Definition Of Randomness Research Articles

A Definition of Randomness and a Model of a Nonlocal Real World

A Definition of Randomness and a Model of a Nonlocal Real World

Definitions of randomness

Randomness is a fundamental but elusive concept in mathematics and physics. Even for the elementary case of a random binary sequence, a generally accepted and operational definition is lacking. However, when ensembles are used for the foundation of probability theory, randomness has to be identified with uncorrelatedness, a neglected notion that yet solves many puzzles surrounding randomness. In any single sequence, a definite amount of correlation is unavoidable. At best, the remaining correlations are of high order or cover a large distance. This is the case for properly defined well-tempered pseudorandom sequences. Reliable algorithms for the generation of pseudorandom bits at rates of 1 GHz result.

Random-Number Generation

In large-scale Monte Carlo simulations, reliable random numbers will soon be needed at bit rates of 1 GHz or more. Therefore, existing recipes for the generation of random numbers have to be improved. This is not easy, due to the many unrelated and laborious statistical tests needed to compensate for the lack of an accepted and operational definition of randomness. When however the notion of randomness as a complete absence of all correlations is made precise, a practical approach results.

Threshold phenomena in random structures

The physical theory of phase transition explains sudden changes of phase in materials that undergo gradual changes of some parameter like temperature. There are analogs of phase transition in the theory of random graphs, initiated by Erdös and Rényi. This paper gives a nontechnical but precise account, without proofs, of some of the beautiful discoveries of Erdös and Rényl about threshold phenomena in graphs, describes an application of their methods to interval graphs, and gives some examples of threshold phenomena under other definitions of randomness and in combinatorial structures other than graphs. The paper offers some speculations on possible applications of random combinatorial structures to telecommunications, neurobiology, and the origin of life. A rich man commissioned three experts, a veterinarian, an engineer, and a theoretical physicist, to find out what made the best race horses. After a few years they reported their results. The vet concluded from genetic studies that brown horses were the fastest. The engineer found that thin legs were optimal for racing. The theoretical physicist asked for more time to study the question because the case of the spherical horse was proving extremely interesting. Aharon Katchalsky No one is exempt from talking nonsense; the only misfortune is to do it solemnly. Montaigne

Randomness conservation inequalities; information and independence in mathematical theories

The article further develops Kolmogorov's algorithmic complexity theory. The definition of randomness is modified to satisfy strong invariance properties (conservation inequalities). This allows definitions of concepts such as mutual information in individual infinite sequences. Applications to several areas, like probability theory, theory of algorithms, intuitionistic logic are considered. These theories are simplified substantially with the postulate that the objects they consider are independent of (have small mutual information with) any sequence specified by a mathematical property.

The Reference Class

The system presented by the author in The Logical Foundations of Statistical Inference (Kyburg 1974) suffered from certain technical difficulties, and from a major practical difficulty; it was hard to be sure, in discussing examples and applications, when you had got hold of the right reference class. The present paper, concerned mainly with the characterization of randomness, resolves the technical difficulties and provides a well structured framework for the choice of a reference class. The definition of randomness that leads to this framework is simplified and clarified in a number of respects. It resolves certain puzzles raised by S. Spielman and W. Harper in their contributions to Profiles: Henry E. Kyburg, Jr. and Isaac Levi (R. Bogdan (ed.) 1982).

Possible Strengthening of the Interpretative Rules of Quantum Mechanics

The usual interpretative rules of quantum mechanics are presented and are shown to be too weak. The reason is that they do not include the intuitive requirement of randomness of the outcome sequence obtained from an infinite repetition of measuring an observable on a state. A strengthening of the rules is proposed which includes, in essence, a precise definition of randomness. The resultant rule is seen to be intuitively more satisfying than the usual rules and to include the expectation value rule and to include essentially all of the spectrum rule of the usual rules. It also suggests that the relationship between the foundations of mathematics and quantum mechanics may be quite deep and complex.

A tutorial on random number generation

A "Random Number Generator" is a program that produces a number each time it is invoked. This number is, for a broad range of uses, "random." More exactly, it is unpredictable and has the same chance of being selected as any other number among those the generator can produce. A random generator is useful in the same way as flipping a coin can be useful: randomness is independent of human motives and desires, it is impartial and fair. Its applications include generating test data, Monte Carlo procedures, testing grammars, and making choices in programs for creating music, poetry, and other art. The definition of randomness is a philosophical and mathematical question, beyond the scope of this discussion. It would not be out of place, however, to explain that there is no such thing as a "random number." While a die may I and five-spot uppermost we would not say that therefore the number 5 is random. Put in a relatively precise way, we would say that a sequence of numbers is random if there is no possible program that could be written to predict what the next member of the sequence was going to be given all previous members of the sequence. We would also require that the sequence had certain statistical properties, for example, that each possible output was equally likely. A random generator is a device (such as an honest roulette wheel) that produces such a sequence. It follows that a computer program cannot generate a random sequence (if it could, then a faster computer with the same program could 'predict' the outputted numbers). This is generally true. The usual "random number generator" or "random function" provided at most installations is more properly a pseudorandom generator. It is, in fact, a program that produces a string of numbers that for most practical purposes appears random. It has good statistical properties and while it will eventually repeat itself and is certainly predictable, it is unlikely that you (or your program) would ever detect these "faults." There is a large and still growing literature on methods of programming pseudorandom functions, and one can probably be written in relatively short order (see for example, the Algorithms department of the Communications of the Association for Computing Machinery ). The random generator at your installation may have any of a number of forms. Usually it is a function or procedure whose name looks like RND, RAN, RANDOM, RANF, IRANF, etc. In APL it is simply a