Luciano Da Fontoura Costa
email: luciano@ifsc.usp.br

Luciano Da

F Costa

Randomness: A Challenging Central Concept

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

One of the tasks most systematically shared between science and typical human experience corresponds to prediction (e.g. [START_REF] Da | Modeling: The human approach to science[END_REF]). Given situations involving the choice between two or more alternatives, it becomes a critical endeavor to predict the implications of each of those alternatives, so that the most suitable choice -given specific requirements -can be taken. Predicting the implications of decisions actually constitutes the main aspect of science, through the activity of modeling, where quantitative constructs are developed (e.g. [START_REF] Da | Statistical modeling[END_REF]) as the means not only for better understanding physical systems, but also for making predictions about the unfolding of their respective dynamics.

Another closely related situation involving prediction is, given two or more possible outcomes, to assign probabilities that can quantify how likely each of those possible results is. This type of problem constitutes the main subject studied in the areas of probability and statistics (e.g. [START_REF] Da | Statistical modeling[END_REF][START_REF] Da | Multivariate statistical modeling[END_REF][START_REF] Bertsekas | Introduction to Probability[END_REF][START_REF] Degroot | Probability and Statistics[END_REF][START_REF]Probability and statistics ebook[END_REF][START_REF] Kreyszig | Advanced Engineering Mathematics[END_REF][START_REF] Mukhopadhyay | Probability and Statistical Inference[END_REF][START_REF] Gallager | Stochastic Processes: Theory for Applications[END_REF]), being often approached in terms of respective statistical models, involving random variables as well as respective models in terms of density probability functions and/or moments.

Thus, prediction can be understood as being directly related to the two important following activities: (a) as-signing a probabilities to possible outcomes, which has been addressed mostly by using concepts and methods from probability and statistics; and/or (b) identifying the implications of specific outcomes or choices, which has been tackled mainly through scientific modeling.

Predictions are necessary because of two main factors: (i) true, intrinsic randomness; and (ii) incomplete knowledge. It should be observed that these two factors can potentially underly either or both the activities (a) and (b) above. In the former case, we have situations such as rolling an ideal dice, in which case it is impossible to known with absolute certainty the respective outcome. Henceforth, 'true' or 'intrinsic' randomness is understood to lead to outcomes that are completely unaccountable for. The latter case is characterized by the fact the observed experiment is actually deterministic, but that lack of knowledge about all its relevant aspects, including its initial condition, implies uncertainty while predicting the unfolding of their respective dynamics.

Prediction is directly motivated by the existence of any of the two above types of randomness, without which it would not exist. Henceforth, we will understand a random event as corresponding to a situation whose outcome is impossible to be predicted with certainty, for any reason. Thus, every measurement from the real-world intrinsically involves a random aspect, because a certain level 1 of uncertainty and incompleteness is always present. The quantities observed when measuring a given property are often called random variables. Events and measurements that are not random, and therefore can be predicted with certainty, can be said to be deterministic. In particular, the classical mechanics approach to the real-world is intrinsically therefore deterministic. Thus, all in all, we shall consider three types of events/measurements: Deterministic, truly random, and random because of incomplete knowledge.

.

The possible existence of random and deterministic events and measurements imply the important problem of devising approaches that can, given a specific system, decide on its randomness. As well shall discuss at more length in Section 5, it turns out that this constitutes a particularly difficult problem, having motivated several multidisciplinary approaches in areas that include probability and statistics, scientific modeling, pattern recognition, dynamical systems, classical and quantum mechanics, among many others. In addition to developing means for identifying randomness, it also becomes important to have means for quantifying its levels and generating sequences of random values or symbols.

One of the few sure aspects regarding prediction about the dynamics of deterministic dynamical system is that prediction would not be required provided one knows everything about the system -in the sense of having a complete model, as well as fully accurate knowledge of the initial conditions. No predictions would be necessary in this case because the dynamics of the system could be directly obtained from the respective complete model, unfolding form the initial condition. Thus, in case our universe were deterministic, the intrinsic randomness could not be verified. However, predictions would still be required for incompletely modeled dynamical systems or when trying to identify randomness experimentally in terms of measurements possibly influenced by noise and interferences.

Despite its critical implication for the activity of predicting, it is not known for certain whether our universe is or not deterministic. While classical mechanics indicated that the universe was deterministic, the advent of quantum mechanics completely changed the perspectives. While currently the odds seem to point toward a nondeterministic universe, the possibility that it eventually turns out to be deterministic cannot completely excluded from the present work. In addition, as it will be argued in this work, true intrinsic randomness is possibly a graded quantity, therefore presenting different levels of intensity, which could tend to decrease as one moves from smaller to larger space-time scales, but which could also be amplified by non-linear effects (e.g. chaos). Though discussing this interesting and centrally challenging question, the present work by no means aims at answering it.

When approached from the perspectives briefly revised above, it becomes hard to overstate the importance of prediction and randomness. Indeed, more effective decisions could be taken provided the probabilities of outcomes, as well as their implications, could be accurately predicted. Ultimately, these prospects have played a major role while developing science and technology. Yet, despite long continuing research efforts, randomness remains an elusive concept, especially when taken at a more strict and comprehensive manner.

The present work aims at presenting and discussing some of the several concepts and methods that underly and/or relate to randomness. In addition, special attention is focused on the issues of identifying, quantifying and generating randomness. The intriguing issue about true randomness is also briefly discussed. Developing in a mostly informal, somewhat unconventional, and hopefully relatively accessible manner, while also involving several figures and examples, the present work should not be understood to be complete, or taken as a basic text on related subjects. However, it is hoped that the discussion of several aspects of randomness will provide some indication about some of the main involved facets of randomness, as well as their respectively implied challenges.

Experimental and Constructive Approaches to Randomness

Figure 1 illustrates a key situation regarding randomness.

Here, we have two agents (natural or artificial) A and B, the former transmitting a sequence of integer digits (from 0 to 9) to the receiving agent B, whose problem is to decide whether the received sequence of integer values is random or not. Let us consider the situation in which the digits being transmitted corresponds to an intermediate portion of the number π, which is irrational and infinite. Two possible outcomes can be expected: (a) the agent B has great knowledge about numbers and identifies the sequence as corresponding to a portion of the number π; and (b) the agent B does not recognize the sequence as a portion of π. In case (a), the sequence will be deemed as being very likely to be deterministic, in which case the agent will always be able to foresee the next digits. However, in case (b), the sequence will be understood as being possibly random, as no prediction could be accurately made. This latter situation provides an example of randomness as a possible consequence of lack of knowledge about the observed system. Another interesting point regarding the possibility to use mathematical irrational constants as π, √ 2 and e is that these values are particularly well-known and prominent, being more likely to be eventually adopted as possible sources of randomness, therefore becoming even less 'random'. A substantially more difficult situation to be considered in the experiment in Figure 1 would arise in case of less known irrational numbers.

The situation in Figure 1 makes it plain that randomness is a concept relative to the agent analyzing the observations, depending critically on the knowledge about the respective random experiments. This knowledge can be effectively approached in terms of respective models (e.g. [START_REF] Da | Modeling: The human approach to science[END_REF]), in the sense that a complete model of the specific random experiment can provide critical information for deciding on its possible randomness. We can summarize this point as follows:

Lack of knowledge can be perceived as randomness. This initial conclusion already poses a substantial challenge for deciding on randomness, as it is not possible to have fully complete models of the physical world (e.g. [START_REF] Da | Modeling: The human approach to science[END_REF]). As a brief illustration of this principle, we have that the motion of a real-world pendulum, when observed at full precision, will ultimately depend of every possible mass in the universe. Thus, in principle, even if there are fully random sequences in the universe, we will not be able to fully identify their nature, as we will always be uncertain if our limitations in the respective prediction are a consequence of intrinsic randomness or lack of full information about the respective system. Other limitations, including the fact that no measurement can be fully accurate and Heisenberg's uncertainty principle, further complicate the experimental identification of randomness.

It follows from the above considerations that the experimental verification of randomness can only be made up to a given level of certainty. In other words, it is in principle impossible to conclude on real intrinsic randomness while making observations, taking measurements and making predictions.

There is, however, a second possibility that can be potentially considered while trying to decide if there is (or not) intrinsic randomness in a system, which consists in devising abstract, logic-mathematic models capable of intrinsic randomness. This alternative approach to randomness, will be henceforth referred to as constructive, involve developing or considering abstract models that can lead or explain randomness.

Though randomness is often illustrated in terms of flipping a coin, this situation actually is potentially influence by several complex factors including the way in which the coin is thrown (e.g. initial speed, direction, rotation, etc.) and falls. In the present work we adopt instead the abstract, ideal situation depicted in Figure 2(a), concerning a hypothetical experiment in which a perfect sphere sits, initially static, onto a perfectly sharp edge. Observe that the center of mass of the sphere is initially in perfect alignment with the edge, and no external influences other than gravity are present.

Figure 2: A perfectly symmetric sphere, initially static, under action of gravity only, lies onto a perfectly sharp edge (a). The center of mass of the sphere is initially aligned with the edge. When approached from the classical mechanics perspective, the sphere can be understood to remain in unstable equilibrium until some additional perturbation (external gravitational fields, vibrations, etc.), even if of infinitesimal intensity, eventually may imply it to fall toward one of the sides and touch the surfaces marked as 0 or 1. Situations involving two or more outcomes can be effectively represented in terms of respective trees, characterized by branches or bifurcations, as illustrated in (b) for the sphere-on-the-edge case. Another situation involving two possible outcomes in a mechanical system is illustrated in (c). Here, substantially more word and time needs to be invested so that the sphere moves to one of the possible upper branches. The outcome of this situation can also be abstracted by a respective tree, shown in (d).

When approached from the classical physics perspective (Newton's laws of motion), the situation depicted in Figure 2(a) will be preserved until some mechanical perturbation, even if infinitesimally small, is applied to the sphere, in which case it will fall to one of the two possible sides until touching, after some time, the surfaces marked as 0 or 1. In other words, if true randomness is to be verified, this is the type of situation in which it would possibly be more prone to happen.

Because mechanical perturbations necessarily involve mass and/or energy, these perturbations need themselves to be consequence of some (classic) mechanics effect that is governed by Newton's laws of motion, therefore being deterministic. It follows from this simple example that, from the classic perspective, it is impossible to have true intrinsic randomness in nature. Actually, by being unaccountable for, true randomness would needed to be caused by something else then an effect involving mass/energy.

Figure 2(b) presents a somewhat distinct situation involving a sphere being eventually displaced until touching surfaces 0 or 1. The sphere, which is initially at rest over the plane at the bottom of the figure and under action of gravity, can be displaced upwards toward any of the two possible sides. Though this situation also involves an intrinsic bilateral symmetry identical to that in Figure 2(a), the complete displacement of the sphere until touching surfaces 0 or 1 now requires substantially larger quantity of work (e.g. [START_REF] Serway | Physics for scientists and engineers[END_REF][START_REF] Da | Single point particle motion: Mass, force, momenta, impulse[END_REF]) than previously during a much longer period of time, implying in much larger energy requirement. Therefore, the cause of this displacement, be it random or not, would typically involve much more 'determination' in the sense of investing larger amounts of energy and time. The randomness of the outcome issuing from situation in Figure 2(b) would therefore depend on how the mechanism required for the complete motion of the sphere is initiated and maintained.

Figure 3 depicts a possible (and relatively naïve) manner to approach the previously discussed sphere equilibrium situation in terms of quantum mechanics.

Given that the mass of the sphere is assumed to be homogeneous, the diversity of wave-function lengths indicates respective projections onto the plan of the image which, as a consequence of the respective symmetry, are enough to be considered in the respective discussion. The equilibrium will be maintained only in case the contributions of all wave-functions are completely symmetric along time and space. Any small bias taking place along some long enough period of time will imply in possible oscillations or even the falling of the sphere.

It should also be observed that the situations illustrated in Figure 2 relate to the phenomenon of symmetry breaking (e.g. [START_REF] Strocchi | Symmetry breaking[END_REF]), in which some overall property of the system changes from a symmetric to asymmetric configura- 3 Randomness, Time, and Causation

Given that randomness depends inexorably of changes, it also strongly related to causation (e.g. [START_REF] Pearl | Causality[END_REF][START_REF] Da | Continuous and event-driven causality: A simple model-based approach[END_REF][START_REF] Granger | Some recent development in a concept of causality[END_REF][START_REF] Hiemstra | Testing for linear and nonlinear Granger causality in the stock price-volume relation[END_REF][START_REF] Bressler | Wiener-Granger causality: a well established methodology[END_REF][START_REF] Salmon | Causality and explanation[END_REF][START_REF] Bunge | Causality and modern science[END_REF]) taking place along time, an issue that is discussed very briefly in the present section. As indicated in [START_REF] Da | Continuous and event-driven causality: A simple model-based approach[END_REF], there are two main types of causations unfolding along time: continuous and event-driven. While the former type involves causal influences taking place continuously along time, the latter is characterized by the observation of discrete events taking place at specific time instants. Interestingly, the latter type of causation is relatively more intricate, being illustrated in Figure 4. [START_REF] Da | Continuous and event-driven causality: A simple model-based approach[END_REF]. The cause C needs to take place for a minimum period of time ∆, which is then followed by a respective delay δ, and the effect E, along which the causation continues but is no longer under control of the agent that initiated it. It is understood that, once the causation period is completed, the effect can no longer be avoided other than by some eventual external influence (not included in the diagram). In continuous causation, we have that ∆ → 0 and δ → 0, so that the effect will always immediately follow the respective immediate cause.

Though time is typically understood naturally by humans, it turns out that it is not an easy concept to be for-malized in physic-mathematical manner. The main issue here is that every physical event is, in principle reversible. In other words, this means that there are no intrinsic constraints in the laws of physics implying the respective dynamics to unfold in only one direction, which is called T-symmetry.

The time irreversibility of physical systems that is typically observed, has been linked to the concept of entropy, in particular the second law of thermodynamics (e.g. [START_REF] Serway | Physics for scientists and engineers[END_REF][START_REF] Newman | Monte Carlo Methods in Statistical Physics[END_REF]). Informally speaking, this law indicates that the entropy of isolated systems cannot decrease, being also related to the fact that heat always flow from hotter to colder objects.

Another interesting perspective involving randomness and time regards the possibility to travel to or observe the future. If this were possible, the outcomes of events could be predicted, therefore preventing true randomness. At the same time, the possibility of having non-local, simultaneous events (as in quantum mechanics EPR, e.g. [START_REF] Blaylock | The EPR paradox, Bell's inequality, and the question of locality[END_REF]) also has potentially critical implications for causality and true randomness.

Randomness and Probability

Random experiments are typically modeled in terms of random variables and respective probabilities, including probability density functions. In this section, we provide a brief overview of these concepts.

Every real-world measurement can be understood and modeled in terms of a respective random variable X. A numeric variable X can be described/modeled in terms of its respective probability density function p(x), which necessarily has to satisfy:

p(x) ≥ 0 (1) ˆ∞ -∞ p(x)dx = 1 (2)
In that case, we have:

P (x 1 ≤ x < x 2 ) = ˆx2 x1 p(x)dx (3)
As an example, the uniform probability density in an interval [a, b], a, b ∈ R and a ≤ b, can be expressed as: understanding that: ˆa a p(x)dx = 0 (5)

p(x) = c = 1 b-a , a ≤ x ≤ b 0, otherwise (4)
The uniform probability density function defined for the interval (0, 1) has special importance, being henceforth expressed as rand(). Each time this function is invoked, it will provide a real value x chosen uniformly from the interval (0, 1).

Discrete random variables can be modeled by using the Dirac delta function (e.g. [START_REF] Da | Statistical modeling[END_REF]). For instance, the perfectly uniform coin experiment can be modeled in terms of the following discrete density probability function:

p(x) = 0.5 δ(0), x = 0 0.5 δ(1), x = 1 (6)
which can be understood as an example of discrete uniform density probability function, being respectively illustrated in Figure 6. Given a set of numbers S M = {1, 2, . . . , M }, they can be drawn with uniform probability by using the rand() function as follows:

x = ⌊rand() M )⌋ + 1, x ∈ S M ( 7
)
where ⌊⌋ is the floor function.

The density function of a constant variable can be expressed as:

p(x) = δ(c), x = c 0, otherwise (8)
One frequent problem in probability and statistics consists of generating events with given probabilities. This problem can be addressed by using the Monte Carlo approach (e.g. [START_REF] Newman | Monte Carlo Methods in Statistical Physics[END_REF][START_REF] Robert | Monte Carlo statistical methods[END_REF]).

In its most basic form, the Monte Carlo method involves performing estimations while drawing values uniformly distributed in a given interval, typically [0, 1], in order to select events with a given probability. Let us illustrate the gist of this approach in terms of a specific example involving drawing an event with probability a given probability p. One value x is uniformly chosen from the interval (0, 1), and the even is taken if only if x ≤ p.

As a simple illustration of a Monte Carlo approach, we mention the Buffon's needle problem (e.g. [START_REF] Badger | Lazzarini's lucky approximation of π[END_REF]), in which the constant π is estimated by dropping needle with length l in a random manner (angles and positions) onto a plane with (infinite) marked parallel lines separated by a distance L (see Fig. 7. In case l < L, we would have:

π ≈ 2l #experiments L #cases in which the needle crosses a line (9)

Identifying Randomness

In this section, we develop a possible approach to understanding of randomness, aiming at a possible respective identification. Given that random variables of several types (e.g. integer, real, complex) can always be represented as (possibly infinite) binary numbers, for generality's sake, these values are henceforth understood as being represented as respective binary numbers, composed only by 0 and 1 digits.

We start by considering flipping a given coin N times (event) and recording the obtained results (outcomes). Such a procedure provides one of the simplest examples of a random experiment, characterized by uncertainty about the possible results. We shall also assume that only two results are possible, corresponding to head and tail, which could be arbitrarily represented by the values 1 and 0, respectively.

After N experiments, the number of respective head and tail observations will be represented as n h (N ) and n t (N ), leading to the following transient probabilities:

p h (N ) = p 1 (N ) = n 1 N ( 10
)
p t (N ) = p 0 (N ) = n 0 N ( 11
)
The probability of the two outcomes can now be defined in terms of the following limits to infinity:

P 1 = p 1 (N = ∞) = lim N →∞ n 1 N ( 12
)
P 0 = p 0 (N = ∞) = lim N →∞ n 0 N (13)
In case the above experiment were fully uniformly random, we would have:

P 1 = P 0 = 0.5 (14)
meaning that the number of observed heads and tails would be identical when N = ∞.

Though the above discussion suggests that the coin could be classified as being truly uniformly random provided P 1 = P 0 , this turns out not to be necessarily the case. That is because the overall probabilities P 1 and P 0 are completely general and do not tell us anything about the specific sequence and arrangements of heads and tails along an observed sequence. Indeed, consider the following sequence of observations, which is known, by hypothesis, to be completely periodic: h,t,h,t,h,t,h,t,h,t,h,t,. . . This sequence is plainly not random, as the next outcome can always be fully accurately predicted. Even so, we will nevertheless have P 1 = P 0 for even values of N tending to infinity:

h, t,
lim N →∞ p 1 (N ) = lim N →∞ p 0 (N ) (15)
It follows from the above example that randomness of the observations constitutes only a necessary condition, not being a sufficient requirement, for it is also required that all the respective sequence of observed values be the most unpredictable as possible.

A possible manner to obtain further indication about the randomness of a given sequence consists of considering the probabilities of respective pairs or adjacent values, of which there are the four following joint possibilities: 00; 01; 10; 11 [START_REF] Hiemstra | Testing for linear and nonlinear Granger causality in the stock price-volume relation[END_REF] A true uniformly random sequence would then be characterized by:

P 00 = p 00 (N = ∞) = lim N →∞ n 00 N = 0.25 (17)
P 01 = p 01 (N = ∞) = lim N →∞ n 01 N = 0.25 (18)
P 10 = p 10 (N = ∞) = lim N →∞ n 10 N = 0.25 (19)
P 11 = p 11 (N = ∞) = lim N →∞ n 11 N = 0.25 (20)
For practical reasons, transient probabilities (N < ∞) would need to be taken.

As an example, we could have the following characterization of the sequence of N = 20 subsequent pairs of observed digits in terms of transient probabilities, here indicated in percentage.

Though the four obtained probabilities present biases, in the sense of not corresponding to 25%, this provides a relatively weak indication whether the sequence is or not uniformly random, because of the small number N of observations of adjacent pairs. In other words, a fully random sequence, from the specific perspective of paired values, would be (partially) characterized by the above probabilities converging to 25% when N → ∞.

Even if a sequence were estimated to be random when its values are observed singly or in adjacent pairs, these two conditions would still be only necessary, and not sufficient, for deeming the sequence as being truly uniformly random.

Parenthetically, observe also that the two tests described above assumed that heads and tails have the same individual probability of 50%. Different probabilities of joint (pair) occurrences would be expected otherwise (see Section 7).

Each specific sub-sequence of heads and tails of length L can have a respective probability assigned in a manner similar to that discussed above. This allows us to consider the following test for fully uniform randomness of the dice rolling experiment: it will be truly random if, after an infinite number N of observations, the probabilities of each possible sub-sequence of length L = 1, 2, . . . , N are all mutually identical. Conveniently, each of these subsequences can be effectively represented by a respective integer decimal value obtained from the respective binary subsequence.

Observe, however, that the number of subsequences decreases with L. More specifically, given a specific value N , the number of sequences of length L will be limited to N -L + 1. In particular, for L = N , we would have just a single sequence of L digits. In order to have statistical relevance, the whole experiment would need to be repeated a large number of times, actually an infinite number in case we wanted a fully precise test.

Even if this were possible, it would still be necessary that the experiment be stationary, in the sense of perfectly preserving all possible single and joint probabilities. In addition, each whole experiment would need to be independent of the previous experiments.

In particular, the test considering N = L would be enough by itself, without requiring the complementary tests for N < L. That is because all possible combinations of digits 0 and 1 are considered in the case N = L. As observed above, this test would require a stationary experiment to be independently repeated an infinite number of times (or a very large number of times in the case of a provisional estimation).

In case the above mentioned test for N = L were viable, assuming fully uniform probabilities, the probability of each possible sequence would be:

P = 1 2 L = 2 -L (21)
as there are 2 L possible distinct binary subsequences with L digits.

Though we have been able to develop a hypothetical test which can, in principle, indicate if the coin experiment is truly random, there is a practical impossibility of its application, as it involves an infinite number of repetitions and observations. This indicates that it is virtually impossible to be completely certain about an observed sequence being fully random or not. What can be aimed instead is to use tests similar to those discussed above for a finite number of experiments of finite length, which can only provide preliminary indications about the respective randomness.

Several interesting aspects underly the discussion developed in the present section, some of which are covered in more detail in the following sections.

Measuring Randomness

Interestingly, even constant variables are often understood as particular cases of random variables, for the sake of generality. This indicates that random variables can have markedly varying levels (or degrees) of randomness, extending from 0 in the case of a constant random variable up to a maximum randomness value.

The level of randomness of a given random variable X can be associated to the dispersion of its respective probability density function p(x). Therefore, the wider the dispersion, the more random the variable can be understood to be.

While the statistical concepts of variance and standard deviation (e.g. [START_REF] Da | Statistical modeling[END_REF][START_REF] Bertsekas | Introduction to Probability[END_REF][START_REF] Degroot | Probability and Statistics[END_REF][START_REF]Probability and statistics ebook[END_REF][START_REF] Kreyszig | Advanced Engineering Mathematics[END_REF]) can be effectively adopted to quantify the level of dispersion of a random variable, there is another related measurement, namely entropy (e.g. [START_REF] Cover | Elements of Information Theory[END_REF]), which has some particularly interesting features. More specifically, the entropy of a continuous random variable X described by a respective density p(x) can be formally expressed as:

ε X = - ˆ∞ -∞ p(x) log [p(x)] dx (22)
In the case of a discrete random variable X taking values x in the set S M = {1, 2, . . . , M }, the respective entropy can be calculated as:

ε X = - M k=1 p i log(p i ). ( 23
)
where p i is the probability of drawing value x = i. One of the interesting features of entropy is that it is directly related to information and statistical mechanics, constituting one of the basis of the second law of thermodynamics (e.g. [START_REF] Serway | Physics for scientists and engineers[END_REF][START_REF] Cover | Elements of Information Theory[END_REF]). In addition, when radix-2 logarithm is employed in Equation 23, the obtained entropy can be understood to have unit of bits, corresponding to the estimated average number of bits necessary to represent the respective random variable.

Given an entropy measurement ε X , it becomes interesting to consider also its respective exponential, giving rise to the respective exponential entropy (e.g. [START_REF] Campbell | Exponential entropy as a measure of extent of a distribution[END_REF][START_REF] Travençolo | Accessibility in complex networks[END_REF][START_REF] Pielou | Shannon's formula as a measure of specific diversity: its use and misuse[END_REF][START_REF] Benatti | Complex networks accessibility symmetry[END_REF][START_REF] Viana | Effective number of accessed nodes in complex networks[END_REF]):

η X = e ε X ( 24
)
The exponential entropy, a real-valued quantity, is of particular interest as it is directly related to the effective number of choices that can be derived from a given random variable X. Thus, in a sense, an analogy can be drawn between the exponential entropy and the concept of fractal dimension (e.g. [START_REF] Peitgen | Chaos and Fractals: New Frontiers of Science[END_REF]), which is another real-valued quantity interpolating between the more traditional topological dimension.

For instance, in the case of the constant density, we would have:

ε X = - 1 k=1 p c log(p c ) = -1 log(1) = -(1) (0) = 0 (25)
This makes complete sense, because the randomness of a constant random variable could indeed be expected to be null.

The respective exponential entropy can then be obtained as: η X = e ε X = e 0 = 1 [START_REF] Travençolo | Accessibility in complex networks[END_REF] confirming that a single value can be effectively derived from X.

In the case of the example described by Equations 17 to 20, we would have: Observe that the effective number of values in this case is substantially greater than 1, but smaller than the total of 4 possible outcomes.

ε X = -[0.
It can be shown that the discrete uniform density probability function is characterized by the discrete density with the largest entropy. Indeed, in case the previous example were perfectly uniform, it would follow that:

ε X = -[4(0.25) log(0.25)] = -log(0.25) η X = e log(1/4) = 4
As could be expected, the effective number of derivable values becomes identical to the number of possible outcomes when the entropy is maximum.

Though the illustrations above consider the application of entropy to characterize the dispersion of a given sequence while taking into account isolated symbols, as discussed in Section 5 this test should also be applied to all respective subsequences in order to characterize more completely the randomness of that sequence.

The quantification of the level of randomness associated to a random variable provides an interesting and not so often considered perspective for appreciating the own nature of randomness, as well as its possible impact. For instance, a measurement involving a small level of randomness can be considered as not being random in specific situations in which the level of randomness does not influence the respective dynamics or outcome in any relevant way.

In addition to taking into account the level of randomness of a random variable, it is also interesting to consider its respective maximum values, which are particularly important in situations involving thresholded activations.

It is important to keep in mind that there are many other approaches to quantifying the randomness of a sequence of observations, including combinations of several tests as the Diehard Battery of Tests (e.g. [START_REF] Marsaglia | Some difficult-topass tests of randomness[END_REF][START_REF]Diehard tests[END_REF]).

Randomness and Hierarchy

One important aspect of randomness concerns whether the observation of one outcome may influence the randomness (probability) of a subsequent outcome. Thus, we have observations of joint random outcomes or variables. For instance, let us assume that we are receiving a sequence of two digits 0 or 1, which can be understood in terms of the two following events: (A) observing the first received digit; and (B) observe the second digit. The probabilities of the outcomes of event (A) will henceforth be expressed as P (0) and P (1), while the outcomes of the second event will be represented as P (0|0), P (0|1), P (1|0), P (1|1).

For the sake of proper normalization, it is henceforth assumed that:

P (0) + P (1) = 1 ( 27
)
P (0|0) + P (0|1) + P (1|0) + P (1|1) = 1 ( 28
)
The blimp character '|' can be read as 'given that', so that B|A means that outcome B follows outcome A, being possibly influenced, or conditioned by that previous occurrence. For this reason, probabilities of type P (B|A) are said to be conditional.

In principle, two situations may occur when observing conditional events: (a) complete independence, in the sense a previous observation by no means affects the probabilities of subsequent observations; and (b) otherwise, i.e. a previous observation influences the successive probabilities.

The conditional probability between two events A and B can be expressed as:

P (B|A) = P (A ∩ B) P (A) (29)
P (A|B) = P (A ∩ B) P (B) (30)
where P (A ∩ B) means the joint observation of A and B.

The Expression 29 can be readily rewritten as follows:

P (A, B) = P (A ∩ B) = P (A) P (B|A) (31)
which more directly reflects eventual interest in the joint, subsequent events, being also more closely related to the following figures presenting trees of probabilities.

Recall that the intersection between two sets A and B is commutative, i.e. A ∩ B = B ∩ A. However, the conditional probability is, in general, not commutative, in the sense that:

In general: P (A|B) ̸ = P (A|B) [START_REF]Diehard tests[END_REF] We will also adopt the alternative representation:

P (A ∩ B) = P (A, B) = P (B, A) (33)
Figure 8 illustrates the above experiment involving an event B taking place subsequently to another event A. Interestingly, the sequence of subsequent events or outcomes therefore defines a respective hierarchy in which each following level is potentially (but not necessarily) conditioned by the outcome of the previous hierarchical levels. A first possibility is that the outcome of the first outcome has no effect whatsoever on the second outcome. In this case, the two outcomes are said to be independent, and we have that: An even more specific situation concerns the case, in which all involved probabilities are identical (and thus mutually independent), is illustrated in Figure 10. In the particular situation in which the events are independent, the order of the digits 0 and 1 will not matter, and the probability of observing a sequence with n 0 digits 0 and n 1 digits 1 can be expressed as:

P (B = 0|A = 0) = P (B =
P = p n0 0 p n1 1 ( 34
)
where p 0 and p 1 are the probabilities of observing a digit 0 and 1, respectively.

Another important situation concerning two subsequent events A and B happens when the events are dependent one another, in the sense that P (B|A) is influenced by P (A). In this case, we have that:

P (A, B) = P (B|A) P (A) (35)
A particular example of dependent events is depicted in Figure 11. The just illustrated situation in which the occurrence of one event influences the probability of a subsequent event is particularly interesting, as it indicate that the randomness of a given random variable is not necessarily independent, also revealing a possible network of influences between random events.

In a sense, in case P (B|A) ̸ = P (B) suggests that the observation of particular outcomes from A 'causes' a modification of the probabilities of the outcomes of B. However, this by no means implies that the event A causes event B, which would otherwise be a deterministic effect.

Generating Randomness

. Now that we have, even though preliminary, got familiarized with some of the important basic concepts underlying randomness, it is time to proceed to the particularly challenging task of generating randomness. Interestingly, the difficulty if generating random events provides an indication that true, inherent randomness is not easily, if ever, to be found in the real-world.

There are two main approaches to generating randomness: (a) by using physical means; and (b) by using computational means. The former method involves performing random experiments and/or measuring properties from the physical world, such as rolling a dice and observing the outcome, measuring the noise in an electronic circuit, or the decay time of particles. The latter family of methods involves conceiving and applying computational algorithms (typically digital) for generating pseudo-random numbers, i.e. random numbers that are obtained in a deterministic manner but which appear to be random (e.g. satisfy some incomplete statistical test for randomness).

The following discussion will focus on the computational approach. The first respective difficulty to be observed concerns the fact that, almost invariably, digital computer algorithms operate in deterministic manner, which is necessary in order a program yields the same results whenever the same input data is supplied. Interestingly, the new area of quantum computing (e.g. [START_REF] Steane | Quantum computing[END_REF][START_REF] Gruska | Quantum computing[END_REF]) presents a completely different nature, in which the computations are performed in stochastic manner.

The challenges of computing random numbers has motivated a wide range of interesting approaches, which cannot be fully reviewed and explained in the present work. Rather, here we provide just a simple approach, in order to give some idea of a possible principle leading to pseudorandom numbers. It should be observed that this is only a didactic example, chosen for its simplicity, being likely to provide biased results.

The approach to be discussed involves the adoption of the function providing the remainder of the integer division between two integer values. For simplicity's sake, we shall be limited to non-negative integer values. The remainder function can be expressed as:

r(m, n) = m mod n, m, n ∈ {1, 2, . . . , M } (36)
necessarily, we have that 0 ≤ r(m, n) ≤ n -1.

Figure 12 illustrates the remainder function for n = 7 and m = 0, 1, 2, . . . , 50. The linear congruential generator family of methods is based on a recurrence expression yielding successive pseudo-random numbers x i , after having started with a specific seed x 0 (e.g. [START_REF] Eichenauer | A non-linear congruential pseudo random number generator[END_REF][START_REF] Marsaglia | The structure of linear congruential sequences[END_REF][START_REF] Impagliazzo | Pseudorandom generation from one-way functions[END_REF][START_REF] Payne | Coding the lehmer pseudo-random number generator[END_REF][START_REF] Fuller | The period of pseudo-random numbers generated by lehmer's congruential method[END_REF]). An example of this type of expression, known as Lehmer generator, is as follows:

x i+1 = [a x i ] mod m (37)
Observe that this recurrence expression involves two parameters, namely a and m, which are typically called multiplier and modulus, respectively. There are several ways in which these parameters can be chosen, and each choice will have specific implications in the generated sequence of pseudo-random values, starting with the seed x 0 . One possible choice of these parameters, known as MINSTD (from 'minimal standard', e.g. [START_REF] Ronald | Random numbers and computers[END_REF]):

m = 2 31 -1 (38) a = 7 5 (39)
A straightforward implementation of the above method in R is provided as follows for didactic purposes only. Starting with the seed x [START_REF] Da | Modeling: The human approach to science[END_REF], it will generate pseudorandom values in the interval (0, 10).

N <-1000 a <-7^5 m <-2^(31)-1 x <-matrix(0,1,N) x[1] <-2 # seed for (i in seq(2,N)) x[i]<-(a*x[i-1] ) %% m x <-x/m * 10
Pseudo-random values in the set {1, 2, . . . , 10) can then be obtained from x[i] as:

X[i] = floor(x[i]) + 1 (40)
As previously observed, exactly the same sequence of pseudo-random values will be obtained as long as the same seed is employed.

It should be noticed that the above implementation is very basic and does not cater for possible 32-bit multiplication overflow, also yielding limited level of randomness. More effective algorithms/codes should be used for practical applications.

A Case-Example

In order to illustrate the difficulties in generating random sequences and quantifying their randomness, let us consider a simple example involving the application of the approach discussed in Section 5 as a means to verify the level of randomness of binary sequences obtained by the MINSTD method as specifically implemented in a straightforwardly manner in Sectionsec:gener.

More specifically, we will estimate the distribution of the observed sub-sequences for increasing values of L = 1, 2, . . . , 9.

The obtained distributions, presented in terms of respective histograms, are moderately uniform, with bars of similar heights, especially for the smallest considered values of L. However, as L increases, implying successively longer subsequences and less samples of a larger number of generated values (2 L ), a smaller number of occurrences of each of the possible values is consequently obtained. As a consequence, larger oscillations along the histograms are observed for the larger considered values of L. This tendency is further illustrated, in more quantitative terms, in Figure 14. In addition, despite the moderately uniform distributions obtained, a closer observation of Figure 13 indicates that the smallest and largest values for each respective L tend to be overrepresented, possibly corresponding to bias in the random values generation.

As already observed, the implemented MINSTD method provides limited randomness, so that other approaches need to be considered in practice for enhanced randomness. Similarly, it should be observe that the test of randomness presented in this section is not only quite limited to small values of L, but also not enough to provide a more substantive and strict quantification of random number generation. More comprehensive test should be considered in practice.

Randomness and Generalization

Let an experiment generating a sequence of measured numbers as illustrated in Figure 15(a), which can be un- derstood as a source ρ 1 . This source could be derived from a radio signal at a given frequency, the temperature at a given point along time, or the pressure taken along several heights at a given location. All these sequences of measurements can be understood, represented, and treated as real-valued signals, which can be have continuous or discrete domains. These sequences are also often called random signals (e.g. [START_REF] Srinath | Introduction to statistical signal processing with applications[END_REF][START_REF] Gray | An introduction to statistical signal processing[END_REF][START_REF] Kay | Fundamentals of statistical signal processing: estimation theory[END_REF]) or stochastic processes (e.g. [START_REF] Cinlar | Introduction to stochastic processes[END_REF][START_REF] Parzen | Stochastic processes[END_REF][START_REF] Ross | Stochastic processes[END_REF]).

Observe that experimental recording of these signals will necessarily imply the observed values as well as their domain (typically time) to be sampled, usually at equal time steps. The measurement of these signals will be affected by limited resolution as well as possible noise and interferences.

We have already seen that as long as possible sequences of numbers would be needed in case we are interested in quantifying and studying the randomness of these signals. More specifically, it would be interesting to have a large number of sequences of a same length N . There are two main manners in which this could be approached: (i) long sequences are obtained serially from the same source along time; and (ii) each of the sequences would be recorded in parallel from distinct sources with same statistical properties. More information could therefore be obtained from the latter situation.

While the former of the above possibilities would require the observed experiment to be stationary, in the sense of preserving all its statistical properties along time, the latter approach assumes all considered sources not only to be stationary, but also have identical statistical properties when observed at a single source or among all the considered sources, a condition known as ergodicity (e.g. [START_REF] Petersen | Ergodic theory[END_REF][START_REF] Gray | Probability, random processes, and ergodic properties[END_REF]). In a sense, ergodicity is therefore related to the concept of two signal sources being statistically equivalent.

It is interesting to observe that the sources do not need to supply identical sequences, but they need to be characterized by all respective statistical properties being identical (e.g. average, moments, etc.). Therefore, though not yielding identical sequences, the three examples (sources ρ 1 , ρ 2 and ρ 3 ) in Figure 15 could still have the same statistical characteristics when taken more completely.

Randomness and Dynamical Systems

One important aspect of randomness that is not very often observed concerns the fact that it is inexorably related to change. Indeed, by remaining static along time, systems cannot be random, because there are not choices or outcomes taking place. More formally speaking, the state of one such system would always remain the same. So, we have that:

Change (and dynamics) is required for randomness.

The phenomenon of change can be effectively described by using concepts from the area of dynamical systems (e.g. [START_REF] Thelen | Dynamic systems theories[END_REF][START_REF] Kailath | Linear systems[END_REF][START_REF] Hannan | The statistical theory of linear systems[END_REF][START_REF] Da | Visualizing the content of differential equations[END_REF]). Given a deterministic system and its respective state space, its dynamical unfolding along time can be fully represented in terms of a respective oriented trajectory, as illustrated in Figure 16(a).

Figure 16: The dynamics of a deterministic system can be represented as a single trajectory along time t in the respective state space (a). Contrariwise, the dynamics of a non-deterministic system, illustrated in (b), involves multifurcations corresponding to two or more outcomes or choices taking place at a respective time instant. As it will be briefly discussed later in this work, multifurcations can also be a consequence of incomplete state space representations of the observed phenomenon (please refer to Section 13).

State spaces are N -dimensional vector spaces having each axis associated to a respective property or measure-ment of the system dynamics. Therefore, at each time instant, the observed properties of the system are mapped into a respective point in the state space. As the system properties undergo successive changes, a respective trajectory unfolds in the associated state space.

A particular example of intuitive state space would correspond to the positions of a point particle moving along time in a respective Euclidean plane. The properties to be mapped in this case would typically include the position and velocity of the particle at each instant of time. More general state space are defined by the several properties required to represent the dynamics of the system of interest. Though the examples of state space in the present work will refer to two-dimensional planes, more general state spaces can have any dimension.

A complete representation of the system state would be so as to allow the complete characterization of the respective dynamics. This requires that enough properties are observed. In case these properties are not enough to completely specify the system dynamics, characterizing lack of information about the system, the representation will be incomplete.

State space representations of dynamic systems are frequently limited by two main factors: (a) it is difficult or impossible to consider measurements of all properties that can influence the respective dynamics; and (b) the limited resolution and noise/interference characterizing experimental measurements. Observe that both these effects can be understood as lack of knowledge about the system, leading to incomplete representations.

The pre-requisite of change for randomness is particularly important because it emphasizes the question of why and how changes can take place, ultimately leading to the concept of causality (e.g. [START_REF] Pearl | Causality[END_REF][START_REF] Da | Continuous and event-driven causality: A simple model-based approach[END_REF][START_REF] Granger | Some recent development in a concept of causality[END_REF][START_REF] Hiemstra | Testing for linear and nonlinear Granger causality in the stock price-volume relation[END_REF][START_REF] Bressler | Wiener-Granger causality: a well established methodology[END_REF][START_REF] Salmon | Causality and explanation[END_REF][START_REF] Bunge | Causality and modern science[END_REF]). Given its own nature, this question can be addressed from the perspective of physics, more specifically according to its classical and quantum approaches.

When approached from the perspective of classical physics, every changes undergone by mass and energy in the real world are explainable in terms of Newton's laws of motion (e.g. [START_REF] Serway | Physics for scientists and engineers[END_REF][START_REF] José | Classical Dynamics: A Contemporary Approach[END_REF][START_REF] Morin | Introduction to classical mechanics: with problems and solutions[END_REF][START_REF] Thornton | Classical dynamics of particles and systems[END_REF][START_REF] Da | Point motion in flat spaces: An ample starting point[END_REF][START_REF] Da | Single point particle motion: Mass, force, momenta, impulse[END_REF]).

For instance, in the case of the point mass particle example mentioned above, the obtained trajectories will necessarily correspond to continuous and smooth curves as a consequence of the inertia implied by the respective particle mass under action of finite forces. Given any point a t specific time along a respective trajectory (e.g. the initial conditions), the whole of the potential trajectory can be determined before and after that time. Indeed, as a consequence of time symmetry of Newton's laws of motion, there is no intrinsic distinction between past and future.

We thus have that, from the classical mechanics per-spective (e.g. [START_REF] Serway | Physics for scientists and engineers[END_REF][START_REF] José | Classical Dynamics: A Contemporary Approach[END_REF][START_REF] Morin | Introduction to classical mechanics: with problems and solutions[END_REF][START_REF] Thornton | Classical dynamics of particles and systems[END_REF][START_REF] Da | Point motion in flat spaces: An ample starting point[END_REF][START_REF] Da | Single point particle motion: Mass, force, momenta, impulse[END_REF]), the universe would be fully deterministic, and therefore predictable, provided one could have access to one of its complete state configurations along time, from which the respective trajectory could then be determined. In a deterministic universe, changes would be the consequences of a chain of causal influences extending along time (e.g. [START_REF] Da | Continuous and event-driven causality: A simple model-based approach[END_REF]). Even so, prediction would still be necessary, as it would very likely be impossible to know the complete state of the universe at any given time. However, in a fully deterministic universe, the predictions could be approached knowing that no intrinsic randomness would be involved.

The phenomenon of choice in a non-deterministic system can be modeled by allowing a trajectory to branch, or multifurcate, into two or more branches, as illustrated in Figure 16(b). This type of here adopted structure should not be confounded with a related concept, also from dynamical systems, in which the trajectories in a state space bifurcate as a consequence of small changes in an involved parameter Each of these multifurcations represents a possibility of the system to change dynamics which, generally speaking, can be controlled by an external effect, or take place in an intrinsic and truly random manner. In addition, as illustrated Section 13, a multifurcation can simply the consequence of incomplete representations of the properties of the system.

In the case of a deterministic system, the branching could controlled by a respectively associated external effect or signal. As a literal example of branching under control of an external effect, we could mention a production line in which products not satisfying given specifications are respectively directed (branch) for rejection.

In non-deterministic, or random systems, the decision to be taken at multifurcations can be a consequence of truly randomness, which cannot be accounted for by any means. From the perspective of the physical universe, it is the quantum physics (e.g. [START_REF] Dirac | The Principles of Quantum Mechanics[END_REF][START_REF] Serway | Modern Physics[END_REF][START_REF] Mandl | Quantum mechanics[END_REF]) approach where true randomness assumes intrinsic importance. According to this approach, particles involving mass and/or energy (and information) are all intrinsically subjected to randomness, at least at the smallest spatial scales (e.g. particles).

More specifically, as implied by the Schrödinger equation, particles are associated to complex-valued wave functions that are associated to real-valued probability density functions. As a consequence, the state (position and momentum) of a given particle is only statistically characterized along a wave of probability before respective measurement. The action of measurement implies the wave function to collapse, resulting a particle with its specific state. Even so, according to Heisenberg un-certainty principle (e.g. [START_REF] Dirac | The Principles of Quantum Mechanics[END_REF][START_REF] Serway | Modern Physics[END_REF][START_REF] Mandl | Quantum mechanics[END_REF]), it is impossible to measure, with full accuracy and simultaneously, both the position and momentum of a particle.

One approach to having a basic unit of information a quantum universe can be abstracted in terms of the concept of qubit, which has intrinsically random properties. For example, measuring a spin along a direction orthogonal to its orientation will yield uniformly distributed values of 1 and -1. Because spins can be modeled as qubits, and particles such as electrons have spin, the universe could potentially incorporate aspects of randomness that would be more intense at the smallest spatial scales. Although macroscopic structures such as a soccer ball ultimately are quantum objects, the implied random variations are extremely small to be typically observed. However, it should be kept in mind that non-linear effects can amplify smaller scale randomness up to larger spatial scales. This possibility is particularly intense in chaotic systems which, even if deterministic, are characterized by potentially exacerbated sensitivity to tiny perturbations.

In case true randomness at mesoscopic or macroscopic scales could indeed emanate from small scale quantum effects, an interesting question remains: given that even quantum effects necessarily take place in terms of energy and/or mass, the determination of a truly random event would possibly need to be immaterial or happen materially in a domain that is not known and/or observable. As the latter situation does not correspond to true randomness, but only lack of knowledge, the only remaining possibility would be to have immaterial causes of randomness. Otherwise, if a random even takes place under effect of material causation, it would be no longer unaccountable because the material causation could be itself tracked back to previous causal effects herein.

It should be kept in mind that, in addition to not answering the great question about true, unaccountable randomness, the above considerations are quite informal and by no means represent a strict or forma approach, especially when considering the quantum perspective, which is particularly challenging and non-intuitive. Nevertheless, the above discussion of physical randomness should provide additional indications about the complexity and challenges implied as one searches for true randomness.

Randomness and Noise

The experimental observation of the properties of a physical system relies on taking respective measurements, which are often subjected not only to limited resolution, but also to several types of noise and interferences. Given that these effects can therefore influence the characterization of randomness of a system, it is important to consider them with particular attention.

One first important aspect concerns the own nature of noise and interference. In principle, noise corresponds to an unwanted component of a measurement or a signal. The concept of interference is closely related, and will actually be understood as a synonym of noise in the present work.

An example of noise is the 60Hz (or 50Hz, depending on the locality) oscillation which is frequently found in electric signals as a consequence of electromagnetic interference from the power line. Observe that, in the case of this example, the noise actually corresponds to a well defined signal, but which corresponds to an unwanted influence. Noise can also have a more intrinsic random nature, such as thermal noise in electrical and electronic circuits, or decaying particles. In addition, observe that noise can be added in several manners to a measurement, including but not limited to addition and multiplication.

Interestingly, in the case of a composed signal corresponding to the sum of a signal and a noise signal, any of these two components can, in principle, be deterministic or fully random. The resulting signal will be deterministic only in case both the source signal and noise are deterministic, otherwise it will be random. The level of randomness of the resulting composed signal will depend on the relative amplitude of the two added components.

Figure 17 illustrates an interesting situation involving two composite signals x(t) and y(t) derived from the same deterministic source signal z(t) (a sinusoidal signal with frequency 1 and amplitude 1) by a respective scalings by a and b and addition of distinct (and mutually independent) uniformly random noises n x (t) and n y (t) with null mean and respective standard deviations σ 1 and σ 2 . The two obtained signals result intrinsically interrelated, in the sense that increase (or decrease) in one of theme will tend to be accompanied by respective increase (decrease) also in the other signal. This tendency for joint variation is illustrated in the scatterplot in Figure 18 in the case of a = 1, b = 2, and σ 1 = σ 2 = 0.4.

The linear interrelationship between the two obtained signals can be quantified in terms of the respective Pear- Several interesting aspects can be observed from the above example. First, we have that two correlated signals may actually derive from the same common source, which causes both signals. Thus, except for the contribution of the added noise, the two signals would actually convey the same original information. Though a relevant level of joint variation can be observed even in presence of noise, the two signals would have no mutual causation in this case.

Then, we have that the level of noise can strongly influence the estimation of the relationship between signals.

As observed in the above example, increased intensity of noise tends to reduce the Pearson correlation coefficient between the two signals. In addition, we also have that a signal composed of a source signal and noise can be deterministic only if both those signals are deterministic. However, signals originating from arithmetic combinations of two more random signals can be deterministic provided the signals derived from identical randomness mutually cancel one another (as in the division between two scaled versions of the same random source).

Randomness and Completeness

Given that randomness is intrinsically associated to changes, it becomes critically important to be able to observe these changes. Complete observation of all involved changes may not always be possible as a consequence of several factors, including the infinite number of points along a continuous dynamical trajectory (in practice, it is only possible to observe (or measure) a finite number of points in the state space), the adoption of insufficient properties to define the respective state space, as well as the existence of states that are unobservable. We shall preliminary assume that the adopted measurements are enough to fully represent the respectively dynamics. This important principle is illustrated in Figures 20(a . Additional independent and simultaneous trajectories are presented in both cases for generality's sake. The enumerated green stars identify the only points of the state spaces that are assumed in this example to be observable (or measurable) during the analysis of the respective system dynamics.

In Figure 20(a), we have a deterministic trajectory (shown in magenta), as well as an additional trajectory (shown in black) respective to some other simultaneous dynamic system taking place in the same state space. It is assumed that these additional trajectories are not preliminary know. In addition, as in a classic mechanics situation, the developed trajectories are assumed to be smooth (inertia) and continuous (though not necessarily smooth). Four available observation points are marked in the state space by green stars, which are assumed to be generate an event whenever the system visit that respective state. Thus, the timing and sequence of these events can be analyzed in order to estimate possible causal effects and/or randomness along the dynamics.

Starting with an observation of an event at point 1, let us now consider some possible hypothetical unfoldings, such as:

(a) The next event takes place at point 2. This suggests that the ordered pair of points (1, 2) belongs to a same trajectory. The eventual repeated observation of these sequence of events can be taken as providing amassing indication about a deterministic dynamics proceeding along a trajectory that contains the points [START_REF] Da | Modeling: The human approach to science[END_REF][START_REF] Da | Statistical modeling[END_REF]. Such repeated observation of this pair of points could be a consequence of the system being restarted at state 1 or could be taken as the trajectory being periodic;

(b) The next event takes place at point 4, which is not related to the dynamics respective to point 1. In case the sequence [START_REF] Da | Modeling: The human approach to science[END_REF][START_REF] Bertsekas | Introduction to Probability[END_REF] is repeated several times (by chance or as a consequence of another effect such as synchronization of the two dynamics), it could be incorrectly inferred that the points [START_REF] Da | Modeling: The human approach to science[END_REF][START_REF] Bertsekas | Introduction to Probability[END_REF] belong to a same deterministic trajectory, which is not the case;

(c) The event 4 often observed, by chance or synchronization, also to follow the event 1. In case this situation repeats itself several times, it could be incorrectly inferred that the non-deterministic trajectory corresponds to a bifurcation involving the sequences (1, 2) and (1, 4).

In the case of the non-deterministic system shown in Figure 20(b), possible observations after the dynamics is observed to pass through point 1 would include:

(i) As above, the next event takes place at point 2. This suggests that the ordered pair of points (1, 2) belongs to a same trajectory. The eventual repeated observation of these sequence would then suggest that the trajectory contains the points (1, 2);

(ii) The event 1 is followed by events 2 or 3 with equal probabilities. In case this happens several times, a possible non-deterministic trajectory could be inferred in which event 1 is followed by equally possible outcomes 2 and 3. Given that the third branch cannot be observed in the case of the specific situation depicted in Figure 20(b), the trifurcation would be incorrectly taken for a bifurcation.

Some important conclusions can be drawn from the above examples and discussion, including but not limited to:

• Causal interactions between two events along a same deterministic trajectory can be statistically suggested, but full confirmation would require either the knowledge of the system structure and respectively implied trajectory, or an infinite number of observation of respective subsequent occurrences;

• Sequence of causal events along a trajectory will go unnoticed when the number and position of the monitoring points are not adequate (e.g. only one point falling on the trajectory);

• The verification of a sequence of events stemming from a specific set of observation points as being fully random could be suggested to the identification of multifurcations, but this can be misled by existence of additional trajectories in the same state space;

• Some of the trajectories branching out of a multifurcation can be overlooked as a consequence of lack of respective monitoring points.

Though the above discussed situations addressed situations in which not enough monitoring points are available, there is another case related to incomplete observation of the dynamical system of interest. More specifically, this situation relates to adopting a set of measurements to define the respective state space that is not enough to fully characterize the respective dynamics.

As could be expected, this situation tends to be even more critical than the lack of monitoring points, because typically more information is lost when one or more essential measurements are left out. Figure 21 illustrate on such case in which a fully deterministic trajectory in an R 3 space leads to a non-deterministic trajectory in a projected space R 3 when the measurement x 3 is missed, therefore implying in lack of information about the respective dynamics

Observe that both true randomness, as well as randomness stemming from incomplete knowledge, lead to multifurcations in respective state spaces. Only a deterministic system completely represented in a state space will necessarily be guaranteed to be devoid of multifurcations. The considerations developed in this section indicate that, though causality and randomness can be estimated by using experimental means in which events are triggered along a state space, the respectively obtained results are not definitive as it would involve infinite repetition of observations and the limited number and positioning of the monitoring points, while assuming that the adopted set of measurements is enough to fully characterize the observed dynamics.

It should also be kept in mind that every experimental measurement is also limited by unavoidably limited resolution/accuracy, as well as eventual presence of noise and interferences, which may incorporate long distance effects that are impossible to incorporate in the respective analysis, such as the influence of the gravity of a yet unknown distant star on the motion of a chaotic pendulum. In this sense, even in a deterministic system, it would be impossible to have a full description of all the influences on the respective dynamics, which indicates that even if the real world is deterministic, we will always have limited knowledge about its properties and states, therefore characterizing a situation of randomness being a consequence of lack of knowledge about the observed dynamics.

Hybrid Random Systems

A particularly interesting possibility that is not often considered relates to having a dynamical system with hybrid dynamics in the sense of incorporating a mix of deterministic and random modules, or presenting levels of randomness, as illustrated in Figures 22 and23, respectively.

Figure 22 depicts one of the simplest possible configurations, involving one deterministic system interconnected to a random counterpart. In such a system, the deterministic module would control, to a limited extent, the dynamics of the random counterpart. Two important aspects can be observed. First, the control needs to be limited, otherwise the random counterpart would become deterministic. At the same time, no net influence can be exerted by the random module on the deterministic counterpart, otherwise the latter could become random. Another possibility of hybrid random dynamic system is illustrated in Figure 23, in which case most of the system is actually random, but at successive, graded respective levels. Hybrid dynamic systems provide an interesting means to try to accommodate concomitant randomness and determinism. As a possible example, we could mention a system aimed at analyzing a set of real-world measurements, intrinsically incorporating randomness (e.g. noise and interferences), which progressively removes this randomness by using filters and other signal processing methods. Another example would correspond to the real physical world, in which randomness tends to be more intense at its smallest scales, then decreasing along the meso-and macroscopic scales.

Randomness and Decision

Randomness has an important implication of taking decisions, in the sense that in a fully deterministic system there would actually be no decisions to be taken, as the complete unfolding of the dynamics is determined from any previous state configuration. As a consequence, and strictly speaking, decisions are only possible in truly random systems or in deterministic systems about which we do not have complete knowledge.

We shall start by considering the situation depicted in Figure 24. Here, we have a system S of interest from which five measurements, indicated by respective distinct colors, are taken as subsidies for taking a specific decision ∆ with binary values 0 or 1.

Figure 24: A decision ∆ has to be taken respectively to a system S, while considering measurements, indicated in respective distinct colors, of five of its properties. The strength of the measurements are indicated in the areas of the respective rectangles. Observe that the adopted five measurements may be not enough to completely characterize the state of the system, characterizing randomness by lack of knowledge.

A possible manner to take decisions is to weight the respective measurements contributing respectively to binary decisions 0 and 1. In the case of the particular example illustrated in Figure 24, we assume the measurements shown in green and blue to support taking decision 0, while the yellow and orange measurements are understood to contribute to taking decision 1.

Figure 25 depicts an abstraction of reaching the decision ∆ by using a scale where the five measurements are placed according to their respective support for the de-cisions 0 and 1. In the case of the illustrated situation, 0 would be taken as the decision as a consequence of the respective support (measurements in green and blue) having greater 'weight'.

Figure 25: Taking a decision on the system S while considering the adopted five respective measurements can be abstractly implemented by using a scale in which the 'weights' of the measurements supporting decisions 0 and 1 are compared. In the case of this particular example, the decision 0 would have been taken.

In case the measurements were fully accurate and provided a complete description of the state of the system, the above procedure would lead to a deterministic and well-defined outcome. However, neither of these two conditions are typically (if ever) met in practice. The implications of these two limitations are briefly discussed in the remainder of this section.

First, we assume that the five measurements do provide a complete representation of the state of the system. However, these measurements are taken with limited resolution and/or in presence of noise, leading to a respective error. Figure 26 illustrates the effect of a relatively small level of noise induced by limited measurement accuracy/resolution in the weighting procedure. Because the noise is is assumed to be small enough, the respective displacements of the scale in this example preserve the decision of taking value 1 as outcome. However, in the case lower resolution and/or higher levels of noise, the maximum error implied in the weight comparison can lead to changing the decision between 0 and 1, as illustrated in Figure 27. The above considerations indicate that taking a decision in a system completely described by the adopted measurements, but with limited accuracy or high levels of noise/interference, can lead to incorrect outcomes. However, provided these unwanted effects can be kept at adequate levels, the outcome can still be deterministic.

Though in the above examples we considered limited and noise as sources of randomness, the respective discussion can be immediately extended to the case in which the measurement errors are consequence of true, intrinsic randomness manifested at specific levels. For instance, if the measurements are influenced by true randomness at small enough intensity, the decision outcome would not be affected. This line of reasoning suggests that deterministic decisions can be taken even in presence of true randomness.

Concluding Remarks

Though not often realized, the concept and phenomenon of randomness constitutes one of (and possibly the most) important and challenging aspects shared by science and human experience. This importance stems mainly from the fact that, in both those areas, there often arise the need to estimate the effects of decisions or outcomes, so that the chances of achieving aimed results can be enhanced. Indeed, the impact of accurately predicting outcomes, as well as the consequence of these outcomes, can hardly be overemphasized regarding respective impact on several of the human dimensions, including but not limited to environmental, social, economic, quality of life, among many other aspects.

Yet, despite all the efforts that have been directed at better understanding randomness, motivated by the above summarized aspects, we still cannot be certain whether the universe is (or not) truly random, or if the limitations of our ability to making predictions are a sole consequence of incomplete information about the state space of an otherwise hypothetically fully deterministic universe. The challenges implied by randomness are further emphasized by the interrelationship among this concept and other several concepts from an intricate multidisciplinary perspective.

The present work has been mainly aimed at presenting, briefly revising, and discussing what constitutes randomness, and whether this property can be truly encountered in the physical world. In order to keep the presentation relatively simple and accessible, only some of the main concepts and methods have been addressed in a relatively informal manner. As such, the present work does not correspond to a formal and complete approach to the covered concepts methods, and therefore should not be understood as a more orthodox basic text.

Given the inter and multidisciplinary nature of randomness and its relationships, our approach unfolded while considering several subsequent, interrelated concepts and issues, including types of randomness, temporality, causation, probability, identification and quantification of randomness, the hierarchical nature between random effects, the challenging issue of generating random values, the generalization of randomness in time and space, a dynamical system approach approach to randomness, noise, completeness of observation, and the task of taking decisions. It is hoped that a more complete picture of randomness and its properties can be achieved when approached from these several perspectives.

In addition to providing some rudiments about the covered topics and their relationship with randomness, the presented material also discusses several respective difficulties that make randomness a concept that is as important as it is challenging. In particular, regarding the core issue about the possible existence of true, unaccountable randomness no definitive answer has been provided, as this is probably an undecidable question. However, the two considered types of randomness, namely involving incomplete knowledge and true randomness, as well as considering hybrid (modular or graded) random dynamic systems, could provide additional subsidies from which to approach the intriguing concept of being random.

Another interesting point is that, if it were possible to know that the randomness underlying a specific case stems from lack of knowledge, and not as a consequence of true randomness, would motivate one to invest further efforts in respective study and research. As it may be impossible to decide on this issue, it remains an interesting prospect to perform as comprehensive as possible experiments and measurements, taking into account as many factors as possible, with ever increasing accuracy.

It is hoped that the presented discussion, illustrations

and numeric examples may motivate the reader to probe further (e.g. starting from the several provided bibliographical references) into the fascinating subject of randomness and its myriad relationships and implications.

Figure 1 :

 1 Figure1: A receiver (B) observes a sequence of digits unknowingly corresponding to an intermediate portion of the the number π and has to decide whether the sequence is random or deterministic. The problem of defining and recognizing randomness is more subtle and complex than it may initially appear, involving the recognition, or not, of the origin of the digits as being part of the number π by the receiver in this particular case.

Figure 3 :

 3 Figure 3: A possible (and naïve) quantum mechanics approach to the situation depicted in Fig. 2(a) consists of considering several uniformly distributed wave functions representing all the involved particles of the object.

  tion.

Figure 4 :

 4 Figure 4: The basic timing diagram typically underlying an eventdriven causation[START_REF] Da | Continuous and event-driven causality: A simple model-based approach[END_REF]. The cause C needs to take place for a minimum period of time ∆, which is then followed by a respective delay δ, and the effect E, along which the causation continues but is no longer under control of the agent that initiated it. It is understood that, once the causation period is completed, the effect can no longer be avoided other than by some eventual external influence (not included in the diagram). In continuous causation, we have that ∆ → 0 and δ → 0, so that the effect will always immediately follow the respective immediate cause.

Figure 5

 5 Figure 5 illustrates the above density for [a, b] = [2, 5] and c = 1/3. Henceforth, we will immediately extend the uniform probability density to intervals (a, b], [a, b) and (a, b), while assuming p(x) to be real-valued and continuous and

Figure 5 :

 5 Figure 5: Illustration of the uniform probability density function p(x) for [a, b] = [2, 5] and c = 1/3. Observe that p(x) ≥ 0, ∀x and ´∞ -∞ p(x)dx = 1.

Figure 6 :

 6 Figure 6: Example of the discrete uniform probability density function a discrete random variable taking values 0 and 1 with identical probabilities 0.5. Observe that, as in Fig. 5, p(x) ≥ 0, ∀x and ´∞ -∞ p(x)dx = 1.

Figure 7 :

 7 Figure 7: Illustration of Buffon's experiment to estimate the constant π.

  1 log(0.1) + 0.70 log(0.35) + 0.2 log(0.2)] = = 1.287... η X = e 1.287 = 3.622...

Figure 8 :

 8 Figure8: The representation of the outcomes, and respective probabilities, involved in the considered two-events experiment. The probabilities of outcomes and subsequence levels may (or not) be conditioned by the outcomes at the previous levels. When a subsequent event B does not depend on the previous event A, they are said to be independent, and we have P (B|A) = P (B), also implying P (A ∩ B) = P (A, B) = P (A) P (B).

Figure 9

 9 Figure 9 depicts an example of a situation in which the two events A and B are independent, also indicating the involved probabilities in this particular case.

Figure 9 :

 9 Figure 9: Illustration of a particular situation in which the two involved events are independent, so that P (A ∩ B) = P (A, B) = P (A) P (B).

Figure 10 :

 10 Figure10:A specific situation in which not only the events A and B are independent, but also involve identical probabilities. As a consequence each of the four possible joint outcomes will have precisely the same probability 0.25 (equiprobable).

Figure 11 :

 11 Figure 11: An example of a situation where the outcomes of even A influence (or condition) the subsequent event B. In this case, the joint probabilities are calculated as P (A, B) = P (B|A) P (A).

Figure 12 :

 12 Figure 12: Illustration of the results of the function that provides the remainder of the integer division between a non-negative integer value m = 0, 1, 2, . . . by the number n = 7. Observe the periodic nature of this function, with period equal to n = 7.

Figure 13 :

 13 Figure 13: Histograms illustrating the distribution of the values obtained for L = 1 to 9 by using the MINSTD method in Section 8 considering N = 30000. The red line in each plot indicates the respectively expected (average) number of counts in each of the histogram bins. The dispersion of counts tends to increase with L as a consequence of respectively reduced sampling of the subsequences. Moderately uniform distributions are obtained, except for the first and last bins tending to presenting slightly higher counts for several of the values of L considered in this figure, which suggests respective bias in the random values generation.

Figure 14 :

 14 Figure 14: The relative error (in %) of the exponential entropy of the number of possible values for L = 1, 2, . . . , 9 respectively to the considered number N of drawn samples (shown in the legend).For each value of N , the error tends to increase monotonically with L, which is caused by the smaller number of samples obtained for each respective possible value. In addition, the overall errors tend to decrease as N increases.

Figure 15 :

 15 Figure15: Three sources (ρ 1 , ρ 2 , and ρ 3 ) of potentially random sequences that are not identical but may (or not) be characterized by presenting full sets of identical statistical properties.

Figure 17 :

 17 Figure 17: Flow diagram illustrating the obtention of two composite signals x(t) and y(t) obtained from a noiseless sinusoidal source signal z(t) by respective scalings by a and b and addition of independent uniform noise nx(t) and ny(t) with null mean and respective standard deviations σ 1 and σ 2 .

Figure 18 :

 18 Figure 18: Scatterplot of the composite signals x(t) and y(t) for a = 1, b = 2, and σ 1 = σ 2 = 0.4. Though the two signals have a tendency for positive joint variation, the added noise makes this relationship less defined. In the case of this particular scatterplot, the Pearson correlation coefficient is P ≈ 0.938.

  Figure 19(b) illustrates the Pearson correlation coefficients obtained from the above composite signals in terms of several values of σ = σ 1 = σ 2 . The presented values were averaged over 1000 simulations.

Figure 19 :

 19 Figure 19: The average Pearson correlation coefficients obtained for the signals in the above example in terms of several standard deviation values.

  ) and (b), involving the deterministic and random trajectories previously shown in Figure16. Equal probabilities are assumed for the three possible outcomes in the latter situation.

Figure 20 :

 20 Figure20: Illustration of the concept of observability of points in a state space respectively to the deterministic and random trajectories respectively shown in magenta in (a) and (b). Additional independent and simultaneous trajectories are presented in both cases for generality's sake. The enumerated green stars identify the only points of the state spaces that are assumed in this example to be observable (or measurable) during the analysis of the respective system dynamics.

Figure 21 :

 21 Figure21: Illustration of a situation in which a deterministic trajectory in a higher dimensional state space (R 3 ) into a smaller dimension (R 2 ) leads to a seemingly non-deterministic trajectory including a bifurcation point P as a consequence of not taking property x 3 into account, which would otherwise ensure that none of the points of the original trajectory overlap.

Figure 22 :

 22 Figure22: A hybrid dynamic system S subdivided into a deterministic component D and a non-deterministic (or random) component N . Some aspects of the non-deterministic part can be influenced by the deterministic component, but the incorporation of random dynamics into an otherwise deterministic system would possible make that system random (unless the random effects mutually cancel one another).

Figure 23 :

 23 Figure 23: Example of a graded random dynamic system, where randomness gradually extends through the whole system.

Figure 26 :

 26 Figure 26: In the situation illustrated in this figure, the presence of a small level of accuracy/resolution in the considered measurements induce a maximum displacement of the scale arms that is not enough to change the outcome of taking 0 as decision.

Figure 27 :

 27 Figure 27: In case the previous measurements are taken with less accuracy and/or in presence of higher levels of noise, the respectively induced displacement of the scale arms can be enough to change the decision.

Acknowledgments Luciano da F. Costa thanks CNPq (grant no. 307085/2018-0) and FAPESP (grant 15/22308-2).

Observations

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