We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is \-complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes.

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

We examine the sequences A that are low for dimension, i.e. those for which the effective dimension relative to A is the same as the unrelativized effective dimension. Lowness for dimension is a weakening of lowness for randomness, a central notion in effective randomness. By considering analogues of characterizations of lowness for randomness, we show that lowness for dimension can be characterized in several ways. It is equivalent to lowishness for randomness, namely, that every Martin-Löf random sequence has effective dimension (...) 1 relative to A, and lowishness for K, namely, that the limit of KA/K is 1. We show that there is a perfect [Formula: see text]-class of low for dimension sequences. Since there are only countably many low for random sequences, many more sequences are low for dimension. Finally, we prove that every low for dimension is jump-traceable in order nε, for any ε > 0. (shrink)

The tower number ${\mathfrak t}$ and the ultrafilter number $\mathfrak {u}$ are cardinal characteristics from set theory. They are based on combinatorial properties of classes of subsets of $\omega $ and the almost inclusion relation $\subseteq ^*$ between such subsets. We consider analogs of these cardinal characteristics in computability theory.We say that a sequence $(G_n)_{n \in {\mathbb N}}$ of computable sets is a tower if $G_0 = {\mathbb N}$, $G_{n+1} \subseteq ^* G_n$, and $G_n\smallsetminus G_{n+1}$ is infinite for each n. (...) A tower is maximal if there is no infinite computable set contained in all $G_n$. A tower ${\left \langle {G_n}\right \rangle }_{n\in \omega }$ is an ultrafilter base if for each computable R, there is n such that $G_n \subseteq ^* R$ or $G_n \subseteq ^* \overline R$ ; this property implies maximality of the tower. A sequence $(G_n)_{n \in {\mathbb N}}$ of sets can be encoded as the “columns” of a set $G\subseteq \mathbb N$. Our analogs of ${\mathfrak t}$ and ${\mathfrak u}$ are the mass problems of sets encoding maximal towers, and of sets encoding towers that are ultrafilter bases, respectively. The relative position of a cardinal characteristic broadly corresponds to the relative computational complexity of the mass problem. We use Medvedev reducibility to formalize relative computational complexity, and thus to compare such mass problems to known ones.We show that the mass problem of ultrafilter bases is equivalent to the mass problem of computing a function that dominates all computable functions, and hence, by Martin’s characterization, it captures highness. On the other hand, the mass problem for maximal towers is below the mass problem of computing a non-low set. We also show that some, but not all, noncomputable low sets compute maximal towers: Every noncomputable (low) c.e. set computes a maximal tower but no 1-generic $\Delta ^0_2$ -set does so.We finally consider the mass problems of maximal almost disjoint, and of maximal independent families. We show that they are Medvedev equivalent to maximal towers, and to ultrafilter bases, respectively. (shrink)

We consider the ower semilattice [MATHEMATICAL SCRIPT CAPITAL D] of differences of c.e. sets under inclusion. It is shown that [MATHEMATICAL SCRIPT CAPITAL D] is not distributive as a semilattice, and that the c.e. sets form a definable subclass.

Combinatorial operations on sets are almost never well defined on Turing degrees, a fact so obvious that counterexamples are worth exhibiting. The case we focus on is the symmetric-difference operator; there are pairs of degrees for which the symmetric-difference operation is well defined. Some examples can be extracted from the literature, e.g. from the existence of nonzero degrees with strong minimal covers. We focus on the case of incomparable r.e. degrees for which the symmetric-difference operation is well defined.

In this paper the structuralist approach of theory reconstruction is applied to International Trade Theory. In the basic element the universal laws of the theory are stated and the general concepts are defined in terms of three sets and seven functions. Ricardo’s Theory of Comparative Advantage and the Factor Proportions Theory by Heckscher and Ohlin are reconstructed as specialisations of the basic element. Two intratheoretical constraints are formulated in order to ensure the consistency of the theory. A number of empirical (...) studies are shown to be paradigmatic, successful or questionable applications of International Trade Theory. The reconstruction allows the assessment of the theory from a metatheoretical angle and illuminates the logical interdependencies of its components. (shrink)

Khutoretskii's Theorem states that the Rogers semilattice of any family of c.e. sets has either at most one or infinitely many elements. A lemma in the inductive step of the proof shows that no Rogers semilattice can be partitioned into a principal ideal and a principal filter. We show that such a partitioning is possible for some family of d.c.e. sets. In fact, we construct a family of c.e. sets which, when viewed as a family of d.c.e. sets, has (up (...) to equivalence) exactly two computable Friedberg numberings ¼ and ν, and ¼ reduces to any computable numbering not equivalent to ν. The question of whether the full statement of Khutoretskii's Theorem fails for families of d.c.e. sets remains open. (shrink)

Substantial literature has investigated the relationship between religiosity and ethical decision-making (the what), while lesser consideration has been given to exploring why decisions are made. As part of a larger study, this paper aims to delve beyond the descriptive relationship between religiosity and ethical decision-making of Muslim employees in Malaysia. We analyse the qualitative data received from 160 employees by using thematic analysis. Our results reveal that, while religious values are important for Muslims in Malaysia, there are other factors that (...) are also pertinent when making decisions at work, namely the organization’s interest and the context of the decision. We also found that individuals’ reasoning based on Islamic teachings is embedded in rules and regulations, which suggests lower levels of ethical reasoning. This study extends the current understanding of the processes of ethical reasoning in highly religious contexts by individuals at their workplace. Implications and suggestions for future research are discussed. (shrink)

We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and effective content of versions of Ramsey's Theorem. In particular, our result implies that every ω-model of RCA 0 + SRT 2 2 must contain a nonlow set.

We prove that the degree structures of the d.c.e. and the 3-c.e. Turing degrees are not elementarily equivalent, thus refuting a conjecture of Downey. More specifically, we show that the following statement fails in the former but holds in the latter structure: There are degrees f > e > d > 0 such that any degree u ≤ f is either comparable with both e and d, or incomparable with both.

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)

We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ03-condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n≥ 1 in ω, there exists a computable tree of finite height which is δ0n+1-categorical but (...) not δ0n-categorical. (shrink)

We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a \Delta^0_2 maximal filter, and there is a computable poset with no \Pi^0_1 or \Sigma^0_1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse mathematics result that the principle (...) "every countable poset has a maximal filter" is equivalent to ACA₀ over RCA₀. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a \Sigma^0_1 unbounded filter, and there is a computable poset with no \Pi^0_1 unbounded filter. We show that there is a computable poset on which every unbounded filter is Turing complete, and the principle "every countable poset has an unbounded filter" is equivalent to ACA₀ over RCA₀. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets. (shrink)

This paper considers reductions between types of numberings; these reductions preserve the Rogers Semilattice of the numberings reduced and also preserve the number of minimal and positive degrees in their semilattice. It is shown how to use these reductions to simplify some constructions of specific semilattices. Furthermore, it is shown that for the basic types of numberings, one can reduce the left-r.e. numberings to the r.e. numberings and the k-r.e. numberings to the k+1-r.e. numberings; all further reductions are obtained by (...) concatenating these reductions. (shrink)

We exhibit a finite lattice without critical triple that cannot be embedded into the enumerable Turing degrees. Our method promises to lead to a full characterization of the finite lattices embeddable into the enumerable Turing degrees.

We characterize the linear order types $\tau $ with the property that given any countable linear order $\mathcal {L}$, $\tau \cdot \mathcal {L}$ is a computable linear order iff $\mathcal {L}$ is a computable linear order, as exactly the finite nonempty order types.

We show that for every nontrivial r.e. wtt-degree a, there are r.e. wtt-degrees b and c incomparable to a such that the infimum of a and b exists but the infimum of a and c fails to exist. This shows in particular that there are no strongly noncappable r.e. wtt-degrees, in contrast to the situation in the r.e. Turing degrees.

We give an algorithm for deciding whether an embedding of a finite partial order [Formula: see text] into the enumeration degrees of the [Formula: see text]-sets can always be extended to an embedding of a finite partial order [Formula: see text].

The degrees of unsolvability were introduced in the ground-breaking papers of Post [20] and Kleene and Post [7] as an attempt to measure theinformation contentof sets of natural numbers. Kleene and Post were interested in the relative complexity of decision problems arising naturally in mathematics; in particular, they wished to know when a solution to one decision problem contained the information necessary to solve a second decision problem. As decision problems can be coded by sets of natural numbers, this question (...) is equivalent to: Given a computer with access to an oracle which will answer membership questions about a setA, can a program be written which will correctly compute the answers to all membership questions about a setB? If the answer is yes, then we say thatBisTuring reducibletoAand writeB≤TA. We say thatB≡TAifB≤TAandA≤TB. ≡Tis an equivalence relation, and ≤Tinduces a partial ordering on the corresponding equivalence classes; the poset obtained in this way is called thedegrees of unsolvability, and elements of this poset are calleddegrees.Post was particularly interested in computability from sets which are partially generated by a computer, namely, those for which the elements of the set can be enumerated by a computer. (shrink)

Work in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X', its Turning jump, is recursive in $\varnothing'$ and high if X' computes $\varnothing''$ . Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep, if for each recursively enumerable set A, the jump of $A \bigoplus W$ is recursive in the jump of A. We prove that there are no (...) deep degrees other than the recursive one. Given a set W, we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turning functionals Φ. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that $(W \bigoplus A)'$ is forced to disagree with Φ(-; A'). The conversion has some ambiguity; in particular, A cannot be found uniformly from W. We also show that there is a "moderately" deep degree: There is a low nonzero degree whose join with any other low degree is not high. (shrink)

We show that the Π 4 -theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.

We show that the $\Pi_4$-theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.

We study the filter ℒ*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ*(A) is still Σ0 3-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ*(A).

We show the decidability of the existential theory of the recursively enumerable degrees in the language of Turing reducibility, Turing reducibility of the Turing jumps, and least and greatest element.

We show that every finite lattice is embeddable into the Σ 0 2 enumeration degrees via a lattice-theoretic embedding which preserves 0 and 1.

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 ${\mathbf 0}_e'$.

We show that the top of any diamond with bottom 0 in the r.e. degrees is also the top of a stack of n diamonds with bottom 0.

An r.e. degree a ≠ 0, 0' is called strongly noncappable if it has no inf with any incomparable r.e. degree. We show the existence of a high strongly noncappable degree.

We describe the motivation for the construction of a general framework for priority arguments, the ideas incorporated into the construction of the framework, and the use of the framework to prove theorems in computability theory which require priority arguments.

In this paper, we investigate the Lindenbaum algebra ℒ of the theory T fin = Th of the class M fin of all finite models of a finite rich signature. We prove that this algebra is an atomic Boolean algebra while its Gödel numeration γ is a [Formula: see text]-numeration. Moreover, the quotient algebra /ℱ, γ/ℱ) modulo the Fréchet ideal ℱ is a [Formula: see text]-algebra, which is universal over the class of all [Formula: see text] Boolean algebras. These conditions (...) characterize uniquely the algebra ℒ; moreover, these conditions characterize up to recursive isomorphism the numerated Boolean quotient algebra /ℱ, γ/ℱ). These results extend the work of Trakhtenbrot [17] and Vaught [18] on the first order theory of the class of all finite models of a finite rich signature. (shrink)

We show that there is a strong minimal pair in the computably enumerable 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.