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)

We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees (...) of categoricity is $\Pi_{1}^{1}$-complete. (shrink)

The Rainbow Ramsey Theorem is essentially an "anti-Ramsey" theorem which states that certain types of colorings must be injective on a large subset (rather than constant on a large subset). Surprisingly, this version follows easily from Ramsey's Theorem, even in the weak system RCA₀ of reverse mathematics. We answer the question of the converse implication for pairs, showing that the Rainbow Ramsey Theorem for pairs is in fact strictly weaker than Ramsey's Theorem for pairs over RCA₀. The separation involves techniques (...) from the theory of randomness by showing that every 2-random bounds an ω-model of the Rainbow Ramsey Theorem for pairs. These results also provide as a corollary a new proof of Martin's theorem that the hyperimmune degrees have measure one. (shrink)

We investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is${\rm{\Delta }}_\alpha ^0 $-complete for someα. To prove this, we extend Montalbán’sη-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinalαand a cone in the Turing degrees such that the exact complexity (...) of computing an isomorphism between the given structure and another copy${\cal B}$in the cone is a c.e. degree in${\rm{\Delta }}_\alpha ^0\left$. In each of our theorems the cone in question is clearly described in the beginning of the proof, so it is easy to see how the theorems can be viewed as general theorems with certain effectiveness conditions. (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 analyze the degree spectra of structures in which different types of immunity conditions are encoded. In particular, we give an example of a structure whose degree spectrum coincides with the hyperimmune degrees. As a corollary, this shows the existence of an almost computable structure of which the complement of the degree spectrum is uncountable.

A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model U of T decidable in X. It is easy to see that $X = 0\prime$ is prime bounding. Denisov claimed that every $X <_{T} 0\prime$ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets $X \leq_{T} 0\prime$ are exactly the sets which are not $low_2$ . Recall that (...) X is $low_2$ if $X\prime\prime$ $\leq_{T} 0\prime$ . To prove that a $low_2$ set X is not prime bounding we use a $0\prime$ -computable listing of the array of sets { Y : Y $\leq_{T}$ X } to build a CAD theory T which diagonalizes against all potential X-decidable prime models U of T. To prove that any $non-low_{2}$ ; X is indeed prime bounding, we fix a function f $\leq_T$ X that is not dominated by a certain $0\prime$ -computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula $\varphi (\bar{x})$ consistent with T, a principal type which contains it, and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory. (shrink)

We consider the Turing degrees of prime models of complete decidable theories. In particular we show that every complete decidable atomic theory has a prime model whose elementary diagram is low. We combine the construction used in the proof with other constructions to show that complete decidable atomic theories have low prime models with added properties. If we have a complete decidable atomic theory with all types of the theory computable, we show that for every degree d with 0 0, (...) T has a d-decidable model omitting the nonprincipal types listed by L. (shrink)

Tarski defined a way of assigning to each Boolean algebra, B, an invariant inv(B) ∈ In, where In is a set of triples from ℕ, such that two Boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a Boolean algebra, there is a computable procedure that decides its elementary theory. If we restrict our attention to dense Boolean algebras, these invariants determine the algebra up to isomorphism. In this paper we (...) analyze the complexity of the question "Does B have invariant x?" For each x ∈ In we define a complexity class Γx that could be either Σⁿ, Πⁿ, Σⁿ ∧ Πⁿ, or Πω+1 depending on x, and we prove that the set of indices for computable Boolean algebras with invariant x is complete for the class Γx. Analogs of many of these results for computably enumerable Boolean algebras were proven in earlier works by Selivanov. In a more recent work, he showed that similar methods can be used to obtain the results for computable ones. Our methods are quite different and give new results as well. As the algebras we construct to witness hardness are all dense, we establish new similar results for the complexity of various isomorphism problems for dense Boolean algebras. (shrink)

A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model A, i.e., the elementary diagram De (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a (...) single CD theory T such that every homogeneous model of T has a PA degree. (shrink)

Topologists Nabutovsky and Weinberger discovered how to embed computably enumerable (c.e.) sets into the geometry of Riemannian metrics modulo diffeomorphisms. They used the complexity of the settling times of the c.e. sets to exhibit a much greater complexity of the depth and density of local minima for the diameter function than previously imagined. Their results depended on the existence of certain sequences of c.e. sets, constructed at their request by Csima and Soare, whose settling times had the necessary dominating (...) properties. Although these computability results had been announced earlier, their proofs have been deferred until this paper. Computably enumerable sets have long been used to prove undecidability of mathematical problems such as the word problem for groups and Hilbert's Tenth Problem. However, this example by Nabutovsky and Weinberger is perhaps the first example of the use of c.e. sets to demonstrate specific mathematical or geometric complexity of a mathematical structure such as the depth and distribution of local minima. (shrink)

In this paper, we give a characterization of the strong degrees of categoricity of computable structures greater or equal to [Formula: see text]. They are precisely the treeable degrees — the least degrees of paths through computable trees — that compute [Formula: see text]. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree [Formula: see text] with [Formula: see text] for [Formula: see text] a computable ordinal greater than [Formula: see (...) text] is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree [Formula: see text] with [Formula: see text] is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree [Formula: see text] with [Formula: see text] that is not the degree of categoricity of a rigid structure. (shrink)

To each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω, there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B >st A) if for every computable function f, for all but finitely many x, mB(x) > f(m₄(x)). This settling-time ordering, which is a natural extension to an ordering of the (...) idea of domination, was first introduced by Nabutovsky and Weinberger in [3] and Soare [6]. They desired a sequence of sets descending in this relationship to give results in differential geometry. In this paper we examine properties of the <st ordering. We show that it is not invariant under computable isomorphism, that any countable partial ordering embeds into it, that there are maximal and minimal sets, and that two c.e. sets need not have an inf or sup in the ordering. We also examine a related ordering, the strong settling-time ordering where we require for all computable f and g, for almost all x, mB(x) > f(mA(g(x))). (shrink)

Fraïssé studied countable structures S through analysis of the age of S i.e., the set of all finitely generated substructures of S. We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by (...) a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal. (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)

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.

A real is called properly n-generic if it is n-generic but not n+1-generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m > n ≥ 2 then an m-generic real cannot compute a properly n-generic real.

Anderson and Csima :245–264, 2014) defined a jump operator, the bounded jump, with respect to bounded Turing reducibility. They showed that the bounded jump is closely related to the Ershov hierarchy and that it satisfies an analogue of Shoenfield jump inversion. We show that there are high bounded low sets and low bounded high sets. Thus, the information coded in the bounded jump is quite different from that of the standard jump. We also consider whether the analogue of the (...) Jump Theorem holds for the bounded jump: do we have A≤bTB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \le _{bT}B$$\end{document} if and only if Ab≤1Bb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^b \le _1 B^b$$\end{document}? We show the forward direction holds but not the reverse. (shrink)

We consider certain linear orders with a function on them, and discuss for which types of functions the resulting structure is or is not computably categorical. Particularly, we consider computable copies of the rationals with a fixed-point free automorphism, and also ω with a non-decreasing function.

The $\Omega $ numbers—the halting probabilities of universal prefix-free machines—are known to be exactly the Martin-Löf random left-c.e. reals. We show that one cannot uniformly produce, from a Martin-Löf random left-c.e. real $\alpha $, a universal prefix-free machine U whose halting probability is $\alpha $. We also answer a question of Barmpalias and Lewis-Pye by showing that given a left-c.e. real $\alpha $, one cannot uniformly produce a left-c.e. real $\beta $ such that $\alpha - \beta $ is neither left-c.e. (...) nor right-c.e. (shrink)

When classes of structures are not first-order definable, we might still try to find a nice description. There are two common ways for doing this. One is to expand the language, leading to notions of pseudo-elementary classes, and the other is to allow infinite conjuncts and disjuncts. In this paper we examine the intersection. Namely, we address the question: Which classes of structures are both pseudo-elementary and ${\mathcal {L}}_{\omega _1, \omega }$ -elementary? We find that these are exactly the classes (...) that can be defined by an infinitary formula that has no infinitary disjunctions. (shrink)

A computable structure $\mathcal {A}$ is $\mathbf {x}$-computably categorical for some Turing degree $\mathbf {x}$ if for every computable structure $\mathcal {B}\cong\mathcal {A}$ there is an isomorphism $f:\mathcal {B}\to\mathcal {A}$ with $f\leq_{T}\mathbf {x}$. A degree $\mathbf {x}$ is a degree of categoricity if there is a computable structure $\mathcal {A}$ such that $\mathcal {A}$ is $\mathbf {x}$-computably categorical, and for all $\mathbf {y}$, if $\mathcal {A}$ is $\mathbf {y}$-computably categorical, then $\mathbf {x}\leq_{T}\mathbf {y}$. We construct a $\Sigma^{0}_{2}$ set whose degree (...) is not a degree of categoricity. We also demonstrate a large class of degrees that are not degrees of categoricity by showing that every degree of a set which is 2-generic relative to some perfect tree is not a degree of categoricity. Finally, we prove that every noncomputable hyperimmune-free degree is not a degree of categoricity. (shrink)

To reduce sex differences in employment outcomes, we must examine them in the context of the sex-gender hierarchy. The conventional explanation for wage gap—job segregation—is incorrect because it ignores men's incentive to preserve their advantages and their ability to do so by establishing the rules that distribute rewards. The primary method through which all dominant groups maintain their hegemony is by differentiating the subordinate group and defining it as inferior and hence meriting inferior treatment. My argument implies that neither sex-integrating (...) jobs nor implementing comparable worth will markedly improve women's employment status because men can subvert these mechanisms or even change the rules by which rewards are allocated. As evidence, I show that occupational integration has failed to advance women appreciably, and I argue that comparable worth is not likely to be much more effective. Instead, we must seek political analyses and political solutions. (shrink)

This study investigates whether and how sex and race affect access to and rewards for job authority, using 1980 survey data for 1,216 employed workers. The authors examine whether, net of human-capital characteristics, sex and race affect access to and compensation for job authority. In addition, the authors examine whether the translation of credentials into authority and earnings varies depending on workers' sex or race.

The role that class and ethnicity played in the self-fashioning of the professional men of letters who shaped fifteenth-century Spain's humanist project has been the subject of intense scholarly scrutiny for over fifty years. It was José Antonio Maravall who first demonstrated that the so-called letrados had a “conciencia estamental,” a class consciousness derived from their indispensable roles as administrators, advisors, diplomats, and chroniclers in the service of the crown. Subsequent scholarship showed that part of letrado self-consciousness resulted from the (...) hostility of Old Christian noblemen toward these ambitious non-noble, university-trained men, many of them New Christians or conversos. One way in which the aristocracy stigmatized the intellectual pursuits of the letrados was in fact to call those endeavors “Jewish.”. (shrink)

Este texto propõe discutir a controvérsia sobre o uso dos transgênicos na agricultura e pergunta também pela possibilidade de uma alternativa melhor a eles. Quando falamos em controvérsia, referimo-nos a uma discussão sobre ideias que refletem posturas opostas. Serão, portanto, apresentadas duas perspectivas distintas a respeito do tema: a perspectiva dos proponentes da transgenia e a dos críticos. A partir da perspectiva crítica, a agroecologia será elencada como possível alternativa para produção de alimentos considerando sua dimensão ética.

In this study, we investigate the use of narrative in online conversations among persons suffering from chronic opiate addiction and evaluate both its positive and negative uses. Illness narratives, as argued by sociologist Arthur Frank and psychiatrist/medical anthropologist Arthur Kleinman, enable patients to give order to life experiences and receive support from others. We wished to explore under what circumstances online support coalesces and breaks apart. The narratives we examined exemplify two topics frequently discussed on the message board: the recovery (...) process and what it means to be ‘clean’. To better understand these narratives from a theoretically based approach, we used the work of rhetorical theorist Kenneth Burke. Burke's description of two human motives, suffering and perfection, led us to an understanding of how unification and division happened within the online community. We found that the recovery narrative primarily embodied the author's suffering and, consequently, received support from other members of the message board. The second narrative centered on what it means to be ‘clean’ through a discussion of the author's desire to court temptation, revealing what Burke calls the rotten nature of perfection. As a result, the author of the narrative provoked disagreements and did not receive support. (shrink)

We performed three experiments to investigate whether adjectives can modulate the sensorimotor activation elicited by nouns. In Experiment 1, nouns of graspable objects were used as stimuli. Participants had to decide if each noun referred to a natural or artifact, by performing either a precision or a power reach‐to‐grasp movement. Response grasp could be compatible or incompatible with the grasp typically used to manipulate the objects to which the nouns referred. The results revealed faster reaction times (RTs) in compatible than (...) in incompatible trials. In Experiment 2, the nouns were combined with adjectives expressing either disadvantageous information about object graspability (e.g., sharp) or information about object color (e.g., reddish). No difference in RTs between compatible and incompatible conditions was found when disadvantageous adjectives were used. Conversely, a compatibility effect occurred when color adjectives were combined with nouns referring to natural objects. Finally, in Experiment 3 the nouns were combined with adjectives expressing tactile or shape proprieties of the objects (e.g., long or smooth). Results revealed faster RTs in compatible than in incompatible condition for both noun categories. Taken together, our findings suggest that adjectives can shape the sensorimotor activation elicited by nouns of graspable objects, highlighting that language simulation goes beyond the single‐word level. (shrink)

Experiments in animal models have shown that running increases neuronal activity in early visual areas in light as well as in darkness. This suggests that visual processing is influenced by locomotion independent of visual input. Combining mobile electroencephalography, motion- and eye-tracking, we investigated the influence of overground free walking on cortical alpha activity and eye movements in healthy humans. Alpha activity has been considered a valuable marker of inhibition of sensory processing and shown to negatively correlate with neuronal firing rates. (...) We found that walking led to a decrease in alpha activity over occipital cortex compared to standing. This decrease was present during walking in darkness as well as during light. Importantly, eye movements could not explain the change in alpha activity. Nevertheless, we found that walking and eye related movements were linked. While the blink rate increased with increasing walking speed independent of light or darkness, saccade rate was only significantly linked to walking speed in the light. Pupil size, on the other hand, was larger during darkness than during light, but only showed a modulation by walking in darkness. Analyzing the effect of walking with respect to the stride cycle, we further found that blinks and saccades preferentially occurred during the double support phase of walking. Alpha power, as shown previously, was lower during the swing phase than during the double support phase. We however could exclude the possibility that the alpha modulation was introduced by a walking movement induced change in electrode impedance. Overall, our work indicates that the human visual system is influenced by the current locomotion state of the body. This influence affects eye movement pattern as well as neuronal activity in sensory areas and might form part of an implicit strategy to optimally extract sensory information during locomotion. (shrink)