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SLIDE 1

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

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SLIDE 2

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❙✉♠♠❛r②

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥ ❬❈❤❛♥❣✲▲❡❡ ❈❤✳ ✺✳✸❪ ❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠ ❬❈❤❛♥❣✲▲❡❡ ❈❤✳ ✺✳✹❪

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SLIDE 3

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❋✐♥❞✐♥❣ ❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧s

♥❡❡❞ ♦❢ ✉♥✐✜❝❛t✐♦♥ ❚♦ ❛♣♣❧② r❡s♦❧✉t✐♦♥ ✇❡ ♥❡❡❞ t♦ ✜♥❞ ❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧s✿ ▲✶ = P, ▲✷ = ¬P ❚❤✐s ✐s ♥♦t ❛ ♣r♦❜❧❡♠ ❢♦r ❣r♦✉♥❞ ♦r ♣r♦♣♦s✐t✐♦♥❛❧ ❝❧❛✉s❡s ❲❤❡♥ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥✈♦❧✈❡❞ t❤✐♥❣s ❣❡t ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ■t ✐s ♥♦t ♦❜✈✐♦✉s t♦ ❞❡❝✐❞❡ ✇❤❡t❤❡r t✇♦ ❧✐t❡r❛❧s ❛r❡ ❝♦♠♣❧❡♠❡♥t❛r②

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SLIDE 4

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧s ✇✐t❤ ✈❛r✐❛❜❧❡s✮ ❈✶ = P(①) ∨ ◗(①), ❈✷ = ¬P(❢ (②)) ∨ ❘(②) ❚❤❡r❡ ✐s ♥♦ ❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧✱ ❜✉t✿ ❈ ′

✶ = P(① = ❢ (❛))∨◗(① = ❢ (❛)), ❈ ′ ✷ = ¬P(❢ (② = ❛))∨❘(② = ❛)

❚❤❡♥ ❈ ′

✶ ❛♥❞ ❈ ′ ✷ ❛r❡ ❣r♦✉♥❞ ✐♥st❛♥❝❡s ♦❢ ❈✶ ❛♥❞ ❈✷✱ ❛♥❞

P(❢ (❛)) ❛♥❞ ¬P(❢ (❛)) ❛r❡ ❝♦♠♣❧❡♠❡♥t❛r②✳

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SLIDE 5

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡✱ ❝♦♥t❞✳

❊①❛♠♣❧❡ ✭❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧s ✇✐t❤ ✈❛r✐❛❜❧❡s✮ ❚❤❡♥ ✇❡ ❝❛♥ ❛♣♣❧② r❡s♦❧✉t✐♦♥ ❛♥❞ ♦❜t❛✐♥✿ P(❢ (❛)) ∨ ◗(❢ (❛)) ¬P(❢ (❛)) ∨ ❘(❛) ◗(❢ (❛)) ∨ ❘(❛) ❲❤❡r❡ ❈ ′

✸ = ◗(❢ (❛)) ∨ ❘(❛) ✐s ❛ r❡s♦❧✈❡♥t ❢♦r ❈ ′ ✶ ❛♥❞ ❈ ′ ✷

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SLIDE 6

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡✱ ❝♦♥t❞✳

❊①❛♠♣❧❡ ✭❝♦♠♣❧❡♠❡♥t❛r② ❧✐t❡r❛❧s ✇✐t❤ ✈❛r✐❛❜❧❡s✮ ▼♦r❡ ✐♥ ❣❡♥❡r❛❧✱ ✇❡ ❝❛♥ s✉❜st✐t✉t❡ ① = ❢ (②) ✐♥ ❈✶ ❛♥❞ ♦❜t❛✐♥ ❈ ∗

✶ = P(❢ (②)) ∨ ◗(❢ (②))

P(❢ (②)) ∨ ◗(❢ (②)) ¬P(❢ (②)) ∨ ❘(②) ◗(❢ (②)) ∨ ❘(②) ❈ ∗

✶ ✐s ❛♥ ✐♥st❛♥❝❡ ♦❢ ❈✶ ❛♥❞ ❈ ′ ✸ ✐s ❛ ✭❣r♦✉♥❞✮ ✐♥st❛♥❝❡ ♦❢

❈✸ = ◗(❢ (②)) ∨ ❘(②) ❇② s✉❜st✐t✉t✐♥❣ ❛♣♣r♦♣✐❛t❡ t❡r♠s ✇❡ ❝❛♥ ❣❡♥❡r❛t❡ ♥❡✇ ❝❧❛✉s❡s ❢♦r ❈✶ ❛♥❞ ❈✷ ❇② ❛♣♣❧②✐♥❣ r❡s♦❧✉t✐♦♥ t♦ s✉❝❤ ❝❧❛✉s❡s ✇❡ ♦❜t❛✐♥ ♦t❤❡r ❝❧❛✉s❡s ✇❤✐❝❤ ✇✐❧❧ ❛❧❧ ❜❡ ✐♥st❛♥❝❡ ♦❢ ❈✸✳ ❈✸ ✐s t❤❡ ♠♦st ❣❡♥❡r❛❧ ❝❧❛✉s❡ ❛♥❞ ✐s ❝❛❧❧❡❞ ❛ r❡s♦❧✈❡♥t ♦❢ ❈✶ ❛♥❞ ❈✷✳

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SLIDE 7

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❙✉❜st✐t✉t✐♦♥s

❙✉❜st✐t✉t✐♦♥s ❚♦ ♦❜t❛✐♥ ❛ r❡s♦❧✈❡♥t ❢r♦♠ ❝❧❛✉s❡s ❝♦♥t❛✐♥✐♥❣ ✈❛r✐❛❜❧❡s ✇❡ ♥❡❡❞ t♦ s✉❜st✐t✉t❡ ✈❛r✐❛❜❧❡s ✇✐t❤ t❡r♠s✱ ❛♥❞ ❛♣♣❧② r❡s♦❧✉t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✭❙✉❜st✐t✉t✐♦♥✮ ❆ s✉❜st✐t✉t✐♦♥ ✐s ❛ ✜♥✐t❡ s❡t {t✶/✈✶, · · · , t♥/✈♥} ✇❤❡r❡ ❡✈❡r② ✈✐ ✐s ❛ ✈❛r✐❛❜❧❡ ❛♥❞ ❡✈❡r② t✐ ✐s ❛ t❡r♠✱ ❞✐✛❡r❡♥t ❢r♦♠ ✈✐✱ ❛♥❞ ♥♦ t✇♦ ❡❧❡♠❡♥ts ✐♥ t❤❡ s❡t ❤❛✈❡ t❤❡ s❛♠❡ ✈❛r✐❛❜❧❡ ❛❢t❡r t❤❡ / s②♠❜♦❧✳

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SLIDE 8

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡✱ ❝♦♥t❞✳

❊①❛♠♣❧❡ ✭❙✉❜st✐t✉t✐♦♥✮ {❢ (③)/①, ②/③} ✐s ❛ s✉❜st✐t✉t✐♦♥ {❛/①, ❣(②)/②, ❢ (❣(❜))/③} ✐s ❛ s✉❜st✐t✉t✐♦♥ {②/①, ❣(❜)/②} ✐s ❛ s✉❜st✐t✉t✐♦♥ {❛/①, ❣(②)/①, ❢ (❣(❜))/③} ✐s ♥♦t ❛ s✉❜st✐t✉t✐♦♥ {❣(②)/①, ③/❢ (❣(❜))} ✐s ♥♦t ❛ s✉❜st✐t✉t✐♦♥

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SLIDE 9

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

  • r♦✉♥❞ ❛♥❞ ❊♠♣t② s✉❜st✐t✉t✐♦♥s

❉❡✜♥✐t✐♦♥ ✭●r♦✉♥❞ s✉❜st✐t✉t✐♦♥✮ ❆ s✉❜st✐t✉t✐♦♥ {t✶/✈✶, · · · , t♥/✈♥} ✐s ❣r♦✉♥❞ ✇❤❡♥ {t✶, · · · , t♥} ❛r❡ ❛❧❧ ❣r♦✉♥❞ t❡r♠s✳ ❊①❛♠♣❧❡ ✭●r♦✉♥❞ ❙✉❜st✐t✉t✐♦♥✮ {❢ (❛)/①, ❜/③} ✐s ❛ ❣r♦✉♥❞ s✉❜st✐t✉t✐♦♥ {❛/①, ❣(❜)/②, ❢ (❣(❜))/③} ✐s ❛ ❣r♦✉♥❞ s✉❜st✐t✉t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭❊♠♣t② s✉❜st✐t✉t✐♦♥✮ ❆ s✉❜st✐t✉t✐♦♥ t❤❛t ❝♦♥t❛✐♥s ♥♦ ❡❧❡♠❡♥t {} ✐s t❤❡ ❡♠♣t② s✉❜st✐t✉t✐♦♥✱ ✇❡ ❞❡♥♦t❡ t❤❡ ❡♠♣t② s✉❜st✐t✉t✐♦♥ ✇✐t❤ ǫ✳

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SLIDE 10

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

■♥st❛♥❝❡s ♦❢ ❝❧❛✉s❡s

❉❡✜♥✐t✐♦♥ ✭■♥st❛♥❝❡✮ ▲❡t θ = {t✶/✈✶, · · · , t♥/✈♥} ❜❡ ❛ s✉❜st✐t✉t✐♦♥ ❛♥❞ ❧❡t ❊ ❜❡ ❛♥ ❡①♣r❡ss✐♦♥✳ ❚❤❡♥ ❊θ ✐s ❛♥ ❡①♣r❡ss✐♦♥ ♦❜t❛✐♥❡❞ ❜② r❡♣❧❛❝✐♥❣ s✐♠✉❧t❛♥❡♦✉s❧② ❛❧❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ❡✈❡r② ✈✐✱ ✶ ≤ ✐ ≤ ♥✱ ✐♥ ❊ ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❡r♠ t✐✳ ❊θ ✐s ❛♥ ✐♥st❛♥❝❡ ♦❢ ❊✳

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SLIDE 11

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭■♥st❛♥❝❡s✮ ▲❡t θ = {❛/①, ❢ (❜)/②, ❝/③} ❛♥❞ ❊ = P(①, ②, ③) t❤❡♥ ❊θ = P(❛, ❢ (❜), ❝) ✐s ❛♥ ✐♥st❛♥❝❡ ♦❢ ❊ ▲❡t λ = {❢ (❢ (❛))/①} ❛♥❞ ❈ = P(①) ∨ ◗(❣(①)) t❤❡♥

  • λ = P(❢ (❢ (❛)) ∨ ◗(❣(❢ (❢ (❛)))) ✐s ❛♥ ✐♥st❛♥❝❡ ♦❢ ❈

▲❡t γ = {②/①, ❢ (❜)/②} ❛♥❞ ❘ = P(①) ∨ ◗(②) t❤❡♥ ❘γ = P(②) ∨ ◗(❢ (❜)) ✐s ❛♥ ✐♥st❛♥❝❡ ♦❢ ❘ ◆♦t❡s ❉❡✜♥✐t✐♦♥ ♦❢ ❣r♦✉♥❞ ✐♥st❛♥❝❡ ♦❢ ❛ ❝❧❛✉s❡ ✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❞❡✜♥✐t✐♦♥ ♦❢ ✐♥st❛♥❝❡ ❣✐✈❡♥ ❤❡r❡✳ s✉❜st✐t✉t✐♦♥ ✐s s✐♠✉❧t❛♥❡♦✉s✳ ■❢ ♥♦t s✐♠✉❧t❛♥❡♦✉s ✇❡ ❝♦✉❧❞ ❤❛✈❡ ❞✐✛❡r❡♥t ♦✉t❝♦♠❡s ❘γ = P(① ← ② ← ❢ (❜)) ∨ ◗(② ← ❢ (❜))

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SLIDE 12

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❈♦♠♣♦s✐t✐♦♥

❈♦♠♣♦s✐t✐♦♥ ▲❡t θ = {t✶/①✶, · · · , t♥/①♥} ❛♥❞ λ = {✉✶/②✶, · · · , ✉♠/②♠} ❜❡ t✇♦ s✉❜st✐t✉t✐♦♥s✳ ❚❤❡♥ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ θ ❛♥❞ λ ✐s ❞❡♥♦t❡❞ ❜② θ ◦ λ✱ ❛♥❞ ✐s ♦❜t❛✐♥❡❞ ❜② ❜✉✐❧❞✐♥❣ t❤❡ s❡t {t✶λ/①✶, · · · , t♥λ/①♥, ✉♥/②✶, · · · , ✉♠/②♠} ❛♥❞ ❞❡❧❡t✐♥❣ t❤❡ ❢♦❧✲ ❧♦✇✐♥❣ ❡❧❡♠❡♥ts✿ ❛♥② ❡❧❡♠❡♥t t❥λ/①❥ s✉❝❤ t❤❛t t❥λ = ①❥ ❛♥② ❡❧❡♠❡♥t ✉✐/②✐ s✉❝❤ t❤❛t ②✐ ✐s ✐♥ {①✶, · · · , ①♥}

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SLIDE 13

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❝♦♠♣♦s✐t✐♦♥✮

  • ✐✈❡♥✿

θ = {t✶/①✶, t✷/①✷} = {❢ (②)/①, ③/②} λ = {✉✶/②✶, ✉✷/②✷, ✉✸/②✸} = {❛/①, ❜/②, ②/③} ❲❡ ❜✉✐❧❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t✿

{t✶λ/①✶, t✷λ/①✷, ✉✶/②✶, ✉✷/②✷, ✉✸/②✸} = {❢ (❜)/①, ②/②, ❛/①, ❜/②, ②/③}

❲❡ r❡♠♦✈❡ t❤❡ ♣r♦♣❡r ❡❧❡♠❡♥ts✿ t❥λ/①❥ s✉❝❤ t❤❛t t❥λ = ①❥ t❤❡r❡❢♦r❡ ✇❡ r❡♠♦✈❡ ②/② ✉✐/②✐ s✉❝❤ t❤❛t ②✐ ✐s ✐♥ {①✶, · · · , ①♥} t❤❡r❡❢♦r❡ ✇❡ r❡♠♦✈❡ ❛/① ❛♥❞ ❜/② ❋✐♥❛❧❧②✱ θ ◦ λ = {❢ (❜)/①, ②/③}

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SLIDE 14

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

Pr♦♣❡rt✐❡s ♦❢ ❝♦♠♣♦s✐t✐♦♥

❛ss♦❝✐❛t✐✈❡♥❡ss ▲❡t θ✱ λ ❛♥❞ µ ❜❡ s✉❜st✐t✉t✐♦♥s ✇❡ ❤❛✈❡ t❤❛t (θ◦λ)◦µ = θ◦(λ◦µ) ❊①❛♠♣❧❡ ▲❡t θ = {❢ (②)/①}✱ λ = {③/②} ❛♥❞ µ = {❛/③}✳ ❲❡ ❤❛✈❡ φ = θ ◦ λ = {❢ (②)λ/①, ③/②} = {❢ (③)/①, ③/②} ❛♥❞ φ ◦ µ = {❢ (③)µ/①, ③µ/②, ❛/③} = {❢ (❛)/①, ❛/②, ❛/③} φ′ = λ ◦ µ = {③µ/②, ❛/③} = {❛/②, ❛/③} ❛♥❞ θ ◦ φ′ = {❢ (②)φ′/①, ❛/②, ❛/③} = {❢ (❛)/①, ❛/②, ❛/③} ■❞❡♥t✐t② ♦❢ ❡♠♣t② s✉❜st✐t✉t✐♦♥ ▲❡t θ ❜❡ ❛ s✉❜st✐t✉t✐♦♥ t❤❡♥ ǫ ◦ θ = θ ◦ ǫ = θ

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SLIDE 15

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❯♥✐✜❝❛t✐♦♥ ❛♥❞ s✉❜st✐t✉t✐♦♥s

❯♥✐❢②✐♥❣ ❡①♣r❡ss✐♦♥s ✉s✐♥❣ s✉❜st✐t✉t✐♦♥s ❲❤❡♥ ✉s✐♥❣ r❡s♦❧✉t✐♦♥ ✇❡ ♥❡❡❞ t♦ ♠❛t❝❤ ♦r ✉♥✐❢② ❡①♣r❡ss✐♦♥s t♦ ✜♥❞ ❝♦♠♣❧❡♠❡♥t❛r② ♣❛✐rs ♦❢ ❧✐t❡r❛❧s✳ ❚❤✐s ❝❛♥ ❜❡ ❞♦♥❡ ❜② ❛♣♣❧②✐♥❣ ♣r♦♣❡r s✉❜st✐t✉t✐♦♥s t♦ r❡❧❡✈❛♥t ❡①♣r❡ss✐♦♥ t♦ ♠❛❦❡ t❤❡♠ ✐❞❡♥t✐❝❛❧ ❊①❛♠♣❧❡ ▲❡t ❈✶ = P(①) ∨ ◗(①) ❛♥❞ ❈✷ = ¬P(❢ (②)) ∨ ◗(②))✱ ❛♥❞ ❧❡t ▲✶ = P(①)✱ ¬▲✷ = P(❢ (②)) ❇② ❛♣♣❧②✐♥❣ θ = {❢ (②)/①}✳ ❲❡ ❤❛✈❡ t❤❛t ▲✶θ = ¬▲✷θ = P(❢ (②))✳

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SLIDE 16

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❯♥✐✜❡r

❉❡✜♥✐t✐♦♥ ✭❯♥✐✜❡r✮ ❆ s✉❜st✐t✉t✐♦♥ θ ✐s ❝❛❧❧❡❞ ❛ ✉♥✐✜❡r ❢♦r ❛ s❡t {❊✶, · · · , ❊❦} ✐✛ ❊✶θ = ❊✷θ = · · · = ❊❦θ✳ ❚❤❡ s❡t {❊✶, · · · , ❊❦} ✐s ✉♥✐✜❛❜❧❡ ✐✛ t❤❡r❡ ❡①✐sts ❛ ✉♥✐✜❡r ❢♦r ✐t✳ ❊①❛♠♣❧❡ ❚❤❡ s❡t {P(①), P(❢ (②))} ✐s ✉♥✐✜❛❜❧❡ ❛♥❞ θ = {❢ (②)/①} ✐s ❛ ✉♥✐✜❡r ❢♦r ✐t✱ ❜❡❝❛✉s❡ P(①)θ = P(❢ (②))θ = P(❢ (②))

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SLIDE 17

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

▼♦st ●❡♥❡r❛❧ ❯♥✐✜❡r

❉❡✜♥✐t✐♦♥ ✭▼●❋✮ ❆ ✉♥✐✜❡r θ ✐s ❛ ♠♦st ❣❡♥❡r❛❧ ✉♥✐✜❡r ❢♦r ❛ s❡t {❊✶, · · · , ❊❦} ✐✛ ❢♦r ❡❛❝❤ ✉♥✐✜❡r λ t❤❡r❡ ❡①✐sts ❛ s✉❜st✐t✉t✐♦♥ µ s✉❝❤ t❤❛t λ = θ ◦ µ✳ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ s❡t {P(①), P(❢ (②))} ❛♥❞ λ = {❢ (❢ (③))/①, ❢ (③)/②}✳ λ ✐s ❛ ✉♥✐✜❡r ❜❡❝❛✉s❡ P(①)λ = P(❢ (②))λ = P(❢ (❢ (③))) θ = {❢ (②)/①} ✐s ❛ ✉♥✐✜❡r ❛♥❞ ✐s ❛ ♠♦st ❣❡♥❡r❛❧ ✉♥✐✜❡r ❢♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ✜♥❞ µ = {❢ (③)/②} s✉❝❤ t❤❛t θ ◦ µ = {❢ (②)µ/①, ❢ (③)/②} = {❢ (❢ (③))/①, ❢ (③)/②} = λ

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SLIDE 18

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡✿ ▼♦st ●❡♥❡r❛❧ ❯♥✐✜❡r

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ s❡t {P(❛, ②), P(①, ❢ (❜))} ❚❤❡ s❡t ✐s ✉♥✐✜❛❜❧❡ θ = {❛/①, ❢ (❜)/②} ✐s ❛ ✭♠♦st ❣❡♥❡r❛❧✮ ✉♥✐✜❡r ❢♦r t❤❡ s❡t✳

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SLIDE 19

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❆♥ ❛❧❣♦r✐t❤♠ ❢♦r ❯♥✐✜❝❛t✐♦♥

❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠ ◆❡❡❞ ❛ ♣r♦❝❡❞✉r❡ t♦ ✜♥❞ ❛ ▼●❯ ❣✐✈❡♥ ❛ s❡t ♦❢ ❡①♣r❡ss✐♦♥s ❘❡q✉✐r❡♠❡♥ts✿

st♦♣ ❛❢t❡r ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ st❡♣s r❡t✉r♥ ❛♥ ▼●❯ ✐❢ t❤❡ s❡t ✐s ✉♥✐✜❛❜❧❡ st❛t❡ t❤❛t t❤❡ s❡t ✐s ♥♦t ✉♥✐✜❛❜❧❡ ♦t❤❡r✇✐s❡

❚❤❡r❡ ❛r❡ ♠❛♥② ♣♦ss✐❜✐❧✐t✐❡s ❲❡ ❣♦ ❢♦r ❛ r❡❝✉rs✐✈❡ ♣r♦❝❡❞✉r❡✳

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SLIDE 20

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❇❛s✐❝ ✐❞❡❛s

❇❛s✐❝ ✐❞❡❛s

  • ✐✈❡♥ ❛ s❡t ♦❢ ❡①♣r❡ss✐♦♥s {❊✶, · · · , ❊❦}

❋✐♥❞ ❛ ❞✐s❛❣r❡❡♠❡♥t s❡t ❇✉✐❧❞ ❛ s✉❜st✐t✉t✐♦♥ t❤❛t ❝❛♥ ❡❧✐♠✐♥❛t❡ t❤❡ ❞✐s❛❣r❡❡♠❡♥t

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SLIDE 21

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❉✐s❛❣r❡❡♠❡♥t ❡❧✐♠✐♥❛t✐♦♥✿ ❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❉✐s❛❣r❡❡♠❡♥t ❡❧✐♠✐♥❛t✐♦♥✮ ❈♦♥s✐❞❡r t❤❡ s❡t {P(❛), P(①)}✳ ❚❤✐s ❡①♣r❡ss✐♦♥s ❛r❡ ♥♦t ✐❞❡♥t✐❝❛❧✳ ❚❤❡② ❞✐s❛❣r❡❡ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❛r❣✉♠❡♥ts ❛ ❛♥❞ ① ❚❤❡ ❞✐s❛❣r❡❡♠❡♥t s❡t ❤❡r❡ ✐s {❛, ①} ❙✐♥❝❡ ① ✐s ❛ ✈❛r✐❛❜❧❡✱ ✇❡ ❝❛♥ ❡❧✐♠✐♥❛t❡ t❤✐s ❞✐s❛❣r❡❡♠❡♥t ❜② ✉s✐♥❣ t❤❡ s✉❜st✐t✉t✐♦♥ θ = {❛/①} P(❛)θ = P(①)θ = P(❛)

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SLIDE 22

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❉✐s❛❣r❡❡♠❡♥t s❡t

❉❡✜♥✐t✐♦♥ ✭❉✐s❛❣r❡❡♠❡♥t ❙❡t✮ ❚❤❡ ❞✐s❛❣r❡❡♠❡♥t s❡t ♦❢ ❛ ♥♦♥❡♠♣t② s❡t ♦❢ ❡①♣r❡ss✐♦♥s ❲ ✐s ♦❜t❛✐♥❡❞ ❜② ✜♥❞✐♥❣ t❤❡ ✜rst ♣♦s✐t✐♦♥ ✭st❛rt✐♥❣ ❢r♦♠ t❤❡ ❧❡❢t✮ ❛t ✇❤✐❝❤ ♥♦t ❛❧❧ t❤❡ ❡①♣r❡ss✐♦♥s ✐♥ t❤❡ ❲ ❤❛✈❡ t❤❡ s❛♠❡ s②♠❜♦❧✳ ❲❡ t❤❡♥ ❡①tr❛❝t✱ ❢r♦♠ ❡❛❝❤ ❡①♣r❡ss✐♦♥✱ t❤❡ s✉❜✲❡①♣r❡ss✐♦♥ t❤❛t ❜❡❣✐♥s ✇✐t❤ t❤❡ s②♠❜♦❧ ♦❝❝✉♣②✐♥❣ t❤❛t ♣♦s✐t✐♦♥✳ ❚❤❡ s❡t ♦❢ t❤❡s❡ s✉❜✲❡①♣r❡ss✐♦♥s ✐s t❤❡ ❉✐s❛❣r❡❡♠❡♥t ❙❡t✳ ❊①❛♠♣❧❡ ✭❉✐s❛❣r❡❡♠❡♥t ❙❡t✮ ❈♦♥s✐❞❡r t❤❡ s❡t {P(❛), P(①)}✱ t❤❡ ❉✐s❛❣r❡❡♠❡♥t ❙❡t ✐s {❛, ①}✳❜❡❝❛✉s❡ t❤❡ ✜rst ♣♦s✐t✐♦♥ ❛t ✇❤✐❝❤ t❤❡ str✐♥❣ ♦❢ s②♠❜♦❧s P(❛) ❛♥❞ P(①) ❞✐✛❡r ✐s t❤❡ ♣♦s✐t✐♦♥ ♥✉♠❜❡r ✸✳ ❚❤❡ s✉❜ ❡①♣r❡ss✐♦♥ st❛rt✐♥❣ ❢r♦♠ ♣♦s✐t✐♦♥ ✸ ✐s ❛ ❛♥❞ ① r❡s♣❡❝t✐✈❡❧②✳

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SLIDE 23

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❉✐s❛❣r❡❡♠❡♥t ❙❡t✮ ❋✐♥❞ t❤❡ ❉✐s❛❣r❡❡♠❡♥t ❙❡t ❢♦r ❲ = {P(①, ❢ (②, ③)), P(①, ❛), P(①, ❣(❤(❦(①))))} ❙♦❧✳ ❉ ❢ ② ③ ❛ ❣ ❤ ❦ ①

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SLIDE 24

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❉✐s❛❣r❡❡♠❡♥t ❙❡t✮ ❋✐♥❞ t❤❡ ❉✐s❛❣r❡❡♠❡♥t ❙❡t ❢♦r ❲ = {P(①, ❢ (②, ③)), P(①, ❛), P(①, ❣(❤(❦(①))))} ❙♦❧✳ ❉ = {❢ (②, ③), ❛, ❣(❤(❦(①)))}

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SLIDE 25

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✿ ❇❛s✐❝ ❙t❡♣s

❇❛s✐❝ ❙t❡♣s

✶ ❙❡t ❦ = ✵✱ ❲✵ = ❲ ❛♥❞ σ✵ = ǫ ✷ ■❢ ❲❦ ✐s ❛ s✐♥❣❧❡t♦♥✱ ❙❚❖P✱ σ❦ ✐s ❛ ▼●❯✳ ❖t❤❡r✇✐s❡✱ ✜♥❞

t❤❡ ❞✐s❛❣r❡❡♠❡♥t s❡t ❉❦ ❢♦r ❲❦✳

✸ ■❢ t❤❡r❡ ✐s ❛ ♣❛✐r ✈❦, t❦ s✉❝❤ t❤❛t ✈❦, t❦ ∈ ❉❦✱ ✈❦ ✐s ❛

✈❛r✐❛❜❧❡ t❤❛t ❞♦❡s ♥♦t ♦❝❝✉r ✐♥ t❦ ❣♦ t♦ st❡♣ ✹✱ ♦t❤❡r✇✐s❡ ❙❚❖P✱ ❲ ✐s ♥♦t ✉♥✐✜❛❜❧❡✳

✹ ▲❡t σ❦+✶ = σ❦ ◦ {t❦/✈❦} ❛♥❞ ❲❦+✶ = ❲❦{t❦/✈❦}✳ ✺ ❙❡t ❦ = ❦ + ✶ ❣♦ t♦ st❡♣ ✷✳

◆♦t❡ ■♥ st❡♣ ✹ ❲❦+✶ = ❲❦{t❦/✈❦} = ❲ σ❦+✶ ❜❡❝❛✉s❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ s✉❜st✐t✉t✐♦♥s ✐s ❛ss♦❝✐❛t✐✈❡✳

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SLIDE 26

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✮ ❋✐♥❞ ❛ ♠♦st ❣❡♥❡r❛❧ ✉♥✐✜❡r ❢♦r t❤❡ s❡t ❲ = {P(❛, ②), P(①, ❢ (❜))} ❙♦❧✳ ❛ ① ❢ ❜ ②

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SLIDE 27

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ✭❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✮ ❋✐♥❞ ❛ ♠♦st ❣❡♥❡r❛❧ ✉♥✐✜❡r ❢♦r t❤❡ s❡t ❲ = {P(❛, ②), P(①, ❢ (❜))} ❙♦❧✳ θ = {❛/①, ❢ (❜)/②}

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SLIDE 28

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡ ■■

❊①❛♠♣❧❡ ✭❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✮ ❋✐♥❞ ❛ ♠♦st ❣❡♥❡r❛❧ ✉♥✐✜❡r ❢♦r t❤❡ s❡t ❲ = {P(❛, ①, ❢ (❣(②))), P(③, ❢ (③), ❢ (✉))} ❙♦❧✳ ❛ ③ ❢ ❛ ① ❣ ② ✉

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SLIDE 29

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡ ■■

❊①❛♠♣❧❡ ✭❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✮ ❋✐♥❞ ❛ ♠♦st ❣❡♥❡r❛❧ ✉♥✐✜❡r ❢♦r t❤❡ s❡t ❲ = {P(❛, ①, ❢ (❣(②))), P(③, ❢ (③), ❢ (✉))} ❙♦❧✳ θ = {❛/③, ❢ (❛)/①, ❣(②)/✉}

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SLIDE 30

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡ ■■■

❊①❛♠♣❧❡ ✭❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✮ ❉❡t❡r♠✐♥❡ ✇❤❡t❤❡r ♦r ♥♦t t❤❡ s❡t ❲ = {◗(❢ (❛), ❣(①)), ◗(②, ②)} ✐s ✉♥✐✜❛❜❧❡✳ ❙♦❧✳ ❲ ✐s ♥♦t ✉♥✐✜❛❜❧❡

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SLIDE 31

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❛♠♣❧❡ ■■■

❊①❛♠♣❧❡ ✭❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✮ ❉❡t❡r♠✐♥❡ ✇❤❡t❤❡r ♦r ♥♦t t❤❡ s❡t ❲ = {◗(❢ (❛), ❣(①)), ◗(②, ②)} ✐s ✉♥✐✜❛❜❧❡✳ ❙♦❧✳ ❲ ✐s ♥♦t ✉♥✐✜❛❜❧❡

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SLIDE 32

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✿ ❚❡r♠✐♥❛t✐♦♥

❚❡r♠✐♥❛t✐♦♥ ❚❤❡ ✉♥✐✜❝❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✇✐❧❧ ❛❧✇❛②s t❡r♠✐♥❛t❡ ❛❢t❡r ❛ ✜♥✐t❡ ♥✉♠✲ ❜❡r ♦❢ st❡♣s✳ ❖t❤❡r✇✐s❡ ✇❡ ✇✐❧❧ ❤❛✈❡ ❛♥ ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ❲ σ✵, ❲ σ✶, ❲ σ✷, · · · ❊❛❝❤ st❡♣ ❡❧✐♠✐♥❛t❡s ♦♥❡ ✈❛r✐❛❜❧❡ ❢r♦♠ ❲ ✭s♣❡❝✐✜❝❛❧❧② ❲ σ❦ ❝♦♥t❛✐♥s ✈❦ ❜✉t ❲ σ❦+✶ ❞♦❡s ♥♦t✮ ❆♥❞ ❲ ❤❛s ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡ ❚❤✉s t❤❡ ❛❧❣♦r✐t❤♠ ✇✐❧❧ ❛❧✇❛②s t❡r♠✐♥❛t❡✿ r❡t✉r♥✐♥❣ ❛ ▼●❯ ♦r st❛t✐♥❣ ❲ ✐s ♥♦t ✉♥✐✜❛❜❧❡

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SLIDE 33

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✿ ❈♦rr❡❝t♥❡ss

❚❤❡♦r❡♠ ✭❈♦rr❡❝t♥❡ss ♦❢ ❯♥✐✜❝❛t✐♦♥ ❆❧❣♦r✐t❤♠✮ ■❢ ❲ ✐s ❛ ✜♥✐t❡✱ ♥♦♥ ❡♠♣t②✱ ✉♥✐✜❛❜❧❡ s❡t ♦❢ ❡①♣r❡ss✐♦♥s✱ t❤❡ ✉♥✐✜❝❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✇✐❧❧ ❛❧✇❛②s t❡r♠✐♥❛t❡ ✇✐t❤ σ❦ ❛ ▼●❯ ❢♦r ❲ ✳ ❜❛s✐❝ ✐❞❡❛✳ ❲❡ ❝❛♥ ♣r♦✈❡ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ ❦ t❤❛t ❢♦r ❛♥② θ ✇❤✐❝❤ ✐s ❛ ✉♥✐✜❡r ✇❡ ❤❛✈❡ θ = σ❦ ◦ λ❦

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SLIDE 34

❙✉❜st✐t✉t✐♦♥ ❛♥❞ ❯♥✐✜❝❛t✐♦♥

❊①❡r❝✐s❡s ❯♥✐✜❝❛t✐♦♥ ❛❧❣♦r✐t❤♠

❊①❡r❝✐s❡ ❉❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ❡①♣r❡ss✐♦♥s ✐s ✉♥✐✜❛❜❧❡✳ ■❢ ②❡s ❣✐✈❡ ❛ ▼●❯

✶ ❲ = {◗(❛, ①, ❢ (①)), ◗(❛, ②, ②)} ✷ ❲ = {◗(①, ②, ③), ◗(✉, ❤(✈, ✈), ✉)}