The infinite monkey theorem states that monkeys who push keys randomly on a typewriter keyboard for an infinite time will almost certainly be typing certain texts, such as William Shakespeare's complete works. In fact, monkeys will almost certainly type any finite text that may be infinite. However, it is likely that the monkey filling the observed universe will type in a complete work such as Shakespeare Hamlet so small that the possibility of it occurring over a period of time is hundreds of thousands of times longer than the age of the universe very low (but technically not zero).
In this context, "almost certain" is a mathematical term with precise meaning, and "monkey" is not a real monkey, but a metaphor for abstract devices that produce endless series of letters and endless symbols. One of the earliest examples of the use of "monkey metaphor" was French mathematics ÃÆ'â € Borel's miles in 1913, but the first example may be earlier.
Variant theorems include many typists and a great many, and the target text varies between the entire library and one sentence. Jorge Luis Borges traces the history of this idea from Aristotle In Generation and Corruption and Cicero De natura deorum (About the Nature of the Gods), through Blaise Pascal and Jonathan Swift, to a modern statement with their iconic simian and typewriter. At the beginning of the 20th century, Borel and Arthur Eddington used the theorem to illustrate the time scale implicit in the foundations of statistical mechanics.
Video Infinite monkey theorem
Solution
Live Evidence
There is direct evidence of this theorem. As an introduction, remember that if two events are statistically independent, then the probability of both occurring is equal to the product of each probability occurring independently. For example, if the chance of rain in Moscow on a certain day in the future is 0.4 and the probability of an earthquake in San Francisco on a given day is 0.00003, then the probability of both occurring on the same day is 0.4 ÃÆ'â € "0,00003 = 0,000012 , assuming that they are indeed independent.
Suppose the typewriter has 50 keys, and the word to type is banana . If a key is randomly and independently, it means that each key has the same chance of being pressed. Then, the likelihood that the first letter typed is 'b' is 1/50, and it is likely that the second letter typed is a also 1/50, and so on. Therefore, the first six letters spelling banana is
- (1/50) ÃÆ'â € "(1/50) ÃÆ'â €" (1/50) ÃÆ'â € "(1/50) ÃÆ'â €" (1/50) ÃÆ'â € "(1/50) ) = (1/50) 6 = 1/15Ã, 625Ã, 000Ã, 000Ã,,
less than one in 15 billion, but not zero, then the chances of the outcome.
Dari atas, kemungkinan tidak mengetik pisang dalam blok 6 huruf yang diberikan adalah 1Â -Â (1/50) 6 . Karena setiap blok diketik secara independen, kemungkinan X n tidak mengetik pisang di salah satu pertama > n blok 6 huruf adalah
As n grows, the X n gets smaller. For n of one million, X n is about 0.9999, but for n 10 billion X n is approximately 0.53 and for n 100 billion is approximately 0 , 0017. When n approaches infinity, the probability X n is close to zero; by making n big enough, X n can be made as small as desired, and possibly typing bananas close to 100%.
The same argument shows why at least one of the many monkeys will produce the text as soon as it will be produced by a very accurate human typist who copies it from the original. In this case X n = (1Ã, - (1/50) 6 ) n where X n represents the probability that there is no first n type monkeys bananas correctly on their first try. When we consider 100 billion monkeys, the probability drops to 0.17%, and as the number of monkeys n increases, the value X n - the possibility of a monkey failing to reproduce the given text - approaches zero arbitrarily. The limit, for n will be unlimited, is zero. So the possibility of the banana word that appears at a certain point in the order of emphasis is less than one.
Unlimited strings
It can be declared more general and compact in terms of strings, which are the selected character sequence of some finite alphabet:
- By providing an unlimited string where each character is chosen uniformly at random, any limited string must almost certainly occur as a substring in some positions.
- Given an unlimited string of infinite strings, in which each character of each string is uniformly randomly selected, a particular finite string almost certainly appears as the prefix of one of these strings.
Both followed easily from the second Borel-Cantelli lemma. For the second theorem, let E k be an event where the string k begins with the given text. Since this has some non-zero probability remains p that occurs, E k is independent, and the number below this deviates,
the infinite probability of E k occurs is 1. The first theorem is shown the same; one can split a random string into non-overlapping blocks matching the desired text size, and create an E k event where k th block is the same as the desired string. Probability
However, for the number of monkeys that are physically significant for a period of physical significance, the result will be reversed. If there are as many monkeys as there are atoms in the universe that are observed typing very rapidly during the trillions of times the life of the universe, the possibility of monkeys replicating even a single page of Shakespeare is very small.
Ignoring punctuation, spacing, and capitalization, randomly typed random letters have the possibility of one of 26 correctly typing the first letter Hamlet. This has one chance at 676 (26Ã, ÃÆ'â € "Ã, 26) typing the first two letters. Since the probability shrinks exponentially, at 20 letters it has only one chance in 26 20 = 19,928,148,895,209,409,152,340,197,376 (almost 2Ã ¢ â,¬ <10 28 ). In the case of the whole text Hamlet , chances are very small and unimaginable. Hamlet's text contains about 130,000 letters. Thus there is a possibility of one in 3.4 Ã,ÃÆ' â € "10 183,946 to get the text right on the first try. The average number of letters that need to be typed until the text appears is also 3.4 Ã,ÃÆ' â € "10 183,946 , or include punctuation, 4.4 Ã,ÃÆ' â €" 10 360.783 .
Even if every proton in the observed universe is a monkey with a typewriter, typing from the Big Bang to the end of the universe (when protons probably do not exist anymore), they still need far more time - more than three hundred and sixty thousand > command magnitude longer - to have even 1 in 10 500 chance of success. In other words, for one in a trillion possible successes, there needs to be a 10 360.641 universe made of atomic monkeys. As Kittel and Kroemer put in their textbook on thermodynamics, the field whose statistical foundations motivate the first known exposition of typing monkeys, "The possibility of Hamlet is zero in the operational meaning of an event.. ", and the assertion that the monkey should ultimately succeed" gives a misleading conclusion about a very, very large sum. "
There is even less than one in a trillion chances of success that such a universe made of monkeys can type certain documents that have only 79 characters.
Almost certainly
The probability that the resulting text string is infinite will contain a certain limited substring is 1. However, this does not mean the absence of the substring is "not possible", even though no one has a prior probability of 0. For example, immortal monkeys can randomly typing G as the first letter, G as the second letter, and G as each letter afterwards, resulting in an infinite string Gs; there is no point monkeys should be "forced" to type anything else. (To assume the opposite implies a gambler error.) However, the randomly generated string length is, there is a small but non-zero chance that will change to consist of the same characters repeated entirely; this opportunity approaches zero as the string length approaches infinity. There is nothing special about such a monotonous sequence except that it is easily illustrated; the same fact applies to any specific sequence which may be mentioned, such as "RGRGRG" repeated forever, or "ab-aa-bb-aaa-bbb -...", or "Three, Six, Nine, Twelve...".
If a hypothetical monkey has a typewriter with 90 possible same keys that include numbers and punctuation marks, the first typed key may be "3.14" (the first three digits of pi) with probability (1/90) 4 along that, much lower: (1/90) 100 . If the length of the text given by the monkeys is infinite, the odds of typing only the pi digit are 0, the same may be maybe (mathematically possible) since it does not type anything but Gs (also the probability of 0).
The same goes for typing a particular version of Hamlet followed by an endless copy of itself; or Hamlet immediately followed by all digit pi; These specific strings are as long as infinite, they are not forbidden by the problem of thought, and they each have a previous possibility of 0. In fact, any an infinite sequence of certain types of enduring monkeys will have i> the previous probability of 0, although the monkey must type something.
This is an extension of the principle that the finite string of random text has a lower and lower probability of being certain longer strings of it (though all certain strings are equally impossible). This probability is close to 0 as the strings approach infinity. Thus, the probability of monkeys typing long strings endlessly, like all digit pi in sequence, on a 90-key keyboard is (1/90) ? is the same (1/?) Which is essentially 0. At the same time, the probability that the sequence contains a certain continuity (such as MONKEY, or the 12th to 999th digit of pi , or King James version) Bible increases when the total string increases. This probability is close to 1 as the total string approaches infinity, and thus the original theorem is true.
Correspondence between string and number
In the simplification of mind experiments, monkeys can have typewriters with only two keys: 1 and 0. The resulting infinitely long string will correspond to the binary digits of a certain real number between 0 and 1. The countless countless number of possible strings with infinite repetition, which means the corresponding real numbers are rational. Examples include strings corresponding to one third (010101...), five sixths (11010101...) and five darkness (1010000...). Only a subset of such real number strings (though unlimited unlimited subset) contains the whole Hamlet (assuming that the text is encoded numerically, like ASCII).
In the meantime, there are countless unlimited string strings that do not end with such repetition; this corresponds to the irrational number. These can be split into two infinite sets: those containing Hamlet and those that are not. However, the "biggest" subset of all real numbers is that not only contains Hamlet , but that contains every possible string with any length, and with the same string distribution. These irrational numbers are called normal. Since almost all numbers are normal, almost all possible strings contain all possible finite substrings. Therefore, the probability of a monkey typing a normal number is 1. The same principle applies regardless of the number of keys the monkey can choose; a 90-key keyboard can be viewed as a numeric generator written in base 90.
Maps Infinite monkey theorem
History
Statistical mechanics
In one form where the probabilist now knows this theorem, with "dactylographic" monkeys (French: singes dactylographes ; the French word < i> singe includes monkeys and monkeys), appeared in Borel's 1913 mile magazine article "Mableique Statistique et Irrà ©  © versibilitÃÆ' © " ( Statistics of mechanics and irreversibility ), and in his book "Le Hasard" in 1914. The "monkey" is not a true monkey; instead, they are a metaphor for imaginary ways to produce a large random sequence of letters. Borel says that if a million monkeys typed ten hours a day, it is highly unlikely that their output would be exactly the same as all the books from the world's richest library; however, when compared, it is even more unlikely that the laws of statistical mechanics will be violated, even briefly.
Physicist Arthur Eddington draws Borel's picture further in The Nature of the Physical World (1928), writes:
If I let my fingers roam over the buttons of the typewriter, my screed might occur making understandable sentences. If a group of monkeys were picking typewriters, they would probably write all the books in the British Museum. Their chances of doing it are clearly more profitable than the possibility of molecules returning to one half of the ship.
These images invite readers to consider the extraordinary impossibility of a large number of monkeys working in large numbers but limited to generating significant work, and comparing it to the greater likelihood of a particular physical event. Any physical process that is even less likely to be compared to such ape's success is unlikely, and it can be said that such a process will never happen. It is clear from the context that Eddington does not suggest that the likelihood of this occurrence should be taken seriously. Rather, it is a rhetorical illustration of the fact that under a certain degree of probability, the term is impossible functionally equivalent to impossible .
Origins and "Total Library"
In the 1939 essay entitled "The Total Library", Argentine writer Jorge Luis Borges traced the monkey-infinite concept back to Aristotle Metaphysics. Explaining the view of Leucippus, who argues that the world emerges through a random combination of atoms, Aristotle notes that atoms themselves are homogeneous and that arrangements may differ only in form, position and order. In Generation and Corruption, the Greek philosopher compares this to the way that tragedy and comedy consist of the same "atom", . , alphabetic characters. Three centuries later, Cicero De natura deorum ( In the Realms of the Gods ) opposes the atomist world view:
He who believes this may also believe that if a large number of letters one and twenty, consisting of gold or other material, are thrown to the ground, they will fall into such a sequence legibly to form the Annals of Ennius. I doubt if luck can make one verse from them.
Borges follows the history of this argument through Blaise Pascal and Jonathan Swift, then observes that in his own time, the vocabulary has changed. In 1939, the phrase was "that half a dozen monkeys equipped with typewriters, in some eternity, will produce all the books in the British Museum." (To which Borges adds, "Actually, one immortal monkey is enough.") Borges then imagines the contents of the Total Library that this company will produce if taken to its full extremes:
Everything will be in a blind volume. Everything: the detailed future history, Aeschylus' the Egyptians , how many times exactly the water in the Ganges reflects the flight of a hawk, the secrets and the true nature of Rome, the Novalis encyclopedia will be built, my dream and the half dream of the moment the dawn of August 14, 1934, the proof of Pierre Fermat's theorem, the unwritten chapters of Edwin Drood, the same chapters were translated into the language spoken by Garamantes, Berkeley's paradoxes of Time but did not publish, Urizen's books on iron, the early enlightenment of Stephen Dedalus, which would be meaningless before the thousand-year cycle, the Gnostic Basilides Gospel, the singing siren song, the complete catalog of the Library, evidence of the catalog's inaccuracies. Everything: but for every line that makes sense or accurate facts, there will be millions of useless cacophonies, verbal farragoes, and babbling. Everything: but all human generations can pass in front of dizzying shelves - shelves that obliterate the day and where chaos lies - ever give them a reward with a tolerable page.
The concept of the total library of Borges is the main theme of his much-read short story of 1941, the "Library of Babel," which describes an unimaginably large library of six-sided connecting rooms, together containing every possible volume that can be arranged from the letters of the alphabet and some punctuation characters.
Original monkey
In 2003, lecturers and students from the University of Plymouth MediaLab Arts used a $ 2,000 grant from the Arts Council to study the original literary results of monkeys. They left computer keyboards at the cage of six Crested Celebes apes at Paignton Zoo in Devon in England for a month, with radio links to broadcast results on a website.
Not only the monkeys that produce anything but the total five pages consist mostly of the S, the lead man begins by banging the keyboard with rocks, and the monkeys continue with urinating and defecating on it. Mike Phillips, director of the Institute for Digital Art and Technology (i-DAT), said the artist-funded project is performing, and they have learned "a lot" from it. He concluded that monkeys "are not random generators, they are more complex than that.... They are quite interested in the screen, and they see that when they type a letter something happens, there is a level of intention there."
Apps and criticism
Evolution
In his 1931 book The Mysterious Universe Eddington's competitor, James Jeans, attributes the monkey parable to "Huxley", which may mean Thomas Henry Huxley. This attribute is incorrect. Today, it is sometimes further reported that Huxley applied the example in the now legendary debate over Charles Darwin on "The Origin of Species" with Oxford Anglican Bishop Samuel Wilberforce, held at the British Association for the Advancement of Science meeting in Oxford at 30 June 1860. This story suffers not only from the lack of evidence, but the fact that in 1860 the typewriter itself had not yet appeared.
Although the mix-up argument, the original monkey-and-typewriter is now common in arguments over evolution. For example, Doug Powell argues as a Christian defender that even if a monkey accidentally typed the letters of Hamlet, he failed to produce Hamlet because he had no intention of communicating. The parallel implication is that natural law can not produce information content in DNA. The more common argument is represented by Reverend John F. MacArthur, who claims that the genetic mutations necessary to produce tapeworms from amoeba are unlikely to be like the monkey typing soliloquy of Hamlet, and hence the possibility of resisting the evolution of all life is impossible. resolve.
Evolutionary biologist Richard Dawkins applies the concept of typing monkeys in his book The Blind Watchmaker to demonstrate the ability of natural selection to produce the biological complexity of random mutations. In a simulated experiment, Dawkins has a ferret program producing the phrase Hamlet METHINKS IT IS LIKE A WEASEL , starting from a randomly typed parent, by "breeding" the next generation and always choosing the closest match of the copying offspring parent, with random mutations. The target phrase opportunity that appears in one step is very small, but Dawkins suggests that it can be produced quickly (in about 40 generations) using the cumulative phrase selection. Random selections produce raw materials, while cumulative selection imparts information. As Dawkins acknowledged, the weasel program is an imperfect analogy for evolution, because the phrase "derivative" is chosen "according to the criteria of similarity to the ideal far target." Instead, Dawkins asserts, evolution has no long-term plan and does not evolve toward some distant destiny (like humans). The civet program is not meant to illustrate the difference between non-random cumulative selection, and one-step random selection. In terms of typing monkey analogy, this means that Romeo and Juliet can be produced relatively quickly if placed under the limits of a nonrandom, Darwinian type option because the fitness function will tend to maintain in any place. letters that match the target text, increasing each generation of typing apes.
A different way to explore the analogy between evolution and unbound monkeys lies in the problem that monkeys typed only one letter at a time, regardless of the other letters. Hugh Petrie argues that more sophisticated settings are needed, in case they are not for biological evolution but the evolution of ideas:
To get the proper analogy, we have to complete the monkey with a more complicated typewriter. It must include all of Elizabeth's sentences and thoughts. It must incorporate Elizabethan beliefs about the pattern of human action and its causes, the morality and science of Elizabethan, and the linguistic pattern for this expression. It may even have to include an explanation of the kind of experience that formed Shakespeare's structures of trust as a special example of an Elizabethan. Then, perhaps, we might allow the monkey to play with such typewriters and produce variants, but the impossibility of getting a Shakespeare game is no longer clear. What varies really summarizes much of the knowledge already achieved.
James W. Valentine, while acknowledging that the classical monkey task is impossible, finds that there is a useful analogy between written English and metazoan genomes in this other sense: both have "combinational, hierarchical structures" that severely limit the large number of combinations at the alphabetical level.
Literary Theory
R. G. Collingwood argued in 1938 that art could not be produced by chance, and wrote as sarcastic in addition to its critics,
... some... have rejected this proposition, pointing out that if a monkey plays with a typewriter... he will produce... the full text of Shakespeare. Any unrelated reader can amuse himself by calculating how long it takes to bet. But the interest of that suggestion lies in the expression of a person's mental state that can identify Shakespeare's 'work' with a series of letters printed on the pages of the book...
Nelson Goodman took the opposite position, illustrating his intent with Catherine Elgin with the example of Borges, "Pierre Menard, Quixote Writer",
What Menard wrote was just another text. Each of us can do the same, just like printing machines and copiers. Indeed, we were told, if there were many monkeys... it would eventually produce a replica of the text. The replica, we hold, will be an example of that work, Don Quixote , as the manuscript of Cervantes, Menard's manuscript, and any copies of the book ever or will be printed.
In another paper, Goodman describes, "That the monkey might have produced a random copy made no difference, it is the same text, and it is open to all the same interpretations...." GÃ © Â © rard Genette dismisses Goodman Arguments as ask questions.
For Jorge J. E. Gracia, the question of text identity leads to a different question, which is about the author. If a monkey is able to type Hamlet , even though it has no intention to decipher it and therefore disqualifies itself as a writer, it appears that the texts do not require the author. Possible solutions include saying that anyone who finds the text and identifies it as Hamlet is the author; or that Shakespeare is the author, his agent monkey, and the inventor is just a text user. These solutions have their own difficulties, because the text seems to have a separate meaning from other agents: what if the monkey operated before Shakespeare was born, or was Shakespeare never born, or if nobody found a monkey manuscript?
Creation of random documents
This theorem discusses mind experiments that can not be fully implemented in practice, since it is thought to require a very limited amount of time and resources. Nonetheless, it has inspired effort in the generation of randomly limited text.
A computer program run by Dan Oliver of Scottsdale, Arizona, according to an article in The New Yorker, came up with a result on August 4, 2004: After the group worked for 42.162.500 billion billion years, one "monkey" is typed, " VALENTINE.Centor: eFLP0FRjWK78aXzVOwm) - '; 8.t " The first 19 letters of this sequence can be found in "The Two Gentlemen of Verona". Another team has reproduced 18 characters from "Timon of Athens", 17 from "Troilus and Cressida", and 16 from "Richard II".
A website titled The Monkey Shakespeare Simulator , launched on July 1, 2003, contains a Java applet that simulates a large population of randomly typed typewriters, with the stated purpose of looking at how long it takes a virtual monkey to produce Shakespeare drama complete from beginning to end. For example, he produced this partial line from Henry IV, Part 2 , reporting that it takes "2,737,850 million billion billion monkey-year" to reach 24 matching characters:
- RUMOR. Open your ears; 9r "5j5 & amp; OWTY Z0d ...
Due to limited processing power, the program uses a probabilistic model (using a random number generator or RNG) rather than actually generating random text and comparing it to Shakespeare. When the simulator "detects a match" (ie, the RNG generates a certain value or value within a certain range), the simulator simulates matches by creating matching text.
More sophisticated methods are used in practice for natural language generation. If instead of just generating random characters one limits the generator to meaningful vocabulary and conservatively follows the rules of grammar, such as using context-free grammar, then random documents generated in this way can even deceive some humans (at least on a cursory reading) as shown in experiments with SCIgen, snarXiv, and Postmodernism Generators.
Test a random number generator
Questions about statistics that explain how often the ideal monkey is expected to type a particular string translate into a practical test for a random number generator; These range from simple to "sophisticated". Computer science professors George Marsaglia and Arif Zaman reported that they used to call one such category of "t-tuple overlap" tests in lectures, as they noticed the overlapping of m-tuples of sequential elements in random order. But they found that calling them "monkey tests" helped motivate the idea with students. They published reports on test classes and their results for various RNGs in 1993.
In popular culture
Unlimited monkey theorems and their associated imagery are seen as popular illustrations and proverbs of probability mathematics, widely known by the general public for their transmission through popular culture rather than through formal education. It is aided by innate humor derived from literal monkey images that crunch away on a set of typewriters, and is a popular visual joke.
In the episode of The Simpsons "Last Exit to Springfield", Mr. Burns showed Homer "a room with a thousand monkeys on a thousand typewriters and soon they will write the greatest novel known to mankind!"
In his 1978 radio drama, Douglas Adams, The Hitchhiker's Guide to the Galaxy, uses the theorem to illustrate the power of the infinite improbability drive supported by spacecraft. From Episode 2: "Ford, there are lots of monkeys outside who want to talk to us about this script for Hamlet they've been working on."
A quote attributed to a 1996 speech by Robert Wilensky states, "We have heard that a million monkeys on a million keyboards can produce Shakespeare's complete work; now, thanks to the Internet, we know it's not true."
The eternal popularity of this theorem was recorded in the introductory paper of 2001, "Monkey, Machine Type and Network: The Internet in the Light of Accidental Advantage Theory". In 2002, an article in The Washington Post said, "Many people have had fun with the famous notion that an infinite number of monkeys with infinite number of typewriters and infinite time could eventually write works of Shakespeare ". In 2003, a previously funded Arts Council experiment involving real monkeys and computer keyboards received extensive press coverage. In 2007, the theorem was listed by Wired magazine in the list of eight classic thought experiments.
See also
- Normal numbers
- The Hilbert Paradox at the Grand Hotel, another thought experiment involving infinity
- The law of a very large number
- Murphy's Law
- The Hidden Reality: Parallel Universes and Deep Laws of the Cosmos , describes a multiverse in which every possible event will occur indefinitely
- The Babylon Library
- Machine
- Boltzmann's Brain
- The Infinite Monkey Cage
Note
References
External links
- Ask the Dr. Math, August 1998, Adam Bridge
- Monkey Parable, bibliography with quotes
- Planck Monkeys, in filling the cosmos with monkey particles
- PixelMonkeys.org - Artist, Matt Kane's Application of Unlimited Monkey Theorem on pixels to create images.
- RFC 2795 - April Fools' Day RFC on the implementation of Infinite monkey theorem.
Source of the article : Wikipedia
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