DOCSLIB.ORG

Average Laws in Analysis Silvius Klein Norwegian University of Science and Technology (NTNU) the Law of Large Numbers: Informal Statement

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Average Laws in Analysis Silvius Klein Norwegian University of Science and Technology (NTNU) the Law of Large Numbers: Informal Statement Average laws in analysis Silvius Klein Norwegian University of Science and Technology (NTNU) The law of large numbers: informal statement The theoretical expected value of an experiment is approximated by the average of a large number of independent samples. theoretical expected value ≈ empirical average The law of large numbers (LLN) Let X1, X2,..., Xn,... be a sequence of jointly independent , identically distributed copies of a scalar random variable X. Assume that X is absolutely integrable, with expectation µ. Define the partial sum process Sn := X1 + X2 + ... + Xn. Then the average process S n ! µ as n ! . n 1 The law of large numbers: formal statements Let X1, X2, . be a sequence of independent, identically distributed random variables with common expectation µ. Let Sn := X1 + X2 + ... + Xn be the corresponding partial sums process. Then Sn 1 (weak LLN) ! µ in probability. n That is, for every > 0, S n - µ > ! 0 as n ! . P n 1 Sn 2 (strong LLN) ! µ almost surely. n It was the best of times, it was the worst of times. Charles Dickens, A tale of two cities (click here) yskpw,qol,all/alkmas;’.a ma;;lal;,qwmswl,;q;[;’ lkle’78623rhbkbads m ,q l;,’;f.w, ’ fwe It was the best of times, it was the worst of times. jllkasjllmk,a s.„,qjwejhns;.2;oi0ppk;q,Qkjkqhjnqnmnmmasi[oqw— qqnkm,sa;l;[ml/w/’q Application of LLN: the infinite monkey theorem Let X1, X2, . be i.i.d. random variables drawn uniformly from a finite alphabet. Then almost surely, every finite phrase (i.e. finite string of symbols in the alphabet) appears (infinitely often) in the string X1X2X3 . .. Application of LLN: the infinite monkey theorem Let X1, X2, . be i.i.d. random variables drawn uniformly from a finite alphabet. Then almost surely, every finite phrase (i.e. finite string of symbols in the alphabet) appears (infinitely often) in the string X1X2X3 . .. yskpw,qol,all/alkmas;’.a ma;;lal;,qwmswl,;q;[;’ lkle’78623rhbkbads m ,q l;,’;f.w, ’ fwe It was the best of times, it was the worst of times. jllkasjllmk,a s.„,qjwejhns;.2;oi0ppk;q,Qkjkqhjnqnmnmmasi[oqw— qqnkm,sa;l;[ml/w/’q Application of LLN: the infinite monkey theorem Let X1, X2, . be i.i.d. random variables drawn uniformly from a finite alphabet. Then almost surely, every finite phrase (i.e. finite string of symbols in the alphabet) appears (infinitely often) in the string X1X2X3 . .. yskpw,qol,all/alkmas;’.a ma;;lal;,qwmswl,;q;[;’ lkle’78623rhbkbads m ,q l;,’;f.w, ’ fwe It was the best of times, it was the worst of times. jllkasjllmk,a s.„,qjwejhns;.2;oi0ppk;q,Qkjkqhjnqnmnmmasi[oqw— qqnkm,sa;l;[ml/w/’q The second Borel-Cantelli lemma Let E1, E2,..., En,... be a sequence of jointly independent events. If P(En) = , 1 = nX1 1 then almost surely, an infinite number of En hold simultaneously. This can be deduced from the strong law of large numbers, applied to the random variables 1 Xk := Ek . The actual proof of the infinite monkey theorem Split every realization of the infinite string of symbols in the alphabet X1X2X3 ... Xn ... into finite strings S1, S2, . of length 52 each. Let En be the event that the phrase It was the best of times, it was the worst of times. is exactly the n-th finite string Sn. These are independent events. They each have the same probability p > 0 to occur. Apply the second Borel-Cantelli lemma. The law of large numbers We have seen that if X1, X2,..., Xn,... is a sequence of jointly independent , identically distributed copies of a scalar random variable X, and if we denote the corresponding sum process by Sn := X1 + X2 + ... + Xn, then the average process S n ! X as n ! . n E 1 A rather deterministic system: circle rotations Let S be the unit circle in the (complex) plane. There is a natural measure λ on S (i.e. the extension of the arc-length). Let 2πα be an angle, and denote by Rα the rotation by 2πα on S. That is, consider the transformation Rα : S ! S, π π α where if z = e2 i x 2 S and if we denote ! := e2 i , then 2π i (x+α) Rα(z) = e = z · !. Note that Rα preserves the measure λ. Iterations of the circle rotation Let 2πα be an angle. π Start with a point z = e2 i x 2 S and consider successive applications of the rotation map Rα: 1 2π i (x+α) Rα(z) = Rα(z) = e 2 2π i (x+2α) Rα(z) = Rα ◦ Rα(z) = e . n 2π i (x+nα) Rα(z) = Rα ◦ ... ◦ Rα(z) = e . 1 2 n The maps Rα, Rα,..., Rα, . are the iterations of Rα. Given a point z 2 S, the set 1 2 n fRα(z), Rα(z),..., Rα(z),... g is called the orbit of z. The orbit of every point is dense on the circle. This transformation satisfies a very weak form of independence called ergodicity. An orbit of a circle rotation Let Rα be the circle rotation by the angle 2πα, where α is an irrational number. Pick a point z on the circle S. The orbit of z (or rather a finite subset of it). An orbit of a circle rotation Let Rα be the circle rotation by the angle 2πα, where α is an irrational number. Pick a point z on the circle S. The orbit of z (or rather a finite subset of it). The orbit of every point is dense on the circle. This transformation satisfies a very weak form of independence called ergodicity. Observables on the unit circle Any measurable function f : S ! R is called a (scalar) observable of the measure space (S, A, λ). We will assume our observables to be absolutely integrable. A basic example of an observable: f = 1I , where I is an arc (or any other measurable set) on the circle. I, | I | ~ e “Observations" of the orbit points of a circle rotation. Average number of orbit points visiting an arc Let Rα be the circle rotation by the angle 2πα, where α is an irrational number. Let I be an arc on the circle. I, | I | ~ e The first n orbit points of a circle rotation and their visits to I. The average number of visits to I: j # j 2 f1, 2, . , ng : Rα(z) 2 I n What does this look like for large enough n? Or in other words, is there a limit of these averages as n ! ? 1 Average number of orbit points visiting an arc Let Rα be the circle rotation by the angle 2πα, where α is an irrational number. Let I be an arc on the circle. I, | I | ~ e The first n orbit points of a circle rotation and their visits to I. As n ! , the average number of visits to I: # j 2 f1, 2, . , ng : Rj (z) 2 I 1 α ! λ(I) , n for all points z 2 S. = 1I dλ . ZS Average number of orbit points visiting an arc Let Rα be the circle rotation by the angle 2πα, where α is an irrational number. Let I be an arc on the circle. The first n orbit points of a circle rotation and their visits to I. I, | I | ~ e n j 1 j # j 2 f1, 2, . , ng : Rα(z) 2 I = I (Rα(z)). = Xj 1 Then the average number of visits to I can be written: 1 (R1 (z)) + 1 (R2 (z)) + ... + 1 (Rn (z)) I α I α I α ! λ(I) n Average number of orbit points visiting an arc Let Rα be the circle rotation by the angle 2πα, where α is an irrational number. Let I be an arc on the circle. The first n orbit points of a circle rotation and their visits to I. I, | I | ~ e n j 1 j # j 2 f1, 2, . , ng : Rα(z) 2 I = I (Rα(z)). = Xj 1 Then the average number of visits to I can be written: 1 (R1 (z)) + 1 (R2 (z)) + ... + 1 (Rn (z)) I α I α I α ! λ(I) = 1 dλ . n I ZS Measure preserving dynamical systems A probability space (X, B, µ) together with a transformation T : X ! X define a measure preserving dynamical system if T is measurable and it preserves the measure of any B-measurable set: µ(T -1A) = µ(A) for all A 2 B. Ergodic dynamical system. For any B-measurable set A with µ(A) > 0, the iterations TA, T 2 A,..., T n A,... fill up the whole space X, except possibly for a set of measure zero. Ergodicity leads to some very, very weak form of independence. Some examples of ergodic dynamical systems 1 The Bernoulli shift, which encodes sequences of independent, identically distributed random variables. 2 The circle rotation by an irrational angle. 3 The doubling map. T :[0, 1] ! [0, 1], Tx = 2x mod 1. The pointwise ergodic theorem Given: an ergodic dynamical system (X, B, µ, T), and an absolutely integrable observable f : X ! R, define the n-th Birkhoff sum 2 n Sn f (x) := f (Tx) + f (T x) + ... + f (T x). Then as n ! 1 S 1f (x) ! f dµ for µ a.e. x 2 X. n n ZX The law of large numbers We have seen that if X1, X2,..., Xn,... is a sequence of jointly independent , identically distributed copies of a scalar random variable X, and if we denote the corresponding sum process by Sn := X1 + X2 + ..
Load more

Recommended publications
  • Probability - Fall 2013 the Infinite Monkey Theorem (Lecture W 9/11)
    MATH 3160 - Fall 2013 Infinite Monkey Theorem MATH 3160 - Probability - Fall 2013 The Infinite Monkey Theorem (Lecture W 9/11) "It was the best of times, it was the blurst of times?! ...You stupid monkey!... Oh, shut up!" | Charles Montgomery Burns The main goal of this handout is to prove the following statement: Theorem 1 (Infinite monkey theorem). If a monkey hits the typewriter keys at random for an infinite amount of time, then he will almost surely produce any type of text, such as the entire collected works of William Shakespeare. To say that an event E happens almost surely means that it happens with probability 1: P (E) = 11. We will state a slightly stronger version of Theorem 1 and give its proof, which is actually not very hard. The key ingredients involve: • basics of convergent/divergent infinite series (which hopefully you already know!); • basic properties of the probability (measure) P (see Ross x2.4, done in lecture F 9/6); • the probability of a monotone increasing/decreasing sequence of events, which is the main content of Ross, Proposition 2.6.1; • the notion of independent events (which we will discuss properly next week, see Ross x3.4, but it's quite easy to state the meaning here). 1 Preliminaries In this section we outline some basic facts which will be needed later. The first two are hopefully familiar (or at least believable) if you have had freshman calculus: 1 While almost surely and surely often mean the same in the context of finite discrete probability problems, they differ when the events considered involve some form of infinity.
    [Show full text]
  • Best Q2 from Project 1, Nov 2011
    Best Q21 from project 1, Nov 2011 Introduction: For the site which demonstrates the imaginative use of random numbers in such a way that it could be used explain the details of the use of random numbers to an entire class, we chose a site that is based on the Infinite Monkey Theorem. This theorem hypothesizes that if you put an infinite number of monkeys at typewriters/computer keyboards, eventually one will type out the script of a Shakespearean work. This theorem asserts nothing about the intelligence of the one random monkey that eventually comes up with the script. It may be referred to semi-seriously when justifying a brute force method (the statement to be proved is split up into a finite number of cases and each case has to be checked to see if the proposition in question holds true); the implication is that, with enough resources thrown at it, any technical challenge becomes a “one-banana problem” (the idea that trained monkeys can do low level jobs). The site we chose is: http://www.vivaria.net/experiments/notes/publication/NOTES_EN.pdf. It was created by researchers at Paignton Zoo and the University of Plymouth, in Devon, England. They carried out an experiment for the infinite monkey theorem. The researchers reported that they had left a computer keyboard in the enclosure of six Sulawesi Crested Macaques for a month; it resulted in the monkeys produce nothing but five pages consisting largely of the letter S (shown on the link stated above). The monkeys began by attacking the keyboard with a stone, and followed by urinating on the keyboard and defecating on it.
    [Show full text]
  • Quantitative Hedge Fund Analysis and the Infinite Monkey Theorem
    Quantitative Hedge Fund Analysis and the Infinite Monkey Theorem Some have asserted that monkeys with typewriters, given enough time, could replicate Shakespeare. Don’t take the wager. While other financial denizens sarcastically note that given enough time there will be a few successes out of all the active money managers, building on the concept that even a blind pig will occasionally find a peanut. Alternatively, our goal in the attached white paper is to more reliably accelerate an investor’s search for sound and safe hedge fund and niche investments. We welcome all comments, suggestions and criticisms, too. Not kidding, really do! A brief abstract follows: Quantitative analysis is an important part of the hedge fund investment process that we divide into four steps: 1) data acquisition and preparation; 2) individual analysis; 3) comparative analysis versus peers and indices; 4) portfolio analysis. Step One: Data Acquisition and Preparation Understand sourced data by carefully reading all of the associated footnotes, considering who prepared it (whether the manager, the administrator or an auditor/review firm), and evaluating the standard of preparation, e.g., is it GIPS compliant? Step Two: Individual Analysis Here, we compare and contrast Compound Annual Growth Rate (CAGR) to Average Annual Return (AAR), discuss volatility measures including time windowing and review the pros and cons of risk/reward ratios. Step Three: Comparative Analysis Any evaluations must be put into context. What are the alternatives, with greatness being a relative value. Step Four: Portfolio Analysis Too often, a single opportunity is considered in isolation, whereas its impact must be measured against a total portfolio of existing positions.
    [Show full text]
  • Do Androids Prove Theorems in Their Sleep?
    DO ANDROIDS PROVE THEOREMS IN THEIR SLEEP? Michael Harris, version of March 21, 2008, (plus one addition) 1. A mathematical dream narrative …antes imagino que todo es ficción, fábula y mentira, y sueños contados por hombres despiertos, o, por mejor decir, medio dormidos. Don Quixote, Book II, Chapter 1. What would later be described as the last of Robert Thomason's "three major results" in mathematics was published as a contribution to the Festschrift in honor of Alexandre Grothendieck's sixtieth birthday, and was cosigned by the ghost of his recently deceased friend Thomas Trobaugh. Thomason described the circumstances of this collaboration in the introduction to their joint article: a rare note of pathos in the corpus of research mathematics, and a brief but, I believe, authentic contribution to world literature. The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom's simulacrum remarked, "The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf." Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K0 obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom's simulacrum had been so insistent, I knew he wouldn't let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper.
    [Show full text]
  • Math Olympiad Hardness Scale (MOHS) Because Arguing About Problem Difficulty Is Fun :P
    Math Olympiad Hardness Scale (MOHS) because arguing about problem difficulty is fun :P Evan Chen April 30, 2021 In this document I provide my personal ratings of difficulties of problems from selected recent contests. This involves defining (rather carefully) a rubric by which I evaluate difficulty. I call this the MOHS hardness scale (pronounced \moez"); I also go sometimes use the unit \M" (for \Mohs"). The scale proceeds in increments of 5M, with a lowest possible rating of 0M and a highest possible rating of 60M; but in practice few problems are rated higher than 50M, so it is better thought of as a scale from 0M to 50M, with a few “off-the-chart" ratings. 1 Warning §1.1 These ratings are subjective Despite everything that's written here, at the end of the day, these ratings are ultimately my personal opinion. I make no claim that these ratings are objective or that they represent some sort of absolute truth. For comedic value: Remark (Warranty statement). The ratings are provided \as is", without warranty of any kind, express or implied, including but not limited to the warranties of merchantability, fitness for a particular purpose, and noninfringement. In no event shall Evan be liable for any claim, damages or other liability, whether in an action of contract, tort or otherwise, arising from, out of, or in connection to, these ratings. §1.2 Suggested usage More important warning: excessive use of these ratings can hinder you. For example, if you end up choosing to not seriously attempt certain problems because their rating is 40M or higher, then you may hurt yourself in your confusion by depriving yourself of occasional exposure to difficult problems.1 If you don't occasionally try IMO3 level problems with real conviction, then you will never get to a point of actually being able to solve them.2 For these purposes, paradoxically, it's often better to not know the problem is hard, so you do not automatically adopt a defeatist attitude.
    [Show full text]
  • Monkeys and Walks
    Monkeys and Walks Muhammad Waliji August 12, 2006 1 Preliminaries We will be dealing with outcomes resulting from random processes. An outcome of the process will sometimes be denoted ω. Note that we will identify the set A with the event that the random outcome ω belongs to A. Hence, A will mean ‘A occurs’, and A ∪ B means ‘either A or B occurs’, whereas A ∩ B means ‘both A and B occur.’ For a sequence of events, A1,A2,..., note that ∞ ∞ \ [ An m=1 n=m means that infinitely many of the An occur, or that An occurs infinitely often (An i.o.). The events A1,...,An are said to be independent if P (A1 ∩ · · · ∩ An) = P (A1) ··· P (An). An infinite collection of events is said to be independent if every finite subcollection is independent. 2 The Infinite-Monkey Theorem Before we state the infinite-monkey theorem, we will prove a useful lemma. Consider the events A1,A2,.... P Lemma 2.1 (Borel-Cantelli). If P (An) < ∞, then P (An i.o.) = 0. Fur- n P thermore, if the events An are independent, then if n P (An) = ∞, we have P (An i.o.) = 1. Proof. Suppose that ∞ X P (An) < ∞. (2.1) n=1 1 Then, ∞ ∞ ! ∞ ! \ [ [ P An = lim P An m→∞ m=1 n=m n=m ∞ X ≤ lim P (An) (2.2) m→∞ n=m and hence by (2.1), the limit in (2.2) must be 0. For the converse, it is enough to show that ∞ ∞ ! [ \ c P An = 0 m=1 n=m and so it is also enough to show that ∞ ! \ c P An = 0 (2.3) n=m Note that since 1 − x ≤ e−x, and by independence, ∞ ! m+k ! \ c \ c P An ≤ P An n=m n=m m+k Y = (1 − P (An)) n=m m+k ! X ≤ exp − P (An) .
    [Show full text]
  • Infinity in Computable Probability
    Infinity in computable probability Logical proof that William Shakespeare probably was not a dactylographic monkey Maarten McKubre-Jordens∗ & Phillip L.Wilson† June 4, 2020 1 Introduction Since at least the time of Aristotle [1], the concept of combining a finite number of objects infinitely many times has been taken to imply certainty of construction of a particular object. In a frequently-encountered modern example of this argument, at least one of infinitely many monkeys, producing a character string equal in length to the collected works of Shakespeare by striking typewriter keys in a uniformly random manner, will with probability one reproduce the collected works. In the following, the term “monkey” can (naturally) refer to some (abstract) device capable of producing sequences of letters of arbitrary (fixed) length at a reasonable speed. Recursive function theory is one possible model for computation; Russian recursive mathematics is a reasonable formalization of this theory [4].1 Here we show that, sur- prisingly, within recursive mathematics it is possible to assign to an infinite number of monkeys probabilities of reproducing Shakespeare’s collected works in such a way that while it is impossible that no monkey reproduces the collected works, the probability of any finite number of monkeys reproducing the works of Shakespeare is arbitrarily small. The method of assigning probabilities depends only on the desired probability of success and not on the size of any finite subset of monkeys. Moreover, the result extends to reproducing all possible texts of any finite given length. However, in the context of implementing an experiment or simulation computationally (such as the small-scale example in [10]; see also [7]), the fraction among all possible probability distributions of such pathological distributions is vanishingly small provided arXiv:1101.3578v3 [math.LO] 3 Jun 2020 sufficiently large samples are taken.
    [Show full text]
  • Zero-One Law for Regular Languages
    論文 / 著書情報 Article / Book Information 題目(和文) 正規言語の零壱則 Title(English) Zero-One Law for Regular Languages 著者(和文) 新屋良磨 Author(English) Ryoma Sin'ya 出典(和文) 学位:博士(理学), 学位授与機関:東京工業大学, 報告番号:甲第10103号, 授与年月日:2016年3月26日, 学位の種別:課程博士, 審査員:鹿島 亮,小島 定吉,南出 靖彦,渡辺 治,寺嶋 郁二,金沢 誠 Citation(English) Degree:Doctor (Science), Conferring organization: Tokyo Institute of Technology, Report number:甲第10103号, Conferred date:2016/3/26, Degree Type:Course doctor, Examiner:,,,,, 学位種別(和文) 博士論文 Type(English) Doctoral Thesis Powered by T2R2 (Tokyo Institute Research Repository) ZERO•ONE LAW FOR REGULAR LANGUAGES 正規⾔語の零壱則 Ryoma Sin’ya Tokyo Institute of Technology, Department of Mathematical and Computing Sciences. This thesis is an exposition of the author’s research on automata theory and zero- one laws which had been done in 2013–2015 at Tokyo Institute of Technology and Télécom ParisTech. Most of the results in the thesis have been already published in [58, 57]. Copyright c 2016 Ryoma Sin’ya. All rights reserved. Copyright c 2015 EPTCS. Reprinted, with permission, from Ryoma Sin’ya, “An Automata Theoretic Approach to the Zero-One Law for Regular Languages: Algo- rithmic and Logical Aspects” [58], In: Proceedings Sixth International Symposium on Games, Automata, Logics and Formal Verification, September 2015, pp.172–185. Copyright c 2014 Springer International Publishing Switzerland. Reprinted, with permission, from Ryoma Sin’ya, “Graph Spectral Properties of Deterministic Finite Automata” [57], In: Developments in Language Theory Volume 8633 of the series Lecture Notes in Computer Science, August 2014, pp.76–83. PROLOGUE The notion of the regularity of a set of words, or regular language, is originally intro- duced by Kleene in 1951 [29].
    [Show full text]
  • Monkeying Around with Copyright – Animals, Ais and Authorship in Law
    CREATe Working Paper 2015/01 (February 2015) Monkeying Around with Copyright – Animals, AIs and Authorship in Law Authors David Komuves Jesus Niebla Zatarain University of Edinburgh University of Edinburgh [email protected] [email protected] Burkhard Schafer Laurence Diver University of Edinburgh University of Edinburgh [email protected] [email protected] 1 This paper was originally presented at the Internationales Rechtsinformatik Symposion (IRIS), Salzburg, 26 – 28 February 2015. It won third place in the LexisNexis best paper award. CREATe Working Paper Series DOI: 10.5281/zenodo.16113. This release was supported by the RCUK funded Centre for Copyright and New Business Models in the Creative Economy (CREATe), AHRC Grant Number AH/K000179/1. 1 MONKEYING AROUND WITH COPYRIGHT – ANIMALS, AIS AND AUTHORSHIP IN LAW David Komuves1, Jesus Niebla Zatarain2, Burkhard Schafer3, Laurence Diver4 1 PhD Fellow, CREATe and University of Edinburgh, SCRIPT Centre for IT and IP Law Old College, EH8 9YL Edinburgh, UK [email protected]; http://www.create.ac.uk 2 PhD Researcher, University of Edinburgh, SCRIPT Centre for IT and IP Law Old College, EH8 9YL Edinburgh, UK [email protected]; http://www.law.ed.ac.uk/research/students/viewstudent?ref=264 3 Professor of Computational Legal Theory and Director, SCRIPT Centre for IT and IP Law, University of Edinburgh Old College, EH8 9YL Edinburgh, UK [email protected]; http://www.law.ed.ac.uk/people/burkhardschafer 4 Research Assistant, CREATe and University of Edinburgh, SCRIPT Centre for IT and IP Law Old College, EH8 9YL Edinburgh, UK [email protected]; http://www.law.ed.ac.uk/people/laurencediver Keywords: copyright, artificial intelligence, computer art Abstract: Advances in artificial intelligence have changed the ways in which computers create “original” work.
    [Show full text]
  • Infinite Monkey Theorem from Wikipedia, the Free Encyclopedia
    Infinite monkey theorem From Wikipedia, the free encyclopedia The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare. In fact the monkey would almost surely type every possible finite text an infinite number of times. However, the probability that monkeys filling the observable universe would type a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). In this context, "almost surely" is a mathematical term with a precise meaning, and the "monkey" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the "monkey metaphor" is that of French mathematician Émile Borel in 1913,[1] but the first instance may have been even earlier. Given an infinite length of time, Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a a chimpanzee punching at single sentence. Jorge Luis Borges traced the history of this idea from Aristotle's On Generation and Corruption and Cicero's De natura random on a typewriter would deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, up to modern statements with their iconic simians and almost surely type out all of typewriters.
    [Show full text]
  • Can Animals Acquire Human Language? Shakespeare's Typewriter
    02-Saxton-3990-CH-02:Saxton-3990-CH-02 02/11/2009 3:44 PM Page 25 2 Can Animals Acquire Human Language? Shakespeare’s Typewriter Contents What is language? 26 The infinite monkey theorem 26 Language, talk and communication 27 The design of language 28 Teaching words to animals 34 The strong, silent types: Gua and Viki 34 Sign language 35 Lexigrams 36 Barking up the wrong tree: A talking dog 37 Alex, the non-parroting parrot 37 Animal grammar 38 Combining words 38 Comprehension of spoken English by Kanzi 40 The linguistic limitations of animals 41 Is speech special? 42 Categorical perception in infants and primates 42 Statistical learning 44 Back to grammar: Infants versus monkeys 45 The language faculty: Broad and narrow 46 02-Saxton-3990-CH-02:Saxton-3990-CH-02 02/11/2009 3:44 PM Page 26 Overview We start this chapter with a basic question: what is language? Being careful to distin - guish between three separate concepts – language, talk and communication – we go on to consider some of the key differences between animal communication systems and language. No animal acquires language spontaneously, but can they be taught? We review evidence of attempts to teach a range of animals, including chimpanzees, monkeys, dogs and parrots. In some ways, animals have surprising linguistic gifts. Yet they fall far short of being able to string words together into grammatical sentences. We will see that attempts to teach the words and rules of a natural human language are overly ambitious. More recent research looks at basic psycholinguistic mecha - nisms, in particular, the processing of speech sounds.
    [Show full text]
  • From Entropy to Information: Biased Typewriters and the Origin of Life
    From Entropy to Information: Biased Typewriters and the Origin of Life Christoph Adami1;2;3 and Thomas LaBar1;3 1 Department of Microbiology and Molecular Genetics, Michigan State University, East Lansing, Michigan, USA. 2 Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan, USA. 3 BEACON Center for the Study of Evolution in Action, Michigan State University, East Lansing, Michigan, USA. ∗ E-mail: [email protected] Introduction So much has been written about the possible origins of life on Earth (see, e.g., the popular books [1{4]) that it sometimes seems that|barring an extraordinary break- through in experimental biochemistry (for example [5]), or else the discovery of the remnants of an ancient biochemistry [6]|nothing new can be said about the problem. But such a point of view does not take into account that perhaps not all the tools of scientific inquiry have been fully utilized in this endeavor to unravel our ultimate origin on this planet. Indeed, Origin- of-Life research has historically been confined to a fairly narrow range of disciplines, such as biochemistry and geochemistry. Today, a much broader set of tools is being unleashed on this problem, including mathematical [7{9] and computational approaches [10{14]. Compu- tational approaches to the study of possible origins of life are often derided because they lack a particular feature of biochemistry, or \because they do not take into account the specific properties of individual organic compounds and polymers" [15]. Such a point of view ignores the possibility that life may not a feature that is dependent on a particular biochemistry [16], arXiv:1506.06988v2 [q-bio.PE] 6 Jan 2017 but could instead be a feature of any chemistry that is capable of encoding information.
    [Show full text]