RELATIVIZING CHAITIN'S HALTING PROBABILITY

2005 ◽  
Vol 05 (02) ◽  
pp. 167-192 ◽  
Author(s):  
ROD DOWNEY ◽  
DENIS R. HIRSCHFELDT ◽  
JOSEPH S. MILLER ◽  
ANDRÉ NIES
Keyword(s):  
Computability Theory ◽  
Algorithmic Randomness ◽  
Jump Operator ◽  
Random Real ◽  
Primary Object ◽  
Object Of Study ◽  
Free Machine ◽  
Computable Functions ◽  
Omega Operator

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory and this Omega operator. But unlike the jump, which is invariant (up to computable permutation) under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness.

scholarly journals DEMUTH’S PATH TO RANDOMNESS

2015 ◽  
Vol 21 (3) ◽  
pp. 270-305 ◽  
Author(s):  
ANTONÍN KUČERA ◽  
ANDRÉ NIES ◽  
CHRISTOPHER P. PORTER
Keyword(s):  
Computability Theory ◽  
Truth Table ◽  
Algorithmic Randomness ◽  
Constructive Mathematics ◽  
Russian School ◽  
Constructive Analysis ◽  
Innovative Work ◽  
Classical Language ◽  
Computable Functions ◽  
Null Set

AbstractOsvald Demuth (1936–1988) studied constructive analysis from the viewpoint of the Russian school of constructive mathematics. In the course of his work he introduced various notions of effective null set which, when phrased in classical language, yield a number of major algorithmic randomness notions. In addition, he proved several results connecting constructive analysis and randomness that were rediscovered only much later.In this paper, we trace the path that took Demuth from his constructivist roots to his deep and innovative work on the interactions between constructive analysis, algorithmic randomness, and computability theory. We will focus specifically on (i) Demuth’s work on the differentiability of Markov computable functions and his study of constructive versions of the Denjoy alternative, (ii) Demuth’s independent discovery of the main notions of algorithmic randomness, as well as the development of Demuth randomness, and (iii) the interactions of truth-table reducibility, algorithmic randomness, and semigenericity in Demuth’s work.

DEEP CLASSES

2016 ◽  
Vol 22 (2) ◽  
pp. 249-286 ◽  
Author(s):  
LAURENT BIENVENU ◽  
CHRISTOPHER P. PORTER
Keyword(s):  
Mutual Information ◽  
Initial Segment ◽  
Computability Theory ◽  
Binary Sequences ◽  
Algorithmic Randomness ◽  
Positive Probability ◽  
Probabilistic Algorithm ◽  
Image Position ◽  
The Relationship ◽  
Mass Problems

AbstractA 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.

Fernando de Alva Ixtlilxochitl

2020 ◽  
Author(s):  
Amber Brian
Keyword(s):  
New Spain ◽  
Material Sources ◽  
Primary Object ◽  
History Of ◽  
New Material ◽  
Magnum Opus ◽  
Object Of Study ◽  
Historical Accounts ◽  
Colonial Government ◽  
The City

Don Fernando de Alva Ixtlilxochitl (b. c. 1578–d. 1650) is a relatively unknown figure outside of specialist academic circles, yet he has been very influential in the development of the historiography of pre-Hispanic central Mexico, or Anahuac. Born in the last quarter of the 16th century, his family had roots in Anahuac and in Spain. His mother was descended from elite native rulers of the city of Tetzcoco, while his father was a Spanish settler who worked as a Nahuatl-Spanish interpreter in the courts of Mexico City. Alva Ixtlilxochitl also served as an interpreter and as a bureaucratic official in the colonial government. During his lifetime, Alva Ixtlilxochitl’s family’s wealth and status were tied to his mother’s and grandmother’s connections to the family’s cacicazgo (landed estate) in San Juan Teotihuacan. Yet it was his ancestors from Tetzcoco who were the primary object of study in his five historical works. In four historical accounts and his magnum opus, the History of the Chichimeca Nation (2019, cited under Manuscripts, Editions, Translations), Alva Ixtlilxochitl recounts the origins, deeds, and exploits of the leaders of Tetzcoco, including the renowned Nezahualcoyotl (r. 1429–1472) and Nezahualpilli (r. 1472–1515). For these histories he relied on native sources. As he says himself in the prefatory materials to the History of the Chichimeca Nation, these sources included “painted histories and annals and the songs with which they preserved them,” and to make sense of these materials he sought out “the elders of New Spain who were renowned for their knowledge and understanding of those stories” (History of the Chichimeca Nation, p. 29). The result of Alva Ixtlilxochitl’s research and writing has left an important legacy in studies of the history of ancient Mexico. Scholars from the 17th century onward drew on Alva Ixtlilxochitl’s representations of pre-Hispanic and conquest-era Mexico to such an extent that his depictions of Tetzcoco as a center of learning and culture and his depictions of Nezahualcoyotl as a revered poet-king became standard in both academic studies and popular culture. Burgeoning scholarly interest in mestizo historians in the 1990s brought renewed attention to Alva Ixtlilxochitl’s writings and to his position as a colonial subject and author, while the rediscovery of Alva Ixtlilxochitl’s original manuscripts in the 1980s provided new material sources with which to study the creation and impact of his works. Alva Ixtlilxochitl’s own projects and his legacy represent an important reminder of how, on occasion, the stories and storytelling of native peoples survived the brutalities of conquest and colonialism.

scholarly journals Denjoy, Demuth and density

2014 ◽  
Vol 14 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Laurent Bienvenu ◽  
Rupert Hölzl ◽  
Joseph S. Miller ◽  
André Nies
Keyword(s):  
Computable Function ◽  
Density Theorem ◽  
Positive Density ◽  
Random Real ◽  
Open Problems ◽  
Covering Problems ◽  
Closed Class ◽  
Random Reals ◽  
Upper And Lower Derivatives ◽  
Computable Functions

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 𝒞 ⊆ [0, 1] containing z has positive density at z. Under the stronger assumption that z is not LR-hard, we show that every such class has density one at z. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Löf random reals and K-trivial sets: the noncupping and covering problems. We say that f : [0, 1] → ℝ satisfies the Denjoy alternative at z ∈ [0, 1] if either the derivative f′(z) exists, or the upper and lower derivatives at z are +∞ and -∞, respectively. The Denjoy–Young–Saks theorem states that every function f : [0, 1] → ℝ satisfies the Denjoy alternative at almost every z ∈ [0, 1]. We answer a question posed by Kučera in 2004 by showing that a real z is computably random if and only if every computable function f satisfies the Denjoy alternative at z. For Markov computable functions, which are only defined on computable reals, we can formulate the Denjoy alternative using pseudo-derivatives. Call a real zDA-random if every Markov computable function satisfies the Denjoy alternative at z. We considerably strengthen a result of Demuth (Comment. Math. Univ. Carolin.24(3) (1983) 391–406) by showing that every Turing incomplete Martin-Löf random real is DA-random. The proof involves the notion of nonporosity, a variant of density, which is the bridge between the two themes of this paper. We finish by showing that DA-randomness is incomparable with Martin-Löf randomness.

scholarly journals USING ALMOST-EVERYWHERE THEOREMS FROM ANALYSIS TO STUDY RANDOMNESS

2016 ◽  
Vol 22 (3) ◽  
pp. 305-331 ◽  
Author(s):  
KENSHI MIYABE ◽  
ANDRÉ NIES ◽  
JING ZHANG
Keyword(s):  
Integrable Function ◽  
Lebesgue Point ◽  
Algorithmic Randomness ◽  
List Type ◽  
Type Number ◽  
Random Real ◽  
Computably Enumerable Set ◽  
Almost Everywhere ◽  
Image Position ◽  
Nondecreasing Functions

AbstractWe study algorithmic randomness notions via effective versions of almost-everywhere theorems from analysis and ergodic theory. The effectivization is in terms of objects described by a computably enumerable set, such as lower semicomputable functions. The corresponding randomness notions are slightly stronger than Martin–Löf (ML) randomness.We establish several equivalences. Given a ML-random realz, the additional randomness strengths needed for the following are equivalent.(1)all effectively closed classes containingzhave density 1 atz.(2)all nondecreasing functions with uniformly left-c.e. increments are differentiable atz.(3)zis a Lebesgue point of each lower semicomputable integrable function.We also consider convergence of left-c.e. martingales, and convergence in the sense of Birkhoff’s pointwise ergodic theorem. Lastly, we study randomness notions related to density of${\rm{\Pi }}_n^0$and${\rm{\Sigma }}_1^1$classes at a real.

The Study of ‘Complex Society’ in the Middle East: A Review Essay

1978 ◽  
Vol 9 (4) ◽  
pp. 539-557 ◽  
Author(s):  
Philip Carl Salzman
Keyword(s):  
Middle East ◽  
Participant Observation ◽  
Good Deal ◽  
The Other ◽  
Small Scale ◽  
Complex Society ◽  
Primary Object ◽  
Organic Solidarity ◽  
Economic Markets ◽  
Object Of Study

Anthropologists have devoted a good deal of attention recently to what they call ‘complex society’. This rather vague concept developed in contrast with ‘primitive’ or ‘simple’ society, the small-scale, isolated, local-oriented, non- literate grouping of like social parts which anthropologists made, or fancied, their primary object of study. This is Tonnies’ gemeinschaft, held together by Durkheim's ‘mechanical solidarity.’ ‘Complex society’ on the other hand, is more similar to Tonnies’ gesellschaft, bases to some degree upon Durkheim's ‘organic solidarity’; it has many differentiated parts, ingeniously interwoven into elaborate structures, with specializations and rankings and overlappings and other imaginative complications. More and more anthropologists found themselves, whatever their original intentions, involved in studies that were manifestly of ‘complex society.’ This was the result of two developments: One was the encapsulation of most ‘simple’ societies by colonial or national societies, and the concomitant engagement with government, economic markets, and development (or under development). This encapsulation was not something completely new that happened during the course of anthropological investigation, but something which had been going on and which anthropologists ‘discovered’ and began to devote attention to. The other development was the carrying of anthropological research to the areas of the ‘great civilizations’ in East Asia, South Asia, and the Middle East. In these areas, long-recorded histories, literate traditions, great states and empires, and sophisticated technologies belied the notion of ‘simple’ society, and raised embarrassing questions about classical anthropological methodology, ‘participant observation’ in a constricted area for one or two years.

scholarly journals The Cosine-Sine Functional Equation on Semigroups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Bruce Ebanks
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Direct Solution ◽  
Continuous Solutions ◽  
Topological Semigroups ◽  
Primary Object ◽  
Object Of Study ◽  
Special Case ◽  
Sine Functional Equation

Abstract The primary object of study is the “cosine-sine” functional equation f(xy) = f(x)g(y)+g(x)f(y)+h(x)h(y) for unknown functions f, g, h : S → ℂ, where S is a semigroup. The name refers to the fact that it contains both the sine and cosine addition laws. This equation has been solved on groups and on semigroups generated by their squares. Here we find the solutions on a larger class of semigroups and discuss the obstacles to finding a general solution for all semigroups. Examples are given to illustrate both the results and the obstacles. We also discuss the special case f(xy) = f(x)g(y) + g(x)f(y) − g(x)g(y) separately, since it has an independent direct solution on a general semigroup. We give the continuous solutions on topological semigroups for both equations.

scholarly journals Vireo: Bayesian demultiplexing of pooled single-cell RNA-seq data without genotype reference

2019 ◽  
Author(s):  
Yuanhua Huang ◽  
Davis J McCarthy ◽  
Oliver Stegle
Keyword(s):  
Single Cell ◽  
Real Data ◽  
Experimental Designs ◽  
Joint Analysis ◽  
Rna Seq ◽  
Computationally Efficient ◽  
Primary Object ◽  
Genotype Information ◽  
Object Of Study ◽  
Multiple Samples

AbstractThe joint analysis of multiple samples using single-cell RNA-seq is a promising experimental design, offering both increased throughput while allowing to account for batch variation. To achieve multi-sample designs, genetic variants that segregate between the samples in the pool have been proposed as natural barcodes for cell demultiplexing. Existing demultiplexing strategies rely on access to complete genotype data from the pooled samples, which greatly limits the applicability of such methods, in particular when genetic variation is not the primary object of study. To address this, we here present Vireo, a computationally efficient Bayesian model to demultiplex single-cell data from pooled experimental designs. Uniquely, our model can be applied in settings when only partial or no genotype information is available. Using simulations based on synthetic mixtures and results on real data, we demonstrate the robustness of our model and illustrate the utility of multi-sample experimental designs for common expression analyses.

scholarly journals Vireo: Bayesian demultiplexing of pooled single-cell RNA-seq data without genotype reference

2019 ◽  
Vol 20 (1) ◽  
Author(s):  
Yuanhua Huang ◽  
Davis J. McCarthy ◽  
Oliver Stegle
Keyword(s):  
Single Cell ◽  
Real Data ◽  
Experimental Designs ◽  
Rna Seq ◽  
Computationally Efficient ◽  
Primary Object ◽  
Genotype Information ◽  
Object Of Study ◽  
Multiple Samples ◽  
Pooled Samples

AbstractMultiplexed single-cell RNA-seq analysis of multiple samples using pooling is a promising experimental design, offering increased throughput while allowing to overcome batch variation. To reconstruct the sample identify of each cell, genetic variants that segregate between the samples in the pool have been proposed as natural barcode for cell demultiplexing. Existing demultiplexing strategies rely on availability of complete genotype data from the pooled samples, which limits the applicability of such methods, in particular when genetic variation is not the primary object of study. To address this, we here present Vireo, a computationally efficient Bayesian model to demultiplex single-cell data from pooled experimental designs. Uniquely, our model can be applied in settings when only partial or no genotype information is available. Using pools based on synthetic mixtures and results on real data, we demonstrate the robustness of Vireo and illustrate the utility of multiplexed experimental designs for common expression analyses.

De Invloed Van Taalstatus en Taaltypologie op T2-Vaardigheid, Attitudes en Motivatie

2009 ◽  
Vol 81 ◽  
pp. 21-30
Author(s):  
E. Schoonheere ◽  
Aurélie Welcomme ◽  
Els Schoonjans ◽  
Sonja Janssens
Keyword(s):  
Social Psychological ◽  
L2 Proficiency ◽  
Young Learners ◽  
Explorative Study ◽  
L2 Motivation ◽  
Primary Object ◽  
Lower Status ◽  
L2 Learning ◽  
Object Of Study ◽  
Complex Matter

The relevance of L1 status and typology for L2 learning is often assumed. These concepts, however, are rarely the primary object of study in SLA research, in part because defining them is a complex matter. In an attempt to redress this imbalance, this explorative study investigates French L2 social-psychological dispositions and global French L2 proficiency in young learners in European schools in Brussels and Luxemburg (n=83; 585h of FSL on average). It was found that pupils with a lower status L1 (German as compared to English) display more positive L2 attitudes, higher L2 motivation and achieve higher levels of global L2 proficiency. A different trend was observed in pupils whose L1 is typologically similar to the L2 (Spanish as compared to German): few significant differences were found concerning social-psychological dispositions as well as global L2 proficiency.

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