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Oct 28, 2023 470 likes •821 views

r r t ssts rs t s t

SLIDE 1

▼❚❆❚✳✵✺✳✶✵✺ ❚②♣❡ ❚❤❡♦r② ❚❤❡ ✕✲❝✉❜❡ ❛♥❞ ♣✉r❡ t②♣❡ s②st❡♠s

SLIDE 2

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡

❚❤❡ s✐♠♣❧② t②♣❡❞ ✕✲❝❛❧❝✉❧✉s ✕✦ ♣r♦✈✐❞❡s ❛ ♣♦ss✐❜✐❧✐t② t♦ ❞❡☞♥❡ t❡r♠s ❞❡♣❡♥❞✐♥❣ ♦♥ t❡r♠s ✉s✐♥❣ ✕✲❛❜str❛❝t✐♦♥ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥✳ ❚❤❡ s❡❝♦♥❞✲♦r❞❡r ✕✲❝❛❧❝✉❧✉s ✕✷ ❛❞❞s t❤❡ ♣♦s✐❜✐❧✐t② t♦ ❞❡☞♥❡ t❡r♠s ❞❡♣❡♥❞✐♥❣ ♦♥ t②♣❡s✳ ❇✉t ✇❤❛t ❛❜♦✉t t②♣❡s ❞❡♣❡♥❞✐♥❣ ♦♥ t②♣❡s ♦r t②♣❡s ❞❡♣❡♥❞✲ ✐♥❣ ♦♥ t❡r♠s❄ ■♥❞❡❡❞ t❤❡r❡ ❛r❡ s❡✈❡r❛❧ ✕✲❝❛❧❝✉❧✐ ✇❤✐❝❤ ❛❧❧♦✇ t❤❡s❡ ❦✐♥❞s ♦❢ ❞❡♣❡♥❞❡♥❝✐❡s✳ ■♥ ✶✾✾✶✱ ❍❡♥❦ ❇❛r❡♥❞r❡❣t ♣r♦♣♦s❡❞ ❛ ❝❧❛ss✐☞❝❛t✐♦♥ ♦❢ ✽ ❞✐❢✲ ❢❡r❡♥t ❝❛❧❝✉❧✐ ✐♥t♦ ✕✲❝✉❜❡ ❛♥❞ ♣r♦✈✐❞❡❞ ❛ ✉♥✐❢♦r♠ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡s❡ ❝❛❧❝✉❧✐✳

SLIDE 3

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡✿ t❤❡ s②st❡♠ ✕✦

✕✦ ❡①t❡♥❞s ✕✦ ✇✐t❤ t②♣❡s ❞❡♣❡♥❞✐♥❣ ♦♥ t②♣❡s✳ ④ ■✳❡✳✱ ♣r♦✈✐❞❡s t②♣❡✲❧❡✈❡❧ ❛❜str❛❝t✐♦♥ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥ ❝♦♥str✉❝ts✳ ◆♦✇✱ t②♣❡s s❤♦✉❧❞ ❛❧s♦ ❤❛✈❡ ✧t②♣❡s✧✳ ④ ✧❚②♣❡s✧ ❢♦r t②♣❡s ❛r❡ ❝❛❧❧❡❞ ❦✐♥❞s✳ K ❂ ✄ ❥ K ✦ K ④ ✧❖r❞✐♥❛r②✧ t②♣❡s ❤❛✈❡ t❤❡ ❦✐♥❞ ✄✱ ✉♥❛r② t②♣❡ ❝♦♥str✉❝t♦rs ❤❛✈❡ t❤❡ ❦✐♥❞ ✄ ✦ ✄✱ ❡t❝✳ ④ ❚❤❡r❡ ✐s ❛❧s♦ ❛ ✧t②♣❡✧ ♦❢ ❦✐♥❞s✱ ✱ ✐✳❡✳✱ ❦ ✷ K ✭ ✮ ❦ ✿ ✳

SLIDE 4

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡✿ t❤❡ s②st❡♠ ✕✦

❙♦rts✿ s ✿✿❂ ✄ st❛r ❥

sq✉❛r❡

❊①♣r❡ss✐♦♥s✿ ❡ ✿✿❂ ① ✈❛r✐❛❜❧❡ ❥ s s♦rt ❝♦♥st❛♥t ❥ ❡✶ ❡✷ ❛♣♣❧✐❝❛t✐♦♥ ❥ ✕① ✿ ❡✶✿ ❡✷ ❛❜str❛❝t✐♦♥ ❥ ❡✶ ✦ ❡✷ ❢✉♥❝t✐♦♥ ❈♦♥t❡①ts ❛r❡ ☞♥✐t❡ ❧✐♥❡❛r❧② ♦r❞❡r❡❞ s❡ts ♦❢ st❛t❡♠❡♥ts ① ✿ ❡ ✇✐t❤ ❞✐st✐♥❝t ✈❛r✐❛❜❧❡s ❛s s✉❜❥❡❝ts✳ ❂ ❬①✶ ✿ ❡✶❀ ①✷ ✿ ❡✷❀ ✿ ✿ ✿ ❀ ①♥ ✿ ❡♥❪

SLIDE 5

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡✿ t❤❡ s②st❡♠ ✕✦

❊✈❡r②t❤✐♥❣ ✐s ❛♥ ❡①♣r❡ss✐♦♥❀ t❤❡r❡ ✐s ♥♦ s❡♣❛r❛t❡ t❡r♠✱ t②♣❡ ♦r ❦✐♥❞ ❣r❛♠♠❛r✳ ❚❤❡ t②♣✐♥❣ r✉❧❡s ❞❡t❡r♠✐♥❡ ✇❤✐❝❤ ❡①♣r❡ss✐♦♥s ❛r❡ t❡r♠s✱ t②♣❡s ♦r ❦✐♥❞s✳ ❑ ✐s ❛ ❦✐♥❞ ✭ ✮ ❵ ❑ ✿ ❚ ✐s ❛ t②♣❡ ✭ ✮ ❵ ❚ ✿ ❑ ✿ ❊ ✐s ❛ t❡r♠ ✭ ✮ ❵ ❊ ✿ ❚ ✿ ❑ ✿ ☞✲r❡❞✉❝t✐♦♥✿ ✭✕①✿❡✶✿ ❡✷✮ ❡✸ ✦☞ ❡✷❬① ✼✦ ❡✸❪

SLIDE 6

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡✿ t❤❡ s②st❡♠ ✕✦

❚②♣✐♥❣ r✉❧❡s✿ ❵ ✄ ✿

❬AXIOM❪

❵ ❡✶ ✿ ❆ ✦ ❇ ❵ ❡✷ ✿ ❆ ❵ ❡✶ ❡✷ ✿ ❇

❬APP❪

❵ ❆ ✿ s ❀ ①✿❆ ❵ ① ✿ ❆

❬START❪

❀ ①✿❆ ❵ ❡ ✿ ❇ ❵ ❆ ✦ ❇ ✿ s ❵ ✭✕①✿❆✿ ❡✮ ✿ ❆ ✦ ❇

❬ABS❪

❵ ❡ ✿ ❆ ❵ ❇ ✿ s ❀ ①✿❇ ❵ ❡ ✿ ❆

❬WEAK❪

❵ ❡ ✿ ❇ ❵ ❆ ✿ s ❆ ❂☞ ❇ ❵ ❡ ✿ ❆

❬CONV❪

❵ ❆ ✿ s ❵ ❇ ✿ s ❵ ❆ ✦ ❇ ✿ s

❬ARROW❪

④ ❍❡r❡ s ✷ ❢✄❀ ❣✳

SLIDE 7

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡✿ t❤❡ s②st❡♠ ✕✦

❊①❛♠♣❧❡s✿ ☛✿✄❀ ☞✿✄ ❵ ☛ ✦ ☞ ✿ ✄ ☛✿✄❀ ☞✿✄❀ ①✿☛ ✦ ☞ ❵ ① ✿ ☛ ✦ ☞ ☛✿✄❀ ☞✿✄ ❵ ✭✕①✿☛ ✦ ☞✿ ①✮ ✿ ✭☛ ✦ ☞✮ ✦ ✭☛ ✦ ☞✮ ❵ ✭✕☛✿✄✿ ☛ ✦ ☛✮ ✿ ✄ ✦ ✄ ☛✿✄ ❵ ✭✕①✿❉☛✿ ①✮ ✿ ❉✭❉☛✮ ✇❤❡r❡ ❉ ✑ ✕☛✿✄✿ ☛ ✦ ☛✳

SLIDE 8

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡✿ t❤❡ s②st❡♠ ✕P

✕P ❡①t❡♥❞s ✕✦ ✇✐t❤ ❞❡♣❡♥❞❡♥t t②♣❡s ✭✐✳❡✳ t②♣❡s ❞❡♣❡♥❞✐♥❣ ♦♥ t❡r♠s✮✳ ❚❤❡ ❞❡♣❡♥❞❡♥t ♣r♦❞✉❝t ✆① ✿ ❆✿ ❇ ✐s t❤❡ t②♣❡ ♦❢ ❢✉♥❝t✐♦♥s ❢r♦♠ ✈❛❧✉❡s ① ♦❢ t②♣❡ ❆ t♦ ✈❛❧✉❡s ♦❢ t②♣❡ ❇✱ ✐♥ ✇❤✐❝❤ t❤❡ t②♣❡ ❇ ♠❛② ❞❡♣❡♥❞ ♦♥ t❤❡ ❛r❣✉♠❡♥t ✈❛❧✉❡ ①✳ ◆♦t❡✿ ❆ ✦ ❇ ✑ ✆ ✿❆✿ ❇✳

SLIDE 9

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡✿ t❤❡ s②st❡♠ ✕P

❊①♣r❡ss✐♦♥s✿ ❡ ✿✿❂ ① ✈❛r✐❛❜❧❡ ❥ ✄ st❛r ❥

sq✉❛r❡

❥ ❡✶ ❡✷ ❛♣♣❧✐❝❛t✐♦♥ ❥ ✕① ✿ ❡✶✿ ❡✷ ❛❜str❛❝t✐♦♥ ❥ ✆① ✿ ❡✶✿ ❡✷ ♣r♦❞✉❝t ☞✲r❡❞✉❝t✐♦♥✿ ✭✕①✿❡✶✿ ❡✷✮ ❡✸ ✦☞ ❡✷❬① ✼✦ ❡✸❪

SLIDE 10

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡✿ t❤❡ s②st❡♠ ✕P

❚②♣✐♥❣ r✉❧❡s✿ ❵ ✄ ✿

❬AXIOM❪

❵ ❡✶ ✿ ✭✆①✿❆✿❇✮ ❵ ❡✷ ✿ ❆ ❵ ❡✶ ❡✷ ✿ ❇❬① ✼✦ ❡✷❪

❬APP❪

❵ ❆ ✿ s ❀ ①✿❆ ❵ ① ✿ ❆

❬START❪

❀ ①✿❆ ❵ ❡ ✿ ❇ ❵ ✭✆①✿❆✿❇ ✿ s✮ ❵ ✭✕①✿❆✿ ❡✮ ✿ ✭✆①✿❆✿❇✮

❬ABS❪

❵ ❡ ✿ ❆ ❵ ❇ ✿ s ❀ ①✿❇ ❵ ❡ ✿ ❆

❬WEAK❪

❵ ❡ ✿ ❇ ❵ ❆ ✿ s ❆ ❂☞ ❇ ❵ ❡ ✿ ❆

❬CONV❪

❵ ❆ ✿ ✄ ❀ ①✿❆ ❵ ❇ ✿ s ❵ ✭✆①✿❆✿❇✮ ✿ s

❬PROD❪

④ ❍❡r❡ s ✷ ❢✄❀ ❣✳

SLIDE 11

❚♦✇❛r❞s t❤❡ ✕✲❝✉❜❡✿ t❤❡ s②st❡♠ ✕P

❊①❛♠♣❧❡s✿ ❆✿✄ ❵ ✭❆ ✦ ✄✮ ✿ ❆✿✄❀ P✿❆✦✄❀ ❛✿❆ ❵ P ❛ ✿ ✄ ❆✿✄❀ P✿❆✦✄❀ ❛✿❆ ❵ ✭P ❛ ✦ ✄✮ ✿ ❆✿✄❀ P✿❆✦✄ ❵ ✭✆❛✿❆✿ P ❛ ✦ ✄✮ ✿ ❆✿✄❀ P✿❆✦✄ ❵ ✭✕❛✿❆✿ ✕①✿P❛✿ ①✮ ✿ ✭✆❛✿❆✿ P ❛ ✦ P ❛✮

SLIDE 12

❚❤❡ ✕✲❝✉❜❡

❊①♣r❡ss✐♦♥s✿ ❡ ✿✿❂ ① ✈❛r✐❛❜❧❡ ❥ ✄ st❛r ❥

sq✉❛r❡

❥ ❡✶ ❡✷ ❛♣♣❧✐❝❛t✐♦♥ ❥ ✕① ✿ ❡✶✿ ❡✷ ❛❜str❛❝t✐♦♥ ❥ ✆① ✿ ❡✶✿ ❡✷ ♣r♦❞✉❝t ☞✲r❡❞✉❝t✐♦♥✿ ✭✕①✿❡✶✿ ❡✷✮ ❡✸ ✦☞ ❡✷❬① ✼✦ ❡✸❪ ❆❜❜r❡✈❛t✐♦♥s✿ ❆ ✦ ❇ ✑ ✆ ✿❆✿ ❇ ✽☛✿❆ ✑ ✆☛✿✄✿ ❆ ❉❡♣❡♥❞❡♥❝② r✉❧❡s✿ ❘ ✒ ❢✭✄❀ ✄✮❀ ✭✄❀ ✮❀ ✭❀ ✄✮❀ ✭❀ ✮❣

SLIDE 13

❚❤❡ ✕✲❝✉❜❡

❚②♣✐♥❣ r✉❧❡s✿ ❵ ✄ ✿

❬AXIOM❪

❵ ❡✶ ✿ ✭✆①✿❆✿❇✮ ❵ ❡✷ ✿ ❆ ❵ ❡✶ ❡✷ ✿ ❇❬① ✼✦ ❡✷❪

❬APP❪

❵ ❆ ✿ s ❀ ①✿❆ ❵ ① ✿ ❆

❬START❪

❀ ①✿❆ ❵ ❡ ✿ ❇ ❵ ✭✆①✿❆✿❇✮ ✿ s ❵ ✭✕①✿❆✿ ❡✮ ✿ ✭✆①✿❆✿❇✮

❬ABS❪

❵ ❡ ✿ ❆ ❵ ❇ ✿ s ❀ ①✿❇ ❵ ❡ ✿ ❆

❬WEAK❪

❵ ❡ ✿ ❇ ❵ ❆ ✿ s ❆ ❂☞ ❇ ❵ ❡ ✿ ❆

❬CONV❪

❵ ❆ ✿ s✶ ❀ ①✿❆ ❵ ❇ ✿ s✷ ❵ ✭✆①✿❆✿❇✮ ✿ s✷

❬PROD❪

④ ❍❡r❡ s ✷ ❢✄❀ ❣ ❛♥❞ ✭s✶❀ s✷✮ ✷ ❘✳

SLIDE 14

❚❤❡ ✕✲❝✉❜❡

❚❤❡ s②st❡♠s ♦❢ t❤❡ ✕✲❝✉❜❡✿

❙②st❡♠ ❙❡t ♦❢ r✉❧❡s ✭❘✮ ✕✦ ✭✄❀ ✄✮ ✕✷ ✭✄❀ ✄✮ ✭❀ ✄✮ ✕P ✭✄❀ ✄✮ ✭✄❀ ✮ ✕P✷ ✭✄❀ ✄✮ ✭❀ ✄✮ ✭✄❀ ✮ ✕✦ ✭✄❀ ✄✮ ✭❀ ✮ ✕✦ ✭✄❀ ✄✮ ✭❀ ✄✮ ✭❀ ✮ ✕P✦ ✭✄❀ ✄✮ ✭✄❀ ✮ ✭❀ ✮ ✕P✦ ❂ ✕❈ ✭✄❀ ✄✮ ✭❀ ✄✮ ✭✄❀ ✮ ✭❀ ✮

SLIDE 15

❚❤❡ ✕✲❝✉❜❡

✕✦ ✕❈ ✕✷ ✕P✷ ✕✦ ✕P✦ ✕✦ ✕P

✭❀ ✮ ✭✄❀ ✮ ✭❀ ✄✮

SLIDE 16

Pr♦♣❡rt✐❡s ♦❢ t❤❡ s②st❡♠s ♦❢ t❤❡ ✕✲❝✉❜❡

❈❤✉r❝❤✲❘♦ss❡r✿ ❋♦r ❛♥② ❡①♣r❡ss✐♦♥s ❆✱ ❇✶ ❛♥❞ ❇✷✱ ❬❆ ։ ❇✶ ❫ ❆ ։ ❇✷❪ ❂ ✮ ✾❈✿ ❬❇✶ ։ ❈ ❫ ❇✷ ։ ❈❪ ❙✉❜❥❡❝t ❘❡❞✉❝t✐♦♥✿ ❋♦r ❛♥② s②st❡♠ ✐♥ t❤❡ ✕✲❝✉❜❡✱ ❵ ❆ ✿ ❇ ❫ ❆ ։ ❆✵ ❂ ✮ ❵ ❆✵ ✿ ❇ ❙tr♦♥❣ ◆♦r♠❛❧✐③❛t✐♦♥✿ ❋♦r ❛♥② s②st❡♠ ✐♥ t❤❡ ✕✲❝✉❜❡✱ ✐❢ ❵ ❆ ✿ ❇✱ t❤❡♥ ❛❧❧ r❡❞✉❝t✐♦♥s st❛rt✐♥❣ ❢r♦♠ ❆ ♦r ❇ t❡r♠✐♥❛t❡✳ ❯♥✐❝✐t② ♦❢ t②♣❡s✿ ❋♦r ❛♥② s②st❡♠ ✐♥ t❤❡ ✕✲❝✉❜❡✱ ❵ ❆ ✿ ❇ ❫ ❵ ❆ ✿ ❇✵ ❂ ✮ ❇ ❂☞ ❇✵

SLIDE 17

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✶✿✶✮ ❆✿✄ ❵ ✭✆①✿❆✿ ❆✮ ✿ ✄ ✭✶✿✷✮ ❆✿✄ ❵ ✭✕❛✿❆✿ ❛✮ ✿ ✭✆①✿❆✿ ❆✮ ✭✶✿✸✮ ❆✿✄❀ ❜✿❆ ❵ ✭✭✕❛✿❆✿ ❛✮ ❜✮ ✿ ❆

SLIDE 18

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✶✿✶✮ ❆✿✄ ❵ ✭✆①✿❆✿ ❆✮ ✿ ✄ ✭✶✿✷✮ ❆✿✄ ❵ ✭✕❛✿❆✿ ❛✮ ✿ ✭✆①✿❆✿ ❆✮ ✭✶✿✸✮ ❆✿✄❀ ❜✿❆ ❵ ✭✭✕❛✿❆✿ ❛✮ ❜✮ ✿ ❆ ❉❡r✐✈❛t✐♦♥ ♦❢ ✭✶✿✶✮✿

❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ❆ ✿ ✄

❬START❪

❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ❆ ✿ ✄

❬START❪

❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ❆ ✿ ✄

❬START❪

❆✿✄❀ ①✿❆ ❵ ❆ ✿ ✄

❬WEAK❪

❆✿✄ ❵ ✭✆①✿❆✿ ❆✮ ✿ ✄

❬PROD✭✄❀✄✮❪

SLIDE 19

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✶✿✶✮ ❆✿✄ ❵ ✭✆①✿❆✿ ❆✮ ✿ ✄ ✭✶✿✷✮ ❆✿✄ ❵ ✭✕❛✿❆✿ ❛✮ ✿ ✭✆①✿❆✿ ❆✮ ✭✶✿✸✮ ❆✿✄❀ ❜✿❆ ❵ ✭✭✕❛✿❆✿ ❛✮ ❜✮ ✿ ❆ ❉❡r✐✈❛t✐♦♥ ♦❢ ✭✶✿✷✮✿

❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ❆ ✿ ✄

❬START❪

❆✿✄❀ ❛✿❆ ❵ ❛ ✿ ❆

❬START❪

✭✶✿✶✮ ❆✿✄ ❵ ✭✆①✿❆✿ ❆✮ ✿ ✄ ❆✿✄ ❵ ✭✕❛✿❆✿ ❛✮ ✿ ✭✆①✿❆✿ ❆✮

❬ABS❪

SLIDE 20

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✶✿✶✮ ❆✿✄ ❵ ✭✆①✿❆✿ ❆✮ ✿ ✄ ✭✶✿✷✮ ❆✿✄ ❵ ✭✕❛✿❆✿ ❛✮ ✿ ✭✆①✿❆✿ ❆✮ ✭✶✿✸✮ ❆✿✄❀ ❜✿❆ ❵ ✭✭✕❛✿❆✿ ❛✮ ❜✮ ✿ ❆ ❉❡r✐✈❛t✐♦♥ ♦❢ ✭✶✿✸✮✿

✭✶✿✷✮ ❆✿✄ ❵ ✭✕❛✿❆✿ ❛✮ ✿ ✭✆①✿❆✿ ❆✮ ❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ❆ ✿ ✄

❬START❪

❆✿✄❀ ❜✿❆ ❵ ✭✕❛✿❆✿ ❛✮ ✿ ✭✆①✿❆✿ ❆✮

❬WEAK❪

❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ❆ ✿ ✄

❬START❪

❆✿✄❀ ❜✿❆ ❵ ❜ ✿ ❆

❬START❪

❆✿✄❀ ❜✿❆ ❵ ✭✭✕❛✿❆✿ ❛✮ ❜✮ ✿ ❆

❬APP❪

SLIDE 21

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✷ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✷✿✶✮ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ✿ ✭✆☛✿✄✿ ☛ ✦ ☛✮ ✭✷✿✷✮ ❆✿✄ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ✿ ❆ ✦ ❆ ✭✷✿✸✮ ❆✿✄❀ ❜✿❆ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ❜ ✿ ❆

SLIDE 22

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✷ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✷✿✶✮ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ✿ ✭✆☛✿✄✿ ☛ ✦ ☛✮ ✭✷✿✷✮ ❆✿✄ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ✿ ❆ ✦ ❆ ✭✷✿✸✮ ❆✿✄❀ ❜✿❆ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ❜ ✿ ❆ ❉❡r✐✈❛t✐♦♥ ♦❢ ✭✷✿✶✮✿

✭✶✿✷✮ ☛✿✄ ❵ ✭✕❛✿☛✿ ❛✮ ✿ ☛ ✦ ☛ ❵ ✄ ✿

❬AXIOM❪

✭✶✿✶✮ ☛✿✄ ❵ ☛ ✦ ☛ ✿ ✄ ❵ ✭✆☛✿✄✿ ☛ ✦ ☛✮ ✿ ✄

❬PROD✭❀✄✮❪

❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ✿ ✭✆☛✿✄✿ ☛ ✦ ☛✮

❬ABS❪

SLIDE 23

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✷ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✷✿✶✮ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ✿ ✭✆☛✿✄✿ ☛ ✦ ☛✮ ✭✷✿✷✮ ❆✿✄ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ✿ ❆ ✦ ❆ ✭✷✿✸✮ ❆✿✄❀ ❜✿❆ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ❜ ✿ ❆ ❉❡r✐✈❛t✐♦♥ ♦❢ ✭✷✿✷✮✿

✭✷✿✶✮ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ✿ ✭✆☛✿✄✿ ☛ ✦ ☛✮ ❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ✿ ✭✆☛✿✄✿ ☛ ✦ ☛✮

❬WEAK❪

❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ❆ ✿ ✄

❬START❪

❆✿✄ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ✿ ❆ ✦ ❆

❬APP❪

SLIDE 24

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✷ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✷✿✶✮ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ✿ ✭✆☛✿✄✿ ☛ ✦ ☛✮ ✭✷✿✷✮ ❆✿✄ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ✿ ❆ ✦ ❆ ✭✷✿✸✮ ❆✿✄❀ ❜✿❆ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ❜ ✿ ❆ ❉❡r✐✈❛t✐♦♥ ♦❢ ✭✷✿✸✮✿

✭✷✿✷✮ ❆✿✄ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ✿ ❆ ✦ ❆ ❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ❆ ✿ ✄

❬START❪

❆✿✄❀ ❜✿❆ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ✿ ❆ ✦ ❆

❬WEAK❪

❵ ✄ ✿

❬AXIOM❪

❆✿✄ ❵ ❆ ✿ ✄

❬START❪

❆✿✄❀ ❜✿❆ ❵ ❜ ✿ ❆

❬START❪

❆✿✄❀ ❜✿❆ ❵ ✭✕☛✿✄✿ ✕❛✿☛✿ ❛✮ ❆ ❜ ✿ ❆

❬APP❪

SLIDE 25

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✸✿✶✮ ❵ ✄ ✦ ✄ ✿ ✭✸✿✷✮ ☛✿✄❀ ☞✿✄ ❵ ☛✫☞ ✿ ✄ ✭✸✿✸✮ ❵ ✭✕☛✿✄✿ ✕☞✿✄✿ ☛✫☞✮ ✿ ✄ ✦ ✄ ✦ ✄ ✇❤❡r❡ ☛✫☞ ✑ ✆✌✿✄✿ ✭☛ ✦ ☞ ✦ ✌✮ ✦ ✌✳

SLIDE 26

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✸✿✶✮ ❵ ✄ ✦ ✄ ✿ ✭✸✿✷✮ ☛✿✄❀ ☞✿✄ ❵ ☛✫☞ ✿ ✄ ✭✸✿✸✮ ❵ ✭✕☛✿✄✿ ✕☞✿✄✿ ☛✫☞✮ ✿ ✄ ✦ ✄ ✦ ✄ ✇❤❡r❡ ☛✫☞ ✑ ✆✌✿✄✿ ✭☛ ✦ ☞ ✦ ✌✮ ✦ ✌✳ ❉❡r✐✈❛t✐♦♥ ♦❢ ✭✸✿✶✮✿

❵ ✄ ✿

❬AXIOM❪

❵ ✄ ✿

❬AXIOM❪

❵ ✄ ✿

❬AXIOM❪

✿ ✄ ❵ ✄ ✿

❬WEAK❪

❵ ✄ ✦ ✄ ✿

❬PROD✭❀✮❪

SLIDE 27

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✸✿✶✮ ❵ ✄ ✦ ✄ ✿ ✭✸✿✷✮ ☛✿✄❀ ☞✿✄ ❵ ☛✫☞ ✿ ✄ ✭✸✿✸✮ ❵ ✭✕☛✿✄✿ ✕☞✿✄✿ ☛✫☞✮ ✿ ✄ ✦ ✄ ✦ ✄ ✇❤❡r❡ ☛✫☞ ✑ ✆✌✿✄✿ ✭☛ ✦ ☞ ✦ ✌✮ ✦ ✌✳ ❉❡r✐✈❛t✐♦♥ ♦❢ ✭✸✿✷✮✿

✳ ✳ ✳ ✳ ☛✿✄❀ ☞✿✄ ❵ ✄ ✿ ✳ ✳ ✳ ✳ ❵ ☛ ✿ ✄ ✳ ✳ ✳ ✳ ✷ ❵ ☞ ✿ ✄ ✳ ✳ ✳ ✳ ✸ ❵ ✌ ✿ ✄ ✷ ❵ ☞ ✦ ✌ ✿ ✄

❬PROD✭✄❀✄✮❪

❵ ☛ ✦ ☞ ✦ ✌ ✿ ✄

❬PROD✭✄❀✄✮❪

✳ ✳ ✳ ✳ ✶ ❵ ✌ ✿ ✄ ❵ ✭☛ ✦ ☞ ✦ ✌✮ ✦ ✌ ✿ ✄

❬PROD✭✄❀✄✮❪

☛✿✄❀ ☞✿✄ ❵ ☛✫☞ ✿ ✄

❬PROD✭❀✄✮❪

✇❤❡r❡ ❂ ☛✿✄❀ ☞✿✄❀ ✌✿✄ ❛♥❞ ✐ ❂ ❀ ✿ ✿ ✿ ✿ ✳

SLIDE 28

❚❤❡ ✕✲❝✉❜❡✿ ❡①❛♠♣❧❡s

■♥ ✕✦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛♥ ❜❡ ❞❡r✐✈❡❞✿ ✭✸✿✶✮ ❵ ✄ ✦ ✄ ✿ ✭✸✿✷✮ ☛✿✄❀ ☞✿✄ ❵ ☛✫☞ ✿ ✄ ✭✸✿✸✮ ❵ ✭✕☛✿✄✿ ✕☞✿✄✿ ☛✫☞✮ ✿ ✄ ✦ ✄ ✦ ✄ ✇❤❡r❡ ☛✫☞ ✑ ✆✌✿✄✿ ✭☛ ✦ ☞ ✦ ✌✮ ✦ ✌✳ ❉❡r✐✈❛t✐♦♥ ♦❢ ✭✸✿✸✮✿

✭✸✿✷✮ ☛✿✄❀ ☞✿✄ ❵ ☛✫☞ ✿ ✄ ✭✸✿✶✮ ✄ ✦ ✄ ✿ ☛✿✄ ❵ ✭✕☞✿✄✿ ☛✫☞✮ ✿ ✄ ✦ ✄

❬ABS❪

❵ ✄ ✿ ✭✸✿✶✮ ❵ ✄ ✦ ✄ ✿ ❵ ✄ ✿ ✿ ✄ ❵ ✄ ✦ ✄ ✿

❬WEAK❪

❵ ✄ ✦ ✄ ✦ ✄ ✿

❬PROD✭❀✮❪

❵ ✭✕☛✿✄✿ ✕☞✿✄✿ ☛✫☞✮ ✿ ✄ ✦ ✄ ✦ ✄

❬ABS❪

SLIDE 29

P✉r❡ t②♣❡ s②st❡♠s

❊①♣r❡ss✐♦♥s✿ ❡ ✿✿❂ ① ✈❛r✐❛❜❧❡ ❥ ❝ ❝♦♥st❛♥t ❥ ❡✶ ❡✷ ❛♣♣❧✐❝❛t✐♦♥ ❥ ✕① ✿ ❡✶✿ ❡✷ ❛❜str❛❝t✐♦♥ ❥ ✆① ✿ ❡✶✿ ❡✷ ♣r♦❞✉❝t ❚❤❡ s♣❡❝✐☞❝❛t✐♦♥ ♦❢ ❛ ♣✉r❡ t②♣❡ s②st❡♠ ❝♦♥s✐sts ♦❢ ❛ tr✐♣❧❡ S ❂ ✭❙❀ ❆❀ ❘✮ ✇❤❡r❡ ④ ❙ ✐s ❛ s✉❜s❡t ♦❢ ❝♦♥st❛♥ts ❈✱ ❝❛❧❧❡❞ t❤❡ s♦rts❀ ④ ❆ ✐s ❛ s❡t ♦❢ ❛①✐♦♠s ♦❢ t❤❡ ❢♦r♠ ❝ ✿ s✱ ✇❤❡r❡ ❝ ✷ ❈ ❛♥❞ s ✷ ❙❀ ④ ❘ ✐s ❛ s❡t ♦❢ r✉❧❡s ♦❢ t❤❡ ❢♦r♠ ✭s✶❀ s✷❀ s✸✮✱ ✇❤❡r❡ s✐ ✷ ❙✳ ◆♦t❛t✐♦♥✿ ✭s✶❀ s✷✮ ❂ ✭s✶❀ s✷❀ s✷✮✳

SLIDE 30

P✉r❡ t②♣❡ s②st❡♠s

❚②♣✐♥❣ r✉❧❡s✿ ✭❝ ✿ s✮ ✷ ❆ ❵ ❝ ✿ s

❬AXIOM❪

❵ ❡✶ ✿ ✭✆①✿❆✿❇✮ ❵ ❡✷ ✿ ❆ ❵ ❡✶ ❡✷ ✿ ❇❬① ✼✦ ❡✷❪

❬APP❪

❵ ❆ ✿ s ❀ ①✿❆ ❵ ① ✿ ❆

❬START❪

❀ ①✿❆ ❵ ❡ ✿ ❇ ❵ ✭✆①✿❆✿❇✮ ✿ s ❵ ✭✕①✿❆✿ ❡✮ ✿ ✭✆①✿❆✿❇✮

❬ABS❪

❵ ❡ ✿ ❆ ❵ ❇ ✿ s ❀ ①✿❇ ❵ ❡ ✿ ❆

❬WEAK❪

❵ ❡ ✿ ❇ ❵ ❆ ✿ s ❆ ❂☞ ❇ ❵ ❡ ✿ ❆

❬CONV❪

❵ ❆ ✿ s✶ ❀ ①✿❆ ❵ ❇ ✿ s✷ ❵ ✭✆①✿❆✿❇✮ ✿ s✸

❬PROD❪

④ ❍❡r❡ s ✷ ❙ ❛♥❞ ✭s✶❀ s✷❀ s✸✮ ✷ ❘✳

SLIDE 31

Pr♦♣❡rt✐❡s ♦❢ ♣✉r❡ t②♣❡ s②st❡♠s

❚❤❡ P❚❙ ✕✭❙❀ ❆❀ ❘✮ ✐s ❢✉❧❧ ✐❢ ❘ ❂ ❢✭s✶❀ s✷✮ ❥ s✶❀ s✷ ✷ ❙❣✳ ❚❤❡ P❚❙ ✕✭❙❀ ❆❀ ❘✮ ✐s s✐♥❣❧② s♦rt❡❞ ✐❢ ✶✿ ✭❝ ✿ s✶✮❀ ✭❝ ✿ s✷✮ ✷ ❆ ❂ ✮ s✶ ✑ s✷ ✷✿ ✭s✶❀ s✷❀ s✸✮❀ ✭s✶❀ s✷❀ s✵

✸✮ ✷ ❘

❂ ✮ s✸ ✑ s✵

✸

❙✉❜❥❡❝t ❘❡❞✉❝t✐♦♥✿ ❋♦r ❛♥② P❚❙✱ ❵ ❆ ✿ ❇ ❫ ❆ ։ ❆✵ ❂ ✮ ❵ ❆✵ ✿ ❇ ❯♥✐❝✐t② ♦❢ t②♣❡s✿ ❋♦r ❛♥② s✐♥❣❧② s♦rt❡❞ P❚❙✱ ❵ ❆ ✿ ❇ ❫ ❵ ❆ ✿ ❇✵ ❂ ✮ ❇ ❂☞ ❇✵

SLIDE 32

P✉r❡ t②♣❡ s②st❡♠s✿ ❡①❛♠♣❧❡s

✕✷ ✐s t❤❡ P❚❙ ❞❡t❡r♠✐♥❡❞ ❜②✿ ✕✷ ❙ ✄❀ ❆ ✄ ✿ ❘ ✭✄❀ ✄✮❀ ✭❀ ✄✮ ✕❈ ✐s t❤❡ ❢✉❧❧ P❚❙ ❞❡t❡r♠✐♥❡❞ ❜②✿ ✕❈ ❙ ✄❀ ❆ ✄ ✿ ❘ ✭✄❀ ✄✮❀ ✭❀ ✄✮❀ ✭✄❀ ✮❀ ✭❀ ✮

SLIDE 33

P✉r❡ t②♣❡ s②st❡♠s✿ ❡①❛♠♣❧❡s

✕✦ ✐s t❤❡ P❚❙ ❞❡t❡r♠✐♥❡❞ ❜②✿ ✕✦ ❙ ✄❀ ❆ ✄ ✿ ❘ ✭✄❀ ✄✮ ✕✜ ✐s t❤❡ P❚❙ ❞❡t❡r♠✐♥❡❞ ❜②✿ ✕✜ ❙ ✄ ❆ ✵ ✿ ✄ ❘ ✭✄❀ ✄✮

SLIDE 34