Automates - banques

O(n)O(n2)
NUMERO (ui)
ui+1 = 3321 ·ui+ 5701
1000 u0u1u2u20 u100 u1000 u0=
NUMERO
A=hQ, δ, F, q0iQ
F⊆Q δ :Q× {0,1} → Q
q0A[q]A q
{0,1}∗{0,1}δ
δ∗:Q×Σ∗→Q
δ∗(p, ) = p
δ∗(p, σ ·a) = δ(δ∗(p, σ), a)

σ σ ∈ {0,1}∗a0 1 σ δ∗(q0, σ)∈
F L(A)A
A=hQ, δ, F, q0i
Q={0, . . . , N −1}
δ
F
q0= 0
ANBNN= 19 97 503
N AN=h{0, . . . , N −1}, δ, F, 0i
δ(q, a) = u2∗q+amod N
q∈F(uqmod N)≤N
3
u0=NUMERO BNu0=
NUMERO + 11
F U ={q∈Q|∃a∈ {0,1}, δ(q, a)∈F}
V={q∈Q|∀a∈ {0,1}, δ(q, a)∈F}ANBNN= 19 97
503
n= 0,1,2,3,4 20
ANBNN= 19 97 503 H(q, n)
n A[q]q∈F H(q, 0) = 1 H(q, 0) = 0
n≥0H(q, n + 1) = H(δ(q, 0), n) + H(δ(q, 1), n)
10e100e1000e10000eANBNN= 19 97
503
A=hQ, δ, F, q0iA=hQ, δ, F , q0iF=Q\F
A A L(A) = L(A)
ANBNN= 19 97 503 n= 0,1,2,3,4
20
B=hQ0, δ0, F 0, q0
0iA∩B=hQ×Q0, δ ⊗δ0, F ×F0,(q0, q0 0)iδ⊗
δ((q, q0), a) = (δ(q, a), δ0(q0, a)) A∪B=hQ×Q0, δ⊗δ0,(F×Q0)∪(Q×F0),(q0, q0
0)i
L(A∩B) = L(A)∩L(B)L(A∪B) = L(A)∪L(B)

AN∩BNAN∪BNN= 19 97 503
n= 0,1,2,3,4 20
A=hQ, δ, F, q0i(A) = {q∈Q|∃x∈ {0,1}∗, q =δ∗(q0, x)}
(A) (A){q0}
δ(q, a)q∈Acc(A)a∈ {0,1}
q∈Acc(A)a∈ {0,1}δ(q, a)∈Acc(A)
(AN) (BN) (AN) (BN) (AN∩BN) (AN∪BN)
N= 19 97 503
(A)A(A)
(A)
Q F U V U V
(AN) (BN) (AN) (BN) (AN∩BN) (AN∪BN)
N= 19 97 503 (A)
L Q L ⊆Q×Q L
A∀q, q0∈Q, ∀a∈ {0,1}
L(q, q0)∧q∈F⇒q0∈F
L(q, q0)⇒L(δ(q, a), δ(q0, a))
L
L A
L
A(A)
P∆(P) = (P×P)∪(P×P)
A
L= ∆(F)
C L a ∈ {0,1}P={q∈Q|δ(q, a)∈C}
L6=L∩∆(P)L L ∩∆(P)
L A L

L|Q|L[q]
q Q ×Q
P L
L∩∆(P)
L0= ∆(F)Li+1 =Li∩∆(Pi)
Pi={q∈Q|Li(δ(q, 0),0)}Lii= 0,...,6
ANBNN= 19 97 503
0
ANBNN= 19 97 503
0
(A)
(AN) (BN) ( (AN)) ( (BN))
( (AN)) ( (BN)) N= 19 97 503
Q F U V U V
( (AN))
( (BN)) ( (AN∩BN)) ( (AN∪BN))
N= 19 97 503
Q F U V U
V N
σ=σ0· · · σk−1σ∈ {0,1}∗
(σ) =
k−1
X
i=0
σi·2i
σ(σ)
n:{0,1}∗→Nnn(σ) = (x0,· · · , xn−1)
xu=X
i:i·n+u<k
σi·n+u·2i
σ= 1011000010 (σ) = 269 2(σ) = (19,2) 3(σ) =
(3,0,5) 4(σ) = (5,0,1,1)

n
n(σ)σ n σ
20 σi= 1 uimod 819 <409 σi= 0 u0=NUMERO
n(σ)n= 1,...,5
A n A
nn(σ)σ∈L(A)
Pn−1
i=0 ci·xi=dZA=hQ, δ, F, q0i
Q={∅} ∪ ({0, . . . , n −1} × Z)
a= 0,1δ(∅, a) = ∅
a= 0,1hi, ei ∈ {0, . . . , n −1} × Z
δ(hi, ei, a) = 
hi+ 1, e +a·ciii < n −1
h0,e+a·cn−1
2ii=n−1∧e+a·cn−1= 0 mod 2
∅
F={0, . . . , n −1}×{0}
q0=h0,−di
(A)
(A)⊆ {∅} ∪ ({0, . . . , n −1} × {−M,...,M})
M= 2 ∗(d+Pn−1
i=0 |ci|)
n Ekn
n−1
X
i=0
ci·xi=d
ci= (uimod 13) −6b= (unmod 13) −6u0=NUMERO + 17 ∗k
E0n= 4,...,8
(A)
E0n= 4,...,8n10e
100e1000e10000e
SkE0, . . . , Ek
Sk
S2, . . . , S5n= 4,...,8
Sk
n= 4,...,8