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❙tr♦♥❣ ◆♦r♠❛❧✐③❛❜✐❧✐t② ❛s ❛ ❋✐♥✐t❡♥❡ss ❙tr✉❝t✉r❡ ✈✐❛ t❤❡ ❚❛②❧♦r ❊①♣❛♥s✐♦♥ ♦❢ λ✲t❡r♠s

▼✐❝❤❡❧❡ P❛❣❛♥✐◦✱ ❈❤r✐st✐♥❡ ❚❛ss♦♥◦ ❛♥❞ ▲✐♦♥❡❧ ❱❛✉①∗

• ■❘■❋✱ ❯♥✐✈❡rs✐té P❛r✐s ❉✐❞❡r♦t✱ ❋r❛♥❝❡

∗ ■✷▼✱ ❆✐①✲▼❛rs❡✐❧❧❡ ❯♥✐✈❡rs✐té✱ ❋r❛♥❝❡

❙✉♣♣♦rt❡❞ ❜② ❋r❡♥❝❤ ❆◆❘ Pr♦❥❡❝t ❈♦q✉❛s ✭❆◆❘ ✶✷ ❏❙✵✷ ✵✵✻ ✵✶✮✳

❋♦❙❙❛❈❙ ❅ ❊❚❆P❙ ✷✵✶✻ ❊✐♥❞❤♦✈❡♥✱ ◆▲✱ ✹✲✼ ❆♣r✐❧ ✷✵✶✻

✶ ✴ ✶✺

❊✈❡r②t❤✐♥❣ ✐s ✐♥ t❤❡ t✐t❧❡ (∀M ∈ Λ+) M ∈ SN ⇐ ⇒ T (M) ∈ F

❲❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ str♦♥❣ ♥♦r♠❛❧✐③❛❜✐❧✐t② ✭SN✮ ♦❢ ✭♥♦♥✲❞❡t❡r♠✐♥✐st✐❝✮ λ✲t❡r♠s ✭Λ+✮ ❛s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ✭F✮ ✈✐❛ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✭T ✮✳

✷ ✴ ✶✺

❊✈❡r②t❤✐♥❣ ✐s ✐♥ t❤❡ t✐t❧❡ (∀M ∈ Λ+) M ∈ SN ⇐ ⇒ T (M) ∈ F

❲❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ str♦♥❣ ♥♦r♠❛❧✐③❛❜✐❧✐t② ✭SN✮ ♦❢ ✭♥♦♥✲❞❡t❡r♠✐♥✐st✐❝✮ λ✲t❡r♠s ✭Λ+✮ ❛s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ✭F✮ ✈✐❛ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✭T ✮✳

✷ ✴ ✶✺

◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s

❆ ♣r✐♠❡ ❛❣❡❞ ✐❞❡❛ ✭●✐r❛r❞✱ ✬✽✵s✱ ❜❡❢♦r❡ ▲▲✮

λ✲t❡r♠s = ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s = ♣♦✇❡r s❡r✐❡s ❖r✐❣✐♥❛❧❧②✿ ❢♦r t❤❡ λ✲❝❛❧❝✉❧✉s✱ ✐♥ ❛♥ ❛❜str❛❝t ❝❛t❡❣♦r✐❝❛❧ s❡tt✐♥❣ ✭❝♦❡✣❝✐❡♥ts ❛r❡ s❡ts✮

❋✐♥✐t♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ❡❛r❧② ✷✵✵✵✬s✮

❘❡❢♦r♠✉❧❛t❡ q✳s✳ ✐♥ ❛ ❧✐♥❡❛r ❧♦❣✐❝ s❡tt✐♥❣ ✉s✐♥❣ st❛♥❞❛r❞ ❛❧❣❡❜r❛✿ t②♣❡s

♣❛rt✐❝✉❧❛r t♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡s✿ ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ str✉❝t✉r❡

❢✉♥❝t✐♦♥ t❡r♠s ♣♦✇❡r s❡r✐❡s

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ✲t❡r♠s ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r ✷✵✵✸✲✷✵✵✹✮

❙♦ ✇❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ✲t❡r♠s✱ ❛♥❞ ❝♦♠♣✉t❡ t❤❡✐r ❚❛②❧♦r ❡①♣❛♥s✐♦♥✦ ❆♥❞ ♦♥❡ ❝❛♥ ♠✐♠✐❝❦ t❤❛t ✐♥ t❤❡ s②♥t❛①✿ ❞✐✛❡r❡♥t✐❛❧ ✲❝❛❧❝✉❧✉s ❛ ✜♥✐t❛r② ❢r❛❣♠❡♥t✿ r❡s♦✉r❝❡ ✲❝❛❧❝✉❧✉s ❂ t❤❡ t❛r❣❡t ♦❢ ❚❛②❧♦r ❡①♣❛♥s✐♦♥

✸ ✴ ✶✺

◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s

❆ ♣r✐♠❡ ❛❣❡❞ ✐❞❡❛ ✭●✐r❛r❞✱ ✬✽✵s✱ ❜❡❢♦r❡ ▲▲✮

λ✲t❡r♠s = ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s = ♣♦✇❡r s❡r✐❡s ❖r✐❣✐♥❛❧❧②✿ ❢♦r t❤❡ λ✲❝❛❧❝✉❧✉s✱ ✐♥ ❛♥ ❛❜str❛❝t ❝❛t❡❣♦r✐❝❛❧ s❡tt✐♥❣ ✭❝♦❡✣❝✐❡♥ts ❛r❡ s❡ts✮

❋✐♥✐t♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ❡❛r❧② ✷✵✵✵✬s✮

❘❡❢♦r♠✉❧❛t❡ q✳s✳ ✐♥ ❛ ❧✐♥❡❛r ❧♦❣✐❝ s❡tt✐♥❣ ✉s✐♥❣ st❛♥❞❛r❞ ❛❧❣❡❜r❛✿

◮ t②♣❡s ♣❛rt✐❝✉❧❛r t♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡s✿

A ⊆ k|A| ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ str✉❝t✉r❡

◮ ❢✉♥❝t✐♦♥ t❡r♠s ♣♦✇❡r s❡r✐❡s

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ✲t❡r♠s ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r ✷✵✵✸✲✷✵✵✹✮

❙♦ ✇❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ✲t❡r♠s✱ ❛♥❞ ❝♦♠♣✉t❡ t❤❡✐r ❚❛②❧♦r ❡①♣❛♥s✐♦♥✦ ❆♥❞ ♦♥❡ ❝❛♥ ♠✐♠✐❝❦ t❤❛t ✐♥ t❤❡ s②♥t❛①✿ ❞✐✛❡r❡♥t✐❛❧ ✲❝❛❧❝✉❧✉s ❛ ✜♥✐t❛r② ❢r❛❣♠❡♥t✿ r❡s♦✉r❝❡ ✲❝❛❧❝✉❧✉s ❂ t❤❡ t❛r❣❡t ♦❢ ❚❛②❧♦r ❡①♣❛♥s✐♦♥

✸ ✴ ✶✺

◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s

❆ ♣r✐♠❡ ❛❣❡❞ ✐❞❡❛ ✭●✐r❛r❞✱ ✬✽✵s✱ ❜❡❢♦r❡ ▲▲✮

λ✲t❡r♠s = ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s = ♣♦✇❡r s❡r✐❡s ❖r✐❣✐♥❛❧❧②✿ ❢♦r t❤❡ λ✲❝❛❧❝✉❧✉s✱ ✐♥ ❛♥ ❛❜str❛❝t ❝❛t❡❣♦r✐❝❛❧ s❡tt✐♥❣ ✭❝♦❡✣❝✐❡♥ts ❛r❡ s❡ts✮

❋✐♥✐t♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ❡❛r❧② ✷✵✵✵✬s✮

❘❡❢♦r♠✉❧❛t❡ q✳s✳ ✐♥ ❛ ❧✐♥❡❛r ❧♦❣✐❝ s❡tt✐♥❣ ✉s✐♥❣ st❛♥❞❛r❞ ❛❧❣❡❜r❛✿

◮ t②♣❡s ♣❛rt✐❝✉❧❛r t♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡s✿

A ⊆ k|A| ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ str✉❝t✉r❡

◮ ❢✉♥❝t✐♦♥ t❡r♠s ♣♦✇❡r s❡r✐❡s

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ λ✲t❡r♠s ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r ✷✵✵✸✲✷✵✵✹✮

❙♦ ✇❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ λ✲t❡r♠s✱ ❛♥❞ ❝♦♠♣✉t❡ t❤❡✐r ❚❛②❧♦r ❡①♣❛♥s✐♦♥✦ ❆♥❞ ♦♥❡ ❝❛♥ ♠✐♠✐❝❦ t❤❛t ✐♥ t❤❡ s②♥t❛①✿

◮ ❞✐✛❡r❡♥t✐❛❧ λ✲❝❛❧❝✉❧✉s ◮ ❛ ✜♥✐t❛r② ❢r❛❣♠❡♥t✿ r❡s♦✉r❝❡ λ✲❝❛❧❝✉❧✉s

❂ t❤❡ t❛r❣❡t ♦❢ ❚❛②❧♦r ❡①♣❛♥s✐♦♥

✸ ✴ ✶✺

❘❡s♦✉r❝❡ λ✲❝❛❧❝✉❧✉s

❘❡s♦✉r❝❡ t❡r♠s

∆ ∋ s, t, . . . ::= x | λx.s | s t ∆! ∋ s, t, . . . ::= [s1, . . . , sn]

▼❡❛♥✐♥❣✿ s [s1, . . . , sn] = (Ds)0 · (s1, . . . , sn)

❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥

λx.s t →ρ ∂xs · t

✭❛♥②✇❤❡r❡✮

∂xs · t =

f∈Sn s

• tf(1), . . . , tf(n)/x1, . . . , xn
• ✐❢ degx(s) = #t = n

♦t❤❡r✇✐s❡

❧✐♥❡❛r✐t②✿ ✱ ✱

❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥ ♣r❡s❡r✈❡s ❢r❡❡ ✈❛r✐❛❜❧❡s✱ ✐s s✐③❡✲❞❡❝r❡❛s✐♥❣✱ str♦♥❣❧② ❝♦♥✢✉❡♥t ❛♥❞ ♥♦r♠❛❧✐③✐♥❣✳

✹ ✴ ✶✺

❘❡s♦✉r❝❡ λ✲❝❛❧❝✉❧✉s

❘❡s♦✉r❝❡ t❡r♠s

∆ ∋ s, t, . . . ::= x | λx.s | s t ∆! ∋ s, t, . . . ::= [s1, . . . , sn]

▼❡❛♥✐♥❣✿ s [s1, . . . , sn] = (Ds)0 · (s1, . . . , sn)

❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥

λx.s t →ρ ∂xs · t

✭❛♥②✇❤❡r❡✮

∂xs · t =

f∈Sn s

• tf(1), . . . , tf(n)/x1, . . . , xn
• ✐❢ degx(s) = #t = n

♦t❤❡r✇✐s❡

❧✐♥❡❛r✐t②✿ λx.0 = 0✱ s [t1 + t2, u] = s [t1, u] + s [t2, u]✱ . . .

❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥ ♣r❡s❡r✈❡s ❢r❡❡ ✈❛r✐❛❜❧❡s✱ ✐s s✐③❡✲❞❡❝r❡❛s✐♥❣✱ str♦♥❣❧② ❝♦♥✢✉❡♥t ❛♥❞ ♥♦r♠❛❧✐③✐♥❣✳

✹ ✴ ✶✺

❘❡s♦✉r❝❡ λ✲❝❛❧❝✉❧✉s

❘❡s♦✉r❝❡ t❡r♠s

∆ ∋ s, t, . . . ::= x | λx.s | s t ∆! ∋ s, t, . . . ::= [s1, . . . , sn]

▼❡❛♥✐♥❣✿ s [s1, . . . , sn] = (Ds)0 · (s1, . . . , sn)

❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥

λx.s t →ρ ∂xs · t

✭❛♥②✇❤❡r❡✮

∂xs · t =

f∈Sn s

• tf(1), . . . , tf(n)/x1, . . . , xn
• ✐❢ degx(s) = #t = n

♦t❤❡r✇✐s❡

❧✐♥❡❛r✐t②✿ λx.0 = 0✱ s [t1 + t2, u] = s [t1, u] + s [t2, u]✱ . . .

◮ ❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥ ♣r❡s❡r✈❡s ❢r❡❡ ✈❛r✐❛❜❧❡s✱ ✐s s✐③❡✲❞❡❝r❡❛s✐♥❣✱

str♦♥❣❧② ❝♦♥✢✉❡♥t ❛♥❞ ♥♦r♠❛❧✐③✐♥❣✳

✹ ✴ ✶✺

❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ λ✲t❡r♠s

❙❡♠❛♥t✐❝❛❧❧②✱ (M) N =

n∈N 1 n! M N n ✇❤❡r❡ N n = [N, . . . , N]✳

❚❛②❧♦r ❡①♣❛♥s✐♦♥✿ ❚❤❡♦r❡♠ ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r✱ ❈✐❊ ✷✵✵✻✮

■❢ ✱ t❤❡♥ ♥♦r♠❛❧✐③❡s t♦ ✳

▼♦r❛❧

■♥ t❤❡ ♦r❞✐♥❛r② ✲❝❛❧❝✉❧✉s ✳

✺ ✴ ✶✺

❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ λ✲t❡r♠s

❙❡♠❛♥t✐❝❛❧❧②✱ (M) N =

n∈N 1 n! M N n ✇❤❡r❡ N n = [N, . . . , N]✳

❚❛②❧♦r ❡①♣❛♥s✐♦♥✿ T (M) ∈ Q+∆

• T ((M) N) =
• n∈N

1 n!

• T (M)
• T (N)

n

• T (x) = x
• T (λx.M) = λx.

T (M)

❚❤❡♦r❡♠ ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r✱ ❈✐❊ ✷✵✵✻✮

■❢ ✱ t❤❡♥ ♥♦r♠❛❧✐③❡s t♦ ✳

▼♦r❛❧

■♥ t❤❡ ♦r❞✐♥❛r② ✲❝❛❧❝✉❧✉s ✳

✺ ✴ ✶✺

❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ λ✲t❡r♠s

❙❡♠❛♥t✐❝❛❧❧②✱ (M) N =

n∈N 1 n! M N n ✇❤❡r❡ N n = [N, . . . , N]✳

❚❛②❧♦r ❡①♣❛♥s✐♦♥✿ T (M) ∈ Q+∆

• T ((M) N) =
• n∈N

1 n!

• T (M)
• T (N)

n

• T (x) = x
• T (λx.M) = λx.

T (M)

❚❤❡♦r❡♠ ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r✱ ❈✐❊ ✷✵✵✻✮

■❢ M ∈ Λ✱ t❤❡♥ T (M) ♥♦r♠❛❧✐③❡s t♦ T (BT (M))✳

▼♦r❛❧

■♥ t❤❡ ♦r❞✐♥❛r② λ✲❝❛❧❝✉❧✉s BT (M) ≃ NF( T (M))✳

✺ ✴ ✶✺

◆♦r♠❛❧✐③✐♥❣ ❚❛②❧♦r ❡①♣❛♥s✐♦♥s ✿ ✉♥✐❢♦r♠✐t② t♦ t❤❡ r❡s❝✉❡✦

❇✉t ❤♦✇ ❝❛♥ T (M) ❡✈❡♥ ♥♦r♠❛❧✐③❡❄

❲❡ ✇❛♥t t♦ s❡t NF

• T (M)
• =
• s∈∆
• T (M)s .NF (s)

✐♥✜♥✐t❡ s✉♠s ✭❛♥❞ ✐♥ ❣❡♥❡r❛❧ ✇❡ ♠✐❣❤t ❝♦♥s✐❞❡r ❛❧❧ ❦✐♥❞s ♦❢ ❝♦❡✣❝✐❡♥ts✮ ❝♦♥✈❡r❣❡♥❝❡❄

❚❤❡♦r❡♠ ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r ✷✵✵✹✱ ♣✉❜❧✐s❤❡❞ ✐♥ ❚❈❙ ✐♥ ✷✵✵✽✮

❲r✐t❡ ✳ ❚❤❡♥ ❢♦r ❛❧❧ ✱ t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡ s✉❝❤ t❤❛t ✳

Pr♦♦❢✳

✲t❡r♠s ❛r❡ ✉♥✐❢♦r♠✿ t❤❡✐r ✜♥✐t❛r② ❛♣♣r♦①✐♠❛♥ts ❛r❡ ♣❛✐r✇✐s❡ ❝♦❤❡r❡♥t✳

❚❤✐s ❢❛✐❧s ✐♥ ❣❡♥❡r❛❧

✻ ✴ ✶✺

◆♦r♠❛❧✐③✐♥❣ ❚❛②❧♦r ❡①♣❛♥s✐♦♥s✿ ✉♥✐❢♦r♠✐t② t♦ t❤❡ r❡s❝✉❡✦

❇✉t ❤♦✇ ❝❛♥ T (M) ❡✈❡♥ ♥♦r♠❛❧✐③❡❄

❲❡ ✇❛♥t t♦ s❡t NF

• T (M)
• =
• s∈∆
• T (M)s .NF (s)

✐♥✜♥✐t❡ s✉♠s ✭❛♥❞ ✐♥ ❣❡♥❡r❛❧ ✇❡ ♠✐❣❤t ❝♦♥s✐❞❡r ❛❧❧ ❦✐♥❞s ♦❢ ❝♦❡✣❝✐❡♥ts✮ ❝♦♥✈❡r❣❡♥❝❡❄

❚❤❡♦r❡♠ ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r ✷✵✵✹✱ ♣✉❜❧✐s❤❡❞ ✐♥ ❚❈❙ ✐♥ ✷✵✵✽✮

❲r✐t❡ T (M) =

• T (M)
• ✳ ❚❤❡♥ ❢♦r ❛❧❧ t ∈ ∆✱ t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡

s ∈ T (M) s✉❝❤ t❤❛t NF (s)t = 0✳

Pr♦♦❢✳

λ✲t❡r♠s ❛r❡ ✉♥✐❢♦r♠✿ t❤❡✐r ✜♥✐t❛r② ❛♣♣r♦①✐♠❛♥ts ❛r❡ ♣❛✐r✇✐s❡ ❝♦❤❡r❡♥t✳

❚❤✐s ❢❛✐❧s ✐♥ ❣❡♥❡r❛❧

✻ ✴ ✶✺

◆♦r♠❛❧✐③✐♥❣ ❚❛②❧♦r ❡①♣❛♥s✐♦♥s✿ ✉♥✐❢♦r♠✐t② t♦ t❤❡ r❡s❝✉❡✦

❇✉t ❤♦✇ ❝❛♥ T (M) ❡✈❡♥ ♥♦r♠❛❧✐③❡❄

❲❡ ✇❛♥t t♦ s❡t NF

• T (M)
• =
• s∈∆
• T (M)s .NF (s)

✐♥✜♥✐t❡ s✉♠s ✭❛♥❞ ✐♥ ❣❡♥❡r❛❧ ✇❡ ♠✐❣❤t ❝♦♥s✐❞❡r ❛❧❧ ❦✐♥❞s ♦❢ ❝♦❡✣❝✐❡♥ts✮ ❝♦♥✈❡r❣❡♥❝❡❄

❚❤❡♦r❡♠ ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r ✷✵✵✹✱ ♣✉❜❧✐s❤❡❞ ✐♥ ❚❈❙ ✐♥ ✷✵✵✽✮

❲r✐t❡ T (M) =

• T (M)
• ✳ ❚❤❡♥ ❢♦r ❛❧❧ t ∈ ∆✱ t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡

s ∈ T (M) s✉❝❤ t❤❛t NF (s)t = 0✳

Pr♦♦❢✳

λ✲t❡r♠s ❛r❡ ✉♥✐❢♦r♠✿ t❤❡✐r ✜♥✐t❛r② ❛♣♣r♦①✐♠❛♥ts ❛r❡ ♣❛✐r✇✐s❡ ❝♦❤❡r❡♥t✳

❚❤✐s ❢❛✐❧s ✐♥ ❣❡♥❡r❛❧

NF

• n∈N λx.xn [y]
• = ?

λx.xn [y] = λx.x [λx.x [· · · [y] · · · ]]

✻ ✴ ✶✺

❆ ♠✐♥✐♠❛❧✐st✐❝ ♥♦♥✲✉♥✐❢♦r♠ ❝❛❧❝✉❧✉s

Λ+ ∋ M, N, . . . ::= x | λx.M | (M) N | M + N (λx.M) N →β M [N/x]

❊①❛♠♣❧❡

▲❡t ❛♥❞ ✿ ✳

❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✐♥ ❛ ♥♦♥ ✉♥✐❢♦r♠ s❡tt✐♥❣

❚❤❡♥

✼ ✴ ✶✺

❆ ♠✐♥✐♠❛❧✐st✐❝ ♥♦♥✲✉♥✐❢♦r♠ ❝❛❧❝✉❧✉s

Λ+ ∋ M, N, . . . ::= x | λx.M | (M) N | M + N (λx.M) N →β M [N/x] (M + N) P = (M) P + (N) P

❊①❛♠♣❧❡

▲❡t ❛♥❞ ✿ ✳

❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✐♥ ❛ ♥♦♥ ✉♥✐❢♦r♠ s❡tt✐♥❣

❚❤❡♥

✼ ✴ ✶✺

❆ ♠✐♥✐♠❛❧✐st✐❝ ♥♦♥✲✉♥✐❢♦r♠ ❝❛❧❝✉❧✉s

Λ+ ∋ M, N, . . . ::= x | λx.M | (M) N | M + N (λx.M) N →β M [N/x] (M + N) P = (M) P + (N) P

❊①❛♠♣❧❡

▲❡t δM = λx. (M + (x) x) ❛♥❞ ∞M = (δM) δM✿ ∞M →∗

β M + ∞M✳

❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✐♥ ❛ ♥♦♥ ✉♥✐❢♦r♠ s❡tt✐♥❣

❚❤❡♥

✼ ✴ ✶✺

❆ ♠✐♥✐♠❛❧✐st✐❝ ♥♦♥✲✉♥✐❢♦r♠ ❝❛❧❝✉❧✉s

Λ+ ∋ M, N, . . . ::= x | λx.M | (M) N | M + N (λx.M) N →β M [N/x] (M + N) P = (M) P + (N) P

❊①❛♠♣❧❡

▲❡t δM = λx. (M + (x) x) ❛♥❞ ∞M = (δM) δM✿ ∞M →∗

β M + ∞M✳

❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✐♥ ❛ ♥♦♥ ✉♥✐❢♦r♠ s❡tt✐♥❣

• T (M + N) =

T (M) + T (N) ❚❤❡♥

✼ ✴ ✶✺

❆ ♠✐♥✐♠❛❧✐st✐❝ ♥♦♥✲✉♥✐❢♦r♠ ❝❛❧❝✉❧✉s

Λ+ ∋ M, N, . . . ::= x | λx.M | (M) N | M + N (λx.M) N →β M [N/x] (M + N) P = (M) P + (N) P

❊①❛♠♣❧❡

▲❡t δM = λx. (M + (x) x) ❛♥❞ ∞M = (δM) δM✿ ∞M →∗

β M + ∞M✳

❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✐♥ ❛ ♥♦♥ ✉♥✐❢♦r♠ s❡tt✐♥❣

• T (M + N) =

T (M) + T (N) ❚❤❡♥ NF

• T (∞M)
• = ?

✼ ✴ ✶✺

❋✐♥✐t❡♥❡ss str✉❝t✉r❡s t♦ t❤❡ r❡s❝✉❡

❚❤❡ ♠❛✐♥ ❛rt✐❢❛❝t ♦❢ ❊❤r❤❛r❞✬s ✜♥✐t❡♥❡ss s♣❛❝❡s✿

❉❡✜♥✐t✐♦♥

◮ ■❢ a, a′ ⊆ A✱ ✇r✐t❡ a ⊥ a′ ✐✛ a ∩ a′ ✐s ✜♥✐t❡✳ ◮ ■❢ S ⊆ P (A)✱ ❧❡t S⊥ := {a′ ⊆ A; ∀a ∈ S, a ⊥ a′}✳ ◮ ❆ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ✐s ❛♥② F = S⊥✳

❲❤❡♥ ✐s T (M) ♥♦r♠❛❧✐③❛❜❧❡❄

◮ ❲r✐t❡ s ≥ t ✐❢ s →∗ ρ t + · · · ✳ ◮ ▲❡t ↑t = {s ∈ ∆; s ≥ t}✳ ◮

T (M) ✐s ♥♦r♠❛❧✐③❛❜❧❡ ✐✛ ❢♦r ❛❧❧ ♥♦r♠❛❧ t ∈ ∆✱ T (M) ⊥ ↑t✳

◮ {↑t ; t ♥♦r♠❛❧ ∈ ∆}⊥ ✐s t❤❡ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ♦❢ ✭s✉♣♣♦rts ♦❢✮

♥♦r♠❛❧✐③❛❜❧❡ ✈❡❝t♦rs✳

✽ ✴ ✶✺

❚②♣❡❞ t❡r♠s ❤❛✈❡ ❛ ✜♥✐t❛r② ❚❛②❧♦r ❡①♣❛♥s✐♦♥

▲❡t s②st❡♠ F+ ❜❡ s②st❡♠ F ♣❧✉s Γ ⊢ M : A Γ ⊢ N : A Γ ⊢ M + N : A ✳

❚❤❡♦r❡♠ ✭❊❤r❤❛r❞✱ ▲■❈❙ ✷✵✶✵✮

■❢ M ∈ Λ+ ✐s t②♣❛❜❧❡ ✐♥ s②st❡♠ F+✱ t❤❡♥ T (M) ∈ {↑t ; t ∈ ∆}⊥✳

Pr♦♦❢✳

▼❛♥❛❣❡ s❡ts ♦❢ r❡s♦✉r❝❡ t❡r♠s ❛s ✐❢ t❤❡② ✇❡r❡ λ✲t❡r♠s✱ ❛♥❞ ❢♦❧❧♦✇ t❤❡ ✉s✉❛❧ r❡❞✉❝✐❜✐❧✐t② t❡❝❤♥✐q✉❡✱ ❛ss♦❝✐❛t✐♥❣ ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ Fin (A) ⊆ {↑t ; t ∈ ∆}⊥ ✇✐t❤ ❡❛❝❤ t②♣❡ A✳

✾ ✴ ✶✺

❖✉r r❡s✉❧ts

◮ ❚②♣❛❜✐❧✐t② ✐♥ F ❝❛♥ ❜❡ r❡❧❛①❡❞ t♦ str♦♥❣ ♥♦r♠❛❧✐③❛❜✐❧✐t②✳ ◮ ❚❤❡♥ t❤❡ ✐♠♣❧✐❝❛t✐♦♥

M ∈ SN ⇒ T (M) ∈ {↑t ; t ∈ ∆}⊥ ❝❛♥ ❜❡ r❡✈❡rs❡❞✳ ✳ ✳

◮ ♣r♦✈✐❞❡❞ t❤❡ ✜♥✐t❡♥❡ss {↑t ; t ∈ ∆}⊥ ✐s r❡✜♥❡❞ t♦ ❛ t✐❣❤t❡r ♦♥❡✳

✶✵ ✴ ✶✺

M ∈ SN ⇒ T (M) ∈ {↑t ; t ∈ ∆}⊥

■♥ t❤❡ ♦r❞✐♥❛r② λ✲❝❛❧❝✉❧✉s✿

◮ SN = t②♣❛❜✐❧✐t② ✐♥ s②st❡♠ D ✭s✐♠♣❧❡ t②♣❡s ✰ ∩✮ ◮ ✏❛♥②✑ ♣r♦♦❢ ❜② r❡❞✉❝✐❜✐❧✐t② ❢♦r s✐♠♣❧❡ t②♣❡s ✐s ✈❛❧✐❞ ❢♦r D

❙♦ ✇❡✿

◮ ✐♥tr♦❞✉❝❡ ❛ s②st❡♠ D+ ♦❢ ✐♥t❡rs❡❝t✐♦♥ t②♣❡s ❢♦r ♥♦♥ ✉♥✐❢♦r♠

t❡r♠s ✭t❤✐s ♥❡❡❞s s♦♠❡ ❝❛r❡✮

◮ ♣r♦✈❡ t❤❛t M ∈ SN ✐♠♣❧✐❡s Γ ⊢ M : A ✐♥ D+ ◮ ❛❞❛♣t ❊❤r❤❛r❞✬s ♣r♦♦❢ t♦ D+

✶✶ ✴ ✶✺

T (M) ∈ {↑t ; t ∈ ∆}⊥ ⇒ M ∈ SN

❋✐♥✐t❡♥❡ss ♣r❡✈❡♥ts ❧♦♦♣s✳ ✳ ✳

❈♦♥s✐❞❡r δn = λx. x [xn]❀ t❤❡♥ ❢♦r ❛❧❧ n ∈ N✱ T (Ω) ∋ δn [δ0, δ0, δ1 . . . , δn−1] ≥ δ0 [] →ρ 0✳ ❍❡♥❝❡ T (Ω) ∈ {↑t ; t ∈ ∆}⊥✳

✳ ✳ ✳ ❜✉t ♥♦t ❞✐✈❡r❣❡♥❝❡

▲❡t ❛♥❞ ✱ t❤❡♥ ❢♦r ❛❧❧ ✳ ❍♦✇❡✈❡r✱ ✇r✐t✐♥❣ ❢♦r t❤❡ ♦♥❧② ❧✐♥❡❛r t❡r♠ ✿

❋✐①✿ ❛❞❞ ♠♦r❡ t❡sts

❈♦♥s✐❞❡r ❛ str✉❝t✉r❡ ❛♥❞ ❧❡t ✇✐t❤ ✳ ■❞❡❛✿ ✐s ❛ s❡t ♦❢ t❡sts✿ ♣❛ss❡s t❤❡ t❡st ✐❢ ✳ ❊❤r❤❛r❞✬s ✜♥✐t❡♥❡ss ✐s ✿ ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ✐♥✜♥✐t❡ t❡sts✳ ❖❢ ❝♦✉rs❡✱ ♥♦t ❛❧❧ ❛r❡ ❛❝❝❡♣t❛❜❧❡✱ ♦t❤❡r✇✐s❡ ✇❡ r❡❥❡❝t t♦♦ ♠❛♥② t❡r♠s ✭❝♦♥s✐❞❡r ✮✳

✶✷ ✴ ✶✺

T (M) ∈ {↑t ; t ∈ ∆}⊥ ⇒ M ∈ SN

❋✐♥✐t❡♥❡ss ♣r❡✈❡♥ts ❧♦♦♣s✳ ✳ ✳

❈♦♥s✐❞❡r δn = λx. x [xn]❀ t❤❡♥ ❢♦r ❛❧❧ n ∈ N✱ T (Ω) ∋ δn [δ0, δ0, δ1 . . . , δn−1] ≥ δ0 [] →ρ 0✳ ❍❡♥❝❡ T (Ω) ∈ {↑t ; t ∈ ∆}⊥✳

✳ ✳ ✳ ❜✉t ♥♦t ❞✐✈❡r❣❡♥❝❡

▲❡t ∆3 := λx. (x) x x ❛♥❞ Ω3 := (∆3) ∆3✱ t❤❡♥ T (Ω3) ⊥ ↑s ❢♦r ❛❧❧ s✳ ❍♦✇❡✈❡r✱ ✇r✐t✐♥❣ ❢♦r t❤❡ ♦♥❧② ❧✐♥❡❛r t❡r♠ ✿

❋✐①✿ ❛❞❞ ♠♦r❡ t❡sts

❈♦♥s✐❞❡r ❛ str✉❝t✉r❡ ❛♥❞ ❧❡t ✇✐t❤ ✳ ■❞❡❛✿ ✐s ❛ s❡t ♦❢ t❡sts✿ ♣❛ss❡s t❤❡ t❡st ✐❢ ✳ ❊❤r❤❛r❞✬s ✜♥✐t❡♥❡ss ✐s ✿ ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ✐♥✜♥✐t❡ t❡sts✳ ❖❢ ❝♦✉rs❡✱ ♥♦t ❛❧❧ ❛r❡ ❛❝❝❡♣t❛❜❧❡✱ ♦t❤❡r✇✐s❡ ✇❡ r❡❥❡❝t t♦♦ ♠❛♥② t❡r♠s ✭❝♦♥s✐❞❡r ✮✳

✶✷ ✴ ✶✺

T (M) ∈ {↑t ; t ∈ ∆}⊥ ⇒ M ∈ SN

❋✐♥✐t❡♥❡ss ♣r❡✈❡♥ts ❧♦♦♣s✳ ✳ ✳

❈♦♥s✐❞❡r δn = λx. x [xn]❀ t❤❡♥ ❢♦r ❛❧❧ n ∈ N✱ T (Ω) ∋ δn [δ0, δ0, δ1 . . . , δn−1] ≥ δ0 [] →ρ 0✳ ❍❡♥❝❡ T (Ω) ∈ {↑t ; t ∈ ∆}⊥✳

✳ ✳ ✳ ❜✉t ♥♦t ❞✐✈❡r❣❡♥❝❡

▲❡t ∆3 := λx. (x) x x ❛♥❞ Ω3 := (∆3) ∆3✱ t❤❡♥ T (Ω3) ⊥ ↑s ❢♦r ❛❧❧ s✳ ❍♦✇❡✈❡r✱ ✇r✐t✐♥❣ ℓ (M) ❢♦r t❤❡ ♦♥❧② ❧✐♥❡❛r t❡r♠ ∈ T (M)✿ T (Ω3) ⊥

• Ω3→∗

βM

• ℓ (M)

❋✐①✿ ❛❞❞ ♠♦r❡ t❡sts

❈♦♥s✐❞❡r ❛ str✉❝t✉r❡ ❛♥❞ ❧❡t ✇✐t❤ ✳ ■❞❡❛✿ ✐s ❛ s❡t ♦❢ t❡sts✿ ♣❛ss❡s t❤❡ t❡st ✐❢ ✳ ❊❤r❤❛r❞✬s ✜♥✐t❡♥❡ss ✐s ✿ ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ✐♥✜♥✐t❡ t❡sts✳ ❖❢ ❝♦✉rs❡✱ ♥♦t ❛❧❧ ❛r❡ ❛❝❝❡♣t❛❜❧❡✱ ♦t❤❡r✇✐s❡ ✇❡ r❡❥❡❝t t♦♦ ♠❛♥② t❡r♠s ✭❝♦♥s✐❞❡r ✮✳

✶✷ ✴ ✶✺

T (M) ∈ {↑t ; t ∈ ∆}⊥ ⇒ M ∈ SN

❋✐♥✐t❡♥❡ss ♣r❡✈❡♥ts ❧♦♦♣s✳ ✳ ✳

❈♦♥s✐❞❡r δn = λx. x [xn]❀ t❤❡♥ ❢♦r ❛❧❧ n ∈ N✱ T (Ω) ∋ δn [δ0, δ0, δ1 . . . , δn−1] ≥ δ0 [] →ρ 0✳ ❍❡♥❝❡ T (Ω) ∈ {↑t ; t ∈ ∆}⊥✳

✳ ✳ ✳ ❜✉t ♥♦t ❞✐✈❡r❣❡♥❝❡

▲❡t ∆3 := λx. (x) x x ❛♥❞ Ω3 := (∆3) ∆3✱ t❤❡♥ T (Ω3) ⊥ ↑s ❢♦r ❛❧❧ s✳ ❍♦✇❡✈❡r✱ ✇r✐t✐♥❣ ℓ (M) ❢♦r t❤❡ ♦♥❧② ❧✐♥❡❛r t❡r♠ ∈ T (M)✿ T (Ω3) ⊥

• Ω3→∗

βM

• ℓ (M)

❋✐①✿ ❛❞❞ ♠♦r❡ t❡sts

◮ ❈♦♥s✐❞❡r ❛ str✉❝t✉r❡ S ⊆ P (∆) ❛♥❞ ❧❡t FS = {↑a ; a ∈ S}⊥

✇✐t❤ ↑a =

s∈a ↑s✳ ◮ ■❞❡❛✿ S ✐s ❛ s❡t ♦❢ t❡sts✿ M ♣❛ss❡s t❤❡ t❡st a ∈ S ✐❢ T (M) ⊥ ↑a✳ ◮ ❊❤r❤❛r❞✬s ✜♥✐t❡♥❡ss ✐s FPf (∆)✿ ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r ✐♥✜♥✐t❡ t❡sts✳ ◮ ❖❢ ❝♦✉rs❡✱ ♥♦t ❛❧❧ S ❛r❡ ❛❝❝❡♣t❛❜❧❡✱ ♦t❤❡r✇✐s❡ ✇❡ r❡❥❡❝t t♦♦ ♠❛♥②

t❡r♠s ✭❝♦♥s✐❞❡r S = P (∆)✮✳

✶✷ ✴ ✶✺

• ❧✉❡✐♥❣ ❡✈❡r②t❤✐♥❣ t♦❣❡t❤❡r

◮ ❲❡ ❝❛♥ ❛❞❛♣t t❤❡ r❡❞✉❝✐❜✐❧✐t② ♣r♦♦❢ ❛♥❞ s❤♦✇ t❤❛t

M ∈ SN ⇒ T (M) ∈ FS ♣r♦✈✐❞❡❞ S s❛t✐s✜❡s✿

◮ ❢♦r ❛❧❧ n ∈ N✱ ❢♦r ❛❧❧ a ∈ S✱ {s ∈ a; height(s) = n} ✐s ✜♥✐t❡✳ ◮ ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧✱ ♣✉r❡❧② t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥s✳

❛s s♦♦♥ ❛s ❝♦♥t❛✐♥s ❛❧❧ s❡ts ♦❢ ❧✐♥❡❛r r❡s♦✉r❝❡ t❡r♠s ❲❡ ✇❛♥t ❢♦r ❛❧❧ ✭♦r ❛t ❧❡❛st t❤♦s❡ ✐♥ ♥♦r♠❛❧ ❢♦r♠✮ s♦ t❤❛t ✐s ❞❡✜♥❡❞✳

❊①❛♠♣❧❡

✐s ❜♦✉♥❞❡❞ ✇❤❡r❡ ❛♥❞ ✐s t❤❡ ♠❛①✐♠✉♠ s✐③❡ ♦❢ ❛ ❜❛❣ ♦❢ ❛r❣✉♠❡♥ts ✐♥ ✳

❚❤❡♦r❡♠ ✭P❛❣❛♥✐✲❚❛ss♦♥✲❱✳✮

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ❀ ✳ ▼♦r❡♦✈❡r✱ ✐♥ t❤✐s ❝❛s❡✱ ✐s ♥♦r♠❛❧✐③❛❜❧❡✳

✶✸ ✴ ✶✺

• ❧✉❡✐♥❣ ❡✈❡r②t❤✐♥❣ t♦❣❡t❤❡r

◮ ❲❡ ❝❛♥ ❛❞❛♣t t❤❡ r❡❞✉❝✐❜✐❧✐t② ♣r♦♦❢ ❛♥❞ s❤♦✇ t❤❛t

M ∈ SN ⇒ T (M) ∈ FS ♣r♦✈✐❞❡❞ S s❛t✐s✜❡s✿

◮ ❢♦r ❛❧❧ n ∈ N✱ ❢♦r ❛❧❧ a ∈ S✱ {s ∈ a; height(s) = n} ✐s ✜♥✐t❡✳ ◮ ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧✱ ♣✉r❡❧② t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥s✳

◮ T (M) ∈ FS ⇒ M ∈ SN ❛s s♦♦♥ ❛s S ❝♦♥t❛✐♥s ❛❧❧ s❡ts ♦❢ ❧✐♥❡❛r

r❡s♦✉r❝❡ t❡r♠s ❲❡ ✇❛♥t ❢♦r ❛❧❧ ✭♦r ❛t ❧❡❛st t❤♦s❡ ✐♥ ♥♦r♠❛❧ ❢♦r♠✮ s♦ t❤❛t ✐s ❞❡✜♥❡❞✳

❊①❛♠♣❧❡

✐s ❜♦✉♥❞❡❞ ✇❤❡r❡ ❛♥❞ ✐s t❤❡ ♠❛①✐♠✉♠ s✐③❡ ♦❢ ❛ ❜❛❣ ♦❢ ❛r❣✉♠❡♥ts ✐♥ ✳

❚❤❡♦r❡♠ ✭P❛❣❛♥✐✲❚❛ss♦♥✲❱✳✮

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ❀ ✳ ▼♦r❡♦✈❡r✱ ✐♥ t❤✐s ❝❛s❡✱ ✐s ♥♦r♠❛❧✐③❛❜❧❡✳

✶✸ ✴ ✶✺

• ❧✉❡✐♥❣ ❡✈❡r②t❤✐♥❣ t♦❣❡t❤❡r

◮ ❲❡ ❝❛♥ ❛❞❛♣t t❤❡ r❡❞✉❝✐❜✐❧✐t② ♣r♦♦❢ ❛♥❞ s❤♦✇ t❤❛t

M ∈ SN ⇒ T (M) ∈ FS ♣r♦✈✐❞❡❞ S s❛t✐s✜❡s✿

◮ ❢♦r ❛❧❧ n ∈ N✱ ❢♦r ❛❧❧ a ∈ S✱ {s ∈ a; height(s) = n} ✐s ✜♥✐t❡✳ ◮ ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧✱ ♣✉r❡❧② t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥s✳

◮ T (M) ∈ FS ⇒ M ∈ SN ❛s s♦♦♥ ❛s S ❝♦♥t❛✐♥s ❛❧❧ s❡ts ♦❢ ❧✐♥❡❛r

r❡s♦✉r❝❡ t❡r♠s

◮ ❲❡ ✇❛♥t {s} ∈ S ❢♦r ❛❧❧ s ∈ ∆ ✭♦r ❛t ❧❡❛st t❤♦s❡ ✐♥ ♥♦r♠❛❧ ❢♦r♠✮

s♦ t❤❛t T (M) ∈ FS ⇒ NF

• T (M)
• ✐s ❞❡✜♥❡❞✳

❊①❛♠♣❧❡

✐s ❜♦✉♥❞❡❞ ✇❤❡r❡ ❛♥❞ ✐s t❤❡ ♠❛①✐♠✉♠ s✐③❡ ♦❢ ❛ ❜❛❣ ♦❢ ❛r❣✉♠❡♥ts ✐♥ ✳

❚❤❡♦r❡♠ ✭P❛❣❛♥✐✲❚❛ss♦♥✲❱✳✮

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ❀ ✳ ▼♦r❡♦✈❡r✱ ✐♥ t❤✐s ❝❛s❡✱ ✐s ♥♦r♠❛❧✐③❛❜❧❡✳

✶✸ ✴ ✶✺

• ❧✉❡✐♥❣ ❡✈❡r②t❤✐♥❣ t♦❣❡t❤❡r

◮ ❲❡ ❝❛♥ ❛❞❛♣t t❤❡ r❡❞✉❝✐❜✐❧✐t② ♣r♦♦❢ ❛♥❞ s❤♦✇ t❤❛t

M ∈ SN ⇒ T (M) ∈ FS ♣r♦✈✐❞❡❞ S s❛t✐s✜❡s✿

◮ ❢♦r ❛❧❧ n ∈ N✱ ❢♦r ❛❧❧ a ∈ S✱ {s ∈ a; height(s) = n} ✐s ✜♥✐t❡✳ ◮ ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧✱ ♣✉r❡❧② t❡❝❤♥✐❝❛❧ ❝♦♥❞✐t✐♦♥s✳

◮ T (M) ∈ FS ⇒ M ∈ SN ❛s s♦♦♥ ❛s S ❝♦♥t❛✐♥s ❛❧❧ s❡ts ♦❢ ❧✐♥❡❛r

r❡s♦✉r❝❡ t❡r♠s

◮ ❲❡ ✇❛♥t {s} ∈ S ❢♦r ❛❧❧ s ∈ ∆ ✭♦r ❛t ❧❡❛st t❤♦s❡ ✐♥ ♥♦r♠❛❧ ❢♦r♠✮

s♦ t❤❛t T (M) ∈ FS ⇒ NF

• T (M)
• ✐s ❞❡✜♥❡❞✳

❊①❛♠♣❧❡

B = {a ⊆ ∆; # (a) ✐s ❜♦✉♥❞❡❞} ✇❤❡r❡ # (a) = {# (s) ; s ∈ a} ❛♥❞ # (s) ✐s t❤❡ ♠❛①✐♠✉♠ s✐③❡ ♦❢ ❛ ❜❛❣ ♦❢ ❛r❣✉♠❡♥ts ✐♥ s✳

❚❤❡♦r❡♠ ✭P❛❣❛♥✐✲❚❛ss♦♥✲❱✳✮

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿

◮ M ∈ SN❀ ◮ T (M) ∈ FB✳

▼♦r❡♦✈❡r✱ ✐♥ t❤✐s ❝❛s❡✱ T (M) ✐s ♥♦r♠❛❧✐③❛❜❧❡✳

✶✸ ✴ ✶✺

❈♦♥❝❧✉s✐♦♥

❲❡ ❛r❡ ❤❛♣♣②✳

❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ ❛ ♥✐❝❡ ❛♥❞ ♥♦✈❡❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ SN✳

❆r❡ ✇❡❄

❚❤❡ r❡❛❧❧② ✉s❡❢✉❧ ❜✐t ✐s t❤❛t✿ t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ ❛ str♦♥❣❧② ♥♦r♠❛❧✐③❛❜❧❡ t❡r♠ ✐s ♥♦r♠❛❧✐③❛❜❧❡ ✇❤✐❝❤ ✐s ❛ ❜✐t ❢r✉str❛t✐♥❣ ✭✇❤② ✏str♦♥❣❧②✑❄✮✳

❨❡s ✇❡ ❛r❡✿ ♣❧❡♥t② ♦❢ ❢✉t✉r❡ ✇♦r❦✦

❖✉r ♠❛❝❤✐♥❡r② ✐s ♠♦❞✉❧❛r ❡♥♦✉❣❤ t❤❛t ✐t ❝❛♥ ❜❡ ❛❞❛♣t❡❞ t♦ ✇❡❛❦✲ ❛♥❞ ❤❡❛❞✲♥♦r♠❛❧✐③❛❜✐❧✐t② ✭❲■P ✇✐t❤ P❛❣❛♥✐ ❛♥❞ ❚❛ss♦♥✮✳ ❚❤❛t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ✇❡ ❝❛♥ tr❛❝❦ ✲r❡❞✉❝t✐♦♥ t❤r♦✉❣❤ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✭❲■P✮✳ ❚♦✇❛r❞s ❛ s❡♠❛♥t✐❝❛❧❧② ❢♦✉♥❞❡❞ ♥♦t✐♦♥ ♦❢ ❇ö❤♠ tr❡❡s ❢♦r ✈❛r✐♦✉s ♥♦♥ ✉♥✐❢♦r♠ s❡tt✐♥❣s ✭q✉❛♥t✐t❛t✐✈❡ ♥♦♥✲❞❡t❡r♠✐♥✐s♠✱ ♣r♦❜❛❜✐❧✐st✐❝ st✉✛✱ ❡t❝✳✮✳

✶✹ ✴ ✶✺

❈♦♥❝❧✉s✐♦♥ ◆♦✱ ✇❛✐t✦

❲❡ ❛r❡ ❤❛♣♣②✳

❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ ❛ ♥✐❝❡ ❛♥❞ ♥♦✈❡❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ SN✳

❆r❡ ✇❡❄

❚❤❡ r❡❛❧❧② ✉s❡❢✉❧ ❜✐t ✐s t❤❛t✿ t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ ❛ str♦♥❣❧② ♥♦r♠❛❧✐③❛❜❧❡ t❡r♠ ✐s ♥♦r♠❛❧✐③❛❜❧❡ ✇❤✐❝❤ ✐s ❛ ❜✐t ❢r✉str❛t✐♥❣ ✭✇❤② ✏str♦♥❣❧②✑❄✮✳

❨❡s ✇❡ ❛r❡✿ ♣❧❡♥t② ♦❢ ❢✉t✉r❡ ✇♦r❦✦

❖✉r ♠❛❝❤✐♥❡r② ✐s ♠♦❞✉❧❛r ❡♥♦✉❣❤ t❤❛t ✐t ❝❛♥ ❜❡ ❛❞❛♣t❡❞ t♦ ✇❡❛❦✲ ❛♥❞ ❤❡❛❞✲♥♦r♠❛❧✐③❛❜✐❧✐t② ✭❲■P ✇✐t❤ P❛❣❛♥✐ ❛♥❞ ❚❛ss♦♥✮✳ ❚❤❛t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ✇❡ ❝❛♥ tr❛❝❦ ✲r❡❞✉❝t✐♦♥ t❤r♦✉❣❤ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✭❲■P✮✳ ❚♦✇❛r❞s ❛ s❡♠❛♥t✐❝❛❧❧② ❢♦✉♥❞❡❞ ♥♦t✐♦♥ ♦❢ ❇ö❤♠ tr❡❡s ❢♦r ✈❛r✐♦✉s ♥♦♥ ✉♥✐❢♦r♠ s❡tt✐♥❣s ✭q✉❛♥t✐t❛t✐✈❡ ♥♦♥✲❞❡t❡r♠✐♥✐s♠✱ ♣r♦❜❛❜✐❧✐st✐❝ st✉✛✱ ❡t❝✳✮✳

✶✹ ✴ ✶✺

❈♦♥❝❧✉s✐♦♥ ◆♦✱ ✇❛✐t✦ ❈♦♥❝❧✉s✐♦♥

❲❡ ❛r❡ ❤❛♣♣②✳

❲❡ ❤❛✈❡ ❡st❛❜❧✐s❤❡❞ ❛ ♥✐❝❡ ❛♥❞ ♥♦✈❡❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ SN✳

❆r❡ ✇❡❄

❚❤❡ r❡❛❧❧② ✉s❡❢✉❧ ❜✐t ✐s t❤❛t✿ t❤❡ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ ❛ str♦♥❣❧② ♥♦r♠❛❧✐③❛❜❧❡ t❡r♠ ✐s ♥♦r♠❛❧✐③❛❜❧❡ ✇❤✐❝❤ ✐s ❛ ❜✐t ❢r✉str❛t✐♥❣ ✭✇❤② ✏str♦♥❣❧②✑❄✮✳

❨❡s ✇❡ ❛r❡✿ ♣❧❡♥t② ♦❢ ❢✉t✉r❡ ✇♦r❦✦

◮ ❖✉r ♠❛❝❤✐♥❡r② ✐s ♠♦❞✉❧❛r ❡♥♦✉❣❤ t❤❛t ✐t ❝❛♥ ❜❡ ❛❞❛♣t❡❞ t♦

✇❡❛❦✲ ❛♥❞ ❤❡❛❞✲♥♦r♠❛❧✐③❛❜✐❧✐t② ✭❲■P ✇✐t❤ P❛❣❛♥✐ ❛♥❞ ❚❛ss♦♥✮✳

◮ ❚❤❛t

T (NF (M)) = NF( T (M)) ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❢❛❝t t❤❛t ✇❡ ❝❛♥ tr❛❝❦ β✲r❡❞✉❝t✐♦♥ t❤r♦✉❣❤ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✭❲■P✮✳

◮ ❚♦✇❛r❞s ❛ s❡♠❛♥t✐❝❛❧❧② ❢♦✉♥❞❡❞ ♥♦t✐♦♥ ♦❢ ❇ö❤♠ tr❡❡s ❢♦r ✈❛r✐♦✉s

♥♦♥ ✉♥✐❢♦r♠ s❡tt✐♥❣s ✭q✉❛♥t✐t❛t✐✈❡ ♥♦♥✲❞❡t❡r♠✐♥✐s♠✱ ♣r♦❜❛❜✐❧✐st✐❝ st✉✛✱ ❡t❝✳✮✳

✶✹ ✴ ✶✺

❚❤❡ ❡♥❞ ❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳ ◗✉❡st✐♦♥s❄

✶✺ ✴ ✶✺

❚❤❡ ❡♥❞ ❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳ ◗✉❡st✐♦♥s❄

✶✺ ✴ ✶✺