exercise in the previous class

Encode ABACABADACABAD by the LZ77 algorithm,

and decode its result.

1

A B AC ABAD ACABAD

back 2 positions copy 1 symbol

back 4 positions copy 3 symbols

back 6 positions copy 6 symbols

(0, 0, A), (0, 0, B), (2, 1, C), (4, 3, D), (6, 6, *)

A B AC ABAD ACABAD

back 2 positions copy 1 symbol

back 4 positions copy 3 symbols

back 6 positions copy 6 symbols

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∈Π(𝑥)|𝑝| .

(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.

K(x3) is about hundreds characters or less

The Kolmogorov complexity K(x)：

measure of the difficultness to construct the sequence x

7

𝑎1 = 𝑎2 = 1

𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2

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)

10

V n V n–1 V n–2 V 1

2n sequence 2n – 1 sequence

px x

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”

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

http://www.fdk.co.jp/

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

14

expected distribution of the number of 1 in s ideal source s

s1 s2

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:

𝜒2 = 𝑛𝑖 − 𝑛𝑝𝑖

2

𝑛𝑝𝑖

𝑘

𝑖=1

15

what is this value?

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

𝜒2 = 𝑛𝑖 − 𝑛𝑝𝑖

2

𝑛𝑝𝑖

𝑘

𝑖=1

𝑛𝑝𝑖...the expected number of symbols in Ci

𝑛𝑖...the observed number of symbols in Ci

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.

19

degree 2

degree 4

degree 6

2 O

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

20

2

O

degree 5

it is quite rare that 2 = 0

4

The 2-values should be interpreted in the 2-distribution.

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

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

PSG seed 0110110101...

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

24

in case you use Xi+1 = 5Xi + 1 mod 7:

1 2 3 4 5 6 O

1

6

5

4

3

2

random sampling is NOT realized

（X1 = 5）

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)

25

Xi

Xi–1 Xi–p Xi–p+1 Xi–p+2

connect or

disconnect

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