Transductions rationnelles

A∗
A∗×B∗
A∗
(M, ·)·
1MM
M
µ:M→N M N
µ(1M)=1N
∀x, y ∈M µ(x·y) = µ(x)·µ(y)
M A∗A

M X ⊂M
N µ :M→N P N
X=µ−1(P)
Rec(M)M
Rec(M)
M Rec(M)
X1, X2∈Rec(M)
µ1:M→N1
µ2:M→N2N1N2P1⊂N1P2⊂N2
X1=µ−1(P1)X2=µ−1(P2)
µ= (µ1, µ2) : M→N=N1×N2
N X1∩X2=µ−1(P1×P2)X1∩X2∈Rec(M)
µ−1
1(N1\P1) = M\X1M\X1∈Rec(M)
X1∪X2=M\((M\X1)∪(M\X2)) X1∪X2∈Rec(M)
M Rat(M)RP(M)
∅ ∈ R∀m∈M, {m} ∈ R
∀X, Y ∈R:X∪Y, X ·Y, X∗∈R
Rat(M)M
A Rec(A∗) = Rat(A∗)
X∈Rat(M)∩Rec(M)
X
Rec(M)Rat(M)
M M0
µ:M→M0∀X0⊂M0
X0∈Rec(M0)⇒µ−1(X0)∈Rec(M)

X0∈Rec(M0)
N ν :M→N P ⊂N
X=ν−1(P)
µ−1(X0)=(ν◦µ)−1(P)∈Rec(M)
µ:M→
M0µ(X)∈Rec(M0)⇒X∈Rec(M)
µ−1(µ(X)) 6=X
Rec(M)
M M0µ:M→M0
∀X⊂M
X∈Rec(M)⇔µ(X)∈Rec(M0)
ν µ ν−1(X) = µ(X)
µ−1(µ(X)) = X
Rat(M)
M M0µ:M→M0
∀X∈Rat(M), µ(X)∈Rat(M0)
A={X∈Rat(M)/ µ(X)∈Rat(M0)}
∅ {m}, m ∈
M Rat(M)
∅ ∈ Rat(M)∀m∈M, µ({m}) = {µ(m)} ∈ Rat(M0){m} ∈
AX, Y ∈ A µ(X∪Y) = µ(X)∪µ(Y), µ(XY ) = µ(X)µ(Y)
µ(X+) = (µ(X))+X∪Y, XY, X+∈ A
µ:M→M0∀X⊂M
X∈Rat(M)⇔µ(X)∈Rat(M0)
Rec(M)Rat(M)
Rec(M)⊂Rat(M)
M
µ:A∗→M A A ={1, ..., n}, µ(k) = mk
{mk, k = 1, ..., n}
X∈Rec(M)µ−1(X)∈Rec(A∗)
µ−1(X)∈Rat(A∗)
µ(µ−1(X)) ∈Rat(M)
µ µ(µ−1(X)) = X

M N
τ:M→ P(N)
τ
dom(τ) = {x∈M/τ(x)6=∅}
im(τ) = {y∈N/∃x∈M, y ∈τ(x)}
τ
R={(x, y)∈M×N/y ∈τ(x)}
τ:M→N τ :M→ P(N)
M×N M N
τ:M→N
τ1, τ2:M→N τ1∪τ2τ1τ2τ+
1
∀x∈M
τ1∪τ2(x) = τ1(x)∪τ2(x)
τ1τ2(x) = τ1(x)τ2(x)
τ+
1(x) =
∞
[
n=0
τ1(x)n
τ1τ2
τ1∪τ2τ1τ2τ+
1
τ1∪τ2τ1τ2τ+
1
R1∪ R2R1R2R+
1R1R2τ1τ2
τ:A∗→B∗A
B
Rec(A∗×B∗) Rat(A∗×B∗)
X={(an, bn)/n ∈N}a∈A, b ∈B
X={(a, b)}∗∈Rat(A∗×B∗)
C{a, b}µ:C∗→A∗×B∗
µ(a)=(a, 1), µ(b) = (1, b) 1
µ−1(X) = {u∈C∗/|u|a=|u|b}/∈Rec(C∗)
X /∈Rec(A∗×B∗)

T=hA, B, Q, q0, F, Ei
A B
Q
q0∈Q
F⊂Q
E⊂Q×A∗×B∗×Q
A B
T=hA, B, Q, q0, F, Ei
c0
(a1,b1)
−−−−→ c1· · · (an,bn) −−−−−→ cn
i= 1, ..., n (ci−1, ai, bi, ci)∈E(a1...an, b1...bn)
c0=q0cn∈F
p, q ∈QΛ(p, q)
TQ0⊂QΛ(p, Q0) = {Λ(p, q)/q ∈Q}
T
τ:A∗→B∗T
τ(x) = {y∈B∗/(x, y)∈Λ(q0, F )}
M M0X M ×M0
C µ :
C∗→M, µ0:C∗→M0K C X =
{(µ(w), µ0(w))/w ∈K}
M M0τ:M→M0
C µ :C∗→
M, µ0:C∗→M0K C ∀x∈M, τ(x) =
µ0(µ−1(x)∩K))