Algorithme d`unification

u v σ
u[σ] = v[σ]
u v σ
σ0u v σ00
σ0=σ00 ◦σ
E={u1∼v1, . . . , un∼vn}ui, vi
σ E
σ←
E6=∅
e∈E
E0←E\ {e}
e=f(u1. . . up)∼g(v1. . . vq)
f=g
E←E0∪ {u1∼v1, . . . , up∼vp}
e=x∼x
E←E0
e=x∼u e =u∼x
x u
σ←[x:= u]◦σ
E←E0[x:= u]
σ
EnσnE σ n
n>1E0σ0

n∈
anEn
bnEn
cnEn
n∈EnEn+1
1an+1 =anbn+1 < bn
2an+1 6anbn+1 =bncn+1 < cn
3an+1 < an
(an+1, bn+1, cn+1)≺(an, bn, cn)
≺3
x[σ] = u[σ]σ=σ◦[x:= u]
σ0=σ◦[x:= u]x[σ0] = x[x:= u][σ] = u[σ] = x[σ]y6=x
y[σ0] = y[x:= u][σ] = y[σ]
σ=σ0
n∈
(Hn) := Enσnσ E σ0En
σ=σ0◦σn
n
n= 0 σ0=E0=E H0
n∈HnEn+1 σn+1
σ E ⇔ ∃σ0, σ =σ0◦σnσ0En
σ E ⇔ ∃σ00, σ =σ00 ◦σn+1 σ00 En+1
∃σ0, σ =σ0◦σnσ0En⇔ ∃σ00, σ =σ00 ◦σn+1 σ00 En+1
En=E0] {f(u1, . . . , up), f(v1, . . . , vp)}En+1 =E0∪ {u1∼v1, . . . , up∼vp}
σn+1 =σn
σ0Enσ0En+1
σ=σ0◦σn=σ0◦σn+1 σn+1 =σn
En+1 =E0σn+1 =σn
En=E0] {x∼u}E0] {u∼x}x6=u x u
”⇐”σ00 σ=σ00 ◦σn+1 σ00 En+1 σ=σ00 ◦[x:= u]◦σn
σ00 ◦[x:= u]Enσ0=σ00 ◦[x:= u]
”⇒”σ0Enx[σ0] = u[σ0]σ0=σ0◦[x:= u]
σ0=σ0◦σn=σ0◦[x:= u]◦σn=σ0◦σn+1 σ00 =σ0
Hnn

N
EN=∅ENHNσN=◦σNσNE
σ E HNσ0
σ=σ0◦σNσNE
n Enf(. . . )∼g(. . . )f6=g
En
HnE
n Enx∼u x u
x6=u u σ
|x[σ]|<|u[σ]|x[σ]6=u[σ]HnE