Problem 0.1 (Solving Equations via Higher-Order Unification)
10min
Let α := (ι → ι) → ι → ι, solve the equation below via higher-order unification.
λXι→ι Yι X(Nα XY ) = λZι→ι Wι Z(Z(ZW ))
Solution: We develop the corresponding unfication problem via the higher-order transformations.
1.
2.
3.
4.
5.
6.
7.
8.
initial problem: λXι→ι Yι X(Nα XY ) =? λZι→ι Wι Z(Z(ZW ))
Sk
simplify with SIM:α: a(Nα ab) =? a(a(ab)) where a ∈ ΣSk
ι→ι and b ∈ Σι .
decompose with SIM:dec to Nα ab =? a(ab)
imitate HOU :fr to Nα =? λXι→ι Yι X(Hα XY ) ∧ Nα ab =? a(ab)
eliminate N with SIM:elim to Nα =? λXι→ι Yι X(Hα XY ) ∧ a(Hα ab) =? a(ab)
decompose with SIM:dec to Nα =? λXι→ι Yι X(Hα XY ) ∧ Hα ab =? ab
imitate HOU :fr to Hα =? λXι→ι Yι X(Kα XY ) ∧ Nα =? λXι→ι Yι X(Hα XY ) ∧ Hα ab =? ab
eliminate H with SIM:elim to
Hα =? λXι→ι Yι X(Kα XY ) ∧ Nα =? λXι→ι Yι X(X(Hα XY )) ∧ a(Kα ab) =? ab
9. decompose with SIM:dec to
Hα =? λXι→ι Yι X(Kα XY ) ∧ Nα =? λXι→ι Yι X(X(Nα XY )) ∧ Kα ab =? b
10. project with HOU :fr to
Kα =? λXι→ι Yι Y ∧ Hα =? λXι→ι Yι X(Kα XY ) ∧ Nα =? λXι→ι Yι X(X(Kα XY )) ∧ Kα ab =? b
11. eliminate K with SIM:elim to
Kα =? λXι→ι Yι Y ∧ Hα =? λXι→ι Yι XY ∧ Nα =? λXι→ι Yι X(XY ) ∧ b =? b
12. decompose SIM:dec to
Kα =? λXι→ι Yι Y ∧ Hα =? λXι→ι Yι XY ∧ Nα =? λXι→ι Yι X(XY )
13. this is a solved equational problem, so the solution is
[(λXι→ι Yι Y )/Kα ], [(λXι→ι Yι XY )/Hα ], [(λXι→ι Yι X(XY ))/Nα ]
1