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Randomness

From Academic Kids

For other meanings, see Random (disambiguation).

In ordinary language, the word random is used to express apparent lack of purpose or cause. This suggests that no matter what the cause of something, its nature is not only unknown but the consequences of its operation are also unknown.

In statistics, the term randomness means some event happens with some probability distribution. This generally implies a lack of bias or correlation unless otherwise specified.

In computing, the term randomness generally refers to generating or using a set of truly random (unbiased) sequence of random numbers within some set range.

In physics the term random means that an event either appears random, or truly is random, such as the ideas behind quantum physics and information theory.

Contents

Randomness versus unpredictability

Randomness should not be confused with unpredictability which is a related idea in ordinary usage. Some mathematically deterministic systems can be unpredictable in practice due to sensitive dependence on initial conditions (See chaos theory). Many random phenomena exhibit organized features at some levels. For example, the increase of the world human population is quite predictable on average, but individual births and deaths cannot be accurately predicted with any precision in most cases. This small-scale randomness is found in almost all real-world systems. Ohm's law and the kinetic theory of gases are statistically reliable descriptions of the 'sum' (ie, the net result or integration) of vast numbers of individual micro event, each of which are random and none of which are individually predictable within practical limits. Theoretically the micro events of gases for example could be predicted if the exact position, velocity, atomic composition, angular momentum and so on was known. All we directly perceive is circuit noise and some bulk gas behaviors.

In some applications, both unbiased randomness (as tested statistically) and unpredictability are required, as for instance in many uses of random numbers in cryptography. In other applications, such as many modeling or simulation applications statistical randomness is essential, but unpredictability is not only unnecessary, but may cause problems as for instance whilst repeating modeling runs during debugging or acceptance tests.

Sensibly dealing with randomness is a hard problem in modern science, mathematics, psychology and philosophy. Merely defining it adequately for the purposes of this or that discipline has been quite difficult. Distinguishing between apparent randomness and actual randomness has been no easier, and additionally assuring unpredictability, especially against a well motivated party (in cryptographic parlance, the "adversary"), has been harder still.

Some philosophers have argued that there is no randomness in the universe, only unpredictability. Others find the distinction meaningless. (See determinism).

Randomness in philosophy

Note that the bias that "everything has a purpose or cause" is actually implicit in the expression "apparent lack of purpose or cause". Humans are always looking for patterns in their experience, and the most basic pattern seems to be cause/effect. This appears to be deeply embedded in the human brain, and perhaps in other animals as well. For example, dogs and cats often have been reported to have apparently made a cause and effect connection that strikes us as amusing or peculiar. (See classical conditioning.) For instance there is a report of a dog who, after a visit to a vet whose clinic had tile floors of a particular kind, refused thereafter to go near such a tiled floor, whether or not it was at a vet's.

It is because of this bias that the absence of a cause seems problematic. See causality.

To solve this 'problem', random events are sometimes said to be caused by chance. Rather than solving the problem of randomness, this opens the gaping hole of defining chance. It is hard to avoid circularity by defining chance in terms of randomness.

Randomness in natural science

Traditionally, randomness takes on an operational meaning in natural science: something is apparently random if its cause cannot be determined or controlled. When an experiment is performed and all the control variables are fixed, the remaining variation is ascribed to uncontrolled (ie, 'random') influences. The assumption, again, is that if it were somehow possible to perfectly control all influences, the result of the experiment would be always the same. Therefore, for most of the history of science, randomness has been interpreted in one way or another as ignorance on the part of the observer.

With the advent of quantum mechanics, however, it appears that the world might be irreducibly random. According to the standard interpretations of the theory, it is possible (and in fact very, very easy) to set up an experiment with total control of all relevant parameters, which will still have a perfectly random outcome. Weak resistance to this idea takes the form of hidden variable theories in which the outcome of the experiment is determined by certain unobservable characteristics (hence the name "hidden variables").

Many physical processes resulting from quantum-mechanical effects are, therefore, believed to be irreducibly random. The best-known example is the timing of radioactive decay events in radioactive substances.

Deviations from randomness are often regarded by parapsychologists as evidence for the theories of parapsychology.

Randomness in mathematics

The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling but soon in connection with situations of interest in physics. Statistics is used to infer the underlying probability distribution of a collection of empirical observations. For the purposes of simulation it is necessary to have a large supply of random numbers, or means to generate them on demand.

Algorithmic information theory studies, among other topics, what constitutes a random sequence. The central idea is that a string of bits is random if and only if it is shorter than any computer program that can produce that string (Chaitin-Kolmogorov randomness). Pioneers of this field include Andrey Kolmogorov, Ray Solomonoff, Gregory Chaitin, Anders Martin-Lf, and others.

Randomness in practical communications and cryptography

Successful communication in the real world depends, at the limit, on understanding and successfully minimizing the deleterious effects of assorted interference sources, many of which are apparently random. Such noise imposes performance limits on any communications channel and it was the study of those limits which led Shannon to develop information theory, make fundamental contributions to communication theory, and establish a theoretical grounding for cryptography.

Access to a source of high-quality randomness is absolutely critical in many applications of cryptography. For example, even a subtly non-random key choice may result in a complete break into a communications channel that was believed to have been secure and was relied upon to be so. See the Enigma machine and one-time pad articles for examples of the consequences of such mis-estimates. Keys used for the Enigma were non-random in many cases which made it possible for Allied cryptanalysts to break into the traffic with substantial consequences for the Nazi war effort. A similar thing happened in the Pacific Theater of WWII with the Japanese PURPLE machine; its key selection was also insufficiently random. The key material used in the theoretically unbreakable one-time pad must be random and unpredictable lest the encryption technique become trivially breakable. Even a slight predictability of the key material used removes the one-time pad from the unbreakable category. The Colossus computer, the world's first programmable digital electronic computer, was developed to attack a mechanical (and subtly non-random) implementation of the one-time pad.

There are techniques for combining biased random data to produce higher quality randomness. The randomness inherent in some computer input/output components allows enough entropy to be extracted for most cryptographic purposes. A greater concern is ensuring that best practices are followed in actual systems and that random number generation is not subverted by attackers. See random number generator attack.

Randomness in gaming

Randomness is central to games of chance and vital to the gambling industry.

Random draws are often used to make a decision where no rational or fair basis exists for making a deterministic decision.

Randomness in music

Randomness in music is deemed postmodern, including John Cage's chance derived Music of Changes, Iannis Xenakis' stochastic music, aleatoric music, indeterminate music, or generative music.

Randomness in Art

The clearest example of randomness occurs in the problem of arranging items in an art exhibit. Usually this is avoided by using a theme. As John Cage pointed out, "While there are many ways that sounds might be produced [ie, in terms of patterns], few are attempted". Similarly, the arrangment of art in exhibits is often deliberately non-random. One case of this was Hitler's attempt to portray modern art in the worst possible light by arranging works in worst possible manner. A case can be made for trying to make art in the worst possible way; ie, either as anti-art, Pop art, Type 2 (ie, social commentary), or as actually random art.

Dadaism as well as many other movements in art and letters have attempted to deal with randomness in various forms. Often people mistake order for randomness based on lack of information; eg, Jackson Pollock's drip paintings, Helen Frankenthaler's abstractions (eg, "For E.M."). Thus, in theory of art, all art is random in that it's "just paint and canvas" (the explanation of Frank Stella's work.

Randomness in Literature

Similarly, the "un-expected" ending is part of the nature of interesting literature. A chief example of this is Dennis Diderot's novel ""Jacques le fataliste"" (Jacques the Fatalist; also sometime published as "Jacques the Servant and his Master"). At one point in the novel, Diderot speaks directly to the reader: Now I, as the author of this novel might have them set upon by thieves, or I might have the rest by a tree until the rain stops, but in fact they kept on walking and then near night-fall they could see the light of an Inn in the distance. (not an exact quote). Diderot's point of making the point that the novel (a new invention then) was in fact random. See also Eugenio Montale, Theatre of the Absurd.

Randomness in humor

Randomness can also be found in humor, as when an event occurs with little or no connection to the plot, context, or sense of reality in the scene. This form of humor has strong historical connections with the Theatre of the Absurd, a modernist drama movement. Shows such as Monty Python's Flying Circus continued this tradition, and it is still popular today in programs like Foxs Family Guy, and Comedy Centrals South Park.

Randomness in finance

The Random walk hypothesis considers that asset prices in an organized market evolve at random.

Quotations

  • "Random numbers should not be generated with a method chosen at random." —Donald E. Knuth
  • "The generation of random numbers is too important to be left to chance." —Robert R. Coveyou, Oak Ridge National Laboratory, 1969
  • "That which is static and repetitive is boring. That which is dynamic and random is confusing. In between lies art." —John A. Locke
  • "Perhaps our thinking exemplifies a selective system. First lots of random scattered ideas compete for survival. Then comes the selection for what works best —one idea dominates, and this is followed by its amplification. Perhaps the moral [...] is that you never learn anything unless you are willing to take a risk and tolerate a little randomness in your life." —Heinz Pagels, The dreams of reason, 1988

Books

See also

External links


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