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Dec 06, 2023 462 likes •1.23k views

t tr tt ts r r r

SLIDE 1

❉◆❆✲❈♦♠♣✉t✐♥❣ ❚❤❡ ❋✉t✉r❡ ♦❢ ❈♦♠♣✉t❛❜✐❧✐t②❄ ❚❤❡ ❋♦✉♥❞❛t✐♦♥s ❜② ❆❞❧❡♠❛♥

❋r❛♥❦ ❇❛✉❡r ✭❜❛✉❡r ❢r❛♥❦❅✇❡❜✳❞❡✮ ❯♥✐✈❡rs✐t② ♦❢ ❑❛✐s❡rs❧❛✉t❡r♥ ❆✉❣✉st ✷✽✱ ✷✵✵✷

SLIDE 2

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ t♦ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s

SLIDE 3

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ t♦ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s
❉❡s❝r✐♣t✐♦♥ ♦❢ t✇♦ ✐♠♣♦rt❛♥t ◆P ♣r♦❜❧❡♠s

SLIDE 4

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ t♦ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s
❉❡s❝r✐♣t✐♦♥ ♦❢ t✇♦ ✐♠♣♦rt❛♥t ◆P ♣r♦❜❧❡♠s

⋆ ❙❆❚ ✭❙❛t✐s❢❛❝t✐♦♥ ♣r♦❜❧❡♠✮ ⋆ ❍P ✭❍❛♠✐❧t♦♥✐❛♥ P❛t❤ Pr♦❜❧❡♠✮

SLIDE 5

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ t♦ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s
❉❡s❝r✐♣t✐♦♥ ♦❢ t✇♦ ✐♠♣♦rt❛♥t ◆P ♣r♦❜❧❡♠s

⋆ ❙❆❚ ✭❙❛t✐s❢❛❝t✐♦♥ ♣r♦❜❧❡♠✮ ⋆ ❍P ✭❍❛♠✐❧t♦♥✐❛♥ P❛t❤ Pr♦❜❧❡♠✮

❯s✐♥❣ ❉◆❆ t♦ ❝♦♠♣✉t❡ s♦❧✉t✐♦♥s t♦ t❤❡s❡ ♣r♦❜❧❡♠s

SLIDE 6

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ t♦ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s
❉❡s❝r✐♣t✐♦♥ ♦❢ t✇♦ ✐♠♣♦rt❛♥t ◆P ♣r♦❜❧❡♠s

⋆ ❙❆❚ ✭❙❛t✐s❢❛❝t✐♦♥ ♣r♦❜❧❡♠✮ ⋆ ❍P ✭❍❛♠✐❧t♦♥✐❛♥ P❛t❤ Pr♦❜❧❡♠✮

❯s✐♥❣ ❉◆❆ t♦ ❝♦♠♣✉t❡ s♦❧✉t✐♦♥s t♦ t❤❡s❡ ♣r♦❜❧❡♠s

⋆ ❆❞❧❡♠❛♥✬s s♦❧✉t✐♦♥ ❢♦r ❍P

SLIDE 7

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ t♦ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s
❉❡s❝r✐♣t✐♦♥ ♦❢ t✇♦ ✐♠♣♦rt❛♥t ◆P ♣r♦❜❧❡♠s

⋆ ❙❆❚ ✭❙❛t✐s❢❛❝t✐♦♥ ♣r♦❜❧❡♠✮ ⋆ ❍P ✭❍❛♠✐❧t♦♥✐❛♥ P❛t❤ Pr♦❜❧❡♠✮

❯s✐♥❣ ❉◆❆ t♦ ❝♦♠♣✉t❡ s♦❧✉t✐♦♥s t♦ t❤❡s❡ ♣r♦❜❧❡♠s

⋆ ❆❞❧❡♠❛♥✬s s♦❧✉t✐♦♥ ❢♦r ❍P ⋆ ▲✐♣t♦♥✬s s♦❧✉t✐♦♥ ❢♦r ❙❆❚

SLIDE 8

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♥t❡♥ts

■♥tr♦❞✉❝t✐♦♥ t♦ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s
❉❡s❝r✐♣t✐♦♥ ♦❢ t✇♦ ✐♠♣♦rt❛♥t ◆P ♣r♦❜❧❡♠s

⋆ ❙❆❚ ✭❙❛t✐s❢❛❝t✐♦♥ ♣r♦❜❧❡♠✮ ⋆ ❍P ✭❍❛♠✐❧t♦♥✐❛♥ P❛t❤ Pr♦❜❧❡♠✮

❯s✐♥❣ ❉◆❆ t♦ ❝♦♠♣✉t❡ s♦❧✉t✐♦♥s t♦ t❤❡s❡ ♣r♦❜❧❡♠s

⋆ ❆❞❧❡♠❛♥✬s s♦❧✉t✐♦♥ ❢♦r ❍P ⋆ ▲✐♣t♦♥✬s s♦❧✉t✐♦♥ ❢♦r ❙❆❚ ⋆ ❉✐s❝✉ss✐♦♥

SLIDE 9

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

▼♦t✐✈❛t✐♦♥

SLIDE 10

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

▼♦t✐✈❛t✐♦♥

❖r❣❛♥✐③❡ t❤❡ r♦✉t❡s ♦❢ ❛ tr❛✈❡❧❧✐♥❣ s❛❧❡s♠❛♥ ✇❤♦ ❤❛s t♦ ✈✐s✐t ✶✵✵ t♦✇♥s✳ ❆❢t❡r ❤✐s ✇♦r❦ ❤❡ s❤♦✉❧❞ r❡t✉r♥ t♦ t❤❡ t♦✇♥ ✇❤❡r❡ ❤❡ st❛rt❡❞ ❢r♦♠✳ ❍❡ s❤♦✉❧❞ s♣❡♥❞ t❤❡ ❧❡❛st t✐♠❡ ♣♦ss✐❜❧❡ ♦♥ t❤❡ r♦❛❞s✳

SLIDE 11

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

▼♦t✐✈❛t✐♦♥

SLIDE 12

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

▼♦t✐✈❛t✐♦♥

SLIDE 13

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

▼♦t✐✈❛t✐♦♥

SLIDE 14

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭▼❡❛s✉r✐♥❣ ❉✐✣❝✉❧t②✮

❉❡✜♥✐t✐♦♥ ✶✳ ▲❡t f : N → R ❛♥❞ g : N → R✳ ❚❤❡♥ f ∈ O(g) ✐✛ t❤❡r❡ ❡①✐sts ❛♥ n ∈ N ❛♥❞ ❛ c ∈ R s✉❝❤ t❤❛t ❢♦r ❛❧❧ N > n ✇❡ ❤❛✈❡ f(N) < c g(N)

SLIDE 15

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭▼❡❛s✉r✐♥❣ ❉✐✣❝✉❧t②✮

❉❡✜♥✐t✐♦♥ ✶✳ ▲❡t f : N → R ❛♥❞ g : N → R✳ ❚❤❡♥ f ∈ O(g) ✐✛ t❤❡r❡ ❡①✐sts ❛♥ n ∈ N ❛♥❞ ❛ c ∈ R s✉❝❤ t❤❛t ❢♦r ❛❧❧ N > n ✇❡ ❤❛✈❡ f(N) < c g(N) ▲❡♠♠❛ ✶✳ ▲❡t f1 ∈ O(f2) ❛♥❞ f2 ∈ O(f3)✳ ❚❤❡♥ ❛❧s♦ f1 ∈ O(f3)✳

SLIDE 16

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭▼❡❛s✉r✐♥❣ ❉✐✣❝✉❧t②✮

❉❡✜♥✐t✐♦♥ ✶✳ ▲❡t f : N → R ❛♥❞ g : N → R✳ ❚❤❡♥ f ∈ O(g) ✐✛ t❤❡r❡ ❡①✐sts ❛♥ n ∈ N ❛♥❞ ❛ c ∈ R s✉❝❤ t❤❛t ❢♦r ❛❧❧ N > n ✇❡ ❤❛✈❡ f(N) < c g(N) ▲❡♠♠❛ ✶✳ ▲❡t f1 ∈ O(f2) ❛♥❞ f2 ∈ O(f3)✳ ❚❤❡♥ ❛❧s♦ f1 ∈ O(f3)✳ ▲❡t f1 ∈ O(g1) ❛♥❞ f2 ∈ O(g2)✳ ❚❤❡♥ ✇❡ ❤❛✈❡ f1f2 ∈ O(g1g2)

SLIDE 17

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭▼❡❛s✉r✐♥❣ ❉✐✣❝✉❧t②✮

❉❡✜♥✐t✐♦♥ ✷✳ ■❢ t❤❡ t✐♠❡ r❡q✉✐r❡❞ ❜② ❛♥ ❛❧❣♦r✐t❤♠ ✐s ✐♥ ❖✭♣✮ ✇❤❡r❡ ♣ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✱ ✇❡ ❝❛❧❧ t❤✐s ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳ ❖t❤❡r✇✐s❡ ✇❡ ❝❛❧❧ ✐t ❛♥ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳

SLIDE 18

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭▼❡❛s✉r✐♥❣ ❉✐✣❝✉❧t②✮

❉❡✜♥✐t✐♦♥ ✷✳ ■❢ t❤❡ t✐♠❡ r❡q✉✐r❡❞ ❜② ❛♥ ❛❧❣♦r✐t❤♠ ✐s ✐♥ ❖✭♣✮ ✇❤❡r❡ ♣ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✱ ✇❡ ❝❛❧❧ t❤✐s ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳ ❖t❤❡r✇✐s❡ ✇❡ ❝❛❧❧ ✐t ❛♥ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳ ❢✉♥❝t✐♦♥ ✴ s✐③❡ ✶✵ ✷✵ ✹✵ ✽✵ n ✳✵✵✵✵✶ s ✳✵✵✵✵✷ s ✳✵✵✵✵✹ s ✳✵✵✵✵✽ s

SLIDE 19

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭▼❡❛s✉r✐♥❣ ❉✐✣❝✉❧t②✮

❉❡✜♥✐t✐♦♥ ✷✳ ■❢ t❤❡ t✐♠❡ r❡q✉✐r❡❞ ❜② ❛♥ ❛❧❣♦r✐t❤♠ ✐s ✐♥ ❖✭♣✮ ✇❤❡r❡ ♣ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✱ ✇❡ ❝❛❧❧ t❤✐s ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳ ❖t❤❡r✇✐s❡ ✇❡ ❝❛❧❧ ✐t ❛♥ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳ ❢✉♥❝t✐♦♥ ✴ s✐③❡ ✶✵ ✷✵ ✹✵ ✽✵ n ✳✵✵✵✵✶ s ✳✵✵✵✵✷ s ✳✵✵✵✵✹ s ✳✵✵✵✵✽ s n2 ✳✵✵✵✶ s ✳✵✵✵✹ s ✳✵✵✶✻ s ✳✵✵✻✹ s

SLIDE 20

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭▼❡❛s✉r✐♥❣ ❉✐✣❝✉❧t②✮

❉❡✜♥✐t✐♦♥ ✷✳ ■❢ t❤❡ t✐♠❡ r❡q✉✐r❡❞ ❜② ❛♥ ❛❧❣♦r✐t❤♠ ✐s ✐♥ ❖✭♣✮ ✇❤❡r❡ ♣ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✱ ✇❡ ❝❛❧❧ t❤✐s ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳ ❖t❤❡r✇✐s❡ ✇❡ ❝❛❧❧ ✐t ❛♥ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳ ❢✉♥❝t✐♦♥ ✴ s✐③❡ ✶✵ ✷✵ ✹✵ ✽✵ n ✳✵✵✵✵✶ s ✳✵✵✵✵✷ s ✳✵✵✵✵✹ s ✳✵✵✵✵✽ s n2 ✳✵✵✵✶ s ✳✵✵✵✹ s ✳✵✵✶✻ s ✳✵✵✻✹ s n4 ✳✵✶ s ✳✶✻ s ✷✳✺✻ s ✹✵✳✾✻ s

SLIDE 21

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭▼❡❛s✉r✐♥❣ ❉✐✣❝✉❧t②✮

❉❡✜♥✐t✐♦♥ ✷✳ ■❢ t❤❡ t✐♠❡ r❡q✉✐r❡❞ ❜② ❛♥ ❛❧❣♦r✐t❤♠ ✐s ✐♥ ❖✭♣✮ ✇❤❡r❡ ♣ ✐s ❛ ♣♦❧②♥♦♠✐❛❧✱ ✇❡ ❝❛❧❧ t❤✐s ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳ ❖t❤❡r✇✐s❡ ✇❡ ❝❛❧❧ ✐t ❛♥ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✳ ❢✉♥❝t✐♦♥ ✴ s✐③❡ ✶✵ ✷✵ ✹✵ ✽✵ n ✳✵✵✵✵✶ s ✳✵✵✵✵✷ s ✳✵✵✵✵✹ s ✳✵✵✵✵✽ s n2 ✳✵✵✵✶ s ✳✵✵✵✹ s ✳✵✵✶✻ s ✳✵✵✻✹ s n4 ✳✵✶ s ✳✶✻ s ✷✳✺✻ s ✹✵✳✾✻ s 2n ✵✳✵✵✶ s ✶✳✵ s ✶✷✱✼ ❞❛②s 3, 8 × 1010 ②❡❛rs

SLIDE 22

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭Pr♦❜❧❡♠ ❈❧❛ss❡s✮

❯♥❞❡❝✐❞❛❜❧❡ ♣r♦❜❧❡♠s

❚❤❡r❡ ❞♦❡s ♥♦t ❡①✐st ❛♥② ❛❧❣♦r✐t❤♠ t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠✳ ❆ ♣r♦♠✐♥❡♥t ❡①❛♠♣❧❡ ✐s t❤❡ ❤❛❧t✐♥❣ ♣r♦❜❧❡♠ ✐♥ ❝♦♠♣✉t❡r s❝✐❡♥❝❡✳

SLIDE 23

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭Pr♦❜❧❡♠ ❈❧❛ss❡s✮

❯♥❞❡❝✐❞❛❜❧❡ ♣r♦❜❧❡♠s

❚❤❡r❡ ❞♦❡s ♥♦t ❡①✐st ❛♥② ❛❧❣♦r✐t❤♠ t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠✳ ❆ ♣r♦♠✐♥❡♥t ❡①❛♠♣❧❡ ✐s t❤❡ ❤❛❧t✐♥❣ ♣r♦❜❧❡♠ ✐♥ ❝♦♠♣✉t❡r s❝✐❡♥❝❡✳

◆♦♥✲♣♦❧②♥♦♠✐❛❧ ♣r♦❜❧❡♠s ♦♥ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝♦♠♣✉t❡r

▼♦st ❡①❛♠♣❧❡s ❛r❡ q✉✐t❡ ❛rt✐✜❝✐❛❧ ❛♥❞ ❛r❡ ♦❢ ♥♦ s✐❣♥✐✜❝❛♥❝❡ ✐♥ r❡❛❧✐t②✳

SLIDE 24

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭Pr♦❜❧❡♠ ❈❧❛ss❡s✮ ✴ ✷

P♦❧②♥♦♠✐❛❧ ♣r♦❜❧❡♠s ♦♥ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝♦♠♣✉t❡r ✭◆P✮

❯s✐♥❣ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝♦♠♣✉t❡r t❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss s♦♠❡ ❡①❛♠♣❧❡s ❢♦r t❤✐s ❝❛s❡ ❧❛t❡r ♦♥✳

SLIDE 25

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭Pr♦❜❧❡♠ ❈❧❛ss❡s✮ ✴ ✷

P♦❧②♥♦♠✐❛❧ ♣r♦❜❧❡♠s ♦♥ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝♦♠♣✉t❡r ✭◆P✮

❯s✐♥❣ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝♦♠♣✉t❡r t❤❡ ♣r♦❜❧❡♠ ✐s s♦❧✈❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss s♦♠❡ ❡①❛♠♣❧❡s ❢♦r t❤✐s ❝❛s❡ ❧❛t❡r ♦♥✳

P♦❧②♥♦♠✐❛❧ ♣r♦❜❧❡♠s ✭P✮

❚❤❡s❡ ♣r♦❜❧❡♠s ❞♦ ♥♦t ❤❛✈❡ t♦♦ ♠❛♥② ❞✐✣❝✉❧t✐❡s✳ ❊✳❣✳ s♦❧✈✐♥❣ ❛ s②st❡♠ ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ♦r s❡❛r❝❤✐♥❣ t❤❡ s❤♦rt❡st ♣❛t❤ ✐♥ ❛ ♥❡t✇♦r❦ ❛r❡ ♣♦❧②♥♦♠✐❛❧ ❛❧❣♦r✐t❤♠s✳

SLIDE 26

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭◆P✲Pr♦❜❧❡♠s✿ ❙❆❚✮

❆ss✉♠❡ ②♦✉ ❤❛✈❡ ❛ ❜♦♦❧❡❛♥ ❢♦r♠✉❧❛ F(x1, . . . , xn) ✱ ❡✳❣✳ F(x1, x2) = (x1 ∨ x2) ∧ (x1 ∨ x2) ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ xi ✇❤❡r❡ F ❣❡ts tr✉❡❄

SLIDE 27

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭◆P✲Pr♦❜❧❡♠s✿ ❙❆❚✮

❆ss✉♠❡ ②♦✉ ❤❛✈❡ ❛ ❜♦♦❧❡❛♥ ❢♦r♠✉❧❛ F(x1, . . . , xn) ✱ ❡✳❣✳ F(x1, x2) = (x1 ∨ x2) ∧ (x1 ∨ x2) ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ xi ✇❤❡r❡ F ❣❡ts tr✉❡❄ ❖❜✈✐♦✉s❧②✱ t❤✐s ♣r♦❜❧❡♠ ✐s ✐♥ ◆P ❜❡❝❛✉s❡ ②♦✉ ❝❛♥ t❡st ❛ ♣❛rt✐❝✉❧❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ xi ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ◆♦✇ ②♦✉ ❝❛♥ ✉s❡ 2n ♣r♦❝❡ss♦rs ❢♦r ❛❧❧ t❤❡ ❞✐✛❡r❡♥t ♣♦ss✐❜✐❧✐t✐❡s ♦❢ t❤❡ xi✳ ❚❤✉s t❤❡ ♣r♦❜❧❡♠ ❤❛s ❥✉st ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ♦♥ t❤✐s ❛rt✐✜❝✐❛❧ ♠❛❝❤✐♥❡✳

SLIDE 28

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭◆P✲Pr♦❜❧❡♠s✿ ❍P✮

❆ss✉♠❡ ②♦✉ ❤❛✈❡ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ n ♥♦❞❡s✳

SLIDE 29

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭◆P✲Pr♦❜❧❡♠s✿ ❍P✮

❆ss✉♠❡ ②♦✉ ❤❛✈❡ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ n ♥♦❞❡s✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ♣❛t❤ ✈✐st✐♥❣ ❡✈❡r② ♥♦❞❡ ❡①❛❝t❧② ♦♥❝❡ ✇❤✐❝❤ st❛rts ❛t vin ❛♥❞ ❡♥❞s ❛t vout❄ ✭■♥ t❤✐s ❝❛s❡ vin = 0 ❛♥❞ vout = 6✮

SLIDE 30

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭◆P✲Pr♦❜❧❡♠s✿ ❍P✮

❆ss✉♠❡ ②♦✉ ❤❛✈❡ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ n ♥♦❞❡s✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ♣❛t❤ ✈✐st✐♥❣ ❡✈❡r② ♥♦❞❡ ❡①❛❝t❧② ♦♥❝❡ ✇❤✐❝❤ st❛rts ❛t vin ❛♥❞ ❡♥❞s ❛t vout❄ ✭■♥ t❤✐s ❝❛s❡ vin = 0 ❛♥❞ vout = 6✮ ❆❣❛✐♥✱ t❤✐s ♣r♦❜❧❡♠ ✐s ✐♥ ◆P✳ ❚❡st✐♥❣ ✐❢ t❤❡ ♣r♦♣❡rt✐❡s ❛r❡ ❢✉❧✜❧❧❡❞ ❜② ❛ ♣❛rt✐❝✉❧❛r ♣❛t❤ ❥✉st t❛❦❡s ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

SLIDE 31

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭◆P✲❝♦♠♣❧❡t❡ Pr♦❜❧❡♠s✮

❉❡✜♥✐t✐♦♥ ✸✳ ❆ ♣r♦❜❧❡♠ P1 ♣♦❧②♥♦♠✐❛❧❧② tr❛♥s❢♦r♠s ✭r❡❞✉❝❡s✮ t♦ ❛ ♣r♦❜❧❡♠ P2✱ ✐❢ t❤❡r❡ ✐s ❛ ✏❢✉♥❝t✐♦♥✑ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤✐❝❤ ♠❛♣s P1 → P2✳ ❋✉rt❤❡r♠♦r❡ s♦❧✈✐♥❣ P2 ❛❧s♦ s♦❧✈❡s P1✳

SLIDE 32

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭◆P✲❝♦♠♣❧❡t❡ Pr♦❜❧❡♠s✮

❉❡✜♥✐t✐♦♥ ✸✳ ❆ ♣r♦❜❧❡♠ P1 ♣♦❧②♥♦♠✐❛❧❧② tr❛♥s❢♦r♠s ✭r❡❞✉❝❡s✮ t♦ ❛ ♣r♦❜❧❡♠ P2✱ ✐❢ t❤❡r❡ ✐s ❛ ✏❢✉♥❝t✐♦♥✑ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤✐❝❤ ♠❛♣s P1 → P2✳ ❋✉rt❤❡r♠♦r❡ s♦❧✈✐♥❣ P2 ❛❧s♦ s♦❧✈❡s P1✳ ❚❤❡♦r❡♠ ✶✳ ❬❈♦♦❦✱ ✶✾✼✶❪ ❊✈❡r② ♣r♦❜❧❡♠ ✐♥ ◆P ♣♦❧②♥♦♠✐❛❧❧② tr❛♥s❢♦r♠s t♦ t❤❡ ❙❆❚ ♣r♦❜❧❡♠✳

SLIDE 33

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♠♣❧❡①✐t② ✭◆P✲❝♦♠♣❧❡t❡ Pr♦❜❧❡♠s✮

❉❡✜♥✐t✐♦♥ ✸✳ ❆ ♣r♦❜❧❡♠ P1 ♣♦❧②♥♦♠✐❛❧❧② tr❛♥s❢♦r♠s ✭r❡❞✉❝❡s✮ t♦ ❛ ♣r♦❜❧❡♠ P2✱ ✐❢ t❤❡r❡ ✐s ❛ ✏❢✉♥❝t✐♦♥✑ ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❝♦♠♣❧❡①✐t② ✇❤✐❝❤ ♠❛♣s P1 → P2✳ ❋✉rt❤❡r♠♦r❡ s♦❧✈✐♥❣ P2 ❛❧s♦ s♦❧✈❡s P1✳ ❚❤❡♦r❡♠ ✶✳ ❬❈♦♦❦✱ ✶✾✼✶❪ ❊✈❡r② ♣r♦❜❧❡♠ ✐♥ ◆P ♣♦❧②♥♦♠✐❛❧❧② tr❛♥s❢♦r♠s t♦ t❤❡ ❙❆❚ ♣r♦❜❧❡♠✳ ❉❡✜♥✐t✐♦♥ ✹✳ ■❢ ❙❆❚ ♣♦❧②♥♦♠✐❛❧❧② tr❛♥s❢♦r♠s t♦ ❛ ♣r♦❜❧❡♠ t❤✐s ♣r♦❜❧❡♠ ✐s ❝❛❧❧❡❞ t♦ ❜❡ ◆P✲❝♦♠♣❧❡t❡✳

SLIDE 34

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❉◆❆✲❈♦♠♣✉t✐♥❣

❘❡❝❛❧❧✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝❧❛ss ♦❢ ◆P✲♣r♦❜❧❡♠s ✇❡ ♦❜s❡r✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛✐♥ ❞✐✣❝✉❧t②✳ ◆♦r♠❛❧❧② ✇❡ ❞♦ ♥♦t ❤❛✈❡ ❛ ✈❡r② ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥t ♣r♦❝❡ss♦rs ❛t ❤❛♥❞✱ ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ ✉s❡❞ t♦ s♦❧✈❡ s✉❝❤ ❛ ❤❛r❞ ♣r♦❜❧❡♠ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

SLIDE 35

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❉◆❆✲❈♦♠♣✉t✐♥❣

❘❡❝❛❧❧✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❝❧❛ss ♦❢ ◆P✲♣r♦❜❧❡♠s ✇❡ ♦❜s❡r✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❛✐♥ ❞✐✣❝✉❧t②✳ ◆♦r♠❛❧❧② ✇❡ ❞♦ ♥♦t ❤❛✈❡ ❛ ✈❡r② ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥t ♣r♦❝❡ss♦rs ❛t ❤❛♥❞✱ ✇❤✐❝❤ ❝♦✉❧❞ ❜❡ ✉s❡❞ t♦ s♦❧✈❡ s✉❝❤ ❛ ❤❛r❞ ♣r♦❜❧❡♠ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❆❞❧❡♠❛♥✬s ✐❞❡❛ ✐s q✉✐t❡ s✐♠♣❧❡✳ ❍❡ ♣r♦♣♦s❡❞ ✉s✐♥❣ str✐♥❣s ♦❢ ❉◆❆ t♦ ❝♦✈❡r ❡✈❡r② ❞✐✛❡r❡♥t ♣♦ss✐❜✐❧✐t② ♦❢ t❤❡ ✐♥✐t✐❛❧ ❞❛t❛✳ ❇❡❝❛✉s❡ ♠❛♥✐♣✉❧❛t✐♦♥s ✇✐t❤ t❤❡s❡ ❉◆❆ ♠♦❧❡❝✉❧❡s ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ ♣❛r❛❧❧❡❧❧②✱ ✇❡ ❝❛♥ ❣❡t ♦✉r r❡s✉❧ts ✐♥ ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

SLIDE 36

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ❆❞❧❡♠❛♥ ✭Pr❡♣❛r❛t✐♦♥s✮

■❞❡♥t✐❢② ❡✈❡r② ♥♦❞❡ ✇✐t❤ ♦❢ t❤❡ ❣r❛♣❤ ✇✐t❤ ❛ r❛♥❞♦♠❧② ❝❤♦s❡♥

k✲♠❡r s❡q✉❡♥❝❡ Oi✳

SLIDE 37

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ❆❞❧❡♠❛♥ ✭Pr❡♣❛r❛t✐♦♥s✮

■❞❡♥t✐❢② ❡✈❡r② ♥♦❞❡ ✇✐t❤ ♦❢ t❤❡ ❣r❛♣❤ ✇✐t❤ ❛ r❛♥❞♦♠❧② ❝❤♦s❡♥

k✲♠❡r s❡q✉❡♥❝❡ Oi✳

◆♦ t✇♦ s❡q✉❡♥❝❡s ♦r t❤❡✐r ❝♦♠♣❧❡♠❡♥t Oi ✜t t♦❣❡t❤❡r✳

SLIDE 38

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ❆❞❧❡♠❛♥ ✭Pr❡♣❛r❛t✐♦♥s✮

■❞❡♥t✐❢② ❡✈❡r② ♥♦❞❡ ✇✐t❤ ♦❢ t❤❡ ❣r❛♣❤ ✇✐t❤ ❛ r❛♥❞♦♠❧② ❝❤♦s❡♥

k✲♠❡r s❡q✉❡♥❝❡ Oi✳

◆♦ t✇♦ s❡q✉❡♥❝❡s ♦r t❤❡✐r ❝♦♠♣❧❡♠❡♥t Oi ✜t t♦❣❡t❤❡r✳
❈♦♥str✉❝t ❛♥ ❡❞❣❡ ❜② ❣❧✉✐♥❣ t♦❣❡t❤❡r t❤❡ k✲♠❡r ♦❢ t❤❡ st❛rt✐♥❣

♥♦❞❡ ❛♥❞ t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ t❤❡ ❡♥❞ ♥♦❞❡✳

SLIDE 39

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ❆❞❧❡♠❛♥ ✭Pr❡♣❛r❛t✐♦♥s✮

■❞❡♥t✐❢② ❡✈❡r② ♥♦❞❡ ✇✐t❤ ♦❢ t❤❡ ❣r❛♣❤ ✇✐t❤ ❛ r❛♥❞♦♠❧② ❝❤♦s❡♥

k✲♠❡r s❡q✉❡♥❝❡ Oi✳

◆♦ t✇♦ s❡q✉❡♥❝❡s ♦r t❤❡✐r ❝♦♠♣❧❡♠❡♥t Oi ✜t t♦❣❡t❤❡r✳
❈♦♥str✉❝t ❛♥ ❡❞❣❡ ❜② ❣❧✉✐♥❣ t♦❣❡t❤❡r t❤❡ k✲♠❡r ♦❢ t❤❡ st❛rt✐♥❣

♥♦❞❡ ❛♥❞ t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ t❤❡ ❡♥❞ ♥♦❞❡✳

❚❤✐s ❝♦♥str✉❝t✐♦♥ ♣r❡s❡r✈❡s ♦r✐❡♥t❛t✐♦♥✳

SLIDE 40

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ❆❞❧❡♠❛♥ ✭❆❧❣♦r✐t❤♠✮

❙t❡♣ ❆❧❣♦r✐t❤♠ ❘❡❛❧✐③❛t✐♦♥ ✶

❡♥❡r❛t❡

r❛♥❞♦♠ ♣❛t❤s t❤r♦✉❣❤ t❤❡ ❣r❛♣❤ ▼✐① t♦❣❡t❤❡r ❛❧❧ ❡❞❣❡s ❛♥❞ ❛❧❧ ♥♦❞❡s ❡①❝❡♣t vin ❛♥❞ vout✳ ❚❤❡ ❧✐❣❛t✐♦♥ r❡❛❝t✐♦♥ ❣❡♥❡r❛t❡s ✭❛❧❧✦❄✮ ♣♦ss✐❜❧❡ ♣❛t❤s ✐♥ t❤❡ ♥❡t✇♦r❦

SLIDE 41

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ❆❞❧❡♠❛♥ ✭❆❧❣♦r✐t❤♠✮

❙t❡♣ ❆❧❣♦r✐t❤♠ ❘❡❛❧✐③❛t✐♦♥ ✶

❡♥❡r❛t❡

r❛♥❞♦♠ ♣❛t❤s t❤r♦✉❣❤ t❤❡ ❣r❛♣❤ ▼✐① t♦❣❡t❤❡r ❛❧❧ ❡❞❣❡s ❛♥❞ ❛❧❧ ♥♦❞❡s ❡①❝❡♣t vin ❛♥❞ vout✳ ❚❤❡ ❧✐❣❛t✐♦♥ r❡❛❝t✐♦♥ ❣❡♥❡r❛t❡s ✭❛❧❧✦❄✮ ♣♦ss✐❜❧❡ ♣❛t❤s ✐♥ t❤❡ ♥❡t✇♦r❦ ✷ ❑❡❡♣ ♦♥❧② t❤♦s❡ ♣❛t❤s ✇❤✐❝❤ ❜❡❣✐♥ ✇✐t❤ vin ❛♥❞ ❡♥❞ ✇✐t❤ vout ❯s❡ P❈❘ ✇✐t❤ ♣r✐♠❡rs ❢♦r Oin ❛♥❞ Oout✳ ❚❤✉s ❥✉st ♠♦❧❡❝✉❧❡s ✇❤✐❝❤ ❜❡❣✐♥ ❛t vin ❛♥❞ ❡♥❞ ❛t vout ❛r❡ ❛♠♣❧✐✜❡❞

SLIDE 42

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

✸ ■❢ t❤❡ ❣r❛♣❤ ❤❛s n ♥♦❞❡s✱ t❤❡♥ ❦❡❡♣ ♦♥❧② t❤♦s❡ ♣❛t❤s ✇❤✐❝❤ ❡①❛❝t❧② ❡♥t❡r n ♥♦❞❡s ❘✉♥ t❤❡ ❛❜♦✈❡ ♣r♦❞✉❝t t❤r♦✉❣❤ ❛♥ ❛❣r♦s❡ ❣❡❧ ❛♥❞ s❡♣❛r❛t❡ t❤❡ (n − 1)k✲ s❡q✉❡♥❝❡s✳ ❋♦r ♣✉r✐✜❝❛t✐♦♥ ②♦✉ ❝❛♥ st❛rt ❛t 2 ❛❣❛✐♥✳

SLIDE 43

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

✸ ■❢ t❤❡ ❣r❛♣❤ ❤❛s n ♥♦❞❡s✱ t❤❡♥ ❦❡❡♣ ♦♥❧② t❤♦s❡ ♣❛t❤s ✇❤✐❝❤ ❡①❛❝t❧② ❡♥t❡r n ♥♦❞❡s ❘✉♥ t❤❡ ❛❜♦✈❡ ♣r♦❞✉❝t t❤r♦✉❣❤ ❛♥ ❛❣r♦s❡ ❣❡❧ ❛♥❞ s❡♣❛r❛t❡ t❤❡ (n − 1)k✲ s❡q✉❡♥❝❡s✳ ❋♦r ♣✉r✐✜❝❛t✐♦♥ ②♦✉ ❝❛♥ st❛rt ❛t 2 ❛❣❛✐♥✳ ✹ ❑❡❡♣ ❛❧❧ ♣❛t❤s ✇❤✐❝❤ ❡♥t❡r ❡❛❝❤ ♥♦❞❡ ❛t ❧❡❛st ♦♥❝❡✳ ❊①tr❛❝t ❛❧❧ s❡q✉❡♥❝❡s ❝♦♥t❛✐♥✐♥❣ Oi ✇✐t❤ ❛ ♠❛❣♥❡t✐❝ ❜❡❛❞ s②st❡♠ ❢♦r ❛❧❧ ❞✐✛❡r❡♥t i

SLIDE 44

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

✸ ■❢ t❤❡ ❣r❛♣❤ ❤❛s n ♥♦❞❡s✱ t❤❡♥ ❦❡❡♣ ♦♥❧② t❤♦s❡ ♣❛t❤s ✇❤✐❝❤ ❡①❛❝t❧② ❡♥t❡r n ♥♦❞❡s ❘✉♥ t❤❡ ❛❜♦✈❡ ♣r♦❞✉❝t t❤r♦✉❣❤ ❛♥ ❛❣r♦s❡ ❣❡❧ ❛♥❞ s❡♣❛r❛t❡ t❤❡ (n − 1)k✲ s❡q✉❡♥❝❡s✳ ❋♦r ♣✉r✐✜❝❛t✐♦♥ ②♦✉ ❝❛♥ st❛rt ❛t 2 ❛❣❛✐♥✳ ✹ ❑❡❡♣ ❛❧❧ ♣❛t❤s ✇❤✐❝❤ ❡♥t❡r ❡❛❝❤ ♥♦❞❡ ❛t ❧❡❛st ♦♥❝❡✳ ❊①tr❛❝t ❛❧❧ s❡q✉❡♥❝❡s ❝♦♥t❛✐♥✐♥❣ Oi ✇✐t❤ ❛ ♠❛❣♥❡t✐❝ ❜❡❛❞ s②st❡♠ ❢♦r ❛❧❧ ❞✐✛❡r❡♥t i ✺ ■❢ ❛♥② ♣❛t❤s r❡♠❛✐♥✱ s❛② ✏❨❡s✑✱ ♦t❤❡r✇✐s❡ ✏◆♦✑✳ ❆♠♣❧✐❢② t❤❡ ❛❜♦✈❡ ♣r♦❞✉❝t ❜② P❈❘ ❛♥❞ r✉♥ ✐t t❤r♦✉❣❤ ❛ ❣❡❧✳

SLIDE 45

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ❆❞❧❡♠❛♥ ✭❘❡❛❧✐③❛t✐♦♥✮

❆❝❝♦r❞✐♥❣ t♦ ❆❞❧❡♠❛♥ ✐t t♦♦❦ ❤✐♠ ❛❜♦✉t ✼ ❞❛②s ♦❢ ❧❛❜✲✇♦r❦ ❢♦r k = 20 ❛♥❞ ✺✵ ♣♠♦❧ ♣❡r ✐♥❣r❡❞✐❡♥t ✐♥ t❤❡ st❛rt✐♥❣ ❦✐t t♦ ❣❡t ❛ s♦❧✉t✐♦♥ ❢♦r t❤✐s ♥❡t✇♦r❦ ✉s✐♥❣ ❤✐s ❛❧❣♦r✐t❤♠✳

SLIDE 46

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❖♣❡r❛t✐♦♥s✮

❡t ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❝♦♣✐❡s

♦❢ ❛♥② s❤♦rt s✐♥❣❧❡ str❛♥❞ P❈❘

SLIDE 47

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❖♣❡r❛t✐♦♥s✮

❡t ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❝♦♣✐❡s

♦❢ ❛♥② s❤♦rt s✐♥❣❧❡ str❛♥❞ P❈❘

❡♥❡r❛t❡

❛ ❞♦✉❜❧❡ str❛♥❞ ♦❢ ❉◆❆ ❢r♦♠ ❝♦♠♣❧❡♠❡♥t❛r② s✐♥❣❧❡ str❛♥❞s P✉t t❤❡♠ t♦❣❡t❤❡r ❛♥❞ ❧❡t t❤❡♠ ❛♥♥❡❛❧

SLIDE 48

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❖♣❡r❛t✐♦♥s✮

❡t ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❝♦♣✐❡s

♦❢ ❛♥② s❤♦rt s✐♥❣❧❡ str❛♥❞ P❈❘

❡♥❡r❛t❡

❛ ❞♦✉❜❧❡ str❛♥❞ ♦❢ ❉◆❆ ❢r♦♠ ❝♦♠♣❧❡♠❡♥t❛r② s✐♥❣❧❡ str❛♥❞s P✉t t❤❡♠ t♦❣❡t❤❡r ❛♥❞ ❧❡t t❤❡♠ ❛♥♥❡❛❧ ❊①tr❛❝t s❡q✉❡♥❝❡s ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ❝❤♦s❡♥ ♣❛tt❡r♥ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ❧❡♥❣t❤ ❯s❡ ❛ ♠❛❣♥❡t✐❝ ❜❡❛❞ s②st❡♠✳ ❨♦✉ ❝❛♥ ❥✉st ❣❡t ❛❜♦✉t 90% ♦❢ t❤❡ ✇❛♥t❡❞ str❛♥❞s✱ s♦ ②♦✉ ❤❛✈❡ t♦ r✉♥ ❛♥ ❛♠♣❧✐✜❝❛t✐♦♥ st❡♣✱ ♣❡r❤❛♣s✳

SLIDE 49

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❖♣❡r❛t✐♦♥s✮

❡t ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❝♦♣✐❡s

♦❢ ❛♥② s❤♦rt s✐♥❣❧❡ str❛♥❞ P❈❘

❡♥❡r❛t❡

❛ ❞♦✉❜❧❡ str❛♥❞ ♦❢ ❉◆❆ ❢r♦♠ ❝♦♠♣❧❡♠❡♥t❛r② s✐♥❣❧❡ str❛♥❞s P✉t t❤❡♠ t♦❣❡t❤❡r ❛♥❞ ❧❡t t❤❡♠ ❛♥♥❡❛❧ ❊①tr❛❝t s❡q✉❡♥❝❡s ✇❤✐❝❤ ❝♦♥t❛✐♥ ❛ ❝❤♦s❡♥ ♣❛tt❡r♥ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ❧❡♥❣t❤ ❯s❡ ❛ ♠❛❣♥❡t✐❝ ❜❡❛❞ s②st❡♠✳ ❨♦✉ ❝❛♥ ❥✉st ❣❡t ❛❜♦✉t 90% ♦❢ t❤❡ ✇❛♥t❡❞ str❛♥❞s✱ s♦ ②♦✉ ❤❛✈❡ t♦ r✉♥ ❛♥ ❛♠♣❧✐✜❝❛t✐♦♥ st❡♣✱ ♣❡r❤❛♣s✳ ❉❡t❡❝t ✐❢ t❤❡r❡ ✐s ❛ str❛♥❞ ✐♥ ❛ t❡st t✉❜❡✳ P❈❘

SLIDE 50

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❈♦❞✐♥❣ ❙❝❤❡♠❡✮

SLIDE 51

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❈♦❞✐♥❣ ❙❝❤❡♠❡✮

r❛♣❤ ❡♥❝♦❞✐♥❣ t✇♦✲❜✐t ♥✉♠❜❡rs✳ ■❢ t❤❡ ❧❛❜❡❧ ✐s ✉♥♣r✐♠❡❞✱ ✐t

r❡♣r❡s❡♥ts ❛ 1✱ ♦t❤❡r✇✐s❡ ❛ 0

SLIDE 52

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❈♦❞✐♥❣ ❙❝❤❡♠❡✮

❊♥❝♦❞❡ t❤❡ ♣❛t❤ ❧✐❦❡ ❜❡❢♦r❡❤❛♥❞ ✐♥ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ❆❞❧❡♠❛♥

SLIDE 53

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❈♦❞✐♥❣ ❙❝❤❡♠❡✮

❊♥❝♦❞❡ t❤❡ ♣❛t❤ ❧✐❦❡ ❜❡❢♦r❡❤❛♥❞ ✐♥ t❤❡ ❛♣♣r♦❛❝❤ ♦❢ ❆❞❧❡♠❛♥
❲❡ ❣❡t ❡✈❡r② ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ xi ✇✐t❤ t❤❡ s❛♠❡

♣r♦❜❛❜✐❧✐t②

❊✈❡r② ❝❤❛✐♥ ❤❛s t❤❡ s❛♠❡ ❧❡♥❣t❤

SLIDE 54

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❇♦♦❧❡❛♥ ❖♣s✮

❖♣❡r❛t✐♦♥ ❘❡❛❧✐③❛t✐♦♥

❡t ❛❧❧ str❛♥❞s ✇❤❡r❡

xi = 1 ♦r xi = 0 r❡s♣❡❝t✐✈❡❧② ❊①tr❛❝t ♦♣❡r❛t✐♦♥ ❢♦r ❡✐t❤❡r xi ♦r xi

SLIDE 55

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❇♦♦❧❡❛♥ ❖♣s✮

❖♣❡r❛t✐♦♥ ❘❡❛❧✐③❛t✐♦♥

❡t ❛❧❧ str❛♥❞s ✇❤❡r❡

xi = 1 ♦r xi = 0 r❡s♣❡❝t✐✈❡❧② ❊①tr❛❝t ♦♣❡r❛t✐♦♥ ❢♦r ❡✐t❤❡r xi ♦r xi

❡t ❛❧❧ str❛♥❞s ❢✉❧✜❧❧✐♥❣

F1 ∧ F2

❡t ❛❧❧ str❛♥❞s ❢✉❧✜❧❧✐♥❣ F1✱ t❤❡♥ ✉s❡ t❤✐s

r❡s✉❧t ❛♥❞ ❣❡t ❛❧❧ str❛♥❞s ❢✉❧✜❧❧✐♥❣ F2✳

SLIDE 56

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❇♦♦❧❡❛♥ ❖♣s✮

❖♣❡r❛t✐♦♥ ❘❡❛❧✐③❛t✐♦♥

❡t ❛❧❧ str❛♥❞s ✇❤❡r❡

xi = 1 ♦r xi = 0 r❡s♣❡❝t✐✈❡❧② ❊①tr❛❝t ♦♣❡r❛t✐♦♥ ❢♦r ❡✐t❤❡r xi ♦r xi

❡t ❛❧❧ str❛♥❞s ❢✉❧✜❧❧✐♥❣

F1 ∧ F2

❡t ❛❧❧ str❛♥❞s ❢✉❧✜❧❧✐♥❣ F1✱ t❤❡♥ ✉s❡ t❤✐s

r❡s✉❧t ❛♥❞ ❣❡t ❛❧❧ str❛♥❞s ❢✉❧✜❧❧✐♥❣ F2✳

❡t ❛❧❧ str❛♥❞s ❢✉❧✜❧❧✐♥❣

F1 ∨ F2

❡t ❛❧❧ str❛♥❞s ❢✉❧✜❧❧✐♥❣ F1 ❛♥❞ ❝❛❧❧ t❤✐s

T1✳ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t ❛❧❧ str❛♥❞s ❢✉❧✜❧❧✐♥❣ F2✱ ❝❛❧❧ t❤✐s T2✳ ▼✐① T1 ❛♥❞ T2 t♦❣❡t❤❡r

SLIDE 57

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶

SLIDE 58

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

SLIDE 59

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

❚❡st x∨y

✳

SLIDE 60

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

❚❡st x∨y

✳

❊①tr❛❝t x = 1 ✶✵✱ ✶✶

SLIDE 61

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

❚❡st x∨y

✳

❊①tr❛❝t x = 1 ✶✵✱ ✶✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✵✶

SLIDE 62

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

❚❡st x∨y

✳

❊①tr❛❝t x = 1 ✶✵✱ ✶✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✵✶ P✉t ✐t t♦❣❡t❤❡r ❛❣❛✐♥ ✵✶✱✶✵✱✶✶

SLIDE 63

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

❚❡st x∨y

✳

❊①tr❛❝t x = 1 ✶✵✱ ✶✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✵✶ P✉t ✐t t♦❣❡t❤❡r ❛❣❛✐♥ ✵✶✱✶✵✱✶✶ ❚❡st x∨y

✳

SLIDE 64

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

❚❡st x∨y

✳

❊①tr❛❝t x = 1 ✶✵✱ ✶✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✵✶ P✉t ✐t t♦❣❡t❤❡r ❛❣❛✐♥ ✵✶✱✶✵✱✶✶ ❚❡st x∨y

✳

❊①tr❛❝t x = 0 ✵✶

SLIDE 65

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

❚❡st x∨y

✳

❊①tr❛❝t x = 1 ✶✵✱ ✶✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✵✶ P✉t ✐t t♦❣❡t❤❡r ❛❣❛✐♥ ✵✶✱✶✵✱✶✶ ❚❡st x∨y

✳

❊①tr❛❝t x = 0 ✵✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✶✵

SLIDE 66

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

❚❡st x∨y

✳

❊①tr❛❝t x = 1 ✶✵✱ ✶✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✵✶ P✉t ✐t t♦❣❡t❤❡r ❛❣❛✐♥ ✵✶✱✶✵✱✶✶ ❚❡st x∨y

✳

❊①tr❛❝t x = 0 ✵✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✶✵ P✉t ✐t t♦❣❡t❤❡r ❛❣❛✐♥ ✵✶✱✶✵

SLIDE 67

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❆♣♣r♦❛❝❤ ♦❢ ▲✐♣t♦♥ ✭❊①❛♠♣❧❡✮

❖♣❡r❛t✐♦♥ ❱❛❧✉❡s ♣r❡s❡♥t ❙t❛rt ✵✵✱ ✵✶✱ ✶✵✱ ✶✶ ❚❡st (x ∨ y)∧(x ∨ y)

✳

❚❡st x∨y

✳

❊①tr❛❝t x = 1 ✶✵✱ ✶✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✵✶ P✉t ✐t t♦❣❡t❤❡r ❛❣❛✐♥ ✵✶✱✶✵✱✶✶ ❚❡st x∨y

✳

❊①tr❛❝t x = 0 ✵✶ ❯s❡ t❤❡ r❡st ❛♥❞ ❡①tr❛❝t y = 1 ✶✵ P✉t ✐t t♦❣❡t❤❡r ❛❣❛✐♥ ✵✶✱✶✵ ❘❡s✉❧t ✵✶✱ ✶✵

SLIDE 68

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❉✐s❝✉ss✐♦♥ ✭Pr♦s✮

❚❤❡ ♠❡t❤♦❞ ✐s ♥❡✇ ❛♥❞ t❤❡r❡ ✐s ♣❡r❤❛♣s s♦♠❡ ♠♦r❡ ♣♦t❡♥t✐❛❧ ❜❡s✐❞❡ t❤❡

❜r✉t❡ ❢♦r❝❡ ❛♣♣r♦❛❝❤

SLIDE 69

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❉✐s❝✉ss✐♦♥ ✭Pr♦s✮

❚❤❡ ♠❡t❤♦❞ ✐s ♥❡✇ ❛♥❞ t❤❡r❡ ✐s ♣❡r❤❛♣s s♦♠❡ ♠♦r❡ ♣♦t❡♥t✐❛❧ ❜❡s✐❞❡ t❤❡

❜r✉t❡ ❢♦r❝❡ ❛♣♣r♦❛❝❤

❉◆❆✲❈♦♠♣✉t✐♥❣ ✐s ✭t❤❡♦r❡t✐❝❛❧❧②✮ ✈❡r② ❡♥❡r❣② ❡✣❝✐❡♥t✳

SLIDE 70

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❉✐s❝✉ss✐♦♥ ✭Pr♦s✮

❚❤❡ ♠❡t❤♦❞ ✐s ♥❡✇ ❛♥❞ t❤❡r❡ ✐s ♣❡r❤❛♣s s♦♠❡ ♠♦r❡ ♣♦t❡♥t✐❛❧ ❜❡s✐❞❡ t❤❡

❜r✉t❡ ❢♦r❝❡ ❛♣♣r♦❛❝❤

❉◆❆✲❈♦♠♣✉t✐♥❣ ✐s ✭t❤❡♦r❡t✐❝❛❧❧②✮ ✈❡r② ❡♥❡r❣② ❡✣❝✐❡♥t✳
P❡r❤❛♣s ❛ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❉◆❆✲❝♦♠♣✉t✐♥❣ ❛♥❞ ❝♦♥✈❡♥t✐♦♥❛❧ ❡❧❡❝tr♦♥✐❝

❝♦♠♣✉t❡rs ❝❛♥ ❜♦♦st t❤❡ s♣❡❡❞ t♦ ❛ ❝♦♠♣❡t✐t✐✈❡ ❧❡✈❡❧

SLIDE 71

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❉✐s❝✉ss✐♦♥ ✭❈♦♥s✮

■♥st❛♥❝❡s ❛❜♦✈❡ t❤❡ s✐③❡ ♦❢ ✶✵✵ ❞♦ ♥♦t s❡❡♠ t♦ ❜❡ ❢❡❛s✐❜❧❡ ❜❡❝❛✉s❡ ♦❢ t❤❡

❛♠♦✉♥t ♦❢ ❉◆❆ ♥❡❡❞❡❞✳

SLIDE 72

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❉✐s❝✉ss✐♦♥ ✭❈♦♥s✮

■♥st❛♥❝❡s ❛❜♦✈❡ t❤❡ s✐③❡ ♦❢ ✶✵✵ ❞♦ ♥♦t s❡❡♠ t♦ ❜❡ ❢❡❛s✐❜❧❡ ❜❡❝❛✉s❡ ♦❢ t❤❡

❛♠♦✉♥t ♦❢ ❉◆❆ ♥❡❡❞❡❞✳

❚❤✐s ❛♣♣r♦❛❝❤ ❥✉st s❤✐❢ts t❤❡ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ t♦ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❛♠♦✉♥t

♦❢ ♠❛ss✱ ✇❤✐❝❤ ♥❡❡❞s ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❢♦r ♣r♦❞✉❝t✐♦♥✳

SLIDE 73

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❉✐s❝✉ss✐♦♥ ✭❈♦♥s✮

■♥st❛♥❝❡s ❛❜♦✈❡ t❤❡ s✐③❡ ♦❢ ✶✵✵ ❞♦ ♥♦t s❡❡♠ t♦ ❜❡ ❢❡❛s✐❜❧❡ ❜❡❝❛✉s❡ ♦❢ t❤❡

❛♠♦✉♥t ♦❢ ❉◆❆ ♥❡❡❞❡❞✳

❚❤✐s ❛♣♣r♦❛❝❤ ❥✉st s❤✐❢ts t❤❡ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ t♦ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❛♠♦✉♥t

♦❢ ♠❛ss✱ ✇❤✐❝❤ ♥❡❡❞s ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❢♦r ♣r♦❞✉❝t✐♦♥✳

❚❤❡ ❛♣♣r♦❛❝❤ ✐s ♣r♦❜❛❜✐❧✐st✐❝✱ ✐✳❡✳ ❣❡tt✐♥❣ ❛ ♥❡❣❛t✐✈❡ r❡s✉❧t ❞♦❡s ♥♦t ✐♠♣❧②

♥♦♥✲❡①✐st❡♥❝❡✳

SLIDE 74

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❉✐s❝✉ss✐♦♥ ✭❈♦♥s✮

■♥st❛♥❝❡s ❛❜♦✈❡ t❤❡ s✐③❡ ♦❢ ✶✵✵ ❞♦ ♥♦t s❡❡♠ t♦ ❜❡ ❢❡❛s✐❜❧❡ ❜❡❝❛✉s❡ ♦❢ t❤❡

❛♠♦✉♥t ♦❢ ❉◆❆ ♥❡❡❞❡❞✳

❚❤✐s ❛♣♣r♦❛❝❤ ❥✉st s❤✐❢ts t❤❡ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ t♦ ❛♥ ❡①♣♦♥❡♥t✐❛❧ ❛♠♦✉♥t

♦❢ ♠❛ss✱ ✇❤✐❝❤ ♥❡❡❞s ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❢♦r ♣r♦❞✉❝t✐♦♥✳

❚❤❡ ❛♣♣r♦❛❝❤ ✐s ♣r♦❜❛❜✐❧✐st✐❝✱ ✐✳❡✳ ❣❡tt✐♥❣ ❛ ♥❡❣❛t✐✈❡ r❡s✉❧t ❞♦❡s ♥♦t ✐♠♣❧②

♥♦♥✲❡①✐st❡♥❝❡✳

❚❤❡r❡ ❡①✐st ✈❡r② ❢❛st ❛♥❞ ✈❡r② ❣♦♦❞ ✭❛♣♣r♦①✐♠❛t✐✈❡✮ s♦❧✈❡rs ❢♦r t❤❡ ♠♦st

✐♠♣♦rt❛♥t ◆P✲♣r♦❜❧❡♠s✳

SLIDE 75

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❈♦♥❝❧✉s✐♦♥

❉◆❆✲❝♦♠♣✉t✐♥❣ s❡❡♠s t♦ ❜❡ ❛ ♥✐❝❡ t♦② ✇❤✐❝❤ ❝❛♥♥♦t ❝♦♣❡ ✇✐t❤ t❤❡ ❦♥♦✇♥ st❛♥❞❛r❞ ♠❡t❤♦❞s ❢♦r t❤❡ s♦❧✉t✐♦♥ ♦❢ ◆P ❛♥❞ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s✳

SLIDE 76

❋r❛♥❦ ❇❛✉❡r✱ ❉◆❆✲❈♦♠♣✉t✐♥❣ ❙t✳ ❏♦❤❛♥♥✱ ✶✳✾✳✷✵✵✷ ✲ ✶✹✳✾✳✷✵✵✷

❘❡❢❡r❡♥❝❡s

❆❞❧❡♠❛♥✱ ▲✳ ▼✳ ▼♦❧❡❝✉❧❛r ❝♦♠♣✉t❛t✐♦♥ ♦❢ s♦❧✉t✐♦♥s t♦ ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦❜❧❡♠s✳ ❙❝✐❡♥❝❡ ✷✻✻ ✭✶✾✾✹✮✱ ✶✵✷✶✕✶✵✷✸✳

❛r❡②✱ ▼✳ ❘✳✱ ❛♥❞ ❏♦❤♥s♦♥✱ ❉✳ ❙✳ ❈♦♠♣✉t❡rs ❛♥❞ ■♥tr❛❝t❛❜✐❧✐t②✱ ❆
✉✐❞❡ t♦ t❤❡ ❚❤❡♦r② ♦❢ ◆P✲❈♦♠♣❧❡t❡♥❡ss✳ ❲✳ ❍✳ ❋r❡❡♠❛♥ ❛♥❞ ❈♦♠♣❛♥②✱