumerable wtt-degree is isolated from side by some computable enumerable wtt-degree. Also it is shown that the same result holds for the low 2-computable enumerable Turing degrees.

A STRUCTURAL DICHOTOMY IN THE ENUMERATION DEGREES

AbstractWe 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.

On Decidable Categoricity and Almost Prime Models

The complexity of isomorphisms for computable and decidable structures plays an important role in computable model theory. Goncharov [26] defined the degree of decidable categoricity of a decidable model $$\mathcal {M} $$ to be the least Turing degree, if it exists, which is capable of computing isomorphisms between arbitrary decidable copies of $$\mathcal {M} $$ . If this degree is $$\mathbf {0} $$ , we say that the structure $$\mathcal {M} $$ is decidably categorical. Goncharov established that every computably enumerable degree is the degree of categoricity of a prime model, and Bazhenov showed that there is a prime model with no degree of categoricity. Here we investigate the degrees of categoricity of various prime models with added constants, also called almost prime models. We relate the degree of decidable categoricity of an almost prime model $$\mathcal {M} $$ to the Turing degree of the set $$C(\mathcal {M}) $$ of complete formulas. We also investigate uniform decidable categoricity, characterizing it by primality of $$\mathcal {M} $$ and Turing reducibility of $$C(\mathcal {M}) $$ to the theory of $$\mathcal {M} $$ .

Quasiminimal pairs for c.e. degrees of generic and coarse reducibilities

Generic approach to algorithmic problems in combinatorial group theory was suggested in 2003 by Kapovich, Myasnikov, Schupp and Shpilrain. This approach deals with algorithms, which solves algorithmic problems on ”most” of the inputs (i.e., on a generic set) instead of the entire domain and output undefined answer (do not halt) on the rest of inputs (a negligible set). Generic analog of Turing reducibility was introduced by Jockusch and Schupp in 2012. Later Hirschfeldt, Jockusch, Kuyper and Schupp defined coarse reducibility. In this paper we prove that there exist quasiminimal pairs for generic and coarse reducibilities of computably enumerable (c.e.) degrees. The work was supported by Russian Science Foundation, grant number 18-71-10028.