Enumerable Turing Degrees Research Articles

Downwards density in the n-computably enumerable Turing degrees and the uniform constructions

Downwards density in the n-computably enumerable Turing degrees and the uniform constructions

Towards characterizing the >ω2-fickle recursively enumerable Turing degrees

Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees 〈RT,≤T〉, it is not known how one can characterize the degrees d∈RT below which L can be embedded. Two important characterizations are of the L7 and M3 lattices, where the lattices are embedded below d if and only if d contains sets of “fickleness” >ω and ≥ωω respectively. We work towards finding a lattice that characterizes the levels above ω2, the first non-trivial level after ω. We considered lattices that are as “short” in height and “narrow” in width as L7 and M3, but the lattices characterize also the >ω or ≥ωω levels, if the lattices are not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some previously considered lattices, but the removals did not change the levels characterized. We discovered three lattices besides M3 that also characterize the ≥ωω-levels. Our search for >ω2-candidates can therefore be reduced to the lattice-theoretic problem of finding lattices that do not contain any of the four ≥ωω-lattices as sublattices.

Nonuniformity of Downward Density in n-Computably Enumerable Turing Degrees

Nonuniformity of Downward Density in n-Computably Enumerable Turing Degrees

Towards Finding a Lattice that Characterizes the -Fickle Recursively Enumerable Turing Degrees

AbstractGiven 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: ko.390@osu.eduURL: http://sites.nd.edu/liling-ko/

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.

Minimal Weak Truth Table Degrees and Computably Enumerable Turing Degrees

Minimal Weak Truth Table Degrees and Computably Enumerable Turing Degrees

On Initial Segments of Turing Degrees Containing Simple T-Mitotic but not wtt-Mitotic Sets

We consider the properties of computably enumerable (c.e.) Turing degrees containing sets, which possess the property of a T-mitotic splitting but don't have a wtt-mitotic splitting. It is proved that for any noncomputable c.e. degree b there exists a degree a, such that a · b and a contains a simple T- itotic set, which is not wtt-mitotic.

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.

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).

2-Computably Enumerable Degrees of Categoricity for Boolean Algebras with Distinguished Automorphisms

We prove that any 2-computably enumerable Turing degree is the degree of categoricity for some computable Boolean algebra with a distinguished automorphism. Bibliography: 20 titles.

Computably enumerable Turing degrees and the meet property

Working in the Turing degree structure, we show that those degrees which contain computably enumerable sets all satisfy the meet property, i.e. if a is c.e. and b < a, then there exists non-zero m < a with b ^m = 0. In fact, more than this is true: m may always be chosen to be a minimal degree. This settles a conjecture of Cooper and Epstein from the 80s.

On the existence of a strong minimal pair

We show that there is a strong minimal pair in the computably enumerable (c.e.) Turing degrees, i.e. a pair of nonzero c.e. degrees a and b such that a∩b = 0 and for any nonzero c.e. degree x ≤ a, b ∪ x ≥ a.

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.

Degrees of autostability relative to strong constructivizations

The spectra of the Turing degrees of autostability of computable models are studied. For almost prime decidable models, it is shown that the autostability spectrum relative to strong constructivizations of such models always contains a certain recursively enumerable Turing degree; moreover, it is shown that for any recursively enumerable Turing degree, there exist prime models in which this degree is the least one in the autostability spectrum relative to strong constructivizations.

The complexity of central series in nilpotent computable groups

The terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders.

Upper bounds on ideals in the computably enumerable Turing degrees

We study ideals in the computably enumerable Turing degrees, and their upper bounds. Every proper Σ 4 0 ideal in the c.e. Turing degrees has an incomplete upper bound. It follows that there is no Σ 4 0 prime ideal in the c.e. Turing degrees. This answers a question of Calhoun (1993) [2]. Every proper Σ 3 0 ideal in the c.e. Turing degrees has a low 2 upper bound. Furthermore, the partial order of Σ 3 0 ideals under inclusion is dense.

ON THE COMPLEXITY OF THE SUCCESSIVITY RELATION IN COMPUTABLE LINEAR ORDERINGS

In this paper, we solve a long-standing open question (see, e.g. Downey [6, Sec. 7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering [Formula: see text] has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing [Formula: see text]-isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications.

A FINITELY PRESENTED GROUP WITH ALMOST SOLVABLE CONJUGACY PROBLEM

The algorithmic unsolvability of the conjugacy problem for finitely presented groups was demonstrated by Novikov in the early 1950s. Various simplifications and alternative proofs were found by later researchers and further questions raised. Recent work by Borovik, Myasnikov and Remeslennikov has considered the question of what proportion of the number of elements of a group (obtained by standard constructions) falls into the realm of unsolvability. In this paper we provide a straightforward construction, as a Britton tower, of a finitely presented group with solvable word problem but unsolvable conjugacy problem of any r.e. (recursively enumerable) Turing degree a. The question of whether two elements are conjugate is bounded truth-table reducible to the question of whether the elements are both conjugate to a single generator of the group. We also define computable normal forms, based on the method of Bokut', that are suitable for the conjugacy problem. We consider (ordered) pairs of normal words U, V for the conjugacy problem whose lengths add to l and show that the proportion of such pairs for which conjugacy is undecidable (in the case a ≠ 0) is strictly less than l2/(2λ - 1)l where λ &gt; 4. The construction is based on modular machines, introduced by Aanderaa and Cohen. For the purposes of this construction it was helpful to extend the notion of configuration to include pairs of m-adic integers. The notion of computation step was also extended and is referred to as s-fold computation where s ∈ ℤ (the usual notion coresponds to s = 1). If gcd (m, s) = 1 then determinism is preserved, i.e., if the modular machine is deterministic then it remains so under the extended notion. Furthermore there is a simple correspondence between s-fold and standard computation in this case. Otherwise computation is non-deterministic and there does not seem to be any straightforward correspondence between s-fold and standard computation.

Degree spectra of the successor relation of computable linear orderings

We establish that for every computably enumerable (c.e.) Turing degree b the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees.