Complexité de Kolmogorov et notion de mot aléatoire 1 Définition

X1, X2, . . .
{0,1}
x1x2. . . {0,1}
A={0,1}
A
B
A|x|x
M x ∈A∗
x M
KM(x) = min{|y||M(y) = x}

min ∅= +∞
M y
x
dM(x)x
|dM(x)|=KM(x)M(dM(x)) = x
dMdM(x) = dM(x0)M(dM(x)) = x M(dM(x0)) = x0
x=x0
M
M
M0
∃C∈N∀x∈A∗KM(x)≤KM0(x) + C
M
M0M
M0
M
y
M
M0C=|< M0, ² > |
M0(y) = x M(< M0, y >) = x|< M0, y > |=C+|y|
KM(x)≤KM0(x) + C
M
x= 011000000 y
|y|= 456 KM(x)≤456
x x

M K(x) = KM(x)
d(x) = dM(x)
∀x∈A∗K(x)≤ |x|+O(1)
f K(f(x)) ≤K(x) + O(1)
∀ncard{x|K(x)≤n}<2n+1
I
∀x∈A∗I(x) = x KI(x) = |x|M
I
N
∀y∈A∗N(y) = f(M(y))
M N
∀x∈A∗K(f(x)) ≤KN(f(x)) + O(1) {y|N(y) = f(x)}⊃{y|M(y) =
x} ∀x∈A∗KN(f(x)) ≤K(x)
n Bn={x|K(x)≤n}d(Bn)⊂(A+²)nd
card(A+²)n= 2n+1 −1
n f :x7−→ xn
K(xn)≤K(x) + O(1)
K
K
f f(m)x
K(x)≥m f
K
m K(f(m)) ≥m f
K(f(m)) ≤K(m) + O(1) K(m)≤ |m|+O(1)
m≤K(f(m)) ≤K(m) + O(1) ≤ |m|+O(1)
K
K(x) = lim
n∞k(x, n)
x K(x)<∞
kn

c∀x K(x)≤ |x|+c
k(x, n)|x|+c M
n x k(x, n) =
|x|+c+ 1
kn K(x)<∞N
M d(x)N x
n≥Nk(x, n) = K(x)
c∈Nx c K(x)<|x| − c
x c
n c ≤n c
n c
n1−2−c
n c ≤n2n−c
n−c−1 2nn
c n c
2n−c/2n= 2−c
c
c
P
P
lim
n∞
card{x|¬P(x)∧ |x| ≤ n}
2n= 0
P
c > 0
P
P
c > 0
f i ieme
P
P

K(f(i)) ≤K(i) + O(1) C
C0∀i K(i)≤ |i|+C0
N∈N
∀n≥Ncard{x|¬P(x)∧ |x| ≤ n}
2n≤2−C−C0−c
P
x P |x| ≥ N n =|x|
∃i∈Nf(i) = x i ≤2n−C−C0−cK(i)≤ |i|+C0≤
n−C−c
K(x) = K(f(i)) ≤K(i) + C≤n−C−c+C=n−c x
P N
P
P
P
n k < 2n
N π(N)π(N) = O(N/ log N)
n
O(1/n) 0
c c
ANA
LANL⊂A∗
µB(AN)µ(LAN) = Px∈L2−|x|
µ(AN) = µ({²}AN) = 2−0= 1
C⊂ANU1, U2,...Un
C⊂Tn∈NUnlimn∞µ(Un) = 0
[0,1]
x∈ANn x|n=x1x2. . . xn
x K(x|n)≥n+O(1)
x
K(x|n)≤n−log n+O(1)