| {"paper_meta":{"paper_id":"arxiv:0704.1043","title":"0704.1043","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"arXiv:0704.1043v5 [cs.CC] 17 Dec 2010\nOctober 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\nChapter 1\nOn the Kolmogorov-Chaitin Complexity for short\nsequences\nJean-Paul Delahaye∗and Hector Zenil†\nLaboratoire d’Informatique Fondamentale de Lille\nCentre National de la Recherche Scientifique (CNRS)\nUniversit ́e des Sciences et Technologies de Lille\nAmong the several new ideas and contributions made by Gregory\nChaitin to mathematics is his strong belief that mathematicians should\ntranscend the millenary theorem-proof paradigm in favor of a quasi-\nempirical method based on current and unprecedented access to compu-\ntational resources.3 In accordance with that dictum, we present in this pa-\nper an experimental approach for defining and measuring the Kolmogorov-\nChaitin complexity, a problem which is known to be quite challenging for\nshort sequences — shorter for example than typical compiler lengths.\nThe Kolmogorov-Chaitin complexity (or algorithmic complexity) of a\nstring s is defined as the length of its shortest description p on a universal\nTuring machine U, formally K(s) = min{l(p) : U(p) = s}. The major\ndrawback of K, as measure, is its uncomputability. So in practical applica-\ntions it must always be approximated by compression algorithms. A string\nis uncompressible if its shorter description is the original string itself. If a\nstring is uncompressible it is said that the string is random since no pat-\nterns were found. Among the 2n different strings of length n, it is easy to\ndeduce by a combinatoric argument that one of them will be completely\nrandom simply because there will be no enough shorter strings so most of\nthem will have a maximal K-C complexity. Therefore many of them will\nremain equal or very close to their original size after the compression. Most\nof them will be therefore random. An important property of K is that it is\nnearly independent of the choice of U. However, when the strings are short\nin length, the dependence of K on a particular universal Turing machine U\n∗delahaye@lifl.fr\n†hector.zenil-chavez@malix.univ-paris1.fr\n1\n\nOctober 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\n2\nJean-Paul Delahaye and Hector Zenil\nis higher producing arbitrary results. In this paper we will suggest an em-\npirical approach to overcome this difficulty and to obtain a stable definition\nof the K-C complexity for short sequences.\nUsing Turing’s model of universal computation, Ray Solomonoff9,10\nand Leonid Levin7 developed a theory about a universal prior distribu-\ntion deeply related to the K-C complexity. This work was later known un-\nder several titles: universal distribution, algorithmic probability, universal\ninference, among others.5,6 This algorithmic probability is the probabil-\nity m(s) that a universal Turing machine U produces the string s when\nprovided with an arbitrary input tape. m(s) can be used as a universal\nsequence predictor that outperforms (in a certain sense) all other predic-\ntors.5 It is easy to see that this distribution is strongly related to the K-C\ncomplexity and that once m(s) is determined so is K(s) since the formula\nm(s) can be written in terms of K as follows m(s) ≈1/2K(s). The distri-\nbution of m(s) predicts that non-random looking strings will appear much\nmore often as the result of a uniform random process, which in our ex-\nperiment is equivalent to running all possible Turing machines and cellular\nautomata of certain small classes according to an acceptable enumeration.\nBy these means, we claim that it might be possible to overcome the problem\nof defining and measuring the K-C complexity of short sequences. Our pro-\nposal consists of measuring the K-C complexity by reconstructing it from\nscratch basically approximating the algorithmic probability of strings to ap-\nproximate the K-C complexity. Particular simple strings are produced with\nhigher probability (i.e. more often produced by the process we will describe\nbelow) than particular complex strings, so they have lower complexity.\nOur experiment proceeded as follows: We took the Turing machine\n(T M) and cellular automata enumerations defined by Stephen Wolfram.11\nWe let run (a) all 2−state 2−symbol Turing machines, and (b) a statistical\nsample of the 3−state 2−symbol ones, both henceforth denoted as T M(2, 2)\nand T M(3, 2).\nThen we examine the frequency distribution of these machines’ outputs\nperforming experiments modifying several parameters: the number of steps,\nthe length of strings, pseudo-random vs. regular inputs, and the sampling\nsizes.\nFor (a) it turns out that there are 4096 different Turing machines accord-\ning to the formula (2sk)sk derived from the traditional 5−tuplet description\nof a Turing machine: d(s{1,2}, k{1,2}) →(s{1,2}, k{1,2}, {1, −1}) where s{1,2}\nare the two possible states, k{1,2} are the two possible symbols and the last\nentry {1,-1} denotes the movement of the head either to the right or to the\n\nOctober 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\nOn the Kolmogorov-Chaitin Complexity for short sequences\n3\nleft. From the same formula it follows that for (b) there are 2985984 so\nwe proceeded by statistical methods taking representative samples of size\n5000, 10000, 20000 and 100000 Turing machines uniformly distributed over\nT M(3, 2). We then let them run 30, 100 and 500 steps each and we pro-\nceeded to feed each one with (1) a (pseudo) random (one per TM) input\nand (2) with a regular input.\nWe proceeded in the same fashion for all one dimensional binary cellular\nautomata (CA), those (1) which their rule depends only on the left and right\nneighbors and those considering two left and one right neighbor, henceforth\ndenoted by CA(t, c)‡ where t and c are the neighbor cells in question, to\nthe left and to the right respectively. These CA were fed with a single 1\nsurrounded by 0s. There are 256 one dimensional nearest-neighbor cellular\nautomata or CA(1, 1), also called Elementary Cellular Automata11) and\n65536 CA(2, 1).\nTo determine the output of the Turing machines we look at the string\nconsisting of all parts of the tape reached by the head. We then partition\nthe output in substrings of length k. For instance, if k=3 and the Turing\nmachine head reached positions 1, 2, 3, 4 and 5 and the tape contains the\nsymbols {0,0,0,1,1} then we increment the counter of the substrings 000,\n001, 011 by one each one. Similar for CA using the ”light cone” of all\npositions reachable from the initial 1 in the time run. Then we perform\nthe above for (1) each different T M and (2) each different CA, giving two\ndistributions over strings of a given length k.\nWe then looked at the frequency distribution of the outputs of both\nclasses T M and CA§, (including ECA) performing experiments modifying\nseveral parameters: the number of steps, the length of strings, (pseudo)\nrandom vs. regular inputs, and the sampling sizes.\nAn important result is that the frequency distribution was very stable\nunder the several variations described above allowing to define a natural\ndistribution m(s) particularly for the top it. We claim that the bottom\nof the distribution, and therefore all of it, will tend to stabilize by taking\nbigger samples. By analyzing the following diagram it can be deduced that\nthe output frequency distribution of each of the independent systems of\n‡A better notation is the 3 −tuplet CA(t, c, j) with j indicating the number of symbols,\nbut because we are only considering 2 −symbol cellular automata we can take it for\ngranted and avoid that complication.\n§Both enumeration schemes are implemented in Mathematica calling the functions\nCelullarAutomaton and TuringMachine, the latter implemented in Mathematica version\n6.0\n\nOctober 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\n4\nJean-Paul Delahaye and Hector Zenil\ncomputation (TM and CA) follow an output frequency distribution. We\nconjecture that these systems of computation and others of equivalent com-\nputational power converge toward a single distribution when bigger samples\nare taken by allowing a greater number of steps and/or bigger classes con-\ntaining more and increasingly sophisticated computational devices. Such\ndistributions should then match the value of m(s) and therefore K(s) by\nmeans of the convergence of what we call their experimental counterparts\nme(s) and Ke(s). If our method succeeds as we claim it could be possible\nto give a stable definition of the K-C complexity for short sequences inde-\npendent of any constant.\nFig. 1.1.\nThe above diagram shows the convergence of the frequency distributions of\nthe outputs of TM and ECA for k = 4. Matching strings are linked by a line. As one\ncan observe, in spite of certain crossings, TM and ECA are strongly correlated and both\nsuccessfully group equivalent output strings. By taking the six groups — marked with\nbrackets — the distribution frequencies only differ by one.\nBy instance, the strings 0101 and 1010 were grouped in second place,\ntherefore they are the second most complex group after the group com-\nposed by the strings of a sequence of zeros or ones but before all the other\n\nOctober 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\nOn the Kolmogorov-Chaitin Complexity for short sequences\n5\n2n strings.\nAnd that is what one would expect since it has a very low\nK-C complexity as prefix of a highly compressible string 0101 . . .. In fa-\nvor of our claims about the nature of these distributions as following m(s)\nand then approaching K(s), notice that all strings were correctly grouped\nwith their equivalent category of complexity under the three possible oper-\nations/symmetries preserving their K-C complexity, namely reversion (sy),\ncomplementation (co) and composition of the two (syco). This also sup-\nports our claim that our procedure is working correctly since it groups all\nstrings by their complexity class.\nThe fact that the method groups all\nthe strings by their complexity category allowed us to apply a well-known\nlemma used in group theory to enumerate actual different cases, which let\nus present a single representative string for each complexity category. So\ninstead of presenting a distribution with 1024 strings of length 10 it allows\nus to compress it to 272 strings.\nWe have also found that the frequency distribution from several real-\nworld data sources also approximates the same distribution, suggesting\nthat they probably come from the same kind of computation, supporting\ncontemporary claims about nature as performing computations.8,11 The\npaper available online contains more detailed results for strings of length\nk = 4, 5, 6, 10 as well as two metrics for measuring the convergence of\nT M(2, 2) and ECA(1, 1) and the real-world data frequency distributions\nextracted from several sources¶. A paper with mathematical formulations\nand further precise conjectures is currently in preparation.\nReferences\n1. G.J. Chaitin, Algorithmic Information Theory, Cambridge University Press,\n1987.\n2. G.J. Chaitin, Information, Randomness and Incompleteness, World Scien-\ntific, 1987.\n3. G.J. Chaitin, Meta-Math! The Quest for Omega, Pantheon Books NY, 2005.\n4. C.S. Calude, Information and Randomness:\nAn Algorithmic Perspective\n(Texts in Theoretical Computer Science. An EATCS Series), Springer; 2nd.\nedition, 2002.\n5. Kirchherr, W., M. Li, and P. Vit ́anyi. The miraculous universal distribution.\nMath. Intelligencer 19(4), 7-15, 1997.\n6. M. Li and P. Vit ́anyi, An Introduction to Kolmogorov Complexity and Its\nApplications, Springer, 1997.\n¶It can be reached at arXiv: http://arxiv.org/abs/0704.1043.\nA\nwebsite\nwith\nthe\ncomplete\nresults\nof\nthe\nwhole\nexperiment\nis\navailable\nat\nhttp://www.mathrix.org/experimentalAIT/\n\nOctober 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\n6\nJean-Paul Delahaye and Hector Zenil\n7. A.K.Zvonkin, L. A. Levin. ”The Complexity of finite objects and the Algo-\nrithmic Concepts of Information and Randomness.”, UMN = Russian Math.\nSurveys, 25(6):83-124, 1970.\n8. S. Lloyd, Programming the Universe, Knopf, 2006.\n9. R. Solomonoff, The Discovery of Algorithmic Probability, Journal of Com-\nputer and System Sciences, Vol. 55, No. 1, pp. 73-88, August 1997.\n10. R. Solomonoff, A Preliminary Report on a General Theory of Inductive In-\nference, (Revision of Report V-131), Contract AF 49(639)-376, Report ZTB-\n138, Zator Co., Cambridge, Mass., Nov, 1960\n11. S. Wolfram, A New Kind of Science, Wolfram Media, 2002.","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0704.1043v5 [cs.CC] 17 Dec 2010\nOctober 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\nChapter 1\nOn the Kolmogorov-Chaitin Complexity for short\nsequences\nJean-Paul Delahaye∗and Hector Zenil†\nLaboratoire d’Informatique Fondamentale de Lille\nCentre National de la Recherche Scientifique (CNRS)\nUniversit ́e des Sciences et Technologies de Lille\nAmong the several new ideas and contributions made by Gregory\nChaitin to mathematics is his strong belief that mathematicians should\ntranscend the millenary theorem-proof paradigm in favor of a quasi-\nempirical method based on current and unprecedented access to compu-\ntational resources.3 In accordance with that dictum, we present in this pa-\nper an experimental approach for defining and measuring the Kolmogorov-\nChaitin complexity, a problem which is known to be quite challenging for\nshort sequences — shorter for example than typical compiler lengths.\nThe Kolmogorov-Chaitin complexity (or algorithmic complexity) of a\nstring s is defined as the length of its shortest description p on a universal\nTuring machine U, formally K(s) = min{l(p) : U(p) = s}. The major\ndrawback of K, as measure, is its uncomputability. So in practical applica-\ntions it must always be approximated by compression algorithms. A string\nis uncompressible if its shorter description is the original string itself. If a\nstring is uncompressible it is said that the string is random since no pat-\nterns were found. Among the 2n different strings of length n, it is easy to\ndeduce by a combinatoric argument that one of them will be completely\nrandom simply because there will be no enough shorter strings so most of\nthem will have a maximal K-C complexity. Therefore many of them will\nremain equal or very close to their original size after the compression. Most\nof them will be therefore random. An important property of K is that it is\nnearly independent of the choice of U. However, when the strings are short\nin length, the dependence of K on a particular universal Turing machine U\n∗delahaye@lifl.fr\n†hector.zenil-chavez@malix.univ-paris1.fr\n1"},{"paragraph_id":"p2","order":2,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\n2\nJean-Paul Delahaye and Hector Zenil\nis higher producing arbitrary results. In this paper we will suggest an em-\npirical approach to overcome this difficulty and to obtain a stable definition\nof the K-C complexity for short sequences.\nUsing Turing’s model of universal computation, Ray Solomonoff9,10\nand Leonid Levin7 developed a theory about a universal prior distribu-\ntion deeply related to the K-C complexity. This work was later known un-\nder several titles: universal distribution, algorithmic probability, universal\ninference, among others.5,6 This algorithmic probability is the probabil-\nity m(s) that a universal Turing machine U produces the string s when\nprovided with an arbitrary input tape. m(s) can be used as a universal\nsequence predictor that outperforms (in a certain sense) all other predic-\ntors.5 It is easy to see that this distribution is strongly related to the K-C\ncomplexity and that once m(s) is determined so is K(s) since the formula\nm(s) can be written in terms of K as follows m(s) ≈1/2K(s). The distri-\nbution of m(s) predicts that non-random looking strings will appear much\nmore often as the result of a uniform random process, which in our ex-\nperiment is equivalent to running all possible Turing machines and cellular\nautomata of certain small classes according to an acceptable enumeration.\nBy these means, we claim that it might be possible to overcome the problem\nof defining and measuring the K-C complexity of short sequences. Our pro-\nposal consists of measuring the K-C complexity by reconstructing it from\nscratch basically approximating the algorithmic probability of strings to ap-\nproximate the K-C complexity. Particular simple strings are produced with\nhigher probability (i.e. more often produced by the process we will describe\nbelow) than particular complex strings, so they have lower complexity.\nOur experiment proceeded as follows: We took the Turing machine\n(T M) and cellular automata enumerations defined by Stephen Wolfram.11\nWe let run (a) all 2−state 2−symbol Turing machines, and (b) a statistical\nsample of the 3−state 2−symbol ones, both henceforth denoted as T M(2, 2)\nand T M(3, 2).\nThen we examine the frequency distribution of these machines’ outputs\nperforming experiments modifying several parameters: the number of steps,\nthe length of strings, pseudo-random vs. regular inputs, and the sampling\nsizes.\nFor (a) it turns out that there are 4096 different Turing machines accord-\ning to the formula (2sk)sk derived from the traditional 5−tuplet description\nof a Turing machine: d(s{1,2}, k{1,2}) →(s{1,2}, k{1,2}, {1, −1}) where s{1,2}\nare the two possible states, k{1,2} are the two possible symbols and the last\nentry {1,-1} denotes the movement of the head either to the right or to the"},{"paragraph_id":"p3","order":3,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\nOn the Kolmogorov-Chaitin Complexity for short sequences\n3\nleft. From the same formula it follows that for (b) there are 2985984 so\nwe proceeded by statistical methods taking representative samples of size\n5000, 10000, 20000 and 100000 Turing machines uniformly distributed over\nT M(3, 2). We then let them run 30, 100 and 500 steps each and we pro-\nceeded to feed each one with (1) a (pseudo) random (one per TM) input\nand (2) with a regular input.\nWe proceeded in the same fashion for all one dimensional binary cellular\nautomata (CA), those (1) which their rule depends only on the left and right\nneighbors and those considering two left and one right neighbor, henceforth\ndenoted by CA(t, c)‡ where t and c are the neighbor cells in question, to\nthe left and to the right respectively. These CA were fed with a single 1\nsurrounded by 0s. There are 256 one dimensional nearest-neighbor cellular\nautomata or CA(1, 1), also called Elementary Cellular Automata11) and\n65536 CA(2, 1).\nTo determine the output of the Turing machines we look at the string\nconsisting of all parts of the tape reached by the head. We then partition\nthe output in substrings of length k. For instance, if k=3 and the Turing\nmachine head reached positions 1, 2, 3, 4 and 5 and the tape contains the\nsymbols {0,0,0,1,1} then we increment the counter of the substrings 000,\n001, 011 by one each one. Similar for CA using the ”light cone” of all\npositions reachable from the initial 1 in the time run. Then we perform\nthe above for (1) each different T M and (2) each different CA, giving two\ndistributions over strings of a given length k.\nWe then looked at the frequency distribution of the outputs of both\nclasses T M and CA§, (including ECA) performing experiments modifying\nseveral parameters: the number of steps, the length of strings, (pseudo)\nrandom vs. regular inputs, and the sampling sizes.\nAn important result is that the frequency distribution was very stable\nunder the several variations described above allowing to define a natural\ndistribution m(s) particularly for the top it. We claim that the bottom\nof the distribution, and therefore all of it, will tend to stabilize by taking\nbigger samples. By analyzing the following diagram it can be deduced that\nthe output frequency distribution of each of the independent systems of\n‡A better notation is the 3 −tuplet CA(t, c, j) with j indicating the number of symbols,\nbut because we are only considering 2 −symbol cellular automata we can take it for\ngranted and avoid that complication.\n§Both enumeration schemes are implemented in Mathematica calling the functions\nCelullarAutomaton and TuringMachine, the latter implemented in Mathematica version\n6.0"},{"paragraph_id":"p4","order":4,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\n4\nJean-Paul Delahaye and Hector Zenil\ncomputation (TM and CA) follow an output frequency distribution. We\nconjecture that these systems of computation and others of equivalent com-\nputational power converge toward a single distribution when bigger samples\nare taken by allowing a greater number of steps and/or bigger classes con-\ntaining more and increasingly sophisticated computational devices. Such\ndistributions should then match the value of m(s) and therefore K(s) by\nmeans of the convergence of what we call their experimental counterparts\nme(s) and Ke(s). If our method succeeds as we claim it could be possible\nto give a stable definition of the K-C complexity for short sequences inde-\npendent of any constant.\nFig. 1.1.\nThe above diagram shows the convergence of the frequency distributions of\nthe outputs of TM and ECA for k = 4. Matching strings are linked by a line. As one\ncan observe, in spite of certain crossings, TM and ECA are strongly correlated and both\nsuccessfully group equivalent output strings. By taking the six groups — marked with\nbrackets — the distribution frequencies only differ by one.\nBy instance, the strings 0101 and 1010 were grouped in second place,\ntherefore they are the second most complex group after the group com-\nposed by the strings of a sequence of zeros or ones but before all the other"},{"paragraph_id":"p5","order":5,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\nOn the Kolmogorov-Chaitin Complexity for short sequences\n5\n2n strings.\nAnd that is what one would expect since it has a very low\nK-C complexity as prefix of a highly compressible string 0101 . . .. In fa-\nvor of our claims about the nature of these distributions as following m(s)\nand then approaching K(s), notice that all strings were correctly grouped\nwith their equivalent category of complexity under the three possible oper-\nations/symmetries preserving their K-C complexity, namely reversion (sy),\ncomplementation (co) and composition of the two (syco). This also sup-\nports our claim that our procedure is working correctly since it groups all\nstrings by their complexity class.\nThe fact that the method groups all\nthe strings by their complexity category allowed us to apply a well-known\nlemma used in group theory to enumerate actual different cases, which let\nus present a single representative string for each complexity category. So\ninstead of presenting a distribution with 1024 strings of length 10 it allows\nus to compress it to 272 strings.\nWe have also found that the frequency distribution from several real-\nworld data sources also approximates the same distribution, suggesting\nthat they probably come from the same kind of computation, supporting\ncontemporary claims about nature as performing computations.8,11 The\npaper available online contains more detailed results for strings of length\nk = 4, 5, 6, 10 as well as two metrics for measuring the convergence of\nT M(2, 2) and ECA(1, 1) and the real-world data frequency distributions\nextracted from several sources¶. A paper with mathematical formulations\nand further precise conjectures is currently in preparation.\nReferences\n1. G.J. Chaitin, Algorithmic Information Theory, Cambridge University Press,\n1987.\n2. G.J. Chaitin, Information, Randomness and Incompleteness, World Scien-\ntific, 1987.\n3. G.J. Chaitin, Meta-Math! The Quest for Omega, Pantheon Books NY, 2005.\n4. C.S. Calude, Information and Randomness:\nAn Algorithmic Perspective\n(Texts in Theoretical Computer Science. An EATCS Series), Springer; 2nd.\nedition, 2002.\n5. Kirchherr, W., M. Li, and P. Vit ́anyi. The miraculous universal distribution.\nMath. Intelligencer 19(4), 7-15, 1997.\n6. M. Li and P. Vit ́anyi, An Introduction to Kolmogorov Complexity and Its\nApplications, Springer, 1997.\n¶It can be reached at arXiv: http://arxiv.org/abs/0704.1043.\nA\nwebsite\nwith\nthe\ncomplete\nresults\nof\nthe\nwhole\nexperiment\nis\navailable\nat\nhttp://www.mathrix.org/experimentalAIT/"},{"paragraph_id":"p6","order":6,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\n6\nJean-Paul Delahaye and Hector Zenil\n7. A.K.Zvonkin, L. A. Levin. ”The Complexity of finite objects and the Algo-\nrithmic Concepts of Information and Randomness.”, UMN = Russian Math.\nSurveys, 25(6):83-124, 1970.\n8. S. Lloyd, Programming the Universe, Knopf, 2006.\n9. R. Solomonoff, The Discovery of Algorithmic Probability, Journal of Com-\nputer and System Sciences, Vol. 55, No. 1, pp. 73-88, August 1997.\n10. R. Solomonoff, A Preliminary Report on a General Theory of Inductive In-\nference, (Revision of Report V-131), Contract AF 49(639)-376, Report ZTB-\n138, Zator Co., Cambridge, Mass., Nov, 1960\n11. S. Wolfram, A New Kind of Science, Wolfram Media, 2002."}],"pages":[{"page":1,"text":"arXiv:0704.1043v5 [cs.CC] 17 Dec 2010\nOctober 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\nChapter 1\nOn the Kolmogorov-Chaitin Complexity for short\nsequences\nJean-Paul Delahaye∗and Hector Zenil†\nLaboratoire d’Informatique Fondamentale de Lille\nCentre National de la Recherche Scientifique (CNRS)\nUniversit ́e des Sciences et Technologies de Lille\nAmong the several new ideas and contributions made by Gregory\nChaitin to mathematics is his strong belief that mathematicians should\ntranscend the millenary theorem-proof paradigm in favor of a quasi-\nempirical method based on current and unprecedented access to compu-\ntational resources.3 In accordance with that dictum, we present in this pa-\nper an experimental approach for defining and measuring the Kolmogorov-\nChaitin complexity, a problem which is known to be quite challenging for\nshort sequences — shorter for example than typical compiler lengths.\nThe Kolmogorov-Chaitin complexity (or algorithmic complexity) of a\nstring s is defined as the length of its shortest description p on a universal\nTuring machine U, formally K(s) = min{l(p) : U(p) = s}. The major\ndrawback of K, as measure, is its uncomputability. So in practical applica-\ntions it must always be approximated by compression algorithms. A string\nis uncompressible if its shorter description is the original string itself. If a\nstring is uncompressible it is said that the string is random since no pat-\nterns were found. Among the 2n different strings of length n, it is easy to\ndeduce by a combinatoric argument that one of them will be completely\nrandom simply because there will be no enough shorter strings so most of\nthem will have a maximal K-C complexity. Therefore many of them will\nremain equal or very close to their original size after the compression. Most\nof them will be therefore random. An important property of K is that it is\nnearly independent of the choice of U. However, when the strings are short\nin length, the dependence of K on a particular universal Turing machine U\n∗delahaye@lifl.fr\n†hector.zenil-chavez@malix.univ-paris1.fr\n1"},{"page":2,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\n2\nJean-Paul Delahaye and Hector Zenil\nis higher producing arbitrary results. In this paper we will suggest an em-\npirical approach to overcome this difficulty and to obtain a stable definition\nof the K-C complexity for short sequences.\nUsing Turing’s model of universal computation, Ray Solomonoff9,10\nand Leonid Levin7 developed a theory about a universal prior distribu-\ntion deeply related to the K-C complexity. This work was later known un-\nder several titles: universal distribution, algorithmic probability, universal\ninference, among others.5,6 This algorithmic probability is the probabil-\nity m(s) that a universal Turing machine U produces the string s when\nprovided with an arbitrary input tape. m(s) can be used as a universal\nsequence predictor that outperforms (in a certain sense) all other predic-\ntors.5 It is easy to see that this distribution is strongly related to the K-C\ncomplexity and that once m(s) is determined so is K(s) since the formula\nm(s) can be written in terms of K as follows m(s) ≈1/2K(s). The distri-\nbution of m(s) predicts that non-random looking strings will appear much\nmore often as the result of a uniform random process, which in our ex-\nperiment is equivalent to running all possible Turing machines and cellular\nautomata of certain small classes according to an acceptable enumeration.\nBy these means, we claim that it might be possible to overcome the problem\nof defining and measuring the K-C complexity of short sequences. Our pro-\nposal consists of measuring the K-C complexity by reconstructing it from\nscratch basically approximating the algorithmic probability of strings to ap-\nproximate the K-C complexity. Particular simple strings are produced with\nhigher probability (i.e. more often produced by the process we will describe\nbelow) than particular complex strings, so they have lower complexity.\nOur experiment proceeded as follows: We took the Turing machine\n(T M) and cellular automata enumerations defined by Stephen Wolfram.11\nWe let run (a) all 2−state 2−symbol Turing machines, and (b) a statistical\nsample of the 3−state 2−symbol ones, both henceforth denoted as T M(2, 2)\nand T M(3, 2).\nThen we examine the frequency distribution of these machines’ outputs\nperforming experiments modifying several parameters: the number of steps,\nthe length of strings, pseudo-random vs. regular inputs, and the sampling\nsizes.\nFor (a) it turns out that there are 4096 different Turing machines accord-\ning to the formula (2sk)sk derived from the traditional 5−tuplet description\nof a Turing machine: d(s{1,2}, k{1,2}) →(s{1,2}, k{1,2}, {1, −1}) where s{1,2}\nare the two possible states, k{1,2} are the two possible symbols and the last\nentry {1,-1} denotes the movement of the head either to the right or to the"},{"page":3,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\nOn the Kolmogorov-Chaitin Complexity for short sequences\n3\nleft. From the same formula it follows that for (b) there are 2985984 so\nwe proceeded by statistical methods taking representative samples of size\n5000, 10000, 20000 and 100000 Turing machines uniformly distributed over\nT M(3, 2). We then let them run 30, 100 and 500 steps each and we pro-\nceeded to feed each one with (1) a (pseudo) random (one per TM) input\nand (2) with a regular input.\nWe proceeded in the same fashion for all one dimensional binary cellular\nautomata (CA), those (1) which their rule depends only on the left and right\nneighbors and those considering two left and one right neighbor, henceforth\ndenoted by CA(t, c)‡ where t and c are the neighbor cells in question, to\nthe left and to the right respectively. These CA were fed with a single 1\nsurrounded by 0s. There are 256 one dimensional nearest-neighbor cellular\nautomata or CA(1, 1), also called Elementary Cellular Automata11) and\n65536 CA(2, 1).\nTo determine the output of the Turing machines we look at the string\nconsisting of all parts of the tape reached by the head. We then partition\nthe output in substrings of length k. For instance, if k=3 and the Turing\nmachine head reached positions 1, 2, 3, 4 and 5 and the tape contains the\nsymbols {0,0,0,1,1} then we increment the counter of the substrings 000,\n001, 011 by one each one. Similar for CA using the ”light cone” of all\npositions reachable from the initial 1 in the time run. Then we perform\nthe above for (1) each different T M and (2) each different CA, giving two\ndistributions over strings of a given length k.\nWe then looked at the frequency distribution of the outputs of both\nclasses T M and CA§, (including ECA) performing experiments modifying\nseveral parameters: the number of steps, the length of strings, (pseudo)\nrandom vs. regular inputs, and the sampling sizes.\nAn important result is that the frequency distribution was very stable\nunder the several variations described above allowing to define a natural\ndistribution m(s) particularly for the top it. We claim that the bottom\nof the distribution, and therefore all of it, will tend to stabilize by taking\nbigger samples. By analyzing the following diagram it can be deduced that\nthe output frequency distribution of each of the independent systems of\n‡A better notation is the 3 −tuplet CA(t, c, j) with j indicating the number of symbols,\nbut because we are only considering 2 −symbol cellular automata we can take it for\ngranted and avoid that complication.\n§Both enumeration schemes are implemented in Mathematica calling the functions\nCelullarAutomaton and TuringMachine, the latter implemented in Mathematica version\n6.0"},{"page":4,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\n4\nJean-Paul Delahaye and Hector Zenil\ncomputation (TM and CA) follow an output frequency distribution. We\nconjecture that these systems of computation and others of equivalent com-\nputational power converge toward a single distribution when bigger samples\nare taken by allowing a greater number of steps and/or bigger classes con-\ntaining more and increasingly sophisticated computational devices. Such\ndistributions should then match the value of m(s) and therefore K(s) by\nmeans of the convergence of what we call their experimental counterparts\nme(s) and Ke(s). If our method succeeds as we claim it could be possible\nto give a stable definition of the K-C complexity for short sequences inde-\npendent of any constant.\nFig. 1.1.\nThe above diagram shows the convergence of the frequency distributions of\nthe outputs of TM and ECA for k = 4. Matching strings are linked by a line. As one\ncan observe, in spite of certain crossings, TM and ECA are strongly correlated and both\nsuccessfully group equivalent output strings. By taking the six groups — marked with\nbrackets — the distribution frequencies only differ by one.\nBy instance, the strings 0101 and 1010 were grouped in second place,\ntherefore they are the second most complex group after the group com-\nposed by the strings of a sequence of zeros or ones but before all the other"},{"page":5,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\nOn the Kolmogorov-Chaitin Complexity for short sequences\n5\n2n strings.\nAnd that is what one would expect since it has a very low\nK-C complexity as prefix of a highly compressible string 0101 . . .. In fa-\nvor of our claims about the nature of these distributions as following m(s)\nand then approaching K(s), notice that all strings were correctly grouped\nwith their equivalent category of complexity under the three possible oper-\nations/symmetries preserving their K-C complexity, namely reversion (sy),\ncomplementation (co) and composition of the two (syco). This also sup-\nports our claim that our procedure is working correctly since it groups all\nstrings by their complexity class.\nThe fact that the method groups all\nthe strings by their complexity category allowed us to apply a well-known\nlemma used in group theory to enumerate actual different cases, which let\nus present a single representative string for each complexity category. So\ninstead of presenting a distribution with 1024 strings of length 10 it allows\nus to compress it to 272 strings.\nWe have also found that the frequency distribution from several real-\nworld data sources also approximates the same distribution, suggesting\nthat they probably come from the same kind of computation, supporting\ncontemporary claims about nature as performing computations.8,11 The\npaper available online contains more detailed results for strings of length\nk = 4, 5, 6, 10 as well as two metrics for measuring the convergence of\nT M(2, 2) and ECA(1, 1) and the real-world data frequency distributions\nextracted from several sources¶. A paper with mathematical formulations\nand further precise conjectures is currently in preparation.\nReferences\n1. G.J. Chaitin, Algorithmic Information Theory, Cambridge University Press,\n1987.\n2. G.J. Chaitin, Information, Randomness and Incompleteness, World Scien-\ntific, 1987.\n3. G.J. Chaitin, Meta-Math! The Quest for Omega, Pantheon Books NY, 2005.\n4. C.S. Calude, Information and Randomness:\nAn Algorithmic Perspective\n(Texts in Theoretical Computer Science. An EATCS Series), Springer; 2nd.\nedition, 2002.\n5. Kirchherr, W., M. Li, and P. Vit ́anyi. The miraculous universal distribution.\nMath. Intelligencer 19(4), 7-15, 1997.\n6. M. Li and P. Vit ́anyi, An Introduction to Kolmogorov Complexity and Its\nApplications, Springer, 1997.\n¶It can be reached at arXiv: http://arxiv.org/abs/0704.1043.\nA\nwebsite\nwith\nthe\ncomplete\nresults\nof\nthe\nwhole\nexperiment\nis\navailable\nat\nhttp://www.mathrix.org/experimentalAIT/"},{"page":6,"text":"October 23, 2018\n8:33\nWorld Scientific Review Volume - 9in x 6in\nLongAbstractDelahaye2\n6\nJean-Paul Delahaye and Hector Zenil\n7. A.K.Zvonkin, L. A. Levin. ”The Complexity of finite objects and the Algo-\nrithmic Concepts of Information and Randomness.”, UMN = Russian Math.\nSurveys, 25(6):83-124, 1970.\n8. S. Lloyd, Programming the Universe, Knopf, 2006.\n9. R. Solomonoff, The Discovery of Algorithmic Probability, Journal of Com-\nputer and System Sciences, Vol. 55, No. 1, pp. 73-88, August 1997.\n10. R. Solomonoff, A Preliminary Report on a General Theory of Inductive In-\nference, (Revision of Report V-131), Contract AF 49(639)-376, Report ZTB-\n138, Zator Co., Cambridge, Mass., Nov, 1960\n11. S. Wolfram, A New Kind of Science, Wolfram Media, 2002."}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"Turing machine U, formally K(s) = min{l(p) : U(p) = s}. The major","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"the output in substrings of length k. 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