An illustrated exploration of fandom that combines academic essays with artist pages and experimental texts. Fandom as Methodology examines fandom as a set of practices for approaching and writing about art. The collection includes experimental texts, autobiography, fiction, and new academic perspectives on fandom in and as art. Key to the idea of “fandom as methodology” is a focus on the potential for fandom in art to create oppositional spaces, communities, and practices, particularly from queer perspectives, but also through transnational, (...) feminist and artist-of-color fandoms. The book provides a range of examples of artists and writers working in this vein, as well as academic essays that explore the ways in which fandom can be theorized as a methodology for art practice and art history. Fandom as Methodology proposes that many artists and art writers already draw on affective strategies found in fandom. With the current focus in many areas of art history, art writing, and performance studies around affective engagement with artworks and imaginative potentials, fandom is a key methodology that has yet to be explored. Interwoven into the academic essays are lavishly designed artist pages in which artists offer an introduction to their use of fandom as methodology. Contributors Taylor J. Acosta, Catherine Grant, Dominic Johnson, Kate Random Love, Maud Lavin, Owen G. Parry, Alice Butler, SooJin Lee, Jenny Lin, Judy Batalion, Ika Willis. Artists featured in the artist pages Jeremy Deller, Ego Ahaiwe Sowinski, Anna Bunting-Branch, Maria Fusco, Cathy Lomax, Kamau Amu Patton, Holly Pester, Dawn Mellor, Michelle Williams Gamaker, The Women of Colour Index Reading Group, Liv Wynter, Zhiyuan Yang. (shrink)

A set $B\subseteq\mathbb{N}$ is called low for Martin-Löf random if every Martin-Löf random set is also Martin-Löf random relative to B . We show that a $\Delta^0_2$ set B is low for Martin-Löf random if and only if the class of oracles which compress less efficiently than B , namely, the class $\mathcal{C}^B=\{A\ |\ \forall n\ K^B(n)\leq^+ K^A(n)\}$ is countable (where K denotes the prefix-free complexity and $\leq^+$ denotes inequality modulo a constant. It follows that $\Delta^0_2$ is the largest arithmetical (...) class with this property and if $\mathcal{C}^B$ is uncountable, it contains a perfect $\Pi^0_1$ set of reals. The proof introduces a new method for constructing nontrivial reals below a $\Delta^0_2$ set which is not low for Martin-Löf random. (shrink)

In this paper we consider conditional random quantities (c.r.q.’s) in the setting of coherence. Based on betting scheme, a c.r.q. X|H is not looked at as a restriction but, in a more extended way, as \({XH + \mathbb{P}(X|H)H^c}\) ; in particular (the indicator of) a conditional event E|H is looked at as EH + P(E|H)H c . This extended notion of c.r.q. allows algebraic developments among c.r.q.’s even if the conditioning events are different; then, for instance, we can give a (...) meaning to the sum X|H + Y|K and we can define the iterated c.r.q. (X|H)|K. We analyze the conjunction of two conditional events, introduced by the authors in a recent work, in the setting of coherence. We show that the conjoined conditional is a conditional random quantity, which may be a conditional event when there are logical dependencies. Moreover, we introduce the negation of the conjunction and by applying De Morgan’s Law we obtain the disjoined conditional. Finally, we give the lower and upper bounds for the conjunction and disjunction of two conditional events, by showing that the usual probabilistic properties continue to hold. (shrink)

We consider two ways one might use algorithmic randomness to characterize a probabilistic law. The first is a generative chance* law. Such laws involve a nonstandard notion of chance. The second is a probabilistic* constraining law. Such laws impose relative frequency and randomness constraints that every physically possible world must satisfy. While each notion has virtues, we argue that the latter has advantages over the former. It supports a unified governing account of non-Humean laws and provides independently motivated (...) solutions to issues in the Humean best-system account. On both notions, we have a much tighter connection between probabilistic laws and their corresponding sets of possible worlds. Certain histories permitted by traditional probabilistic laws are ruled out as physically impossible. As a result, such laws avoid one variety of empirical underdetermination, but the approach reveals other varieties of underdetermination that are typically overlooked. (shrink)

According to a traditional view, scientific laws and theories constitute algorithmic compressions of empirical data sets collected from observations and measurements. This article defends the thesis that, to the contrary, empirical data sets are algorithmically incompressible. The reason is that individual data points are determined partly by perturbations, or causal factors that cannot be reduced to any pattern. If empirical data sets are incompressible, then they exhibit maximal algorithmic complexity, maximal entropy and zero redundancy. They are therefore maximally efficient carriers (...) of information about the world. Since, on algorithmic information theory, a string is algorithmically random just if it is incompressible, the thesis entails that empirical data sets consist of algorithmically random strings of digits. Rather than constituting compressions of empirical data, scientific laws and theories pick out patterns that data sets exhibit with a certain noise. (shrink)

It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Löf randomness? This question is discussed in Section 6.Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore likely (...) to be hard, we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers. Some questions are close contenders here; these are marked by ▷. With few exceptions, the questions are precise. They mostly have a yes/no answer. However, there are often more general questions of an intuitive or even philosophical nature behind. We give an outline, indicating the more general questions.All sets will be sets of natural numbers, unless otherwise stated. These sets are identified with infinite strings over {0, 1}. Other terms used in the literature are sequence and real. (shrink)

We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0'. This contrasts with a result of Kučera and Terwijn [5] on sets that are low for the class of Martin-Löf random reals.

How realistic is it to adopt a quantum random walk model to account for decisions involving two choices? Here, we discuss the neural plausibility and the effect of initial state and boundary thresholds on such a model and contrast it with various features of the classical random walk model of decision making.

A randomly selected microcosm of the people can usefully play an official role in the lawmaking process. However, there are serious issues to be confronted if such a random sample were to take on the role of a full-scale, full-time second chamber. Some skeptical considerations are detailed. There are also advantages to short convenings of such a sample to take on some of the roles of a second chamber. This article provides a response to the skeptical considerations. Precedents from ancient (...) Athens show how such short-term convenings of a deliberating microcosm can be positioned before, during, or after other elements of the lawmaking process. The article draws on experience from Deliberative Polling to show how this is both practical and productive for the lawmaking process. (shrink)

Belief in free will is widespread. The present research considered one reason why people may believe that actions are freely chosen rather than determined: they attribute randomness in behavior to free will. Experiment 1 found that participants who were prompted to perform a random sequence of actions experienced their behavior as more freely chosen than those who were prompted to perform a deterministic sequence. Likewise, Experiment 2 found that, all else equal, the behavior of animated agents was perceived to (...) be more freely chosen if it consisted of a random sequence of actions than if it consisted of a deterministic sequence; this was true even when the degree of randomness in agents’ behavior was largely a product of their environments. Together, these findings suggest that randomness in behavior—one’s own or another’s—can be mistaken for free will. (shrink)

We investigate the strength of a randomness notion ${\cal R}$ as a set-existence principle in second-order arithmetic: for each Z there is an X that is ${\cal R}$-random relative to Z. We show that the equivalence between 2-randomness and being infinitely often C-incompressible is provable in $RC{A_0}$. We verify that $RC{A_0}$ proves the basic implications among randomness notions: 2-random $\Rightarrow$ weakly 2-random $\Rightarrow$ Martin-Löf random $\Rightarrow$ computably random $\Rightarrow$ Schnorr random. Also, over $RC{A_0}$ the existence of computable (...) randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-Löf randoms, and we describe a sense in which this result is nearly optimal. (shrink)

We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; higher level randomness (...) notions going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less naturalrequirement-freesolution to Post's Problem, much along the lines of the Dekker deficiency set. (shrink)

The book is intended to explain the larger and intuitive concept of randomness by means of computation, particularly through algorithmic complexity and recursion theory. It also includes the transcriptions (by A. German) of two panel discussion on the topics: Is The Universe Random?, held at the University of Vermont in 2007; and What is Computation? (How) Does Nature Compute?, held at the University of Indiana Bloomington in 2008. The book is intended to the general public, undergraduate and graduate students (...) in math, computer science, physics and other sciences, but also to philosophers of science and researchers. (shrink)

We study randomness beyond${\rm{\Pi }}_1^1$-randomness and its Martin-Löf type variant, which was introduced in [16] and further studied in [3]. Here we focus on a class strictly between${\rm{\Pi }}_1^1$and${\rm{\Sigma }}_2^1$that is given by the infinite time Turing machines introduced by Hamkins and Kidder. The main results show that the randomness notions associated with this class have several desirable properties, which resemble those of classical random notions such as Martin-Löf randomness and randomness notions defined via effective (...) descriptive set theory such as${\rm{\Pi }}_1^1$-randomness. For instance, mutual randoms do not share information and a version of van Lambalgen’s theorem holds.Towards these results, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool towards this result, we prove facts of independent interest about random forcing over increasing unions of admissible sets, which allow efficient proofs of some classical results about hyperarithmetic sets. (shrink)

Chance inevitably plays a role in law but it is not often that we consciously try to import an element of randomness into a legal process. Random Justice: On Lotteries and Legal Decision-Making explores the potential for the use of lotteries in social, and particularly legal, decision-making contexts. Utilizing a variety of disciplines and materials, Neil Duxbury considers in detail the history, advantages, and drawbacks of deciding issues of social significance by lot and argues that the value of the (...) lottery as a legal decision-making device has generally been underestimated. The very fact that there exists widespread resistance to the use of lotteries for legal decision-making purposes betrays a commonly held belief that legal processes are generally more important than are legal outcomes. Where, owing to the existence of indeterminacy, the process of reasoning is likely to be excessively protracted and the reasons provided strongly contestable, the most cost-efficient and impartial decision-making strategy may well be recourse to lot. Aversion to this strategy, while generally understandable, is not necessarily rational. Yet in law, as Professor Duxbury demonstrates, reason is generally valued more highly than is rationality. The lottery is often conceived to be a decision-making device that operates in isolation. Yet lotteries can frequently and profitably be incorporated into other decision-frameworks. The book concludes by controversially considering how lotteries might be so incorporated and also advances the thesis that it may sometimes be sensible to require that adjudication takes place in the shadow of a lottery. (shrink)

We study model theory of random variables using finitary integral logic. We prove definability of some probability concepts such as having F as distribution function, independence and martingale property. We then deduce Kolmogorov's existence theorem from the compactness theorem.

A widely accepted tenet of evolutionary biology is that spontaneous mutations occur randomly with regard to their fitness effect. However, since the mutation rate varies along a genome and this variation can be subject to selection, organisms might evolve lower mutation rates at loci where mutations are most deleterious or increased rates where mutations are most needed. In fact, mechanisms of targeted hypermutation are known in organisms ranging from bacteria to humans. Here we review the main forces driving the evolution (...) of local mutation rates and identify the main limiting factors. Both targeted hyper‐ and hypomutation can evolve, although the former is restricted to loci under very frequent positive selection and the latter is severely limited by genetic drift. Nevertheless, we show how an association of repair with transcription or chromatin‐associated proteins could overcome the drift limit and lead to non‐random hypomutation along the genome in most organisms.Editor's suggested further reading in BioEssays Stress‐induced mutation via DNA breaks in Escherichia coli: A molecular mechanism with implications for evolution and medicine Abstract. (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)

We investigate the question “To what extent can random reals be used as a tool to establish number theoretic facts?” Let $\text{2-\textit{RAN\/}}$ be the principle that for every real $X$ there is a real $R$ which is 2-random relative to $X$. In Section 2, we observe that the arguments of Csima and Mileti [3] can be implemented in the base theory $\text{\textit{RCA}}_0$ and so $\text{\textit{RCA}}_0+\text{2-\textit{RAN\/}}$ implies the Rainbow Ramsey Theorem. In Section 3, we show that the Rainbow Ramsey Theorem is (...) not conservative over $\text{\textit{RCA}}_0$ for arithmetic sentences. Thus, from the Csima—Mileti fact that the existence of random reals has infinitary-combinatorial consequences we can conclude that $\text{2-\textit{RAN\/}}$ has non-trivial arithmetic consequences. In Section 4, we show that $\text{2-\textit{RAN\/}}$ is conservative over $\text{\textit{RCA}}_0+\text{\textit{B\/}$\,\Sigma$}_2$ for $\Pi^1_1$-sentences. Thus, the set of first-order consequences of $\text{2-\textit{RAN\/}}$ is strictly stronger than $P^-+I\Sigma_1$ and no stronger than $P^-+\text{\textit{B\/}$\,\Sigma$}_2$. (shrink)

We develop and knit together several theodicies in order to find a more complete picture of why certain forms of animal suffering might be permitted by a perfect being. We focus on an especially potent form of the problem of evil, which arises from considering why a perfectly good, wise, and powerful God might use evolutionary mechanisms that predictably result in so much animal suffering and loss of life. There are many existing theodicies on the market, and although they offer (...) helpful resources, we combine and further develop several proposals to produce a composite theodicy that avoids certain shortcomings of the individual theodicies. An important element of our project is locating a role for randomness in cosmic and biological evolution. In particular, we show how randomness might enhance or enable certain goods, including everlasting goods, at the risk of temporary evils. (shrink)

Using possibly infinite computations on universal monotone Turing machines, we prove Martin-Löf randomness in ∅' of the probability that the output be in some set.

We investigate the class of m-hypergraphs whose substructures with l elements have more than sm-element subsets that do not form a hyperedge. The class will have the free amalgamation property if s is small, but it does not if s is large. We find the boundary of s. Suppose the class has the free amalgamation property. In the case $$m \ge 3$$ m ≥ 3, we demonstrate that the random structure for the class has continuum-many automorphisms with a single orbit. (...) The situation differs from the case of Henson graphs. In the case of generic hypergraphs constructed by Hrushovski’s method using a predimension function, we also demonstrate that they have no automorphisms with a single orbit. (shrink)

We examine the randomness and triviality of reals using notions arising from martingales and prefix-free machines.

Are there any truly ontologically random events? This paper argues that randomness is an unavoidably epistemic concept and therefore ascription of ontological randomness to any particular event or series of events can never be justified.

Peter Urbach; X*—Random Sampling and the Principles of Estimation, Proceedings of the Aristotelian Society, Volume 89, Issue 1, 1 June 1989, Pages 143–164, http.

The latter half of the twentieth century has been marked by debates in evolutionary biology over the relative significance of natural selection and random drift: the so-called “neutralist/selectionist” debates. Yet John Beatty has argued that it is difficult, if not impossible, to distinguish the concept of random drift from the concept of natural selection, a claim that has been accepted by many philosophers of biology. If this claim is correct, then the neutralist/selectionist debates seem at best futile, and at worst, (...) meaningless. I reexamine the issues that Beatty raises, and argue that random drift and natural selection, conceived as processes, can be distinguished from one another. (shrink)

We show that there exists a real α such that, for all reals β, if α is linear reducible to β then β≤Tα. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no ℓ-complete Δ2 real. Upon realizing that quasi-maximality does not characterize the random reals–there exist reals which are not random but which are of quasi-maximal ℓ-degree–it is then natural to ask whether maximality could provide such a characterization. Such hopes, (...) however, are in vain since no real is of maximal ℓ-degree. (shrink)

We consider the nature of quantum randomness and how one might have empirical evidence for it. We will see why, depending on one’s computational resources, it may be impossible to determine whether...

Libertarians claim that human behaviour is undetermined and cannot be predicted from knowledge of past history even in principle since it is based on the random movements of quantum mechanics. Determinists on the other hand deny thatmacroscopic phenomena can be activated bysub-microscopic events, and assert that if human action is unpredictable in the way claimed by libertarians, it must be aimless and irrational. This is not true of some types of random behaviour described in this paper. Random behaviour may make (...) one unpredictable to opponents and may therefore be rational. Similarly, playing a game with a mixed strategy may have an unpredictable outcome in every single play, but the strategy is rational, in that it is meant to maximize the expected value of an objective, be it private or social. As to whether the outcome of such behaviour is genuinely unpredictable as in quantum mechanics, or predictable by a hypothetical outside observer knowing all natural laws, it is argued that it makes no difference in practice, as long as it is not humanly predictable. Thus we have a new version of libertarianism which is compatible with determinism. (shrink)

Of course, the rationale of PME is so different from what has been taught in “orthodox” statistics courses for fifty years, that it causes conceptual hangups for many with conventional training. But beginning students have no difficulty with it, for it is just a mathematical model of the natural, common sense way in which anybody does conduct his inferences in problems of everyday life.The difficulties that seem so prominent in the literature today are, therefore, only transient phenomena that will disappear (...) automatically in time. Indeed, this revolution in our attitude toward inference is already an accomplished fact among those concerned with a few specialized applications; with a little familarity in its use its advantages are apparent and it no longer seems strange. It is the idea that inference was once thought to be tied to frequencies in random experiments, that will seem strange to future generations. (shrink)

Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin–Löf. We extend this result and demonstrate that QM is algorithmic $$\omega$$ -random and generic, precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo–Fraenkel Solovay random (...) on infinite-dimensional Hilbert spaces. Moreover, it is more likely that there exists a standard transitive ZFC model M, where QM is expressed in reality, than in the universe V of sets. Then every generic quantum measurement adds to M the infinite sequence, i.e. random real $$r\in 2^{\omega }$$, and the model undergoes random forcing extensions M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit, therefore showing the structural resemblance of both in this limit. We discuss several questions regarding measurability and possible practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC; however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself, the issues considered here remain hidden to a great extent. (shrink)

The study of Martin‐Löf randomness on a computable metric space with a computable measure has seen much progress recently. In this paper we study Martin‐Löf randomness on a more general space, that is, a computable topological space with a computable measure. On such a space, Martin‐Löf randomness may not be a natural notion because there is no universal test, and Martin‐Löf randomness and complexity randomness (defined in this paper) do not coincide in general. We show (...) that SCT3 is a sufficient condition for the existence and coincidence, and study how much we can weaken this condition. (shrink)

The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in the classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations: Δq>0 and Δp>0, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is (...) always positive (non zero).A “functional” formulation of classical mechanics was suggested. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. An interpretation of quantum mechanics is attempted in which both classical and quantum mechanics contain fundamental randomness. Instead of an ensemble of events one introduces an ensemble of observers. (shrink)

We first relate the random matrix model to a Fokker-Planck Hamiltonian system, such that the correlation functions of the model are expressed as the vacuum expectation values of equal-time products of density operators. We then analyze the universality of the random matrix model by solving the Focker-Planck Hamiltonian system for large N. We use two equivalent methods to do this, namely the method of relating it to a system of interacting fermions in one space dimension and the method of collective (...) fields for large N matrix quantum mechanics. The final result using both these methods is the same Hamiltonian system of chiral bosons on a circle, which manifestly exhibits the universality of the random matrix model. (shrink)

We say that A ≤LR B if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ. In other words. B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumberable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α in not GL₂ the LR degree of (...) α bounds $2^{\aleph _{0}}$ degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. (shrink)

Randomness, a core concept of gambling, is seen in problem gambling as responsible for the formation of the math-related cognitive distortions, especially the Gambler’s Fallacy. In problem-gambling research, the concept of randomness was traditionally referred to as having a mathematical nature and categorized and approached as such. Randomness is not a mathematical concept, and I argue that its weak mathematical dimension is not decisive at all for the randomness-related issues in gambling and problem gambling, including the (...) correction of the misconceptions and fallacies about probability and statistical concepts applied in gambling. I distinguish between mathematical and nonmathematical dimensions of randomness (the epistemic, the theoretical-methodological, the functional, and the ethical) falling within the general concept, and I argue that both the studies having as object the math-related cognitive distortions among gamblers and the educational programs aiming at correcting them should employ this distinction in their design and content. (shrink)

We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.

The modern mathematical theory of dynamical systems proposes a new model of mechanical motion. In this model the deterministic unstable systems can behave in a statistical manner. Both kinds of motion are inseparably connected, they depend on the point of view and researcher's approach to the system. This mathematical fact solves in a new way the old problem of statistical laws in the world which is essentially deterministic. The classical opposition: deterministic‐statistical, disappears in random dynamics. The main thesis of the (...) paper is that the new theory of motion is a revolution in the research programme of classical mechanics. It is the revolution brought about by the development of mathematics. (shrink)

Research in evolutionary ecology on random foraging seems to ignore the possibility that some random foraging is an adaptation not to environmental randomness, but to what Wimsatt called “perceived randomness”. This occurs when environmental features are unpredictable, whether physically random or not. Mere perceived randomness may occur, for example, due to effects of climate change or certain kinds of static landscape variation. I argue that an important mathematical model concerning random foraging doesn’t depend on environmental randomness, (...) despite contrary remarks by researchers. I use computer simulations to illustrate the idea that random foraging is an adaptation to mere perceived randomness. (shrink)

One can consider μ‐Martin‐Löf randomness for a probability measure μ on 2ω, such as the Bernoulli measure given. We study Bernoulli randomness of sequences in with parameters, and we reintroduce Bernoulli normality, where the uniform distribution of digits is replaced with a Bernoulli distribution. We prove the equivalence of three characterizations of Bernoulli normality. We show that every Bernoulli random real is Bernoulli normal, and this has the corollary that the set of Bernoulli normal reals has full Bernoulli (...) measure in. We give an algorithm for computing Bernoulli normal sequences from normal sequences so that we can give explicit examples of Bernoulli normal reals. We investigate an application of randomness to iterated function systems. Finally, we list a few further questions relating to Bernoulli randomness and Bernoulli normality. (shrink)

We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [34]). 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.

This paper argues, first, that biological evolution can be both random and divinely guided at the same time. Next it discusses the idea that the claim that evolution is unguided is not part of the science of evolution, and defends it against a number of objections.

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

n this text, we revisit part of the analysis of anti-entropy in Bailly and Longo (2009} and develop further theoretical reflections. In particular, we analyze how randomness, an essential component of biological variability, is associated to the growth of biological organization, both in ontogenesis and in evolution. This approach, in particular, focuses on the role of global entropy production and provides a tool for a mathematical understanding of some fundamental observations by Gould on the increasing phenotypic complexity along evolution. (...) Lastly, we analyze the situation in terms of theoretical symmetries, in order to further specify the biological meaning of anti-entropy as well as its strong link with randomness. (shrink)