3(pdf) - KOMORI, Yuichi

情報論理学 II 試験 3
2003 年 1 月 31 日 (金)
1. CL 項 λ∗ xy.xyy
を求めよ (S, K で表せ)。
2. CL 項 (λ∗ xy.xyy)xy
3. CL 項 λ∗ xy.xyy
を weak reduction せよ。
が持つ型 δ を求め,
λ∗ xy.xyy ∈ δ
に至る TAC の証明
4. ラムダ項 λxy.(λuz.y(xz))(λy.yz) が持つ型 δ を求め,
λxy.(λuz.y(xz))(λy.yz) ∈ δ に至る TAλ の証明図をかけ。
5. ラムダ項 λxy.(λuz.y(xz))(λy.yz) の β 正規形 (β-nf) M を求め, 4 で求め
た δ に対して M ∈ δ に至る TAλ の証明図をかけ。
定義 0.1 (Weak reduction) Any CL-term KXY or SXY Z is called a (weak)
redex. Contracting an occurence of a redex in a term U means replacing one
occurence of
by X,
SXY Z by XZ(Y Z).
If this changes U to U ′ , we say that U (weakly) contracts to U ′ , or
U ✄1w U ′ .
We say that U (weakly) reduces to V , or
U ✄w V
iff V is obtained from U by a finite (perhaps empty) series of weak contractions.
定義 0.2 (Abstraction) For each CL-term M and each variable x, a CL-term
λ∗ x.M is defined by induction on M , thus:
λ∗ x.M
λ∗ x.x
λ∗ x.U x
λ∗ x.U V
S(λ∗ x.U )(λ∗ x.V )
if x ∈
/ F V (M );
if x ∈
/ F V (U );
if neither (a) nor (c) applies.
定義 0.3 (The type-assignment system TAC ) It has two axiom-schemes; they
(→ K) K ∈ α → β → α,
(→ S) S ∈ (α → β → γ) → (α → β) → α → γ.
Its only rule is called the →-elimination rule or (→e);
X∈α→β Y ∈α
(→ e)
XY ∈ β
定義 0.4 (The type-assignment system TAλ ) It has no axiom-schemes and
two rules the →-elimination rule (→e) and the →-introduction rule (→i);
X∈α→β Y ∈α
(→ e)
XY ∈ β
[x ∈ α]
M ∈β
(→ i)
λx.M ∈ α → β