CHAITIN RANDOMNESS AND MATHEMATICAL PROOF PDF

Watson Research Center in New York and remains an emeritus researcher. He has written more than 10 books that have been translated to about 15 languages. He is today interested in questions of metabiology and information-theoretic formalizations of the theory of evolution. Other scholarly contributions[ edit ] Chaitin also writes about philosophy , especially metaphysics and philosophy of mathematics particularly about epistemological matters in mathematics. In recent writings, he defends a position known as digital philosophy.

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Statistics In the physical sciences In the 19th century, scientists used the idea of random motions of molecules in the development of statistical mechanics to explain phenomena in thermodynamics and the properties of gases. According to several standard interpretations of quantum mechanics , microscopic phenomena are objectively random.

For example, if a single unstable atom is placed in a controlled environment, it cannot be predicted how long it will take for the atom to decay—only the probability of decay in a given time. Hidden variable theories reject the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are at work behind the scenes, determining the outcome in each case.

In biology The modern evolutionary synthesis ascribes the observed diversity of life to random genetic mutations followed by natural selection. The latter retains some random mutations in the gene pool due to the systematically improved chance for survival and reproduction that those mutated genes confer on individuals who possess them. Several authors also claim that evolution and sometimes development requires a specific form of randomness, namely the introduction of qualitatively new behaviors.

Instead of the choice of one possibility among several pre-given ones, this randomness corresponds to the formation of new possibilities. For instance, insects in flight tend to move about with random changes in direction, making it difficult for pursuing predators to predict their trajectories.

In mathematics The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling , but later in connection with 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 Kolmogorov randomness , which means that random strings are those that cannot be compressed. That is, an infinite sequence is random if and only if it withstands all recursively enumerable null sets. The other notions of random sequences include, among others, recursive randomness and Schnorr randomness, which are based on recursively computable martingales.

It was shown by Yongge Wang that these randomness notions are generally different. The decimal digits of pi constitute an infinite sequence and "never repeat in a cyclical fashion. Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times.

Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases. This allows surveys of completely random groups of people to provide realistic data that is reflective of the population. Common methods of doing this include drawing names out of a hat, or using a random digit chart a large table of random digits.

In information science In information science, irrelevant or meaningless data is considered noise. Noise consists of numerous transient disturbances, with a statistically randomized time distribution. In communication theory , randomness in a signal is called "noise", and is opposed to that component of its variation that is causally attributable to the source, the signal.

More generally, asset prices are influenced by a variety of unpredictable events in the general economic environment. In politics Random selection can be an official method to resolve tied elections in some jurisdictions. Randomness and religion Randomness can be seen as conflicting with the deterministic ideas of some religions, such as those where the universe is created by an omniscient deity who is aware of all past and future events.

If the universe is regarded to have a purpose, then randomness can be seen as impossible. This is one of the rationales for religious opposition to evolution , which states that non-random selection is applied to the results of random genetic variation. Hindu and Buddhist philosophies state that any event is the result of previous events, as is reflected in the concept of karma. As such, this conception is at odd with the idea of randomness, and any reconciliation between both of them would require an explanation.

Cleromancy uses the casting of bones or dice to reveal what is seen as the will of the gods. Applications Main article: Applications of randomness In most of its mathematical, political, social and religious uses, randomness is used for its innate "fairness" and lack of bias. Politics: Athenian democracy was based on the concept of isonomia equality of political rights , and used complex allotment machines to ensure that the positions on the ruling committees that ran Athens were fairly allocated.

Allotment is now restricted to selecting jurors in Anglo-Saxon legal systems, and in situations where "fairness" is approximated by randomization , such as selecting jurors and military draft lotteries.

Games: Random numbers were first investigated in the context of gambling , and many randomizing devices, such as dice , shuffling playing cards , and roulette wheels, were first developed for use in gambling. The ability to produce random numbers fairly is vital to electronic gambling, and, as such, the methods used to create them are usually regulated by government Gaming Control Boards. Random drawings are also used to determine lottery winners. In fact, randomness has been used for games of chance throughout history, and to select out individuals for an unwanted task in a fair way see drawing straws.

Sports: Some sports, including American football , use coin tosses to randomly select starting conditions for games or seed tied teams for postseason play. The National Basketball Association uses a weighted lottery to order teams in its draft. Mathematics: Random numbers are also employed where their use is mathematically important, such as sampling for opinion polls and for statistical sampling in quality control systems. Computational solutions for some types of problems use random numbers extensively, such as in the Monte Carlo method and in genetic algorithms.

Medicine: Random allocation of a clinical intervention is used to reduce bias in controlled trials e. Religion: Although not intended to be random, various forms of divination such as cleromancy see what appears to be a random event as a means for a divine being to communicate their will see also Free will and Determinism for more.

Generation The ball in a roulette can be used as a source of apparent randomness, because its behavior is very sensitive to the initial conditions. It is generally accepted that there exist three mechanisms responsible for apparently random behavior in systems: Randomness coming from the environment for example, Brownian motion , but also hardware random number generators. Randomness coming from the initial conditions. This aspect is studied by chaos theory , and is observed in systems whose behavior is very sensitive to small variations in initial conditions such as pachinko machines and dice.

Randomness intrinsically generated by the system. This is also called pseudorandomness , and is the kind used in pseudo-random number generators. There are many algorithms based on arithmetics or cellular automaton for generating pseudorandom numbers. The behavior of the system can be determined by knowing the seed state and the algorithm used. These methods are often quicker than getting "true" randomness from the environment. The many applications of randomness have led to many different methods for generating random data.

These methods may vary as to how unpredictable or statistically random they are, and how quickly they can generate random numbers.

Before the advent of computational random number generators , generating large amounts of sufficiently random numbers which is important in statistics required a lot of work. Results would sometimes be collected and distributed as random number tables. Measures and tests Main article: Randomness tests There are many practical measures of randomness for a binary sequence. These include measures based on frequency, discrete transforms , complexity , or a mixture of these, such as the tests by Kak, Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman.

In this case, once a jack is removed from the deck, the next draw is less likely to be a jack and more likely to be some other card. However, if the jack is returned to the deck, and the deck is thoroughly reshuffled, a jack is as likely to be drawn as any other card.

The same applies in any other process where objects are selected independently, and none are removed after each event, such as the roll of a die, a coin toss, or most lottery number selection schemes. Truly random processes such as these do not have memory, which makes it impossible for past outcomes to affect future outcomes. In fact, there is no finite number of trials that can guarantee a success. A number may be assumed to be blessed because it has occurred more often than others in the past, and so it is thought likely to come up more often in the future.

This logic is valid only if the randomisation is biased, for example with a loaded die. If the die is fair, then previous rolls can give no indication of future events. In nature, events rarely occur with perfectly equal frequency, so observing outcomes to determine which events are more probable makes sense.

However, it is fallacious to apply this logic to systems designed to make all outcomes equally likely, such as shuffled cards, dice, and roulette wheels. Odds are never dynamic In the beginning of a scenario, one might calculate the probability of a certain event. However, as soon as one gains more information about the scenario, one may need to re-calculate the probability accordingly. In the Monty Hall problem , when the host reveals one door that contains a goat, this provides new information that needs to be factored into the calculation of probabilities.

For example, when being told that a woman has two children, one might be interested in knowing if either of them is a girl, and if yes, what is probability that the other child is also a girl.

To be sure, the probability space does illustrate four ways of having these two children: boy-boy, girl-boy, boy-girl, and girl-girl. But once it is known that at least one of the children is female, this rules out the boy-boy scenario, leaving only three ways of have the two children: boy-girl, girl-boy, girl-girl. In general, by using a probability space, one is less likely to miss out on possible scenarios, or to neglect the importance of new information.

This technique can be used to provide insights in other situations such as the Monty Hall problem , a game show scenario in which a car is hidden behind one of three doors, and two goats are hidden as booby prizes behind the others.

Once the contestant has chosen a door, the host opens one of the remaining doors to reveal a goat, eliminating that door as an option. With only two doors left one with the car, the other with another goat , the player must decide to either keep their decision, or to switch and select the other door. Intuitively, one might think the player is choosing between two doors with equal probability, and that the opportunity to choose another door makes no difference.

However, an analysis of the probability spaces would reveal that the contestant has received new information, and that changing to the other door would increase their chances of winning.

EVOLINE PORT PDF

The Mathematical Foundations of Randomness

However, we can also imagine a relatively simple program for U, call it P, which checks through all the proofs in FAS and when it finds a proof that some positive integer, I, requires a program of N bits to specify it, prints out I and halts. Which is to say that a lower bound on program-size complexity cannot be derived in any FAS which itself has a complexity cFAS much lower than that bound. To prove that a particular object has program-size complexity of N, you need more or less "N bits of axioms", as Chaitin is fond of saying, meaning that the size of the shortest proof-checker for the formal system used must be not less than N. Where Turing and Godel say "undecidable! Kolmogorov had defined an infinite binary sequence to be random where all their "prefixes" are incompressible the "prefixes" are just the finite sequences you find at the beginning of any infinite sequence - their initial n elements.

CEYLON GHAT TRACER PDF

random number

Statistics In the physical sciences In the 19th century, scientists used the idea of random motions of molecules in the development of statistical mechanics to explain phenomena in thermodynamics and the properties of gases. According to several standard interpretations of quantum mechanics , microscopic phenomena are objectively random. For example, if a single unstable atom is placed in a controlled environment, it cannot be predicted how long it will take for the atom to decay—only the probability of decay in a given time. Hidden variable theories reject the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are at work behind the scenes, determining the outcome in each case.

BRYAN MAGEE CONFESSIONS OF A PHILOSOPHER PDF

Gregory Chaitin

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