We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare our results with similar results in computable topology.

We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$ -c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $ -c.a. as shown in [9]. We also show that if cone avoidance is added then there (...) is no level below $\varepsilon _0$ which can be used to characterize the complexity of $A_1$ and $A_2$. (shrink)

We study for each computably bounded $\Pi ^0_1$ class P the set of degrees of c.e. paths in P. We show, amongst other results, that for every c.e. degree a there is a perfect $\Pi ^0_1$ class where all c.e. members have degree a. We also show that every $\Pi ^0_1$ set of c.e. indices is realized in some perfect $\Pi ^0_1$ class, and classify the sets of c.e. degrees which can be realized in some $\Pi ^0_1$ class as exactly (...) those with a computable representation. (shrink)

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 study the degree structure of the ω-c.e., n-c.e., and Π10 equivalence relations under the computable many-one reducibility. In particular, we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-computably enumerable equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e., and Π10 equivalence relations. We prove that for all the degree classes considered, upward density (...) holds and downward density fails. (shrink)

We solve a problem posed by Goncharov and Knight 639–681, 757]). More specifically, we produce an effective Friedberg enumeration of computable equivalence structures, up to isomorphism. We also prove that there exists an effective Friedberg enumeration of all isomorphism types of infinite computable equivalence structures.

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 give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on (...) the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture. (shrink)

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 complexity of equivalence relations has received much attention in the recent literature. The main tool for such endeavour is the following reducibility: given equivalence relations R and S on natural numbers, R is computably reducible to S if there is a computable function f:ω→ω that induces an injective map from R-equivalence classes to S-equivalence classes. In order to compare the complexity of equivalence relations which are computable, researchers considered also feasible variants of computable reducibility, such as the polynomial-time reducibility. (...) In this work, we explore Peq, the degree structure generated by primitive recursive reducibility on primitive recursive equivalence relations with infinitely many equivalence classes. In contrast with all other known degree structures on equivalence relations, we show that Peq has much more structure: e.g., we show that it is a dense distributive lattice. On the other hand, we also offer evidence of the intricacy of Peq, proving, e.g., that the structure is neither rigid nor homogeneous. (shrink)

We show that there is a cuppable c.e. degree, all of whose cupping partners are high. In particular, not all cuppable degrees are${\operatorname {\mathrm {low}}}_3$-cuppable, or indeed${\operatorname {\mathrm {low}}}_n$cuppable for anyn, refuting a conjecture by Li. On the other hand, we show that one cannot improve highness to superhighness. We also show that the${\operatorname {\mathrm {low}}}_2$-cuppable degrees coincide with the array computable-cuppable degrees, giving a full understanding of the latter class.

We study the relationship between effective domination properties and the bounded jump. We answer two open questions about the bounded jump: We prove that the analogue of Sacks jump inversion fails for the bounded jump and the wtt-reducibility. We prove that no c.e. bounded high set can be low by showing that they all have to be Turing complete. We characterize the class of c.e. bounded high sets as being those sets computing the Halting problem via a reduction with use (...) bounded by an ω-c.e. function. We define several notions of a c.e. set being effectively dominant, and show that together with the bounded high sets they form a proper hierarchy. (shrink)

In this paper we show that there is no minimal bound for jump traceability. In particular, there is no single order function such that strong jump traceability is equivalent to jump traceability for that order. The uniformity of the proof method allows us to adapt the technique to showing that the index set of the c.e. strongly jump traceables is image-complete.

In this paper we show that there is a pair of superhigh r.e. degree that forms a minimal pair. An analysis of the proof shows that a critical ingredient is the growth rates of certain order functions. This leads us to investigate certain high r.e. degrees, which resemble ∅′ very closely in terms of ∅′-jump traceability. In particular, we will construct an ultrahigh degree which is cappable.

Semi-hyperhypersimple c.e. sets, also known as diagonals, were introduced by Kummer. He showed that by considering an analogue of hyperhypersimplicity, one could characterize the sets which are the Halting problem relative to arbitrary computable numberings. One could also consider half of splittings of maximal or hyperhypersimple sets and get another variant of maximality and hyperhypersimplicity, which are closely related to the study of automorphisms of the c.e. sets. We investigate the Turing degrees of these classes of c.e. sets. In particular, (...) we show that the analogue of a theorem of Martin fails for these classes. (shrink)

We show that for every intermediate \ s-degree there exists an incomparable \ s-degree. As a consequence, for every intermediate \ Q-degree there exists an incomparable \ Q-degree. We also show how these results can be applied to provide proofs or new proofs of upper density results in local structures of s-degrees and Q-degrees.

We study the Turing degrees which contain a real of effective packing dimension one. Downey and Greenberg showed that a c.e. degree has effective packing dimension one if and only if it is not c.e. traceable. In this paper, we show that this characterization fails in general. We construct a real $A\leq_T\emptyset''$ which is hyperimmune-free and not c.e. traceable such that every real $\alpha\leq_T A$ has effective packing dimension 0. We construct a real $B\leq_T\emptyset'$ which is not c.e. traceable such (...) that every real $\alpha\leq_T B$ has effective packing dimension 0. (shrink)

A strong degree of categoricity is a Turing degree [Formula: see text] such that there is a computable structure [Formula: see text] that is [Formula: see text]-computably categorical (there is a [Formula: see text]-computable isomorphism between any two computable copies of [Formula: see text]), and such that there exist two computable copies of [Formula: see text] between which every isomorphism computes [Formula: see text]. The question of whether every [Formula: see text] degree is a strong degree of categoricity has been (...) of interest since the first paper on this subject. We answer the question in the affirmative, by constructing an example. (shrink)

We introduce a natural strengthening of prompt simplicity which we call strong promptness, and study its relationship with existing lowness classes. This notion provides a ≤ wtt version of superlow cuppability. We show that every strongly prompt c.e. set is superlow cuppable. Unfortunately, strong promptness is not a Turing degree notion, and so cannot characterize the sets which are superlow cuppable. However, it is a wtt-degree notion, and we show that it characterizes the degrees which satisfy a wtt-degree notion very (...) close to the definition of superlow cuppability. Further, we study the strongly prompt c.e. sets in the context of other notions related promptness, superlowness, and cupping. In particular, we show that every benign cost function has a strongly prompt set which obeys it, providing an analogue to the known result that every cost function with the limit condition has a prompt set which obeys it. We also study the effect that lowness properties have on the behaviour of a set under the join operator. In particular we construct an array noncomputable c.e. set whose join with every low c.e. set is low. (shrink)

A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma (...) }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel. (shrink)

We prove a number of results in effective randomness, using methods in which Π⁰₁ classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relationsR,S, a componentwise reducibility is defined byR≤S⇔ ∃f∀x, y[x R y↔fS f].Here,fis taken from a suitable class of effective functions. For us the relations will be on natural numbers, andfmust be computable. We show that there is a${\rm{\Pi }}_1^0$-complete equivalence relation, but no${\rm{\Pi }}_k^0$-complete fork≥ 2. We show that${\rm{\Sigma }}_k^0$preorders arising naturally in the above-mentioned areas are${\rm{\Sigma }}_k^0$-complete. This includes polynomial timem-reducibility on (...) exponential time sets, which is${\rm{\Sigma }}_2^0$, almost inclusion on r.e. sets, which is${\rm{\Sigma }}_3^0$, and Turing reducibility on r.e. sets, which is${\rm{\Sigma }}_4^0$. (shrink)

We show that Sacks' and Shoenfield's analogs of jump inversion fail for both tt- and wtt-reducibilities in a strong way. In particular we show that there is a ${\mathrm{\Delta }}_{2}^{0}$ set B > tt ∅′ such that there is no c.e. set A with A′ ≡ wtt B. We also show that there is a ${\mathrm{\Sigma }}_{2}^{0}$ set C > tt ∅′ such that there is no ${\mathrm{\Delta }}_{2}^{0}$ set D with D′ ≡ wtt C.

We study inversions of the jump operator on ${\mathrm{\Pi }}_{1}^{0}$ classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0′ sets which are not 2-random. Both of the classes coincide with the degrees above 0′ which are not 0′-dominated. A (...) further application is the complete solution of [24, Problem 3.6.9]: one direction of van Lambalgen's theorem holds for weak 2-randomness, while the other fails. Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [24, Problem 8.2.14]: not all weakly 2-random sets are array computable. In fact, given any oracle X, there is a weakly 2-random which is not array computable relative to X. This contrasts with the fact that all 2-random sets are array computable. (shrink)

Are words like ‘woman’ or ‘man’ sex terms that we use to talk about biological features of individuals? Are they gender terms that we use to talk about non-biological features e.g. social roles? Contextualists answer both questions affirmatively, arguing that these terms concern biological or non-biological features depending on context. I argue that a recent version of contextualism from Jennifer Saul that Esa Diaz-Leon develops doesn't exhibit the right kind of flexibility to capture our theoretical intuitions or moral and political (...) practices concerning our uses of these words. I then float the view that terms like 'woman' or 'man' are polysemous, arguing that it makes better sense of the significance of some forms of criticisms of mainstream gender ideology. (shrink)

Upon doing something generous for someone with whom you are close, some kind of reciprocity may be appropriate. But it often seems wrong to actually request reciprocity. This chapter explores the wrongness in making these requests, and why they can nevertheless appear appropriate. After considering several explanations for the wrongness at issue (involving, e.g. distinguishing oughts from obligation, the suberogatory, imperfect duties, and gift-giving norms), a novel proposal is advanced. The requests are disrespectful; they express that their agent insufficiently trusts (...) the hearer to recognize their own reasons to reciprocate, and the close relationship between them morally requires this kind of trust. This proposal is articulated and situated in the recent discussion of the normativity of requests. The chapter concludes by explaining how these requests may appear appropriate. Agents in these relationships often have standing to make requests, though the standing is lacking in these particular cases. (shrink)

A critical survey of various positions on the nature, use, possession, and analysis of normative concepts. We frame our treatment around G.E. Moore’s Open Question Argument, and the ways metaethicists have responded by departing from a Classical Theory of concepts. In addition to the Classical Theory, we discuss synthetic naturalism, noncognitivism (expressivist and inferentialist), prototype theory, network theory, and empirical linguistic approaches. Although written for a general philosophical audience, we attempt to provide a new perspective and highlight some underappreciated problems (...) about normative concepts. (shrink)

Ethicists struggle to take reductive views seriously. They also have trouble conceiving of some supervenience failures. Understanding why provides further evidence for a kind of hybrid view of normative concept use.

The revolution in science and technology is significantly changing the character of production and work. One of the important aspects of this change is that labor becomes a scientifically organized process that makes increased demands on development of the human factor in production. The most productive scientific ideas, the most perfect technologies, when combined with production, may fail to produce the desired "yield" unless a sufficient number of well-trained personnel, or of the required organizational structures corresponding to the level of (...) societal relationships, the habits of collective work, and human psychology, are available. Implementation of a truly scientific organization of work presumes as a condition that a considerable level of development will already have been achieved in socialist labor discipline, as well as habits of collectivism and ethical motivations for work, and not merely occupational skills and qualities. More precisely, the list of qualities that, while they may not be occupational in the strict sense are nonetheless necessary features of the person employed in socialist society, includes general cultural and moral development, social activity, and political consciousness. The transformation of work into a scientifically organized process presumes an increasing demand for a highly skilled labor force with comprehensive development. Moreover, the development of the labor force and the rise in its skills should run ahead of the development of the means of production. It is precisely this that determines the need for conscious management of the development of the human being. For comprehensive development of the labor force to run ahead of the means of production requires considerable outlays for occupational and general cultural training of the direct participants in production. A worker of a new type, with a higher level of general and occupational education, will emerge in the course of implementation of a complex of economic, social, and scientific-technological measures. (shrink)

On one way of talking about a traditional metaethical topic, realists accept that some items appear on the list of what exists in the moral or more broadly normative domain of inquiry. They then divide over whether those items are like what science and experience suggest that all other items on the list of what exists across all domains are like – naturalistic and secular. Reductive naturalists answer this further question affirmatively. Why don’t nonnaturalists? I explore the answer that it’s (...) because normative entities are “just too different” in the sense that they are countably different things. However, I argue that this answer rests on the subtle presupposition that the normative domain doesn’t also contain uncountable entities of the sort that analytic metaphysicians call “stuff”. Taking intuitions of difference seriously points the way to a novel form of metaethical reductive naturalism. The key is to identify the stuff that matters. (shrink)

The Priestley duality for Wajsberg algebras is developed. The Wajsberg space is a De Morgan space endowed with a family of functions that are obtained in rather natural way.As a first application of this duality, a theorem about unicity of the structure is given.

Elementos de juicio: reflejos psicosociales, tendencias de los elementos.--El compromiso en el arte: dinámica de las relaciones estructurales en la mímesis artística.