Com plex Syst ems 9 (1995) 1- 10
Sequences of Pseudorandom N umb e rs w it h
Arbitrarily Large Perio d s
P. S. Joag
Departm en t of Pbysics, University of Poon a,
Ganesllkllind, P une-411007, India
A b st ract . We show t hat t he rest riction of th e Berno ulli shift map
x <- 1' X (mod L) to t he set of rati onals between 0 and 1 can generate sequences of uniformly dist ributed pseudorandom numbers whose
periods exceed any given k > 0, however large. Here r > 1 is an
integer.
Introduction
1.
In t his p ap er , we present an algorit hm (Theor em 1) t hat gene rates sequences
of pseudorandom nu mb ers, com prising ration al number s b et ween 0 and 1,
wit h a rbit rarily large p erio ds. For any given p osit ive int eger k , however large,
t his algorit hm ca n b e used to generate pseud orandom number sequences wit h
p eri od greate r t han k. Mor eover , at least for large pe riods, these pseudor an dom numbers are approx imately uni for ml y distrib ut ed , t he approximati on
gettin g b et t er as t he period in creases. T hese pseudorandom seque nces are
gene rated usin g fini t e strings of r-ary digit s which are Kolmogorov random ,
that is, whose Kolm ogorov complexity is close, in some sense, to t he len gt h
of t he st ring . We explain this conce pt precisely in sect ion 2.
In ord er for the m ain result of T heorem 1 to b e of a ny utility, we need
Lemma 1 stated b elow, which is probably well known , but I can fin d no exact
reference. Before stat ing t he lemma , we give m eaning to t he phrase "a lmos t
all ration als."
Let S den ot e a n infinitely denumerabl e set wit h an op eration of multiplica tion defined on it . Define a real-valued m ap ping on S, denot ed IIall, a E S
such t hat :
Ilabll < Il all llbll
(i)
'Va, b E S;
(ii) t he to tal number Ns( x) of eleme nts a E S wit h
eac h real x > o.
Ii all :::;
x is finite for
Let
S[x]
=
{a
E
S [ ]« ] :::; x}
Den ot e t he cardinalit y of S[x] by card S[x]. Obviously, card S[x]
Ns(x) .
P. S. Joag
2
Consider E
c
S and let
E [x] = { a E E l
lIall ::;
x} .
Let card E [x] = NE(x) . We define the relati ve density of E [x] in S[x] as
NE(x )/Ns( x) . T he asy mptotic relative density of E in 5 is t hen defined as
.
NE(x)
ds(E) = hm -SN
(X ) '
X
---tCX)
provided that this limi t exists .
A proper ty is said to hold for almost all elements of 5 if it is valid for all
elements in some subse t of S havin g asy mpt ot ic relat ive density 1 in S. We
specialize this to the set of ration al numb ers between 0 and 1. Let
Q = {x l x = p/q where 0 < P < q are int egers}.
Define a real-valued map on Q t o be
II ~II
=
(1)
q.
Ob viously, (1) sa t isfies t he pr op erti es (i) and (ii) above.
We kn ow that every ration al number has eit her t erminat ing or recurring
representation with respect t o any base r > 1. We now state and pro ve
Lemma 1.
Le m m a 1. Giv en any in teger r > 1; alm ost all rationals have recurring
r-epresen tati on with r-egard t o th e base r (r -a r-y representa tion) .
P roof. We prove this by showing t hat t he set of rati on als having term inating
r-ary repr esenta tion have zero asympt otic density in Q. We call such a set T.
A given rational number p/q has a terminating r -a ry represent at ion provided
(2)
where a I , az, . .. , as are the pr ime factors of r an d ml , m z, . . . , rn, are integers
~
O. Let at. and at denote the largest and smallest prime factors of r ,
respect ively. For a given integer m > 0, t he set M of integers q ::; aT
which can be fact orized as in equation (2) , is equinumerous t o the set of all
s tuples sa tisfying
ml
+ ... + m s
::;
aLl
log-- .
m log at
The cardinality of this set is given by
'
mr~
lo g at l
L
k= l
(
k+s s- l
1)
(3)
Sequences of Pseudorandom N um bers with Arbitrarily Large Periods
3
where t he summand stands for the number of s t uples [6] sa t isfying t he
equat ion
ml
+ m 2 + ... + m s =
k.
Expression (3) is a po lynom ial in m , say P (m), and is an upp er bound on
t he numb er of int egers q :::; aT satisfying equation (2).
Now let ,
Q[y]
where
I xii
=
{x
E
Q
I llxll <
y}
is defined by (1) and y
> 1 is real. We easily get
card Q[aT]
where c is a constant and ¢(q) is the Euler fun ct ion: [8] . T he last inequality
is prov ed in [2].
Next we define
T [y] = {x
E
Q n
Tll lxli <
y}.
Obviously,
card T [aT]
NT(aT)
=
L
¢(q) < aT P (m).
qEM
Now, given y > 1, choose m such t hat aT- 1
< y < aT.
T his gives,
NT(y) = card T [y] < NT(aT) < aT P(m)
Ndy ) = card Q[y] > NdaT- 1) 2: c a i (m - l ).
T hus t he relati ve density of T [y] in Q[y] is bo und ed above by
NT(y) <
NQ(y) -
P(m)
c aT- 2 '
Taking t he limit as y ----+ 00, t he left- hand side of the previous inequ ali ty gives
t he asy mpt ot ic relati ve densit y ddT) . Since m increases monotonically wit h
y and P (m) diverge s as a polynomial in m while the denomi nator diverges
exp onent ially, the right-h an d side of the ab ove inequality te nds to zero as
y ----+ 00 . T hus dQ(T) -; 0. Since by definit ion dQ(T) 2: 0, we mu st have
ddT) = 0. •
T hu s almost all rati on als have recurring r-a ry represent ati ons. T here ar e
two cases of recurring repr esent ation : purely and mixed . A pur ely recurring
repr esentat ion consists of an infini t e periodi c sequence of digit s wit h period
n, A mixed recurring repr esent ati on consists of finit ely man y nonperiodi c
digit s followed by an infini t e periodic sequence of digits.
For a given r , we call any st ring cons tructe d out of {O , 1, . .. , r - I} a
r-ar y st ring . T he digit s are called r-ary digit s. A st ring x havin g n digit s is
said to be of length n , denot ed Ixl.
P. S. Joag
4
2.
Random strings of digits: Kolmogorov complexity
vVe br iefly review the concep t of Kolmo gorov comp lexity of a finit e string x of
length n and use it to define a finit e random st ring of r-ar y digit s [3,5,10- 15].
W itho ut losing any generality, we t ake r = 2 while discussing this con cept.
Kolmo gorov complexity concerns t he problem of describing a finit e obj ect
x. Since a finit e obj ect can be coded in terms of a finit e binary string, we can
t ake t his st ring to be our ob ject. It is useful t o t hink that the complexity
of specifying an ob ject can be facilit ated when another obj ect is already
spec ified . Thus we define the complexity of an ob ject x given an object y.
Let p E {0,1 }*. We call p a p rogram. Any computable fun ct ion f together
with st rings p and y such that f(p , y) = x is a description of x . We call f
the interpre ter or decoding fun ct ion. The complexity K f of x wit h resp ect t o
f , condit ional to y , is defined by
K f( xIY)
= min{lp[ : p
E {O,l}* andf(p ,y)
= x }.
If there is no such p , then K f( xIY ) = 00 .
T he invari an ce theorem [3,10,11,15] asserts that each finit e object has
an int rinsic complexity that is indep end ent of the means of descripti on .
Na mely, there exist asymptotically optimal functions (universal Turing machines ) such that t he descrip tion length wit h resp ect to them minorizes t he
description length wit h respe ct to any other fun ctio n , apart from an additive
constant , for all finit e object s.
Invariance Theorem. Th ere exis ts a pa rtial rec ursive fu n ction fo , suc h
that , for ' any other- par ti al r-ecu r-sive funct ion I, ther-e is a constan t cf suc h
that fo r all strings x , y , K fo(xly) ::; K f( x ly) + cf .
Clea rly, any fun ct ion fo that satisfies the invariance t heorem is optimal in
the sense discussed pr eviou sly. Therefore, we are justified to fix a particular
partial recur sive function fo and drop the subscripts on K . We define the con ditional K olmoqorou complexi t y K (x Iy) of x under condit ion of y to be equal
to K fo(xly) for t his fixed optim al f o. We can now define t he unconditional
K olmoqorou com p lexi t y (or Kolmogorov comp lexity) of x as K( x) = K(x IE)
where E denotes the empty st ring (lE I = 0) .
We are basically concerne d with Lemma 2, which is the most important
consequence of the invari ance theorem [14].
Lemma 2. Th ere is a fi x ed cons tant c' su ch that fo r all x of length n ,
K( :];) < n+ cl
(4)
T hus K( x) is bo unded above by t he length of x modulo an addit ive
constant . The cons tant c' turns out to be the number corr esponding to
the machine T that just copies it s input to its output , in some st andard
enumeration of Turing machines and can be conveniently chosen . We ar e
int erested in th e bin ar y strings x of length n for wh ich K (x) 2: n - c where
c 2: 0 is a cons tant . If cf n is understood to hav e a fixed fract ional value,
Sequences of Pseudorandom N um bers with Arbitrarily Large Periods
5
we call such st rings c-incomp ressible, or simply incom pressible. Lemma 3 [4]
answers the qu est ion : How man y strings are incompressible?
Lemma 3 . For a fixed c < n , out of all possi ble bin ary strings of length n
at m ost one in 2 C has K (x ) < n - c.
T hus, if we fix cf n to be a small fraction , t hen the fract ion of c-incom pressible binar y st rings increases expo ne nt ially with n as (1 - 2 - (c/ n )n ). Generally, let g(n ) be an integer fun cti on. Ca ll a string x of length n g-incompressible
if K (x ) ~ n - g(n) . T here are 2n bin ary st rings of length n , and only
2n - g (n ) - 1 possibl e descrip tions shorter than n - g(n) . Thus t he rati o
between the number of strings x of length n wit h K (x ) < n - g (n) and t he
total number of st rings of length n is at most 2- g (n ) , a vanis hing fun ction
when g( n) increases unb oundedly with n .
In tuiti vely, incompressib ilit y implies the absence of regu lari ties, since regulari ties can be used to compress descriptions . Accordingly, we identify incomp ressibility with t he absen ce of regularities or randomness. In par ticular ,
we call c-incompressible st rings c-ran dom . We do not deal here wit h the infinite ran dom sequences of digits.
3.
Main r e sults
We now state and prove our m ain results.
Lemma 4 . Suppo se an r-ary string x of length n has comp lexi ty K (x ). Cut
th is string aft er th e sth digit so that the conca tenation of the two resu lting
su bstrings (partitions), say Xl and Xz, can give the original string. Let K (XI)
an d K (x z) deno te th e comp lexi ti es of thes e partit ions. Th en K (XI )+ K (x z) ~
K (x ).
Proof. Suppose K (XI) + K (x z) < K (x ). However , t he st ring X can be
pro duced by using t he minim al pr ogram s corresponding to Xl and Xz in
success ion . T his gives the complexity of X to be :S K( XI) + K (x z) < K (x ),
which contradicts our premise that the complexity of X is K( x ). •
Corollary 1. If a string x of length n is c-ran dom , th en any two partitions of
x, say Xl and x z, are at least 2c-ran dom . A string gen erated by concaten ating
Xl and Xz is c-ran dom .
Proof Let IXII = n l , Ixzl = n z, wit h n l + n z = n . Wi th ou t losing generality
we can choose t he constant c' ap pearing in the inequalit ies K( Xl) :S nl + c',
K (x z) :S nz + c', and K (x ) :S n + c' , which are true by virtue of Lemma 2,
to sa t isfy c > c' . Now ass ume that Xl is not 2c-random so that K (Xl ) <
nl - 2c. Subtracti ng K( Xl ) from t he left-hand side and n l - 2c from t he righthan d side of t he inequality K (XI ) + K( x z) ~ K (x ) ~ n l + n z - c we get
K (xz) ~ n z + c > nz + c! which contradicts the requirement K (xz) :S nz + c'
imp osed by Lemma 2.
P. S.
6
Joag
As for the seco nd par t of the corollary, if the two par t iti ons are concatenat ed to produce the original st ring , there is nothing to pr ove. Now suppose
t hat the par t it ions Xl and X2 are concatenated in t he reverse orde r , giving a
st ring Xinv = X2XI wit h complexity
(5)
which mean s that Xinv is not c-rando m. Then the string X can b e pr inte d
in the following way. P roduce Xinv and pr int t he last n l digits first and t he
first n 2 digit s next . This will enhance t he lengt h of the minimal program
producing X by mi n (log nl , log n2) which we take to be log nl' T hu s
K (x )
:s;
K (Xinv) + log n, < n - c.
(6)
The las t inequ ality follows from (5) and log n ; « n - c. In equ ali ty (6)
mean s t hat x is not c-rando m , which cont radicts our pr em ise an d com pletes
the pro of. _
Definition. A string x of length n is said to be ran dom if K (x ) 2: n O( logn).
Corollar y 1 can b e eas ily exte nded t o the case of a ran dom st ring of lengt h
n , as is done in Corollary 2.
C or o llary 2 . If a string x of length n is random, th en any tw o partitions of
x , say Xl and X2, are random . A string gen erated by concatenating Xl and X2
is random.
P roof. In the previous state me nt an d in th e proof of the Corollary 1, if we
replace c by a funct ion f(n ) = O (log n ) , then we ca n also replace 2c by
f (n ). -
In Theorem 1 we make use of Corollary 2.
D e finition. W e call the m ap X <-- r x (mod 1) on Q, that is, th e B ernoulli
shift [7], restricted to rationals a shift ov er rati on als (SOR) . H ere r > 1 is
an integ er .
Theorem 1. Let n = n(r, xo) deno t e the peri od of the recurrinq r-ary rep resentation of Xo E Q. Th en , f or alm ost all Xo E Q, at least th e fi rst n
ite rati ons of th e S OR gen erat e a sequ en ce of pseudora ndom numbers wit h
period { x o, Xl , . . . , x n- d suc h that th e fi rst (m ost signific an t) n digits of the
r-ar y representation of each Xi (i = 0,1 , . . . , n - 1) is a ran dom string, provi ded th e fir st n digit s of the r-ary represen t ation of Xo is a ran dom string.
P roof. This proof applies to Xo E Q havin g a recurr ing r-ary represent ation.
By virtue of Lemma 1, t his means t hat t he pro of applies to almost all Xo E Q.
T here are two p ossib ilities: Xo may have eit her purely rec urr ing or mixed
recur ring represent ation. We deal with these two cases separately.
Sequences of Pseudorandom Numbers with Arbitrarily Large Periods
7
Case 1: Purely recurring representation.
Let {d I, d2 , · . . , dn } b e the first (most sign ificant) n digit s (t he first p eriod) of the r-a ry represent ati on of x o . Op er ate by SOR on Xo to get X l
whose first n digits are { d2 , d3 , . . . , dn, dd, b ecau se SOR removes d l from
the r-ary repr esent at ion of X o to generate t hat of Xl . Thus t he string of t he
first n digits of the r-ary represen ta tion of X l is obtained by par ti tioning that
of X o afte r d l and concatenating the two par ti ti ons in reverse order. Therefore, by corollary to Lemma 4, the string formed by the first n digit s in t he
r-a ry repr esentati on of X l is random , pr ovided t he corresp onding st ring for X o
was random. By the sa me argument, fur ther it eration s of the SOR, X t+l <Xt (mo d 1) (t = 1, 2, . . . , n - 2) generate rational numbers X 2 , · . . , Xn - l having r-ary representations who se fir t (mos t significant ) n digits for m a random
string , provided the corres p onding st ring for X o was random . Aft er the n t h
iter ati on , the sa me cycle { x o , XI, . . . , X n - l } repeats du e to p eriod icity of t he
represent ati on .
Case II: Mixed recurring representation.
Let the r-a ry rep resent ati on of X o b e p er iodic aft er t he first m nonperiodic
digit s and let t he pe riod b e n. Consider th e most sign ificant m + n - 1 digit s
of Xl = r xo (mod 1). T his is t he par ti tion of size m + n - 1 of t he string of
most significant m + n digi t s in t he r-ary rep resentation of xo . By corollary
to Lemma 4, this partit ion is a random string, provided th e corres p onding
string for Xo was random and the next m - 1 iterat ions generate m - 1
more rati on als havin g a random string of digits of lengt h > n as t he most
significant digits in their r-ary repr esent ati ons, provided the corresp onding
st ring for X o was random . Aft er m it erati ons, r-ary repr esent ati on of X m is
pu rely recurring and Case I ap plies. _
We now show t hat t he sequence of pseudorandom numb ers produced
by the SOR as in Theorem 1, is un iformly distributed [9].
ote that t he
frequ ency of occ urre nce of all t he r digits {a, 1, ... , r - 1} in a string of r-ary
digit s of lengt h n is close t o nlr , provid ed n is large. In fact , the probability
that t his frequ ency deviates from n l r by an amount grea ter than a fraction
8 of n is b ounded above by
L
nexp ( ~1 K 8 n)
2
(7)
where L and K are constants that dep end on r [8] . Since rati on als are
dense in (0, 1) , those havin g r- ary repr esent ations wit h very large p eriods
are a bundantly ava ila ble (see the b eginnin g of sect ion 4) .
Now divide [0, 1) int o r int ervals
s
s
+1
- :S y < - - (s = 0, 1, . .. , T
r
r
-
1) .
(8)
The (s + 1)th int erval cont ains just those numbers wh ose r-ary represent ation b egins wit h s . Suppose we cons truct the sequence of pseudorandom
numbers X o , X I, .. . , X n - l using Theor em 1. It is eas ily seen that ea ch one
P. S. Joag
8
of t he first n = n( 1', xo) digits {d 1 , d2 , ... , dn } of t he r-ary represent ati on of
Xo successively becomes t he most significant digit of the r-ary repr esenta tion
of {X1, X2, ' " , xn- r} . Since all digits occur wit h equal frequ ency nfr , (neglectin g very small fluct uat ions decaying exponent ially wit h n ), each of the r
intervals define d above will contain ti]« numb ers out of the pseudorandom
sequence {xo, Xl, . . . , xn- r}. We now divide each of the int ervals given by
(8) int o r subintervals so that the (s + 1)t h subinterval contains numbers
whose r-ary representations have s as t heir second digit . T he fract ion of
pseud orandom nu mbers that fall in th e interval corresponding to the ord ered
pair of digit s (s, t) equals the number of pairs (s, t) occur ring in the string
of the first n digits in the r-a ry representat ion of xo. T he number of such
pairs is (n - 1) / 1'2 provided tha t this string has n = n( Xo,1') large enough
to make the fluctu ations given by (7) negligibl e. Thus the expe cte d number
of rati onals from t he pseud orandom sequence in each of the 1'2 int ervals is
(n - 1)/ 1'2 :::::; n / T2 We can cont inue dividing [0, 1) in the same way, each
tim e getting 1'3 , 1'4, . .. int ervals wit h the expect ed equal occupancy given by
(n - 2)/1'3 :::::; n /T 3, (n - 3)/1'4 :::::; n/T 4 . .. rat ion als from the pseudorand om
sequence . T his shows tha t t he distribu tion of pseudor an dom num bers is un iform over t he pattern of int ervals described , provided n = n (xo, 1') is lar ge
enough to m ake t he fluctua tions given by (7) negligibl e. Since the ration als
are dense in (0, 1), for any given k, however lar ge, th ere exist infin it e Xo E Q
wit h n( xo,T) > k. T herefore, we can always choose Xo with n( xo,T) > k
for any given k .
It is not t he case t hat we are get ting the uniform distrib ution of {xo, . . . ,
xn- r} due to some spec ial characterist ics of t he way we are dividing [0, 1)
into subinte rvals. In fact , any division of [0, 1) into sub int ervals of equal
lengt h satisfies inequ ali ty (8) for some h , (h = 2,3 ,4, . .. ). Suppose h i=
T.
Divide t he int erval [0, 1) int o h£ subintervals of equal length for some
> O. For every E > 0, it is possib le to find integers m and k such
tha t 0 < (h-£ - m T - k) < E. On the given mesh of he int ervals we now
supe rimpose a mesh of Tj inter vals such that Tj- 1 < m h£ < Tj . We divide
t his supe rimposed mesh int o par t itions of approximate ly T j / h£ intervals such
that each par ti tion closely overlaps each of the he subint ervals of the original
mesh . Since the distribution of pseudo random numb ers is uniform over the
above par titions it is un iform over t he h£ subint ervals, each wit h length h:" ,
e
4.
Discu ssi o n
T heor em 1 gives an algorit hm to generate sequences of pseu doran dom numb ers {x o, . . . , xn- r} whose period is not less than t hat of t he r-ary represent ation of Xo nam ely n = n (T, xO)' A very imp ort an t advant age of this method
lies in the fact t hat we can choo se n = n(T, xo) as large as we please. If we
choose Xo = ]J/ q such t hat q is relat ively pr ime to both ]J and 1' , t hen t he
p eriod n = n (T , xo) is given by the smallest v satisfying TV == 1 (mo d q), that
is, t he smallest v such t hat (TV - 1) is divisible by q [8J. For instan ce, wit h
r = 3 and Xo = 0.187500000000 0001 (q = 10 16 ) , t his period is of the order
Sequences of Pseudorandom Num bers with Arbitrarily Ltuge Periods
9
of 1014 . In fac t, given any values k a nd r , it is possible to choose Xo = vt«
t hat has a recurring r-ary represent a ti on wit h p eriod n greater t han k. Since
ra tionals are den se in (0, 1) , t he ra tionals havi ng r-ary represent a tion wit h
pe riod greater t han a given fixed integer are infinit ely abundant . T hus the algorit hm given by T heorem 1 can gene rate seq uences of un iformly distribut ed
pseudor andom numbe rs wit h arbitrarily large pe riods .
Suppo se we wa nt to choose rand Xo = vt« such t hat n = n(r,xo) > k
whe re k is some fixed integer. T his can be do ne , for exam ple, as follows .
Choose r to be a pr im e and Xo = piq such t hat p and q are re lat ively
prim e and q = r k + 1. Thus q and r are also relati vely prime and the p eriod
n(r, xo) is given by t he smallest 1/ such t hat r V
1 is divisible by q = r k + 1.
k
T he refore, r '/ - 1 > r + 1 giving n = 1/ > k. T he actual dig its of p ca n
b e chosen from a table of random nu mb ers [1].
In order t hat T heorem 1 applies in p racti ce, we have to com pute t he
successive values of rx (m od 1) usin g full y ra tion al arit hmet ic a nd no t by
t he floati ng p oint arithme t ic t hat is ava ila ble on m ost comp uters . T his facility is offered by all t he computer algebra sys te ms . T his slows dow n t he
comp utation , but is not reall y a bo ttleneck as one ca n gene rate a huge b an k
of pseudorandom numb er s, an d issu e t he m to various comput ing pro cesses
(eit her parallel or seq ue nt ial) as and whe n required.
We have carr ied out statist ical tests for randomness as given in [9] for a
large sa mp le of pseudorandom seque nces generate d using T heorem 1, eac h
of t he lengt h of a few t ho usand . E ach of these seque nces has passed t he
frequ en cy t est and t he serial tes ts satisfactorily. T he serial tes t was rep eated
by choosing t he pairs of number s wit h t he ga p be twee n t he m increasing from
o t o 23 in ste ps of 1. T hu s, in t hese seque nces , t he pairs of numbers wit h
ga ps up to 23 are uncorr ela ted .
As we know, SO R cuts t he r-ary represent at ion of a ration al number
afte r t he first digit and cho ps it off. It is p ossible to thin k of a large class
of maps tha t chop the st ring of r-ary digits at different pl aces in succ ess ive
it era ti on s. Alt hough t hese m aps will generate pse udo random seque nces in
t he sense of T heor em 1, t he p eriod s of t hese sequences will not , in general,
b e equal to t he p eri od of t he r-ary represent a t ion of t he seed xo, in which
case t he pseudor andom numbers m ay not be uniformly distribut ed .
Acknowledgement s
I t hank P rofessor S. R. Adke for t he perusal of t he m anu script an d m an y
useful suggestions . I a m grateful to Dr . Mohan Na ir and Dr . Mr s. Mangala
Na ralikar for p ointing out mistakes in the proof of Lem ma 1, and to Dr.
Moh an Na ir for suggesting a generalization of this pro of. F inally, it is a
pl easur e t o thank Dr. Do m inic Welsh for some illu m inati ng discu ssion s.
10
P. S. Joag
R eferences
[1] M. Abramowit z and 1. A. St egan (ed ito rs) , Handbook of Mathema tical Fun ctions (Dover P ublica t ions , ew York , 1965).
[2] T. M . Ap ostol, Analytic Num ber Th eory (Narosa, New Delhi, 1972) .
[3] G. J. Chaiti n , "On t he Length of Progr ams for Computing Finit e Bin ar y
Sequ en ces: Stati stical Considerations ," Journ al of the Association f or Com put ing Mac hina ry, 16 (1969) 145-159.
[4] G. J . Chaitin , "R andomness and Mat hemat ica l P roof," Scientific American,
232 (1975) 47- 52.
[5] G . J. Ch ait in , Algorithmic Inf orm ation Th eory (Cambridge Un iversity Press ,
Cambridge, 1987).
[6] W. Feller , An Introduction to Probability Th eory and its App lication s ( Vol. I)
(3rd editi on) (W iley E ast ern Lt d. , New Delhi , 1968)
[7] J. Ford, "Chaos: Solving the Un solvab le , P red ict ing the Un predict able!" in
Chaotic Dyn amics and Fractals, ed ite d by M . F. Barn sley and S. G. Demeo
(Acad emic Press , London , 1986) .
[8] G . H. Hardy and E. M . Wri ght , Th eory of Numb ers (5th editi on) (Oxford
Universit y Press , Ox ford , 1978).
[9] D. E . Knuth , Th e A rt of Computer Programming Volume 2: S eminumerical
Algorithms (2n d editi on) (Ad dison-Wesley, Reading , 1980 )
[10] A. N. Kolm ogorov , "On Tab les of Rando m Numbe rs ," Sankhya Series A 25
(1963) 369- 376.
[11] A. . Kolm ogor ov , "T hree Approaches to the Qu an tita tive Tr an smi ssion of
Information ," Problems Inf orm ation Transmission 1 (1965) 1-7.
[12] A. . Kolmo goro v, "Combina torial Foundations of Informat ion Theor y a nd
the Calculus of P robabilities ," Russian Math ematical Surveys 3 8 (1983)
29-40.
[13] L. A . Levin , "R andom ness Conse rva t ion Inequ aliti es: Inform ati on and Indep endence in Mathematical T heor ies," Information and Cont rol 61 (1984)
15-37.
[14] M . Li and P . M. B. Vit an yi , "Kolmogorov Co mplexity and it s Applications, "
in Handbook of Th eoretical Computer Science, J. van Leeuwen , editor (E lsevier Scien ce, 1990) 189- 253.
[15] R. J. Solomonoff, "A Formal Theory of Indu cti ve Inferen ce, P art 1 and Part
2," Inf orm ation and Cont rol 7 (1964) 1- 22 and 224-254.