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SLIDE 1

❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t

PP❙✴πr2

✷✶st ▼❛r❝❤ ✷✵✶✹

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶ ✴ ✹✶

SLIDE 2

❖♥❝❡ ✉♣♦♥ ❛ t✐♠❡✳✳✳

❈❛t❛❝❧②s♠✿ ●ö❞❡❧✬s ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ✭✶✾✸✶✮ ❲❡ ❞♦ ♥♦t ✜❣❤t ❛❧✐❡♥❛t✐♦♥ ✇✐t❤ ❛♥ ❛❧✐❡♥❛t❡❞ ❧♦❣✐❝✳ ❏✉st✐❢②✐♥❣ ❛r✐t❤♠❡t✐❝ ❞✐✛❡r❡♥t❧② ✳✳✳ ■♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝✦

❉♦✉❜❧❡✲♥❡❣❛t✐♦♥ tr❛♥s❧❛t✐♦♥ ✭✶✾✸✸✮ ❉✐❛❧❡❝t✐❝❛ ✭✸✵✬s✱ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✾✺✽✮

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷ ✴ ✹✶

SLIDE 3

❖♥❝❡ ✉♣♦♥ ❛ t✐♠❡✳✳✳

❈❛t❛❝❧②s♠✿ ●ö❞❡❧✬s ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ✭✶✾✸✶✮ ❲❡ ❞♦ ♥♦t ✜❣❤t ❛❧✐❡♥❛t✐♦♥ ✇✐t❤ ❛♥ ❛❧✐❡♥❛t❡❞ ❧♦❣✐❝✳ ❏✉st✐❢②✐♥❣ ❛r✐t❤♠❡t✐❝ ❞✐✛❡r❡♥t❧② ✳✳✳ ■♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝✦

❉♦✉❜❧❡✲♥❡❣❛t✐♦♥ tr❛♥s❧❛t✐♦♥ ✭✶✾✸✸✮ ❉✐❛❧❡❝t✐❝❛ ✭✸✵✬s✱ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✾✺✽✮

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷ ✴ ✹✶

SLIDE 4

❖♥❝❡ ✉♣♦♥ ❛ t✐♠❡✳✳✳

❈❛t❛❝❧②s♠✿ ●ö❞❡❧✬s ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ✭✶✾✸✶✮ ❲❡ ❞♦ ♥♦t ✜❣❤t ❛❧✐❡♥❛t✐♦♥ ✇✐t❤ ❛♥ ❛❧✐❡♥❛t❡❞ ❧♦❣✐❝✳ ❏✉st✐❢②✐♥❣ ❛r✐t❤♠❡t✐❝ ❞✐✛❡r❡♥t❧② ✳✳✳ ■♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝✦

❉♦✉❜❧❡✲♥❡❣❛t✐♦♥ tr❛♥s❧❛t✐♦♥ ✭✶✾✸✸✮ ❉✐❛❧❡❝t✐❝❛ ✭✸✵✬s✱ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✾✺✽✮

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷ ✴ ✹✶

SLIDE 5

P❧❛♥

✶

❍✐st♦r✐❝❛❧ ♣r❡s❡♥t❛t✐♦♥

✷

❆ st❡♣ ✐♥t♦ ♠♦❞❡r♥✐t②

✸

❊♥t❡rs ▲✐♥❡❛r ▲♦❣✐❝

✹

❆ s②♥t❛❝t✐❝ ♣r❡s❡♥t❛t✐♦♥

✺

❚♦✇❛r❞s CCω

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✸ ✴ ✹✶

SLIDE 6

❖✈❡r✈✐❡✇

❲❤❛t ✐s ❉✐❛❧❡❝t✐❝❛❄ ❆ tr❛♥s❧❛t✐♦♥ ❢r♦♠ ❍❆ ✐♥t♦ ❍❆ ❚❤❛t ♣r❡s❡r✈❡s ✐♥t✉✐t✐♦♥✐st✐❝ ❝♦♥t❡♥t ❇✉t ♦✛❡rs t✇♦ s❡♠✐✲❝❧❛ss✐❝❛❧ ♣r✐♥❝✐♣❧❡s✿ ▼P ■P

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✹ ✴ ✹✶

SLIDE 7

❖✈❡r✈✐❡✇

❲❤❛t ✐s ❉✐❛❧❡❝t✐❝❛❄ ❆ tr❛♥s❧❛t✐♦♥ ❢r♦♠ ❍❆ ✐♥t♦ ❍❆ω ❚❤❛t ♣r❡s❡r✈❡s ✐♥t✉✐t✐♦♥✐st✐❝ ❝♦♥t❡♥t ❇✉t ♦✛❡rs t✇♦ s❡♠✐✲❝❧❛ss✐❝❛❧ ♣r✐♥❝✐♣❧❡s✿ ▼P ■P

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✹ ✴ ✹✶

SLIDE 8

❖✈❡r✈✐❡✇

❲❤❛t ✐s ❉✐❛❧❡❝t✐❝❛❄ ❆ tr❛♥s❧❛t✐♦♥ ❢r♦♠ ❍❆ ✐♥t♦ ❍❆ω ❚❤❛t ♣r❡s❡r✈❡s ✐♥t✉✐t✐♦♥✐st✐❝ ❝♦♥t❡♥t ❇✉t ♦✛❡rs t✇♦ s❡♠✐✲❝❧❛ss✐❝❛❧ ♣r✐♥❝✐♣❧❡s✿ ¬(∀n ∈ N. ¬P n) ▼P ∃n ∈ N. P n (∀n ∈ N. P n) → ∃m ∈ N. Q m ■P ∃m ∈ N. (∀n ∈ N. P n) → Q m

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✹ ✴ ✹✶

SLIDE 9

P❛r❡♥t❛❧ ❛❞✈✐s♦r② r❡q✉✐r❡❞

❋♦r t❤❡ s❛❦❡ ♦❢ ❡①❤❛✉st✐✈✐t②✱ ✇❡✬❧❧ t❛❦❡ ❛ ❣❧✐♠♣s❡ ❛t t❤❡ ❤✐st♦r✐❝❛❧ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❉✐❛❧❡❝t✐❝❛✳ ❲❛r♥✐♥❣✦ ❉✉st② ❧♦❣✐❝ ✐♥s✐❞❡ ❚r❛♥s❧❛t✐♦♥ ❛❝t✐♥❣ ♦♥ ❢♦r♠✉❧æ Pr❡✈❛❧❡♥❝❡ ♦❢ ♥❡❣❛t✐✈❡ ❝♦♥♥❡❝t✐✈❡s ❋✐rst✲♦r❞❡r ❧♦❣✐❝ ▲♦ts ♦❢ ❛r✐t❤♠❡t✐❝ ❡♥❝♦❞✐♥❣ ❉♦❡s ♥♦t ♣r❡s❡r✈❡ ✲r❡❞✉❝t✐♦♥

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✺ ✴ ✹✶

SLIDE 10

P❛r❡♥t❛❧ ❛❞✈✐s♦r② r❡q✉✐r❡❞

❋♦r t❤❡ s❛❦❡ ♦❢ ❡①❤❛✉st✐✈✐t②✱ ✇❡✬❧❧ t❛❦❡ ❛ ❣❧✐♠♣s❡ ❛t t❤❡ ❤✐st♦r✐❝❛❧ ♣r❡s❡♥t❛t✐♦♥ ♦❢ ❉✐❛❧❡❝t✐❝❛✳ ❲❛r♥✐♥❣✦ ❉✉st② ❧♦❣✐❝ ✐♥s✐❞❡ ❚r❛♥s❧❛t✐♦♥ ❛❝t✐♥❣ ♦♥ ❢♦r♠✉❧æ Pr❡✈❛❧❡♥❝❡ ♦❢ ♥❡❣❛t✐✈❡ ❝♦♥♥❡❝t✐✈❡s ❋✐rst✲♦r❞❡r ❧♦❣✐❝ ▲♦ts ♦❢ ❛r✐t❤♠❡t✐❝ ❡♥❝♦❞✐♥❣ ❉♦❡s ♥♦t ♣r❡s❡r✈❡ β✲r❡❞✉❝t✐♦♥

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✺ ✴ ✹✶

SLIDE 11

❉✉st② ❧♦❣✐❝s

❉✐❛❧❡❝t✐❝❛✱ ❉❛✇♥ ♦❢ ❈✉rr②✲❍♦✇❛r❞✿ ⊢ A → ⊢ AD ≡ ∃

u. ∀
x. AD[

u, x]

A ∧ B ∃ u v. ∀ x y. AD[ u, x] ∧ BD[ v, y] A ∨ B ∃ u v b. ∀ x y. (b = 0 ∧ AD[ u, x]) ∨ (b = 1 ∧ BD[ v, y]) A → B ∃ ϕ ψ. ∀ u y. AD[ u, ψ( u, y)] → BD[ ϕ( u), y] ∀n. A[n] ∃ ϕ. ∀ x n. AD[ ϕ(n), x, n] ∃n. A[n] ∃ u n. ∀ x. AD[ u, n, x]

❙♦✉♥❞ tr❛♥s❧❛t✐♦♥✱ ❜❧❛❤ ❜❧❛❤ ❜❧❛❤✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✻ ✴ ✹✶

SLIDE 12

❆ st❡♣ ✐♥t♦ ♠♦❞❡r♥✐t②

▲❡t ✉s ❢♦r❣❡t t❤❡ ✺✵✬s✱ ❛♥❞ r❛t❤❡r ❥✉♠♣ ❞✐r❡❝t❧② t♦ t❤❡ ✾✵✬s✳ ❚❛❦❡ s❡r✐♦✉s❧② t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♥t❡♥t ❉✐❛❧❡❝t✐❝❛ ❛s ❛ t②♣❡❞ ♦❜❥❡❝t ❲♦r❦s ♦❢ ❉❡ P❛✐✈❛✱ ❍②❧❛♥❞✱ ❡t❝✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✼ ✴ ✹✶

SLIDE 13

ö❞❡❧✬s ❛♥❛t♦♠②

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✽ ✴ ✹✶

SLIDE 14

❚❤❡ s❛♠❡✱ ✇✐t❤ t②♣❡s

❆ ♣r♦♦❢ ⊢ u : A ✐s ❛ t❡r♠ ⊢ u : W(A) s✉❝❤ t❤❛t✿ ∀x : C(A). u ⊥A x ■❢ ✇❡ ✇✐s❤ t♦ ♣✉t ♠♦r❡ t②♣❡s ✐♥ t❤❡r❡✿

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✾ ✴ ✹✶

SLIDE 15

❚❤❡ s❛♠❡✱ ✇✐t❤ t②♣❡s

❆ ♣r♦♦❢ ⊢ u : A ✐s ❛ t❡r♠ ⊢ u : W(A) s✉❝❤ t❤❛t✿ ∀x : C(A). u ⊥A x ■❢ ✇❡ ✇✐s❤ t♦ ♣✉t ♠♦r❡ t②♣❡s ✐♥ t❤❡r❡✿

W C A ∧ B ∃ u v. ∀ x y. A × B W(A) × W(B) C(A) × C(B) A ∨ B ∃b u v. ∀ x y. A + B bool × W(A) × W(B) C(A) × C(B) A → B ∃ ϕ ψ. ∀ u y. A → B W(A) → W(B) C(B) → W(A) → C(A) W(A) × C(B)

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✾ ✴ ✹✶

SLIDE 16

❚❤❡ s❛♠❡✱ ✇✐t❤ t②♣❡s

❆ ♣r♦♦❢ ⊢ u : A ✐s ❛ t❡r♠ ⊢ u : W(A) s✉❝❤ t❤❛t✿ ∀x : C(A). u ⊥A x ■❢ ✇❡ ✇✐s❤ t♦ ♣✉t ♠♦r❡ t②♣❡s ✐♥ t❤❡r❡✿

W C A ∧ B ∃ u v. ∀ x y. A × B W(A) × W(B) C(A) × C(B) A ∨ B ∃b u v. ∀ x y. A + B bool × W(A) × W(B) C(A) × C(B) A → B ∃ ϕ ψ. ∀ u y. A → B W(A) → W(B) C(B) → W(A) → C(A) W(A) × C(B)

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✾ ✴ ✹✶

SLIDE 17

❆ ❉é❥à ✈✉

❇✉t✱ ❣r❛♥❞♠♦t❤❡r✱ ❤♦✇ ❢❛♠✐❧✐❛r ②♦✉ ❧♦♦❦✳✳✳ ❈❧❛ss✐❝❛❧ r❡❛❧✐③❛❜✐❧✐t②✿ ♣r♦♦❢s ✱ st❛❝❦s ❉♦✉❜❧❡✲♦rt❤♦❣♦♥❛❧✐t② ❜❛s❡❞ ♠♦❞❡❧s ❉♦✉❜❧❡✲❣❧✉❡✐♥❣ ❘❡❞✉❝✐❜✐❧✐t② ❝❛♥❞✐❞❛t❡s

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✵ ✴ ✹✶

SLIDE 18

❆ ❉é❥à ✈✉

❇✉t✱ ❣r❛♥❞♠♦t❤❡r✱ ❤♦✇ ❢❛♠✐❧✐❛r ②♦✉ ❧♦♦❦✳✳✳ ❈❧❛ss✐❝❛❧ r❡❛❧✐③❛❜✐❧✐t②✿ W(A) ♣r♦♦❢s |A|✱ C(A) st❛❝❦s ||A|| ❉♦✉❜❧❡✲♦rt❤♦❣♦♥❛❧✐t② ❜❛s❡❞ ♠♦❞❡❧s ❉♦✉❜❧❡✲❣❧✉❡✐♥❣ ❘❡❞✉❝✐❜✐❧✐t② ❝❛♥❞✐❞❛t❡s . . .

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✵ ✴ ✹✶

SLIDE 19

◆♦t t♦♦ ❤❛st✐❧②

❲❡ ❝♦✉❧❞ ❣✐✈❡ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♥t❡♥t r✐❣❤t ♥♦✇ ❇✉t ✐t ✇♦✉❧❞ ❜❡ ❛❞✲❤♦❝✱ ✐♥❤❡r✐t✐♥❣ ❢r♦♠ t❤❡ ❡♥❝♦❞✐♥❣s ♦❢ ❉✐❛❧❡❝t✐❝❛ ▲❡t ✉s ✉s❡ ♦✉r ♦✉r ❢❛✈♦r✐t❡ t♦♦❧✿ ▲✐♥❡❛r ▲♦❣✐❝✦

❆ ❣❡♥✉✐♥❡ ❡①♣♦♥❡♥t✐❛❧✦ ❲✐t❤ r❡❛❧ ❝❤✉♥❦s ♦❢ s✉♠ t②♣❡s✦

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✶ ✴ ✹✶

SLIDE 20

◆♦t t♦♦ ❤❛st✐❧②

❲❡ ❝♦✉❧❞ ❣✐✈❡ ❛ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♥t❡♥t r✐❣❤t ♥♦✇ ❇✉t ✐t ✇♦✉❧❞ ❜❡ ❛❞✲❤♦❝✱ ✐♥❤❡r✐t✐♥❣ ❢r♦♠ t❤❡ ❡♥❝♦❞✐♥❣s ♦❢ ❉✐❛❧❡❝t✐❝❛ ▲❡t ✉s ✉s❡ ♦✉r ♦✉r ❢❛✈♦r✐t❡ t♦♦❧✿ ▲✐♥❡❛r ▲♦❣✐❝✦

❆ ❣❡♥✉✐♥❡ ❡①♣♦♥❡♥t✐❛❧✦ ❲✐t❤ r❡❛❧ ❝❤✉♥❦s ♦❢ s✉♠ t②♣❡s✦

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✶ ✴ ✹✶

SLIDE 21

▲✐♥❡❛r✐③❡❞ ❉✐❛❧❡❝t✐❝❛

❆s ❢♦r❡❝❛st❡❞ ♦♥ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡✱ ✇❡ ❡ss❡♥t✐❛❧❧② ❛♣♣❧② t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞✐✜❝❛t✐♦♥s✿ ■♥tr♦❞✉❝t✐♦♥ ♦❢ ❞✉❛❧✐t② ✇✐t❤ s✉♠ t②♣❡s ❈❛❧❧✲❜②✲♥❛♠❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❛rr♦✇✿ A → B ≡ !A ⊸ B ◆♦✇ ✇❡ ✇✐❧❧ ❜❡ tr❛♥s❧❛t✐♥❣ ❢♦r♠✉❧æ ✐♥t♦ ♦♥❡s✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✷ ✴ ✹✶

SLIDE 22

▲✐♥❡❛r✐③❡❞ ❉✐❛❧❡❝t✐❝❛

❆s ❢♦r❡❝❛st❡❞ ♦♥ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡✱ ✇❡ ❡ss❡♥t✐❛❧❧② ❛♣♣❧② t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞✐✜❝❛t✐♦♥s✿ ■♥tr♦❞✉❝t✐♦♥ ♦❢ ❞✉❛❧✐t② ✇✐t❤ s✉♠ t②♣❡s ❈❛❧❧✲❜②✲♥❛♠❡ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❛rr♦✇✿ A → B ≡ !A ⊸ B ◆♦✇ ✇❡ ✇✐❧❧ ❜❡ tr❛♥s❧❛t✐♥❣ LL ❢♦r♠✉❧æ ✐♥t♦ LJ ♦♥❡s✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✷ ✴ ✹✶

SLIDE 23

❘❡q✉✐r❡♠❡♥ts

❲❡ ✇✐❧❧ ❜❡ ✐♥t❡r♣r❡t✐♥❣ t❤❡ ❢♦r♠✉❧æ ♦❢ ❧✐♥❡❛r ❧♦❣✐❝✿ A, B ::= A ⊗ B | A ` B | A ⊕ B | A & B | !A | ?A ■t ✐s t❤❡r❡❢♦r❡ s✉✣❝✐❡♥t t♦ ❞❡✜♥❡ W(A)✱ C(A) ❛♥❞ ⊥A ❢♦r ❡❛❝❤ A✱ ✇❤❡r❡✿ ⊥A ⊆ W(A) × C(A)

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✸ ✴ ✹✶

SLIDE 24

❋♦r❣❡t t❤❡ ❞✉❛❧

❚❛❦✐♥❣ ✐♥s♣✐r❛t✐♦♥ ❢r♦♠ t❤❡ ❞♦✉❜❧❡✲♦rt❤♦❣♦♥❛❧✐t② ♠♦❞❡❧s✱ ✇❡ r❡q✉✐r❡✿ W(A⊥) ≡ C(A) ❛♥❞ ❝♦♥✈❡rs❡❧②❀ ■t ✐s s✉✣❝✐❡♥t t♦ ❞❡✜♥❡ ♦✉r str✉❝t✉r❡s ♦♥ ♣♦s✐t✐✈❡ t②♣❡s ❲❡ ✇✐❧❧ ❣❡t t❤❡♠ ❢♦r ❞✉❛❧ ❝♦♥♥❡❝t✐✈❡s✳✳✳ ❜② ❞✉❛❧✐t②✳ ❲❡ ❞❡✜♥❡ t❤❡r❡❢♦r❡✿

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✹ ✴ ✹✶

SLIDE 25

❋♦r❣❡t t❤❡ ❞✉❛❧

❚❛❦✐♥❣ ✐♥s♣✐r❛t✐♦♥ ❢r♦♠ t❤❡ ❞♦✉❜❧❡✲♦rt❤♦❣♦♥❛❧✐t② ♠♦❞❡❧s✱ ✇❡ r❡q✉✐r❡✿ W(A⊥) ≡ C(A) ❛♥❞ ❝♦♥✈❡rs❡❧②❀ ■t ✐s s✉✣❝✐❡♥t t♦ ❞❡✜♥❡ ♦✉r str✉❝t✉r❡s ♦♥ ♣♦s✐t✐✈❡ t②♣❡s ❲❡ ✇✐❧❧ ❣❡t t❤❡♠ ❢♦r ❞✉❛❧ ❝♦♥♥❡❝t✐✈❡s✳✳✳ ❜② ❞✉❛❧✐t②✳ ❲❡ ❞❡✜♥❡ t❤❡r❡❢♦r❡✿ u ⊥A x x ⊥A⊥ u

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✹ ✴ ✹✶

SLIDE 26

❙✉♠ t②♣❡s

W C A × B W(A) × W(B) C(A) × C(B) A & B W(A) × W(B) C(A) + C(B) A + B bool × W(A) × W(B) C(A) × C(B) A ⊕ B W(A) + W(B) C(A) × C(B)

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✺ ✴ ✹✶

SLIDE 27

❙✉♠ t②♣❡s

W C A × B W(A) × W(B) C(A) × C(B) A & B W(A) × W(B) C(A) + C(B) A + B bool × W(A) × W(B) C(A) × C(B) A ⊕ B W(A) + W(B) C(A) × C(B) v ⊥A z2 inr v ⊥A⊕B (z1, z2) u ⊥A z1 inl u ⊥A⊕B (z1, z2)

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✺ ✴ ✹✶

SLIDE 28

▲✐♥❡❛r ❞❡❝♦♠♣♦s✐t✐♦♥

W C A → B

W(A) → W(B)

C(B) → W(A) → C(A) C(A) × C(B) A ⊸ B W(A) → W(B) C(B) → C(A) W(A) × C(B) !A W(A) W(A) → C(A)

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✻ ✴ ✹✶

SLIDE 29

▲✐♥❡❛r ❞❡❝♦♠♣♦s✐t✐♦♥

W C A → B

W(A) → W(B)

C(B) → W(A) → C(A) C(A) × C(B) A ⊸ B W(A) → W(B) C(B) → C(A) W(A) × C(B) !A W(A) W(A) → C(A) u ⊥A ψ y → ϕ u ⊥B y (ϕ, ψ) ⊥A⊸B (u, y) u ⊥A z u u ⊥!A z

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✻ ✴ ✹✶

SLIDE 30

❍❛♥❞✇❛✈✐♥❣ ❥✉st✐✜❝❛t✐♦♥

❚❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❛rr♦✇ ❢♦r❝❡s ✐ts r❡✈❡rs✐❜✐❧✐t②✿ A ⊸ B ∼ = B⊥ ⊸ A⊥

▲✐❦❡ t❤❡ t✇♦✲✇❛② ♣r♦♦❢♥❡t ✇✐r❡s

❚❤❡ ❜❛♥❣ ❝♦♥♥❡❝t✐✈❡ ✐s ❛ s❤✐❢t ✿

❖♣♣♦♥❡♥t ♠❛② ✇❛✐t ❢♦r t❤❡ ♣❧❛②❡r t♦ ♣❧❛② ❛♥❞ ✐♥s♣❡❝t ✐ts ❛♥s✇❡r

❉✉❛❧✐t② ✐s rô❧❡ s✇❛♣♣✐♥❣

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✼ ✴ ✹✶

SLIDE 31

❍❛♥❞✇❛✈✐♥❣ ❥✉st✐✜❝❛t✐♦♥

❚❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❛rr♦✇ ❢♦r❝❡s ✐ts r❡✈❡rs✐❜✐❧✐t②✿ A ⊸ B ∼ = B⊥ ⊸ A⊥

▲✐❦❡ t❤❡ t✇♦✲✇❛② ♣r♦♦❢♥❡t ✇✐r❡s

❚❤❡ ❜❛♥❣ ❝♦♥♥❡❝t✐✈❡ ✐s ❛ s❤✐❢t ✿

❖♣♣♦♥❡♥t ♠❛② ✇❛✐t ❢♦r t❤❡ ♣❧❛②❡r t♦ ♣❧❛② ❛♥❞ ✐♥s♣❡❝t ✐ts ❛♥s✇❡r

❉✉❛❧✐t② ✐s rô❧❡ s✇❛♣♣✐♥❣

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✼ ✴ ✹✶

SLIDE 32

❆❜♦✉t ❧✐♥❡❛r✐t②

❲❡✬r❡ ♥♦t ❧✐♥❡❛r ❜② ❝❤❛♥❝❡✳ ■♥❞❡❡❞✱ ✐♥ ❉✐❛❧❡❝t✐❝❛✱ ✇❡ ❞♦ ♥♦t ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦r♣❤✐s♠s✿

✶

❍❡♥❝❡ ✇❡ ❤❛✈❡ tr✉❡ ❧✐♥❡❛r ❝♦♥str❛✐♥ts✦✷

✶❆ss✉♠✐♥❣ ✇❡✬✈❡ ❞❡✜♥❡❞ 1✳ ✷▼❛② ❝♦♥t❛✐♥ ♥✉ts✳ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✽ ✴ ✹✶

SLIDE 33

❆❜♦✉t ❧✐♥❡❛r✐t②

❲❡✬r❡ ♥♦t ❧✐♥❡❛r ❜② ❝❤❛♥❝❡✳ ■♥❞❡❡❞✱ ✐♥ ❉✐❛❧❡❝t✐❝❛✱ ✇❡ ❞♦ ♥♦t ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦r♣❤✐s♠s✿ ⊢ A ⊸ 1✶ ⊢ A ⊸ A ⊗ A ❍❡♥❝❡ ✇❡ ❤❛✈❡ tr✉❡ ❧✐♥❡❛r ❝♦♥str❛✐♥ts✦✷

✶❆ss✉♠✐♥❣ ✇❡✬✈❡ ❞❡✜♥❡❞ 1✳ ✷▼❛② ❝♦♥t❛✐♥ ♥✉ts✳ P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✽ ✴ ✹✶

SLIDE 34

■♥t❡♣r❡t❛t✐♦♥ ♦❢ t❤❡ ❝❛❧❧✲❜②✲♥❛♠❡ λ✲❝❛❧❝✉❧✉s

❲❡✬r❡ ♥♦✇ tr②✐♥❣ t♦ tr❛♥s❧❛t❡ t❤❡ λ✲❝❛❧❝✉❧✉s t❤r♦✉❣❤ ❉✐❛❧❡❝t✐❝❛✳ ❋✐rst t❤r♦✉❣❤ t❤❡ ❝❛❧❧✲❜②✲♥❛♠❡ ❧✐♥❡❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ✐♥t♦ ▲▲❀ ❚❤❡♥ ✐♥t♦ ▲❏ ✇✐t❤ t❤❡ ❧✐♥❡❛r ❉✐❛❧❡❝t✐❝❛✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✶✾ ✴ ✹✶

SLIDE 35

❇r✐❡❢ r❡♠✐♥❞❡r

❲❡ r❡❝❛❧❧ ❤❡r❡ t❤❡ ❝❛❧❧✲❜②✲♥❛♠❡ tr❛♥s❧❛t✐♦♥ ♦❢ t❤❡ λ✲❝❛❧❝✉❧✉s ✐♥t♦ ▲▲✿ [ [A → B] ] ≡ ![ [A] ] ⊸ [ [B] ] [ [A × B] ] ≡ ![ [A] ] ⊗ ![ [B] ] [ [A + B] ] ≡ ![ [A] ] ⊕ ![ [B] ] [ [Γ ⊢ A] ] ≡

![

[Γ] ] ⊢ [ [A] ]

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✵ ✴ ✹✶

SLIDE 36

Pr♦❧❡❣♦♠❡♥❛

■♥ ♦r❞❡r t♦ ✐♥t❡r♣r❡t t❤❡ λ✲❝❛❧❝✉❧✉s✱ ✇❡ ♥❡❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❉✉♠♠② t❡r♠

❋♦r ❛❧❧ t②♣❡ A✱ t❤❡r❡ ❡①✐sts ⊢ ∅A : W(A)✳

❉❡❝✐❞❛❜✐❧✐t② ♦❢ t❤❡ ♦rt❤♦❣♦♥❛❧✐t②

❚❤❡ ⊥A r❡❧❛t✐♦♥ ✐s ❞❡❝✐❞❛❜❧❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡r❡ ♠✉st ❡①✐st s♦♠❡ λ✲t❡r♠ @A : W(A) → W(A) → C(A) → W(A) ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❜❡❤❛✈✐♦✉r✿ u1@A

x u2 ∼

= if u1 ⊥A x then u2 else u1

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✶ ✴ ✹✶

SLIDE 37

❉✐❞ ②♦✉ s♦❧✈❡ t❤❡ ♦r❣❛♥✐③❛t✐♦♥ ✐ss✉❡❄

■❢ ✇❡ ✇❡r❡ t♦ ✉s❡ t❤❡ tr❛♥s❧❛t✐♦♥ ❛s ✐s✱ ✇❡ ✇♦✉❧❞ ❜✉♠♣ ✉♣ ✐♥t♦ ❛♥ ✉♥❜❡❛r❛❜❧❡ ❜✉r❡❛✉❝r❛❝②✳ ■♥st❡❛❞✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s♦♠♦r♣❤✐s♠✳ [ [x1 : Γ1, . . . xn : Γn ⊢ t : A] ] ∼ = W(Γ) →          W(A) C(A) → C(Γ1) ✳ ✳ ✳ C(A) → C(Γn) ❲❤✐❝❤ r❡s✉❧ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛♥s❧❛t✐♦♥s✿ ✳ ✳ ✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✷ ✴ ✹✶

SLIDE 38

❉✐❞ ②♦✉ s♦❧✈❡ t❤❡ ♦r❣❛♥✐③❛t✐♦♥ ✐ss✉❡❄

■❢ ✇❡ ✇❡r❡ t♦ ✉s❡ t❤❡ tr❛♥s❧❛t✐♦♥ ❛s ✐s✱ ✇❡ ✇♦✉❧❞ ❜✉♠♣ ✉♣ ✐♥t♦ ❛♥ ✉♥❜❡❛r❛❜❧❡ ❜✉r❡❛✉❝r❛❝②✳ ■♥st❡❛❞✱ ✇❡ ❛r❡ ❣♦✐♥❣ t♦ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s♦♠♦r♣❤✐s♠✳ [ [x1 : Γ1, . . . xn : Γn ⊢ t : A] ] ∼ = W(Γ) →          W(A) C(A) → C(Γ1) ✳ ✳ ✳ C(A) → C(Γn) ❲❤✐❝❤ r❡s✉❧ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ tr❛♥s❧❛t✐♦♥s✿ [ [ x : Γ ⊢ t : A] ] ≡               

x : W(Γ) ⊢ t• : W(A)
x : W(Γ) ⊢ tx1 : C(A) → C(Γ1)

✳ ✳ ✳

x : W(Γ) ⊢ txn : C(A) → C(Γn)

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✷ ✴ ✹✶

SLIDE 39

❚r❛♥s❧❛t✐♦♥

❋♦r (−)• ✿ x• ≡ x (λx. t)• ≡ λx. t• λπx. tx π (t u)• ≡ (fst t•) u•

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✸ ✴ ✹✶

SLIDE 40

❚r❛♥s❧❛t✐♦♥ ■■

❋♦r tx ✿ xx ≡ λπ. π : C(A) → C(A) yx ≡ λπ. ∅ : C(A) → C(Γi) (λy. t)x ≡ λ(y, π). tx π : W(A) × C(B) → C(Γi) (t u)x ≡ λπ. ux ((snd t•) π u•) @π tx (u•, π) : C(B) → C(Γi)

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✹ ✴ ✹✶

SLIDE 41

■t ❥✉st ✇♦r❦s✳✳✳ ❉♦❡s ✐t❄

❙♦✉♥❞♥❡ss

■❢ ⊢ t : A✱ t❤❡♥ ⊢ [ [t] ] : W(A)✱ ❛♥❞ ✐♥ ❛❞❞✐t✐♦♥✱ ❢♦r ❛❧❧ π : C(A)✱ t ⊥A π✳ ❙❛❞♥❡ss ❚❤❡ tr❛♥s❧❛t✐♦♥ ✐s st✐❧❧ ♥♦t st❛❜❧❡ ❜② ✲r❡❞✉❝t✐♦♥✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✺ ✴ ✹✶

SLIDE 42

■t ❥✉st ✇♦r❦s✳✳✳ ❉♦❡s ✐t❄

❙♦✉♥❞♥❡ss

■❢ ⊢ t : A✱ t❤❡♥ ⊢ [ [t] ] : W(A)✱ ❛♥❞ ✐♥ ❛❞❞✐t✐♦♥✱ ❢♦r ❛❧❧ π : C(A)✱ t ⊥A π✳ ❙❛❞♥❡ss ❚❤❡ tr❛♥s❧❛t✐♦♥ ✐s st✐❧❧ ♥♦t st❛❜❧❡ ❜② β✲r❡❞✉❝t✐♦♥✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✺ ✴ ✹✶

SLIDE 43

❆❧♠♦st t❤❡r❡

❯s✐♥❣ ∅ ❛♥❞ @ ✐s ❛♥♦t❤❡r ❡♥❝♦❞✐♥❣ ♦❢ ❉✐❛❧❡❝t✐❝❛✳ ❲❡ ✇❛♥t ❧✐sts✦

❛❧♠♦st✳✳✳

❲❡ ❥✉st ❝❤❛♥❣❡✿ ❚❡r♠ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ❛❧♠♦st ✉♥❝❤❛♥❣❡❞✿

❜❡❝♦♠❡s t❤❡ ❡♠♣t② ❧✐st❀ ❜❡❝♦♠❡s ❝♦♥❝❛t❡♥❛t✐♦♥ ♣❧✉s ❛ ❜✐t ♦❢ ♠♦♥❛❞✐❝ ❜♦✐❧❡r♣❧❛t❡

❲❡ ❞♦ ♥♦t ♥❡❡❞ ♦rt❤♦❣♦♥❛❧✐t② ❛♥②♠♦r❡✳✳✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✻ ✴ ✹✶

SLIDE 44

❆❧♠♦st t❤❡r❡

❯s✐♥❣ ∅ ❛♥❞ @ ✐s ❛♥♦t❤❡r ❡♥❝♦❞✐♥❣ ♦❢ ❉✐❛❧❡❝t✐❝❛✳ ❲❡ ✇❛♥t ❧✐sts✦

❛❧♠♦st✳✳✳

❲❡ ❥✉st ❝❤❛♥❣❡✿ C(!A) ≡ W(A) → C(A) C(!A) ≡ W(A) → list C(A) ❚❡r♠ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ❛❧♠♦st ✉♥❝❤❛♥❣❡❞✿

∅ ❜❡❝♦♠❡s t❤❡ ❡♠♣t② ❧✐st❀ @ ❜❡❝♦♠❡s ❝♦♥❝❛t❡♥❛t✐♦♥ . . . ♣❧✉s ❛ ❜✐t ♦❢ ♠♦♥❛❞✐❝ ❜♦✐❧❡r♣❧❛t❡

❲❡ ❞♦ ♥♦t ♥❡❡❞ ♦rt❤♦❣♦♥❛❧✐t② ❛♥②♠♦r❡✳✳✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✻ ✴ ✹✶

SLIDE 45

❲❤❛t ❛❜♦✉t t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♥t❡♥t❄

❚❤✐s ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t②♣❡s ❢♦r t❤❡ tr❛♥s❧❛t✐♦♥✿ [ [ x : Γ ⊢ t : A] ] ≡               

x : W(Γ) ⊢ t• : W(A)
x : W(Γ) ⊢ tx1 : C(A) → list C(Γ1)

✳ ✳ ✳

x : W(Γ) ⊢ txn : C(A) → list C(Γn)

✐s ❝❧❡❛r❧② t❤❡ ❧✐❢t✐♥❣ ♦❢ ❀ ❲❤❛t ♦♥ ❡❛rt❤ ✐s ❄

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✼ ✴ ✹✶

SLIDE 46

❲❤❛t ❛❜♦✉t t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♥t❡♥t❄

❚❤✐s ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t②♣❡s ❢♦r t❤❡ tr❛♥s❧❛t✐♦♥✿ [ [ x : Γ ⊢ t : A] ] ≡               

x : W(Γ) ⊢ t• : W(A)
x : W(Γ) ⊢ tx1 : C(A) → list C(Γ1)

✳ ✳ ✳

x : W(Γ) ⊢ txn : C(A) → list C(Γn)

t• ✐s ❝❧❡❛r❧② t❤❡ ❧✐❢t✐♥❣ ♦❢ t❀ ❲❤❛t ♦♥ ❡❛rt❤ ✐s ❄

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✼ ✴ ✹✶

SLIDE 47

❲❤❛t ❛❜♦✉t t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♥t❡♥t❄

❚❤✐s ❣✐✈❡s ✉s t❤❡ ❢♦❧❧♦✇✐♥❣ t②♣❡s ❢♦r t❤❡ tr❛♥s❧❛t✐♦♥✿ [ [ x : Γ ⊢ t : A] ] ≡               

x : W(Γ) ⊢ t• : W(A)
x : W(Γ) ⊢ tx1 : C(A) → list C(Γ1)

✳ ✳ ✳

x : W(Γ) ⊢ txn : C(A) → list C(Γn)

t• ✐s ❝❧❡❛r❧② t❤❡ ❧✐❢t✐♥❣ ♦❢ t❀ ❲❤❛t ♦♥ ❡❛rt❤ ✐s txi❄

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✼ ✴ ✹✶

SLIDE 48

❆♥ ✉♥❜❡❛r❛❜❧❡ s✉s♣❡♥s❡

❆ s♠❛❧❧ ✐♥t❡r❧✉❞❡ ♦❢ ❛❞✈❡rt✐s❡♠❡♥t ❞❡✜♥✐t✐♦♥s t♦ ✐♥tr♦❞✉❝❡ ②♦✉ t♦ t❤❡ ❑❆▼✳

❈❧♦s✉r❡s ❊♥✈✐r♦♥♠❡♥ts ❙t❛❝❦s Pr♦❝❡ss❡s P✉s❤ P♦♣

r❛❜
❛r❜❛❣❡

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✽ ✴ ✹✶

SLIDE 49

❆♥ ✉♥❜❡❛r❛❜❧❡ s✉s♣❡♥s❡

❆ s♠❛❧❧ ✐♥t❡r❧✉❞❡ ♦❢ ❛❞✈❡rt✐s❡♠❡♥t ❞❡✜♥✐t✐♦♥s t♦ ✐♥tr♦❞✉❝❡ ②♦✉ t♦ t❤❡ ❑❆▼✳

r❛❜

(x, σ + (x := c)) | π → c | π

❛r❜❛❣❡

(x, σ + (y := c)) | π → (x, σ) | π The Krivine Machine™

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✽ ✴ ✹✶

SLIDE 50

❈❧♦s✉r❡s ❛❧❧ t❤❡ ✇❛② ❞♦✇♥

▲❡t✿

❛ t❡r♠ x : Γ ⊢ t : A ❛ ❝❧♦s✉r❡ σ ⊢ Γ ❛ st❛❝❦ ⊢ π : A⊥ ✭✐✳❡✳ [ [π] ] : C(A)✮

❚❤❡♥ ✐s t❤❡ ❧✐st ♠❛❞❡ ♦❢ t❤❡ st❛❝❦s ❡♥❝♦✉♥t❡r❡❞ ❜② ✇❤✐❧❡ ❡✈❛❧✉❛t✐♥❣ ✱ ✐✳❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖t❤❡r✇✐s❡ s❛✐❞✱ ❉✐❛❧❡❝t✐❝❛ tr❛❝❦s t❤❡ ●r❛❜ r✉❧❡✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✾ ✴ ✹✶

SLIDE 51

❈❧♦s✉r❡s ❛❧❧ t❤❡ ✇❛② ❞♦✇♥

▲❡t✿

❛ t❡r♠ x : Γ ⊢ t : A ❛ ❝❧♦s✉r❡ σ ⊢ Γ ❛ st❛❝❦ ⊢ π : A⊥ ✭✐✳❡✳ [ [π] ] : C(A)✮

❚❤❡♥ txi π ✐s t❤❡ ❧✐st ♠❛❞❡ ♦❢ t❤❡ st❛❝❦s ❡♥❝♦✉♥t❡r❡❞ ❜② xi ✇❤✐❧❡ ❡✈❛❧✉❛t✐♥❣ (t, σ) | π✱ ✐✳❡✳ (txi{ x := σ}) π = [ρ1; . . . ; ρm] (t, σ) | π − →∗ (xi, σ1) | ρ1 ✳ ✳ ✳ ✳ ✳ ✳ − →∗ (xi, σm) | ρm ❖t❤❡r✇✐s❡ s❛✐❞✱ ❉✐❛❧❡❝t✐❝❛ tr❛❝❦s t❤❡ ●r❛❜ r✉❧❡✳

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✷✾ ✴ ✹✶

SLIDE 52

▲♦♦❦

xx ≡ λπ. [π] : C(A) → list C(A) yx ≡ λπ. [ ] : C(A) → list C(Γi) (λy. t)x ≡ λ(y, π). tx π : W(A) × C(B) → list C(Γi) (t u)x ≡ λπ. (((snd t•) π u•) ❃ ❃❂ ux) @ tx (u•, π) : C(B) → list C(Γi)

✭❲❡ ❝❛♥ ❣❡♥❡r❛❧✐③❡ t❤✐s ✐♥t❡r♣r❡t❛t✐♦♥ t♦ ❛❧❣❡❜r❛✐❝ ❞❛t❛t②♣❡s✳✮

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✸✵ ✴ ✹✶

SLIDE 53

❉✐❛❧❡❝t✐❝❛ ❘❡❧♦❛❞❡❞

❚❤❡ st❛♥❞❛r❞ ❉✐❛❧❡❝t✐❝❛ ♦♥❧② r❡t✉r♥s ♦♥❡ st❛❝❦

t❤❡ ✜rst ❝♦rr❡❝t st❛❝❦✱ ❞②♥❛♠✐❝❛❧❧② t❡st❡❞

❚❤✐s ✐s s♦♠❡❤♦✇ ❛ ✇❡❛❦ ❢♦r♠ ♦❢ ❞❡❧✐♠✐t❡❞ ❝♦♥tr♦❧

■♥s♣❡❝t❛❜❧❡ st❛❝❦s✿ ✈s✳ ❋✐rst ❝❧❛ss ❛❝❝❡ss t♦ t❤♦s❡ st❛❝❦s ✇✐t❤ ❖r t❤r♦✉❣❤ ❛ ❝♦♥tr♦❧ ♦♣❡r❛t♦r

❲❡ ❝❛♥ ❞♦ t❤❡ s❛♠❡ t❤✐♥❣ ✇✐t❤ ♦t❤❡r ❝❛❧❧✐♥❣ ❝♦♥✈❡♥t✐♦♥s

❚❤❡ ♣r♦t♦❤✐st♦r✐❝ ❉✐❛❧❡❝t✐❝❛ ✇❛s ❝❛❧❧✲❜②✲♥❛♠❡ ❈❤♦♦s❡ ②♦✉r ❢❛✈♦r✐t❡ tr❛♥s❧❛t✐♦♥ ✐♥t♦ ▲▲✦

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✸✶ ✴ ✹✶

SLIDE 54

❉✐❛❧❡❝t✐❝❛ ❘❡❧♦❛❞❡❞

❚❤❡ st❛♥❞❛r❞ ❉✐❛❧❡❝t✐❝❛ ♦♥❧② r❡t✉r♥s ♦♥❡ st❛❝❦

t❤❡ ✜rst ❝♦rr❡❝t st❛❝❦✱ ❞②♥❛♠✐❝❛❧❧② t❡st❡❞

❚❤✐s ✐s s♦♠❡❤♦✇ ❛ ✇❡❛❦ ❢♦r♠ ♦❢ ❞❡❧✐♠✐t❡❞ ❝♦♥tr♦❧

■♥s♣❡❝t❛❜❧❡ st❛❝❦s✿ ∼A ✈s✳ ¬A ❋✐rst ❝❧❛ss ❛❝❝❡ss t♦ t❤♦s❡ st❛❝❦s ✇✐t❤ (−)x ❖r t❤r♦✉❣❤ ❛ ❝♦♥tr♦❧ ♦♣❡r❛t♦r D : (A → B) → A → ∼B → list(∼A)

❲❡ ❝❛♥ ❞♦ t❤❡ s❛♠❡ t❤✐♥❣ ✇✐t❤ ♦t❤❡r ❝❛❧❧✐♥❣ ❝♦♥✈❡♥t✐♦♥s

❚❤❡ ♣r♦t♦❤✐st♦r✐❝ ❉✐❛❧❡❝t✐❝❛ ✇❛s ❝❛❧❧✲❜②✲♥❛♠❡ ❈❤♦♦s❡ ②♦✉r ❢❛✈♦r✐t❡ tr❛♥s❧❛t✐♦♥ ✐♥t♦ ▲▲✦

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✸✶ ✴ ✹✶

SLIDE 55

❉✐❛❧❡❝t✐❝❛ ❘❡❧♦❛❞❡❞

❚❤❡ st❛♥❞❛r❞ ❉✐❛❧❡❝t✐❝❛ ♦♥❧② r❡t✉r♥s ♦♥❡ st❛❝❦

t❤❡ ✜rst ❝♦rr❡❝t st❛❝❦✱ ❞②♥❛♠✐❝❛❧❧② t❡st❡❞

❚❤✐s ✐s s♦♠❡❤♦✇ ❛ ✇❡❛❦ ❢♦r♠ ♦❢ ❞❡❧✐♠✐t❡❞ ❝♦♥tr♦❧

■♥s♣❡❝t❛❜❧❡ st❛❝❦s✿ ∼A ✈s✳ ¬A ❋✐rst ❝❧❛ss ❛❝❝❡ss t♦ t❤♦s❡ st❛❝❦s ✇✐t❤ (−)x ❖r t❤r♦✉❣❤ ❛ ❝♦♥tr♦❧ ♦♣❡r❛t♦r D : (A → B) → A → ∼B → list(∼A)

❲❡ ❝❛♥ ❞♦ t❤❡ s❛♠❡ t❤✐♥❣ ✇✐t❤ ♦t❤❡r ❝❛❧❧✐♥❣ ❝♦♥✈❡♥t✐♦♥s

❚❤❡ ♣r♦t♦❤✐st♦r✐❝ ❉✐❛❧❡❝t✐❝❛ ✇❛s ❝❛❧❧✲❜②✲♥❛♠❡ ❈❤♦♦s❡ ②♦✉r ❢❛✈♦r✐t❡ tr❛♥s❧❛t✐♦♥ ✐♥t♦ ▲▲✦

P✐❡rr❡✲▼❛r✐❡ Pé❞r♦t ✭PP❙✴πr2✮ ❈❛♥ ❉✐❛❧❡❝t✐❝❛ ❜r❡❛❦ ❜r✐❝❦s❄ ✷✶✴✵✸✴✷✵✶✹ ✸✶ ✴ ✹✶

SLIDE 56

■ ❧✐❡❞ ✭t❤❛t ✇♦♥✬t ♦❝❝✉r ❛♥②♠♦r❡✱ ■ s✇❡❛r✮

❆❝t✉❛❧❧②✱ t❤❡r❡ ✐s s♦♠❡t❤✐♥❣ ✇r♦♥❣✳ Pr♦❞✉❝❡❞ st❛❝❦s ❛r❡ t❤❡ r✐❣❤t ♦♥❡s✳✳✳ ❚❤❡② ❤❛✈❡ t❤❡ r✐❣❤t ♠✉❧t✐♣❧✐❝✐t②✳✳✳ ❇✉t t❤❡② ❛r❡ ♥♦t r❡s♣❡❝t✐♥❣ t❤❡ ❑❆▼ ♦r❞❡r✦ ❙t✐❧❧ ♥♦t st❛❜❧❡ ❜② ❲❡ ❤❛✈❡ t♦ ✉s❡ ✜♥✐t❡ ♠✉❧t✐s❡ts ❢♦r ✐t t♦ ✇♦r❦ ❚❤❡ ❢❛✉❧t② ♦♥❡ ✐s t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ❝❛s❡ ✭♠♦r❡ ❣❡♥❡r❛❧❧② ❞✉♣❧✐❝❛t✐♦♥✮✳ ❃ ❃❂