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5wj/Martin-Lof35rffandom*hj&ML-random+j'K(Ff18 nUR7!w̱nmj 냀m$݀nj'KH*3#(hV5wj/35ML-rffandomj'*Ff19 lim.l(w̱nmo?j.50j.klՎSL=jwRj)5j 1=2j+mē넀j*H*3#(hVj/dML-rffandomj+/ *I*Ff20 nUR7!2n5냀gj.3#(X@Ej 냀m$݀nj'KH*3#(hVj/5ML-rffandomj'/*ML-random5j']*3#(hVj/*jK4gJj'+oj+l2j*(냀ܓ݀j mhj.ohj'Kpr B21 fxUR2 1 :x5j/ML-rffandomYӽg=1Er(3.2){*w]xg{+test5j/fno݀m$kj*.'testQj/݀m$kj'Kj #&!9_fxUR2 1 :x5j/ML-randomj'*~+g /M[allXtests *\̱iX̱id 1 |([allXtests *\̱iX̱id 1 ) /X 8獒MallXtests04(\̱idڼX̱i 1 ) 8덍 =X 8獒MallXtests04limsup I捒8Bñi!1V6h(X̱id 1 ) /X 8獒MallXtests04limsup I捒8Bñi!1V6h2 i =M0 VC15 1 : j(H!*K.'5ML-randomj'H!*3#(hVj.mݓkj/hPmݓkj(*KjH#&5fxvȽ221b: x5j/ML-randomT}zgUR=(21 ) 0UR=15j'KE2j$>J;(S)9&.g\k3#(hVj/5(Martin-L ofj.4gJj')IS @j'K!qэU3.2%@ML-randomness . b> ff
cmmi10K+HK6^b##ML-random5j*-3#(hVj/pre xj.Kolmogororvcomplexityj.-Otoj*(n#j+H#&mhj.H+n6o^ibjILK Cpr B22(ML-randomnessj(BKolmogoro vcomplexity)jxj OS(1)j lh jK(/n$o&j'Kj*O!p"7wis35ML-rffandomQ( )UR9c8nK ܞ(w̱nP)>n c{*w]2xg{+ (= ))BNQN29j.imݘk5Xut=f(i;x)A?:K ܞ(x)KmkX 20=URf(jxj~ i;x)3:(2i;x)UR2X g5j+ & bX w(`;x)2X ҟ0 2 ` := wRX =ʱi hX "% x2X8:2i 1~2 jxj+i := wRX =ʱi 2 i
P'X "%dرx2X8:2i =2 jxj Zc ! wRX =ʱi 2 i
R2 2i ! wR1 VC16 L1 : j(5Kraftj.#oianmwj lh jK5X 20j/fדno݀ߓm$ݓkj*.'oh8j+HJKolc20nj #&5xUR2X2i
jj*I0K ܞ(x)jxj/G i+c20nj(*Kj&nu4j.nj+넀j&K ܞ(w̱nP)UR>n/G cj':(K(K5iۀ=c!+c20nj(:(
X2i
jj+/wj.pre xj fj>L&K j .H*w̱n2URX2i
jj+$&wn cURn OS(1)j+/*I*E2 VC17 1 |a4}R- aComplexitry~. & G
dgjkh10릀~ & G
dgjkata103#( & G
dgjkl10V~. & G
dgjki106^ & G
dgjkj10b~@ j.lRj g%hij.mgj*냀݀j'KM j'lUf j eMartin-L ofrandomnessj+H#& pre xj ykjKolmogororvcomplexityjRyB#&K*KMe>pn OS(1)+yg\k3#(hVj.n6o^j (IIL j'/Ivpre xj ojIvKolmogororvcomplexity5*KN<n+jRB#&KH*3#(hVj+$&.Kn6o^ibjRmoj"5VU4.1%@( b>
cmmi10f Č( fU-test##f
4jR5njHJBF#J넀݀jKflj (Kj.(
g\kj.nj+넀j&K ܞ(w̱nP)UR >'/'
*j> ).H*4gJj K. *j*. +j>J/#
J&*j> Martin-L ofrandomnessj.5testj+넀݀j&mhj.H*mkjRkzjKp룀 B24(fGB-test)^fmk5XFս2URBNo29j )j.lu냀jRgG넀jK(
+XjRfG-testj(kj6(+K _Xj/fno݀m$k _X̱i,=URfx:(i;x)2X̱idڽg5j(K(35s2X̱i= )jsj=i VC18 ʠ1 :Z _on j*olՖ5cj(9&.ij+넀j&̱fw (X̱id)UR22 K (i)+c NfG-testj.53i gj.lu냀j'+cj(iafGojR+j ./jB(B(5Kj.ofj++OS(1) j .iafGoj fj>L&K A'K!U4.2%@2($.6^b##pre xj.Kolmogororvcomplexityj jKpre xj.opj5nj.flՑ5f
4j'lhj IgjIL/)&Kg\k3#(hVjRkzjKj.H*g\k3#(hVj/5̱fiX̱j+?j'K{*w]xg{+(= ))5mhj.H*fno݀m$kjRkzjKsVXF=URf(i;s):jsj=iand|K ܞ(s)fG(i)+c 09gEr(4.2)j%'oh513jRg֓jK(#X̱i\$22f (i) K (i)+c+c-:0@j(*Kj cj/oh13j+'&Ko lj'Kmj#& ̱fw (X̱id) А3= uHX "% ̢s2X8:i 2 f (jsj)Es(4.3) Zc А3= uHX "% ̢s2X8:i 2 f (i)Es(4.4) Nƍ А3= ̢#X̱id2 f (i)Es(4.5) wK ̢2 K (i)+cEs(4.6)j(Z*J5Xj/fG-testj(Z*KjH#&W1 **9 ꩼnK ܞ(w̱nP)f(n)+OS(1)j*ZI05w[2limsup'Ti,X̱id21j'K(( =)5jKfG-testX+j+넀j&w2URlimsup'WTi,X̱id21 =j'K(K ̱fw (X̱id) А3= uHX "% ̢s2X8:i 2 f (jsj) Zc А3= uHX "% ̢s2X8:i 2 f (i) Nƍ А3= ̢#X̱id2 f (i) wK ̢2 K (i)+c+c-:0 VC19 1 : j I ?#X̱i,UR2 f (i) K (i)+cEr(4.7) j'KXj/Ffדno݀ߓm$ݓkj*.'5Xj.kjRmh2j&냀ȓm$f֖5Aj FK5ij.minimalprogram
xjR5pj(SX̱i*j+Aj.mh2l j.l jRnqjL (
.x2X̱i*j.ikjR5q3nj(K(4.7)5mwjHJq3nj.opjRfG(i)& K ܞ(i)+c5j.kopj+'
Kj> Kj.ofjHJjpjUR=K(i)j'Kj #&5jpqn9jUR=K ܞ(i)+fG(i) K(i)+cUR=fG(i)+c5j(*Kmhj.H+&5pq3njRnqh2j(&xjRmh2jK냀m$f֓Mj kjlj'
K" \h1. _nqh2xj.pre xj9&+$&gEn냀m$f֓U+jRihVmkjK\h2. _U+j omHj xj.pre xjRp5U@(p)j.o&jRij(K\h3. _xUR=pqXj(jqn9j6=fG(i) jpj+cj # Ig\kK 7j+nqjK\h4. _AjRmkj&X̱iOj.qXigj.o&jRmh2jKjK(5Mj.domainj/pre x-freej+*K _xz1;1xz2AR:*KM%.domain.(Kxz1ʫ=
pz1qz1;1xz2=
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I=Epz2 _**I0qz1A(qz2.*p/xz1;1xz2 :*#&K.'qz1 c6=_qz2(*JD/ _Jxz1A(xz2/pre x-free'K" mj#&oh56j I7K ܞ(x)vfG(i)+c+c̱M
'j(*Kc̱M I%j/mݓkX*냀ȓm$f֘M@+j+H#&o j>K5i;xj+/g\f냀j*olj'K݀٘ojHJg\kj.ij+$&wl2¼X̱id21 =j'# Ig\kj.5nj+넀j&K ܞ(w̱nP)URfG(n)+OS(1)5j(*KE2"5VU4.3%@늀4.뢀##jBY(B(/5pre xj.Kolmogororvcomplexityj 5njHYJojg\k3#(hVj+$&Martin-L of randomnessj+넀˓݀jKH*n6o^ibjR # .'K >J> * # o.h525jRfh,jK*I0f-testj.ofj l8ojKH+m@jOLK.'LRFK
AK(. 2($.ikMj'Kj(+5X̱i0ǽ22i*j(lu냀jRfhj&? j> 5̱f hFj0j.H*fGh oj*f
m> |j gj(M'K VC20 1 b먀aA- b> G
cmmi10K0~. & G
dgjkc10뫀d$a7 & G
dgjka10멀}쎟? flՖ5mjRm(x)=minfK ܞ(yn9):yeҽxg5j(ojAKjK(mj/o
k넀݀ޘj&g\k넀j+nޘm jKQGflj(*K _#fK ȁ(yd)`:y_m(x)j(*K{*w]xg{+o
kj+넀݀j&g\k넀j+nm jKfnoflՑpfoj Jg\kj.5xj+넀j&fG(x)URm(x)5j'DV# (&g^ljRn&jf
4j.domainj/fדno݀ߓm$j*g\kmݓkj'Kfדno݀m$j*g\kmkj/fnoj*g\kmkjRfj@ _AR\똀렀럀ۀ$란(K똀렀럀ۀ$.럀똀 IA/럀 *똀렀럀똀뗀Փ .do-
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4j.domainj+fj>LKfnog\kmkjRB`;j(K# Ygn9(x)UR=f\(30cfG(x); Y(x2B ) cfG(y) yË=minꚽfz52BX:z g\kj.xj+넀˓j &5gn9(x)URm(x)j'K _xx=A;2DB**I0럀 *y(<A;x)2B #&gd(x)A;=f-(y)m(y)m(x)(*K.
_' VC21 A1 : jf'5M@(a)o=maxMfx:K ܞ(x)agj(fflMjRkzjKKj.݀דkj/g\k넀j+nޓm j0K.'5Mj/nu4j.xj+$&ofjLKj> M@(a)+1s=minfx:m(x)>agj(*#&Kmj#&g\kj.5aj+$&# 7ѼM@(a)URagmax3|fx:gn9(x)a+1gV(A.1)j(ianmwj ljJh j$ _\:.x+:$&gd(x)Zm(x)(:*K.H*x+뙀˓&a+1Z=m(x)*m
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k뗀 I.H*a/\+K+(
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K$ pr B27. g\LKfnoj*g\kmkj(KjK(gn9(E0(x))UR=minHfyH:K ܞ(yn9)x;Fy2Ag5j/"fדnoflՓj(*K*Alhj'/K=f
4j I+j #&g\kj.5xj+$&5K ܞ(gn9(E0(x)))mxj(*KjL/K ܞ(gn9(E0(x)))mslog >x+loglog%:x+UN+2log-KWi%x@gn9(x)_!=sjxj+`(jxj)+`(`(jxj))+UN+` (i)
R(jxj)_!2x+log(2log-G2x)+UN+log(2log-I((2log-G2x)))31| 31 2{z 31 2}'Í<i!ٍj(lh݀j IgjILKoh528j(P
SpxU1=xj nm jP
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UR00#yj*.'>Uz 1 =s1 1 UZ=UR0031UR| 31 {z 31 }t+ ߭`1* 1t q1#B1 1 [0031| 31 {z 31 }t+ .5`1* 2
it q2"j1 1[UN\0131|{z}
Ɋti?`1* 1I 1[131|{z}
Ɋ>=ti?`1 1+j(if
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v X T/i=1m2 jt8:i,rj ꦞ 2 jtX.N n kj+1 Zj* .'5`̱n+1v/cjṯN n ]Yj+1>jṯN n ]Yj5j' Kmj#&ic=minfj6:`̱n+1v/jṯjf
jgj i9j(LK a`̱n+1Jj+(넀˓݀jKiqkj/5s̱n+1=ݍۤ` qn+1#DE:::uD>jṯi 1AVj>쾘m` qn+1Ah z 31
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{ jṯi01!$̽j2R>:::uD>jṯi1j>jṯi+1AVj>:::uD>jṯN n ]Yj \j(XY*K I5ft1;:::ʜ;ṯi 1AV;ݍ f` qn+1#DEz 31 {}| 31 {{
ṯid01'c ;:::;ṯid1;ṯi+1AV;:::;ṯN n ]Yg5jRXYfjA&ft1;:::ʜ;ṯN qn+1ngj(
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5fs1;s2;:::ʜ;s̱n+1̼;t1;t2;:::ʜ;ṯN qn+1ng5j/Qxpre x-freej*mݕ kj+*K If냀5(C.2)jB ljJh j#&Kmj#&.KJ@j/g\kj+넀jILKE2iqkj.OJ&.݀ojRiqkgj'hFmojK(l5C.1j.H+*KwqGW ꍍ\. bC
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cmr101 C
t(1bDpn 5 pa 366 -208pa 85 0fp pn 5 pa 366 -208pa 659 0fp ĕx
t(2bEpn 5 pa 752 -208pa 366 0fp pn 5 pa 752 -208 pa 1429 0fp `1V=UR2;s1=00&\%Ѝ bC
s(1 }C
s(2 "C
t(1}Dpn 5 pa 297 -277pa 85 0fp pn 5 pa 297 -277pa 520 0fp bDpn 5 pa 486 -346pa 85 0fp pn 5 pa 486 -346pa 674 0fp x
t(2bEpn 5 pa 871 -208pa 486 0fp pn 5 pa 871 -208 pa 1668 0fp `2V=UR3;s2=010D\% bC
s(1 }C
s(2 "C
t(1}Dpn 5 pa 297 -277pa 85 0fp pn 5 pa 297 -277pa 520 0fp bDpn 5 pa 486 -346pa 85 0fp pn 5 pa 486 -346pa 674 0fp x
s(3bDpn 5 pa 882 -208pa 486 0fp pn 5 pa 882 -208 pa 1679 0fp `3V=UR1;s3=1 bC
s(1sC
s(2 "C
s(4 bC
t(1 "Dpn 5 pa 228 -277pa 85 0fp pn 5 pa 228 -277pa 382 0fp x
t(2 "Epn 5 pa 406 -277pa 228 0fp pn 5 pa 406 -277pa 738 0fp sCpn 5 pa 525 -346pa 85 0fp pn 5 pa 525 -346pa 644 0fp bCpn 5 pa 644 -346pa 85 0fp pn 5 pa 644 -346pa 764 0fp ȭ
s(3bCpn 5 pa 902 -208pa 644 0fp pn 5 pa 902 -208 pa 1720 0fp `4V=UR5;s4=01100 c clC.1:8iqkj.OJ& VC26 31 dz몀뫀? [1]' MingpsLi,PraulM.B.Vit anyi,KolmogorovComplexityanditsApplications,S ' 4E f`h/S)6#5I KJ@j(imj,fN,1994.! [2]' MingJLi,bPraulVit anyi,bA2nIntrffoductiontoKolmogorovComplexityandItsAppli-' cffations,Springer-VVerlag,1993.[3]' G.cJ.Chaitin,A2lgorithmicfInformationTheffory,CamrbridgecUniversityPress,1987.[4]' G.J.Chaitin, IncompletenessTheoremsforRandomReals,Affdvancesqinappliad' mathematics8:8119-146,1987.[5]' G.J.Chaitin,Information35Rffandomness&Incompleteness,WVorldScienrti c,1987.[6]' H.pRogers.Jr.,HTheffory ofRecursiveFunctionsandE ectiveComputability,HMaGrarw-' Hill,1967.[7]' O.pWVatanabSe(Ed.),HKolmoffgorov ComplexityandComputatinalComplexity,Springer-' VVerlag,1992. VC27
; 1l~ & G
dgjsy10 & G
dgjkd10 & G
dgjkc10 & G
dgjkj10 & G
dgjki10 & G
dgjkl10 & G
dgjkata10 & G
dgjkh10 &
3
dmjkk10 &
3
dgjkl10 . dgjkb12 & ff
dgjkh10 & ff
dgjki10 &
3
dmjki10 &
3
dmjkh10 &
3
dmjkd10 &
3
dmjkata10 &
3
dmjkj10 &
3
dmjsy10 &
3
dmjke10 &
3
dmjkg10 &
3
dmjkb10 &
3
dmjkf10 &
3
dmjhira10 &
3
dmjkl10 &
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dmjkc10"V
3
cmbx10K`y
3
cmr10 . dmjko12 & ff
dgjke10 & ff
dgjkf10 & ff
dgjkd10 & ff
dgjhira10 & ff
dgjkj10 & ff
dgjkl10 . dmjkq12 . dmjroma12 . dmjkg12 . dmjkc12 . dmjkata12 . dmjsy12 & ff
dgjkk10 & ff
dgjkb10~ & G
dgjhira10} & G
dgjkf10| & G
dgjkg10{ . dgjsy12z . dgjkg12y . dgjkd12x . dgjkk12w . dgjkf12v . dgjke12u . dgjkc12t . dgjkj12s . dgjki12r . dgjkl12q . dgjkata12p . dgjkh12o . dmjkh12n . dmjki12m . dmjke12l . dmjkf12k . dmjkd12j . dmjhira12i . dmjkj12h . dmjkl12g . dmjkk12f . dmjkb12e . dgjhira12d & G
dgjke10c & G
dgjkk10a"V G
cmbx10_ & ff
dmjkh10^ & ff
dmjke10] & ff
dmjkj10\ & ff
dmjkf10[ & ff
dmjkl10Z & ff
dmjkb10Y & ff
dmjkg10X & ff
dmjkd10W & ff
dmjkc10V & ff
dmjki10U"V ff
cmbx10T & p
dmjkj10S & p
dmjki10R & p
dmjkl10Q & p
dmjkata10P & p
dmjkd10O & p
dmjkk10N & p
dmjke10M & p
dmjhira10L & p
dmjkh10IK`y ff
cmr10HK`y p
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