EveryK-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.

We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy–Young–Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class.

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets by higher (...) random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma. (shrink)

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences. In what follows, (...) we study the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness and Kurtz randomness, together with the relations of equivalence and consistency from probability theory. We show that all these relations coincide when restricted to the class of computable strongly positive generalized Bernoulli measures. For the case of arbitrary computable measures, we obtain a complete and somewhat surprising picture of the implications between these relations that hold in general. (shrink)

We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15]. We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence. We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence and that every Demuth random sequence forms a minimal pair with every pb-generic sequence. Moreover, we prove that for every (...) comeager${\cal G} \subseteq {2^\omega }$, there is some weakly 2-random sequenceXthat computes some$Y \in {\cal G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity. (shrink)

A semimeasure is a generalization of a probability measure obtained by relaxing the additivity requirement to superadditivity. We introduce and study several randomness notions for left-c.e. semimeasures, a natural class of effectively approximable semimeasures induced by Turing functionals. Among the randomness notions we consider, the generalization of weak 2-randomness to left-c.e. semimeasures is the most compelling, as it best reflects Martin-Löf randomness with respect to a computable measure. Additionally, we analyze a question of Shen, a positive answer to which would (...) also have yielded a reasonable randomness notion for left-c.e. semimeasures. Unfortunately, though, we find a negative answer, except for some special cases. (shrink)

A set of infinite binary sequences ${\cal C} \subseteq 2$ℕ is negligible if there is no partial probabilistic algorithm that produces an element of this set with positive probability. The study of negligibility is of particular interest in the context of ${\rm{\Pi }}_1^0 $ classes. In this paper, we introduce the notion of depth for ${\rm{\Pi }}_1^0 $ classes, which is a stronger form of negligibility. Whereas a negligible ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot (...) probabilistically compute a member of ${\cal C}$ with positive probability, a deep ${\rm{\Pi }}_1^0 $ class ${\cal C}$ has the property that one cannot probabilistically compute an initial segment of a member of ${\cal C}$ with high probability. That is, the probability of computing a length n initial segment of a deep ${\rm{\Pi }}_1^0 $ class converges to 0 effectively in n.We prove a number of basic results about depth, negligibility, and a variant of negligibility that we call tt-negligibility. We provide a number of examples of deep ${\rm{\Pi }}_1^0 $ classes that occur naturally in computability theory and algorithmic randomness. We also study deep classes in the context of mass problems, examine the relationship between deep classes and certain lowness notions in algorithmic randomness, and establish a relationship between members of deep classes and the amount of mutual information with Chaitin’s Ω. (shrink)

Cet article a pour but de présenter la première traduction française de deux textes de Helmholtz de 1878. À la lumière du développement des géométries non euclidiennes, il y critique la conception kantienne de l’espace. Par là même, il expose une redéfinition purement empiriste de la construction des déterminations spatiales qui, sous le nom de « géométrie physique », joua un rôle important chez Poincaré (qui la révise) et chez Einstein.

RÉSUMÉ: En 1938, Hans Reichenbach soutenait, dans Experience and Prediction, que la théorie de la signification défendue par le positivisme logique, à savoir le «verificationnisme strict», n’avait pas réussi à formuler un critère satisfaisant de la signification cognitive des énoncés. Il le remplaça par un critère purement probabiliste qui lui permettait de restaurer la connexion pragmatique entre le langage et l’action. Mais de sérieuses difficultés grèvent la justification de cette théorie. Le but de cet article est de se pencher sur (...) certaines de ces difficultés et de tenter d’y répondre d’une façon «quasi-reichenbachienne».ABSTRACT: Hans Reichenbach argued in Experience and Prediction (1938) that the theory of meaning defended by Logical Positivism, i.e., “strict verificationism,” failed to give a relevant criterion for cognitive meaningfulness. He replaces it with a wholly probabilistic criterion that allows us to establish a pragmatic connection between language and action. But serious difficulties arise in the justification of his theory. The aim of this article is to assess some of these difficulties and to indicate a way to solve them in a “reichenbachian” way. (shrink)

RÉSUMÉ: En 1938, Hans Reichenbach soutenait, dans Experience and Prediction, que la théorie de la signification défendue par le positivisme logique, à savoir le «verificationnisme strict», n’avait pas réussi à formuler un critère satisfaisant de la signification cognitive des énoncés. Il le remplaça par un critère purement probabiliste qui lui permettait de restaurer la connexion pragmatique entre le langage et l’action. Mais de sérieuses difficultés grèvent la justification de cette théorie. Le but de cet article est de se pencher sur (...) certaines de ces difficultés et de tenter d’y répondre d’une façon «quasi-reichenbachienne».ABSTRACT: Hans Reichenbach argued in Experience and Prediction (1938) that the theory of meaning defended by Logical Positivism, i.e., “strict verificationism,” failed to give a relevant criterion for cognitive meaningfulness. He replaces it with a wholly probabilistic criterion that allows us to establish a pragmatic connection between language and action. But serious difficulties arise in the justification of his theory. The aim of this article is to assess some of these difficulties and to indicate a way to solve them in a “reichenbachian” way. (shrink)

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)

Since the mid-1980s, there has been a gradual ethical drift in the provision of maternal care in African health facilities in general, and in Cameroon in particular, despite government efforts. In fact, in Cameroon, an increasing number of caregivers are reportedly not providing compassionate care in maternity services. Consequently, many women, particularly the financially vulnerable, experience numerous difficulties in accessing these health services. In this article, we highlight the unequal access to care in public maternity services in Cameroon in general (...) and the Noun Division in particular. For this study, in addition to documentary review, two qualitative data collection techniques were used: direct observation and individual interviews. Following the field work, the observation data were categorized and analyzed to assess their relevance and significance in relation to the topics listed in the observation checklist. Interviews were recorded using a dictaphone; they were subsequently transcribed and the data categorized and coded. After this stage, an analysis grid was constructed for content analysis of the transcripts, to study the frequency of topics addressed during the interviews, as well as divergences and convergences among the respondents. The results of this data analysis showed that money has become the driving force in service provision. As such, it is the patient’s economic capital that counts. Considered “clients”, pregnant women without sufficient financial resources wait long hours in corridors; some die in pain under the indifferent gaze of the professionals who are supposed to take care of them. In sharp contrast, the findings revealed that financially privileged patients are able to bribe caregivers to attract their favour and obtain prompt, careful, and effective care. These ethical abuses observed in the Noun public health facilities drive women to use, from the beginning of their pregnancies to the delivery, only healthcare delivered by traditional health attendants. (shrink)

The New Evil Demon problem has been hotly debated since the case was introduced in the early 1980’s (e.g. Lehrer and Cohen 1983; Cohen 1984), and there seems to be recent increased interest in the topic. In a forthcoming collection of papers on the New Evil Demon problem (Dutant and Dorsch, forthcoming), at least two of the papers, both by prominent epistemologists, attempt to resist the problem by appealing to the distinction between justification and excuses. My primary aim here is (...) to critically evaluate this new excuse maneuver as a response to the New Evil Demon problem. -/- Their response attempts to give us reason to reject the idea that victims of the New Evil Demon have justification for believing as they do. I shall argue that this approach is ultimately unsuccessful, however much of value can be learned from these attempts. In particular, progress in the debate can be made by following those who advance the excuse maneuver and make explicit the connection between epistemic justification and epistemic norms. By doing so, the questions being debated are clarified, as is the methodology being used to attempt to answer them. (shrink)

There are various equivalent formulations of the Church-Turing thesis. A common one is that every effective computation can be carried out by a Turing machine. The Church-Turing thesis is often misunderstood, particularly in recent writing in the philosophy of mind.

Crop genetic resources constitute a ‘new’ global commons, characterized by multiple layers of activities of farmers, genebanks, public and private research and development organizations, and regulatory agencies operating from local to global levels. This paper presents sui generis biocultural community protocols that were developed by four communities in Benin and Madagascar to improve their ability to contribute to, and benefit from, the crop commons. The communities were motivated in part by the fact that their national governments’ had recently ratified the (...) Plant Treaty and the Nagoya Protocol, which make commitments to promoting the rights of indigenous peoples, local communities and farmers, without being prescriptive as to how Contracting Parties should implement those commitments. The communities identified the protocols as useful means to advance their interests and/or rights under both the Plant Treaty and the Nagoya Protocol to be recognized as managers of local socio-ecological systems, to access genetic resources from outside the communities, and to control others’ access to resources managed by the community. (shrink)