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)

Kuera, A., On relative randomness, Annals of Pure and Applied Logic 63 61–67. It is the aim of the paper to answer a question raised by M. van Lambalgen and D. Zambella whether there can be a nonrecursive set A having the property that there is a set B such that B is 1-random relative to A and simultaneously A is recursive in B. We give apositive answer to this question as well as further information about (...) class='Hi'>relative randomness. (shrink)

We show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field. With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also (...) proved nearly simultaneously and independently by Ng (Keng Meng Ng. Master's Thesis. National University of Singapore, in preparation). Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility). (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)

Van Lambalgen's Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen's Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of the reals in the range of Omega operators. It is known that $\Omega^{\phi'}$ is high. We extend this result to that $\Omega^{\phi^{(n)}}$ is $\textrm{high}_n$ . We also prove that there exists A such (...) that, for each n , the real $\Omega^A_M$ is $\textrm{high}_n$ for some universal Turing machine M by using the extended van Lambalgen's Theorem. (shrink)

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)

The concept of randomness has been unjustly neglected in recent philosophical literature, and when philosophers have thought about it, they have usually acquiesced in views about the concept that are fundamentally flawed. After indicating the ways in which these accounts are flawed, I propose that randomness is to be understood as a special case of the epistemic concept of the unpredictability of a process. This proposal arguably captures the intuitive desiderata for the concept of randomness; at least (...) it should suggest that the commonly accepted accounts cannot be the whole story and more philosophical attention needs to be paid. 1. Randomness in science1.1Random systems1.2Random behaviour1.3Random sampling1.4Caprice, arbitrariness and noise2. Concepts of randomness2.1Von Mises/church/martin-löf randomness2.2KCS-randomness3. Randomness is unpredictability: preliminaries3.1Process and product randomness3.2Randomness is indeterminism?4. Predictability4.1Epistemic constraints on prediction4.2Computational constraints on prediction4.3Pragmatic constraints on prediction4.4Prediction defined5. Unpredictability6. Randomness is unpredictability6.1Clarification of the definition of randomness6.2Randomness and probability6.3Subjectivity and context sensitivity of randomness7. Evaluating the analysis[R]andomness … is going to be a concept which is relative to our body of knowledge, which will somehow reflect what we know and what we don't know. Henry E. Kyburg, Jr ([1974], p. 217)Phenomena that we cannot predict must be judged random. Patrick Suppes ([1984], p. 32). (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)

Reestablishing feelings of control after experiencing uncertainty has long been considered a fundamental motive for human behavior. We propose that rituals (i.e., socially stipulated, causally opaque practices) provide a means for coping with the aversive feelings associated with randomness due to the perception of a connection between ritual action and a desired outcome. Two experiments were conducted (one in Brazil [n = 40] and another in the United States [n = 94]) to evaluate how the perceived efficacy of rituals (...) is affected by feelings of randomness. In a between-subjects design, the Scramble Sentence Task was used as a priming procedure in three conditions (i.e., randomness, negativity, and neutral) and participants were then asked to rate the efficacy of rituals used for problem-solving purposes. The results demonstrate that priming randomness increased participants' perception of ritual efficacy relative to negativity and neutral conditions. Implications for increasing our understanding of the relationship between perceived control and ritualistic behavior are discussed. (shrink)

We analyze the pointwise convergence of a sequence of computable elements of L1 in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA0, each is equivalent to the assertion that every Gδ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Königʼs lemma relativized to the Turing jump of any set. It is also equivalent to (...) the conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Σ2 collection. (shrink)

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.

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 survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on reducibilities that measure the initial segment complexity of reals and 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.

We use Demuth randomness to study strong lowness properties of computably enumerable sets, and sometimes of Δ20 sets. A set A⊆N is called a base for Demuth randomness if some set Y Turing above A is Demuth random relative to A. We show that there is an incomputable, computably enumerable base for Demuth randomness, and that each base for Demuth randomness is strongly jump-traceable. We obtain new proofs that each computably enumerable set below all superlow (...) Martin-Löf random sets is strongly jump traceable, using Demuth tests. The sets Turing below each ω2-computably approximable Martin-Löf random set form a proper subclass of the bases for Demuth randomness, and hence of the strongly jump traceable sets. The c.e. sets Turing below each ω2-computably approximable Martin-Löf random set satisfy a new, very strong combinatorial lowness property called ω-traceability. (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)

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a (...) real for which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

We formalise the notion of those infinite binary sequences z that admit a single program P which expresses the entire algorithmical structure of z. Such a program P minimizes the information which must be used in a relative computation for z. We propose two concepts with different strength for this notion, the learnable and the super-learnable sequences. We establish three different equivalent characterizations of learnable (super-learnable, resp.) sequences. In particular, we prove that a sequences z is learnable (super-learnable, resp.) (...) if and only if there is a computable probability measure p such that p is Schnorr (Martin-Lof, resp.) p-random. There is a recursively enumerable sequence which is not learnable. The learnable sequences are invariant with respect to all total and effective transformations of infinite binary sequences. (shrink)

This article explores the connection between objective chance and the randomness of a sequence of outcomes. Discussion is focussed around the claim that something happens by chance iff it is random. This claim is subject to many objections. Attempts to save it by providing alternative theories of chance and randomness, involving indeterminism, unpredictability, and reductionism about chance, are canvassed. The article is largely expository, with particular attention being paid to the details of algorithmic randomness, a topic relatively (...) unfamiliar to philosophers. (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 consider the following problem for various infinite-time machines. If a real is computable relative to a large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent of ZFC for ordinal Turing machines with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider infinite-time Turing machines, unresetting (...) and resetting infinite-time register machines, and α-Turing machines for countable admissible ordinals α. (shrink)

We investigate the initial segment complexity of random reals. Let K denote prefix-free Kolmogorov complexity. A natural measure of the relative randomness of two reals α and β is to compare complexity K and K. It is well-known that a real α is 1-random iff there is a constant c such that for all n, Kn−c. We ask the question, what else can be said about the initial segment complexity of random reals. Thus, we study the fine behaviour (...) of K for random α. Following work of Downey, Hirschfeldt and LaForte, we say that αK β iff there is a constant such that for all n, . We call the equivalence classes under this measure of relative randomness K-degrees. We give proofs that there is a random real α so that lim supn K−K=∞ where Ω is Chaitin's random real. One is based upon work of Solovay, and the other exploits a new idea. Further, based on this new idea, we prove there are uncountably many K-degrees of random reals by proving that μ=0. As a corollary to the proof we can prove there is no largest K-degree. Finally we prove that if n ≠ m then the initial segment complexities of the natural n- and m-random sets and Ω︀) are different. The techniques introduced in this paper have already found a number of other applications. (shrink)

Using randomly generated sequences of binary events we asked participants to make predictions about the next event. It turned out that while predicting uncertain events, people do not behave unsystematically. Our research identifies four types of relatively consistent strategies for predicting uncertain binary events: a strategy immune to short-run sequential dependencies consisting of the persistent prediction of long-run majority events, hereafter called the long-run momentum strategy ; a strategy immune to short-run sequential dependencies consisting of the persistent prediction of long-run (...) minority events, called the long-run contrarian strategy ; a strategy sensitive to short-run sequential dependencies consisting of the prediction of short-run majority events, called the short-run momentum strategy ; and a strategy sensitive to short-run sequential dependencies consisting of the prediction of short-run minority events, called the short-run contrarian strategy . When the character of events remains unknown, the most common strategy is the short-run momentum strategy. With the increase of a perceived randomness of the situation, people tend more often to use the short-run contrarian strategy. People differ in their general beliefs about the continuation or reversal of a trend in various natural and social processes. Trend believers, when facing sequences of binary events commonly perceived as random, tend to use momentum strategies, whereas those who believe in the trend's reversal tend to use contrarian strategies. (shrink)

A seminal theorem due to Weyl [14] states that if $(a_n)$ is any sequence of distinct integers, then, for almost every $x \in \mathbb{R}$, the sequence $(a_n x)$ is uniformly distributed modulo one. In particular, for almost every $x$ in the unit interval, the sequence $(a_n x)$ is uniformly distributed modulo one for every computable sequence $(a_n)$ of distinct integers. Call such an $x$ UD random. Here it is shown that every Schnorr random real is UD random, but there are (...) Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random. (shrink)

Let f be a computable function from finite sequences of 0ʼs and 1ʼs to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We (...) prove that f-randomness relative to a PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree. (shrink)

Gödel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the incompleteness phenomenon unveiled by an information-theoretic approach to randomness and recent developments in quantum computing.

Suppose various observers are divided randomly into two groups, a large and a small. Not knowing into which group anyone has been sent, each can have strong grounds for believing in being in the large group, although recognizing that every observer in the other group has equally powerful reasons for thinking of this other group as the large one. Justified belief can therefore be observer-relative in a rather paradoxical way. Appreciating this allows one to reject an intriguing new objection (...) against Brandon Carter's 'doomsday argument'. Carter encourages us to doubt that we are among only the first hundredth, say, or first millionth, of all humans who will ever have existed. He thereby reinforces whatever reasons we may have for suspecting that, unless we take great care, the human race will not survive long. Admittedly his argument is weakened if our world is indeterministic, so that there is no suitably guaranteed 'fact of the matter' of how many humans will ever have existed. But even then, it can caution us against believing that a lengthy future for humankind 'is as good as determined'. Of all the objections the argument has yet faced, the new one is the most interesting. (shrink)

The aim of this study was to find out whether the achievement of high school social studies students differs significantly according to whether they are taught by the reflective teaching method or the lecture approach method. A stratified random sample of 215 year three social studies students was randomly assigned to two groups: 107 students into the reflective approach group and 108 into the lecture approach group. An achievement test was used as the pre‐ and the post‐test. The means of (...) the aggregate mean performance scores were compared by the use of Student's t‐test. There was no significant difference between the two groups on the pre‐test scores but there was a significant difference between the post‐test scores, in favour of the group taught by the reflective approach method. (shrink)

To compare two randomness notions with each other, we ask whether a given randomness notion can be defined via another randomness notion. Inspired by Yu's pioneering study, we formalize our question using the concept of relativization of randomness. We give some solutions to our formalized questions. Also, our results include the affirmative answer to the problem asked by Yu in a discussion with the second author, i.e., whether Schnorr randomness relative to the halting problem (...) is equivalent to Martin‐Löf randomness relative to all low 1‐generic reals. (shrink)

We describe an Aldous–Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman \)-invariant measures: part I, 2015. arXiv:1509.06170) and Crane and Towsner and hierarchical exchangeability results due to Austin and Panchenko :809–823, 2014).

Sudden, broad-scale shifts in public opinion about social problems are relatively rare. Until recently, social scientists were forced to conduct post-hoc case studies of such unusual events that ignore the broader universe of possible shifts in public opinion that do not materialize. The vast amount of data that has recently become available via social media sites such as Facebook and Twitter—as well as the mass-digitization of qualitative archives provide an unprecedented opportunity for scholars to avoid such selection on the dependent (...) variable. Yet the sheer scale of these new data creates a new set of methodological challenges. Conventional linear models, for example, minimize the influence of rare events as “outliers”—especially within analyses of large samples. While more advanced regression models exist to analyze outliers, they suffer from an even more daunting challenge: equifinality, or the likelihood that rare events may occur via different causal pathways. I discuss a variety of possible solutions to these problems—including recent advances in fuzzy set theory and machine learning—but ultimately advocate an ecumenical approach that combines multiple techniques in iterative fashion. (shrink)

We study the coupled dynamics of two populations of random replicators by means of statistical mechanics methods, and focus on the effects of relative population size, strategy correlations and heterogeneities in the respective cooperation pressures. To this end we generalise existing path-integral approaches to replicator systems with random asymmetric couplings. This technique allows one to formulate an effective dynamical theory, which is exact in the thermodynamic limit and which can be solved for persistent order parameters in a fixed-point regime (...) regardless of the symmetry of the interactions. The onset of instability can be determined self-consistently. We calculate quantities such as the diversity of the respective populations and their fitnesses in the stationary state, and compare results with data from a numerical integration of the replicator equations. -/- . (shrink)

Compared the use of unit regression weights with least squared error multiple regression weights. 100 sample correlation matrices computed from randomly drawn samples from a multivariate normal distribution were employed, with the sample size varying from 25-1,000 and number of variables from 2-10. In addition, correlation matrices were also sampled from published data. The results of comparing the efficiency of the 2 methods of weighting are presented for correlation matrices with and without the effect of suppressor variables. Use of these (...) findings is discussed. (shrink)

We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first‐order structure. We argue that the notion parallels Martin‐Löf (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of “small” cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero sets. It is shown that if the domain M satisfies, then there exist typical elements and (...) only non‐typical ones. In particular this is true for the standard model of second‐order arithmetic. By allowing parameters in the defining formulas, we are led to relative typicality, which satisfies most of van Lambalgen's axioms for relative randomness. However van Lambalgen's theorem is false for relative typicality. The class of typical reals is incomparable (with respect to ⊆) with the classes of ML‐random, Schnorr random and computably random reals. Also the class of typical reals is closed under Turing degrees and under the jump operation (both ways). (shrink)

I. INTRODUCTION The notion of randomness has always been rather perplexing. Altho it is frequently used in natural and social science, both technically a informally, it seems to have been somewhat neglected by philosophers o science ever since the discussion of the foundations of the so-called fre- quency theory of probability, in which it was assigned a basic role, faded. Yet this discussion is of such significance that any attempt clarifying the notion of randomness will have to relate (...) to it. After a fe preliminary remarks on some of the problems and puzzles of randomne I shall, therefore, expound and discuss a concept of random distribution of a property in classes and sequences, defined in terms of relative f quencies and their limits. Because of certain shortcomings of this conce it appears advisable to turn to probabilities, in terms of which a qu different concept, viz., that of random conjunction of properties, can rea ily be defined as stochastic independence. This concept still has featu clashing with the ordinary sense of 'randomness' which become ma fest in cases where certain probabilities assume extreme values. However when we take measures defined in information theory as measuring the degree of randomness, to which purpose they lend themselves parti larly well, we find that these seemingly troublesome cases are rath harmless. A by-product of the discussion of measures of randomnes the concept of primitive randomness. The conclusion points out som further problems. (shrink)

Let ${2-\textsf{RAN}}$ be the statement that for each real X a real 2-random relative to X exists. We apply program extraction techniques we developed in Kreuzer and Kohlenbach (J. Symb. Log. 77(3):853–895, 2012. doi:10.2178/jsl/1344862165), Kreuzer (Notre Dame J. Formal Log. 53(2):245–265, 2012. doi:10.1215/00294527-1715716) to this principle. Let ${{\textsf{WKL}_0^\omega}}$ be the finite type extension of ${\textsf{WKL}_0}$ . We obtain that one can extract primitive recursive realizers from proofs in ${{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} + 2-\textsf{RAN}}$ , i.e., if ${{\textsf{WKL}_0^\omega} + \Pi^0_1-{\textsf{CP}} + (...) 2-\textsf{RAN} \, {\vdash} \, \forall{f}\, {\exists}{x} A_{qf}(f,x)}$ then one can extract from the proof a primitive recursive term t(f) such that ${A_{qf}(f,t(f))}$ . As a consequence, we obtain that ${{\textsf{WKL}_0}+ \Pi^0_1 - {\textsf{CP}} + 2-\textsf{RAN}}$ is ${\Pi^0_3}$ -conservative over ${\textsf{RCA}_0}$. (shrink)

This thesis is an attempt to reconstruct the conceptual foundations of quantum mechanics. First, we argue that the wave function in quantum mechanics is a description of random discontinuous motion of particles, and the modulus square of the wave function gives the probability density of the particles being in certain locations in space. Next, we show that the linear non-relativistic evolution of the wave function of an isolated system obeys the free Schrödinger equation due to the requirements of spacetime translation (...) invariance and relativistic invariance. Thirdly, we argue that the random discontinuous motion of particles may lead to a stochastic, nonlinear collapse evolution of the wave function. A discrete model of energy-conserved wavefunction collapse is proposed and shown to be consistent with existing experiments and our macroscopic experience. In addition, we also give a critical analysis of the de Broglie-Bohm theory, the many-worlds interpretation and other dynamical collapse theories, and briefly discuss the issue of unifying quantum mechanics and special relativity. (shrink)

It was pointed out in the first part of this study (Herbut in Found. Phys. 38:1046–1064, 2008) that EPR-type entanglement is defined by the possibility of performing any of two mutually incompatible distant, i.e., direct-interaction-free, measurements. They go together under the term ‘EPR-type disentanglement’. In this second part, quantum-mechanical insight is gained in the real random delayed-choice erasure experiment of Kim et al. (Phys. Rev. Lett. 84:1–5, 2000) by a relative-reality-of-unitarily-evolving-state (RRUES) approach (explained in the first part). Finally, it (...) is shown that this remarkable experiment, which performs, by random choice, two incompatible measurements at the same time, is actually an EPR-type disentanglement experiment, closely related to the micromaser experiment discussed in the first part. (shrink)

The quantization error in a fixed-size Self-Organizing Map (SOM) with unsupervised winner-take-all learning has previously been used successfully to detect, in minimal computation time, highly meaningful changes across images in medical time series and in time series of satellite images. Here, the functional properties of the quantization error in SOM are explored further to show that the metric is capable of reliably discriminating between the finest differences in local contrast intensities and contrast signs. While this capability of the QE is (...) akin to functional characteristics of a specific class of retinal ganglion cells (the so-called Y-cells) in the visual systems of the primate and the cat, the sensitivity of the QE surpasses the capacity limits of human visual detection. Here, the quantization error in the SOM is found to reliably signal changes in contrast or colour when contrast information is removed from or added to the image, but not when the amount and relative weight of contrast information is constant and only the local spatial position of contrast elements in the pattern changes. While the RGB Mean reflects coarser changes in colour or contrast well enough, the SOM-QE is shown to outperform the RGB Mean in the detection of single-pixel changes in images with up to five million pixels. This could have important implications in the context of unsupervised image learning and computational building block approaches to large sets of image data (big data). (shrink)

To ground judgments of credal probability on knowledge of chance via direct inference, one should appeal to the information about chances available relative to the most specific description known to be true of the trial event.Thus, to obtain a judgment of credal probability concerning the hypothesis that coin a landed heads at t given that it is known that at t it is known that a was tossed by Levi in 728 Philosophy Hall, the pertinent knowledge of chances concerns (...) the chances of coin a landing heads on a toss of coin a which is a toss at t, a toss in 728 Philosophy Hall and a toss by Levi. In many contexts, appeal is made to knowledge of the chance of coin a landing heads on a toss. This appeal, when legitimate, takes for granted that the additional information about the specific tossing listed in, and is stochastically irrelevant to landing heads and, hence, may be ignored. Such an assumption is a substantive presupposition about chances and not an a priori, self evident or pragmatic principle of methodology. (shrink)

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)

Discriminative feature-based methods are widely used in natural language processing, but sentence parsing is still dominated by generative methods. While prior feature-based dynamic programming parsers have restricted training and evaluation to artificially short sentences, we present the first general, featurerich discriminative parser, based on a conditional random field model, which has been successfully scaled to the full WSJ parsing data. Our efficiency is primarily due to the use of stochastic optimization techniques, as well as parallelization and chart prefiltering. On WSJ15, (...) we attain a state-of-the-art F-score of 90.9%, a 14% relative reduction in error over previous models, while being two orders of magnitude faster. On sentences of length 40, our system achieves an F-score of 89.0%, a 36% relative reduction in error over a generative baseline. (shrink)

Advances in multiagent simulation techniques make it possible to study more realistic dynamics of complex systems and allow evolutionary theories to be tested. Here I use simulations to assess the relative importance of reproductive systems (haplodiploidy vs. diploidy), mate selection (assortative mating vs. random mating) and social economics (pay-off matrices of evolutionary games) in the evolutionary dynamics leading to the emergence of social cooperation in the provision of parental care. The simulations confirm that haplo-diploid organisms and organisms mating assortatively (...) have a higher probability for fixing alleles and require less favorable conditions for their fixation, than diploids or organisms mating randomly. The simulations showed that social behavior was most likely to emerge a) when the cost for parental investment was much lower than the benefits to the offspring, b) when cooperation improved synergistically the fitness of offspring compared to the corresponding egoistic behavior and c) when alleles coding for altruistic or social behavior could be rapidly fixed in the population, thanks to mechanisms such as haplo-diploidy and/or assortative mating. Cooperative social behavior always appeared if sociality conferred much higher fitness gains compared to non cooperative alternatives suggesting that the most important factors for the emergence and maintenance of social behavior are those based on energetic or efficiency considerations. The simulations, in congruence with the scant experimental evidence available, suggest that economic considerations rather than genetic ones are critical in explaining the emergence and maintenance of sociality. (shrink)

We give a partial answer to the question whether the Schrödinger equation can be derived from the Newtonian mechanics of a particle in a potential subject to a random force. We show that the fluctuations around the classical motion of a one dimensional harmonic oscillator subject to a random force can be described by the Schrödinger equation for a period of time depending on the frequency and the energy of the oscillator. We achieve this by deriving the postulates of Nelson’s (...) stochastic formulation of quantum mechanics for a random force depending on a small parameter. We show that the same result applies to small potential perturbations around the harmonic oscillator. We also show that the noise spectrum can be chosen to obtain the result for all oscillator frequencies for fixed mass. We discuss heuristics to generalize the result for a particle in one dimension in a potential where the motion can be described using action-angle variables. The main motivation of this paper is to provide a step for constructing a Newtonian theory which would approximately reproduce quantum mechanics both in unitary evolution and measurement regimes. (shrink)

Introduction: Falls are the leading cause of fatal and nonfatal injuries among older adults. Studies showed that older adults can reduce the risk of falls after participation in an unexpected perturbation-based balance training, a relatively novel approach that challenged reactive balance control. This study aims to investigate the effect of the practice schedule on reactive balance function and its transfer to proactive balance function. Our primary hypothesis is that improvements in reactive balance control following block PBBT will be not inferior (...) to the improvements following random PBBT.Methods and Analysis: This is a double-blind randomized controlled trial. Fifty community-dwelling older adults will be recruited and randomly allocated to a random PBBT group or a block PBBT group. The random PBBT group will receive eight training sessions over 4 weeks that include unexpected machine-induced perturbations of balance during hands-free treadmill walking. The block PBBT group will be trained by the same perturbation treadmill system, but only one direction will be trained in each training session, and the direction of the external perturbations will be announced. Both PBBT groups will receive a similar perturbation intensity during training, the same training period, and the same concurrent cognitive tasks during training. The generalization and transfer of learning effects will be measured by assessing the reactive and proactive balance control during standing and walking before and after 1 month of PBBT, for example, step and multiple steps and fall thresholds, Berg balance test, and fear of falls. The dependent variable will be rank transformed prior to conducting the analysis of covariance to allow for nonparametric analysis.Discussion: This research will explore which of the balance retraining paradigms is more effective to improve reactive balance and proactive balance control in older adults over 1 month. The research will address key issues concerning balance retraining: older adults’ neuromotor capacities to optimize training responses and their applicability to real-life challenges.Clinical Trial Registration: Helsinki research ethics approval has been received. (shrink)

Background Health care professionals have to judge the appropriateness of treatment in critical care on a daily basis. There is general consensus that critical care interventions should not be performed when they are inappropriate. It is not yet clear which chances of survival are considered necessary or which risk for serious disabilities is acceptable in quantitative terms for different stakeholders to start intensive care treatment. Methods We performed an anonymous online survey in a random sample of 1,052 participants recruited via (...) email invitation and social media. Age, sex, nationality, education, professional involvement in health care, critical care medicine and treatment decisions in critical care medicine as well as personal experience with critical illness were assessed as potential influencing variables. Participants provided their opinion on the necessary chances of survival and the acceptable risk for serious disabilities to start a high-risk or uncomfortable therapy for themselves, relatives or for their patients on a scale of 0–100%. Results Answers ranged from 0 to 100% for all questions. A three-peak pattern with different distributions of the peaks was observed. Sex, education, being a health care professional, being involved in treatment decisions and religiosity influence these opinions. Male respondents and those with a university education would agree that a risky and uncomfortable treatment should be started even with a low chance of survival for themselves, relatives and patients. More respondents would choose a lower necessary chance of survival when deciding for patients compared to themselves or relatives to start a risky and uncomfortable treatment. On the other hand, the majority of respondents would accept only a low risk of severe disability for both themselves and their patients. Conclusion No cut-off can be identified for the necessary chances of survival or the acceptable risk of disability to help quantify the “inappropriateness” of critical care treatment. Sex and education are the strongest influencing factors on this opinion. The large variation in personal opinions, depending on demographic and personality variables and education needs to be considered in the communication between health care professionals and patients or surrogates. (shrink)

Many experiments have been conducted over the past eight decades to explore whether the ostensible psychic ability of psychokinesis (PK, or "mind over matter") might be a genuine human potential, and the most extensive of these have involved attempts to mentally influence the output of electronic, binary-bit random number generators (RNGs). Research of this type can generally be divided into two lines: proof-oriented (concerned with the accumulation and statistical evaluation of data from controlled experiments designed specifically to test for the (...) presence of PK effects on the microscopic scale) and process-oriented (concerned with conducting exploratory experiments designed to systematically vary certain test conditions in order to search for and identify any physical, biological, and psychological factors which might have a role in improving or moderating PK effects). To help orient novice investigators and cross-disciplinary researchers who may be considering work along these lines (as well as offer some initial guiding insight on possible directions for future research), this paper provides a general review of some of the notable proof- and process-oriented findings that have been obtained to date in experimental microscopic PK research using RNGs. The review generally indicates that although a considerable amount of proof-oriented data for micro-PK has accumulated over the years, the relatively sparse amount of process-oriented data available at present leaves a lot of open questions regarding the underlying factors involved, providing ample opportunity for novice investigators and cross-disciplinary researchers to make valuable research contributions in the future. (shrink)

We discuss the problems of quantum theory complicating its merging with general relativity. QT is treated as a general theory of micro-phenomena—a bunch of models. Quantum mechanics and quantum field theory are the most widely known. The basic problems of QM and QFT are considered in interrelation. For QM, we stress its nonrelativistic character and the presence of spooky action at a distance. For QFT, we highlight the old problem of infinities. And this is the main point of the paper: (...) it is meaningless to try to unify QFT so heavily suffering of infinities with GR. We also highlight difficulties of the QFT-treatment of entanglement. We compare the QFT and QM based measurement theories by presenting both theoretical and experimental viewpoints. Then we discuss two basic mathematical constraints of both QM and QFT, namely, the use of real numbers and the Hilbert state space. We briefly present non-archimedean and non-hilbertian approaches to QT and their consequences. Finally, we claim that, in spite of the Bell theorem, it is still possible to treat quantum phenomena on the basis of a classical-like causal theory. We present a random field model generating the QM and QFT formalisms. This emergence viewpoint can serve as the basis for unification of novel QT and GR. (shrink)

This is an extended essay review of Tanya and Jeffrey Bub’s Totally Random: Why Nobody Understands Quantum Mechanics: A serious comic on entanglement. We review the philosophical aspects of the book, provide suggestions for instructors on how to use the book in a class setting, and evaluate the authors’ artistic choices in the context of comics theory. Although Totally Random does not defend any particular interpretation of quantum mechanics, we find that, in its mode of presentation, Totally Random is a (...) beautiful expression and illustration of the information-theoretic interpretation and its value. (shrink)