Notion Of Randomness Research Articles

Computability in infinite Galois theory and algorithmically random algebraic fields

AbstractWe introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of that are random according to our definition. We show that there are noncomputable algebraic fields which are not random. We also partially characterize the index set, relative to an oracle, of the set of random algebraic fields computable relative to that oracle.In order to carry out this investigation of randomness for fields, we develop computability in the context of the infinite Galois theory (where the relevant Galois groups are uncountable), including definitions of computable and computably enumerable Galois groups and computability of Haar measure on the Galois groups.

Rebounds in football: A systematic investigation of characteristics of goals scored after rebounded balls in English Premier League seasons 2012/2013 to 2018/2019

The literature on systematic aspects of goal scoring in football is rich, while little is known about random influences. This study focuses on an unsystematic element of goal scoring, namely rebounds, where a shot that bounces back from another player or the goal creates contested opportunities. Aspects including causes, match situation, player positioning and pitch zones were investigated for 554 rebound goals from seven seasons (2012/2013–2018/2019) of the English Premier League. Rebounds were predominantly caused by short distance and defensive errors, with goalkeepers playing a crucial role. Interestingly, the superiority of home teams and stronger teams to score significantly more goals than away teams and weaker teams can also be observed for rebound goals. Rebound goals are particularly prevalent in zones close to the goal, more prevalent after corners than non-rebound goals and associated with a high number of players in the penalty box. Overall, high similarities exist between rebound and non-rebound goals, while differences in match situations and spatial patterns suggest unique characteristics of rebound goals. These results are in line with the notion of systematicity and randomness concurrently contributing to success in football. More research appears useful to further disentangle the influence of both factors.

From Wald to Schnorr: von Mises’ definition of randomness in the aftermath of Ville’s Theorem

The first formal definition of randomness, seen as a property of sequences of events or experimental outcomes, dates back to Richard von Mises’ work in the foundations of probability and statistics. The randomness notion introduced by von Mises is nowadays widely regarded as being too weak. This is, to a large extent, due to the work of Jean Ville, which is often described as having dealt the death blow to von Mises’ approach, and which was integral to the development of algorithmic randomness—the now-standard theory of randomness for elements of a probability space. The main goal of this article is to trace the history and provide an in-depth appraisal of two lesser-known, yet historically and methodologically notable proposals for how to modify von Mises’ definition so as to avoid Ville’s objection. The first proposal is due to Abraham Wald, while the second one is due to Claus-Peter Schnorr. We show that, once made precise in a natural way using computability theory, Wald’s proposal constitutes a much more radical departure from von Mises’ framework than intended. Schnorr’s proposal, on the other hand, does provide a partial vindication of von Mises’ approach: it demonstrates that it is possible to obtain a satisfactory randomness notion—indeed, a canonical algorithmic randomness notion—by characterizing randomness in terms of the invariance of limiting relative frequencies. More generally, we argue that Schnorr’s proposal, together with a number of little-known related results, reveals that there is more continuity than typically acknowledged between von Mises’ approach and algorithmic randomness. Even though von Mises’ exclusive focus on limiting relative frequencies did not survive the passage to the theory of algorithmic randomness, another crucial aspect of his conception of randomness did endure; namely, the idea that randomness amounts to a certain type of stability or invariance under an appropriate class of transformations.

Imprecision in martingale- and test-theoretic prequential randomness

In a prequential approach to algorithmic randomness, probabilities for the next outcome can be forecast ‘on the fly’ without the need for fully specifying a probability measure on all possible sequences of outcomes, as is the case in the more standard approach. We take the first steps in allowing for probability intervals instead of precise probabilities on this prequential approach, based on ideas borrowed from our earlier imprecise-probabilistic, standard account of algorithmic randomness. We define what it means for an infinite sequence (I1,x1,I2,x2,…) of successive interval forecasts Ik and subsequent binary outcomes xk to be random, both in a martingale-theoretic and a test-theoretic sense. We prove that these two versions of prequential randomness coincide, we compare the resulting prequential randomness notions with the more standard ones, and we investigate where the prequential and standard randomness notions coincide.

The reality of backdoored S-Boxes—An eye opener

The real-life incidents researched in academia have revealed that (possibly) state-level efforts are made to camouflage the intentional flaws in the mathematical layer of an S-Box for exploiting the information-theoretic properties, i.e., Kuznyechik. To investigate the common features in the intentionally weakened mappings, this research thoroughly examines the backdoored structures from the perspective of 24 cryptanalytic attack vectors available in the open literature. We have debunked the earlier claims by the backdoor engineers that the proposed designs are stealthy against statistical distinguishers. Although a backdoored architecture fulfils the notions of randomness, it lacks the strength to resist sophisticated cryptanalytic attacks. Our analysis has revealed that during the backdoor insertion phase, a malicious designer compromises vital cryptographic properties, prominently the algebraic degree, differential trails, avalanche characteristics and leaving the open ground for hybrid cryptanalytic attacks. It is also observed that these mappings attain the upper bound of BCT, FBCT and DLCT, paving the way for hybrid attacks with high probability and lower data complexity.

Algorithmically random series

Rademacher (Mathematische Annalen 87 (1922) 112–138), Steinhaus (Mathematische Zeitschrift 31 (1930) 408–416) and Paley and Zygmund (Mathematical Proceedings of the Cambridge Philosophical Society 26 (1930) 337–257, Mathematical Proceedings of the Cambridge Philosophical Society 26 (1930) 458–474, Mathematical Proceedings of the Cambridge Philosophical Society 28 (1932) 190–205) initiated the extensive study of random series. Using the theory of algorithmic randomness, which is a mix of computability theory and probability theory, we investigate the effective content of some classical theorems. We discuss how this is related to an old question of Kahane and Bollobás. We also discuss how considerations of such algorithmic questions about random series seem to lead to new notions of algorithmic randomness.

The Interplay between Error, Total Variation, Alpha-Entropy and Guessing: Fano and Pinsker Direct and Reverse Inequalities.

Using majorization theory via "Robin Hood" elementary operations, optimal lower and upper bounds are derived on Rényi and guessing entropies with respect to either error probability (yielding reverse-Fano and Fano inequalities) or total variation distance to the uniform (yielding reverse-Pinsker and Pinsker inequalities). This gives a general picture of how the notion of randomness can be measured in many areas of computer science.

Randomness extraction in computability theory

In this article, we study a notion of the extraction rate of Turing functionals that translate between notions of randomness with respect to different underlying probability measures. We analyze several classes of extraction procedures: (1) a class that generalizes von Neumann’s trick for extracting unbiased randomness from the tosses of a biased coin, (2) a class based on work of by Knuth and Yao (which more properly can be characterized as extracting biased randomness from unbiased randomness), and (3) a class independently developed by Levin and Kautz that generalizes the data compression technique of arithmetic coding. For the first two classes of extraction procedures, we identify a level of algorithmic randomness for an input that guarantees that we attain the extraction rate along that input, while for the third class, we calculate the rate attained along sufficiently random input sequences.

On the (dis)similarities between stationary imprecise and non-stationary precise uncertainty models in algorithmic randomness

The field of algorithmic randomness studies, amongst other things, what it means for infinite binary sequences to be random for some given uncertainty model. Classically, martingale-theoretic notions of randomness involve precise uncertainty models, and it is only recently that imprecision has been introduced into this context. As a consequence, the investigation into how imprecision alters our view on martingale-theoretic random sequences has only just begun. In this contribution, where we allow for non-computable uncertainty models, we establish a close and surprising connection between precise and imprecise uncertainty models in this randomness context. In particular, we show that there are stationary imprecise models and non-computable non-stationary precise models that have the exact same set of random sequences. We also give a preliminary discussion of the possible implications of our result for a statistics based on imprecise probabilities, and shed some light on the practical (ir)relevance of both imprecise and non-computable precise uncertainty models in that context.

Locally Random Groups

In this work, we introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls, and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we develop a Littlewood–Paley decomposition and explore its connection to the existence of a spectral gap for random walks. Moreover, under the dimension condition alone, we prove a multi-scale entropy gain result à la Bourgain–Gamburd and Tao.

Бродский и Клио: профиль скептической философии истории

In the wake of mounting disasters of the last century, poets and philosophers de­veloped the notion of the total arbitrariness of history. In the wording of Karl Löwith, “the skeptic and the believer have a common cause against the easy reading of history and its meaning. Their wisdom, like all wisdom consists not the least in disillusion and resignation, in freedom from illusions and presump­tions”. The Hegelian claim that history has an ultimate meaning, or final pur­pose, or goal was vehemently rejected, as well as Plato’s utopia of a “just city”. In his essays and poems, Brodsky postulates the randomness of history. He bor­rowed the notion of randomness from the book by Lev Shestov “The Apotheosis of Groundlessness”. Shestov promoted the idea of mental nomadism, or freedom from causal thinking and scientism. For Brodsky, being a nomad mentally meant to escape from both rationalist interpretation of history and the theological idea of Providence. He believed that rationalism’s greatest casualty was individual­ism. Besides, the doctrine of historical determinism and the notion of Provi­dence’s general benevolence translated itself into a patient waiting for a Storm Trooper. Seeing problems with such patient waiting in place, such as deportation of Jews to death camps, Brodsky suggested that it would be much better to be­come a nomad. Brodsky developed his ideas against the foil of the Bible, the Re­public of Plato and Hamlet by Shakespeare. He described nomadism as Israel’s Exodus from Egypt and the expulsion of poets from Plato’s Just City.

Against Identification of Contextuality with Violation of the Bell Inequalities: Lessons from Theory of Randomness

Nowadays, contextuality is the hottest topic of quantum foundations and, especially, foundations of quantum information theory. This notion is characterized by the huge diversity of approaches and interpretations. One of the strongest trends in contextual research is to identify contextuality with Bell test contextuality (BTC). In this paper, we criticize the BTC approach. It can be compared with an attempt to identify the complex and theoretically nontrivial notion of randomness with a test for randomness (or a batch of tests, as the NIST test). We advertise Bohr contextuality taking into account all experimental conditions (context). In the simplest case, the measurement context of an observable A is reduced to joint measurement with a compatible observable B. The latter definition was originally considered by Bell in relation to his inequality. We call it joint measurement contextuality (JMC). Although JMC is based on the use of counterfactuals, by considering it within the general Bohr’s framework, it is possible to handle JMC on physical grounds. We suggest (similarly to randomness) to certify JMC in experimental data with Bell tests, but only certify and not reduce.

From Randomness in Two Symbols to Randomness in Three Symbols

Abstract In 1909 Borel defined normality as a notion of randomness of the digits of the representation of a real number over certain base (fractional expansion). If we think of the representation of a number over a base as an infinite sequence of symbols from a finite alphabet A, we can define normality directly for words of symbols of A: A word x is normal to the alphabet A if every finite block of symbols from A appears with the same asymptotic frequency in x as every other block of the same length. Many examples of normal words have been found since its definition, being Champernowne in 1933 the first to show an explicit and simple instance. Moreover, it has been characterized how we can select subsequences of a normal word x preserving its normality, always leaving the alphabet A fixed. In this work we consider the dual problem which consists of inserting symbols in infinitely many positions of a given word, in such a way that normality is preserved. Specifically, given a symbol b that is not present in the original alphabet A and given a word x that is normal to the alphabet A we solve how to insert the symbol b in infinitely many positions of the word x such that the resulting word is normal to the expanded alphabet A ∪{b}.

Randomness and initial segment complexity for measures

We study algorithmic randomness properties for probability measures on Cantor space. We say that a measure μ on the space of infinite bit sequences is Martin-Löf absolutely continuous if the non-Martin-Löf random bit sequences form a null set with respect to μ. We think of this as a weak randomness notion for measures.We begin with examples, and provide a robustness property related to Solovay tests. The initial segment complexity of a measure μ at a length n is defined as the μ-average over the descriptive complexity of strings of length n, in the sense of either C or K. We relate this weak randomness notion for a measure to the growth of its initial segment complexity. We show that a maximal growth implies the weak randomness property, but also that both implications of the Levin-Schnorr theorem fail. We discuss C-triviality and K-triviality for measures and relate these two notions with each other. Here, triviality means that the initial segment complexity grows as slowly as possible.We show that every measure that is Martin-Löf random in the sense of Hoyrup and Rojas is Martin-Löf absolutely continuous; the converse fails because only the latter property is compatible with having atoms. In a final section we consider weak randomness relative to a general ergodic computable measure. We seek appropriate effective versions of the Shannon-McMillan-Breiman theorem and the Brudno theorem where the bit sequences are replaced by measures. We conclude with several open questions.

DEGREES OF RANDOMIZED COMPUTABILITY

AbstractIn this survey we discuss work of Levin and V’yugin on collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. More precisely, Levin and V’yugin introduced an ordering on collections of sequences that are closed under Turing equivalence. Roughly speaking, given two such collections $\mathcal {A}$ and $\mathcal {B}$ , $\mathcal {A}$ is below $\mathcal {B}$ in this ordering if $\mathcal {A}\setminus \mathcal {B}$ is negligible. The degree structure associated with this ordering, the Levin–V’yugin degrees (or $\mathrm {LV}$ -degrees), can be shown to be a Boolean algebra, and in fact a measure algebra. We demonstrate the interactions of this work with recent results in computability theory and algorithmic randomness: First, we recall the definition of the Levin–V’yugin algebra and identify connections between its properties and classical properties from computability theory. In particular, we apply results on the interactions between notions of randomness and Turing reducibility to establish new facts about specific LV-degrees, such as the LV-degree of the collection of 1-generic sequences, that of the collection of sequences of hyperimmune degree, and those collections corresponding to various notions of effective randomness. Next, we provide a detailed explanation of a complex technique developed by V’yugin that allows the construction of semi-measures into which computability-theoretic properties can be encoded. We provide two examples of the use of this technique by explicating a result of V’yugin’s about the LV-degree of the collection of Martin-Löf random sequences and extending the result to the LV-degree of the collection of sequences of DNC degree.

Randomness is inherently imprecise

We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into the study of randomness. In particular, we define several notions of randomness associated with interval, rather than precise, forecasting systems, and study their properties. The richer mathematical structure that thus arises lets us, amongst other things, better understand and place existing results for the precise limit. When we focus on constant interval forecasts, we find that every sequence of binary outcomes has an associated filter of intervals it is random for. It may happen that none of these intervals is precise—a single real number—which justifies the title of this paper. We illustrate this by showing that randomness associated with non-stationary precise forecasting systems can be captured by a constant interval forecast, which must then be less precise: a gain in model simplicity is thus paid for by a loss in precision. But imprecise randomness can't always be explained away as a result of oversimplification: we show that there are sequences that are random for a constant interval forecast, but never random for any computable (more) precise forecasting system. We also show that the set of sequences that are random for a non-vacuous interval forecasting system is meagre, as it is for precise forecasting systems.

Variability and Randomness of the Instantaneous Firing Rate.

The apparent stochastic nature of neuronal activity significantly affects the reliability of neuronal coding. To quantify the encountered fluctuations, both in neural data and simulations, the notions of variability and randomness of inter-spike intervals have been proposed and studied. In this article we focus on the concept of the instantaneous firing rate, which is also based on the spike timing. We use several classical statistical models of neuronal activity and we study the corresponding probability distributions of the instantaneous firing rate. To characterize the firing rate variability and randomness under different spiking regimes, we use different indices of statistical dispersion. We find that the relationship between the variability of interspike intervals and the instantaneous firing rate is not straightforward in general. Counter-intuitively, an increase in the randomness (based on entropy) of spike times may either decrease or increase the randomness of instantaneous firing rate, in dependence on the neuronal firing model. Finally, we apply our methods to experimental data, establishing that instantaneous rate analysis can indeed provide additional information about the spiking activity.

A NOTE ON THE LEARNING-THEORETIC CHARACTERIZATIONS OF RANDOMNESS AND CONVERGENCE

AbstractRecently, a connection has been established between two branches of computability theory, namely between algorithmic randomness and algorithmic learning theory. Learning-theoretical characterizations of several notions of randomness were discovered. We study such characterizations based on the asymptotic density of positive answers. In particular, this note provides a new learning-theoretic definition of weak 2-randomness, solving the problem posed by (Zaffora Blando, Rev. Symb. Log. 2019). The note also highlights the close connection between these characterizations and the problem of convergence on random sequences.

Illustrating Randomness in Statistics Courses With Spatial Experiments

Understanding the concept of randomness is fundamental for students in introductory statistics courses, but the notion of randomness is deceivingly complex, so it is often emphasized less than the mechanics of probability and inference. The most commonly used classroom tools to assess students’ production or perception of randomness are binary choices, such as coin tosses, and number sequences, such as dice rolls. The field of psychology has a long history of research on random choice, and we have replicated some experiments that support results seen there regarding the collective distribution of individual choices in spatial geometries. The data from these experiments can easily be incorporated into the undergraduate classroom to visually illustrate the concepts of random choice, complete spatial randomness (CSR), and Poisson processes. Furthermore, spatial statistics classes can use this point pattern data in exploring hypothesis tests for CSR along with simulation. To foster student engagement, it is simple to collect additional data from students to assess agreement with existing data or to develop related, unique experiments. All R code and data to duplicate results are provided.

Quantum algorithmic randomness

Quantum Martin-Löf randomness (q-MLR) for infinite qubit sequences was introduced by Nies and Scholz [J. Math. Phys. 60(9), 092201 (2019)]. We define a notion of quantum Solovay randomness, which is equivalent to q-MLR. The proof of this goes through a purely linear algebraic result about approximating density matrices by subspaces. We then show that random states form a convex set. Martin-Löf absolute continuity is shown to be a special case of q-MLR. Quantum Schnorr randomness is introduced. A quantum analog of the law of large numbers is shown to hold for quantum Schnorr random states.