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Pr trt s t - - PowerPoint PPT Presentation

Oct 27, 2022 275 likes •1.45k views

Pr trt s t ts rt tt Pr trt

slide-1
SLIDE 1

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s✿ t❤❡ ❝❛s❡ ♦❢ ①♦r✲t❡r♠s

❏✳✲❋✳ ▼♦♥✐♥ ❏✳ ❈♦✉r❛♥t

❱❊❘■▼❆●

  • r❡♥♦❜❧❡✱ ❋r❛♥❝❡

❚r❡♥❞s ✐♥ ❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣✱ ◆♦tt✐♥❣❤❛♠✱ ✷✵✵✻

slide-2
SLIDE 2

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-3
SLIDE 3

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋♦r♠❛❧ ♠♦❞❡❧s ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s

Pr♦t♦❝♦❧s ❙❡❝✉r✐t② ❆P■s ❳♦r ✐s ✉❜✐q✉✐t♦✉s ❊①❛♠♣❧❡s ❢r♦♠ ❛ s❡❝✉r✐t② ❆P■ ❝❛❧❧❡❞ ❈❈❆ ✭❈♦♠♠♦♥ ❈r②♣t♦❣r❛♣❤✐❝ ❆r❝❤✐t❡❝t✉r❡✮✿ ① ② ③

① ❑P ❑▼

③ ②

① ❑P ❑▼

① ② ③

① ❑P ❑▼

③ ②

① ❑▼

❘❡❛s♦♥✐♥❣ ✐♥✈♦❧✈❡s✿ ❈♦♠♠✉t❛t✐✈✐t②✿ ① ② ② ① ❆ss♦❝✐❛t✐✈✐t②✿ ① ② ③ ① ② ③ ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ✵ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ① ✵

slide-4
SLIDE 4

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋♦r♠❛❧ ♠♦❞❡❧s ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s

◮ Pr♦t♦❝♦❧s ◮ ❙❡❝✉r✐t② ❆P■s

❳♦r ✐s ✉❜✐q✉✐t♦✉s ❊①❛♠♣❧❡s ❢r♦♠ ❛ s❡❝✉r✐t② ❆P■ ❝❛❧❧❡❞ ❈❈❆ ✭❈♦♠♠♦♥ ❈r②♣t♦❣r❛♣❤✐❝ ❆r❝❤✐t❡❝t✉r❡✮✿ ① ② ③

① ❑P ❑▼

③ ②

① ❑P ❑▼

① ② ③

① ❑P ❑▼

③ ②

① ❑▼

❘❡❛s♦♥✐♥❣ ✐♥✈♦❧✈❡s✿ ❈♦♠♠✉t❛t✐✈✐t②✿ ① ② ② ① ❆ss♦❝✐❛t✐✈✐t②✿ ① ② ③ ① ② ③ ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ✵ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ① ✵

slide-5
SLIDE 5

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋♦r♠❛❧ ♠♦❞❡❧s ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s

◮ Pr♦t♦❝♦❧s ◮ ❙❡❝✉r✐t② ❆P■s

❳♦r ✐s ✉❜✐q✉✐t♦✉s ❊①❛♠♣❧❡s ❢r♦♠ ❛ s❡❝✉r✐t② ❆P■ ❝❛❧❧❡❞ ❈❈❆ ✭❈♦♠♠♦♥ ❈r②♣t♦❣r❛♣❤✐❝ ❆r❝❤✐t❡❝t✉r❡✮✿ ① ② ③

① ❑P ❑▼

③ ②

① ❑P ❑▼

① ② ③

① ❑P ❑▼

③ ②

① ❑▼

❘❡❛s♦♥✐♥❣ ✐♥✈♦❧✈❡s✿ ❈♦♠♠✉t❛t✐✈✐t②✿ ① ② ② ① ❆ss♦❝✐❛t✐✈✐t②✿ ① ② ③ ① ② ③ ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ✵ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ① ✵

slide-6
SLIDE 6

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋♦r♠❛❧ ♠♦❞❡❧s ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s

◮ Pr♦t♦❝♦❧s ◮ ❙❡❝✉r✐t② ❆P■s

❳♦r ✐s ✉❜✐q✉✐t♦✉s ❊①❛♠♣❧❡s ❢r♦♠ ❛ s❡❝✉r✐t② ❆P■ ❝❛❧❧❡❞ ❈❈❆ ✭❈♦♠♠♦♥ ❈r②♣t♦❣r❛♣❤✐❝ ❆r❝❤✐t❡❝t✉r❡✮✿ ①, ②, {③}①⊕❑P⊕❑▼ → {③ ⊕ ②}①⊕❑P⊕❑▼ ①, ②, {③}①⊕❑P⊕❑▼ → {③ ⊕ ②}①⊕❑▼ ❘❡❛s♦♥✐♥❣ ✐♥✈♦❧✈❡s✿ ❈♦♠♠✉t❛t✐✈✐t②✿ ① ② ② ① ❆ss♦❝✐❛t✐✈✐t②✿ ① ② ③ ① ② ③ ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ✵ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ① ✵

slide-7
SLIDE 7

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋♦r♠❛❧ ♠♦❞❡❧s ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s

◮ Pr♦t♦❝♦❧s ◮ ❙❡❝✉r✐t② ❆P■s

❳♦r ✐s ✉❜✐q✉✐t♦✉s ❊①❛♠♣❧❡s ❢r♦♠ ❛ s❡❝✉r✐t② ❆P■ ❝❛❧❧❡❞ ❈❈❆ ✭❈♦♠♠♦♥ ❈r②♣t♦❣r❛♣❤✐❝ ❆r❝❤✐t❡❝t✉r❡✮✿ ①, ②, {③}①⊕❑P⊕❑▼ → {③ ⊕ ②}①⊕❑P⊕❑▼ ①, ②, {③}①⊕❑P⊕❑▼ → {③ ⊕ ②}①⊕❑▼ ❘❡❛s♦♥✐♥❣ ✐♥✈♦❧✈❡s✿ ❈♦♠♠✉t❛t✐✈✐t②✿ ① ⊕ ② ≃ ② ⊕ ① ❆ss♦❝✐❛t✐✈✐t②✿ (① ⊕ ②) ⊕ ③ ≃ ① ⊕ (② ⊕ ③) ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ⊕ ✵ ≃ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ⊕ ① ≃ ✵

slide-8
SLIDE 8

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-9
SLIDE 9

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

  • ❡♥❡r❛❧ s❡tt✐♥❣✿ q✉♦t✐❡♥t❡❞ ✜rst ♦r❞❡r✲t❡r♠s

❲❡ ❛r❡ ❣✐✈❡♥

◮ ❆ t②♣❡ ♦❢ t❡r♠s T ✇✐t❤ ❝♦♥str✉❝t♦rs ❈❦✿

■♥❞✉❝t✐✈❡ T ✿ ❙❡t ✿❂ | ❈✶ : T ✳ ✳ ✳ | ❈❦ : . . . → T . . . → T . . . → T ✳ ✳ ✳ ❆ ❝♦♥❣r✉❡♥❝❡ Pr♦♣

❋♦r ❡❛❝❤ ❝♦♥str✉❝t♦r ❈❦ ❛ ①✶ ②✶ ❜ ①✷ ②✷ ❝ ①✶ ②✶ ①✷ ②✷ ❈❦ ❛ ①✶ ❜ ②✶ ❝ ❈❦ ❛ ①✷ ❜ ②✷ ❝ s♣❡❝✐✜❝ ❧❛✇s✱ ❡✳❣✳ ①② ❈✷ ① ❈✶ ② ❈✷ ② ①

❲❡ ✇❛♥t t♦ r❡❛s♦♥ ♦♥ ✉♣ t♦

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SLIDE 10

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

  • ❡♥❡r❛❧ s❡tt✐♥❣✿ q✉♦t✐❡♥t❡❞ ✜rst ♦r❞❡r✲t❡r♠s

❲❡ ❛r❡ ❣✐✈❡♥

◮ ❆ t②♣❡ ♦❢ t❡r♠s T ✇✐t❤ ❝♦♥str✉❝t♦rs ❈❦✿

■♥❞✉❝t✐✈❡ T ✿ ❙❡t ✿❂ | ❈✶ : T ✳ ✳ ✳ | ❈❦ : . . . → T . . . → T . . . → T ✳ ✳ ✳

◮ ❆ ❝♦♥❣r✉❡♥❝❡ ≃ : T → T → Pr♦♣

❋♦r ❡❛❝❤ ❝♦♥str✉❝t♦r ❈❦ ❛ ①✶ ②✶ ❜ ①✷ ②✷ ❝ ①✶ ②✶ ①✷ ②✷ ❈❦ ❛ ①✶ ❜ ②✶ ❝ ❈❦ ❛ ①✷ ❜ ②✷ ❝ s♣❡❝✐✜❝ ❧❛✇s✱ ❡✳❣✳ ①② ❈✷ ① ❈✶ ② ❈✷ ② ①

❲❡ ✇❛♥t t♦ r❡❛s♦♥ ♦♥ ✉♣ t♦

slide-11
SLIDE 11

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

  • ❡♥❡r❛❧ s❡tt✐♥❣✿ q✉♦t✐❡♥t❡❞ ✜rst ♦r❞❡r✲t❡r♠s

❲❡ ❛r❡ ❣✐✈❡♥

◮ ❆ t②♣❡ ♦❢ t❡r♠s T ✇✐t❤ ❝♦♥str✉❝t♦rs ❈❦✿

■♥❞✉❝t✐✈❡ T ✿ ❙❡t ✿❂ | ❈✶ : T ✳ ✳ ✳ | ❈❦ : . . . → T . . . → T . . . → T ✳ ✳ ✳

◮ ❆ ❝♦♥❣r✉❡♥❝❡ ≃ : T → T → Pr♦♣

◮ ❋♦r ❡❛❝❤ ❝♦♥str✉❝t♦r ❈❦

∀❛, . . . ①✶, ②✶, ❜, . . . ①✷, ②✷, . . . ❝, ①✶ ≃ ②✶ → ①✷ ≃ ②✷ → ❈❦ ❛ . . . ①✶ ❜ . . . ②✶ ❝ ≃ ❈❦ ❛ . . . ①✷ ❜ . . . ②✷ ❝ s♣❡❝✐✜❝ ❧❛✇s✱ ❡✳❣✳ ①② ❈✷ ① ❈✶ ② ❈✷ ② ①

❲❡ ✇❛♥t t♦ r❡❛s♦♥ ♦♥ ✉♣ t♦

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SLIDE 12

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

  • ❡♥❡r❛❧ s❡tt✐♥❣✿ q✉♦t✐❡♥t❡❞ ✜rst ♦r❞❡r✲t❡r♠s

❲❡ ❛r❡ ❣✐✈❡♥

◮ ❆ t②♣❡ ♦❢ t❡r♠s T ✇✐t❤ ❝♦♥str✉❝t♦rs ❈❦✿

■♥❞✉❝t✐✈❡ T ✿ ❙❡t ✿❂ | ❈✶ : T ✳ ✳ ✳ | ❈❦ : . . . → T . . . → T . . . → T ✳ ✳ ✳

◮ ❆ ❝♦♥❣r✉❡♥❝❡ ≃ : T → T → Pr♦♣

◮ ❋♦r ❡❛❝❤ ❝♦♥str✉❝t♦r ❈❦

∀❛, . . . ①✶, ②✶, ❜, . . . ①✷, ②✷, . . . ❝, ①✶ ≃ ②✶ → ①✷ ≃ ②✷ → ❈❦ ❛ . . . ①✶ ❜ . . . ②✶ ❝ ≃ ❈❦ ❛ . . . ①✷ ❜ . . . ②✷ ❝

◮ s♣❡❝✐✜❝ ❧❛✇s✱ ❡✳❣✳ ∀①②, ❈✷ ① ❈✶ ② ≃ ❈✷ ② ①

❲❡ ✇❛♥t t♦ r❡❛s♦♥ ♦♥ ✉♣ t♦

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SLIDE 13

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

  • ❡♥❡r❛❧ s❡tt✐♥❣✿ q✉♦t✐❡♥t❡❞ ✜rst ♦r❞❡r✲t❡r♠s

❲❡ ❛r❡ ❣✐✈❡♥

◮ ❆ t②♣❡ ♦❢ t❡r♠s T ✇✐t❤ ❝♦♥str✉❝t♦rs ❈❦✿

■♥❞✉❝t✐✈❡ T ✿ ❙❡t ✿❂ | ❈✶ : T ✳ ✳ ✳ | ❈❦ : . . . → T . . . → T . . . → T ✳ ✳ ✳

◮ ❆ ❝♦♥❣r✉❡♥❝❡ ≃ : T → T → Pr♦♣

◮ ❋♦r ❡❛❝❤ ❝♦♥str✉❝t♦r ❈❦

∀❛, . . . ①✶, ②✶, ❜, . . . ①✷, ②✷, . . . ❝, ①✶ ≃ ②✶ → ①✷ ≃ ②✷ → ❈❦ ❛ . . . ①✶ ❜ . . . ②✶ ❝ ≃ ❈❦ ❛ . . . ①✷ ❜ . . . ②✷ ❝

◮ s♣❡❝✐✜❝ ❧❛✇s✱ ❡✳❣✳ ∀①②, ❈✷ ① ❈✶ ② ≃ ❈✷ ② ①

❲❡ ✇❛♥t t♦ r❡❛s♦♥ ♦♥ ✉♣ t♦

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SLIDE 14

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

  • ❡♥❡r❛❧ s❡tt✐♥❣✿ q✉♦t✐❡♥t❡❞ ✜rst ♦r❞❡r✲t❡r♠s

❲❡ ❛r❡ ❣✐✈❡♥

◮ ❆ t②♣❡ ♦❢ t❡r♠s T ✇✐t❤ ❝♦♥str✉❝t♦rs ❈❦✿

■♥❞✉❝t✐✈❡ T ✿ ❙❡t ✿❂ | ❈✶ : T ✳ ✳ ✳ | ❈❦ : . . . → T . . . → T . . . → T ✳ ✳ ✳

◮ ❆ ❝♦♥❣r✉❡♥❝❡ ≃ : T → T → Pr♦♣

◮ ❋♦r ❡❛❝❤ ❝♦♥str✉❝t♦r ❈❦

∀❛, . . . ①✶, ②✶, ❜, . . . ①✷, ②✷, . . . ❝, ①✶ ≃ ②✶ → ①✷ ≃ ②✷ → ❈❦ ❛ . . . ①✶ ❜ . . . ②✶ ❝ ≃ ❈❦ ❛ . . . ①✷ ❜ . . . ②✷ ❝

◮ s♣❡❝✐✜❝ ❧❛✇s✱ ❡✳❣✳ ∀①②, ❈✷ ① ❈✶ ② ≃ ❈✷ ② ①

❲❡ ✇❛♥t t♦ r❡❛s♦♥ ♦♥ T ✉♣ t♦ ≃

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SLIDE 15

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❆❧r❡❛❞② ✇❡❧❧✲❦♥♦✇♥ ❡①❛♠♣❧❡s

◮ ✜♥✐t❡ ❜❛❣s r❡♣r❡s❡♥t❡❞ ❜② ✜♥✐t❡ ❧✐sts

❛❧❣❡❜r❛ ♦❢ ❢♦r♠❛❧ ❛r✐t❤♠❡t✐❝ ❡①♣r❡ss✐♦♥s ✐s ❛ss♦❝✐❛t✐✈❡✱ ❝♦♠♠✉t❛t✐✈❡✱ ✵ ✐s ♥❡✉tr❛❧ ✐s ❛ss♦❝✐❛t✐✈❡✱ ❝♦♠♠✉t❛t✐✈❡✱ ✶ ✐s ♥❡✉tr❛❧ ❞✐str✐❜✉t❡s ♦✈❡r ✭♠♦❜✐❧❡✮ ♣r♦❝❡ss ❝❛❧❝✉❧✐✱ ❝❤❡♠✐❝❛❧ ❛❜str❛❝t ♠❛❝❤✐♥❡s ♣❛r❛❧❧❡❧ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❝❤♦✐❝❡ ♦♣❡r❛t♦rs ❛r❡ ❆❈

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SLIDE 16

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❆❧r❡❛❞② ✇❡❧❧✲❦♥♦✇♥ ❡①❛♠♣❧❡s

◮ ✜♥✐t❡ ❜❛❣s r❡♣r❡s❡♥t❡❞ ❜② ✜♥✐t❡ ❧✐sts ◮ ❛❧❣❡❜r❛ ♦❢ ❢♦r♠❛❧ ❛r✐t❤♠❡t✐❝ ❡①♣r❡ss✐♦♥s

✐s ❛ss♦❝✐❛t✐✈❡✱ ❝♦♠♠✉t❛t✐✈❡✱ ✵ ✐s ♥❡✉tr❛❧ ✐s ❛ss♦❝✐❛t✐✈❡✱ ❝♦♠♠✉t❛t✐✈❡✱ ✶ ✐s ♥❡✉tr❛❧ ❞✐str✐❜✉t❡s ♦✈❡r ✭♠♦❜✐❧❡✮ ♣r♦❝❡ss ❝❛❧❝✉❧✐✱ ❝❤❡♠✐❝❛❧ ❛❜str❛❝t ♠❛❝❤✐♥❡s ♣❛r❛❧❧❡❧ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❝❤♦✐❝❡ ♦♣❡r❛t♦rs ❛r❡ ❆❈

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SLIDE 17

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❆❧r❡❛❞② ✇❡❧❧✲❦♥♦✇♥ ❡①❛♠♣❧❡s

◮ ✜♥✐t❡ ❜❛❣s r❡♣r❡s❡♥t❡❞ ❜② ✜♥✐t❡ ❧✐sts ◮ ❛❧❣❡❜r❛ ♦❢ ❢♦r♠❛❧ ❛r✐t❤♠❡t✐❝ ❡①♣r❡ss✐♦♥s

✐s ❛ss♦❝✐❛t✐✈❡✱ ❝♦♠♠✉t❛t✐✈❡✱ ✵ ✐s ♥❡✉tr❛❧ ✐s ❛ss♦❝✐❛t✐✈❡✱ ❝♦♠♠✉t❛t✐✈❡✱ ✶ ✐s ♥❡✉tr❛❧ ❞✐str✐❜✉t❡s ♦✈❡r

◮ ✭♠♦❜✐❧❡✮ ♣r♦❝❡ss ❝❛❧❝✉❧✐✱ ❝❤❡♠✐❝❛❧ ❛❜str❛❝t ♠❛❝❤✐♥❡s

♣❛r❛❧❧❡❧ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❝❤♦✐❝❡ ♦♣❡r❛t♦rs ❛r❡ ❆❈

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SLIDE 18

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❆❧r❡❛❞② ✇❡❧❧✲❦♥♦✇♥ ❡①❛♠♣❧❡s

◮ ✜♥✐t❡ ❜❛❣s r❡♣r❡s❡♥t❡❞ ❜② ✜♥✐t❡ ❧✐sts ◮ ❛❧❣❡❜r❛ ♦❢ ❢♦r♠❛❧ ❛r✐t❤♠❡t✐❝ ❡①♣r❡ss✐♦♥s

+ ✐s ❛ss♦❝✐❛t✐✈❡✱ ❝♦♠♠✉t❛t✐✈❡✱ ✵ ✐s ♥❡✉tr❛❧ × ✐s ❛ss♦❝✐❛t✐✈❡✱ ❝♦♠♠✉t❛t✐✈❡✱ ✶ ✐s ♥❡✉tr❛❧ × ❞✐str✐❜✉t❡s ♦✈❡r +

◮ ✭♠♦❜✐❧❡✮ ♣r♦❝❡ss ❝❛❧❝✉❧✐✱ ❝❤❡♠✐❝❛❧ ❛❜str❛❝t ♠❛❝❤✐♥❡s

♣❛r❛❧❧❡❧ ❝♦♠♣♦s✐t✐♦♥ ❛♥❞ ❝❤♦✐❝❡ ♦♣❡r❛t♦rs ❛r❡ ❆❈

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SLIDE 19

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

◗✉♦t✐❡♥ts ✐♥ t②♣❡ t❤❡♦r②

◮ ❍✐❣❤ ❧❡✈❡❧ ❛♣♣r♦❛❝❤ ✿ s❡t♦✐❞s

❊①♣❧✐❝✐t ❛♣♣r♦❛❝❤ ✿

❉❡✜♥❡ ❛ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥ ◆ ♦♥ ❈♦♠♣❛r❡ t❡r♠s ✉s✐♥❣ s②♥t❛❝t✐❝ ❡q✉❛❧✐t② ♦♥ t❤❡✐r ♥♦r♠s ✿ ① ② ✐✛ ◆ ① ◆ ②

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SLIDE 20

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

◗✉♦t✐❡♥ts ✐♥ t②♣❡ t❤❡♦r②

◮ ❍✐❣❤ ❧❡✈❡❧ ❛♣♣r♦❛❝❤ ✿ s❡t♦✐❞s ◮ ❊①♣❧✐❝✐t ❛♣♣r♦❛❝❤ ✿

❉❡✜♥❡ ❛ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥ ◆ ♦♥ ❈♦♠♣❛r❡ t❡r♠s ✉s✐♥❣ s②♥t❛❝t✐❝ ❡q✉❛❧✐t② ♦♥ t❤❡✐r ♥♦r♠s ✿ ① ② ✐✛ ◆ ① ◆ ②

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SLIDE 21

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

◗✉♦t✐❡♥ts ✐♥ t②♣❡ t❤❡♦r②

◮ ❍✐❣❤ ❧❡✈❡❧ ❛♣♣r♦❛❝❤ ✿ s❡t♦✐❞s ◮ ❊①♣❧✐❝✐t ❛♣♣r♦❛❝❤ ✿

◮ ❉❡✜♥❡ ❛ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥ ◆ ♦♥ T

❈♦♠♣❛r❡ t❡r♠s ✉s✐♥❣ s②♥t❛❝t✐❝ ❡q✉❛❧✐t② ♦♥ t❤❡✐r ♥♦r♠s ✿ ① ② ✐✛ ◆ ① ◆ ②

slide-22
SLIDE 22

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

◗✉♦t✐❡♥ts ✐♥ t②♣❡ t❤❡♦r②

◮ ❍✐❣❤ ❧❡✈❡❧ ❛♣♣r♦❛❝❤ ✿ s❡t♦✐❞s ◮ ❊①♣❧✐❝✐t ❛♣♣r♦❛❝❤ ✿

◮ ❉❡✜♥❡ ❛ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥ ◆ ♦♥ T ◮ ❈♦♠♣❛r❡ t❡r♠s ✉s✐♥❣ s②♥t❛❝t✐❝ ❡q✉❛❧✐t② ♦♥ t❤❡✐r ♥♦r♠s ✿

① ≃ ② ✐✛ ◆ ① = ◆ ②

slide-23
SLIDE 23

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-24
SLIDE 24

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ♥❡❡❞ ♠♦r❡

❘❡❛s♦♥✐♥❣ ♦♥ s✉❝❤ s②st❡♠s ✐♥✈♦❧✈❡s

◮ ❝♦♠♣❛r✐♥❣ t❡r♠s ✉♣ t♦ ❆❈ ✰ ✐♥✈♦❧✉t✐✈✐t② ♦❢ ⊕✿

❈♦♠♠✉t❛t✐✈✐t②✿ ① ⊕ ② ≃ ② ⊕ ① ❆ss♦❝✐❛t✐✈✐t②✿ (① ⊕ ②) ⊕ ③ ≃ ① ⊕ (② ⊕ ③) ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ⊕ ✵ ≃ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ⊕ ① ≃ ✵ ❛ r❡❧❛t✐♦♥ ❢♦r ♦❝❝✉rr❡♥❝❡✿ ✐❢ ①✱ ② ❛♥❞ ③ ❛r❡ ❞✐✛❡r❡♥t t❡r♠s✱

② ♦❝❝✉rs ✐♥ ① ② ③ ❜✉t ② ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ ① ② ③ ② ① ② ✐❢ ① ② ① t ✐❢ t ① ②✵ ②♥ ❛♥❞ ① ②✐ ❢♦r ❛❧❧ ✐✱ ✵ ✐ ♥

♥♦r♠❛❧✐③❛t✐♦♥ ✐s ♥❡❡❞❡❞✦

slide-25
SLIDE 25

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ♥❡❡❞ ♠♦r❡

❘❡❛s♦♥✐♥❣ ♦♥ s✉❝❤ s②st❡♠s ✐♥✈♦❧✈❡s

◮ ❝♦♠♣❛r✐♥❣ t❡r♠s ✉♣ t♦ ❆❈ ✰ ✐♥✈♦❧✉t✐✈✐t② ♦❢ ⊕✿

❈♦♠♠✉t❛t✐✈✐t②✿ ① ⊕ ② ≃ ② ⊕ ① ❆ss♦❝✐❛t✐✈✐t②✿ (① ⊕ ②) ⊕ ③ ≃ ① ⊕ (② ⊕ ③) ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ⊕ ✵ ≃ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ⊕ ① ≃ ✵

◮ ❛ r❡❧❛t✐♦♥ ❢♦r ♦❝❝✉rr❡♥❝❡✿

✐❢ ①✱ ② ❛♥❞ ③ ❛r❡ ❞✐✛❡r❡♥t t❡r♠s✱

② ♦❝❝✉rs ✐♥ ① ② ③ ❜✉t ② ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ ① ② ③ ② ① ② ✐❢ ① ② ① t ✐❢ t ① ②✵ ②♥ ❛♥❞ ① ②✐ ❢♦r ❛❧❧ ✐✱ ✵ ✐ ♥

♥♦r♠❛❧✐③❛t✐♦♥ ✐s ♥❡❡❞❡❞✦

slide-26
SLIDE 26

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ♥❡❡❞ ♠♦r❡

❘❡❛s♦♥✐♥❣ ♦♥ s✉❝❤ s②st❡♠s ✐♥✈♦❧✈❡s

◮ ❝♦♠♣❛r✐♥❣ t❡r♠s ✉♣ t♦ ❆❈ ✰ ✐♥✈♦❧✉t✐✈✐t② ♦❢ ⊕✿

❈♦♠♠✉t❛t✐✈✐t②✿ ① ⊕ ② ≃ ② ⊕ ① ❆ss♦❝✐❛t✐✈✐t②✿ (① ⊕ ②) ⊕ ③ ≃ ① ⊕ (② ⊕ ③) ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ⊕ ✵ ≃ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ⊕ ① ≃ ✵

◮ ❛ r❡❧❛t✐♦♥ ❢♦r ♦❝❝✉rr❡♥❝❡✿

✐❢ ①✱ ② ❛♥❞ ③ ❛r❡ ❞✐✛❡r❡♥t t❡r♠s✱

◮ ② ♦❝❝✉rs ✐♥ ① ⊕ ② ⊕ ③

❜✉t ② ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ ① ② ③ ② ① ② ✐❢ ① ② ① t ✐❢ t ① ②✵ ②♥ ❛♥❞ ① ②✐ ❢♦r ❛❧❧ ✐✱ ✵ ✐ ♥

♥♦r♠❛❧✐③❛t✐♦♥ ✐s ♥❡❡❞❡❞✦

slide-27
SLIDE 27

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ♥❡❡❞ ♠♦r❡

❘❡❛s♦♥✐♥❣ ♦♥ s✉❝❤ s②st❡♠s ✐♥✈♦❧✈❡s

◮ ❝♦♠♣❛r✐♥❣ t❡r♠s ✉♣ t♦ ❆❈ ✰ ✐♥✈♦❧✉t✐✈✐t② ♦❢ ⊕✿

❈♦♠♠✉t❛t✐✈✐t②✿ ① ⊕ ② ≃ ② ⊕ ① ❆ss♦❝✐❛t✐✈✐t②✿ (① ⊕ ②) ⊕ ③ ≃ ① ⊕ (② ⊕ ③) ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ⊕ ✵ ≃ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ⊕ ① ≃ ✵

◮ ❛ r❡❧❛t✐♦♥ ❢♦r ♦❝❝✉rr❡♥❝❡✿

✐❢ ①✱ ② ❛♥❞ ③ ❛r❡ ❞✐✛❡r❡♥t t❡r♠s✱

◮ ② ♦❝❝✉rs ✐♥ ① ⊕ ② ⊕ ③ ◮ ❜✉t ② ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ ① ⊕ ② ⊕ ③ ⊕ ②

① ② ✐❢ ① ② ① t ✐❢ t ① ②✵ ②♥ ❛♥❞ ① ②✐ ❢♦r ❛❧❧ ✐✱ ✵ ✐ ♥

♥♦r♠❛❧✐③❛t✐♦♥ ✐s ♥❡❡❞❡❞✦

slide-28
SLIDE 28

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ♥❡❡❞ ♠♦r❡

❘❡❛s♦♥✐♥❣ ♦♥ s✉❝❤ s②st❡♠s ✐♥✈♦❧✈❡s

◮ ❝♦♠♣❛r✐♥❣ t❡r♠s ✉♣ t♦ ❆❈ ✰ ✐♥✈♦❧✉t✐✈✐t② ♦❢ ⊕✿

❈♦♠♠✉t❛t✐✈✐t②✿ ① ⊕ ② ≃ ② ⊕ ① ❆ss♦❝✐❛t✐✈✐t②✿ (① ⊕ ②) ⊕ ③ ≃ ① ⊕ (② ⊕ ③) ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ⊕ ✵ ≃ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ⊕ ① ≃ ✵

◮ ❛ r❡❧❛t✐♦♥ ❢♦r ♦❝❝✉rr❡♥❝❡✿

✐❢ ①✱ ② ❛♥❞ ③ ❛r❡ ❞✐✛❡r❡♥t t❡r♠s✱

◮ ② ♦❝❝✉rs ✐♥ ① ⊕ ② ⊕ ③ ◮ ❜✉t ② ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ ① ⊕ ② ⊕ ③ ⊕ ②

① ② ✐❢ ① ② ① t ✐❢ t ① ②✵ ②♥ ❛♥❞ ① ②✐ ❢♦r ❛❧❧ ✐✱ ✵ ✐ ♥

♥♦r♠❛❧✐③❛t✐♦♥ ✐s ♥❡❡❞❡❞✦

slide-29
SLIDE 29

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ♥❡❡❞ ♠♦r❡

❘❡❛s♦♥✐♥❣ ♦♥ s✉❝❤ s②st❡♠s ✐♥✈♦❧✈❡s

◮ ❝♦♠♣❛r✐♥❣ t❡r♠s ✉♣ t♦ ❆❈ ✰ ✐♥✈♦❧✉t✐✈✐t② ♦❢ ⊕✿

❈♦♠♠✉t❛t✐✈✐t②✿ ① ⊕ ② ≃ ② ⊕ ① ❆ss♦❝✐❛t✐✈✐t②✿ (① ⊕ ②) ⊕ ③ ≃ ① ⊕ (② ⊕ ③) ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ⊕ ✵ ≃ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ⊕ ① ≃ ✵

◮ ❛ r❡❧❛t✐♦♥ ❢♦r ♦❝❝✉rr❡♥❝❡✿

✐❢ ①✱ ② ❛♥❞ ③ ❛r❡ ❞✐✛❡r❡♥t t❡r♠s✱

◮ ② ♦❝❝✉rs ✐♥ ① ⊕ ② ⊕ ③ ◮ ❜✉t ② ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ ① ⊕ ② ⊕ ③ ⊕ ②

① ② ✐❢ ① ≃ ② ① t ✐❢ t ≃ ① ⊕ ②✵ . . . ⊕ ②♥ ❛♥❞ ① ≃ ②✐ ❢♦r ❛❧❧ ✐✱ ✵ ≤ ✐ ≤ ♥

♥♦r♠❛❧✐③❛t✐♦♥ ✐s ♥❡❡❞❡❞✦

slide-30
SLIDE 30

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ♥❡❡❞ ♠♦r❡

❘❡❛s♦♥✐♥❣ ♦♥ s✉❝❤ s②st❡♠s ✐♥✈♦❧✈❡s

◮ ❝♦♠♣❛r✐♥❣ t❡r♠s ✉♣ t♦ ❆❈ ✰ ✐♥✈♦❧✉t✐✈✐t② ♦❢ ⊕✿

❈♦♠♠✉t❛t✐✈✐t②✿ ① ⊕ ② ≃ ② ⊕ ① ❆ss♦❝✐❛t✐✈✐t②✿ (① ⊕ ②) ⊕ ③ ≃ ① ⊕ (② ⊕ ③) ◆❡✉tr❛❧ ❡❧❡♠❡♥t✿ ① ⊕ ✵ ≃ ① ■♥✈♦❧✉t✐✈✐t②✿ ① ⊕ ① ≃ ✵

◮ ❛ r❡❧❛t✐♦♥ ❢♦r ♦❝❝✉rr❡♥❝❡✿

✐❢ ①✱ ② ❛♥❞ ③ ❛r❡ ❞✐✛❡r❡♥t t❡r♠s✱

◮ ② ♦❝❝✉rs ✐♥ ① ⊕ ② ⊕ ③ ◮ ❜✉t ② ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ ① ⊕ ② ⊕ ③ ⊕ ②

① ② ✐❢ ① ≃ ② ① t ✐❢ t ≃ ① ⊕ ②✵ . . . ⊕ ②♥ ❛♥❞ ① ≃ ②✐ ❢♦r ❛❧❧ ✐✱ ✵ ≤ ✐ ≤ ♥

→ ♥♦r♠❛❧✐③❛t✐♦♥ ✐s ♥❡❡❞❡❞✦

slide-31
SLIDE 31

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-32
SLIDE 32

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋✐rst ❛tt❡♠♣t✿ r❡✇r✐t❡✱ r❡✇r✐t❡✱ r❡✇r✐t❡. . .

❘❡♣❧❛❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ r❡✇r✐t❡ r✉❧❡s ❈♦♠♠✉t❛t✐✈✐t②✿ ✜♥❞ ❛♥ s✉✐t❛❜❧❡ ✇❡❧❧ ♦r❞❡r✐♥❣ ♦♥ t❡r♠s ❋✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✿ ◆♦t ✈❡r② ❞✐✣❝✉❧t ✕ ✉s❡ ❣❡♥❡r❛❧ r❡❝✉rs✐♦♥ ❏✉st ❜♦r✐♥❣ ■♥ ❛ t②♣❡ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦✱ t❡r♠✐♥❛t✐♦♥ ♣r♦♦❢ ♠❛♥❞❛t♦r② ❛♥❞ ♥♦♥✲tr✐✈✐❛❧✿ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ♦t❤❡r ❛♣♣r♦❛❝❤❡s ✭❧♣♦✱ r♣♦✱ ✮✿ ♦✈❡r❦✐❧❧❄ ❆❈ ♠❛t❝❤✐♥❣✿ ❛ ♥♦♥ tr✐✈✐❛❧ ♠❛tt❡r

slide-33
SLIDE 33

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋✐rst ❛tt❡♠♣t✿ r❡✇r✐t❡✱ r❡✇r✐t❡✱ r❡✇r✐t❡. . .

❘❡♣❧❛❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ r❡✇r✐t❡ r✉❧❡s ❈♦♠♠✉t❛t✐✈✐t②✿ ✜♥❞ ❛♥ s✉✐t❛❜❧❡ ✇❡❧❧ ♦r❞❡r✐♥❣ ♦♥ t❡r♠s ❋✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✿ ◆♦t ✈❡r② ❞✐✣❝✉❧t ✕ ✉s❡ ❣❡♥❡r❛❧ r❡❝✉rs✐♦♥ ❏✉st ❜♦r✐♥❣ ■♥ ❛ t②♣❡ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦✱ t❡r♠✐♥❛t✐♦♥ ♣r♦♦❢ ♠❛♥❞❛t♦r② ❛♥❞ ♥♦♥✲tr✐✈✐❛❧✿ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ♦t❤❡r ❛♣♣r♦❛❝❤❡s ✭❧♣♦✱ r♣♦✱ ✮✿ ♦✈❡r❦✐❧❧❄ ❆❈ ♠❛t❝❤✐♥❣✿ ❛ ♥♦♥ tr✐✈✐❛❧ ♠❛tt❡r

slide-34
SLIDE 34

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋✐rst ❛tt❡♠♣t✿ r❡✇r✐t❡✱ r❡✇r✐t❡✱ r❡✇r✐t❡. . .

❘❡♣❧❛❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ r❡✇r✐t❡ r✉❧❡s ❈♦♠♠✉t❛t✐✈✐t②✿ ✜♥❞ ❛♥ s✉✐t❛❜❧❡ ✇❡❧❧ ♦r❞❡r✐♥❣ ♦♥ t❡r♠s ❋✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✿ ◆♦t ✈❡r② ❞✐✣❝✉❧t ✕ ✉s❡ ❣❡♥❡r❛❧ r❡❝✉rs✐♦♥ ❏✉st ❜♦r✐♥❣ ■♥ ❛ t②♣❡ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦✱ t❡r♠✐♥❛t✐♦♥ ♣r♦♦❢ ♠❛♥❞❛t♦r② ❛♥❞ ♥♦♥✲tr✐✈✐❛❧✿ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ♦t❤❡r ❛♣♣r♦❛❝❤❡s ✭❧♣♦✱ r♣♦✱ ✮✿ ♦✈❡r❦✐❧❧❄ ❆❈ ♠❛t❝❤✐♥❣✿ ❛ ♥♦♥ tr✐✈✐❛❧ ♠❛tt❡r

slide-35
SLIDE 35

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋✐rst ❛tt❡♠♣t✿ r❡✇r✐t❡✱ r❡✇r✐t❡✱ r❡✇r✐t❡. . .

❘❡♣❧❛❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ r❡✇r✐t❡ r✉❧❡s ❈♦♠♠✉t❛t✐✈✐t②✿ ✜♥❞ ❛♥ s✉✐t❛❜❧❡ ✇❡❧❧ ♦r❞❡r✐♥❣ ♦♥ t❡r♠s ❋✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✿

◮ ◆♦t ✈❡r② ❞✐✣❝✉❧t ✕ ✉s❡ ❣❡♥❡r❛❧ r❡❝✉rs✐♦♥

❏✉st ❜♦r✐♥❣ ■♥ ❛ t②♣❡ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦✱ t❡r♠✐♥❛t✐♦♥ ♣r♦♦❢ ♠❛♥❞❛t♦r② ❛♥❞ ♥♦♥✲tr✐✈✐❛❧✿ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ♦t❤❡r ❛♣♣r♦❛❝❤❡s ✭❧♣♦✱ r♣♦✱ ✮✿ ♦✈❡r❦✐❧❧❄ ❆❈ ♠❛t❝❤✐♥❣✿ ❛ ♥♦♥ tr✐✈✐❛❧ ♠❛tt❡r

slide-36
SLIDE 36

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋✐rst ❛tt❡♠♣t✿ r❡✇r✐t❡✱ r❡✇r✐t❡✱ r❡✇r✐t❡. . .

❘❡♣❧❛❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ r❡✇r✐t❡ r✉❧❡s ❈♦♠♠✉t❛t✐✈✐t②✿ ✜♥❞ ❛♥ s✉✐t❛❜❧❡ ✇❡❧❧ ♦r❞❡r✐♥❣ ♦♥ t❡r♠s ❋✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✿

◮ ◆♦t ✈❡r② ❞✐✣❝✉❧t ✕ ✉s❡ ❣❡♥❡r❛❧ r❡❝✉rs✐♦♥ ◮ ❏✉st ❜♦r✐♥❣

■♥ ❛ t②♣❡ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦✱ t❡r♠✐♥❛t✐♦♥ ♣r♦♦❢ ♠❛♥❞❛t♦r② ❛♥❞ ♥♦♥✲tr✐✈✐❛❧✿ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ♦t❤❡r ❛♣♣r♦❛❝❤❡s ✭❧♣♦✱ r♣♦✱ ✮✿ ♦✈❡r❦✐❧❧❄ ❆❈ ♠❛t❝❤✐♥❣✿ ❛ ♥♦♥ tr✐✈✐❛❧ ♠❛tt❡r

slide-37
SLIDE 37

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋✐rst ❛tt❡♠♣t✿ r❡✇r✐t❡✱ r❡✇r✐t❡✱ r❡✇r✐t❡. . .

❘❡♣❧❛❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ r❡✇r✐t❡ r✉❧❡s ❈♦♠♠✉t❛t✐✈✐t②✿ ✜♥❞ ❛♥ s✉✐t❛❜❧❡ ✇❡❧❧ ♦r❞❡r✐♥❣ ♦♥ t❡r♠s ❋✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✿

◮ ◆♦t ✈❡r② ❞✐✣❝✉❧t ✕ ✉s❡ ❣❡♥❡r❛❧ r❡❝✉rs✐♦♥ ◮ ❏✉st ❜♦r✐♥❣

■♥ ❛ t②♣❡ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦✱ t❡r♠✐♥❛t✐♦♥ ♣r♦♦❢ ♠❛♥❞❛t♦r② ❛♥❞ ♥♦♥✲tr✐✈✐❛❧✿ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ♦t❤❡r ❛♣♣r♦❛❝❤❡s ✭❧♣♦✱ r♣♦✱ ✮✿ ♦✈❡r❦✐❧❧❄ ❆❈ ♠❛t❝❤✐♥❣✿ ❛ ♥♦♥ tr✐✈✐❛❧ ♠❛tt❡r

slide-38
SLIDE 38

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋✐rst ❛tt❡♠♣t✿ r❡✇r✐t❡✱ r❡✇r✐t❡✱ r❡✇r✐t❡. . .

❘❡♣❧❛❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ r❡✇r✐t❡ r✉❧❡s ❈♦♠♠✉t❛t✐✈✐t②✿ ✜♥❞ ❛♥ s✉✐t❛❜❧❡ ✇❡❧❧ ♦r❞❡r✐♥❣ ♦♥ t❡r♠s ❋✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✿

◮ ◆♦t ✈❡r② ❞✐✣❝✉❧t ✕ ✉s❡ ❣❡♥❡r❛❧ r❡❝✉rs✐♦♥ ◮ ❏✉st ❜♦r✐♥❣

■♥ ❛ t②♣❡ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦✱ t❡r♠✐♥❛t✐♦♥ ♣r♦♦❢ ♠❛♥❞❛t♦r② ❛♥❞ ♥♦♥✲tr✐✈✐❛❧✿

◮ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣

♦t❤❡r ❛♣♣r♦❛❝❤❡s ✭❧♣♦✱ r♣♦✱ ✮✿ ♦✈❡r❦✐❧❧❄ ❆❈ ♠❛t❝❤✐♥❣✿ ❛ ♥♦♥ tr✐✈✐❛❧ ♠❛tt❡r

slide-39
SLIDE 39

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋✐rst ❛tt❡♠♣t✿ r❡✇r✐t❡✱ r❡✇r✐t❡✱ r❡✇r✐t❡. . .

❘❡♣❧❛❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ r❡✇r✐t❡ r✉❧❡s ❈♦♠♠✉t❛t✐✈✐t②✿ ✜♥❞ ❛♥ s✉✐t❛❜❧❡ ✇❡❧❧ ♦r❞❡r✐♥❣ ♦♥ t❡r♠s ❋✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✿

◮ ◆♦t ✈❡r② ❞✐✣❝✉❧t ✕ ✉s❡ ❣❡♥❡r❛❧ r❡❝✉rs✐♦♥ ◮ ❏✉st ❜♦r✐♥❣

■♥ ❛ t②♣❡ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦✱ t❡r♠✐♥❛t✐♦♥ ♣r♦♦❢ ♠❛♥❞❛t♦r② ❛♥❞ ♥♦♥✲tr✐✈✐❛❧✿

◮ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ◮ ♦t❤❡r ❛♣♣r♦❛❝❤❡s ✭❧♣♦✱ r♣♦✱. . .✮✿ ♦✈❡r❦✐❧❧❄

❆❈ ♠❛t❝❤✐♥❣✿ ❛ ♥♦♥ tr✐✈✐❛❧ ♠❛tt❡r

slide-40
SLIDE 40

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋✐rst ❛tt❡♠♣t✿ r❡✇r✐t❡✱ r❡✇r✐t❡✱ r❡✇r✐t❡. . .

❘❡♣❧❛❝❡ ❡q✉❛t✐♦♥s ✇✐t❤ r❡✇r✐t❡ r✉❧❡s ❈♦♠♠✉t❛t✐✈✐t②✿ ✜♥❞ ❛♥ s✉✐t❛❜❧❡ ✇❡❧❧ ♦r❞❡r✐♥❣ ♦♥ t❡r♠s ❋✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤✿

◮ ◆♦t ✈❡r② ❞✐✣❝✉❧t ✕ ✉s❡ ❣❡♥❡r❛❧ r❡❝✉rs✐♦♥ ◮ ❏✉st ❜♦r✐♥❣

■♥ ❛ t②♣❡ t❤❡♦r❡t✐❝ ❢r❛♠❡✇♦r❦✱ t❡r♠✐♥❛t✐♦♥ ♣r♦♦❢ ♠❛♥❞❛t♦r② ❛♥❞ ♥♦♥✲tr✐✈✐❛❧✿

◮ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❧❡①✐❝♦❣r❛♣❤✐❝ ♦r❞❡r✐♥❣ ◮ ♦t❤❡r ❛♣♣r♦❛❝❤❡s ✭❧♣♦✱ r♣♦✱. . .✮✿ ♦✈❡r❦✐❧❧❄ ◮ ❆❈ ♠❛t❝❤✐♥❣✿ ❛ ♥♦♥ tr✐✈✐❛❧ ♠❛tt❡r

slide-41
SLIDE 41

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

✭❉❡♣❡♥❞❡♥t✮ t②♣❡ t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤

❙t❡♣ ✶ ❈♦♥s✐❞❡r ❛ ♠♦r❡ str✉❝t✉r❡❞ ✈❡rs✐♦♥ ♦❢ t ♣r♦✈✐❞❡ ❛♥ ❛❝❝✉r❛t❡ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ t②♣✐♥❣ t♦ t ❙t❡♣ ✷ ◆♦r♠❛❧✐③❡ ❜② str✉❝t✉r❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥❡✇❧② t②♣❡❞ ✈❡rs✐♦♥ ♦❢ t ❙t❡♣ ✶ ♠❛❦❡s st❡♣ ✷ ❡❛s②✳ ❇❡tt❡r ❢♦r♠✉❧❛t✐♦♥✿ t tr❛♥s❢♦r♠❡❞ ✐♥t♦ t ❚ ❡♥r✐❝❤❡❞ ✈❡rs✐♦♥ ♦❢ ✱ tr✐✈✐❛❧ ❢♦r❣❡t❢✉❧ ♠♦r♣❤✐s♠ ✳ ■♥t❡r❡st✐♥❣ ♣❛rt ❂

slide-42
SLIDE 42

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

✭❉❡♣❡♥❞❡♥t✮ t②♣❡ t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤

❙t❡♣ ✶

◮ ❈♦♥s✐❞❡r ❛ ♠♦r❡ str✉❝t✉r❡❞ ✈❡rs✐♦♥ ♦❢ t

♣r♦✈✐❞❡ ❛♥ ❛❝❝✉r❛t❡ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ t②♣✐♥❣ t♦ t ❙t❡♣ ✷ ◆♦r♠❛❧✐③❡ ❜② str✉❝t✉r❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥❡✇❧② t②♣❡❞ ✈❡rs✐♦♥ ♦❢ t ❙t❡♣ ✶ ♠❛❦❡s st❡♣ ✷ ❡❛s②✳ ❇❡tt❡r ❢♦r♠✉❧❛t✐♦♥✿ t tr❛♥s❢♦r♠❡❞ ✐♥t♦ t ❚ ❡♥r✐❝❤❡❞ ✈❡rs✐♦♥ ♦❢ ✱ tr✐✈✐❛❧ ❢♦r❣❡t❢✉❧ ♠♦r♣❤✐s♠ ✳ ■♥t❡r❡st✐♥❣ ♣❛rt ❂

slide-43
SLIDE 43

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

✭❉❡♣❡♥❞❡♥t✮ t②♣❡ t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤

❙t❡♣ ✶

◮ ❈♦♥s✐❞❡r ❛ ♠♦r❡ str✉❝t✉r❡❞ ✈❡rs✐♦♥ ♦❢ t

♣r♦✈✐❞❡ ❛♥ ❛❝❝✉r❛t❡ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ t②♣✐♥❣ t♦ t ❙t❡♣ ✷ ◆♦r♠❛❧✐③❡ ❜② str✉❝t✉r❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥❡✇❧② t②♣❡❞ ✈❡rs✐♦♥ ♦❢ t ❙t❡♣ ✶ ♠❛❦❡s st❡♣ ✷ ❡❛s②✳ ❇❡tt❡r ❢♦r♠✉❧❛t✐♦♥✿ t tr❛♥s❢♦r♠❡❞ ✐♥t♦ t ❚ ❡♥r✐❝❤❡❞ ✈❡rs✐♦♥ ♦❢ ✱ tr✐✈✐❛❧ ❢♦r❣❡t❢✉❧ ♠♦r♣❤✐s♠ ✳ ■♥t❡r❡st✐♥❣ ♣❛rt ❂

slide-44
SLIDE 44

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

✭❉❡♣❡♥❞❡♥t✮ t②♣❡ t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤

❙t❡♣ ✶

◮ ❈♦♥s✐❞❡r ❛ ♠♦r❡ str✉❝t✉r❡❞ ✈❡rs✐♦♥ ♦❢ t

= ♣r♦✈✐❞❡ ❛♥ ❛❝❝✉r❛t❡ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ t②♣✐♥❣ t♦ t ❙t❡♣ ✷ ◆♦r♠❛❧✐③❡ ❜② str✉❝t✉r❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥❡✇❧② t②♣❡❞ ✈❡rs✐♦♥ ♦❢ t ❙t❡♣ ✶ ♠❛❦❡s st❡♣ ✷ ❡❛s②✳ ❇❡tt❡r ❢♦r♠✉❧❛t✐♦♥✿ t tr❛♥s❢♦r♠❡❞ ✐♥t♦ t ❚ ❡♥r✐❝❤❡❞ ✈❡rs✐♦♥ ♦❢ ✱ tr✐✈✐❛❧ ❢♦r❣❡t❢✉❧ ♠♦r♣❤✐s♠ ✳ ■♥t❡r❡st✐♥❣ ♣❛rt ❂

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SLIDE 45

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

✭❉❡♣❡♥❞❡♥t✮ t②♣❡ t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤

❙t❡♣ ✶

◮ ❈♦♥s✐❞❡r ❛ ♠♦r❡ str✉❝t✉r❡❞ ✈❡rs✐♦♥ ♦❢ t

= ♣r♦✈✐❞❡ ❛♥ ❛❝❝✉r❛t❡ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ t②♣✐♥❣ t♦ t ❙t❡♣ ✷

◮ ◆♦r♠❛❧✐③❡ ❜② str✉❝t✉r❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥❡✇❧② t②♣❡❞

✈❡rs✐♦♥ ♦❢ t ❙t❡♣ ✶ ♠❛❦❡s st❡♣ ✷ ❡❛s②✳ ❇❡tt❡r ❢♦r♠✉❧❛t✐♦♥✿ t tr❛♥s❢♦r♠❡❞ ✐♥t♦ t ❚ ❡♥r✐❝❤❡❞ ✈❡rs✐♦♥ ♦❢ ✱ tr✐✈✐❛❧ ❢♦r❣❡t❢✉❧ ♠♦r♣❤✐s♠ ✳ ■♥t❡r❡st✐♥❣ ♣❛rt ❂

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SLIDE 46

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

✭❉❡♣❡♥❞❡♥t✮ t②♣❡ t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤

❙t❡♣ ✶

◮ ❈♦♥s✐❞❡r ❛ ♠♦r❡ str✉❝t✉r❡❞ ✈❡rs✐♦♥ ♦❢ t

= ♣r♦✈✐❞❡ ❛♥ ❛❝❝✉r❛t❡ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ t②♣✐♥❣ t♦ t ❙t❡♣ ✷

◮ ◆♦r♠❛❧✐③❡ ❜② str✉❝t✉r❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥❡✇❧② t②♣❡❞

✈❡rs✐♦♥ ♦❢ t ❙t❡♣ ✶ ♠❛❦❡s st❡♣ ✷ ❡❛s②✳ ❇❡tt❡r ❢♦r♠✉❧❛t✐♦♥✿ t tr❛♥s❢♦r♠❡❞ ✐♥t♦ t ❚ ❡♥r✐❝❤❡❞ ✈❡rs✐♦♥ ♦❢ ✱ tr✐✈✐❛❧ ❢♦r❣❡t❢✉❧ ♠♦r♣❤✐s♠ ✳ ■♥t❡r❡st✐♥❣ ♣❛rt ❂

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SLIDE 47

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

✭❉❡♣❡♥❞❡♥t✮ t②♣❡ t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤

❙t❡♣ ✶

◮ ❈♦♥s✐❞❡r ❛ ♠♦r❡ str✉❝t✉r❡❞ ✈❡rs✐♦♥ ♦❢ t

= ♣r♦✈✐❞❡ ❛♥ ❛❝❝✉r❛t❡ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ t②♣✐♥❣ t♦ t ❙t❡♣ ✷

◮ ◆♦r♠❛❧✐③❡ ❜② str✉❝t✉r❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥❡✇❧② t②♣❡❞

✈❡rs✐♦♥ ♦❢ t ❙t❡♣ ✶ ♠❛❦❡s st❡♣ ✷ ❡❛s②✳ ❇❡tt❡r ❢♦r♠✉❧❛t✐♦♥✿ t tr❛♥s❢♦r♠❡❞ ✐♥t♦ t ❚ ❡♥r✐❝❤❡❞ ✈❡rs✐♦♥ ♦❢ ✱ tr✐✈✐❛❧ ❢♦r❣❡t❢✉❧ ♠♦r♣❤✐s♠ ✳ ■♥t❡r❡st✐♥❣ ♣❛rt ❂

slide-48
SLIDE 48

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

✭❉❡♣❡♥❞❡♥t✮ t②♣❡ t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤

❙t❡♣ ✶

◮ ❈♦♥s✐❞❡r ❛ ♠♦r❡ str✉❝t✉r❡❞ ✈❡rs✐♦♥ ♦❢ t

= ♣r♦✈✐❞❡ ❛♥ ❛❝❝✉r❛t❡ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ t②♣✐♥❣ t♦ t ❙t❡♣ ✷

◮ ◆♦r♠❛❧✐③❡ ❜② str✉❝t✉r❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥❡✇❧② t②♣❡❞

✈❡rs✐♦♥ ♦❢ t ❙t❡♣ ✶ ♠❛❦❡s st❡♣ ✷ ❡❛s②✳ ❇❡tt❡r ❢♦r♠✉❧❛t✐♦♥✿ t : T tr❛♥s❢♦r♠❡❞ ✐♥t♦ t′ : T ′ ❚ ′ ❡♥r✐❝❤❡❞ ✈❡rs✐♦♥ ♦❢ T ✱ tr✐✈✐❛❧ ❢♦r❣❡t❢✉❧ ♠♦r♣❤✐s♠ T ′ → T ✳ ■♥t❡r❡st✐♥❣ ♣❛rt ❂

slide-49
SLIDE 49

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

✭❉❡♣❡♥❞❡♥t✮ t②♣❡ t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤

❙t❡♣ ✶

◮ ❈♦♥s✐❞❡r ❛ ♠♦r❡ str✉❝t✉r❡❞ ✈❡rs✐♦♥ ♦❢ t

= ♣r♦✈✐❞❡ ❛♥ ❛❝❝✉r❛t❡ ❛♥❞ ✐♥❢♦r♠❛t✐✈❡ t②♣✐♥❣ t♦ t ❙t❡♣ ✷

◮ ◆♦r♠❛❧✐③❡ ❜② str✉❝t✉r❛❧ ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ ♥❡✇❧② t②♣❡❞

✈❡rs✐♦♥ ♦❢ t ❙t❡♣ ✶ ♠❛❦❡s st❡♣ ✷ ❡❛s②✳ ❇❡tt❡r ❢♦r♠✉❧❛t✐♦♥✿ t : T tr❛♥s❢♦r♠❡❞ ✐♥t♦ t′ : T ′ ❚ ′ ❡♥r✐❝❤❡❞ ✈❡rs✐♦♥ ♦❢ T ✱ tr✐✈✐❛❧ ❢♦r❣❡t❢✉❧ ♠♦r♣❤✐s♠ T ′ → T ✳ ■♥t❡r❡st✐♥❣ ♣❛rt ❂ T → T ′

slide-50
SLIDE 50

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

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SLIDE 51

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✉♥❝❤ t✐♠❡✦

❧❛②❡r✐♥❣ ❛ t❡r♠ ❧❛②❡rs ❞♦ ♥♦t ❝♦♠♠✉♥✐❝❛t❡✿ ❡❛❝❤ ❧❛②❡r ♣♦ss❡ss❡s ✐ts ♦✇♥ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥ ✐♥ ♦✉r ❝❛s❡✿ ♥❡❡❞ ✷ ❧❛②❡rs✱ ♣❛st❛ ❛♥❞ s❛✉❝❡ ♥♦r♠❛❧✐③✐♥❣ ♣❛st❛ ✐❞❡♥t✐t② ♥♦r♠❛❧✐③✐♥❣ s❛✉❝❡ r❡❛rr❛♥❣✐♥❣ ✰ r❡♠♦✈✐♥❣ ❞✉♣❧✐❝❛t❡s

slide-52
SLIDE 52

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲❛s❛❣♥❛s r❡✈❡❛❧ t❤❡ tr✉t❤

❧❛②❡r✐♥❣ ❛ t❡r♠ ❧❛②❡rs ❞♦ ♥♦t ❝♦♠♠✉♥✐❝❛t❡✿ ❡❛❝❤ ❧❛②❡r ♣♦ss❡ss❡s ✐ts ♦✇♥ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥ ✐♥ ♦✉r ❝❛s❡✿ ♥❡❡❞ ✷ ❧❛②❡rs✱ ♣❛st❛ ❛♥❞ s❛✉❝❡ ♥♦r♠❛❧✐③✐♥❣ ♣❛st❛ ✐❞❡♥t✐t② ♥♦r♠❛❧✐③✐♥❣ s❛✉❝❡ r❡❛rr❛♥❣✐♥❣ ✰ r❡♠♦✈✐♥❣ ❞✉♣❧✐❝❛t❡s

slide-53
SLIDE 53

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲❛s❛❣♥❛s r❡✈❡❛❧ t❤❡ tr✉t❤

◮ ❧❛②❡r✐♥❣ ❛ t❡r♠

❧❛②❡rs ❞♦ ♥♦t ❝♦♠♠✉♥✐❝❛t❡✿ ❡❛❝❤ ❧❛②❡r ♣♦ss❡ss❡s ✐ts ♦✇♥ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥ ✐♥ ♦✉r ❝❛s❡✿ ♥❡❡❞ ✷ ❧❛②❡rs✱ ♣❛st❛ ❛♥❞ s❛✉❝❡ ♥♦r♠❛❧✐③✐♥❣ ♣❛st❛ ✐❞❡♥t✐t② ♥♦r♠❛❧✐③✐♥❣ s❛✉❝❡ r❡❛rr❛♥❣✐♥❣ ✰ r❡♠♦✈✐♥❣ ❞✉♣❧✐❝❛t❡s

slide-54
SLIDE 54

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲❛s❛❣♥❛s r❡✈❡❛❧ t❤❡ tr✉t❤

◮ ❧❛②❡r✐♥❣ ❛ t❡r♠ ◮ ❧❛②❡rs ❞♦ ♥♦t ❝♦♠♠✉♥✐❝❛t❡✿

❡❛❝❤ ❧❛②❡r ♣♦ss❡ss❡s ✐ts ♦✇♥ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥ ✐♥ ♦✉r ❝❛s❡✿ ♥❡❡❞ ✷ ❧❛②❡rs✱ ♣❛st❛ ❛♥❞ s❛✉❝❡ ♥♦r♠❛❧✐③✐♥❣ ♣❛st❛ ✐❞❡♥t✐t② ♥♦r♠❛❧✐③✐♥❣ s❛✉❝❡ r❡❛rr❛♥❣✐♥❣ ✰ r❡♠♦✈✐♥❣ ❞✉♣❧✐❝❛t❡s

slide-55
SLIDE 55

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲❛s❛❣♥❛s r❡✈❡❛❧ t❤❡ tr✉t❤

◮ ❧❛②❡r✐♥❣ ❛ t❡r♠ ◮ ❧❛②❡rs ❞♦ ♥♦t ❝♦♠♠✉♥✐❝❛t❡✿

❡❛❝❤ ❧❛②❡r ♣♦ss❡ss❡s ✐ts ♦✇♥ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥

◮ ✐♥ ♦✉r ❝❛s❡✿ ♥❡❡❞ ✷ ❧❛②❡rs✱ ♣❛st❛ ❛♥❞ s❛✉❝❡

♥♦r♠❛❧✐③✐♥❣ ♣❛st❛ ✐❞❡♥t✐t② ♥♦r♠❛❧✐③✐♥❣ s❛✉❝❡ r❡❛rr❛♥❣✐♥❣ ✰ r❡♠♦✈✐♥❣ ❞✉♣❧✐❝❛t❡s

slide-56
SLIDE 56

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲❛s❛❣♥❛s r❡✈❡❛❧ t❤❡ tr✉t❤

◮ ❧❛②❡r✐♥❣ ❛ t❡r♠ ◮ ❧❛②❡rs ❞♦ ♥♦t ❝♦♠♠✉♥✐❝❛t❡✿

❡❛❝❤ ❧❛②❡r ♣♦ss❡ss❡s ✐ts ♦✇♥ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥

◮ ✐♥ ♦✉r ❝❛s❡✿ ♥❡❡❞ ✷ ❧❛②❡rs✱ ♣❛st❛ ❛♥❞ s❛✉❝❡ ◮ ♥♦r♠❛❧✐③✐♥❣ ♣❛st❛ = ✐❞❡♥t✐t②

♥♦r♠❛❧✐③✐♥❣ s❛✉❝❡ r❡❛rr❛♥❣✐♥❣ ✰ r❡♠♦✈✐♥❣ ❞✉♣❧✐❝❛t❡s

slide-57
SLIDE 57

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲❛s❛❣♥❛s r❡✈❡❛❧ t❤❡ tr✉t❤

◮ ❧❛②❡r✐♥❣ ❛ t❡r♠ ◮ ❧❛②❡rs ❞♦ ♥♦t ❝♦♠♠✉♥✐❝❛t❡✿

❡❛❝❤ ❧❛②❡r ♣♦ss❡ss❡s ✐ts ♦✇♥ ♥♦r♠❛❧✐③❛t✐♦♥ ❢✉♥❝t✐♦♥

◮ ✐♥ ♦✉r ❝❛s❡✿ ♥❡❡❞ ✷ ❧❛②❡rs✱ ♣❛st❛ ❛♥❞ s❛✉❝❡ ◮ ♥♦r♠❛❧✐③✐♥❣ ♣❛st❛ = ✐❞❡♥t✐t② ◮ ♥♦r♠❛❧✐③✐♥❣ s❛✉❝❡ = r❡❛rr❛♥❣✐♥❣ ✰ r❡♠♦✈✐♥❣ ❞✉♣❧✐❝❛t❡s

slide-58
SLIDE 58

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-59
SLIDE 59

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

T ❛s ❛ ❧❛s❛❣♥❛

■♥❞✉❝t✐✈❡ ✿ ❙❡t ✿❂ ❩❡r♦✿ P❈✿ ♣✉❜❧✐❝ ❝♦♥st ❙❈✿ s❡❝r❡t ❝♦♥st ❊✿ ❳♦r✿ ❍❛s❤✿ ✳

slide-60
SLIDE 60

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

T ❛s ❛ ❧❛s❛❣♥❛

■♥❞✉❝t✐✈❡ T ✿ ❙❡t ✿❂ | ❩❡r♦✿ T | P❈✿ ♣✉❜❧✐❝ ❝♦♥st → T | ❙❈✿ s❡❝r❡t ❝♦♥st → T | ❊✿ T → T → T | ❳♦r✿ T → T → T | ❍❛s❤✿ T → T → T ✳

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SLIDE 61

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

T ❛s ❛ ❧❛s❛❣♥❛

■♥❞✉❝t✐✈❡ T ✿ ❙❡t ✿❂ | ❩❡r♦✿ T | P❈✿ ♣✉❜❧✐❝ ❝♦♥st → T | ❙❈✿ s❡❝r❡t ❝♦♥st → T | ❊✿ T → T → T | ❳♦r✿ T → T → T | ❍❛s❤✿ T → T → T ✳

E H P ⊕ ⊕ S ⊕ E P S ⊕ S

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SLIDE 62

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

T ❛s ❛ ❧❛s❛❣♥❛

■♥❞✉❝t✐✈❡ T ✿ ❙❡t ✿❂ | ❩❡r♦✿ T | P❈✿ ♣✉❜❧✐❝ ❝♦♥st → T | ❙❈✿ s❡❝r❡t ❝♦♥st → T | ❊✿ T → T → T | ❳♦r✿ T → T → T | ❍❛s❤✿ T → T → T ✳

E H P ⊕ ⊕ S ⊕ E P S ⊕ S

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SLIDE 63

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-64
SLIDE 64

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❉❡❝♦♠♣♦s✐♥❣ T

❱❛r✐❛❜❧❡ ❆ ✿ ❙❡t✳ ■♥❞✉❝t✐✈❡ T①✿❙❡t ✿❂ | ❳ ❩❡r♦ ✿ T① | ❳ ❳♦r ✿ T① → T① → T① ❳ ♥s ✿ ❆

①

■♥❞✉❝t✐✈❡ T♥✿ ❙❡t ✿❂ | ◆❳ P❈ ✿ ♣✉❜❧✐❝ ❝♦♥st → T♥ | ◆❳ ❙❈ ✿ s❡❝r❡t ❝♦♥st → T♥ | ◆❳ ❊ ✿ T♥ → T♥ → T♥ | ◆❳ ❍❛s❤ ✿ T♥ → T♥ → T♥ ◆❳ s✉♠ ✿ ❆

♥

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SLIDE 65

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❉❡❝♦♠♣♦s✐♥❣ T

❱❛r✐❛❜❧❡ ❆ ✿ ❙❡t✳ ■♥❞✉❝t✐✈❡ T①✿❙❡t ✿❂ | ❳ ❩❡r♦ ✿ T① | ❳ ❳♦r ✿ T① → T① → T① | ❳ ♥s ✿ ❆ → T① ■♥❞✉❝t✐✈❡ T♥✿ ❙❡t ✿❂ | ◆❳ P❈ ✿ ♣✉❜❧✐❝ ❝♦♥st → T♥ | ◆❳ ❙❈ ✿ s❡❝r❡t ❝♦♥st → T♥ | ◆❳ ❊ ✿ T♥ → T♥ → T♥ | ◆❳ ❍❛s❤ ✿ T♥ → T♥ → T♥ | ◆❳ s✉♠ ✿ ❆ → T♥

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SLIDE 66

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-67
SLIDE 67

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠

❙t❡♣ ✶ ❚r❛♥s❧❛t❡ ❛ t❡r♠ t ✐♥t♦ t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♠❛♣♣✐♥❣ ✵ ❳ ❩❡r♦✱ ❳♦r ❳ ❳♦r✱ P❈ ◆❳ P❈✱ ❡t❝✳ ❚❤❡ t②♣✐♥❣ ♦❢ t ✐s

① ♥ ① ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳ ❙t❡♣ ✷ ❆ t②♣❡ ✐s s♦rt❛❜❧❡ ✐❢ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❞❡❝✐❞❛❜❧❡ ❡q✉❛❧✐t② ❛♥❞ ❛ ❞❡❝✐❞❛❜❧❡ t♦t❛❧ ♦r❞❡r✐♥❣✳ ■❢ ❆ ✐s s♦rt❛❜❧❡✱ t❤❡♥

♥ ❆ ✐s s♦rt❛❜❧❡ ❛s ✇❡❧❧❀

t❤❡ ♠✉❧t✐s❡t ♦❢ ❆✲❧❡❛✈❡s ♦❢ ❛

① ❆ ✲t❡r♠ ❝❛♥ ❜❡ s♦rt❡❞

✭❛♥❞ r❡♠♦✈❡❞ ✇❤❡♥ ♣♦ss✐❜❧❡✮ ✐♥t♦ ❛ ❧✐st❀ ❧✐st ❆ ✐s s♦rt❛❜❧❡✳

slide-68
SLIDE 68

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠

❙t❡♣ ✶ ❚r❛♥s❧❛t❡ ❛ t❡r♠ t ✐♥t♦ t′ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♠❛♣♣✐♥❣ ✵ → ❳ ❩❡r♦✱ ❳♦r → ❳ ❳♦r✱ P❈ → ◆❳ P❈✱ ❡t❝✳ ❚❤❡ t②♣✐♥❣ ♦❢ t ✐s

① ♥ ① ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳ ❙t❡♣ ✷ ❆ t②♣❡ ✐s s♦rt❛❜❧❡ ✐❢ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❞❡❝✐❞❛❜❧❡ ❡q✉❛❧✐t② ❛♥❞ ❛ ❞❡❝✐❞❛❜❧❡ t♦t❛❧ ♦r❞❡r✐♥❣✳ ■❢ ❆ ✐s s♦rt❛❜❧❡✱ t❤❡♥

♥ ❆ ✐s s♦rt❛❜❧❡ ❛s ✇❡❧❧❀

t❤❡ ♠✉❧t✐s❡t ♦❢ ❆✲❧❡❛✈❡s ♦❢ ❛

① ❆ ✲t❡r♠ ❝❛♥ ❜❡ s♦rt❡❞

✭❛♥❞ r❡♠♦✈❡❞ ✇❤❡♥ ♣♦ss✐❜❧❡✮ ✐♥t♦ ❛ ❧✐st❀ ❧✐st ❆ ✐s s♦rt❛❜❧❡✳

slide-69
SLIDE 69

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠

❙t❡♣ ✶ ❚r❛♥s❧❛t❡ ❛ t❡r♠ t ✐♥t♦ t′ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♠❛♣♣✐♥❣ ✵ → ❳ ❩❡r♦✱ ❳♦r → ❳ ❳♦r✱ P❈ → ◆❳ P❈✱ ❡t❝✳ ❚❤❡ t②♣✐♥❣ ♦❢ t ✐s

① ♥ ① ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳ ❙t❡♣ ✷ ❆ t②♣❡ ✐s s♦rt❛❜❧❡ ✐❢ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❞❡❝✐❞❛❜❧❡ ❡q✉❛❧✐t② ❛♥❞ ❛ ❞❡❝✐❞❛❜❧❡ t♦t❛❧ ♦r❞❡r✐♥❣✳ ■❢ ❆ ✐s s♦rt❛❜❧❡✱ t❤❡♥

♥ ❆ ✐s s♦rt❛❜❧❡ ❛s ✇❡❧❧❀

t❤❡ ♠✉❧t✐s❡t ♦❢ ❆✲❧❡❛✈❡s ♦❢ ❛

① ❆ ✲t❡r♠ ❝❛♥ ❜❡ s♦rt❡❞

✭❛♥❞ r❡♠♦✈❡❞ ✇❤❡♥ ♣♦ss✐❜❧❡✮ ✐♥t♦ ❛ ❧✐st❀ ❧✐st ❆ ✐s s♦rt❛❜❧❡✳

slide-70
SLIDE 70

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠

❙t❡♣ ✶ ❚r❛♥s❧❛t❡ ❛ t❡r♠ t ✐♥t♦ t′ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♠❛♣♣✐♥❣ ✵ → ❳ ❩❡r♦✱ ❳♦r → ❳ ❳♦r✱ P❈ → ◆❳ P❈✱ ❡t❝✳ ❚❤❡ t②♣✐♥❣ ♦❢ t ✐s

① ♥ ① ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳ ❙t❡♣ ✷ ❆ t②♣❡ ✐s s♦rt❛❜❧❡ ✐❢ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❞❡❝✐❞❛❜❧❡ ❡q✉❛❧✐t② ❛♥❞ ❛ ❞❡❝✐❞❛❜❧❡ t♦t❛❧ ♦r❞❡r✐♥❣✳ ■❢ ❆ ✐s s♦rt❛❜❧❡✱ t❤❡♥

◮ T♥(❆) ✐s s♦rt❛❜❧❡ ❛s ✇❡❧❧❀

t❤❡ ♠✉❧t✐s❡t ♦❢ ❆✲❧❡❛✈❡s ♦❢ ❛

① ❆ ✲t❡r♠ ❝❛♥ ❜❡ s♦rt❡❞

✭❛♥❞ r❡♠♦✈❡❞ ✇❤❡♥ ♣♦ss✐❜❧❡✮ ✐♥t♦ ❛ ❧✐st❀ ❧✐st ❆ ✐s s♦rt❛❜❧❡✳

slide-71
SLIDE 71

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠

❙t❡♣ ✶ ❚r❛♥s❧❛t❡ ❛ t❡r♠ t ✐♥t♦ t′ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♠❛♣♣✐♥❣ ✵ → ❳ ❩❡r♦✱ ❳♦r → ❳ ❳♦r✱ P❈ → ◆❳ P❈✱ ❡t❝✳ ❚❤❡ t②♣✐♥❣ ♦❢ t ✐s

① ♥ ① ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳ ❙t❡♣ ✷ ❆ t②♣❡ ✐s s♦rt❛❜❧❡ ✐❢ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❞❡❝✐❞❛❜❧❡ ❡q✉❛❧✐t② ❛♥❞ ❛ ❞❡❝✐❞❛❜❧❡ t♦t❛❧ ♦r❞❡r✐♥❣✳ ■❢ ❆ ✐s s♦rt❛❜❧❡✱ t❤❡♥

◮ T♥(❆) ✐s s♦rt❛❜❧❡ ❛s ✇❡❧❧❀ ◮ t❤❡ ♠✉❧t✐s❡t ♦❢ ❆✲❧❡❛✈❡s ♦❢ ❛ T①(❆)✲t❡r♠ ❝❛♥ ❜❡ s♦rt❡❞

✭❛♥❞ r❡♠♦✈❡❞ ✇❤❡♥ ♣♦ss✐❜❧❡✮ ✐♥t♦ ❛ ❧✐st❀ ❧✐st ❆ ✐s s♦rt❛❜❧❡✳

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SLIDE 72

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠

❙t❡♣ ✶ ❚r❛♥s❧❛t❡ ❛ t❡r♠ t ✐♥t♦ t′ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♠❛♣♣✐♥❣ ✵ → ❳ ❩❡r♦✱ ❳♦r → ❳ ❳♦r✱ P❈ → ◆❳ P❈✱ ❡t❝✳ ❚❤❡ t②♣✐♥❣ ♦❢ t ✐s

① ♥ ① ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳ ❙t❡♣ ✷ ❆ t②♣❡ ✐s s♦rt❛❜❧❡ ✐❢ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❞❡❝✐❞❛❜❧❡ ❡q✉❛❧✐t② ❛♥❞ ❛ ❞❡❝✐❞❛❜❧❡ t♦t❛❧ ♦r❞❡r✐♥❣✳ ■❢ ❆ ✐s s♦rt❛❜❧❡✱ t❤❡♥

◮ T♥(❆) ✐s s♦rt❛❜❧❡ ❛s ✇❡❧❧❀ ◮ t❤❡ ♠✉❧t✐s❡t ♦❢ ❆✲❧❡❛✈❡s ♦❢ ❛ T①(❆)✲t❡r♠ ❝❛♥ ❜❡ s♦rt❡❞

✭❛♥❞ r❡♠♦✈❡❞ ✇❤❡♥ ♣♦ss✐❜❧❡✮ ✐♥t♦ ❛ ❧✐st❀

◮ ❧✐st(❆) ✐s s♦rt❛❜❧❡✳

slide-73
SLIDE 73

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠

❙t❡♣ ✶ ❚r❛♥s❧❛t❡ ❛ t❡r♠ t ✐♥t♦ t′ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♠❛♣♣✐♥❣ ✵ → ❳ ❩❡r♦✱ ❳♦r → ❳ ❳♦r✱ P❈ → ◆❳ P❈✱ ❡t❝✳ ❚❤❡ t②♣✐♥❣ ♦❢ t′ ✐s T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳ ❙t❡♣ ✷ ❆ t②♣❡ ✐s s♦rt❛❜❧❡ ✐❢ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❞❡❝✐❞❛❜❧❡ ❡q✉❛❧✐t② ❛♥❞ ❛ ❞❡❝✐❞❛❜❧❡ t♦t❛❧ ♦r❞❡r✐♥❣✳ ■❢ ❆ ✐s s♦rt❛❜❧❡✱ t❤❡♥

◮ T♥(❆) ✐s s♦rt❛❜❧❡ ❛s ✇❡❧❧❀ ◮ t❤❡ ♠✉❧t✐s❡t ♦❢ ❆✲❧❡❛✈❡s ♦❢ ❛ T①(❆)✲t❡r♠ ❝❛♥ ❜❡ s♦rt❡❞

✭❛♥❞ r❡♠♦✈❡❞ ✇❤❡♥ ♣♦ss✐❜❧❡✮ ✐♥t♦ ❛ ❧✐st❀

◮ ❧✐st(❆) ✐s s♦rt❛❜❧❡✳

slide-74
SLIDE 74

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠

❙t❡♣ ✶ ❚r❛♥s❧❛t❡ ❛ t❡r♠ t ✐♥t♦ t′ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♠❛♣♣✐♥❣ ✵ → ❳ ❩❡r♦✱ ❳♦r → ❳ ❳♦r✱ P❈ → ◆❳ P❈✱ ❡t❝✳ ❚❤❡ t②♣✐♥❣ ♦❢ t′ ✐s T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳ ❙t❡♣ ✷ ❆ t②♣❡ ✐s s♦rt❛❜❧❡ ✐❢ ✐t ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ ❛ ❞❡❝✐❞❛❜❧❡ ❡q✉❛❧✐t② ❛♥❞ ❛ ❞❡❝✐❞❛❜❧❡ t♦t❛❧ ♦r❞❡r✐♥❣✳ ■❢ ❆ ✐s s♦rt❛❜❧❡✱ t❤❡♥

◮ T♥(❆) ✐s s♦rt❛❜❧❡ ❛s ✇❡❧❧❀ ◮ t❤❡ ♠✉❧t✐s❡t ♦❢ ❆✲❧❡❛✈❡s ♦❢ ❛ T①(❆)✲t❡r♠ ❝❛♥ ❜❡ s♦rt❡❞

✭❛♥❞ r❡♠♦✈❡❞ ✇❤❡♥ ♣♦ss✐❜❧❡✮ ✐♥t♦ ❛ ❧✐st❀

◮ ❧✐st(❆) ✐s s♦rt❛❜❧❡✳

slide-75
SLIDE 75

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-76
SLIDE 76

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳ ❲❤❛t ✐s ❦❄ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡ r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛① ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ ♥ ♦r ♥ ♠ ♥❡❡❞ t♦ ❧✐❢t

① ♥ ❛♥❞ ① ♠ t♦ ① ♠❛① ♥ ♠

▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢ ♠ ♥ ♠

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-77
SLIDE 77

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄

❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡ r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛① ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ ♥ ♦r ♥ ♠ ♥❡❡❞ t♦ ❧✐❢t

① ♥ ❛♥❞ ① ♠ t♦ ① ♠❛① ♥ ♠

▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢ ♠ ♥ ♠

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-78
SLIDE 78

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛① ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ ♥ ♦r ♥ ♠ ♥❡❡❞ t♦ ❧✐❢t

① ♥ ❛♥❞ ① ♠ t♦ ① ♠❛① ♥ ♠

▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢ ♠ ♥ ♠

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-79
SLIDE 79

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳

E H P ⊕ ⊕ S ⊕ E P S ⊕ S

❚❛❦❡ t❤❡ ♠❛① ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ ♥ ♦r ♥ ♠ ♥❡❡❞ t♦ ❧✐❢t

① ♥ ❛♥❞ ① ♠ t♦ ① ♠❛① ♥ ♠

▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢ ♠ ♥ ♠

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-80
SLIDE 80

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛① ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ ♥ ♦r ♥ ♠ ♥❡❡❞ t♦ ❧✐❢t

① ♥ ❛♥❞ ① ♠ t♦ ① ♠❛① ♥ ♠

▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢ ♠ ♥ ♠

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-81
SLIDE 81

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛①

◮ ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ ♥ ♦r ♥ ♠ ♥❡❡❞ t♦ ❧✐❢t

① ♥ ❛♥❞ ① ♠ t♦ ① ♠❛① ♥ ♠

▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢ ♠ ♥ ♠

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-82
SLIDE 82

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛①

◮ ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

◮ ✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠

❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ ♥ ♦r ♥ ♠ ♥❡❡❞ t♦ ❧✐❢t

① ♥ ❛♥❞ ① ♠ t♦ ① ♠❛① ♥ ♠

▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢ ♠ ♥ ♠

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-83
SLIDE 83

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛①

◮ ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

◮ ✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ◮ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ − ♥ ♦r ♥ − ♠

♥❡❡❞ t♦ ❧✐❢t

① ♥ ❛♥❞ ① ♠ t♦ ① ♠❛① ♥ ♠

▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢ ♠ ♥ ♠

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-84
SLIDE 84

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛①

◮ ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

◮ ✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ◮ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ − ♥ ♦r ♥ − ♠ ◮ ♥❡❡❞ t♦ ❧✐❢t L① ♥ ❛♥❞ L① ♠ t♦ L① (♠❛① ♥ ♠)

▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢ ♠ ♥ ♠

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-85
SLIDE 85

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛①

◮ ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

◮ ✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ◮ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ − ♥ ♦r ♥ − ♠ ◮ ♥❡❡❞ t♦ ❧✐❢t L① ♥ ❛♥❞ L① ♠ t♦ L① (♠❛① ♥ ♠)

◮ ▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢

= = ♠ + (♥ − ♠)

❧✐❢t①

① ❦ ① ❦

❞ ✱ ❧✐❢t♥

♥ ❦ ♥ ❦

❞ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-86
SLIDE 86

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛①

◮ ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

◮ ✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ◮ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ − ♥ ♦r ♥ − ♠ ◮ ♥❡❡❞ t♦ ❧✐❢t L① ♥ ❛♥❞ L① ♠ t♦ L① (♠❛① ♥ ♠)

◮ ▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢

= = ♠ + (♥ − ♠)

◮ ❧✐❢t① : L① ❦ → L① (❦ + ❞)✱ ❧✐❢t♥ : L♥ ❦ → L♥ (❦ + ❞)

◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-87
SLIDE 87

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▲✐❢t✐♥❣ ❧❛s❛❣♥❛

L① ❦

❞❡❢

= = T①(T♥(T①(. . . (∅))))

  • ❦ ❧❛②❡rs

❢♦r ❦ ❧❛r❣❡ ❡♥♦✉❣❤✳

◮ ❲❤❛t ✐s ❦❄ ◮ ❚❤❡ ♥✉♠❜❡r ♦❢ ❧❛②❡rs ♦♥ t❤❡ ❧❡❢t s✉❜t❡r♠ ❛♥❞ ♦♥ t❤❡

r✐❣❤t s✉❜t❡r♠ ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❣❡♥❡r❛❧✳ ❚❛❦❡ t❤❡ ♠❛①

◮ ❙t❛♥❞❛r❞ s♦❧✉t✐♦♥✿ ④❧❡ ♥ ♠⑥ ✰ ④❧❡ ♠ ♥⑥

◮ ✐♥t❡r❛❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ ❧❛r❣❡ ♣r♦♦❢ t❡r♠ ◮ ❤❡❛✈② ❡♥❝♦❞✐♥❣ ♦❢ ♠ − ♥ ♦r ♥ − ♠ ◮ ♥❡❡❞ t♦ ❧✐❢t L① ♥ ❛♥❞ L① ♠ t♦ L① (♠❛① ♥ ♠)

◮ ▲✐❣❤t✇❡✐❣❤t ❛♣♣r♦❛❝❤✿ ♠❛① ♥ ♠ ❞❡❢

= = ♠ + (♥ − ♠)

◮ ❧✐❢t① : L① ❦ → L① (❦ + ❞)✱ ❧✐❢t♥ : L♥ ❦ → L♥ (❦ + ❞) ◮ ◆♦ ♥❡❡❞ t♦ ♣r♦♦❢ t❤❛t ♠❛① ✐s t❤❡ ♠❛①✳

slide-88
SLIDE 88

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-89
SLIDE 89

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

■♥t❡r♥❛❧✐③✐♥❣ ❛❧t❡r♥❛t✐♦♥

❲❡❧❧ ❞❡s✐❣♥❡❞ t②♣❡s ❤❡❧♣ ✉s t♦ ❞❡s✐❣♥ ♣r♦❣r❛♠s ▼❛♥② ❢✉♥❝t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ❜② ♠✉t✉❛❧ ✐♥❞✉❝t✐♦♥✱ ❡✳❣✳ ❧✐❢t① ❛♥❞ ❧✐❢t♥ ❈♦♥tr♦❧ t❤❡♠ ✉s✐♥❣ ❛❧t❡r♥❛t✐♥❣ ♥❛t✉r❛❧ ♥✉♠❜❡rs ■♥❞✉❝t✐✈❡ ❛❧t❡✈❡♥✿ ❙❡t ✿❂ ✵❡✿ ❛❧t❡✈❡♥ ❙♦

❡✿ ❛❧t♦❞❞

❛❧t❡✈❡♥ ✇✐t❤ ❛❧t♦❞❞✿ ❙❡t ✿❂ ❙❡

♦✿ ❛❧t❡✈❡♥

❛❧t♦❞❞

slide-90
SLIDE 90

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

■♥t❡r♥❛❧✐③✐♥❣ ❛❧t❡r♥❛t✐♦♥

❲❡❧❧ ❞❡s✐❣♥❡❞ t②♣❡s ❤❡❧♣ ✉s t♦ ❞❡s✐❣♥ ♣r♦❣r❛♠s ▼❛♥② ❢✉♥❝t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ❜② ♠✉t✉❛❧ ✐♥❞✉❝t✐♦♥✱ ❡✳❣✳ ❧✐❢t① ❛♥❞ ❧✐❢t♥ ❈♦♥tr♦❧ t❤❡♠ ✉s✐♥❣ ❛❧t❡r♥❛t✐♥❣ ♥❛t✉r❛❧ ♥✉♠❜❡rs ■♥❞✉❝t✐✈❡ ❛❧t❡✈❡♥✿ ❙❡t ✿❂ ✵❡✿ ❛❧t❡✈❡♥ ❙♦

❡✿ ❛❧t♦❞❞

❛❧t❡✈❡♥ ✇✐t❤ ❛❧t♦❞❞✿ ❙❡t ✿❂ ❙❡

♦✿ ❛❧t❡✈❡♥

❛❧t♦❞❞

slide-91
SLIDE 91

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

■♥t❡r♥❛❧✐③✐♥❣ ❛❧t❡r♥❛t✐♦♥

❲❡❧❧ ❞❡s✐❣♥❡❞ t②♣❡s ❤❡❧♣ ✉s t♦ ❞❡s✐❣♥ ♣r♦❣r❛♠s ▼❛♥② ❢✉♥❝t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ❜② ♠✉t✉❛❧ ✐♥❞✉❝t✐♦♥✱ ❡✳❣✳ ❧✐❢t① ❛♥❞ ❧✐❢t♥ ❈♦♥tr♦❧ t❤❡♠ ✉s✐♥❣ ❛❧t❡r♥❛t✐♥❣ ♥❛t✉r❛❧ ♥✉♠❜❡rs ■♥❞✉❝t✐✈❡ ❛❧t❡✈❡♥✿ ❙❡t ✿❂ ✵❡✿ ❛❧t❡✈❡♥ ❙♦

❡✿ ❛❧t♦❞❞

❛❧t❡✈❡♥ ✇✐t❤ ❛❧t♦❞❞✿ ❙❡t ✿❂ ❙❡

♦✿ ❛❧t❡✈❡♥

❛❧t♦❞❞

slide-92
SLIDE 92

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

■♥t❡r♥❛❧✐③✐♥❣ ❛❧t❡r♥❛t✐♦♥

❲❡❧❧ ❞❡s✐❣♥❡❞ t②♣❡s ❤❡❧♣ ✉s t♦ ❞❡s✐❣♥ ♣r♦❣r❛♠s ▼❛♥② ❢✉♥❝t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ❜② ♠✉t✉❛❧ ✐♥❞✉❝t✐♦♥✱ ❡✳❣✳ ❧✐❢t① ❛♥❞ ❧✐❢t♥ ❈♦♥tr♦❧ t❤❡♠ ✉s✐♥❣ ❛❧t❡r♥❛t✐♥❣ ♥❛t✉r❛❧ ♥✉♠❜❡rs ■♥❞✉❝t✐✈❡ ❛❧t❡✈❡♥✿ ❙❡t ✿❂ ✵❡✿ ❛❧t❡✈❡♥ ❙♦

❡✿ ❛❧t♦❞❞

❛❧t❡✈❡♥ ✇✐t❤ ❛❧t♦❞❞✿ ❙❡t ✿❂ ❙❡

♦✿ ❛❧t❡✈❡♥

❛❧t♦❞❞

slide-93
SLIDE 93

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

■♥t❡r♥❛❧✐③✐♥❣ ❛❧t❡r♥❛t✐♦♥

❲❡❧❧ ❞❡s✐❣♥❡❞ t②♣❡s ❤❡❧♣ ✉s t♦ ❞❡s✐❣♥ ♣r♦❣r❛♠s ▼❛♥② ❢✉♥❝t✐♦♥s ❛r❡ ❞❡✜♥❡❞ ❜② ♠✉t✉❛❧ ✐♥❞✉❝t✐♦♥✱ ❡✳❣✳ ❧✐❢t① ❛♥❞ ❧✐❢t♥ ❈♦♥tr♦❧ t❤❡♠ ✉s✐♥❣ ❛❧t❡r♥❛t✐♥❣ ♥❛t✉r❛❧ ♥✉♠❜❡rs ■♥❞✉❝t✐✈❡ ❛❧t❡✈❡♥✿ ❙❡t ✿❂ | ✵❡✿ ❛❧t❡✈❡♥ | ❙♦→❡✿ ❛❧t♦❞❞ → ❛❧t❡✈❡♥ ✇✐t❤ ❛❧t♦❞❞✿ ❙❡t ✿❂ | ❙❡→♦✿ ❛❧t❡✈❡♥ → ❛❧t♦❞❞

slide-94
SLIDE 94

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-95
SLIDE 95

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s

■♥❞✉❝t✐✈❡

①✿

❙❡t ✿❂ ❳ ❩❡r♦ ✿

①

❳ ♥s ✿ ❆

①

❳ ❳♦r ✿

① ① ①

■♥❞✉❝t✐✈❡

♥✿

❙❡t ✿❂ ◆❳ P❈ ✿ ♣✉❜❧✐❝ ❝♦♥st

♥

◆❳ ❙❈ ✿ s❡❝r❡t ❝♦♥st

♥

◆❳ s✉♠ ✿ ❆

♥

◆❳ ❊ ✿

♥ ♥ ♥

◆❳ ❍❛s❤ ✿

♥ ♥ ♥

❳ ♥s ✭◆❳ s✉♠ ✭ ❳ ♥s ✭◆❳ s✉♠ ✭✳ ✳ ✳ ✮✮✮✮

slide-96
SLIDE 96

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s

■♥❞✉❝t✐✈❡ T①✿ ❙❡t ✿❂ | ❳ ❩❡r♦ ✿ T① | ❳ ♥s ✿ ❆ → T① | ❳ ❳♦r ✿ T① → T① → T① ■♥❞✉❝t✐✈❡ T♥✿ ❙❡t ✿❂ | ◆❳ P❈ ✿ ♣✉❜❧✐❝ ❝♦♥st → T♥ | ◆❳ ❙❈ ✿ s❡❝r❡t ❝♦♥st → T♥ | ◆❳ s✉♠ ✿ ❆ → T♥ | ◆❳ ❊ ✿ T♥ → T♥ → T♥ | ◆❳ ❍❛s❤ ✿ T♥ → T♥ → T♥ ❳ ♥s ✭◆❳ s✉♠ ✭ ❳ ♥s ✭◆❳ s✉♠ ✭✳ ✳ ✳ ✮✮✮✮

slide-97
SLIDE 97

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s

■♥❞✉❝t✐✈❡ T①✿ ❙❡t ✿❂ | ❳ ❩❡r♦ ✿ T① | ❳ ♥s ✿ ❆ → T① | ❳ ❳♦r ✿ T① → T① → T① ■♥❞✉❝t✐✈❡ T♥✿ ❙❡t ✿❂ | ◆❳ P❈ ✿ ♣✉❜❧✐❝ ❝♦♥st → T♥ | ◆❳ ❙❈ ✿ s❡❝r❡t ❝♦♥st → T♥ | ◆❳ s✉♠ ✿ ❆ → T♥ | ◆❳ ❊ ✿ T♥ → T♥ → T♥ | ◆❳ ❍❛s❤ ✿ T♥ → T♥ → T♥ ❳ ♥s ✭◆❳ s✉♠ ✭ ❳ ♥s ✭◆❳ s✉♠ ✭✳ ✳ ✳ ✮✮✮✮

slide-98
SLIDE 98

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s

■♥❞✉❝t✐✈❡ T①✿ ❜♦♦❧ → ❙❡t ✿❂ | ❳ ❩❡r♦ ✿ ∀ ❜✱ T① ❜ | ❳ ♥s ✿ ∀ ❜✱ ■s tr✉❡ ❜ → ❆ → T① ❜ | ❳ ❳♦r ✿ ∀ ❜✱ T① tr✉❡ → T① tr✉❡ → T① ❜ ■♥❞✉❝t✐✈❡ T♥✿ ❜♦♦❧ → ❙❡t ✿❂ | ◆❳ P❈ ✿ ∀ ❜✱ ♣✉❜❧✐❝ ❝♦♥st → T♥ ❜ | ◆❳ ❙❈ ✿ ∀ ❜✱ s❡❝r❡t ❝♦♥st → T♥ ❜ | ◆❳ s✉♠ ✿ ∀ ❜✱ ■s tr✉❡ ❜ → ❆ → T♥ ❜ | ◆❳ ❊ ✿ ∀ ❜✱ T♥ tr✉❡ → T♥ tr✉❡ → T♥ ❜ | ◆❳ ❍❛s❤ ✿ ∀ ❜✱ T♥ tr✉❡ → T♥ tr✉❡ → T♥ ❜ ❳ ♥s ✭◆❳ s✉♠ ✭ ❳ ♥s ✭◆❳ s✉♠ ✭✳ ✳ ✳ ✮✮✮✮

slide-99
SLIDE 99

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-100
SLIDE 100

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▼✉t✉❛❧ ✐♥❞✉❝t✐♦♥

◮ Pr❡❢❡r ✜①♣♦✐♥ts✿ ❜✉✐❧t✲✐♥ ❝♦♠♣✉t❛t✐♦♥✱ ♥♦ ✐♥✈❡rs✐♦♥

❯s❡ ♠❛♣ ❝♦♠❜✐♥❛t♦rs ▼❛♥② ✶✵ ❧✐♥❡s ❞❡✜♥✐t✐♦♥s✱ ❛❧♠♦st ♥♦ t❤❡♦r❡♠ ❋✐①♣♦✐♥t ❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷ ④str✉❝t ❡✶⑥ ✿

① ❡✶ ① ✭❡✶ ✰ ❡✷✮ ✿❂

♠❛t❝❤ ❡✶ r❡t✉r♥

① ❡✶ ① ✭❡✶ ✰ ❡✷✮ ✇✐t❤

✵❡ ❢✉♥ ❡♠♣ ♠❛t❝❤ ❡♠♣ ✇✐t❤ ❡♥❞ ❙♦

❡ ♦✶

♠❛♣① ✭❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷✮ ❢❛❧s❡ ❡♥❞ ✇✐t❤ ❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷ ④str✉❝t ♦✶⑥ ✿

♥ ♦✶ ♥ ✭♦✶ ✰ ❡✷✮ ✿❂

♠❛t❝❤ ♦✶ r❡t✉r♥

♥ ♦✶ ♥ ✭♦✶ ✰ ❡✷✮ ✇✐t❤

❙❡

♦ ❡✶

♠❛♣♥ ✭❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷✮ ❢❛❧s❡ ❡♥❞✳

slide-101
SLIDE 101

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▼✉t✉❛❧ ✐♥❞✉❝t✐♦♥

◮ Pr❡❢❡r ✜①♣♦✐♥ts✿ ❜✉✐❧t✲✐♥ ❝♦♠♣✉t❛t✐♦♥✱ ♥♦ ✐♥✈❡rs✐♦♥ ◮ ❯s❡ ♠❛♣ ❝♦♠❜✐♥❛t♦rs

▼❛♥② ✶✵ ❧✐♥❡s ❞❡✜♥✐t✐♦♥s✱ ❛❧♠♦st ♥♦ t❤❡♦r❡♠ ❋✐①♣♦✐♥t ❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷ ④str✉❝t ❡✶⑥ ✿

① ❡✶ ① ✭❡✶ ✰ ❡✷✮ ✿❂

♠❛t❝❤ ❡✶ r❡t✉r♥

① ❡✶ ① ✭❡✶ ✰ ❡✷✮ ✇✐t❤

✵❡ ❢✉♥ ❡♠♣ ♠❛t❝❤ ❡♠♣ ✇✐t❤ ❡♥❞ ❙♦

❡ ♦✶

♠❛♣① ✭❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷✮ ❢❛❧s❡ ❡♥❞ ✇✐t❤ ❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷ ④str✉❝t ♦✶⑥ ✿

♥ ♦✶ ♥ ✭♦✶ ✰ ❡✷✮ ✿❂

♠❛t❝❤ ♦✶ r❡t✉r♥

♥ ♦✶ ♥ ✭♦✶ ✰ ❡✷✮ ✇✐t❤

❙❡

♦ ❡✶

♠❛♣♥ ✭❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷✮ ❢❛❧s❡ ❡♥❞✳

slide-102
SLIDE 102

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▼✉t✉❛❧ ✐♥❞✉❝t✐♦♥

◮ Pr❡❢❡r ✜①♣♦✐♥ts✿ ❜✉✐❧t✲✐♥ ❝♦♠♣✉t❛t✐♦♥✱ ♥♦ ✐♥✈❡rs✐♦♥ ◮ ❯s❡ ♠❛♣ ❝♦♠❜✐♥❛t♦rs

▼❛♥② ✶✵ ❧✐♥❡s ❞❡✜♥✐t✐♦♥s✱ ❛❧♠♦st ♥♦ t❤❡♦r❡♠ ❋✐①♣♦✐♥t ❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷ ④str✉❝t ❡✶⑥ ✿

① ❡✶ ① ✭❡✶ ✰ ❡✷✮ ✿❂

♠❛t❝❤ ❡✶ r❡t✉r♥

① ❡✶ ① ✭❡✶ ✰ ❡✷✮ ✇✐t❤

✵❡ ❢✉♥ ❡♠♣ ♠❛t❝❤ ❡♠♣ ✇✐t❤ ❡♥❞ ❙♦

❡ ♦✶

♠❛♣① ✭❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷✮ ❢❛❧s❡ ❡♥❞ ✇✐t❤ ❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷ ④str✉❝t ♦✶⑥ ✿

♥ ♦✶ ♥ ✭♦✶ ✰ ❡✷✮ ✿❂

♠❛t❝❤ ♦✶ r❡t✉r♥

♥ ♦✶ ♥ ✭♦✶ ✰ ❡✷✮ ✇✐t❤

❙❡

♦ ❡✶

♠❛♣♥ ✭❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷✮ ❢❛❧s❡ ❡♥❞✳

slide-103
SLIDE 103

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▼✉t✉❛❧ ✐♥❞✉❝t✐♦♥

◮ Pr❡❢❡r ✜①♣♦✐♥ts✿ ❜✉✐❧t✲✐♥ ❝♦♠♣✉t❛t✐♦♥✱ ♥♦ ✐♥✈❡rs✐♦♥ ◮ ❯s❡ ♠❛♣ ❝♦♠❜✐♥❛t♦rs

▼❛♥② ✶✵ ❧✐♥❡s ❞❡✜♥✐t✐♦♥s✱ ❛❧♠♦st ♥♦ t❤❡♦r❡♠ ❋✐①♣♦✐♥t ❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷ ④str✉❝t ❡✶⑥ ✿

① ❡✶ ① ✭❡✶ ✰ ❡✷✮ ✿❂

♠❛t❝❤ ❡✶ r❡t✉r♥

① ❡✶ ① ✭❡✶ ✰ ❡✷✮ ✇✐t❤

✵❡ ❢✉♥ ❡♠♣ ♠❛t❝❤ ❡♠♣ ✇✐t❤ ❡♥❞ ❙♦

❡ ♦✶

♠❛♣① ✭❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷✮ ❢❛❧s❡ ❡♥❞ ✇✐t❤ ❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷ ④str✉❝t ♦✶⑥ ✿

♥ ♦✶ ♥ ✭♦✶ ✰ ❡✷✮ ✿❂

♠❛t❝❤ ♦✶ r❡t✉r♥

♥ ♦✶ ♥ ✭♦✶ ✰ ❡✷✮ ✇✐t❤

❙❡

♦ ❡✶

♠❛♣♥ ✭❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷✮ ❢❛❧s❡ ❡♥❞✳

slide-104
SLIDE 104

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

▼✉t✉❛❧ ✐♥❞✉❝t✐♦♥

◮ Pr❡❢❡r ✜①♣♦✐♥ts✿ ❜✉✐❧t✲✐♥ ❝♦♠♣✉t❛t✐♦♥✱ ♥♦ ✐♥✈❡rs✐♦♥ ◮ ❯s❡ ♠❛♣ ❝♦♠❜✐♥❛t♦rs

▼❛♥② ✶✵ ❧✐♥❡s ❞❡✜♥✐t✐♦♥s✱ ❛❧♠♦st ♥♦ t❤❡♦r❡♠ ❋✐①♣♦✐♥t ❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷ ④str✉❝t ❡✶⑥ ✿ L① ❡✶ → L① ✭❡✶ ✰ ❡✷✮ ✿❂ ♠❛t❝❤ ❡✶ r❡t✉r♥ L① ❡✶ → L① ✭❡✶ ✰ ❡✷✮ ✇✐t❤ | ✵❡ ⇒ ❢✉♥ ❡♠♣ ⇒ ♠❛t❝❤ ❡♠♣ ✇✐t❤ ❡♥❞ | ❙♦→❡ ♦✶ ⇒ ♠❛♣① ✭❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷✮ ❢❛❧s❡ ❡♥❞ ✇✐t❤ ❧✐❢t ❧❛s❛❣♥❛ ♥ ♦✶ ❡✷ ④str✉❝t ♦✶⑥ ✿ L♥ ♦✶ → L♥ ✭♦✶ ✰ ❡✷✮ ✿❂ ♠❛t❝❤ ♦✶ r❡t✉r♥ L♥ ♦✶ → L♥ ✭♦✶ ✰ ❡✷✮ ✇✐t❤ | ❙❡→♦ ❡✶ ⇒ ♠❛♣♥ ✭❧✐❢t ❧❛s❛❣♥❛ ① ❡✶ ❡✷✮ ❢❛❧s❡ ❡♥❞✳

slide-105
SLIDE 105

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❖✉t❧✐♥❡

▼♦t✐✈❛t✐♦♥ ❚❤❡ ❝❛s❡ ♦❢ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦❣r❛♣❤✐❝ s②st❡♠s ❙♦❧✈✐♥❣ str❛t❡❣✐❡s ❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮ ❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐❢②✐♥❣ ❛♥❞ ♥♦r♠❛❧✐③✐♥❣ ❛ t❡r♠ ■ss✉❡s ▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋♦r❜✐❞ ❢❛❦❡ ✐♥❝❧✉s✐♦♥s ❋✐①♣♦✐♥ts ❈♦♥✈❡rs✐♦♥ r✉❧❡ ❈♦♥❝❧✉s✐♦♥

slide-106
SLIDE 106

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥✈❡rs✐♦♥ r✉❧❡

❯s❡❞ ❡✈❡r②✇❤❡r❡ ❉❡✜♥✐t✐♦♥ ❜✐♥ ①♦r ✭❜✐♥ ✿ ❆ ❜✱

① ❆ tr✉❡ ① ❆ tr✉❡ ① ❆ ❜✮ ♦✶ ♦✷ ❜

✭❧✶ ✿ ❧❛s❛❣♥❛ ❝❛♥❞ ① ♦✶ tr✉❡✮ ✭❧✷ ✿ ❧❛s❛❣♥❛ ❝❛♥❞ ① ♦✷ tr✉❡✮ ✿ ❧❛s❛❣♥❛ ❝❛♥❞ ① ✭♠❛① ♦♦ ♦✶ ♦✷✮ ❜ ✿❂ ❜✐♥ ✭

♥ ✭♠❛① ♦♦ ♦✶ ♦✷✮✮ ❜

✭❧✐❢t ❧❛s❛❣♥❛ ❝❛♥❞ ① tr✉❡ ♦✶ ✭♦✷ ✲ ♦✶✮ ❧✶✮ ✭❝♦❡r❝❡ ♠❛① ❝♦♠♠ ✭❧✐❢t ❧❛s❛❣♥❛ ❝❛♥❞ ① tr✉❡ ♦✷ ✭♦✶ ✲ ♦✷✮ ❧✷✮✮✳

slide-107
SLIDE 107

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥✈❡rs✐♦♥ r✉❧❡

❯s❡❞ ❡✈❡r②✇❤❡r❡ ❉❡✜♥✐t✐♦♥ ❜✐♥ ①♦r ✭❜✐♥ ✿ ❆ ❜✱

① ❆ tr✉❡ ① ❆ tr✉❡ ① ❆ ❜✮ ♦✶ ♦✷ ❜

✭❧✶ ✿ ❧❛s❛❣♥❛ ❝❛♥❞ ① ♦✶ tr✉❡✮ ✭❧✷ ✿ ❧❛s❛❣♥❛ ❝❛♥❞ ① ♦✷ tr✉❡✮ ✿ ❧❛s❛❣♥❛ ❝❛♥❞ ① ✭♠❛① ♦♦ ♦✶ ♦✷✮ ❜ ✿❂ ❜✐♥ ✭

♥ ✭♠❛① ♦♦ ♦✶ ♦✷✮✮ ❜

✭❧✐❢t ❧❛s❛❣♥❛ ❝❛♥❞ ① tr✉❡ ♦✶ ✭♦✷ ✲ ♦✶✮ ❧✶✮ ✭❝♦❡r❝❡ ♠❛① ❝♦♠♠ ✭❧✐❢t ❧❛s❛❣♥❛ ❝❛♥❞ ① tr✉❡ ♦✷ ✭♦✶ ✲ ♦✷✮ ❧✷✮✮✳

slide-108
SLIDE 108

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥✈❡rs✐♦♥ r✉❧❡

❯s❡❞ ❡✈❡r②✇❤❡r❡ ❉❡✜♥✐t✐♦♥ ❜✐♥ ①♦r ✭❜✐♥ ✿ ∀ ❆ ❜✱ T① ❆ tr✉❡ → T① ❆ tr✉❡ → T① ❆ ❜✮ ♦✶ ♦✷ ❜ ✭❧✶ ✿ ❧❛s❛❣♥❛ ❝❛♥❞ ① ♦✶ tr✉❡✮ ✭❧✷ ✿ ❧❛s❛❣♥❛ ❝❛♥❞ ① ♦✷ tr✉❡✮ ✿ ❧❛s❛❣♥❛ ❝❛♥❞ ① ✭♠❛① ♦♦ ♦✶ ♦✷✮ ❜ ✿❂ ❜✐♥ ✭L♥ ✭♠❛① ♦♦ ♦✶ ♦✷✮✮ ❜ ✭❧✐❢t ❧❛s❛❣♥❛ ❝❛♥❞ ① tr✉❡ ♦✶ ✭♦✷ ✲ ♦✶✮ ❧✶✮ ✭❝♦❡r❝❡ ♠❛① ❝♦♠♠ ✭❧✐❢t ❧❛s❛❣♥❛ ❝❛♥❞ ① tr✉❡ ♦✷ ✭♦✶ ✲ ♦✷✮ ❧✷✮✮✳

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SLIDE 109

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❚②♣❡ t❤❡♦r② ✐s ✢❡①✐❜❧❡

◮ P♦❧②♠♦r♣❤✐s♠

▼✉t✉❛❧❧② ✐♥❞✉❝t✐✈❡ t②♣❡s ❉❡♣❡♥❞❡♥t t②♣❡s ❈♦♥✈❡rs✐♦♥ r✉❧❡ ◆♦ ❏▼❊◗ ❋✉t✉r❡ ✇♦r❦ ❙t✉❞② ✈❛r✐❛♥ts✱ ❝♦♠♣❛r❡ ❛♥❞ ❝❤♦♦s❡ t❤❡ ❜❡st ❛♣♣❧✐❝❛t✐♦♥ t♦ ❈❈❆

slide-110
SLIDE 110

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❚②♣❡ t❤❡♦r② ✐s ✢❡①✐❜❧❡

◮ P♦❧②♠♦r♣❤✐s♠ ◮ ▼✉t✉❛❧❧② ✐♥❞✉❝t✐✈❡ t②♣❡s

❉❡♣❡♥❞❡♥t t②♣❡s ❈♦♥✈❡rs✐♦♥ r✉❧❡ ◆♦ ❏▼❊◗ ❋✉t✉r❡ ✇♦r❦ ❙t✉❞② ✈❛r✐❛♥ts✱ ❝♦♠♣❛r❡ ❛♥❞ ❝❤♦♦s❡ t❤❡ ❜❡st ❛♣♣❧✐❝❛t✐♦♥ t♦ ❈❈❆

slide-111
SLIDE 111

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❚②♣❡ t❤❡♦r② ✐s ✢❡①✐❜❧❡

◮ P♦❧②♠♦r♣❤✐s♠ ◮ ▼✉t✉❛❧❧② ✐♥❞✉❝t✐✈❡ t②♣❡s ◮ ❉❡♣❡♥❞❡♥t t②♣❡s

❈♦♥✈❡rs✐♦♥ r✉❧❡ ◆♦ ❏▼❊◗ ❋✉t✉r❡ ✇♦r❦ ❙t✉❞② ✈❛r✐❛♥ts✱ ❝♦♠♣❛r❡ ❛♥❞ ❝❤♦♦s❡ t❤❡ ❜❡st ❛♣♣❧✐❝❛t✐♦♥ t♦ ❈❈❆

slide-112
SLIDE 112

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❚②♣❡ t❤❡♦r② ✐s ✢❡①✐❜❧❡

◮ P♦❧②♠♦r♣❤✐s♠ ◮ ▼✉t✉❛❧❧② ✐♥❞✉❝t✐✈❡ t②♣❡s ◮ ❉❡♣❡♥❞❡♥t t②♣❡s ◮ ❈♦♥✈❡rs✐♦♥ r✉❧❡

◆♦ ❏▼❊◗ ❋✉t✉r❡ ✇♦r❦ ❙t✉❞② ✈❛r✐❛♥ts✱ ❝♦♠♣❛r❡ ❛♥❞ ❝❤♦♦s❡ t❤❡ ❜❡st ❛♣♣❧✐❝❛t✐♦♥ t♦ ❈❈❆

slide-113
SLIDE 113

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❚②♣❡ t❤❡♦r② ✐s ✢❡①✐❜❧❡

◮ P♦❧②♠♦r♣❤✐s♠ ◮ ▼✉t✉❛❧❧② ✐♥❞✉❝t✐✈❡ t②♣❡s ◮ ❉❡♣❡♥❞❡♥t t②♣❡s ◮ ❈♦♥✈❡rs✐♦♥ r✉❧❡ ◮ ◆♦ ❏▼❊◗

❋✉t✉r❡ ✇♦r❦ ❙t✉❞② ✈❛r✐❛♥ts✱ ❝♦♠♣❛r❡ ❛♥❞ ❝❤♦♦s❡ t❤❡ ❜❡st ❛♣♣❧✐❝❛t✐♦♥ t♦ ❈❈❆

slide-114
SLIDE 114

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❚②♣❡ t❤❡♦r② ✐s ✢❡①✐❜❧❡

◮ P♦❧②♠♦r♣❤✐s♠ ◮ ▼✉t✉❛❧❧② ✐♥❞✉❝t✐✈❡ t②♣❡s ◮ ❉❡♣❡♥❞❡♥t t②♣❡s ◮ ❈♦♥✈❡rs✐♦♥ r✉❧❡ ◮ ◆♦ ❏▼❊◗

❋✉t✉r❡ ✇♦r❦ ❙t✉❞② ✈❛r✐❛♥ts✱ ❝♦♠♣❛r❡ ❛♥❞ ❝❤♦♦s❡ t❤❡ ❜❡st ❛♣♣❧✐❝❛t✐♦♥ t♦ ❈❈❆

slide-115
SLIDE 115

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❚②♣❡ t❤❡♦r② ✐s ✢❡①✐❜❧❡

◮ P♦❧②♠♦r♣❤✐s♠ ◮ ▼✉t✉❛❧❧② ✐♥❞✉❝t✐✈❡ t②♣❡s ◮ ❉❡♣❡♥❞❡♥t t②♣❡s ◮ ❈♦♥✈❡rs✐♦♥ r✉❧❡ ◮ ◆♦ ❏▼❊◗

❋✉t✉r❡ ✇♦r❦

◮ ❙t✉❞② ✈❛r✐❛♥ts✱ ❝♦♠♣❛r❡ ❛♥❞ ❝❤♦♦s❡ t❤❡ ❜❡st

❛♣♣❧✐❝❛t✐♦♥ t♦ ❈❈❆

slide-116
SLIDE 116

Pr♦✈✐♥❣ t❡r♠✐♥❛t✐♦♥ ✉s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s ❏✳✲❋✳ ▼♦♥✐♥✱ ❏✳ ❈♦✉r❛♥t ▼♦t✐✈❛t✐♦♥

❈r②♣t♦✳ s②st✳ ❙t❛t❡ ♦❢ t❤❡ ❛rt ❇❛❝❦ t♦ ❝r②♣t♦ ❙♦❧✈✐♥❣ str❛t❡❣✐❡s

❙♦❧✉t✐♦♥ ✭✐♥t✉✐t✐✈❡✮

❇❛s✐❝ ✐❞❡❛ ❆♥❛❧②s❡ ♦❢ T ❉❡❝♦♠♣♦s✐♥❣ T ❙tr❛t✐✜❝❛t✐♦♥

■ss✉❡s

▲✐❢t✐♥❣ ❆❧t❡r♥❛t✐♦♥ ❋❛❦❡ ✐♥❝❧ ❋✐①♣♦✐♥ts ❈♦♥✈ r✉❧❡

❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❚②♣❡ t❤❡♦r② ✐s ✢❡①✐❜❧❡

◮ P♦❧②♠♦r♣❤✐s♠ ◮ ▼✉t✉❛❧❧② ✐♥❞✉❝t✐✈❡ t②♣❡s ◮ ❉❡♣❡♥❞❡♥t t②♣❡s ◮ ❈♦♥✈❡rs✐♦♥ r✉❧❡ ◮ ◆♦ ❏▼❊◗

❋✉t✉r❡ ✇♦r❦

◮ ❙t✉❞② ✈❛r✐❛♥ts✱ ❝♦♠♣❛r❡ ❛♥❞ ❝❤♦♦s❡ t❤❡ ❜❡st ◮ ❛♣♣❧✐❝❛t✐♦♥ t♦ ❈❈❆