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Dec 10, 2022 36 likes •270 views

r Pr trts r r r r Pr trts r rt

SLIDE 1

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s

SLIDE 2

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❙✉♠♠❛r②

❚❤❡♦r❡♠✲♣r♦✈✐♥❣ ❛♥❞ s❡❛r❝❤ ❋❛✐r ❞❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s ❬❆♠❜r♦s✐✉s✲❏♦❤❛♥♥ ✼✳✺❪

SLIDE 3

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❖❜❥❡❝t✐✈❡ ♦❢ ❚❤❡♦r❡♠ Pr♦✈✐♥❣

Pr♦✈❡ ✈❛❧✐❞✐t②

✐✈❡♥ ❛ s❡t ♦❢ ❛ss✉♠♣t✐♦♥ ❍
✐✈❡♥ ❛ ❝♦♥❥❡❝t✉r❡ ψ

Pr♦✈❡ ✇❤❡t❤❡r ❍ | = ψ✱ ✐✳❡✳ ♣r♦✈❡ ❍ ∪ {¬ψ} ✉♥s❛t✐s✜❛❜❧❡

SLIDE 4

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❆✉t♦♠❛t❡❞ ❚❤❡♦r❡♠ Pr♦✈✐♥❣

❇✉✐❧❞ ❝♦♠♣✉t❡r ♣r♦❣r❛♠s t❤❛t ♣r♦✈❡ ✈❛❧✐❞✐t② ❍ ❛♥❞ ψ ✇r✐tt❡♥ ✐♥ ❛ ❢♦r♠❛❧ ❧❛♥❣✉❛❣❡✱ ❡✳❣✳ ❋❖▲ ❉❡❞✉❝t✐♦♥ ♦❢ ❢r♦♠ ❍ ∪ {¬ψ} ❆ ❞❡❞✉❝t✐♦♥ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ st❛t❡♠❡♥ts ✐♥ t❤❡ ❢♦r♠❛❧ ❧❛♥❣✉❛❣❡ ✭❡✳❣✳ ❋❖▲ ❢♦r♠✉❧❛s✮ ❧♦❣✐❝❛❧❧② ❝♦♥♥❡❝t❡❞ ❜② ✐♥❢❡r❡♥❝❡ r✉❧❡s

SLIDE 5

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

■♥❢❡r❡♥❝❡ ❘✉❧❡

■♥❢❡r❡♥❝❡ ❘✉❧❡ ❢ : ψ✶, · · · , ψ♥ ψ ❢ ✐♥❢❡r❡♥❝❡ r✉❧❡ ψ✶, · · · , ψ♥ ♣r❡♠✐s❡s ψ ❝♦♥s❡q✉❡♥❝❡ ■♥❢❡r❡♥❝❡ s②st❡♠✿ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✐♥❢❡r❡♥❝❡ r✉❧❡s ❊①❛♠♣❧❡ ✭❇✐♥❛r② ❘❡s♦❧✉t✐♦♥✮ ■♥❢❡r❡♥❝❡ r✉❧❡ ▲✶ ∨ ❈ ▲✷ ∨ ❉ (❈ ∨ ❉)σ ▲✶σ = ¬▲✷σ σ ▼♦st ●❡♥❡r❛❧ ❯♥✐✜❡r

SLIDE 6

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

■♥❢❡r❡♥❝❡ ❙②st❡♠ ■

❊①❛♠♣❧❡ ✭■♥❢❡r❡♥❝❡ ❙②st❡♠✮ ❇✐♥❛r② r❡s♦❧✉t✐♦♥ ❋❛❝t♦r✐♥❣ ❚❛✉t♦❧♦❣② ❡❧✐♠✐♥❛t✐♦♥ ❙✉❜s✉♠♣t✐♦♥ ❡❧✐♠✐♥❛t✐♦♥

SLIDE 7

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❡♥❡r❛❧ ♣r♦♣❡rt✐❡s ♦❢ ■♥❢❡r❡♥❝❡ ❙②st❡♠s

❈♦rr❡❝t♥❡ss ❛♥❞ ❈♦♠♣❧❡t❡♥❡ss ❈♦rr❡❝t❡♥❡ss ♦❢ ✐♥❢❡r❡♥❝❡ r✉❧❡s

❝♦♥s❡q✉❡♥❝❡s ❛r❡ ❧♦❣✐❝❛❧ ❝♦♥s❡q✉❡♥❝❡s ♦❢ ♣r❡♠✐s❡s✿ ψ✶, · · · , ψ♥ | = ψ

❈♦♠♣❧❡t❡♥❡ss

✐❢ ❍ | = ψ t❤❡r❡ ✐s ❛ ❞❡❞✉❝t✐♦♥ ♦❢ ψ ❢r♦♠ ❍

❘❡❢✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡t❡♥❡ss

t❤❡r❡ ✐s ❛ ❞❡❞✉❝t✐♦♥ ♦❢ ✐❢ ❍ ∪ {ψ} ✐s ✉♥s❛t✐s✜❛❜❧❡

SLIDE 8

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❛s ❛ ❙❡❛r❝❤ Pr♦❜❧❡♠

❙t❛t❡s ❛♥❞ Pr♦❞✉❝t✐♦♥ ❘✉❧❡s ❈♦♠♣❧❡t❡♥❡ss ♠❡❛♥s ✐❢ t❤❡r❡ ✐s ❛ ♣r♦♦✈❡ ✇❡ ✇✐❧❧ ✜♥❞ ✐t✱ ✇❡ st✐❧❧ ❞♦ ♥♦t ❦♥♦✇ ❤♦✇ ❲❡ ❝❛♥ s❡❡ t❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛s ❛ s❡❛r❝❤ ♣r♦❜❧❡♠ ❙t❛t❡s✿ s❡ts ♦❢ ♣♦ss✐❜❧❡ ❢♦r♠✉❧❛s ✭❡✳❣✳ s❡ts ♦❢ ❝❧❛✉s❡s✮ ❚r❛♥s❢♦r♠❛t✐♦♥ ♦r ♣r♦❞✉❝t✐♦♥ r✉❧❡s✿ ✐♥❢❡r❡♥❝❡ r✉❧❡s ❙✉❝❝❡ss❢✉❧ st❛t❡s✿ ❝♦♥t❛✐♥✐♥❣ ❝♦♠♣❧❡t❡ ♣r♦♦❢s ✭❡✳❣✳✱ st❛t❡s ❝♦♥t❛✐♥✐♥❣ ✮

SLIDE 9

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❙❡❛r❝❤ P❧❛♥

❙❡❛r❝❤ ♣❧❛♥ Σ ❘✉❧❡ s❡❧❡❝t✐♥❣ ❢✉♥❝t✐♦♥✿ ●✐✈❡♥ ❤✐st♦r② ♦❢ st❛t❡s r❡t✉r♥s ✇❤✐❝❤ ✐♥❢❡r❡♥❝❡ r✉❧❡ t♦ ✉s❡ Pr❡♠✐s❡s s❡❧❡❝t✐♦♥ ❢✉♥❝t✐♦♥✿ ●✐✈❡♥ ❤✐st♦r② ♦❢ st❛t❡s r❡t✉r♥s ✇❤✐❝❤ ♣r❡♠✐s❡s t♦ ✉s❡ ❢♦r t❤❡ ✐♥❢❡r❡♥❝❡ r✉❧❡ ❚❡r♠✐♥❛t✐♦♥ ❞❡t❡❝t✐♦♥✿ ●✐✈❡♥ t❤❡ ❝✉rr❡♥t st❛t❡ r❡t✉r♥ tr✉❡ ✐✛ st❛t❡ ✐s s✉❝❝❡ss❢✉❧ ❚❤❡ s❡q✉❡♥❝❡ ♦❢ st❛t❡s ♦❜t❛✐♥❡❞ ❜② ❛♣♣❧②✐♥❣ Σ t♦ ■ ✐s ❛ ❞❡r✐✈❛t✐♦♥

SLIDE 10

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❚❤❡♦r❡♠✲Pr♦✈✐♥❣ ❙tr❛t❡❣②

❙❡❛r❝❤ P❧❛♥ ❛♣♣❧②✐♥❣ r✉❧❡s ❢r♦♠ ■ r❡s✉❧ts ✐♥ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❞❡r✐✈❛t✐♦♥ ❆♣♣❧②✐♥❣ Σ t♦ ■ ✇❡ ❤❛✈❡ ❛ ❞❡t❡r♠✐♥✐st✐❝ ❞❡r✐✈❛t✐♦♥ ■ ✰ Σ ❂ t❤❡♦r❡♠ ♣r♦✈✐♥❣ str❛t❡❣② ❲❡ ✇❛♥t Σ t♦ ❜❡ ❢❛✐r Σ ✐s ❢❛✐r✿ ✐❢ t❤❡r❡ ✐s ❛ s✉❝❝❡ss❢✉❧ st❛t❡ Σ ✇✐❧❧ ✜♥❞ ✐t

SLIDE 11

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❙tr❛t❡❣✐❡s

❈❧❛ss✐✜❝❛t✐♦♥ ❖r❞❡r✐♥❣ ❇❛s❡❞✿

✇♦r❦ ♦♥ ❛ s❡t ♦❢ ♦❜❥❡❝ts ✐♠♣❧✐❝✐t❧② ❣❡♥❡r❛t❡ ♠❛♥② ♣r♦♦❢ ❛tt❡♠♣ts ❖r❞❡r✐♥❣ ❛♥❞ ❈♦♥tr❛❝t✐♦♥ ✈❡r② ✐♠♣♦rt❛♥t ❖r❞❡r❡❞ r❡s♦❧✉t✐♦♥ ✇✐t❤ ▲❡✈❡❧ ❙❛t✉r❛t✐♦♥

♦❛❧ ❇❛s❡❞✿

✇♦r❦ ♦♥ ♦♥❡ ♦❜❥❡❝t ❡①♣❧✐❝✐t❧② ❣❡♥❡r❛t❡ ♦♥❡ ♣r♦♦❢ ❛tt❡♠♣t ❜❛❝❦tr❛❝❦ ✐❢ t❤❡ ❝✉rr❡♥t ♣r♦♦❢ ❛tt❡♠♣t ❝❛♥♥♦t ❜❡ ❝♦♠♣❧❡t❡❞ ✐♥t♦ ❛ ♣r♦♦❢ ▲✐♥❡❛r r❡s♦❧✉t✐♦♥ ✇✐t❤ ♦r❞❡r❡❞ ❝❧❛✉s❡ ❛♥❞ tr❡❡ ❡①♣❛♥s✐♦♥ ♣♦❧✐❝✐❡s

SLIDE 12

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❈❤❛r❛❝t❡r✐st✐❝s ♦❢ ❙tr❛t❡❣✐❡s

♠❛✐♥ ❢❡❛t✉r❡s ❖r❞❡r✐♥❣ ❇❛s❡❞

♦❛❧ ❇❛s❡❞

❞❛t❛ s❡t ♦❢ ♦❜❥❡❝ts ♦♥❡ ♦❜❥❡❝t ♣r♦♦❢ ❛tt❡♠♣t ♠❛♥② ✐♠♣❧✐❝✐t ♦♥❡ ❡①♣❧✐❝✐t ❜❛❝❦tr❛❝❦✐♥❣ ◆♦ ❨❡s

SLIDE 13

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❖r❞❡r✐♥❣ ❇❛s❡❞ ❙tr❛t❡❣✐❡s

❇❛s✐❝ ❝♦♥❝❡♣ts tr❛♥s❢♦r♠ ❍ ∪ ¬ψ ✐♥t♦ ❈❧❛✉s❡s ❙ ✐s t❤❡ ❝❧❛✉s❛❧ ❢♦r♠ ♦❢ t❤❡♦r❡♠ ♣r♦✈✐♥❣ ♣r♦❜❧❡♠ ¬ψ ✐s ❛♥ ❛❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥ ■♥❢❡r❡♥❝❡ r✉❧❡s ✇♦r❦ ♦♥ ❙

SLIDE 14

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

■♥❢❡r❡♥❝❡ ❙②st❡♠s ❢♦r ❖r❞❡r✐♥❣ ❇❛s❡❞ ❙tr❛t❡❣✐❡s

■♥❢❡r❡♥❝❡ ❘✉❧❡s

❡♥❡r❛❧ ❢♦r♠✿

❢ : ❙ ❙′ ❊①❛♠♣❧❡ ❙ ∪ {▲✶ ∨ ❉, ▲✷ ∨ ❈} ❙ ∪ {▲✶ ∨ ❉, ▲✷ ∨ ❈, (❈ ∨ ❉)σ}▲✶σ = ¬▲✷σ

SLIDE 15

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❚❤❡ ■♥❢❡r❡♥❝❡ ❙②st❡♠s R

■♥❢❡r❡♥❝❡ ❘✉❧❡s ❊①♣❛♥s✐♦♥ r✉❧❡s ✭❇✐♥❛r② ❘❡s♦❧✉t✐♦♥ ✰ ❋❛❝t♦r✐♥❣✮ ❢ : ❙ ❙′ ❙ ⊆ ❙′ ❈♦♥tr❛❝t✐♦♥ r✉❧❡s ✭❚❛✉t♦❧♦❣② ❡❧✐♠✐♥❛t✐♦♥ ✰ ❙✉❜s✉♠♣t✐♦♥✮ ❢ : ❙ ❙′ ❙′ ⊆ ❙ ❖r❞❡r✐♥❣s ♦♥ ❝❧❛✉s❡s ✭❡✳❣✳ s✐♠♣❧✐✜❝❛t✐♦♥ ♦r❞❡r✐♥❣s✮ ❛r❡ ❢r❡q✉❡♥t❧② ✉s❡❞ t♦✿

r❡str✐❝t ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ✐♥❢❡r❡♥❝❡ r✉❧❡ ❝♦♥t❛✐♥✐♥❣ ❡①♣❛♥s✐♦♥ ♦❢ ❙ ❞❡❝✐❞❡ ✇❤✐❝❤ ❝❧❛✉s❡ ❝❛♥ ❜❡ ❞❡❧❡t❡❞✳✭❡✳❣✳✱ ❝❧❛✉s❡ ❡♥t❛✐❧❡❞ ❜② s♠❛❧❧❡r ❝❧❛✉s❡s✮

SLIDE 16

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❈♦♥tr❛❝t✐♦♥ ❛♥❞ ❘❡❞✉♥❞❛♥❝②

❈♦♥tr❛❝t✐♦♥ r✉❧❡s ❋♦r✇❛r❞✿ r❡❞✉❝❡s ♥❡✇❧② ❣❡♥❡r❛t❡❞ ❝❧❛✉s❡s ❇❛❝❦✇❛r❞✿ ✉s❡ ♥❡✇ ❝❧❛✉s❡s t♦ r❡❞✉❝❡ ❡①✐st✐♥❣ ♦♥❡s ❈♦♥tr❛❝t✐♦♥ ❦❡② ✐♥ ♦r❞❡r✐♥❣ ❜❛s❡❞ ♠❡t❤♦❞s ❉❡❧❡t❡ ❡①✐st✐♥❣ ❝❧❛✉s❡s Pr❡✈❡♥ts ❣❡♥❡r❛t✐♦♥ ♦❢ ✉s❡❧❡ss ❝❧❛✉s❡s ❆✐♠✿ ❞❡❧❡t❡ r❡❞✉♥❞❛♥t ❝❧❛✉s❡s

SLIDE 17

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❋❛✐r ❉❡r✐✈❛t✐♦♥

❇❛s✐❝ ✐❞❡❛ ❈♦♠♣❧❡t❡♥❡ss ♦❢ R ❞❡♣❡♥❞s ✉♣♦♥ ❞❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s ✉s❡❞ ❲❛♥t t♦ ❣❡♥❡r❛t❡ ❛❧❧ ✉s❡❢✉❧ ❝❧❛✉s❡s ❋❛✐r ❞❡r✐✈❛t✐♦♥ str❛t❡❣②✿ ❡✈❡r② r✉❧❡ ✐♥ R t❤❛t ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❝❧❛✉s❡s ✐♥ t❤❡ ❞❡r✐✈❛t✐♦♥ ✐s ❛♣♣❧✐❡❞ ❡✈❡♥t✉❛❧❧② ■♥ ❛ ❢❛✐r ❞❡r✐✈❛t✐♦♥✱ ❡✈❡r② r✉❧❡ ✐♥ R ❡✈❡♥t✉❛❧❧② ❢❛✐❧s t♦ ❛❞❞ ♥❡✇ ❝❧❛✉s❡s

SLIDE 18

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❋❛✐r ❉❡r✐✈❛t✐♦♥✿ ❞❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❋❛✐r ❉❡r✐✈❛t✐♦♥✮ ❆♥ R✲❞❡r✐✈❛t✐♦♥ ❙✵, ❙✶, ❙✷, · · · ✐s ❢❛✐r ✐❢ ❙∞ =

❦≥✵
❥≥❦

❙❥ ✐s ❝❧♦s❡❞ ✉♥❞❡r R✳ ❙∞ ✐s ❝❛❧❧❡❞ t❤❡ s❡t ♦❢ ♣❡rs✐st❡♥t ❝❧❛✉s❡s

SLIDE 19

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❙❡t ♦❢ ♣❡rs✐st❡♥t ❝❧❛✉s❡s

❊①❛♠♣❧❡

✐✈❡♥ ❙ = {¬P ∨ ◗, P, ¬◗}✱ ❝♦♠♣✉t❡ ❙∞ ❝♦♥s✐❞❡r✐♥❣

❙✵, ❙✶, ❙✷, · · · ❛s ❢♦❧❧♦✇✿ ❙✵ = { } ❙✶ = ❙ · · · ❙♥+✷ = ❙♥+✶ ∪ {❘❡s♦❧✈❡♥ts ♦❢ ❈✶ ❛♥❞ ❈✷|❈✶ ∈ ❙♥+✶ ❛♥❞ ❈✷ ∈ ❙♥+✶ \ ❙♥}

SLIDE 20

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

▲❡✈❡❧ ❙❛t✉r❛t✐♦♥ ✐s ❋❛✐r

❋❛✐r♥❡ss ♦❢ ▲❡✈❡❧ ❙❛t✉r❛t✐♦♥ ❆ss✉♠❡ ♥♦ ❝♦♥tr❛❝t✐♦♥ ❆t ❡❛❝❤ st❛❣❡ ❣❡♥❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ r❡s♦❧✈❡♥ts ▼❛✐♥t❛✐♥s ❛❧❧ r❡s♦❧✈❡♥ts ✐♥ t❤❡ s❡t ❊✈❡♥t✉❛❧❧② ❛❧❧ ♣♦ss✐❜❧❡ r❡s♦❧✈❡♥ts ✇✐❧❧ ❜❡ ❣❡♥❡r❛t❡❞

SLIDE 21

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

❆ r❡❞✉♥❞❛♥❝② ❝r✐t❡r✐♦♥

r❡❞✉♥❞❛♥t ❝❧❛✉s❡ ❆ ❝❧❛✉s❡ ❈ ✐s r❡❞✉♥❞❛♥t ✇✐t❤ r❡s♣❡❝t t♦ ❙✐ ✐❢ ❈ ✐s ❛ ❚❛✉t♦❧♦❣② ❈ ✐s s✉❜s✉♠❡❞ ❜② ❛ ❝❧❛✉s❡ ❉ ∈ ❙✐

SLIDE 22

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

▲❡✈❡❧ ❙❛t✉r❛t✐♦♥ ✇✐t❤ s♦❧✉t✐♦♥

▲❡✈❡❧ ❙❛t✉r❛t✐♦♥ ✇✐t❤ s♦❧✉t✐♦♥ ❙✵ = { } ❙✶ = ❙ · · · ❖♥❡ s❛t✉r❛t✐♦♥ st❡♣

❙✷♥+✶ = ❙✷♥ ∪ {❘❡s♦❧✈❡♥ts ♦❢ ❈✶ ❛♥❞ ❈✷|❈✶ ∈ ❙✷♥ ❛♥❞ ❈✷ ∈ ❙✷♥ \ ❙✷♥−✷}

❖♥❡ r❡❞✉❝t✐♦♥ st❡♣

❙✷♥+✷ = ❙✷♥+✶ \ {❈ ∈ ❙✷♥+✶|❈ ✐s r❡❞✉♥❞❛♥t ✇✐t❤ r❡s♣❡❝t t♦ ❙✷♥}

❙♦❧✉t✐♦♥✿ r❡❞✉❝t✐♦♥ st❡♣

SLIDE 23

❚❤❡♦r❡♠ Pr♦✈✐♥❣ ❙tr❛t❡❣✐❡s ❚❤❡♦r❡♠ ♣r♦✈✐♥❣ ❛♥❞ ❙❡❛r❝❤ ❋❛✐r ❉❡r✐✈❛t✐♦♥ str❛t❡❣✐❡s

▲❡✈❡❧ ❙❛t✉r❛t✐♦♥ ✇✐t❤ s♦❧✉t✐♦♥ ✐s ❢❛✐r

❋❛✐r♥❡ss ❊✈❡r② ❝❧❛✉s❡ ❈ ♥♦t ✐♥ ❙∞ ✐s r❡❞✉♥❞❛♥t ✇✐t❤ r❡s♣❡❝t t♦ ❙∞ ❙✉♣♣♦s❡ ❈ ❛♥❞ ❉ ❛r❡ ✐♥ ❙∞ ❚❤❡♥ ❈ ❛♥❞ ❉ ❛r❡ ♥♦t r❡❞✉♥❞❛♥t ♦t❤❡r✇✐s❡ ❡❧✐♠✐♥❛t❡❞ ❜❡❢♦r❡ ▲❡t ❘ ❜❡ ❛ r❡s♦❧✈❡♥t ♦❢ ❈ ❛♥❞ ❉✳ ❚❤❡♥ ❘ ✇❛s ♥❡❝❡ss❛r✐❧② ❣❡♥❡r❛t❡❞ ✐♥ s♦♠❡ ❙✷♥ ❜② ❝♦♥str✉❝t✐♦♥ ■❢ ❘ ✇❛s r❡♠♦✈❡❞ t❤❡♥ ❘ ✐s s✉❜s✉♠❡❞ ❜② s♦♠❡ ♦t❤❡r ❝❧❛✉s❡ ✐♥ s♦♠❡ ❙❥ ❥ > ♥✳ ❚❤❡r❡❢♦r❡ ❘ ✐s r❡❞✉♥❞❛♥t ✇✐t❤ r❡s♣❡❝t t♦ ❙∞