Schnorr - A Unified Approach to the Definition of Random Sequences

8/11/2019 Schnorr - A Unified Approach to the Definition of Random Sequences

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Unified pproach to the Definition of Random Sequences

b y

C P SCHNORR

Institut fiir Angew andte Mathematik

Universit/it Saarbriicken, West Germany

BSTR CT

Usin g th e conce pt of test functions, we develop a general fram ewo rk within w hich

man y recent approaches to the definition of rando m sequences can be described.

Using this c oncep t we give some definitions of rand om sequences tha t are narrowe r

than those proposed in the literature. We form ulate an objection to som e of these

conce pts of randomness. Using the n otio n of effective test function, w e form ulate a

thesis on the true concep t of randomness.

1. The Concept of Test Function Let X [X*] be the set of al l infini te [f ini te]

b ina ry sequences . A e X* deno tes t he e mp ty sequence . Ixl deno tes t he l eng th

o f x s X * . T h e c o n c a t e n a t i o n o f s eq u e n c es x a n d y is d e s cr i b ed a s t h e p r o d u c t

x y . T h i s i n a n o b v i o u s w a y d e f i n e s a p r o d u c t AB c X* u X ~ of se t s A c X*

an d B c X* u X ~ . F or a seque nce x ~ X* u X ~ we den ote by x n) the ini t ia l

s e g m e n t o f l e n g th n x n) = x i f Ixl < n) . The m ap ~ : 2 x* -+ 2 x~ i s def ined by

~ A) = A X ~ A c X*) . T h r o u g h o u t t h e p a p e r ~ w i l l b e t h e p r o d u c t m e a s u r e

on X re l a ti ve t o t he p roba b i l i t y fo r I and 0 .

Le t us f ir s t exp l a in t he i n tu i t i ve i dea o f r a nd om sequences , w hich wi ll be

d i scussed he re . An in f in it e sequence i s cons ide red a r a nd om sequence i f i t

w i ths t ands a l l cons t ruc t i ve s tochas t i c i t y t e s t s . Our ma in as sumpt ion i s t ha t any

s t o c h a s ti c i ty t e s t c a n b e e x p r es s e d b y a f u n c t i o n F : X * - + R , w h e re F x )

i nd i ca t es t he ex t en t t o which t he sequence x i s suscep t ib l e t o t he s t ochas t i c i t y

tes t F (R i s the se t of al l rea ls ) . I t seems na tura l to t hin k th at F x ) i s h igh

when x i s suscep t ib l e t o t he t e s t and l ow o the rwi se . However , t h i s i s no t i n -

ev i t ab l e . Le t us g ive some examples .

Con s ide r a func t ion V: X * --~ R + t ha t i nd i ca t es t he cap i t a l o f a gam ble r

whe n p l ay ing on b ina ry sequences (R is the se t o f al l non-neg a t ive r eal s ).

V x ) denotes t he cap i t a l a f t e r t he Ix ls t t r i al wh en the sequence o f t he gamb l ing

sys t em has t he i n i t i a l s egment x . In a f a i r gambl ing sys t em the p l aye r ' s ga in has

to sa t i s fy t he r e l a t i on V x ) = 2 - l ( V ( x l ) + l~ (x 0)) . I t is n a t u r a l t o t h i n k t h a t

l i ra sup, V z n) ) < ~ i f z is a r a nd om sequence . Con seq uen t ly h igh va lues

V z n) ) m e a n t h a t t h e s e q u en c e s z n) a re suscep t ib l e t o t es t V . Ga mb l ing sys t ems

of t h is k ind w ere i n t rod uce d by J . V i ll e [15] . V i ll e p ro ved tha t fo r eve ry fun c t ion

V: X * -+ R + sa t i s fy ing V x ) = 2 -1 V xO)+ V(x l ) ) the se t {z e X~[ lim su p.

V z n) )

= m } is a n ull set.

246


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A Unified Approach to the Definition of Random Sequences 247

To g ive ano ther example , we define a se t U c N x X* to be a sequent ia l t e s t

if

Ui

=aef {x ~ X * [ i, x ) ~ U} satisfies

1) U, = U,X ;

:2) U~+~ = U~;

3) ~ ( w , ) N

is def ined by my x ) = ma x {mix ~ Urn . Conseq uent ly z wi ths tands the sequen-

t ia l tes t U i f and only i f

l im sup

mv z n)) < oo.

He nce m v ref lects ou r intu i t ion of a tes t fun ct ion. A sequent ia l tes t U is cal led

recurs ive i f U c N x X* is recurs ively enu me rable r .e. ) . These tes ts were

in t rodu ced by M ar t in -L6 f [6] . A sequence z e X is ran do m in the sense of

M a r t i n - L 6 f i f z Ni~N p Ui) for every recurs ive sequent ia l tes t .

In order to genera l i ze the above ment ioned examples , we def ine a func t ion

F: X*- -> R to be a c onst ru ct ive tes t i f F sa t is f ies the fol low ing propert ies ,

which wi l l be s ta ted in an informal way.

T1) F has to be const ruct ive , i .e . , F is to be given by a lgori thms.

T2) Th er e is a ru le w hich assigns to F a nu ll set gtF C X , the set of

infin it e sequences which do not wi ths tand the t e s t F . W hether z s ~ e

has to depend only on the sequence

F z n)) ln e N) .

The different def ini t ions of random sequences we shal l discuss here are

mere ly d i s tingui shed by spec ify ing the a bove men t ioned axioms mo re prec ise ly .

W e shall essent ia lly consid er two different rules accord ing to T2) , nam ely:

a) 9lF = {Z ~ X ~ llim su p F z n )) = oe },

b) 91F = {Z ~ X~ liim

infF z n)) = oo}.

Because of T1) the se t of t e st func t ions o f any f ixed concep t of t e st func t ions

can be enum era ted . Thi s implies tha t re l a tive to an y conc ept of t e st func t ions

sa t i s fy ing the above ment ioned axioms the fo l lowing theorem i s t rue .

T H E O R E M 1 . 1 . The set of random sequences has measure 1.

Give n a f ixed con cep t of tes t funct ions , a tes t F is cal led universal if 91e D

gte , for any other tes t F~.

2 . Mar t in-Li i f Random Sequences Described by Mar t inga les . Le t Q be the

se t of a l l ra t iona l numbers .

Defini tion 2.1. A tota l ) func t ion F : X* --> R is cal led weakly com pu tabl e

i f the re i s a recurs ive func t ion g : N x X* ~ Q such tha t

g i, x)


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248 C P SCHNORR

A f u n c t i o n F : X * - + R is c o m p u t a b l e in t h e u s u a l se n se i f F a n d - F a r e w e a k ly

c o m p u t a b l e .

A f u n c t i o n F : X * - + R i s s ai d t o h a v e t h e m a r t i n g a l e p r o p e r t y r e l a ti v e t o

t h e p r o b a b i l i ti e s f o r 0 a n d I i f i t s at is fi es t h e c o n d i t i o n

2.2)

F ( x ) = F ( xO ) +

F x l ) x e X * ) .

T h e s e f u n c t i o n s a r e c a l l e d m a r t i n g a l e s . I n t h e a c t u a l c a s e t h i s m e a n s t h a t t h e

e x t e n t t o w h i c h x w i t h s t a n d s t h e t e s t F i s t h e w e i g h t e d a v e r a g e o f t h e e x t e n t

t o w h i c h x 0 a n d x l w i t h s t a n d t h e s a m e t e s t .

T h e f o l l o w i n g l e m m a w a s p r o v e d b y J . V i l le [1 5].

L E M M A 2 . 3 .

I f F : X * - + R + s a t i s fi e s ( 2 . 2) , t h e n t h e s e t 9 1 = { z e X ~ [

l i m s u p . F ( z ( n ) ) = o o} is a n u l l s e t .

P r o o f

W e d e f i n e f o r k s N :

F k = { x e X I F ( x ) > k }

F k = { x e r k I x F k X X * } .

T h i s i m p l i e s

F k n l r k X X * = o . I rk

c o n s i s ts o f a ll t h o s e s e q u e n c e s i n F k w h i c h

ha ve no in i t i a l seg m en t in Fk . W e ha ve / z~0 F~)=/z~o I rk) = ~x~r~ 2-1xl . I t

f o l lo w s f r o m 2 . 2) t h a t

F A ) > ~

r ( x ) 2 - Ix l > k Z

2-1xl >

k t z g ( r k )

x~lek x~Fk

C o n s e q u e n t l y ,

2 . 4 ) ~ r k ) X * u { I} s u c h t h a t

A = h ( N ) n X * .

T h e s y m b o l I h a s t o b e u s e d i f A i s e m p t y . ) W e d e n o t e A , =

U i < , h ( i ) (~ x * . r (n ) i s t o b e t h e m a x i m a l l e n g t h o f s e q u e n c e s i n A , , r (n ) = 0

i f A , = ~ . T h e i n d i c a t o r f u n c t i o n I , : X * --> { 0, 1 } o f t h e s e t B is d e f i n e d a s


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A Unified Approach to the Definit ion of Random Sequences 249

f o l l o w s : IB(x) = 0 f o r a l l x E X * s u c h t h a t Ix I ~ r ( n ) + n f o r a l l n ~ N . I B ( x )

w i t h I x l = r ( n ) + n i s de f i ned r ecu r s i ve l y . Fo r [ x l = r 0 ) :

/~ (x ) = {10 x = h(O)

o t h e r w i s e ,

and f o r I x l =

r ( n ) + n , n >

1:

{10 x s h ( n ) X * - A - l X *

l ~ ( x )

o t he r wi s e .

T h e c o n s t r u c t i o n i m p l i e s t h a t B is p re f ix - f re e . I n a d d i t i o n w e h a v e

A n X * =

(B n {x I Ixj 2 ' k } .

Si nce F is weak l y co m pu t ab l e , V i s r . e . Bec aus e o f ( 2 . 4 ) we have / ~o ( V i ) _< 2 - i .

H e n c e a r e c u r s i v e s e q u e n t i a l t e s t U c a n b e d e f i n e d b y U~ = V~X* . F r o m

z N ~ u ~ ( U ~ ) i t f o l l o w s t h a t z w i t h s t a n d s t h e t e s t F .

P e r h a p s i t i s i n t e r e s t i n g t o n o t e t h a t t h e e x i s t e n c e o f a u n i v e r s a l ( 1 ) - t e s t

f o l l o w s f r o m a s i m p l e a r g u m e n t . L e t (F ~ li ~ N ) b e a r e c u r s i v e e n u m e r a t i o n o f a ll

(1) - t es t s wi th Fi (A) _< 1 . Hence F = ~ i ~ u 2 - i F i i s a universa l (1) - t es t .

I t s h o u l d b e n o t e d t h a t w e c a n u s e li m i n f a s w e ll a s l ir a s u p i n t h e d e f i n i t io n

o f 9 2 v f o r a ( 0 - t e s t F .

L E M M A 2 .8 .

Le t F be a universal (1)- tes t. Then the fo l low ing equivalence holds

fo r a l l z ~ X~:

l i ra i n f

F(z(n)) = ~


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250 C .P . ScrINOgg

F a s d e f i n e d i n p a r t 1 ) o f th e p r o o f o f 2 . 6) . I t w il l s u ff ic e t o s h o w t h a t

z ~ Ai~N~o Vi) i m p l i e s l i m , F z n ) ) = o o.

Su ppo s e z ~ A i~N~ Vi). Th en t o i ~ N t he r e ex i s ts n E N s uch t ha t z n ) ~ V i .

T h i s i m p l i es

F z m ) ) >_ i

fo r a l l m _> n . S ince th i s h old s fo r eve ry i ~ N, 2 .8) is

p r o v e d .

3 . An bjection to Randomness in the Sense of M a r t i n - L i i f . T h e a l g o r i t h m i c

s t r u c t u r e o f a 1 ) - te s t F i s n o t s y m m e t r i c a l. T h e r e i s n o r e a s o n w h y a m a r t i n g a l e

F s h o u l d b e w e a k l y c o m p u t a b l e a n d - F s h o u l d n o t b e so . T a k i n g t h is i n t o

c o n s i d e r a t i o n w e m a k e t h e f o l l o w i n g d e f i n i t i o n .

De f i n i t i on 3 . 1 . A f un c t i o n F : X * ~ R is a 2 ) - t e s t i f i t s a ti s fi e s 2 . 2 ) an d

i f - F is w e a k l y c o m p u t a b l e . T h e s et o f s e q u e n ce s t h a t d o n o t w i t h s t a n d t h e

t e s t F i s de f i ned t o be

9~F = {z e X 1 im su p F z n ) ) = o o } .

i1

W e c o n s id e r t h e q u e s t i o n w h e t h e r 1 ) - ra n d o m n e s s i s e q u i v a l e n t t o 2 )-

r a n d o m n e s s . T h e r e s e em s t o b e a n o b j e c t i o n t o e i t h e r o f t h e se c o n c e p t s o f

r a n d o m n e s s b e c a u s e t h i s i s n o t t r u e .

T H E O R E M 3 . 2 . T h e r e e x i s t s e q u e n c e s w h i c h a r e 2 ) - r a n d o m a n d w h ic h a r e

n o t 1 ) - r a n d o m .

W e w i l l f i r s t p r o v e a l e m m a .

L E M M A 3 . 3 . L e t F b e a 2 ) - t e s t a n d a > 0 a r a t i o n a l n u m b e r . T h e n t h e r e

e x i s t s a r e c u r s i v e z ~ X ~ s u c h t h a t F z n ) ) < F A ) + a n E N ) . T h i s m e a n s t h a t

z ~

P r o o f . L e t t h e 2 ) -t e st F b e g iv e n b y a r e c u r s i v e g : N x X * ~ Q s u c h t h a t

g i , x ) >_ g i + 1 , x ) a n d l i m s u p , g i , x ) = F x ) . L e t b b e r a t i o n a l w i th F A )

- a / 2 < b < F A ) . T h e s e q u e n c e z w i ll b e c o n s t r u c t e d r e c u r s i v e l y a s f o ll o w s .

A s s u m e t h a t z n ) h a s b e e n c o n s t r u c t e d s u c h t h a t

F z n ) )


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I t w i l l su f fi c e t o de f i ne a s eque nce z w h i ch i s no t ( 1 ) - r an do m an d s a ti s fi e s

l i r a sup , Fi ( z ( n ) ) < ~ f o r a ll i ~ N .

L e t F b e a un i ve r sa l ( 1 ) - te s t , i .e . X - ~ F cons i s t s p r ec i s e l y o f a ll ( 1 ) - r a nd om

sequenc es , z s X ~ w i l l be de f i ned i nduc t i ve l y . A ssu m e t ha t

Z ( n k )

w i t h

n k ~ N

i s a l r e a d y d e f i n e d s u c h t h a t

F ( z ( n k ) ) > k ,

a n d t h a t

k k

2 - ' - ' F , ( z ( j ) ) < ~ 2 - ' + 1 ( j < n k).

i = 0 i = 0

( N o t e t h a t t h e i n d u c t i o n h y p o t h e s i s is t r iv i a l f o r k = 0 , n o = 0 a n d

F o ( A ) < 1 .)

T o p e r f o r m t h e i n d u c t i o n s t e p , w e c o n s i d e r

F k + l .

W e o b v i o u s l y h a v e

2 - k - k F k + l ( z ( n k ) )

< 2 - k . T h i s f o l lo w s f r o m

F k + I ( A )

< 1 a n d t h e m a r t i n g a l e

p r o p e r t y . C o n s e q u e n t l y t h e r e i s a r e c u r s i v e y ~ X s u c h t h a t

y ( n k ) = z ( n ~

sa t i s f y i ng f o r eve r y nk+ 1 >- nk t he r e l a t i o n

k l k l

2 - ' - ' F i ( y ( j ) ) n k such t ha t

F ( y ( n k + l ) )

> k + l .

D e f i n e

z(nk+ 1) = Y(nk+ 1) .

T h e d e f i n i ti o n o f z i m p l ie s t h a t z ~ 9 ~F . O n t h e o t h e r h a n d w e h a v e

l im s u p ~ 2 - ' -

f F i ( z ( j ) ) < 4 ( i ~ N ) .

j =0

C o n s e q u e n t l y z ~ e ~ ( i ~ N ) .

W e r e m a r k t h a t i t is n o t d if fi cu lt t o p r o v e t h a t e v e r y ( 1 ) - r a n d o m s e q u e n c e

is a ls o ( 2 ) - ra n d o m . ( 1 ) - ra n d o m n e s s is a n a r r o w e r c o n c e p t o f r a n d o m s e q u e nc e s

t h a n ( 2 ) - r a n d o m n e s s . I t s e e m s su r p r i s in g t h a t f o r a m a r t i n g a l e F i t i s i m p o r t a n t

w h e t h e r w e c h o o s e F o r - F t o b e w e a k l y c o m p u t a b le .

N o w w e a im a t d e v e l o p i n g a c o n c e p t o f r a n d o m n e s s b a s e d o n m a r t i n g a le s

w h o s e a l g o r i t h m i c s t r u c t u r e i s s y m m e t r i c a l .

D e f i n i t i on 3 . 4 . A m ar t i ng a l e F : X * ~ R + is a ( 3 ) -t e s t i f t he r e i s a r ecu r s i ve

f u n c t i o n g : N x X * ---> Q s u c h t h a t l im i g ( i, x ) = F ( x ) ( x ~ X * ) . T h e s e t o f

s e q u e n c e s t h a t d o n o t w i t h s t a n d a ( 3 ) -t e st F i s d e f in e d to b e

~R~, = {z ~ X~ llim

F( z ( n ) ) = 0 0 } .

W e w i l l p r o v e t h a t ( 3 ) - r a n d o m n e s s i s c o n s i d e r a b l y n a r r o w e r t h a n ( 1 ) -

r a n d o m n e s s . O b v i o u s l y e v e r y ( 1 )- te s t is a l s o a ( 3 )- te s t. T h e r e f o r e e v e r y ( 3) -

r a n d o m s e q u e n c e is a ( 1 ) -r a n d o m s e q u e n ce . T o p r o v e t h a t t h e c o n v e r s e d o e s n o t

h o l d , w e c o n s i d e r t h e K l e e n e h i e r a r c h y o f s e t s .

T h e K l e e n e h i e r a r c h y o f p r e d i c a t e s c la s si fi es t h e a r i t h m e t i c a l s e ts i n

c l a sse s Z . , 17. ( n = 0 , 1 , . . ) de f i ned a s f o l l ow s . ~ . i s t he c l a s s o f a l l s e ts A o f

t h e f o r m A =

{ a l (Q a x a ) ( Q 2 x 2 ) . . . ( Q . x . ) P ( a , x . x z , . ' . , x . ) } ,

w her e P i s a

r e c u r s iv e p r e d i c a t e , t h e Q 2 k + l a r e e x i s t e n ti a l q u a n t if i e r s a n d t h e Q 2k a r e

un i v e r sa l qu an t i f i e r s . I I . i s the c l a s s o f a ll s e ts a s abo ve , e xcep t t ha t t he Q 2k+ 1


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252 C .P . Scn~oP,

a r e u n i v e r s a l q u a n t i f ie r s a n d t h e Q 2k a r e e x i s t e n t ia l q u a n t i f ie r s . T h e f o l l o w i n g

f a c t s a r e k n o w n ( s e e D a v i s [ 1 7 ] ) .

(1 ) X o = rl o = z 1 c~ 1-I1 i s th e c o l l e c t i o n o f a l l r e c u r s i v e s e ts .

( 2 ) A e E . ~ A c ~ l-I,, ( A c i s t h e c o m p l e m e n t o f A . )

( 3) Z , u I I , c Z . + l ~ I I , + ~ f o r a l l n > 0 a n d c o n t a i n m e n t is p r o p e r f o r n > 0 .

( 4 ) A E Y ~.+ 1 ~ A i s r e c u r s i v e l y e n u m e r a b l e r e l a t i v e t o a s e t B s I I . .

( 5 ) A e Z . + 1 c~ I I , + 1 ~ A i s a r e c u r s iv e r e l a t i v e t o a s e t B s I I , .

T h e c l a ss I -I . n Z , i s u s u a l l y d e n o t e d b y A . A s e q u e n c e z E X ~ i s t o b e i n Z .

[1-I,] if { n [ z n = 1} is in Z , [ I-I , ] .

L E M M A 3 . 5 . T h e r e e x i s t s e q u e n c e s i n A 2 w h i c h a r e ( 1 ) -r a n d o m .

P r o o f L e t F : X * - + R + b e a u n i v e r s a l ( 1 )- te s t t h a t i s g i v e n b y a r e c u r s i v e

f u n c t i o n g : N x X * - + Q . W e s u p p o s e t h a t F ( A ) < 1. T h e n t h e f o ll o w i n g

p r e d i c a te P : X * ~ { 0 , 1 } i s i n H a :

P ( x ) = 1 ~ V i ~ N g (i ,

x ) < 1 . G i v e n P o n e

c a n c o n s t r u c t z s X ~ r e c u r s i v e l y a s f o l l o w s : z i + 1 = 1 ~ : . P ( z ( i ) l ) = 1 . H e n c e

i t f o l l o w s f r o m t h e a b o v e p r o p e r t y ( 5) t h a t z is i n A z. T h e c o n s t r u c t i o n i m p l i e s

tha t z 9~F .

T o c o m p l e t e t h e p r o o f o f o u r a s s e r t io n t h a t ( 3 ) - r a n d o m n e s s is c o n s i d e r a b l y

n a r r o w e r t h a n ( 1 ) - r a n d o m n e s s , w e e s ta b l is h t h e f o ll o w i n g t h e o r e m .

T H E O R E M 3 .6 .

T h e r e d o n o t e x i s t s e q u e n c e s i n

Z 2 u I I 2

w h i c h a r e ( 3 ) -

r a n d o m .

P r o o f ( I ) L e t z b e a s e q u e n c e i n Z 2 . A c c o r d i n g t o t h e d e f i n i t io n t h i s m e a n s

t h a t

{ n l z .

= 1 } i s i n ~ 2 . T h e n t h e r e e x i s t s a r e c u r s i v e p r e d i c a t e P : N 3 - + { 0 , 1 }

s u c h t h a t , f o r a l l n e N , z , = 1 ~

3 j~NVi~NP( j ,

i , n ) = 1 . W e def i ne a (3 ) - te s t F

s a t i s f y in g z e 9 2v b y s p e c i f y i n g a r e c u rs i v e f u n c t i o n g : N x X * ~ Q . W e d e n o t e

f ( i , n ) = { j [ V P ( L r , n ) = 1 ; j < i }.

r i

T h e f i n i t e s e t f ( i , x ) c a n b e e f fe c ti v el y d e t e r m i n e d . T h e n w e c o m p u t e g ( i , x )

r e c u r s i v e l y a s f o l l o w s : g ( i , A ) = 1 ( i e N ) . I f ( f ( i , n ) ~ A y , = 1) v ( f ( i , n )

= ~ A y , = 0 ) w e d e f i n e g ( i, y ( n )) = 2 g ( i , y ( n - 1) ) (y e X ) . I f ( f ( i , n ) # ~ A

y , = O ) V ( f ( i ,

n ) = e A y . = 1 ) w e d e fi n e

g (i , y (n )) = 0 ( y e X ~ ) .

T h i s

i m p l i e s t h a t

2 l i m g (i , y ( n - 1 )) z , = y , .

l i ra g ( i , y ( n ) ) = i 0 z , ~ y , .

H e n c e

F i s a ( 3 ) - te s t a n d i t f o l l o w s t h a t

F ( y ( n ) ) = { 2 0 y ( n ) = z ( n )

y ( n ) # z ( n ) .

( I I ) L e t z b e a n e l e m e n t o f 1 12 . T h i s m e a n s t h a t

{ n l z ~

= 0 ) is i n Z v H e n c e

t h e a b o v e a r g u m e n t a l so i m p l ie s t h a t z is n o t ( 3 ) - r a n d o m .

4 Random Sequences and the Concept of Minimal Program Complexity W e

s h a ll p ro v e t h a t t h e c o n c e p t o f r a n d o m n e s s t h a t h a s b e e n p r o p o s e d b y L o v e l a n d

[5 ] is n a r r o w e r t h a n t h e d e f in i ti o n o f r a n d o m s e q u e n c es b y M a r t i n - L 6 f .


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A Unified Ap proa ch to the Definition of Ra ndo m Sequences 253

L e t A : X * N ~ X * b e a p a r t i a l r e c u r s i v e f u n c t i o n s a t i s fy i n g A ( x , n ) ~ X

f o r a l l ( x , n ) i n th e d o m a i n o f A . K o l m o g o r o f f [4] d e f in e d t h e c o n d i t i o n a l

c o m p l e x i t y

K a ( x l n )

o f a s e q u e n c e x w i t h l e n g th n r e l a ti v e t o t h e a l g o r i t h m A :

' ~ i f A ( y , n ) ~ x f o r a l l ~ X * ,

Y

K a ( x [ n ) = ~m in { [y] I A ( y , n) = x} o the rw i se .

I f

A ( y , n ) = x :

t h e n y is ca l le d a p r o g r a m t o c o m p u t e t h e s e q u e n c e x b y t h e

a l g o r i t h m A .

Def in i t i on 4 .1 . A f un c t i o n F : X - -~ Q i s c a l l ed a ( 4 ) - t e s t i f t he r e i s a p . r .

f u n c t i o n A :

X * N - - + X *

a s a b o v e , s u c h t h a t

F ( x ) = n - K a ( x ] n )

f o r a l l

s e q u e n c e s x o f le n g t h n . T t v = { z ~ X ~ l li m i n f ,

F ( z ( n ) )

> ~ } i s t h e s e t o f

s e q u e n c e s t h a t d o n o t w i t h s t a n d t h e ( 4 )- te s t F .

T h i s d e f i n it io n o f r a n d o m n e s s h a s b e e n e x p l ic i tl y p r o p o s e d b y L o v e l a n d

i n t e r m s o f t h e u n i f o r m c o m p l e x i t y [ 5 ]. ( 4 ) - ra n d o m n e s s m e a n s t h a t f o r e v e r y

a l g o r i t h m A t h e r e e x i s t i n fi n it e ly m a n y i n it ia l s e g m e n t s w i t h h i g h p r o g r a m

c o m p l e x i t y . T h e o r i g i n a l id e a w a s t h a t e v e r y r a n d o m s e q u e n c e z w o u l d s a t is f y

l im SUpn ( n - - K A ( z ( n ) l n ) ) < ~ f o r e v e ry a l g o r i t h m A . H o w e v e r i t w a s s h o w n b y

M a r t i n - L 6 f [ 7 ] t h a t t h e r e e x i s t n o s e q u e n c e s t h a t s a t i s f y t h i s p r o p e r t y .

M a r t i n - L 6 f [ 7] p r o v e d t h a t ~Re i s a n u l l s e t f o r e v e r y ( 4 ) -t e s t F . H e n c e ,

D e f i n i t i o n 4 .1 s a ti sf ie s o u r a x i o m s ( T 1 ) a n d ( T 2 ) o f a t e s t f u n c t i o n . M o r e o v e r ,

i t i s k n o w n f r o m [7 ] t h a t e v e r y ( 4 ) - r a n d o m s e q u e n c e is r a n d o m i n th e s e n s e o f

M a r t i n - L 6 f . W e s h a ll p r o v e t h a t t h e c o n v e r s e o f t h is t h e o r e m d o e s n o t h o l d .

T H E O R E M 4 . 2 .

T h e r e e x & t s a s e q u e n c e z ~ X a n d a p . r . f u n c t i o n A :

X * x N --> X * s u c h t h a t l i m , ( n - K a ( z ( n ) ln ) ) = oo a n d z is a M a r t i n - L 6 f r a n d o m

s e q u e n c e .

P r o o f L e t F : X * --> R + b e a u n i v e r s a l ( 1 ) -t e s t w i t h F ( A ) < 1 t h a t i s g i v e n

b y t h e r e c u r si v e f u n c t io n g : N x X * - + Q . W e c o n s i d e r t h e ( 1 ) - r a n d o m s e q u e n c e

z

E A

d e f i n e d a s i n t h e p r o o f o f L e m m a 3 .5 . P : X * ~ { 0, 1 } i s t h e f o l l o w i n g

pr ed ica te in 17 a :

P (x ) = 1 .~> Vi~Ng( i,

x ) < 1. G i v e n P w e d e fi n e z r e c u r s i v e l y

a s f o l l o w s : z i + l = m i n { j ~ X l P ( z ( i ) j ) = 1 }. z i s ( 1 ) - r a n d o m a n d w e c o n s t r u c t

a p . r . f u n c t i o n A : X * x N - - - > X * s u c h t h a t l i m , ( n - K a ( z ( n ) ] n ) ) = o o.

Let h: N --->N x X * x Q b e t h e r e c u r s i v e f u n c t i o n d e f i n e d b y h ( N ) = {(i, x,

q) [g ( i , x ) = q }. T h e r e e x is ts a p . r . f u n c t i o n A : X * x N - + X * s u c h t h a t , f o r a l l

x ~ X * , A ( x , i + Ix[) = r ( i , x ) x . H e r e b y w e d e f i n e r ( i , x ) ~ X i r e c u r s i v e l y a s

f o l l o w s :

r O , x ) = A x ~ x * ) ,

w i t h

r ( i + 1 , x ) = r ( i , x ) s ( i , x )

a n d rn _< Ix[ =~ q _< 1

H e r e b y w e s u p p o s e t h a t m i n ~ = 1 .

U s i n g t h is c o n s t r u c t i o n i t c a n e a s il y b e p r o v e d t h a t f o r e v e r y i ~ N t h e r e

ex i s t s a n n , ~ N such t ha t , f o r a l l x ~ X * w i th [x [ >_ n , , A ( x , i + [x]) = z ( i ) x .

C o n s e q u e n t l y l i m ,

( n - K a ( z (n )

]n)) = oo.


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254 C P SCHNORR

I t i s o b v i o u s t h a t t h e r e i s a n i n c o n c e i v a b l e m u l t ip l i c i ty o f p o s si b il it ie s t o

d e f in e te s t f u n c t i o n s s a t i s fy i n g c o n d i t i o n s ( T 1 ) , ( T 2 ). T h e r e f o r e o n e p r o b l e m

s t i l l r em a i ns uns o l ved , wh i ch we s ha l l d i s cus s l a t e r . I t dea l s wi t h t he ques t i on

w h i c h c o n c e p t a m o n g a l l p o s s i b le c o n c e p t s o f t e s t f u n c t i o n s is t h e r e a ll y t r u e

o n e a n d i f su c h a d i s t in g u i s h e d c o n c e p t e x i st s a t a ll .

5 . A n O b j e c t i o n t o th e C o n c e p t o f ( 4 ) -R a n d o m n e s s . W e s h a ll f o r m u l a t e a n

o b j e c t i o n to t h e c o n c e p t o f (4 ) - ra n d o m n e s s , a l t h o u g h ( 4 ) - r a n d o m s e q u e n c e s

p o s s e s s a ll s t a n d a r d s t a ti s ti c a l p r o p e r t i e s o f ra n d o m n e s s s u c h a s t h e l a w o f

l a rg e n u m b e r s a n d t h e l a w o f th e i t e r a te d l o g a r i th m . O u r m a i n o b j e c t i o n sh a ll

b e d i s c u s se d l a te r . I t c o n c e r n s o u r f e el i n g t h a t p r o p e r t i e s o f r a n d o m n e s s a r e

i m p o s e d o n ( 4 ) - r a n d o m s e q u e n c e s t h a t h a v e n o p h y s i c a l m e a n i n g .

A n o t h e r d i f fi c u lt y a r is e s f r o m t h e f a c t t h a t t h e r e i s n o a n a l o g u e t o t h e

m a r t i n g a l e p r o p e r t y ( 2 . 2 ) r e g a r d i n g ( 4 ) - t e s t s .

T h i s l a c k o f a n a n a l o g u e t o ( 2 .2 ) h a s t h e c o n s e q u e n c e t h a t a ( 4 ) - r a n d o m

s e q u e n c e z h a s i n f i n i te l y m a n y i n it ia l s e q u e n c e s z n ) w i t h h i g h v a l u e s F z n ) ) re l -

a t i ve t o a un i ve r s a l ( 4 ) - t e s t F ( h i gh va l ue s F z n ) ) m e a n l ow c o m p l e x it y o f z n) ) .

T h i s f o l lo w s f r o m a n a r g u m e n t b y M a r t i n - L 6 f [ 7] w h i c h , w h e n a p p l i e d t o (4 )-

r a n d o m n e s s , s h o w s t h a t t h e r e e x i s t s n o i n f i n i t e s e q u e n c e z s u c h t h a t l i r a s u p n

F z n ) ) < oo . Be caus e o f Le m m a 2 . 8 t h i s e f fec t is exc l uded a s r ega r ds ( 1 )-

r a n d o m n e s s .

I t s e e m s n a t u r a l t o d e f in e a h i e r a r c h y o f c o m p l e x i t y f o r i n f in i te s e q u e n c e s

a s f o l l o w s ( i n a s i m i l a r m a n n e r t h i s h a s b e e n d o n e b y L o v e l a n d w h o u s e d t h e

un i f o r m c om pl e x i t y m ea s u r e [ 5] ). Le t F be a un i ve r s a l ( 4 ) - te s t . I t s ex i s t ence is

p r o v e d i n [ 4] a n d [1 4]. F o r e v e r y n o n - d e c r e a s i n g f u n c t i o n f w e d e n o t e

( 5 . 1 ) Cs = d a {Z ~ X ~ [ lim i n f F z n ) ) - f n ) ) < oo }.

I f f is b o u n d e d , t h e n C y is e x a c t l y t h e s e t o f a ll ( 4 ) - r a n d o m s e q u e n c e s . F o r a

s l o w l y i n c r e a s in g u n b o u n d e d f u n c t i o n f t h e s e q u e n c e s in C y a r e e x p e c t e d to b e

a p p r o x i m a t e l y r a n d o m . T h i s , h o w e v e r , is b y n o m e a n s t ru e . O n t h e c o n t r a r y ,

w e c a n p r o v e t h e f o l l o w i n g t h e o r e m w h i c h a l s o h o l d s r e l a ti v e t o ( 4 ) - r a n d o m

s e q u e nc e s a n d f o r t h e c o n c e p t s o f r a n d o m n e s s d e r i v e d f r o m t h e u n c o n d i t i o n a l

( c f. [4 ]) a n d a l s o f o r u n i f o r m c o m p l e x i t y m e a s u r e s .

T H E O R E M 5 . 2 . Le t f : N- - + N be a non- decreasi ng unb ounded f unc t i on .

Then t here ex i s t s a s equence i n C f t ha t does no t sa t is f y t he l aw o f l a rge num bers .

F r o m t h e s t a t is t ic a l p o i n t o f v ie w t h e l a w o f l a r g e n u m b e r s is o n e o f t h e

m o s t b a s i c la w s o f r a n d o m n e s s . T h e r e a r e v e r y s im p l e s t a ti s ti c a l te s t s w h i c h

r e j e c t s e q u e n c e s n o t s a t is f y in g th i s l aw . T h e r e f o r e t h e s e q u e n c e s i n C f , e v e n

w i t h a v e r y s lo w l y i n c re a s in g u n b o u n d e d f , c a n n o t b e v ie w e d a s a p p r o x i m a t e l y

r a n d o m . A n d t h i s i s a n o b j e c t i o n t o ( 4 ) - r a n d o m n e s s .

W e r e m a r k t h a t T h e o r e m 5 . 2 s o l v e s t h e q u e s t i o n o f L o v e l a n d [ 5 ] w h e t h e r

t h e r e e x i st s a n o n - d e c r e a s i n g u n b o u n d e d f s u c h t h a t C ; i s p r e c i s e ly t h e s e t o f

a ll r a n d o m s e q ue n c es . B e c a u s e o f T h e o r e m 5 .2 s u c h a f u n c t i o n c a n n o t e x i st .

Proo f . L e t f : N - + N b e a n u n b o u n d e d n o n - d e c re a s in g f u n c ti o n a n d l e t

F be a un i ve r s a l ( 4 ) - t e s t. W e d e f i ne z = z l z 2 . . . z i . . . G X ~ b y i n d u c t i o n .

S u p p o s e t h a t z n i ) i s a l r e a d y d e f i n e d . T h e n w e s e t Zk = 1 f o r n i < k


10/13


s e q u e n c e

x(2ni)

is e q u a l t o z (2 n i) . T h e n t h e r e is a n m > 2 h i s u c h t h a t

F(x(m)) 1 ). O n t h e o t h e r h a n d , T h e o r e m 5 . 2 is n o t a s u i t ab l e d e f i n i t io n re l a t iv e

t o a u n i v e rs a l ( 4) -t es t F . I t f o ll o w s f ro m a n a r g u m e n t b y M a r t i n - L 6 f t h a t i n

t h i s c a se K : is e m p t y f o r a l l s l o w l y i n c r e a s in g f u n c t i o n s f F o r i n s t a n c e , t h e

r e l a t i o n ~ , % 1 2 - : ( ) = m i m p l i e s t h a t

K :

= ~ [7] .

6 . O n t h e T r u e

Concept o f Randomness

T h e d e f i c ie n c y r e s i d in g i n t h e

p r e v i o u s c o n c e p t s o f r a n d o m n e s s is , i n o u r o p i n i o n , t h a t p r o p e r t ie s o f r a n d o m

s e q u e n c e s a r e p o s t u l a t e d w h i c h a r e o f n o s i g n i fi c a n ce t o s t a ti s ti c s. M a n y

i n s u ff i ci e n t a p p r o a c h e s h a v e b e e n m a d e u n t i l a d e f i n i ti o n o f r a n d o m s e q u e n c e s

w a s p r o p o s e d b y M a r t i n - L 6 f w h i c h f o r t h e f i rs t t i m e i n c lu d e d a l l s t a n d a r d

s t a ti s ti c a l p r o p e r t i e s o f r a n d o m n e s s . H o w e v e r , t h e i n v e r s e p o s t u l a t e n o w s e e m s

t o h a v e b e e n v i o l a t e d .

T h e a c c e p t a b le d e fi n it io n o f r a n d o m s e q u e nc e s c a n n o t b e a n y f o r m u l a t i o n

o f r e c u r s iv e f u n c t i o n t h e o r y w h i c h c o n t a i n s a ll r e l e v a n t s t a ti s ti c a l p r o p e r t i e s

o f r a n d o m n e s s , b u t i t h a s t o b e p r e c i s e ly a c h a r a c t e r i z a t i o n o f a ll t h o s e p r o p e r t i e s

o f r a n d o m n e s s t h a t h a v e a p h y s i c a l m e a n i n g . T h e s e a r e i n t u i ti v e l y t h o s e

p r o p e r t i e s t h a t c a n b e e s t a b l i s h e d b y s t a t i s t i c a l e x p e r i e n c e . T h i s m e a n s t h a t a

s e q u e n c e f a i ls t o b e r a n d o m i n t h is s e n s e i f a n d o n l y i f t h e r e i s a n e f f ec t iv e

p r o c e s s i n w h i c h t h i s f a i l u r e b e c o m e s e v i d e n t . O n t h e o t h e r h a n d , i t i s c l e a r

t h a t i f t h e r e i s n o e f f ec t iv e p r o c e s s i n w h i c h t h e f a i l u r e o f t h e s e q u e n c e t o b e

r a n d o m a p p e a r s , t h e n t h e s e q u e n c e b e h a v e s l i k e a r a n d o m s e q u e n c e . T h e r e f o r e

t h e d e f i n it io n o f r a n d o m s e q u e n c e h a s t o b e m a d e i n s u c h a w a y t h a t t h is

s e q u e n c e i s r a n d o m b y d e f i n i t i o n .

I n a s e r i e s o f pape r s [ 9, 10 , 11 , 12 ] w e t r i ed t o r en de r c l ea r t h is i n t u i t i ve

n o t i o n o f ra n d o m n e s s . I t tu r n s o u t t h a t t h e r e a r e r a t h e r d i ff e re n t a p p r o a c h e s

t o t h i s c o n c e p t o f w h i c h a l l l e a d t o e q u i v a l e n t d e f i n it io n s . T h i s p a p e r h a s b e e n

w r i t t e n t o g i v e a c o m p r e h e n s i v e a p p r o a c h b y t e s t f u n c t i o n s .

F r o m t h e p o i n t o f v ie w o f te s t f u n c t i o n s o n e w o u l d c o n s i d e r a s e q u e n c e t o

be r a nd om i f i t s t ands a l l e f f ec ti ve t e s ts . B u t ho w a r e t he e f f ec t i ve t e s t s t o be

d e f i n e d ? It is n a t u r a l t o p o s t u l a t e t h a t a n e f fe c t iv e t e s t F : X * ~ R is c o m p u t a b l e

i n t h e o r d i n a r y s e n se i n s t e a d o f b e i n g m e r e l y c o n s t r u c t i v e .


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256 C P SCHNORR

D e f i n i ti o n 6 . 1 . A f u n c t i o n F : X * ~ R i s c o m p u t a b l e i f t h e r e i s a r e c u r s i v e

f u n c t io n g : N x X * - + Q s u c h th a t [ g(n , x ) - F ( x ) [ < 2 - ( x ~ X * , n ~ N ) .

O u r c o n s i d e r a t i o n s i n S e c t i o n 5 s h o u l d h a v e m a d e c l e a r t h a t a r e a s o n a b l e

c o n c e p t o f te s t f u n c t i o n h a s t o i n c lu d e t h e m a r t i n g a l e p r o p e r t y (2 .2 ). C o m -

p u t a b i l i t y a n d t h e m a r t i n g a l e p r o p e r t y s u f fi ce t o c h a r a c t e r i z e e f fe c t iv e t e st s.

B u t w h i c h s e q u e n c e s a r e r e f u s e d b y a n e f f ec t iv e t e s t ? I n a n a l o g y t o ( 2 .3 ) o n e

w o u l d d e f i n e t h a t a s e q u e n c e z d o e s n o t w i t h s t a n d t h e t e s t F i f a n d o n l y i f

l i m s u p , F ( z ( n ) ) = ~ . H o w e v e r , i f t h e s e q u e n c e F ( z ( n ) ) i n c r e as e s s o s l o w l y t h a t

n o o n e w o r k i n g w i t h ef fe c ti v e m e t h o d s o n l y w o u l d o b s e r v e it s g r o w t h , t h e n

t h e s e q u e n c e z b e h a v e s a s i f i t w i t h s t a n d s t h e t e s t F . T h e d e f i n i t io n o f ~RF h a s

t o r e f l e c t t h is f a c t . T h a t i s, w e h a v e t o m a k e c o n s t r u c t i v e th e n o t i o n l i m s u p ,

F ( z ( n ) ) = o o.

D e f i n i ti o n 6 . 2 . L e t f : N - + N b e a f u n c t i o n . W e w r i te k l i m s u p ,

f ( n ) = 0 %

i f a n d o n l y i f t h e r e e x is ts a r ec u r s iv e , m o n o t o n e a n d u n b o u n d e d f u n c t i o n

g : N ~ N s u c h t h a t l i m s u p , ( f ( n ) - g ( n ) ) > O.

N o w w e p r e s e n t o u r c o n c e p t o f e ff e c ti v e t e s ts [ 10 ].

D e f i n i ti o n 6 . 3 . A f u n c t i o n F : X * ~ R + is a n e f f e c t iv e t e s t if i t i s c o m p u t a b l e

a n d s a ti sf ie s t h e m a r t i n g a l e p r o p e r t y . T h e s e t o f se q u e n c e s t h a t d o n o t w i t h -

s t an d t he t e s t F i s de f i ned t o be ~ l l r = { z e X ] k l i m s u p , F ( z ( n ) ) = o o } .

A sequ ence i s c a l l ed ( 0 ) - r and om i f i t w i t h s t an ds a l l e f f ec ti ve t e s ts . I t i s ou r

t h e s i s t h a t ( 0 ) - r a n d o m n e s s c h a r a c t e r i z e s a l l r e l e v a n t s t a t i s t i c a l p r o p e r t i e s o f

r a n d o m s e q u e n c e s . T o c o n f i r m t h is t h e si s w e sh a l l r e s t a t e s o m e r e s u l ts f r o m

ea r l i e r pape r s .

I n [ 1 1 ] w e e s t a b l i s h e d t w o i n t e r e s t in g c l a s s if i c at io n s o f th e p r o p e r t i e s o f

r a n d o m s e q u e n c e s . A l a w o f s t o c h a s t i c i ty is c a l l e d o f o r d e r ] ( f : N ~ N is a

n o n - d e c r e a s i n g f u n c t i o n ) i f t h e r e i s a n e f f e c ti v e t e s t F : X * ~ R + s u c h t h a t

( 6 . 4 ) l i m sup ( r ( z ( n ) ) / f ( n ) ) > 0

f o r a l l z e X ~ t ha t do no t s a t i s f y t h is l aw .

T h e g r o w t h o f t h e f u n c t i o n f i n d i ca t e s th e i m p o r t a n c e o f th e l a w u n d e r

c o n s i d e r a t i o n . I t is s h o w n i n [1 1] t h a t t h e l a w o f l a r g e n u m b e r s h a s a r a p i d l y

g r o w i n g o r d e r f u n c t i o n . T h i s is in h a r m o n y w i t h t h e f a c t t h a t t h e l aw o f la r g e

n u m b e r s i s c e r t a i n l y o n e o f t h e m o s t b a s i c la w s o f p r o b a b i l i ty . I t i s a l s o s h o w n

i n [1 1] t h a t t h e c l a ss o f l aw s h a v i n g t h e s a m e o r d e r a s t h e l a w o f l a r g e n u m b e r s

is a v e r y r e a s o n a b l e c l as s . A s is th e c a s e w i t h s o m e o t h e r c o n c e p t s o f r a n d o m n e s s ,

t h e s e t o f r e c u r s iv e s e q u e n c e s c a n b e c h a r a c t e r i z e d i n t e r m s o f t e s t f u n c t i o n s .

A s e q u e n c e i s r e c u rs i v e i f a n d o n l y i f i t d o e s n o t s a t is f y so m e l a w o f o r d e r

f ( n ) = 2 (as wi l l be pr ov ed in [16] ).

T h e a b o v e r e s u l ts r e l a t iv e t o t h e o r d e r o f a l aw h o l d f o r e f fe c t iv e t e st s , a s

w e l l a s f o r ( 1 ) -t e st s, i f a s u i t a b le f o r m u l a t i o n i s c h o s e n . H o w e v e r , t h e s e c o n d

c l a ss i fi c a ti o n o f t e st s w h i c h i s b a s e d o n t h e c o m p l e x i t y o f t e s t f u n c t i o n s ( r e g a r d -

i n g t h e a m o u n t o f ti m e a n d s p a ce t o c o m p u t e t h e m [ 3] ) is m e a n i n g f u l f o r

ef fec t ive t es t s only .

T he r e i s no un i v e r sa l e f f ec t i ve t e s t. T o eve r y e f f ec ti ve t e s t F a r ec u r s i ve

s e q u e n c e n o t i n T t~ c a n b e c o n s t r u c t e d . T o e v e r y e ff e c ti v e t e s t F t h e r e e x is ts


12/13

A U n i f i e d A p p r o a c h t o t h e D e f i n i t i o n o f R a n d o m S e q u e n c e s 2 57

an equ iv a len t e ffec tive t e s t F , * ~ Z(2 ) th a t i s r e cu rs ive (Z(2 ) i s the s e t o f

a l l f in i te dua l f rac t ions in R+ ) . L e t u s con s ide r a f ixed com plex i ty cl a ss (acco rd -

i n g th e a m o u n t o f t i m e o r s p a c e ) o f e f fe c ti v e t e s t f u n c t i o n s F : Y * ~ Z ( 2 ) .

O ne ca n co ns t ru c t r ecu rs ive s equences w h ich wi l l s t an d eve ry e f fec t ive t e s t o f

a f ixed complexi ty c lass [11] .

A n in f in it e s equence z ha s a ce r t a in deg ree o f (0 ) - rand om ness i f i t s t ands a l l

t e st s o f a c o r r e s p o n d i n g c o m p l e x i t y cl as s. T h e d e g r e e o f ( 0 ) - r a n d o m n e s s y i e ld s

a c l a s s i f i c a t ion o f r ecu rs ive s equences . Sequences tha t have a ce r t a in deg ree o f

(0 ) - rando m ness hav e v e ry in te re s t ing p rop e r t i e s ( [11 ], [12 ]) .

I t i s th e o p i n i o n o f t h e a u t h o r t h a t t h i s c l a s si f ic a t io n o f r e c u r si v e s e q u e n ce s

r e l a t i v e t o t h e i r d e g r e e o f ( 0 ) - r a n d o m n e s s i s a n i m p o r t a n t a r g u m e n t f o r t h e

con cep t o f e f fec tive t e s ts . Th e ex i s t ence o f (1 ) - rand om sequences a s we l l a s tha t

o f ( 0 ) - r a n d o m s e q u e nc e s c a n b e p r o v e d b y n o n - c o n s t r u c t i v e m e t h o d s o n l y .

T h e s e s e q u e n c e s e x i s t o n l y b y v i r t u e o f t h e a x i o m o f c h o i c e . H o w e v e r , w e c a n

a p p r o x i m a t e t h e b e h a v i o u r o f ( 0 ) - ra n d o m s e q ue n c e s b y c o n s t r u c t iv e m e t h o d s .

T h i s d o e s n o t h o l d f o r ( I ) - r a n d o m s e q u en c e s. B e c a u s e o f T h e o r e m 3 .2 , (1 )-

randomness i s no t equ iva len t to (0 ) - randomness ( s ee a l so [9 ] ) .

A n o t h e r i m p o r t a n t a r g u m e n t f o r o u r t he si s w h i ch p r o p o se s ( 0 ) - ra n d o m n e s s

a s t h e r e a l l y t r u e c o n c e p t o f r a n d o m n e s s is t h a t s o m e d if f e re n t a p p r o a c h e s

lead to an e qu iv a len t de f in i t ion . I t is p roved in [10 , Pa r t I ] t ha t a s equence i s

( 0 ) - r a n d o m i f a n d o n l y i f i t is n o t c o n t a i n e d i n a n y n u l l s et i n th e s e ns e o f

Brouwer . Th i s concep t o f nu l l s e t i s cu r ren t in cons t ruc t ive ana lys i s ( s ee , fo r

i n s ta n c e , [1 ]) a n d d a t e s b a c k t o a n i n t u i t io n i s t i c f o r m u l a t i o n o f L . E . J. B r o u w e r

[ 2 ] , A r e a s o n a b l e c h a r a c t e r i z a t i o n o f ( 0 ) - r a n d o m s e q u e n c e s b y t h e i r p r o g r a m

c o m p l e x i t y c a n b e f o u n d i n [1 6]. B u t w h a t i s m o s t s u r p r i s in g i s t h a t t h e o r i g i n a l

i d e a s o n w h i c h v o n M i s e s b a s e d h i s c o n c e p t o f co l le c t iv e c a n b e m o d i f i e d t o

cha rac te r i ze exac t ly the (0 ) - ra ndo m sequences . I t is sho w n in [10 ] tha t a s equen ce

z is ( 0 ) - r a n d o m i f a n d o n l y i f e v e r y s eq u e n c e y w h i c h is a n i m a g e o f z u n d e r a

cons t ru c t ive me asu re -p re se rv ing m ap H : X Q ~ X ~ s a ti s fi e s the l aw o f l a rge

n u m b e r s . T h i s m e a n s t h a t t h e c o n c e p t o f S t e l le n a u s w a h l in [ 8] h a s t o b e r e p l a c e d

b y t h e n o t i o n o f c o n s t r u c t i v e m e a s u r e - p r e se r v i n g m a p . A l l t h e s e a n d s o m e

add i t iona l r e su l t s w i l l be inc luded in [16 ] .

REFERENCES

[1] E. BISHOP,

Foundations o f Constructive Analysis

M c G r a w - H i l l , N e w Y o r k , 1 96 7.

[2 ] L . E . J . BROU WER, Beg r i indung de r M e ngen leh re una bh / ing ig vo m log i s chen S a tz vom

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602-619.

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13/13

25 8 C P SCHNOIU~

[8] R. VON MiseS, Gru ndlag en der Wahrscheinlichkeitstheorie,

Math .

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theorie Verw. Gebiete

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Z. Wahrscheinlich-

keitstheorie Verw. Gebiete

15 (1970), 297-312, 313-328.

[11] C. P. SCHNORR, Klassifikation der Zufallsgesetze nac h Kom plexit~it und O rdnu ng,

Z . Wah rscheinlichkeitstheorie V erw. Gebiete 16 (1970), 1-21.

[12] C. P . SCHNORR, 13b er die Zufiilligkeit un d den Zufallsgrad von Folgen, Symposium on

Formale Sprachen und Automatentheorie , Oberwolfach, 1969.

[13] E. SPECKER,N icht kon struk tiv beweisbare S~itze der A nalysis, 3 . symb. Logic 14 (1949),

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[14] R. J. SOLOMONOFF,A fo rm al theory o f inductive inference, I,

Information and Control

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[15] J. VILLE, ~tude Critique de la Notion de Collectif, Gauthiers-Villars, Paris, 1939.

[16] C. P. SCHNORR, Zuf~illigkeit un d W ahrscheinlichkeit,

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preparation).

[17] M. DAvis,

Computability and Unsolvability,

McGraw-Hill , N ew York, 1958.

Received 24 Apr i l 1970, and

in rev i sed far m 11 Augus t 1970)

Schnorr - A Unified Approach to the Definition of Random Sequences

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