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exercise in the previous class Encode ABACABADACABAD by the LZ77 algorithm, and decode its result. back 4 positions copy 3 symbols A B AC back 2 positions copy 1 symbol ABAD ACABAD (0, 0, A), (0, 0, B), (2, 1, C), (4, 3, D), (6, 6, *) back 6 positions copy 6 symbols back 4 positions copy 3 symbols A B AC back 2 positions copy 1 symbol ABAD ACABAD back 6 positions copy 6 symbols 1 exercise in the previous class Survey what has happened concerning LZW algorithm. UNISYS had the patent. The patent was granted free of charge for non-commercial use. Many people used LZW, for example in GIF format. in 1990s, the Web was born, and GIF can be a “business”: UNISYS changed the policy; everybody needs to pay. Much confusion in late 90’s... today: The patent has been expired. 2 today’s class think of “uncertainty” outside of the Information Theory “randomness” random numbers, pseudo-random numbers (乱数,疑似乱数) Kolmogorov complexity statistical (統計的) test of pseudo-random numbers 2-test (chi-square test) algorithms for pseudo-random number generation 3 random numbers A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities (Wikipedia). recognizable? regularities? approach from statistics ( mid part of today’s talk) approach from computation ( first part of today’s talk) 4 approach from computation For a finite sequence x, let (x) denote the set of programs which outputs x. program...deterministic, no input, written as a sequence “#include <stdio.h>; main(){printf(“hello”);}” (“hello”) |p|...the size of the program p (in bytes, lines, etc.) The Kolmogorov complexity (コルモゴロフ複雑さ) of x is 𝐾 𝑥 = min |𝑝| . 𝑝∈Π(𝑥) (the size of the shortest program which outputs x) 5 example (1) x1 = “0101010101010101010101010101010101010101”, 40chars. (x1) contains; program p1 : printf(“010101...01”); ...51chars. program p2 : for(i=0;i<20;i++)printf(“01”); ...30chars. K(x1) 30 < 40 x2 = “0110100010101101001011010110100100100010”, 40chars. (x2) contains; program p3 : printf(“011010...10”); ...51chars. K(x2) 51 6 example (2) x3 = “11235813213455891442333776109871597...”, million chars. (x3) contains; program p4 : printf(“1123...”); ... million + 11chars. program p5 : compute & print the Fibonacci sequence ... hundreds chars. 𝑎1 = 𝑎2 = 1 𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2 K(x3) is about hundreds characters or less The Kolmogorov complexity K(x): measure of the difficultness to construct the sequence x 7 the relation to entropy TWO measures of uncertainty entropy measure of uncertainty respect to the statistical property contributes to measure information Kolmogorov complexity measure of uncertainty respect to the mechanistic property contributes to mathematical discussion 8 randomness, from the viewpoint of Kolmogorov A sequence x is random (in the sense of Kolmogorov) if K(x) ≥ |x|. “to write down x, write x down directly” “there is no alternative way (他の手段) to write x” “x does not have more compact representation than itself” theorem: There exists a random sequence. before go to the proof, we assume that... sequences (programs) are represented over {0, 1}. the set of sequences of length n is written by Vn. 9 the proof theorem: There exists a random sequence. proof (by contradiction, 背理法) Assume that there is no random sequence of length n, then... for each xVn, there is a program px with px (x) and |px|<n. |V n|=2n |V 1|+|V 2|+ ... +|V n–1| = 2n – 1 contradiction, because (# of programs) < (# of sequences) V1 V n–2 2n – 1 sequence V n–1 px Vn x 2n sequence 10 can we have a random sequence? There are random sequences, but how can we have them? approach 1: make use of physical phenomena toss coins, catch thermal noise, wait for quantum events... we have “true” random sequences (真性乱数) “expensive” http://www.fdk.co.jp/ approach 2: construct a sequence using a certain procedure use equations or computer program cheap and efficient not “true” random, but pseudo-random (疑似乱数) 11 pseudo-random sequence (numbers) pseudo-random sequence (numbers): a sequence of numbers generated by a deterministic rule (決定性の規則) looks like random, but not “random in Kolmogorov’s sense” easily constructible by computer programs Before discussing the algorithm, we should learn how to evaluate the randomness. 12 criteria of randomness there are two criteria to evaluate the randomness unpredictability how difficult is it to predict the “next” symbol? important in cryptography and games statistical bias is there any anomalous (特異な) bias in the sequence? sufficient for many applications In general, anomalous bias help prediction... unpredictability is more favorable, but difficult to obtain... we discuss statistical tests in this class. 13 2-test: idea 2-test (chi-square test, カイ2乗検定) one of the most basic statistical tests evaluate the distance between “a given sequence” and “a typical (ideal) sequence” sketch of idea: generate a sequence by rolling a dice s1 = 1625341625163412 random like s2 = 1115121121131116 NOT random like, because the number of “1” seems too many ideal source s expected distribution of the number of 1 in s s1 s2 14 2-test: definition prepararation (準備)... partition the set of possible symbols to classes C1, ..., Ck pi: probability that a symbol in Ci is generated from ideal source in the dice roll, C1 = {1}, ..., C6={6}, pi = 1/6, for example You are given a sequence s of length n... ni: the number of symbols in Ci occuring in the sequence s the 2 (chi-square) value of s: 𝑘 𝜒2 = 𝑖=1 𝑛𝑖 − 𝑛𝑝𝑖 𝑛𝑝𝑖 2 15 what is this value? 𝑘 𝜒2 = 𝑖=1 𝑛𝑖 − 𝑛𝑝𝑖 𝑛𝑝𝑖 2 𝑛𝑝𝑖 ...the expected number of symbols in Ci 𝑛𝑖 ...the observed number of symbols in Ci If the sequence s is a typical output of an ideal source... 𝑛𝑖 → 𝑛𝑝𝑖 : the numerator (分子) → 0 the 2-value → 0 The 2-value is the “distance” (not strict sense) of the given sequence to ideal sequences. smaller 2-value more close to the ideal output 16 example consider sequences s1 and s2 over {1, ..., 6} of length n = 42: if “ideal source = fair dice”, then pi = 1/6 ⇒ npi = 7 s1 = 145325415432115341662126421535631153154363 n1 = 10, n2 = 5, n3 = 8, n4 = 6, n5 =8, n6 = 5 2 = 32/7 + 22/7 + 12/7 + 12/7 + 12/7 + 22/7 = 20/7 s2 = 112111421115331111544111544111134411151114 n1 = 25, n2 = 2, n3 = 3, n4 = 8, n5 =4, n6 = 0 2 = 182/7 + 52/7 + 42/7 + 12/7 + 32/7 + 72/7 = 424/7 s1 is closer to the ideal output (true random) than s2 17 example (cnt’d) s3 = 111111111111222222...666666 (assume length n = 72) n1 = 12, n2 = 12, n3 = 12, n4 = 12, n5 =12, n6 = 12 2 = 02/12 + 02/12 + 02/12 + 02/12 + 02/12 + 02/12 = 0 Is s3 random? NEVER! consider a block of length 2...n = 36 blocks ideal: “11”, ..., “66” occur with probability 1/36, npi = 1 n11 = 6, n12 = 0, ..., n22 = 6, ... 2 = (6 – 1)2/1 + (0 – 1)2/1 + ... = 180 ⇒ large! lesson learned : use as many different class partitions as possible 18 small 2-values good? Small 2-value is good, but the discussion is not so simple... It is “rare” that the 2-value of the “real dice roll” becomes 0. a sequence of n = 60000, n1 = ...= n6 = 10000 exactly too good, rather strange We need to know the distribution of 2-values of an ideal source. Theorem: If there are k classes C1, ..., Ck, then 2-values of an ideal source obey the 2-distribution of degree k – 1. O degree 4 degree 6 degree 2 2 19 interpretation of 2-values s3 = 111111111111222222...666666 (n = 72) 2 = 02/12 + 02/12 + 02/12 + 02/12 + 02/12 + 02/12 = 0 6 classes ⇒ should obey 2-distribution of degree 5 degree 5 O 2 The 2-values should be interpreted in the 2-distribution. 4 it is quite rare that 2 = 0 20 other statistical tests KS test (Kolmogorov-Smirnov test) “continuous” version of x2 test run-length test 2-test for the length of runs ( # of runs of length l )= 0.5×( #runs of length l – 1 ) for a binary random sequence porker test, collision test, interval test, etc. There is no simple yes/no answer. The interpretation of scores must be discussed. 21 generating pseudo-random sequences pseudo-random sequence generator (PSG) procedure which produces a sequence from a given seed. there are many different algorithms 0110110101... PSG seed linear congruent method (線形合同法) poor but simple example of PSG determine numbers in a sequence according to a recurrence typically, Xi+1 = aXi + c mod M, with a, c, M parameters used in early implementations of rand( ) of C language 22 properties of linear congruent method Xi+1 = aXi + c mod M The period of the sequence cannot be more than M M must be chosen sufficiently large. If the choice of M is bad, then the randomness is degraded. The relation between a and M is important. Choosing M from prime numbers is safe option. There is some heuristics on the choice of a and c, also. 23 bad usage of linear congruent method You want to sample points on a plane uniformly and randomly. If you use (Xi, Xi+1) as sampled points, then... the value of Xi uniquely determines the value of Xi+1 all points are on the line y = ax + c mod M (X1 = 5) 6 in case you use 5 Xi+1 = 5Xi + 1 mod 7: random sampling is NOT realized 4 3 2 1 O 1 2 3 4 5 6 24 M-sequence method M-sequence method (M系列法) generate a sequence using a linear-feedback register if you use p registers ⇒ there are 2p internal states with carefully setting the feedback connection, we can go through all of 2p – 1 nonzero states. a sequence with period 2p – 1 (the Maximum with p registers) connect or disconnect Xi–p Xi Xi–p+1 Xi–p+2 Xi–1 25 about M-sequence the connection is determined by a primitive polynomial the generated sequence show good score for statistical tests the difference of initial seed phase-shift “shift additive” property good “self-correlation” property applications in digital communication like as in CDMA 26 other PSG algorithms Mersenne Twister algorithm (メルセンヌ・ツイスタ法) M. Matsumoto (U. Tokyo), T. Nishimura (Yamagata U.) make use of Mersenne numbers efficiently generates a high-quality sequence PSG with unpredictable property important in cryptography and games Blum method The PSG algorithms introduced in this class are predictable: don’t use them in security or game applications. 27 summary “randomness” Kolmogorov complexity and randomness statistical tests of pseudo-random numbers 2-test (chi-square test) algorithms for pseudo-random number generation linear congruent, M-sequence 28 exercise For s = 110010111110001000110100101011101100 (|s|=36), compute 2-values of s for block length with 1, 2, 3 and 4. Implement the linear congruent method as computer program. Generate a random number sequence with the program, and plot sampled points as in slide 24. 29 have nice holidays! NO CLASS on May 1 (TUE) / 5月1日(火)の講義は休講 repot assignment (レポート課題): http://apal.naist.jp/~kaji/lecture/report.pdf available from the above URL by tomorrow, due May 8 (TUE) 明日までに公開予定,5/8 (火)までに提出 30