Notion Of Randomness Research Articles

Randomness notions and partial relativization

We study the computational complexity of an oracle set using a number of notions of randomness that lie between Martin-Löf randomness and 2-randomness in terms of strength. These notions are weak 2-randomness, weak randomness relative to ∅′, Demuth randomness and Schnorr randomness relative to ∅′. We characterize the oracles A such that ML[A] ⊆ C, where C is such a randomness notion and ML[A] denotes the Martin-Löf random reals relative to A, using a new meta-concept called partial relativization. We study the reducibility associated with weak 2-randomness and relate it with LR-reducibility.

How to build a probability-free casino

Casinos operate by generating sequences of outcomes which appear unpredictable, or random, to effective gamblers. We investigate relative notions of randomness for gamblers whose wagers are restricted to a finite set. Some sequences which appear unpredictable to gamblers using wager amounts in one set permit unbounded profits for gamblers using different wager values. In particular, we show that for non-empty finite sets A and B, every A-valued random is B-valued random if and only if there exists a k⩾0 such that B⊆A⋅k.

Fixed point theorems on partial randomness

In our former work [K. Tadaki, A statistical mechanical interpretation of algorithmic information theory, in: Local Proceedings of Computability in Europe 2008, CiE 2008, pp. 425–434, June 15–20, 2008, University of Athens, Greece. Extended Version Available at arXiv:0801.4194v1], we developed a statistical mechanical interpretation of algorithmic information theory by introducing the notion of thermodynamic quantities at temperature T, such as free energy F(T), energy E(T), and statistical mechanical entropy S(T), into the theory. These quantities are real functions of real argument T>0. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by program-size complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity. Namely, the computability of the value of partition function Z(T) gives a sufficient condition for T∈(0,1) to be a fixed point on partial randomness. In this paper, we show that the computability of each of all the thermodynamic quantities above gives the sufficient condition also. Moreover, we show that the computability of F(T) gives completely different fixed points from the computability of Z(T).

Algorithmic tests and randomness with respect to a class of measures

This paper offers some new results on randomness with respect to classes of measures, along with a didactic exposition of their context based on results that appeared elsewhere. We start with the reformulation of the Martin-Löf definition of randomness (with respect to computable measures) in terms of randomness deficiency functions. A formula that expresses the randomness deficiency in terms of prefix complexity is given (in two forms). Some approaches that go in another direction (from deficiency to complexity) are considered. The notion of Bernoulli randomness (independent coin tosses for an asymmetric coin with some probability p of head) is defined. It is shown that a sequence is Bernoulli if it is random with respect to some Bernoulli measure B p . A notion of “uniform test” for Bernoulli sequences is introduced which allows a quantitative strengthening of this result. Uniform tests are then generalized to arbitrary measures. Bernoulli measures B p have the important property that p can be recovered from each random sequence of B p . The paper studies some important consequences of this orthogonality property (as well as most other questions mentioned above) also in the more general setting of constructive metric spaces.

Characterizing strong randomness via Martin-Löf randomness

We introduce two methods for characterizing strong randomness notions via Martin-Löf randomness. We apply these methods to investigate Schnorr randomness relative to 0̸ ′ .

Randomness and lowness notions via open covers

One of the main lines of research in algorithmic randomness is that of lowness notions. Given a randomness notion ℛ, we ask for which sequences A does relativization to A leave ℛ unchanged (i.e., ℛA=ℛ)? Such sequences are called low forℛ. This question extends to a pair of randomness notions ℛ and S, where S is weaker: for which A is SA still weaker than ℛ? In the last few years, many results have characterized the sequences that are low for randomness by their low computational strength. A few results have also given measure-theoretic characterizations of low sequences. For example, Kjos-Hanssen (following Kučera) proved that A is low for Martin-Löf randomness if and only if every A-c.e. open set of measure less than 1 can be covered by a c.e. open set of measure less than 1.In this paper, we give a series of results showing that a wide variety of lowness notions can be expressed in a similar way, i.e., via the ability to cover open sets of a certain type by open sets of some other type. This provides a unified framework that clarifies the study of lowness for randomness notions, and allows us to give simple proofs of a number of known results. We also use this framework to prove new results, including showing that the classes Low(MLR,SR) and Low(W2R,SR) coincide, answering a question of Nies. Other applications include characterizations of highness notions, a broadly applicable explanation for why low for randomness is the same as low for tests, and a simple proof that Low(W2R,S)=Low(MLR,S), where S is the class of Martin-Löf, computable, or Schnorr random sequences.The final section gives characterizations of lowness notions using summable functions and convergent measure machines instead of open covers. We finish with a simple proof of a result of Nies, that Low(MLR)=Low(MLR,CR).

Leftover Hashing Against Quantum Side Information

The Leftover Hash Lemma states that the output of a two-universal hash function applied to an input with sufficiently high entropy is almost uniformly random. In its standard formulation, the lemma refers to a notion of randomness that is (usually implicitly) defined with respect to classical side information. Here, we prove a (strictly) more general version of the Leftover Hash Lemma that is valid even if side information is represented by the state of a quantum system. Furthermore, our result applies to arbitrary delta-almost two-universal families of hash functions. The generalized Leftover Hash Lemma has applications in cryptography, e.g., for key agreement in the presence of an adversary who is not restricted to classical information processing.

Aesthetics of Serendipity

Creativity is stochastic and assumptive in nature. The importance of randomness in the creative process must not be ignored, underestimated, or intentionally disregarded in a condescending way. Notions of chance, randomness, or unpredictability are important, especially when it comes to artistic creation. In addition to these notions, serendipity can be seen as the expected contribution for making expedient discoveries by coincidence or by chance. To put serendipity into work, a need exists to accumulate a list of questions that need solving, acquaintance with already existing answers, and their use in daily life. Only when this knowledge is present, ‘chance’ can take its part in establishing the perfect milieu for the ‘problem’ and the ‘solution’ to find each other. If there is a great deal of knowledge accrued about the problem and the requisites for the solution, chance adds the final piece to the puzzle. Traditional ‘prescriptive, authoritarian and rather conventional’ aesthetics vs. a new ‘generative, irregular, unprescribed’ aesthetics can then be examined.

Jump inversions inside effectively closed sets and applications to randomness

AbstractWe study inversions of the jump operator on 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.

Relativizations of randomness and genericity notions

A set A is a base for Schnorr randomness if it is Turing reducible to a set R that is Schnorr random relative to A, and the notion of a base for weak 1-genericity can be defined similarly. We show that A is a base for Schnorr randomness if and only if A is a base for weak 1-genericity if and only if the halting set K is not Turing reducible to A. Furthermore, we define a set A to be high for Schnorr randomness versus Martin-Löf randomness if and only if every set that is Schnorr random relative to A is also Martin-Löf random unrelativized, and we show that A is high for Schnorr randomness versus Martin-Löf randomness if and only if K is Turing reducible to A. Results concerning highness for other pairs of randomness notions are also presented.

NON-LINEAR DYNAMICS, COMPLEXITY AND RANDOMNESS: ALGORITHMIC FOUNDATIONS

Abstract In this paper I make an attempt to describe, discuss and extend a few aspects of the rich mathematical tapestry that can be woven with rigorous notions of non-linear dynamics, complexity and randomness, in terms of algorithmic mathematics. It is a tapestry that I try to weave with economic analysis, economic theory and economic modelling in mind. All three notions – that is, non-linear dynamics, complexity and randomness – have a rich conceptual, modelling or analytic tradition in core areas of economic theory, both at the micro and macro levels. It is the algorithmic foundation I try to provide for them that could be considered the novel contribution in this paper. Once the algorithmic foundations are in place, it is, for example, almost natural to consider the famed difficulties of obtaining closed form solutions for non-linear, complex or random dynamic models in economics almost a trivial vestige of a pre-simulation era in mathematical modelling.

Difference randomness

In this paper, we define new notions of randomness based on the difference hierarchy. We consider various ways in which a real can avoid all effectively given tests consisting of $n$-r.e. sets for some given $n$. In each case, the $n$-r.e. randomness hierarchy collapses for $n\geq 2$. In one case, we call the resulting notion difference randomness and show that it results in a class of random reals that is a strict subclass of the Martin-Löf random reals and a proper superclass of both the Demuth random and weakly 2-random reals. In particular, we are able to characterize the difference random reals as the Turing incomplete Martin-Löf random reals. We also provide a martingale characterization for difference randomness.

Descriptive set theoretical complexity of randomness notions

Descriptive set theoretical complexity of randomness notions

Neuronal Jitter: Can We Measure the Spike Timing Dispersion Differently?

We propose a novel measure of statistical dispersion of a positive continuous random variable: the entropy-based dispersion (ED). We discuss the properties of ED and contrast them with the widely employed standard deviation (SD) measure. We show that the properties of SD and ED are different: while SD is a second moment characteristics measuring the dispersion relative to the mean value, ED measures an effective spread of the probability distribution and is more closely related to the notion of randomness of spiking activity. We apply both SD and ED to analyze the temporal precision of neuronal spiking activity of the perfect integrate-and-fire model, which is a plausible neural model under the assumption of high input synaptic activity. We show that SD and ED may give strikingly different results for some widely used models of presynaptic activity.

Separations of non-monotonic randomness notions

In the theory of algorithmic randomness, several notions of random sequence are defined via a game-theoretic approach, and the notions that received the most attention are perhaps Martin-Lof (ML) randomness and computable randomness. The latter notion was introduced by Schnorr and is rather natural: an infinite binary sequence is computably random if no total computable strategy succeeds on it by betting on bits in order. However, computably random sequences can have properties that one may consider to be incompatible with being random, in particular, there are computably random sequences that are highly compressible. The concept of ML randomness is much better behaved in this and other respects, on the other hand its definition in terms of martingales is considerably less natural. Muchnik, elaborating on ideas of Kolmogorov and Loveland, refined Schnorr’s model by also allowing non-monotonic strategies, i.e. strategies that do not bet on bits in order. The subsequent ‘non-monotonic’ notion of randomness, now called Kolmogorov–Loveland randomness, has been shown to be quite close to ML randomness, but whether these two classes coincide remains a fundamental open question. In order to get a better understanding of non-monotonic randomness notions, Miller and Nies introduced some interesting intermediate concepts, where one only allows non-adaptive strategies, i.e. strategies that can still bet non-monotonically, but such that the sequence of betting positions is known in advance (and computable). Recently, these notions were shown by Kastermans and Lempp to differ from ML randomness. We continue the study of the non-monotonic randomness notions introduced by Miller and Nies and obtain results about the Kolmogorov complexities of initial segments that may and may not occur for such sequences, where these results then imply a complete classification of these randomness notions by order of strength.

Prequential randomness and probability

This paper studies Dawid’s prequential framework from the point of view of the algorithmic theory of randomness. Our first main result is that two natural notions of randomness coincide. One notion is the prequential version of the measure-theoretic definition due to Martin-Löf, and the other is the prequential version of the game-theoretic definition due to Schnorr and Levin. This is another manifestation of the close relation between the two main paradigms of randomness. The algorithmic theory of randomness can be stripped of its algorithmic aspect and still give meaningful results; the measure-theoretic paradigm then corresponds to Kolmogorov’s measure-theoretic probability and the game-theoretic paradigm corresponds to game-theoretic probability. Our second main result is that measure-theoretic probability coincides with game-theoretic probability on all analytic (in particular, Borel) sets.

Randomness on Computable Probability Spaces—A Dynamical Point of View

We extend the notion of randomness (in the version introduced by Schnorr) to computable probability spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff’s pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications.

HESS Opinions "A random walk on water"

Abstract. According to the traditional notion of randomness and uncertainty, natural phenomena are separated into two mutually exclusive components, random (or stochastic) and deterministic. Within this dichotomous logic, the deterministic part supposedly represents cause-effect relationships and, thus, is physics and science (the "good"), whereas randomness has little relationship with science and no relationship with understanding (the "evil"). Here I argue that such views should be reconsidered by admitting that uncertainty is an intrinsic property of nature, that causality implies dependence of natural processes in time, thus suggesting predictability, but even the tiniest uncertainty (e.g. in initial conditions) may result in unpredictability after a certain time horizon. On these premises it is possible to shape a consistent stochastic representation of natural processes, in which predictability (suggested by deterministic laws) and unpredictability (randomness) coexist and are not separable or additive components. Deciding which of the two dominates is simply a matter of specifying the time horizon and scale of the prediction. Long horizons of prediction are inevitably associated with high uncertainty, whose quantification relies on the long-term stochastic properties of the processes.

A statistical mechanical interpretation of algorithmic information theory: Total statistical mechanical interpretation based on physical argument

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed by our former works [K. Tadaki, Local Proceedings of CiE 2008, pp. 425–434, 2008] and [K. Tadaki, Proceedings of LFCS'09, Springer's LNCS, vol. 5407, pp. 422–440, 2009], where we introduced the notion of thermodynamic quantities, such as partition function Z(T), free energy F(T), energy E(T), statistical mechanical entropy S(T), and specific heat C(T), into AIT. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. Furthermore, we showed that this situation holds for the temperature T itself, which is one of the most typical thermodynamic quantities. Namely, we showed that, for each of the thermodynamic quantities Z(T), F(T), E(T), and S(T) above, the computability of its value at temperature T gives a sufficient condition for T (0,1) to satisfy the condition that the partial randomness of T equals to T. In this paper, based on a physical argument on the samelevel of mathematical strictness as normal statistical mechanics in physics, we develop a totalstatistical mechanical interpretation of AIT which actualizes a perfect correspondence to normalstatistical mechanics. We do this by identifying a microcanonical ensemble in the framework ofAIT. As a result, we clarify the statistical mechanical meaning of the thermodynamic quantitiesof AIT.

Promptness does not imply superlow cuppability

AbstractA classical theorem in computability is that every promptly simple set can be cupped in the Turing degrees to some complete set by a low c.e. set. A related question due to A. Nies is whether every promptly simple set can be cupped by a superlow c.e. set, i.e. one whose Turing jump is truth-table reducible to the halting problem ∅′. A negative answer to this question is provided by giving an explicit construction of a promptly simple set that is not superlow cuppable. This problem relates to effective randomness and various lowness notions.