Liste d`exercices

w∈Σ∗w
w=ε wR=ε σ ∈Σv∈Σ∗w=σv
wR=vRσ w =ε
wR=ε σ ∈Σv∈Σ∗w=vσ wR=σvR
u, v, w ∈Σ∗
(wR)R=w(uv)R=vRuR.
u, v, w ∈Σ∗
uw =vw ⇒u=v
wu =wv ⇒u=v.
x, y, u, v Σxy =uv
• |x|>|u|w6=ε x =uw v =wy
• |x|=|u|x=u y =v
• |x|<|u|w6=ε u =xw y =wv
w#(Pref(w)) = |w|+ 1
#(Fac(w))
Σ = {a, b}w∈Σ∗w2=w3
Σ = {a, b}w∈Σ∗
v∈Σ∗w3=v2
w w =wR
L⊆Σ∗L=LR
Σ
L⊆ {a, b}∗
•ε∈L
•u L aaub L
•L
L
ΣL⊆Σ∗
•ε∈L σ ∈Σσ∈L
•w L σ ∈Σσwσ L
•L
L
L⊆Σ∗
(L∗)∗=L∗, L∗L∗=L∗, L∗L∪ {ε}=L∗=LL∗∪ {ε}.
LL∗=L∗

L, M, N ΣL(M∩N)⊆LM ∩LN
L={aa, bb}M={ε, b, ab}
•LM L tt M
•M∗
•6L∗
•n∈NL∗
L M
#L.#M= #(LM)
Σ Γ m, n ∈N
t(m, n) := #(utt v), u ∈Σm, v ∈Γn.
t(m, n) = t(m, n −1) + t(m−1, n), m, n > 0
t(m, 0) = t(0, m)=1
#(abba tt cd).
t(m, n)
Σ = {a, b}w∈Σ∗
xx x Σ∗
Σw∈Σ
w=ui, u ∈Σ∗
i≥2w
u∈Σ∗w=uii≥1
u v Σx, y ∈Σ∗u=xy
v=yx abbaa
Σ∗w
{→,←,↑,↓}
(0,0) (1,1)
2
{a, b}a b
{a, b}
ba
{a, b}
aa bb
{a, b}
aa bb

{a, b}
aa aaa
{a, b, c}
a
{a, b}
aa
{a, b, c}
a b cc
{a, b, c}
a3
{a, b}
bb
{a, b}
a
{a, b, c}
ab
3
10
5
Σ
ϕ+ϕ∈ RΣL(ϕ+) = (L(ϕ))+=∪i>0(L(ϕ))i
•(ba)+(a∗b∗+a∗)=(ba)∗b a+(b∗+e)
•b+(a∗b∗+e)b=b(b∗a∗+e)b+
•(a+b)∗= (a+b)∗b∗
•(a+b)∗= (a∗+ba∗)∗
•(a+b)∗= (b∗(a+e)b∗)∗
Σ = {a, b}
(ab)+= (aΣ∗∩Σ∗b)\(Σ∗aaΣ∗+ Σ∗bbΣ∗).
|L(ϕ)|
•ϕ= (ab)∗+bbb
•ϕ= (a(ba∗) + a∗)∗
•ϕ= (ab)(ac(a+b))∗
•ϕ=ab(bbc)∗
ϕ
• |L(ϕ)|= 5 + N.3
• |L(ϕ)|=N.7∪(4 + N.5)
{an3+2n+1 |n∈N}
{a2n+1 |n∈N}
{an|n∈ P} P
a b (a, b) = 1
a+N.b

A= (Q, 1, F, Σ, δ)Q={1,2,3}Σ = {a, b}F={3}
δ a b
1 1 2
2 3 2
3 3 1
.
A A
•abba
•bbbabb
•bababa
•bbbaa
A
a∗b∗{c}a∗b∗tt{c}
{a, b}a∗b∗{c}
{c}ρLn∈N
n L
ρL:N→N:n7→ #(L∩Σn).
ρa∗b∗(n)ρa∗b∗tt {c}(n)
b
b
a
a
b
aa
b
a
b
1
3
a
b
a
a
a
2
b

•∆
•aaabb
•aaabb
•
•
ε
12
3
4
,a
ε
ε
b
bab
b
a
a
a
b
a
b
b
aa
(ab)∗+a∗.
n≥1 (a+b)∗b(a+b)n−1
n+ 1
2n
(abc)∗a∗.
((abc)∗a∗)R.