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❈✉❜✐❝❛❧ ❆ss❡♠❜❧✐❡s ❛♥❞ t❤❡ ■♥❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ Pr♦♣♦s✐t✐♦♥❛❧ ❘❡s✐③✐♥❣ ❆①✐♦♠

❚❛✐❝❤✐ ❯❡♠✉r❛ ❯♥✐✈❡rs✐t② ♦❢ ❆♠st❡r❞❛♠ ❍♦❚❚✴❯❋ ✷✵✶✽✱ ✼t❤ ❏✉❧②✱ ✷✵✶✽

❖✈❡r✈✐❡✇

■ ♣r❡s❡♥t ❛ ♠♦❞❡❧ ♦❢ ❝✉❜✐❝❛❧ t②♣❡ t❤❡♦r② ❬❈♦❤❡♥ ❡t ❛❧✳✱ ✷✵✶✽❪ ✐♥ t❤❡ ❝❛t❡❣♦r② ♦❢ ✐♥t❡r♥❛❧ ❝✉❜✐❝❛❧ ♦❜❥❡❝ts ✐♥ t❤❡ ❝❛t❡❣♦r② ♦❢ ❛ss❡♠❜❧✐❡s✳ ❚❤✐s ♠♦❞❡❧

◮ ❤❛s ❛ ✉♥✐✈❛❧❡♥t ❛♥❞ ✐♠♣r❡❞✐❝❛t✐✈❡ ✉♥✐✈❡rs❡❀ ◮ ❞♦❡s ◆❖❚ s❛t✐s❢② t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ r❡s✐③✐♥❣ ❛①✐♦♠✳

❉r❛❢t ❛✈❛✐❧❛❜❧❡ ♦♥ ❛r❳✐✈✿✶✽✵✸✳✵✻✻✹✾✳

❈✉❜✐❝❛❧ ❚②♣❡ ❚❤❡♦r②

❈✉❜✐❝❛❧ t②♣❡ t❤❡♦r② ❬❈♦❤❡♥ ❡t ❛❧✳✱ ✷✵✶✽❪ ✐s ❛ t②♣❡ t❤❡♦r② ✐♥t❡♥❞❡❞ ❢♦r ❝♦♥str✉❝t✐✈❡ ❥✉st✐✜❝❛t✐♦♥s ♦❢ t❤❡ ✉♥✐✈❛❧❡♥❝❡ ❛①✐♦♠ ❛♥❞ ❤✐❣❤❡r ✐♥❞✉❝t✐✈❡ t②♣❡s✳ ❚❤❡ ✉♥✐✈❛❧❡♥❝❡ ❛①✐♦♠ ✐s ♣r♦✈❛❜❧❡ ✐♥ ❝✉❜✐❝❛❧ t②♣❡ t❤❡♦r②✳

■♠♣r❡❞✐❝❛t✐✈❡ ❯♥✐✈❡rs❡s

❲❡ s❛② ❛ ✉♥✐✈❡rs❡ U ✐s ✐♠♣r❡❞✐❝❛t✐✈❡ ✐❢✱ ❢♦r ❛♥② t②♣❡ A ❛♥❞ t②♣❡ ❢❛♠✐❧② B : A → U✱ t❤❡ ❞❡♣❡♥❞❡♥t ♣r♦❞✉❝t t②♣❡

x:A B(x) ❜❡❧♦♥❣s

t♦ U✳

❊①❛♠♣❧❡s

◮ ( X:U X → X) : U ◮ ( X:U X → (X → X) → X) : U ◮ ( X:U

• x:X x = x → X) : U

■♠♣r❡❞✐❝❛t✐✈❡ ❊♥❝♦❞✐♥❣s ♦❢ ✭❍✐❣❤❡r✮ ■♥❞✉❝t✐✈❡ ❚②♣❡s

❆♥ ✐♥t❡r❡st✐♥❣ ✉s❡ ♦❢ s✉❝❤ ❛♥ ✐♠♣r❡❞✐❝❛t✐✈❡ ✉♥✐✈❡rs❡ ✐s t♦ r❡♣r❡s❡♥t ❞❛t❛ t②♣❡s ❛s ♣♦❧②♠♦r♣❤✐❝ ❢✉♥❝t✐♦♥ t②♣❡s ❬❙❤✉❧♠❛♥✱ ✷✵✶✶❪✳

◮ N :≡ X:U X → (X → X) → X ◮ S✶ :≡ X:U

• x:X x = x → X

◮ A :≡ X:❤Pr♦♣(A → X) → X ✇❤❡r❡ ❤Pr♦♣ ✐s t❤❡ ✉♥✐✈❡rs❡ ♦❢

❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥s ✐♥ U✳ ❚❤❡② ❤❛✈❡ ❝♦♥str✉❝t♦rs ❛♥❞ r❡❝✉rs♦rs ✐♥ t❤❡ s❡♥s❡ ♦❢ t❤❡ ❍♦❚❚ ❇♦♦❦✱ ❜✉t ❞♦ ♥♦t s❛t✐s❢② t❤❡ ✐♥❞✉❝t✐♦♥ ♣r✐♥❝✐♣❧❡ ✐♥ ❣❡♥❡r❛❧✳ ❘❡✜♥❡♠❡♥ts ♦❢ ✐♠♣r❡❞✐❝❛t✐✈❡ ❡♥❝♦❞✐♥❣s ❛r❡ st✉❞✐❡❞ ❜② ❆✇♦❞❡②✱ ❋r❡② ❛♥❞ ❙♣❡✐❣❤t ❬❆✇♦❞❡② ❡t ❛❧✳✱ ✷✵✶✽❪✳

Pr♦♣♦s✐t✐♦♥❛❧ ❘❡s✐③✐♥❣

❲❡ s❛② ❛ ❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥ A ❛❞♠✐ts ♣r♦♣♦s✐t✐♦♥❛❧ r❡s✐③✐♥❣ ✐❢ t❤❡r❡ ❡①✐st ❛ ❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥ A′ ✐♥ U ❛♥❞ ❛♥ ❡q✉✐✈❛❧❡♥❝❡ A ≃ A′✳

◮ ❖r✐❣✐♥❛❧❧② ♣r♦♣♦s❡❞ ❜② ❱♦❡✈♦❞s❦② ❬❱♦❡✈♦❞s❦②✱ ✷✵✶✷❪✳ ◮ ❆ ❢♦r♠ ♦❢ ✐♠♣r❡❞✐❝❛t✐✈✐t② ❢♦r ❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥s✳

■❢ U ✐s ✐♠♣r❡❞✐❝❛t✐✈❡✱ ✇❡ ❤❛✈❡ ❛ ❝❛♥❞✐❞❛t❡ ❢♦r ♣r♦♣♦s✐t✐♦♥❛❧ r❡s✐③✐♥❣✳ ❋♦r ❛ ❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥ A✱ ❞❡✜♥❡ A∗ :≡

X:❤Pr♦♣(A → X) → X ❛♥❞ ηA :≡ λa✳λXh✳ha : A → A∗✳

Pr♦♣♦s✐t✐♦♥

❆ ❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥ A ❛❞♠✐ts ♣r♦♣♦s✐t✐♦♥❛❧ r❡s✐③✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢✉♥❝t✐♦♥ ηA : A → A∗ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡✳

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❈✉❜✐❝❛❧ ❆ss❡♠❜❧② ▼♦❞❡❧ ❚❤❡ ❈♦✉♥t❡r❡①❛♠♣❧❡ t♦ Pr♦♣♦s✐t✐♦♥❛❧ ❘❡s✐③✐♥❣ ❈♦♥❝❧✉s✐♦♥

❖✈❡r✈✐❡✇

■♥t❡r♥❛❧✐③✐♥❣ ❝✉❜✐❝❛❧ s❡t ♠♦❞❡❧s ❬❇❡③❡♠ ❡t ❛❧✳✱ ✷✵✶✹✱ ❈♦❤❡♥ ❡t ❛❧✳✱ ✷✵✶✽❪ ✐♥ t❤❡ ❝❛t❡❣♦r② ❆s♠ ♦❢ ❛ss❡♠❜❧✐❡s✳

◮ ❆s♠ ✐s ❛ ♠♦❞❡❧ ♦❢ ❡①t❡♥s✐♦♥❛❧ ❞❡♣❡♥❞❡♥t t②♣❡ t❤❡♦r② ✇✐t❤

♣r♦♣♦s✐t✐♦♥❛❧ tr✉♥❝❛t✐♦♥✳

◮ ❆s♠ ❤❛s ❛♥ ✐♠♣r❡❞✐❝❛t✐✈❡ ✉♥✐✈❡rs❡ P❊❘✳ ◮ ❲❡ ❞❡✜♥❡ ❛♥ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r② ♦❢ ❝✉❜❡s ✐♥ ❆s♠✳ ◮ ▲❡t ❈❆s♠ ❜❡ t❤❡ ❝❛t❡❣♦r② ♦❢ ✐♥t❡r♥❛❧ ♣r❡s❤❡❛✈❡s ♦♥ ✇❤✐❝❤

✇❡ ❝❛❧❧ ❝✉❜✐❝❛❧ ❛ss❡♠❜❧✐❡s✳

❚❤❡♦r❡♠

❈❆s♠ ✐s ❛ ♠♦❞❡❧ ♦❢ ❝✉❜✐❝❛❧ t②♣❡ t❤❡♦r② ✇✐t❤ ❛♥ ✐♠♣r❡❞✐❝❛t✐✈❡ ✉♥✐✈❡rs❡✳

❚❤❡ ❖rt♦♥✲P✐tts ❆①✐♦♠s

❈❆s♠ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛♥ ✐♥st❛♥❝❡ ♦❢ ♠♦r❡ ❣❡♥❡r❛❧ ❝♦♥str✉❝t✐♦♥ ♦❢ ♠♦❞❡❧s ♦❢ ❝✉❜✐❝❛❧ t②♣❡ t❤❡♦r② ❣✐✈❡♥ ❜② ❖rt♦♥ ❛♥❞ P✐tts ❬❖rt♦♥ ❛♥❞ P✐tts✱ ✷✵✶✻❪✳ ❚❤❡② ❝♦♥str✉❝t ❛ ♠♦❞❡❧ ♦❢ ❝✉❜✐❝❛❧ t②♣❡ t❤❡♦r② ✐♥ ❛♥ ❡❧❡♠❡♥t❛r② t♦♣♦s ✇✐t❤

◮ ❛♥ ✐♥t❡r✈❛❧ ♦❜❥❡❝t I❀ ❛♥❞ ◮ ❛♥ ♦❜❥❡❝t ❈♦❢ ֌ Ω ♦❢ ❝♦✜❜r❛♥t ♣r♦♣♦s✐t✐♦♥s

s❛t✐s❢②✐♥❣ s❡✈❡r❛❧ ❛①✐♦♠s✳ ■♥ t❤❛t ♠♦❞❡❧✱

◮ ❛ ❝♦♥t❡①t Γ ✐s ❛♥ ♦❜❥❡❝t ♦❢ t❤❡ t♦♣♦s❀ ◮ ❛ t②♣❡ Γ ⊢ A ✐s ❛ ♠♦r♣❤✐s♠ A → Γ ❡q✉✐♣♣❡❞ ✇✐t❤ ❛

❝♦♠♣♦s✐t✐♦♥ str✉❝t✉r❡❀ ❛♥❞

◮ ❛ t❡r♠ Γ ⊢ a : A ✐s ❛ s❡❝t✐♦♥ ♦❢ t❤❡ ♠♦r♣❤✐s♠ A → Γ✳

❚❤❡ ❛①✐♦♠s ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❡①t❡♥s✐♦♥❛❧ ❞❡♣❡♥❞❡♥t t②♣❡ t❤❡♦r② ✇✐t❤ ♣r♦♣♦s✐t✐♦♥❛❧ tr✉♥❝❛t✐♦♥ ❜✉t ✇✐t❤♦✉t t❤❡ s✉❜♦❜❥❡❝t ❝❧❛ss✐✜❡r✳ ❙♦ ✇❡ ❝❛♥ ✉s❡ t❤❡✐r ❝♦♥str✉❝t✐♦♥ ❢♦r ♠❛❦✐♥❣ ❈❆s♠ ❛ ♠♦❞❡❧ ♦❢ ❝✉❜✐❝❛❧ t②♣❡ t❤❡♦r②✱ ❛❧t❤♦✉❣❤ ❈❆s♠ ✐s ♥♦t ❛♥ ❡❧❡♠❡♥t❛r② t♦♣♦s✳

❯♥✐✈❡rs❡s ♦❢ ❋✐❜r❛♥t ❚②♣❡s

❚❤❡ ❖rt♦♥✲P✐tts ❝♦♥str✉❝t✐♦♥ ❞♦❡s ♥♦t ❣✉❛r❛♥t❡❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ✉♥✐✈❡rs❡s ♦❢ ✜❜r❛♥t t②♣❡s✳ ❲❡ ❝❛♥ ✉s❡ r❡❝❡♥t ✇♦r❦ ❜② ▲✐❝❛t❛✱ ❖rt♦♥✱ P✐tts ❛♥❞ ❙♣✐tt❡rs ❬▲✐❝❛t❛ ❡t ❛❧✳✱ ✷✵✶✽❪ ♦♥ ❝♦♥str✉❝t✐♥❣ ✉♥✐✈❡rs❡s ♦❢ ✜❜r❛♥t t②♣❡s ✉s✐♥❣ t❤❡ r✐❣❤t ❛❞❥♦✐♥t t♦ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t♦r (−)I✳

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ❈✉❜✐❝❛❧ ❆ss❡♠❜❧② ▼♦❞❡❧ ❚❤❡ ❈♦✉♥t❡r❡①❛♠♣❧❡ t♦ Pr♦♣♦s✐t✐♦♥❛❧ ❘❡s✐③✐♥❣ ❈♦♥❝❧✉s✐♦♥

Pr♦♣♦s✐t✐♦♥❛❧ ❘❡s✐③✐♥❣

❘❡❝❛❧❧✿

Pr♦♣♦s✐t✐♦♥

■♥ ❝✉❜✐❝❛❧ t②♣❡ t❤❡♦r② ✇✐t❤ ❛♥ ✐♠♣r❡❞✐❝❛t✐✈❡ ✉♥✐✈❡rs❡✱ ❛ ❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥ Γ ⊢ A ❛❞♠✐ts ♣r♦♣♦s✐t✐♦♥❛❧ r❡s✐③✐♥❣ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢✉♥❝t✐♦♥ Γ ⊢ ηA : A → A∗ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡✱ ✇❤❡r❡ A∗ :≡

X:❤Pr♦♣(A → X) → X✳

❚❤❡♦r❡♠

■♥ t❤❡ ❝✉❜✐❝❛❧ ❛ss❡♠❜❧② ♠♦❞❡❧✱ t❤❡r❡ ❡①✐sts ❛ ❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥ Γ ⊢ A s✉❝❤ t❤❛t A∗ ✐s ✐♥❤❛❜✐t❡❞ ❜✉t A ✐s ♥♦t✳

❖✈❡r✈✐❡✇

✶✳ ❋✐♥❞ ❛ ❢❛♠✐❧② Γ ⊢ A ♦❢ ❛ss❡♠❜❧✐❡s t❤❛t ✐s ✉♥✐❢♦r♠ ❛♥❞ ✇❡❧❧✲s✉♣♣♦rt❡❞ ❜✉t ♥♦t ✐♥❤❛❜✐t❡❞✳ ✷✳ ❚❤❡ ❝♦❞✐s❝r❡t❡ ♣r❡s❤❡❛❢ ∆Γ ⊢ ∇A ♦✈❡r t❤❡ ❝♦♥st❛♥t ♣r❡s❤❡❛❢ ∆Γ ✐s ✉♥✐❢♦r♠ ❛♥❞ ✇❡❧❧✲s✉♣♣♦rt❡❞ ❜✉t ♥♦t ✐♥❤❛❜✐t❡❞✳ ✸✳ ∇A ✐s ❛ ❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥✳ ✹✳ ■❢ ❛ ❢❛♠✐❧② ∆ ⊢ B ✐♥ ❈❆s♠ ✐s ✉♥✐❢♦r♠ ❛♥❞ ✇❡❧❧✲s✉♣♣♦rt❡❞✱ t❤❡♥ B∗ ✐s ✐♥❤❛❜✐t❡❞✳

❆ss❡♠❜❧✐❡s

◮ ❆♥ ❛ss❡♠❜❧② ✐s ❛ s❡t A ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ♥♦♥✲❡♠♣t② s❡t EA(a)

♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ❢♦r ❡❛❝❤ a ∈ A✳ ❊❧❡♠❡♥ts ♦❢ EA(a) ❛r❡ ❝❛❧❧❡❞ r❡❛❧✐③❡rs ♦❢ a✳

◮ ❆ ♠♦r♣❤✐s♠ f : A → B ♦❢ ❛ss❡♠❜❧✐❡s ✐s ❛ ❢✉♥❝t✐♦♥ f : A → B

❜❡t✇❡❡♥ t❤❡ ✉♥❞❡r❧②✐♥❣ s❡ts s✉❝❤ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♣❛rt✐❛❧ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥ e s✉❝❤ t❤❛t✱ ❢♦r ❛♥② a ∈ A ❛♥❞ n ∈ EA(a)✱ t❤❡ ❛♣♣❧✐❝❛t✐♦♥ en ✐s ❞❡✜♥❡❞ ❛♥❞ ❜❡❧♦♥❣s t♦ EB(f(a))✳

◮ ❚❤❡ ❝❛t❡❣♦r② ❆s♠ ♦❢ ❛ss❡♠❜❧✐❡s ❤❛s ❛♥ ✐♠♣r❡❞✐❝❛t✐✈❡ ✉♥✐✈❡rs❡

P❊❘ ♦❢ ♣❛rt✐❛❧ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s ♦♥ N✳

◮ ❆ ♠♦r♣❤✐s♠ f : A → P❊❘ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠♦❞❡st ❢❛♠✐❧②

B → A✿ ❢♦r ❛♥② a ∈ A ❛♥❞ b✱ b′ ∈ Ba✱ ✐❢ E(b) ∩ E(b′) = ∅ t❤❡♥ b = b′✳

❯♥✐❢♦r♠ ❖❜❥❡❝ts

◮ ❆♥ ❛ss❡♠❜❧② A ✐s ✉♥✐❢♦r♠ ✐❢ a∈A E(a) ✐s ♥♦♥✲❡♠♣t②✳ ◮ ■♥ ❛ r❡❣✉❧❛r ❝❛t❡❣♦r②✱ ❛♥ ♦❜❥❡❝t A ✐s ✇❡❧❧✲s✉♣♣♦rt❡❞ ✐❢ t❤❡

✉♥✐q✉❡ ♠♦r♣❤✐s♠ A → ✶ ✐s r❡❣✉❧❛r ❡♣✐✳

Pr♦♣♦s✐t✐♦♥

❋♦r ❛ss❡♠❜❧✐❡s A ❛♥❞ X✱ ✐❢ A ✐s ✉♥✐❢♦r♠ ❛♥❞ ✇❡❧❧✲s✉♣♣♦rt❡❞ ❛♥❞ X ✐s ♠♦❞❡st✱ t❤❡♥ t❤❡ ♠♦r♣❤✐s♠ λxa✳x : X → (A → X) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳

❯♥✐❢♦r♠ Pr❡s❤❡❛✈❡s

❯♥✐❢♦r♠ ❛♥❞ ♠♦❞❡st ✐♥t❡r♥❛❧ ♣r❡s❤❡❛✈❡s ❛r❡ ❞❡✜♥❡❞ ♣♦✐♥t✇✐s❡✳

Pr♦♣♦s✐t✐♦♥

❋♦r ✐♥t❡r♥❛❧ ♣r❡s❤❡❛✈❡s A ❛♥❞ X✱ ✐❢ A ✐s ✉♥✐❢♦r♠ ❛♥❞ ✇❡❧❧✲s✉♣♣♦rt❡❞ ❛♥❞ X ✐s ♠♦❞❡st✱ t❤❡♥ t❤❡ ♠♦r♣❤✐s♠ λxa✳x : X → (A → X) ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳

❈♦r♦❧❧❛r②

■❢ ❛ t②♣❡ Γ ⊢ A ✐♥ ❈❆s♠ ✐s ✉♥✐❢♦r♠ ❛♥❞ ✇❡❧❧✲s✉♣♣♦rt❡❞✱ t❤❡♥ A∗ ≡

X:❤Pr♦♣(A → X) → X ✐s ✐♥❤❛❜✐t❡❞✳

❈♦❞✐s❝r❡t❡ ❈✉❜✐❝❛❧ ❆ss❡♠❜❧✐❡s

❋♦r ❛♥ ❛ss❡♠❜❧② Γ✱ ❧❡t ∆Γ ❞❡♥♦t❡ t❤❡ ❝♦♥st❛♥t ♣r❡s❤❡❛❢✳ ❋♦r ❛ ❢❛♠✐❧② Γ ⊢ A ♦❢ ❛ss❡♠❜❧✐❡s✱ ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ❝♦❞✐s❝r❡t❡ ❝✉❜✐❝❛❧ ❛ss❡♠❜❧② ∆Γ ⊢ ∇A ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t

◮ t❤❡ ♣♦✐♥ts ♦❢ ∇A ✐s ❡❧❡♠❡♥ts ♦❢ A❀ ❛♥❞ ◮ ❛♥② t✇♦ ♣♦✐♥ts ❛r❡ ❝♦♥♥❡❝t❡❞ ❜② ❛ ✉♥✐q✉❡ ♣❛t❤✳

Pr♦♣♦s✐t✐♦♥

◮ ■❢ A ✐s ✉♥✐❢♦r♠ t❤❡♥ s♦ ✐s ∇A✳ ◮ ■❢ A ✐s ✇❡❧❧✲s✉♣♣♦rt❡❞ t❤❡♥ s♦ ✐s ∇A✳ ◮ ■❢ ∇A ✐s ✐♥❤❛❜✐t❡❞ t❤❡♥ s♦ ✐s A✳

❙♦ ♦✉r ❣♦❛❧ ✐s t♦ ✜♥❞ ❛ ❢❛♠✐❧② Γ ⊢ A ♦❢ ❛ss❡♠❜❧✐❡s t❤❛t ✐s ✉♥✐❢♦r♠ ❛♥❞ ✇❡❧❧✲s✉♣♣♦rt❡❞ ❜✉t ♥♦t ✐♥❤❛❜✐t❡❞✳

❚❤❡ ❈♦✉♥t❡r❡①❛♠♣❧❡

❉❡✜♥❡ ❛ ❢❛♠✐❧② Γ ⊢ A ♦❢ ❛ss❡♠❜❧✐❡s ❛s ❢♦❧❧♦✇s✳

◮ Γ = (N✱ n → {m ∈ N | m > n}) ◮ A(n) = ({m ∈ N | m > n}✱ m → {n✱ m})

❚❤❡♥ A ✐s ✉♥✐❢♦r♠ ❛♥❞ ✇❡❧❧✲s✉♣♣♦rt❡❞ ❜✉t ♥♦t ✐♥❤❛❜✐t❡❞✱ ❛♥❞ t❤✉s ∆Γ ⊢ ∇A ✐s ❛ ❤♦♠♦t♦♣② ♣r♦♣♦s✐t✐♦♥ ✐♥ ❈❆s♠ t❤❛t ❞♦❡s ♥♦t ❛❞♠✐t ♣r♦♣♦s✐t✐♦♥❛❧ r❡s✐③✐♥❣✳

❈♦♥❝❧✉s✐♦♥

❲❡ ❤❛✈❡ ❣♦t ❛ ♠♦❞❡❧ ♦❢ ❝✉❜✐❝❛❧ t②♣❡ t❤❡♦r② ✐♥ t❤❡ ❝❛t❡❣♦r② ♦❢ ❝✉❜✐❝❛❧ ❛ss❡♠❜❧✐❡s t❤❛t

◮ ❤❛s ❛ ✉♥✐✈❛❧❡♥t ❛♥❞ ✐♠♣r❡❞✐❝❛t✐✈❡ ✉♥✐✈❡rs❡❀ ❜✉t ◮ ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ r❡s✐③✐♥❣ ❛①✐♦♠✳

❘❡❧❛t❡❞ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ■

◮ ■s t❤❡r❡ ❛ ♠♦❞❡❧ ♦❢ t②♣❡ t❤❡♦r② ✇✐t❤ ❛ ✉♥✐✈❛❧❡♥t ❛♥❞

✐♠♣r❡❞✐❝❛t✐✈❡ ✉♥✐✈❡rs❡ t❤❛t s❛t✐s✜❡s t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ r❡s✐③✐♥❣ ❛①✐♦♠❄

◮ ❇❡♥♥♦ ✈❛♥ ❞❡♥ ❇❡r❣ ❬❇❡r❣✱ ✷✵✶✽❪ ❤❛s ❝♦♥str✉❝t❡❞ ❛ ♠♦❞❡❧ ♦❢

t②♣❡ t❤❡♦r② ✇✐t❤ ❛ ✉♥✐✈❛❧❡♥t ❛♥❞ ✐♠♣r❡❞✐❝❛t✐✈❡ ✉♥✐✈❡rs❡ ♦❢ ✵✲t②♣❡s t❤❛t s❛t✐s✜❡s t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ r❡s✐③✐♥❣ ❛①✐♦♠✱ ❜✉t ✐♥ ❤✐s ♠♦❞❡❧ ❝♦♠♣✉t❛t✐♦♥❛❧ r✉❧❡s ❢♦r ✐❞❡♥t✐t② t②♣❡s ❛♥❞ ❞❡♣❡♥❞❡♥t ♣r♦❞✉❝t t②♣❡s ❤♦❧❞ ♦♥❧② ✉♣ t♦ ♣r♦♣♦s✐t✐♦♥❛❧ ❡q✉❛❧✐t②✳

◮ ❍✐❣❤❡r ✐♥❞✉❝t✐✈❡ t②♣❡s ✈✐❛ W✲t②♣❡s ✇✐t❤ r❡❞✉❝t✐♦♥s

❬❙✇❛♥✱ ✷✵✶✽❜❪✱ ✐♥❝❧✉❞✐♥❣ ♣r♦♣♦s✐t✐♦♥❛❧ tr✉♥❝❛t✐♦♥✳

◮ ❈❛♥ ✇❡ s❛② t❤❛t ❈❆s♠ ✐s ❛ ✏r❡❛❧✐③❛❜✐❧✐t② ❤✐❣❤❡r ❝❛t❡❣♦r②✑❄

◮ ■♥ ❆s♠ ❡✈❡r② ♣r♦♣♦s✐t✐♦♥ ✭♠♦♥♦♠♦r♣❤✐s♠✮ ✐s ♠♦❞❡st✳ ◮ ❈❤✉r❝❤✬s ❚❤❡s✐s✿ ∀f:N→N∃e:Nϕe = f✱ ✇❤❡r❡ ϕe ✐s t❤❡ e✲t❤

♣❛rt✐❛❧ r❡❝✉rs✐✈❡ ❢✉♥❝t✐♦♥✳ ■t ❞♦❡s ♥♦t ❤♦❧❞ ✐♥ ❈❆s♠ ❜❡❝❛✉s❡ t❤❡ t②♣❡ ♦❢ ♣♦✐♥ts ✐s

f:N→N

• e:N ϕe = f ✐♥ ❆s♠✳ P♦ss✐❜❧②

s♦♠❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ❈❆s♠ s❛t✐s✜❡s ❈❤✉r❝❤✬s ❚❤❡s✐s ✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆♥❞r❡✇ ❙✇❛♥✮✳

❘❡❧❛t❡❞ ❛♥❞ ❋✉t✉r❡ ❲♦r❦ ■■

◮ ▼♦❞❡❧ str✉❝t✉r❡s

◮ ❋r✉♠✐♥ ❛♥❞ ✈❛♥ ❞❡♥ ❇❡r❣ ❬❋r✉♠✐♥ ❛♥❞ ❇❡r❣✱ ✷✵✶✽❪ ❤❛s

❝♦♥str✉❝t❡❞ ❛ ♠♦❞❡❧ str✉❝t✉r❡ ♦♥ ❛ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ❛♥ ❡❧❡♠❡♥t❛r② t♦♣♦s t❤❛t s❛t✐s✜❡s s✐♠✐❧❛r ❝♦♥❞✐t✐♦♥s t♦ t❤♦s❡ ♦❢ ❖rt♦♥ ❛♥❞ P✐tts✳ ❚❤❡✐r ❝♦♥str✉❝t✐♦♥ ✇✐❧❧ ✇♦r❦ ❢♦r ❝✉❜✐❝❛❧ ❛ss❡♠❜❧✐❡s✳

◮ ❚r✐✈✐❛❧ ❝♦✜❜r❛t✐♦♥✲✜❜r❛t✐♦♥ ❢❛❝t♦r✐③❛t✐♦♥ ♦♥ ❈❆s♠ ✉s✐♥❣

❙✇❛♥✬s s♠❛❧❧ ♦❜❥❡❝t ❛r❣✉♠❡♥t ✐♥ ●r♦t❤❡♥❞✐❡❝❦ ✜❜r❛t✐♦♥s ❬❙✇❛♥✱ ✷✵✶✽❛✱ ❙✇❛♥✱ ✷✵✶✽❜❪✳

◮ ❍♦✇ t♦ r❡❧❛t❡ ❙t❡❦❡❧❡♥❜✉r❣✬s ❝♦♥str✉❝t✐✈❡ s✐♠♣❧✐❝✐❛❧ ❤♦♠♦t♦♣②

❬❙t❡❦❡❧❡♥❜✉r❣✱ ✷✵✶✻❪❄

❘❡❢❡r❡♥❝❡s ■

❆✇♦❞❡②✱ ❙✳✱ ❋r❡②✱ ❏✳✱ ❛♥❞ ❙♣❡✐❣❤t✱ ❙✳ ✭✷✵✶✽✮✳ ■♠♣r❡❞✐❝❛t✐✈❡ ❊♥❝♦❞✐♥❣s ♦❢ ✭❍✐❣❤❡r✮ ■♥❞✉❝t✐✈❡ ❚②♣❡s✳ ■♥ ✷✵✶✽ ✸✸r❞ ❆♥♥✉❛❧ ❆❈▼✴■❊❊❊ ❙②♠♣♦s✐✉♠ ♦♥ ▲♦❣✐❝ ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ✭▲■❈❙✮✳ t♦ ❛♣♣❡❛r✳ ❇❡r❣✱ ❇✳ ✈✳ ❞✳ ✭✷✵✶✽✮✳ ❯♥✐✈❛❧❡♥t ♣♦❧②♠♦r♣❤✐s♠✳ ❇❡③❡♠✱ ▼✳✱ ❈♦q✉❛♥❞✱ ❚✳✱ ❛♥❞ ❍✉❜❡r✱ ❙✳ ✭✷✵✶✹✮✳ ❆ ▼♦❞❡❧ ♦❢ ❚②♣❡ ❚❤❡♦r② ✐♥ ❈✉❜✐❝❛❧ ❙❡ts✳ ■♥ ▼❛tt❤❡s✱ ❘✳ ❛♥❞ ❙❝❤✉❜❡rt✱ ❆✳✱ ❡❞✐t♦rs✱ ✶✾t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❚②♣❡s ❢♦r Pr♦♦❢s ❛♥❞ Pr♦❣r❛♠s ✭❚❨P❊❙ ✷✵✶✸✮✱ ✈♦❧✉♠❡ ✷✻ ♦❢ ▲❡✐❜♥✐③ ■♥t❡r♥❛t✐♦♥❛❧ Pr♦❝❡❡❞✐♥❣s ✐♥ ■♥❢♦r♠❛t✐❝s ✭▲■P■❝s✮✱ ♣❛❣❡s ✶✵✼✕✶✷✽✱ ❉❛❣st✉❤❧✱ ●❡r♠❛♥②✳ ❙❝❤❧♦ss ❉❛❣st✉❤❧✕▲❡✐❜♥✐③✲❩❡♥tr✉♠ ❢✉❡r ■♥❢♦r♠❛t✐❦✳

❘❡❢❡r❡♥❝❡s ■■

❈♦❤❡♥✱ ❈✳✱ ❈♦q✉❛♥❞✱ ❚✳✱ ❍✉❜❡r✱ ❙✳✱ ❛♥❞ ▼ört❜❡r❣✱ ❆✳ ✭✷✵✶✽✮✳ ❈✉❜✐❝❛❧ ❚②♣❡ ❚❤❡♦r②✿ ❆ ❈♦♥str✉❝t✐✈❡ ■♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❯♥✐✈❛❧❡♥❝❡ ❆①✐♦♠✳ ■♥ ❯✉st❛❧✉✱ ❚✳✱ ❡❞✐t♦r✱ ✷✶st ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❚②♣❡s ❢♦r Pr♦♦❢s ❛♥❞ Pr♦❣r❛♠s ✭❚❨P❊❙ ✷✵✶✺✮✱ ✈♦❧✉♠❡ ✻✾ ♦❢ ▲❡✐❜♥✐③ ■♥t❡r♥❛t✐♦♥❛❧ Pr♦❝❡❡❞✐♥❣s ✐♥ ■♥❢♦r♠❛t✐❝s ✭▲■P■❝s✮✱ ♣❛❣❡s ✺✿✶✕✺✿✸✹✱ ❉❛❣st✉❤❧✱ ●❡r♠❛♥②✳ ❙❝❤❧♦ss ❉❛❣st✉❤❧✕▲❡✐❜♥✐③✲❩❡♥tr✉♠ ❢✉❡r ■♥❢♦r♠❛t✐❦✳ ❋r✉♠✐♥✱ ❉✳ ❛♥❞ ❇❡r❣✱ ❇✳ ✈✳ ❞✳ ✭✷✵✶✽✮✳ ❆ ❤♦♠♦t♦♣②✲t❤❡♦r❡t✐❝ ♠♦❞❡❧ ♦❢ ❢✉♥❝t✐♦♥ ❡①t❡♥s✐♦♥❛❧✐t② ✐♥ t❤❡ ❡✛❡❝t✐✈❡ t♦♣♦s✳ ▲✐❝❛t❛✱ ❉✳ ❘✳✱ ❖rt♦♥✱ ■✳✱ P✐tts✱ ❆✳ ▼✳✱ ❛♥❞ ❙♣✐tt❡rs✱ ❇✳ ✭✷✵✶✽✮✳ ■♥t❡r♥❛❧ ❯♥✐✈❡rs❡s ✐♥ ▼♦❞❡❧s ♦❢ ❍♦♠♦t♦♣② ❚②♣❡ ❚❤❡♦r②✳ ■♥ ✸r❞ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❋♦r♠❛❧ ❙tr✉❝t✉r❡s ❢♦r ❈♦♠♣✉t❛t✐♦♥ ❛♥❞ ❉❡❞✉❝t✐♦♥ ✭❋❙❈❉ ✷✵✶✽✮✳ t♦ ❛♣♣❡❛r✳

❘❡❢❡r❡♥❝❡s ■■■

❖rt♦♥✱ ■✳ ❛♥❞ P✐tts✱ ❆✳ ▼✳ ✭✷✵✶✻✮✳ ❆①✐♦♠s ❢♦r ▼♦❞❡❧❧✐♥❣ ❈✉❜✐❝❛❧ ❚②♣❡ ❚❤❡♦r② ✐♥ ❛ ❚♦♣♦s✳ ■♥ ❚❛❧❜♦t✱ ❏✳✲▼✳ ❛♥❞ ❘❡❣♥✐❡r✱ ▲✳✱ ❡❞✐t♦rs✱ ✷✺t❤ ❊❆❈❙▲ ❆♥♥✉❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ▲♦❣✐❝ ✭❈❙▲ ✷✵✶✻✮✱ ✈♦❧✉♠❡ ✻✷ ♦❢ ▲❡✐❜♥✐③ ■♥t❡r♥❛t✐♦♥❛❧ Pr♦❝❡❡❞✐♥❣s ✐♥ ■♥❢♦r♠❛t✐❝s ✭▲■P■❝s✮✱ ♣❛❣❡s ✷✹✿✶✕✷✹✿✶✾✱ ❉❛❣st✉❤❧✱ ●❡r♠❛♥②✳ ❙❝❤❧♦ss ❉❛❣st✉❤❧✕▲❡✐❜♥✐③✲❩❡♥tr✉♠ ❢✉❡r ■♥❢♦r♠❛t✐❦✳ ❙❤✉❧♠❛♥✱ ▼✳ ✭✷✵✶✶✮✳ ❍✐❣❤❡r ✐♥❞✉❝t✐✈❡ t②♣❡s ✈✐❛ ✐♠♣r❡❞✐❝❛t✐✈❡ ♣♦❧②♠♦r♣❤✐s♠✳ ❙t❡❦❡❧❡♥❜✉r❣✱ ❲✳ P✳ ✭✷✵✶✻✮✳ ❈♦♥str✉❝t✐✈❡ ❙✐♠♣❧✐❝✐❛❧ ❍♦♠♦t♦♣②✳ ❙✇❛♥✱ ❆✳ ❲✳ ✭✷✵✶✽❛✮✳ ▲✐❢t✐♥❣ Pr♦❜❧❡♠s ✐♥ ●r♦t❤❡♥❞✐❡❝❦ ❋✐❜r❛t✐♦♥s✳ ❙✇❛♥✱ ❆✳ ❲✳ ✭✷✵✶✽❜✮✳ W✲❚②♣❡s ✇✐t❤ ❘❡❞✉❝t✐♦♥s ❛♥❞ t❤❡ ❙♠❛❧❧ ❖❜❥❡❝t ❆r❣✉♠❡♥t✳

❘❡❢❡r❡♥❝❡s ■❱

❱♦❡✈♦❞s❦②✱ ❱✳ ✭✷✵✶✷✮✳ ❆ ✉♥✐✈❡rs❡ ♣♦❧②♠♦r♣❤✐❝ t②♣❡ s②st❡♠✳