Computably Enumerable Degrees Research Articles

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.

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.

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.

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.

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.

Degrees of Autostability Relative to Strong Constructivizations of Graphs

We show that each computably enumerable Turing degree is a degree of autostability relative to strong constructivizations for a decidable directed graph. We construct a decidable undirected graph whose autostability spectrum relative to strong constructivizations is equal to the set of all PA-degrees.

On the Problem of Definability of the Computably Enumerable Degrees in the Difference Hierarchy

Questions of definability of computably enumerable degrees in the difference hierarchy (degrees of sets from finite levels of the Ershov difference hierarchy) are studied. Several approaches to the solution of this problem are outlined.

A HIERARCHY OF COMPUTABLY ENUMERABLE DEGREES

AbstractWe 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.

Cone Avoidance of Some Turing Degrees

We study the cone avoidance and the upper cone avoidance of two substructures of m-introimmune Turing degrees. We show that the substructure of the m-introimmune Turing degrees satisfies the cone avoidance property, and that the substructure of the computably enumerable m-introimmune Turing degrees satisfies the upper cone avoidance property.

TRIAL AND ERROR MATHEMATICS II: DIALECTICAL SETS AND QUASIDIALECTICAL SETS, THEIR DEGREES, AND THEIR DISTRIBUTION WITHIN THE CLASS OF LIMIT SETS

AbstractThis paper is a continuation of Amidei, Pianigiani, San Mauro, Simi, & Sorbi (2016), where we have introduced the quasidialectical systems, which are abstract deductive systems designed to provide, in line with Lakatos’ views, a formalization of trial and error mathematics more adherent to the real mathematical practice of revision than Magari’s original dialectical systems. In this paper we prove that the two models of deductive systems (dialectical systems and quasidialectical systems) have in some sense the same information content, in that they represent two classes of sets (the dialectical sets and the quasidialectical sets, respectively), which have the same Turing degrees (namely, the computably enumerable Turing degrees), and the same enumeration degrees (namely, the ${\rm{\Pi }}_1^0$ enumeration degrees). Nonetheless, dialectical sets and quasidialectical sets do not coincide. Even restricting our attention to the so-called loopless quasidialectical sets, we show that the quasidialectical sets properly extend the dialectical sets. As both classes consist of ${\rm{\Delta }}_2^0$ sets, the extent to which the two classes differ is conveniently measured using the Ershov hierarchy: indeed, the dialectical sets are ω-computably enumerable (close inspection also shows that there are dialectical sets which do not lie in any finite level; and in every finite level n ≥ 2 of the Ershov hierarchy there is a dialectical set which does not lie in the previous level); on the other hand, the quasidialectical sets spread out throughout all classes of the hierarchy (close inspection shows that for every ordinal notation a of a nonzero computable ordinal, there is a quasidialectical set lying in ${\rm{\Sigma }}_a^{ - 1}$, but in none of the preceding levels).

Integer valued betting strategies and Turing degrees

Betting strategies are often expressed formally as martingales. A martingale is called integer-valued if each bet must be an integer value. Integer-valued strategies correspond to the fact that in most betting situations, there is a minimum amount that a player can bet. According to a well known paradigm, algorithmic randomness can be founded on the notion of betting strategies. A real X is called integer-valued random if no effective integer-valued martingale succeeds on X. It turns out that this notion of randomness has interesting interactions with genericity and the computably enumerable degrees. We investigate the computational power of the integer-valued random reals in terms of standard notions from computability theory.

Computable categoricity of the Boolean algebra $ \mathfrak{B}\left( \omega \right) $ with a distinguished automorphism

It is proved that every computably enumerable Turing degree is a degree of categoricity of some computable Boolean algebra with a distinguished automorphism. We construct an example of a computably categorical Boolean algebra with a distinguished automorphism, having a set of atoms in a given computably enumerable Turing degree.

A semilattice generated by superlow computably enumerable degrees

We prove that a partially ordered set of all computably enumerable (c. e.) degrees that are the least upper bounds of two superlow c. e. degrees is an upper semilattice not elementary equivalent to the semilattice of all c. e. degrees.

A superhigh diamond in the c.e. tt-degrees

The notion of superhigh computably enumerable (c.e.) degrees was first introduced by (Mohrherr in Z Math Logik Grundlag Math 32: 5---12, 1986) where she proved the existence of incomplete superhigh c.e. degrees, and high, but not superhigh, c.e. degrees. Recent research shows that the notion of superhighness is closely related to algorithmic randomness and effective measure theory. Jockusch and Mohrherr proved in (Proc Amer Math Soc 94:123---128, 1985) that the diamond lattice can be embedded into the c.e. tt-degrees preserving 0 and 1 and that the two atoms can be low. In this paper, we prove that the two atoms in such embeddings can also be superhigh.

Presentations of the successor relation of computable linear ordering

We prove that a nontrivial degree spectrum of the successor relation of either strongly η-like or non-η-like computable linear orderings is closed upwards in the class of all computably enumerable degrees. We also show that the degree spectrum contains 0 if and only if either it is trivial or it contains all computably enumerable degrees.

Isolated 2-computably enumerable Q-degrees

In this work we prove that for every pair of computably enumerable degrees a<Q b there exists a properly 2-computably enumerable degree d such that a <Q d <Q b, a isolates d from below, and b isolates d from above. Two corollaries follow from this result. First, there exists a 2-computably enumerable degree which is Q-incomparable with any nontrivial (different from 0 and 0′) computably enumerable degree. Second, every nontrivial computably enumerable degree isolates some 2-computably enumerable degree from below and some 2-computably enumerable degree from above.

Strong noncuppability in low computably enumerable degrees

We prove the existence of noncomputable low computably enumerable degrees b < a such that b is strongly noncuppable to a in the class R.

Joining to high degrees via noncuppables

Cholak, Groszek and Slaman proved in J Symb Log 66:881–901, 2001 that there is a nonzero computably enumerable (c.e.) degree cupping every low c.e. degree to a low c.e. degree. In the same paper, they pointed out that every nonzero c.e. degree can cup a low2 c.e. degree to a nonlow2 degree. In Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) improved the latter result by showing that every nonzero c.e. degree c is cuppable to a high c.e. degree by a low2 c.e. degree b. It is natural to ask in which subclass of low2 c.e. degrees can b in Jockusch et al. (Trans Am Math Soc 356:2557–2568, 2004) be located. Wu proved in Math Log Quart 50:189–201, 2004 that b can be cappable. We prove in this paper that b in Jockusch, Li and Yang’s result can be noncuppable, improving both Jockusch, Li and Yang, and Wu’s results.

Prime models of computably enumerable degree

AbstractWe examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b′ = 0′), This extends Csima's result that every CAD theory has a low prime model. We also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d &lt; c and c not low2 (c″ &gt; 0″), then for any CAD theory T, there exists a c.e. degree b with d &lt; b &lt; c such that T has a prime model of degree b, where b can be chosen so that b′ is any degree c.e. in c with d′ ≤ b′. As a corollary, we show that for any degree c with 0 &lt; c &lt; 0′, every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models.