Unification(単一化)とAnti-Unification(半単一化)のアルゴリズムの概説

1. UNIFICATION / ANTI-UNIFICATION

2. OVERVIEW
¡ Basics
¡ Unification
¡ Anti-Unification

3. BASICS
¡ term: variable, constant symbol, function symbol
¡ equation: <term> = <term>
¡ ex. f(x, g(z)) = f(a, y)
¡ function symbol (arity)
¡ constant symbol arity 0 function symbol
¡ equation set E
¡ grounding substitution: variable ground term
¡ ground term: variable
¡ ex. {x <- f(g(a)), y <- b}
¡ solution: E equation grounding
substitution
¡ ex. E={f(x, g(z)) = f(a, y)} {x <- a, y <- g(b), z <- b} solution

4. UNIFICATION

5. UNIFICATION PROBLEM
¡ Input: equation E
¡ Output: E equivalent Solved form equation set
¡ equation
¡ Solved form: equation variable variable
equation
¡ equivalent: solution
g(x) = g(g(z))
f(a, z) = f(a, y)
x = g(y)
z = y
Problem E Solved Form
equivalent

6. UNIFICATION ALGORITHM [LASSEZ ET AL.]
Input: equation set E
E equation
s, t , x variable, f, g function symbol
1. f(t1, ..., tn) = f(s1, ..., sn)
→ t1=s1, ..., tn=sn
2. f(t1, ..., tn) = g(s1, ..., sm) f ≠g
→
3. x = x
→ equation
( )

7. UNIFICATION ALGORITHM [LASSEZ ET AL.]
4. t = x t variable
→ x = t
5. x = t t x equation x
i. x t
→
ii. x t
→ equation x t

8. EXAMPLE
g(x) = g(g(z))
f(a, z) = f(a, y)
x = g(z)
f(a, z) = f(a, y)
1
x = g(z)
a = a
z = y
1
1x = g(z)
z = y
5.ii
x = g(y)
z = y

9. 1. f(t1, ..., tn) = f(s1, ..., sn)
→ t1=s1, ..., tn=sn
2. f(t1, ..., tn) = g(s1, ..., sm) f ≠g
→
3. x = x
→ equation
4. t = x t variable
→ x = t
f
equation

10. 5. x = t t x equation
x
i. x t
→
ii. x t
→ equation x t
variable function symbol
1 variable 1

11. 1, 3, 4 5
5.ii

12. ANTI-UNIFICATION

13. ANTI-UNIFICATION PROBLEM
¡ θ = { v1 -> t1, ..., vn -> tn }
¡ tθ t vi ti term
¡ t s instance ⇔ s t anti-instance ⇔ s ≧ t
⇔ inst(s) ⊇ inst(t) ⇔ θ t = sθ
¡ inst(t): t instance
¡ variable renaming
≧ complete lattice (inst(⊥)=∅ ⊥ )
¡ ex. f(x, y) f(y, z)
¡ lattice least upper bound anti-unification
complete lattice
poset subset
greatest lower bound least
upper bound

14. EXAMPLE
f(a, g(a, y), a)
f(x, g(x, y), a)
f(x, g(x, b), a)
f(b, g(b, b), a)
x=a y=b
x=b
f(x, g(x, x), a)
y=x
x=b
anti-unification
f(x, g(x, x), y)
y=a
f(x, g(x, y), x)
x=a
... ...
...

15. ANTI-UNIFICATION ALGORITHM
¡ φ: T × T -> V': bijection term variable
¡ T: term , V': variable
¡ λ: Anti-unification
¡ λ( f(s1, ..., sm), f(t1, ..., tm) ) = f( λ(s1, t1), ..., λ(sm, tm) )
¡ λ( s, t ) = φ( s, t )

16. EXAMPLE
λ( f(a, g(a, y), a), f(b, g(b, b), a) )
= f( λ(a, b), λ(g(a, y), g(b, b)), g(a, a) )
= f( x, g( λ(a, b), λ(y, b) ), a )
= f( x, g( x, y ), a )
φ(a, b) = x
φ(y, b) = y