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SLIDE 1

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❍✲■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲❙❛t✐s✜❛❜✐❧✐t②

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SLIDE 2

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❙✉♠♠❛r②

❍✲■♥t❡r♣r❡t❛t♦♥ ❛♥❞ ❍✲❙❛t✐s✜❛❜✐❧✐t② ❬❈❤❛♥❣✲▲❡❡ ❈❤✳ ✹✳✸❪ ❙❡♠❛♥t✐❝ ❚r❡❡s ❬❈❤❛♥❣✲▲❡❡ ❈❤✳ ✹✳✹❪

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SLIDE 3

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

■♥t❡r♣r❡t❛t✐♦♥s ♦✈❡r t❤❡ ❍❡r❜r❛♥❞ ❯♥✐✈❡rs❡

■♥t❡r♣r❡t❛t✐♦♥s ❛♥❞ t❤❡ ❍❡r❜r❛♥❞ ❯♥✐✈❡rs❡ ▲❡t ✉s ❝♦♥s✐❞❡r ■♥t❡r♣r❡t❛t✐♦♥s ♦✈❡r t❤❡ ❍❡r❜r❛♥❞ ✉♥✐✈❡rs❡✳

  • ✐✈❡♥ ❛ s❡t ♦❢ ❝❧❛✉s❡s ❙ ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ♠✉st ♣r♦✈✐❞❡✿

❛ss✐❣♥♠❡♥t ❢♦r ❝♦st❛♥ts t♦ ❡❧❡♠❡♥t ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥ ❛ss✐❣♥♠❡♥t ❢♦r ❢✉♥❝t✐♦♥ s②♠❜♦❧s t♦ ❡❧❡♠❡♥t ♦❢ t❤❡ ❞♦♠❛✐♥ ❛♥ ❛ss✐❣♥♠❡♥t ❢♦r ♣r❡❞✐❝❛t❡ s②♠❜♦❧s t♦ ⊤, ⊥ ❲❤❡r❡ t❤❡ ❞♦♠❛✐♥ ✐s t❤❡ ❍❡r❜r❛♥❞ ❯♥✐✈❡rs❡ ❢♦r ❙

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SLIDE 4

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❍ ■♥t❡r♣r❡t❛t✐♦♥s

❉❡✜♥✐t✐♦♥ ✭❍ ■♥t❡r♣r❡t❛t✐♦♥✮ ▲❡t ❙ ❜❡ ❛ s❡t ♦❢ ❝❧❛✉s❡s✱ ❍ t❤❡ ❍❡r❜r❛♥❞ ❯♥✐✈❡rs❡ ♦❢ ❙ ❛♥❞ ■ = ❉, ❆ ❛♥ ■♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❙✳ ■ ✐s ❛♥ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❙ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ❉ = ❍ t❆ = t ❢♦r ❛❧❧ t❡r♠s t ■♥ ♠♦r❡ ❞❡t❛✐❧ ▲❡t ❝ ❜❡ ❛ ❝♦st❛♥t s②♠❜♦❧ ❝❆ = ❝✳ ▲❡t ❢ ❜❡ ❛ ♥✲❛r② ❢✉♥❝t✐♦♥ s②♠❜♦❧ ❢ ❆ ♠❛♣s (❤✶, · · · , ❤♥) ∈ ❍♥ t♦ ❢ (❤✶, · · · , ❤♥) ∈ ❍

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SLIDE 5

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❍ ■♥t❡r♣r❡t❛t✐♦♥s✿ Pr❡❞✐❝❛t❡ s②♠❜♦❧s

❍✲■♥t❡r♣r❡t❛t✐♦♥s✿ Pr❡❞✐❝❛t❡ ◆♦ r❡str✐❝t✐♦♥s ❢♦r ♣r❡❞✐❝❛t❡ s②♠❜♦❧s

  • ✐✈❡♥ ❙✱ ❧❡t ❆ = {❆✶, · · · ❆♥, · · · } ❜❡ t❤❡ ❍❡r❜r❛♥❞ ❜❛s❡ ✭♦r ❛t♦♠

s❡t✮ ♦❢ ❙✱ ❛♥ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s✿ ■ = {♠✶, · · · ♠♥, · · · } ✇❤❡r❡ ♠❥ = ❆❥ ♦r ♠❥ = ¬❆❥ ❢♦r ❥ = ✶, · · · , ♥, · · ·

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SLIDE 6

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ♦❢ ❍✲■♥t❡r♣r❡t❛t✐♦♥

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ s❡t ❙ = {P(①) ∨ ◗(①), ❘(❢ (②))} ❍✲■♥t❡r♣r❡t❛t✐♦♥ ❍ ❛ ❢ ❛ ❢ ❢ ❛ ❆ P ❛ ◗ ❛ ❘ ❛ P ❢ ❛ ◗ ❢ ❛ ❘ ❢ ❛ P♦ss✐❜❧❡ ❍✲■♥t❡r♣r❡t❛t✐♦♥s✿

■✶ P ❛ ◗ ❛ ❘ ❛ P ❢ ❛ ◗ ❢ ❛ ❘ ❢ ❛ ■✷ P ❛ ◗ ❛ ❘ ❛ P ❢ ❛ ◗ ❢ ❛ ❘ ❢ ❛ ■✸ P ❛ ◗ ❛ ❘ ❛ P ❢ ❛ ◗ ❢ ❛ ❘ ❢ ❛

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SLIDE 7

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ♦❢ ❍✲■♥t❡r♣r❡t❛t✐♦♥

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ s❡t ❙ = {P(①) ∨ ◗(①), ❘(❢ (②))} ❍✲■♥t❡r♣r❡t❛t✐♦♥ ❍ = {❛, ❢ (❛), ❢ (❢ (❛)), · · · } ❆ = {P(❛), ◗(❛), ❘(❛), P(❢ (❛)), ◗(❢ (❛)), ❘(❢ (❛)), · · · } P♦ss✐❜❧❡ ❍✲■♥t❡r♣r❡t❛t✐♦♥s✿

■✶ = {P(❛), ◗(❛), ❘(❛), P(❢ (❛)), ◗(❢ (❛)), ❘(❢ (❛)), · · · } ■✷ = {¬P(❛), ◗(❛), ❘(❛), ¬P(❢ (❛)), ◗(❢ (❛)), ❘(❢ (❛)), · · · } ■✸ = {¬P(❛), ¬◗(❛), ¬❘(❛), ¬P(❢ (❛)), ¬◗(❢ (❛)), ¬❘(❢ (❛)), · · · }

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SLIDE 8

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ♦❢ ♥♦t ❍✲■♥t❡r♣r❡t❛t✐♦♥

❊①❛♠♣❧❡ ✭♥♦t ❍✲■♥t❡r♣r❡t❛t✐♦♥✮ ❈♦♥s✐❞❡r t❤❡ s❡t ❙ = {P(①) ∨ ◗(①), ❘(❢ (②))}✳ ◆❍■ = ❉, ❆ ❉ = {✶, ✷} ❢ ❆(✶) = ✶, ❢ ❆(✷) = ✷ {P(✶), ¬P(✷), ◗(✶), ¬◗(✷), ❘(✶), ¬❘(✷)}

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SLIDE 9

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

▼❛♣♣✐♥❣ ❛♠♦♥❣ ■♥t❡r♣r❡t❛t✐♦♥s

♠❛♣♣✐♥❣ t♦ ❍✲■♥t❡r♣r❡t❛t✐♦♥s

  • ✐✈❡♥ ❛♥ ■♥t❡r♣r❡t❛t✐♦♥ ■ ✇❡ ❝❛♥ ❛❧✇❛②s ✜♥❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ■ ∗

❍✲■♥t❡r♣r❡t❛t✐♦♥

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SLIDE 10

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ♦❢ ♥♦t ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■■

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ s❡t ❙ = {P(①), ◗(②, ❢ (②, ❛))}✳ ■ = ❉, ❆ ❉ = {✶, ✷} ❛❆ = ✷ ❢ ❆(✶, ✶) = ✶, ❢ ❆(✶, ✷) = ✷, ❢ ❆(✷, ✶) = ✷, ❢ ❆(✷, ✷) = ✶ {P(✶), ¬P(✷), ¬◗(✶, ✶), ◗(✶, ✷), ¬◗(✷, ✶), ◗(✷, ✷)}

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SLIDE 11

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ♦❢ ♠❛♣♣✐♥❣ ❜❡t✇❡❡♥ ❍✲■♥t❡r♣r❡t❛t✐♦♥

❊①❛♠♣❧❡

  • ✐✈❡♥ ❙ = {P(①), ◗(②, ❢ (②, ❛))} ❛♥❞ ■ ✇❡ ❝❛♥ ❞❡✜♥❡ ■ ∗ ❛s

❢♦❧❧♦✇s✿

✶ ❍ =

{❛, ❢ (❛, ❛), ❢ (❛, ❢ (❛, ❛)), ❢ (❢ (❛, ❛), ❛), ❢ (❢ (❛, ❛), ❢ (❛, ❛)), · · · }

✷ ❆ = {P(❛), ◗(❛, ❛), ◗(❛, ❢ (❛, ❛)), ◗(❢ (❛, ❛), ❛),

P(❢ (❛, ❛)), ◗(❢ (❛, ❛), ❢ (❛, ❛)), · · · }

✸ ■ ∗ = {¬P(❛), ◗(❛, ❛), P(❢ (❛, ❛)), ¬◗(❛, ❢ (❛, ❛)), · · · }

P(❛) = P(✷) = ⊥ ◗(❛, ❛) = ◗(✷, ✷) = ⊤ P(❢ (❛, ❛)) = P(✶) = ⊤ ◗(❛, ❢ (❛, ❛)) = ◗(✷, ❢ (✷, ✷)) = ◗(✷, ✶) = ⊥ · · ·

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SLIDE 12

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

▼✉❧t✐♣❧✐❝✐t② ♦❢ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ♠❛♣♣✐♥❣

▼✉❧t✐♣❧❡ ❍✲■♥t❡r♣r❡t✐♦♥s ❈♦♥s✐❞❡r ❛♥ ■♥t❡r♣r❡t❛t✐♦♥ ■ ■❢ t❤❡r❡ ✐s ♥♦ ❝♦♥st❛♥t ❛♣♣❡❛r✐♥❣ ✐♥ ❙ t❤❡♥ t❤❡ ❛❞❞❡❞ ❝♦st❛♥t ❛ ✐♥ t❤❡ ❍❡r❜r❛♥❞ ❯♥✐✈❡rs❡ ❝❛♥ ❜❡ ♠❛♣♣❡❞ t♦ ❛♥② ❡❧❡♠❡♥t ✐♥ ❉✳ ❚❤❡r❡❢♦r❡ t❤❡r❡ ❛r❡ ♠♦r❡ t❤❛t ♦♥❡ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■ ∗ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ■ ❞❡♣❡♥❞✐♥❣ ♦♥ ✈❛❧✉❡s ❣✐✈❡♥ t♦ ❛

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SLIDE 13

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ♦❢ ▼✉❧t✐♣❧❡ ❍✲■♥t❡r♣r❡t❛t✐♦♥s

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ s❡t ❙ = {P(①), ◗(②, ❢ (②, ③))}✳ ■ = ❉, ❆ ❉ = {✶, ✷} ❢ ❆(✶, ✶) = ✶, ❢ ❆(✶, ✷) = ✷, ❢ ❆(✷, ✶) = ✷, ❢ ❆(✷, ✷) = ✶ {P(✶), ¬P(✷), ¬◗(✶, ✶), ◗(✶, ✷), ¬◗(✷, ✶), ◗(✷, ✷)} ❊①❛♠♣❧❡ ❈♦rr❡s♣♦♥❞✐♥❣ ❍✲■♥t❡r♣r❡t❛t✐♦♥s ■ ∗

✶ = {¬P(❛), ◗(❛, ❛), P(❢ (❛, ❛)), ¬◗(❛, ❢ (❛, ❛)), · · · } ✐❢

❛ = ✷ ■ ∗

✷ = {P(❛), ¬◗(❛, ❛), P(❢ (❛, ❛)), ¬◗(❛, ❢ (❛, ❛)), · · · } ✐❢

❛ = ✶

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SLIDE 14

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ♦❢ ▼✉❧t✐♣❧❡ ❍✲■♥t❡r♣r❡t❛t✐♦♥s ■■

❊①❛♠♣❧❡

  • ✐✈❡♥ ❙ = {P(①) ∨ ◗(①), ❘(❢ (②))} ❛♥❞ ◆❍■ = ❉, ❆

❉ = {✶, ✷} ❢ ❆(✶) = ✶, ❢ ❆(✷) = ✷ {P(✶), ¬P(✷), ◗(✶), ¬◗(✷), ❘(✶), ¬❘(✷)} ✇❡ ❝❛♥ ❞❡✜♥❡ ■ ∗

✶ ❛s ❢♦❧❧♦✇s✿ ✶ ❍ = {❛, ❢ (❛), ❢ (❢ (❛)), · · · } ✷ ❆ = {P(❛), ◗(❛), ❘(❛), P(❢ (❛)), ◗(❢ (❛)), ❘(❢ (❛)), · · · } ✸ ❛❆ = ✶ ✹ {P(❛) = P(✶) = ⊤, ◗(❛) = ◗(✶) = ⊤, ❘(❛) = ❘(✶) =

⊤, P(❢ (❛)) = P(✶) = ⊤ · · · }

slide-15
SLIDE 15

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ♦❢ ▼✉❧t✐♣❧❡ ❍✲■♥t❡r♣r❡t❛t✐♦♥s ■■

❊①❛♠♣❧❡ ✭❝♦♥t✳ ❢r♦♠ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡✮ ✇❡ ❝❛♥ ❛❧s♦ ❞❡✜♥❡ ■ ∗

✷ ❛s ❢♦❧❧♦✇s✿ ✶ ❍ = {❛, ❢ (❛), ❢ (❢ (❛)), · · · } ✷ ❆ = {P(❛), ◗(❛), ❘(❛), P(❢ (❛)), ◗(❢ (❛)), ❘(❢ (❛)), · · · } ✸ ❛❆ = ✷ ✹ {P(❛) = P(✷) = ⊥, ◗(❛) = ◗(✶) = ⊥, ❘(❛) = ❘(✶) =

⊥, P(❢ (❛)) = P(✶) = ⊥ · · · }

slide-16
SLIDE 16

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

▼❛♣♣✐♥❣ t♦ ❍✲■♥t❡r♣r❡t❛t✐♦♥

❉❡✜♥✐t✐♦♥ ✭▼❛♣♣✐♥❣ t♦ ❍✲■♥t❡r♣r❡t❛t✐♦♥✮

  • ✐✈❡♥ ■ = ❉, ❆ ✐♥t❡r♣r❡t❛t✐♦♥ ♦✈❡r ❉✱ ❛♥ ❍✲✐♥t❡r♣r❡t❛t✐♦♥

■ ∗❍, ❆∗ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ■ ✐s ❛♥ ❍✲✐♥t❡r♣r❡t❛t✐♦♥ t❤❛t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥✿ ▲❡t ❤✶, · · · , ❤♥ ❜❡ ❡❧❡♠❡♥ts ♦❢ ❍ ❛♥❞ ❧❡t ♠ : ❍ → ❉ ❜❡ ❛ ♠❛♣♣✐♥❣ ❢r♦♠ ❍ t♦ ❉✱ t❤❡♥ P❆∗(❤✶, · · · , ❤♥) = P❆(♠(❤✶), · · · , ♠(❤♥))

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SLIDE 17

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

Pr❡s❡r✈✐♥❣ ❙❛t✐s✜❛❜✐❧✐t②

▲❡♠♠❛ ■❢ ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ■ ♦✈❡r ❛ ❞♦♠❛✐♥ ❉ s❛t✐s✜❡s ❛ s❡t ♦❢ ❝❧❛✉s❡s ❙✱ t❤❡♥ ❛♥② ♦❢ t❤❡ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■ ∗ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ■ s❛t✐s✜❡s ❙✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢✳ ❙✉♣♣♦s❡ ■ | = ❙ ❜✉t ■ ∗ | = ❙✳ ❙✐♥❝❡ ■ ∗ | = ❙ t❤❡♥ ∃❈ ∗ ❣r♦✉♥❞ t❤❛t ✐s ♥♦t s❛t✐s✜❡❞ ❜② ■ ∗ ❙✐♥❝❡ ■ ∗ ✐s ❛♥ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ■✱ ❢♦r ❡❛❝❤ ❡❧❡♠❡♥t ✐♥ ■ ∗ ✇❡ ❝❛♥ ✜♥❞ ❛♥ ❡❧❡♠❡♥t ✐♥ ■ ✇✐t❤ t❤❡ s❛♠❡ tr✉t❤ ✈❛❧✉❡✳ ❚❤❡r❡❢♦r❡ ✇❡ ❤❛✈❡ ❛ ❣r♦✉♥❞ ❝❧❛✉s❡ ❈ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❈ ∗ t❤❛t ✐s ♥♦t s❛t✐s✜❡❞ ❜② ■✱ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts t❤❡ ❤②♣♦t❤❡s✐s

slide-18
SLIDE 18

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

Pr❡s❡r✈✐♥❣ ❙❛t✐s❢❛✐❜✐❧✐t② ❊①❛♠♣❧❡

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ s❡t ♦❢ ❝❧❛✉s❡s ❙ = {P(①, ❢ (①))}✳ ❈♦♥s✐❞❡r t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ■✿ ❉ = ✶, ✷ ❢ (✶) = ✶, ❢ (✷) = ✷ P(✶, ✶) = ⊤, P(✶, ✷) = ⊥, P(✷, ✶) = ⊥, P(✷, ✷) = ⊤, ■ | = ❙ ❜❡❝❛✉s❡ ❛❧❧ ❣r♦✉♥❞ ❝❧❛✉s❡s {P(✶, ✶), P(✷, ✷)} ❛r❡ s❛t✐s✜❡❞ ❜② ■✳ ❆ss✉♠❡ ■ ∗ ✐s t❤❡ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ■ ✇✐t❤ ❛ = ✶✳ ❍✵ = {❛}, ❍✶ = {❛, ❢ (❛)}, ❍✸ = {❛, ❢ (❛), ❢ (❢ (❛)), · · · } ❆ = {P(❛, ❛), P(❛, ❢ (❛)), P(❢ (❛), ❛), P(❢ (❛), ❢ (❛)), · · · } P(❛, ❛) = P(✶, ✶) = ⊤, P(❛, ❢ (❛)) = P(✶, ✶) = ⊤, P(❢ (❛), ❛) = P(✶, ✶) = ⊤P(❢ (❛), ❢ (❛)) = P(✶, ✶) = ⊤ ■ ∗ | = ❙ ❛s ✇❡❧❧✳

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SLIDE 19

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❍✲❙❛t✐s✜❛❜✐❧✐t②

❚❤❡♦r❡♠ ✭❍✲❙❛t✐s✜❛❜✐❧✐t②✮ ❆ s❡t ❙ ♦❢ ❝❧❛✉s❡s ✐s ✉♥s❛t✐s✜❛❜❧❡ ✐✛ ❙ ✐s ❢❛❧s❡ ✉♥❞❡r ❛❧❧ t❤❡ ❍✲■♥t❡r♣r❡t❛t✐♦♥s Pr♦♦❢✳ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ⇒ ■❢ ✉♥s❛t✐s✜❛❜❧❡ t❤❡♥ ♠✉st ❜❡ ❢❛❧s❡ ✉♥❞❡r ❛❧❧ ✐♥t❡r♣r❡t❛t✐♦♥s ❛♥❞ t❤✉s s♣❡❝✐✜❝❛❧❧② ✉♥❞❡r ❛❧❧ ❍✲■♥t❡r♣r❡t❛t✐♦♥s ⇐ ❆ss✉♠❡ ❙ ✐s ❢❛❧s❡ ✉♥❞❡r ❛❧❧ ❍✲■♥t❡r♣r❡t❛t✐♦♥s ❜✉t ❙ ✐s s❛t✐s✜❛❜❧❡✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ■ | = ❙✳ ❚❤❡♥ ❢♦r t❤❡ ❛❜♦✈❡ ❧❡♠♠❛ t❤❡r❡ ❡①✐sts ❛♥ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■ ∗ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ■ s✉❝❤ t❤❛t ■ ∗ | = ❙ ✇❤✐❝❤ ❝♦♥tr❛❞❞✐❝ts t❤❡ ❤②♣♦t❤❡s✐s

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SLIDE 20

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

■♠♣♦rt❛♥❝❡ ♦❢ ❍✲❙❛t✐s✜❛❜✐❧✐t②

❍✲■♥t❡r♣r❡t❛t✐♦♥s ❛r❡ ❛❧❧ ✇❡ ♥❡❡❞

  • ✐✈❡♥ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ t♦ ♣r♦♦✈❡ ✉♥s❛t✐s✜❛❜✐❧✐t② ♦❢ ❙ ✇❡

♥❡❡❞ ♦♥❧② t♦ ❝♦♥s✐❞❡r ❍ − ■♥t❡r♣r❡t❛t✐♦♥ ❲❡ ❝❛♥ t❤✉s r❡str✐❝t ♦✉r ❛tt❡♥t✐♦♥ ♦♥❧② t♦ t❤❡ ❍❡r❜r❛♥❞ ✉♥✐✈❡rs❡ ❋r♦♠ ♥♦✇ ♦♥✇❛r❞s ✇❡ ❝♦♥s✐❞❡r ♦♥❧② ❍✲■♥t❡r♣r❡t❛t✐♦♥s✳

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SLIDE 21

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❖❜s❡r✈❛t✐♦♥s ♦♥ ❙❛t✐s✜❛❜✐❧✐t② ■

❖❜s❡r✈❛t✐♦♥ ■ ❆ ❣r♦✉♥❞ ✐♥st❛♥❝❡ ❈ ′ ❢♦ ❛ ❝❧❛✉s❡ ❈ ✐s s❛t✐s✜❡❞ ❜② ❛♥ ❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ■ ✐✛ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ❧✐t❡r❛❧ ▲′ ∈ ❈ ′ s✉❝❤ t❤❛t ▲′ ∈ ■✱ ✇❤✐❝❤ ✐s ❈ ′ ∩ ■ = { }✳ ❊①❛♠♣❧❡

  • ✐✈❡♥ ❈ ¬P(①) ∨ ◗(❢ (①)) ❛♥❞ ❈ ′ ¬P(❛) ∨ ◗(❢ (❛)) ❛

❣r♦✉♥❞ ✐♥st❛♥❝❡✱ ❛♥❞ ■ = {P(❛), ¬◗(❛), P(❢ (❛)), ◗(❢ (❛)), ¬◗(❢ (❢ (❛))) · · · }✳ ❉♦❡s ■ | = ❈ ′ ❄ ❙♦❧✳ ■ ❈ ◗ ❢ ❛ t❤❡r❡❢♦r❡ ■ ❈

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SLIDE 22

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❖❜s❡r✈❛t✐♦♥s ♦♥ ❙❛t✐s✜❛❜✐❧✐t② ■

❖❜s❡r✈❛t✐♦♥ ■ ❆ ❣r♦✉♥❞ ✐♥st❛♥❝❡ ❈ ′ ❢♦ ❛ ❝❧❛✉s❡ ❈ ✐s s❛t✐s✜❡❞ ❜② ❛♥ ❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ■ ✐✛ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ❧✐t❡r❛❧ ▲′ ∈ ❈ ′ s✉❝❤ t❤❛t ▲′ ∈ ■✱ ✇❤✐❝❤ ✐s ❈ ′ ∩ ■ = { }✳ ❊①❛♠♣❧❡

  • ✐✈❡♥ ❈ ¬P(①) ∨ ◗(❢ (①)) ❛♥❞ ❈ ′ ¬P(❛) ∨ ◗(❢ (❛)) ❛

❣r♦✉♥❞ ✐♥st❛♥❝❡✱ ❛♥❞ ■ = {P(❛), ¬◗(❛), P(❢ (❛)), ◗(❢ (❛)), ¬◗(❢ (❢ (❛))) · · · }✳ ❉♦❡s ■ | = ❈ ′ ❄ ❙♦❧✳ ■ ∩ ❈ ′ = ◗(❢ (❛)) = { } t❤❡r❡❢♦r❡ ■ | = ❈ ′

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SLIDE 23

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❖❜s❡r✈❛t✐♦♥ ♦♥ ❙❛t✐✜❛❜✐❧✐t② ■■

❖❜s❡r✈❛t✐♦♥ ■■

  • ✐✈❡♥ ❛ ❝❧❛✉s❡ ❈ ❛♥❞ ❛♥ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■✱ ■ |

= ❈ ✐✛ ❢♦r ❡✈❡r② ❈ ′ ❣r♦✉♥❞ ✐♥st❛♥❝❡ ■ | = ❈ ′ ❖❜s❡r✈❛t✐♦♥ ■■■ ❆ ❝❧❛✉s❡ ❈ ✐s ❢❛❧s✐✜❡❞ ❜② ❛♥ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■ ✐✛ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ❈ ′ ❣r♦✉♥❞ ✐♥st❛♥❝❡ s✉❝❤ t❤❛t ■ | = ❈ ′ ❊①❛♠♣❧❡

  • ✐✈❡♥ ❈ ¬P(①) ∨ ◗(❢ (①))✱ ❛♥❞

■ = {P(❛), ¬◗(❛), P(❢ (❛)), ◗(❢ (❛)), ¬◗(❢ (❢ (❛))) · · · }✳ ❉♦❡s ■ | = ❈ ❄ ❙♦❧✳ ❈ P ❢ ❛ ◗ ❢ ❢ ❛ ■ ❈ t❤❡r❡❢♦r❡ ■ ❈

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SLIDE 24

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❖❜s❡r✈❛t✐♦♥ ♦♥ ❙❛t✐✜❛❜✐❧✐t② ■■

❖❜s❡r✈❛t✐♦♥ ■■

  • ✐✈❡♥ ❛ ❝❧❛✉s❡ ❈ ❛♥❞ ❛♥ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■✱ ■ |

= ❈ ✐✛ ❢♦r ❡✈❡r② ❈ ′ ❣r♦✉♥❞ ✐♥st❛♥❝❡ ■ | = ❈ ′ ❖❜s❡r✈❛t✐♦♥ ■■■ ❆ ❝❧❛✉s❡ ❈ ✐s ❢❛❧s✐✜❡❞ ❜② ❛♥ ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■ ✐✛ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ❈ ′ ❣r♦✉♥❞ ✐♥st❛♥❝❡ s✉❝❤ t❤❛t ■ | = ❈ ′ ❊①❛♠♣❧❡

  • ✐✈❡♥ ❈ ¬P(①) ∨ ◗(❢ (①))✱ ❛♥❞

■ = {P(❛), ¬◗(❛), P(❢ (❛)), ◗(❢ (❛)), ¬◗(❢ (❢ (❛))) · · · }✳ ❉♦❡s ■ | = ❈ ❄ ❙♦❧✳ ❈ ′′ = ¬P(❢ (❛)) ∨ ◗(❢ (❢ (❛))) ■ ∩ ❈ ′′ = { } t❤❡r❡❢♦r❡ ■ | = ❈

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SLIDE 25

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❖❜s❡r✈❛t✐♦♥s ♦♥ ❙❛t✐s✜❛❜✐❧✐t② ■■■

❖❜s❡r✈❛t✐♦♥ ■❱ ❆ s❡t ♦❢ ❝❧❛✉s❡ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡ ✐✛ ❢♦r ❡✈❡r② ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ❈ ′ ❣r♦✉♥❞ ❝❧❛✉s❡ ♦❢ s♦♠❡ ❈ ∈ ❙ s✉❝❤ t❤❛t ■ | = ❈ ′ ❊①❛♠♣❧❡

  • ✐✈❡♥ ❙ {¬P(①), P(❛)} ✐s ❙ ✉♥s❛t✐s✜❛❜❧❡ ❄

❙♦❧✳

❍ ❛ ✱ ❆ P ❛ ❖♥❧② t✇♦ ❍✲■♥t❡r♣r❡t❛t✐♦♥s ■✶ P ❛ ❛♥❞ ■✷ P ❛ ■✶ ❙ ✿ ❈ P ❛ ❣r♦✉♥❞ ✐♥st❛♥❝❡ ♦❢ ❈ P ① ❛♥❞ ■✶ ❈ ■✷ ❙ ✿ ❈ P ❛ ❣r♦✉♥❞ ✐♥st❛♥❝❡ ♦❢ ❈ P ❛ ❛♥❞ ■✷ ❈ ❚❤❡r❡❢♦r❡ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡✳

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SLIDE 26

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❖❜s❡r✈❛t✐♦♥s ♦♥ ❙❛t✐s✜❛❜✐❧✐t② ■■■

❖❜s❡r✈❛t✐♦♥ ■❱ ❆ s❡t ♦❢ ❝❧❛✉s❡ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡ ✐✛ ❢♦r ❡✈❡r② ❍✲■♥t❡r♣r❡t❛t✐♦♥ ■ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ❈ ′ ❣r♦✉♥❞ ❝❧❛✉s❡ ♦❢ s♦♠❡ ❈ ∈ ❙ s✉❝❤ t❤❛t ■ | = ❈ ′ ❊①❛♠♣❧❡

  • ✐✈❡♥ ❙ {¬P(①), P(❛)} ✐s ❙ ✉♥s❛t✐s✜❛❜❧❡ ❄

❙♦❧✳

❍ = {❛}✱ ❆ = {P(❛)} ❖♥❧② t✇♦ ❍✲■♥t❡r♣r❡t❛t✐♦♥s ■✶ = {P(❛)} ❛♥❞ ■✷ = {¬P(❛)} ■✶ | = ❙ ✿ ❈ ′ = ¬P(❛) ❣r♦✉♥❞ ✐♥st❛♥❝❡ ♦❢ ❈ = ¬P(①) ❛♥❞ ■✶ | = ❈ ′ ■✷ | = ❙ ✿ ❈ ′′ = P(❛) ❣r♦✉♥❞ ✐♥st❛♥❝❡ ♦❢ ❈ = P(❛) ❛♥❞ ■✷ | = ❈ ′′ ❚❤❡r❡❢♦r❡ ❙ ✐s ✉♥s❛t✐s✜❛❜❧❡✳

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SLIDE 27

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ♦♥ ❙❛t✐s✜❛❜✐❧✐t②

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❝❧❛✉s❡ ❈ = ¬P(①) ∨ ◗(❢ (①))✳ ❍ = {❛, ❢ (❛), ❢ (❢ (❛)), · · · } ❛♥❞ ❆ = {P(❛), ◗(❛), P(❢ (❛)), ◗(❢ (❛)), · · · } ■✶ = {¬P(❛), ¬◗(❛), ¬P(❢ (❛)), ¬◗(❢ (❛)), · · · } ■✷ = {P(❛), ◗(❛), P(❢ (❛)), ◗(❢ (❛)), · · · } ■✸ = {P(❛), ¬◗(❛), P(❢ (❛)), ¬◗(❢ (❛)), · · · } ❚❤❡♥ ■✶ | = ❈✱ ■✷ | = ❈ ❜✉t ■✸ | = ❈✳ ◆♦t❡ ❲❡ ❛r❡ ❛ss✉♠✐♥❣ ❛ ♣❛tt❡r♥ ♦♥ t❤❡ ■♥t❡r♣r❡t❛t✐♦♥s ♦t❤❡r✇✐s❡ ✇❡ ❝♦✉❧❞ ♥♦t ❞❡❝✐❞❡ ♦♥ s❛t✐s✜❛❜✐❧✐t②

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SLIDE 28

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❡r❝✐s❡s

❊①❡r❝✐s❡ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛✉s❡ ❈ : P(①) ∨ ◗(①, ❢ (①)) ■ : {¬P(❛), ¬P(❢ (❛)), ¬P(❢ (❢ (❛))), · · · ¬◗(❛, ❛), ◗(❛, ❢ (❛)), ¬◗(❛, ❢ (❢ (❛))), · · · ¬◗(❢ (❛), ❛), ◗(❢ (❛), ❢ (❛)), ¬◗(❢ (❛), ❢ (❢ (❛))), · · · } ❉♦❡s ■ | = ❈ ❄ ❬❈❤❛♥❣✲▲❡❡ ✽ ♣❛❣❡ ✻✽❪ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ❝❧❛✉s❡s ❙ : {P(①), ◗(❢ (②))} ■ : {P(❛), P(❢ (❛)), P(❢ (❢ (❛))), · · · ◗(❛), ¬◗(❢ (❛)), ◗(❢ (❢ (❛))), · · · } ❉♦❡s ■ | = ❙ ❄ ❬❈❤❛♥❣✲▲❡❡ ✾ ♣❛❣❡ ✻✽❪ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ❝❧❛✉s❡s ❙ : {P(①), ¬P(❢ (②))}

✶ ●✐✈❡ ❍✵✱ ❍✶✱ ❍✷ ❛♥❞ ❍✸✳ ✷ ■s ✐t ♣♦ss✐❜❧❡ t♦ ✜♥❞ ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ t❤❛t s❛t✐s✜❡s ❙ ❄ ■❢

②❡s ♣r♦✈✐❞❡ ♦♥❡✳ ■❢ ♥♦ ❡①♣❧❛✐♥ ✇❤② ❬❈❤❛♥❣✲▲❡❡ ✶✵ ♣❛❣❡ ✻✽❪✳

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SLIDE 29

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❙❡♠❛♥t✐❝ ❚r❡❡s

❇❛s✐❝ ❈♦♥❝❡♣t ❚r❡❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛ s❡t ♦❢ ❝❧❛✉s❡s Pr♦✈✐❞❡s ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ s❛t✐s✜❛❜✐❧✐t② ♦❢ t❤❡ s❡t ♦❢ ❝❧❛✉s❡s ❊①❛♠♣❧❡ ❙✐♠♣❧❡ ❊①❛♠♣❧❡ ❢♦r Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝

¬◗ ◗ P ◗ ¬◗ ¬P

❋✐❣✉r❡✿ ❙❡♠❛♥t✐❝ tr❡❡ ❢♦r ❙ = P ∨ ◗

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SLIDE 30

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❙❡♠❛♥t✐❝ ❚r❡❡s✿ ❉❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❙❡♠❛♥t✐❝ ❚r❡❡✮

  • ✐✈❡♥ ❛ s❡t ♦❢ ❈❧❛✉s❡s ❙ ❧❡t ❆ ❜❡ t❤❡ ❍❡r❜r❛♥❞ ❜❛s❡ ✭♦r ❛t♦♠

s❡t✮ ♦❢ ❙ ❛ ❙❡♠❛♥t✐❝ ❚r❡❡ ❢♦r ❙ ✐s ❛ tr❡❡ ❚✱ ✇❤❡r❡ ❡❛❝❤ ❧✐♥❦ ♦❢ t❤❡ tr❡❡ ✐s ❛♥♥♦t❛t❡❞ ✇✐t❤ ❛ s❡t ♦❢ ❛t♦♠s ♦r ♥❡❣❛t✐♦♥ ♦❢ ❛t♦♠s ❢r♦♠ ❆ s✉❝❤ t❤❛t

✶ ♣r♦♣❡rt② ■ ❋♦r ❡❛❝❤ ♥♦❞❡ ◆ t❤❡r❡ ❛r❡ ♦♥❧② ✜♥✐t❡❧② ♠❛♥②

✐♠♠❡❞✐❛t❡ ❧✐♥❦s {▲✶, · · · , ▲♠} ❢r♦♠ ◆✳ ▲❡t ◗✐ ❜❡ t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❛❧❧ ❧✐t❡r❛❧s ❛tt❛❝❤❡❞ t♦ t❤❡ ❧✐♥❦ ▲✐✱ t❤❡♥ ◗✶ ∨ ◗✷ ∨ · · · ∨ ◗♥ ✐s ❛ ✈❛❧✐❞ ♣r♦♣♦s✐t✐♦♥❛❧ ❢♦r♠✉❧❛✳

✷ ♣r♦♣❡rt② ■■ ❋♦r ❡❛❝❤ ♥♦❞❡ ◆ ❧❡t ■(◆) ❜❡ t❤❡ ✉♥✐♦♥ ♦❢ ❛❧❧

s❡ts ❛tt❛❝❤❡❞ t♦ t❤❡ ❧✐♥❦s ♦❢ t❤❡ ❜r❛♥❝❤ ❝♦♥♥❡❝t✐♥❣ ◆ ✉♣ t♦ t❤❡ r♦♦t ❛♥❞ ✐♥❝❧✉❞✐♥❣ ◆✳ ❚❤❡♥ ■(◆) ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② ❝♦♠♣❧❡♠❡♥t❛r② ♣❛✐r✳

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SLIDE 31

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❈♦♠♣❧❡♠❡♥t❛r② P❛✐r✿ ❉❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❈♦♠♣❧❡♠❡♥t❛r② P❛✐r✮ ■❢ ❆ ✐s ❛♥ ❛t♦♠ t❤❡♥ t❤❡ t✇♦ ❧✐t❡r❛❧s ❆ ❛♥❞ ¬❆ ❛r❡ s❛✐❞ t♦ ❜❡ ❡❛❝❤ ♦t❤❡r✬s ❝♦♠♣❧❡♠❡♥t ❛♥❞ t❤❡ s❡t {❆, ¬❆} ✐s s❛✐❞ t♦ ❜❡ ❛ ❝♦♠♣❧❡♠❡♥t❛r② ♣❛✐r✳ ◆♦t❡ ❆ ❈❧❛✉s❡ t❤❛t ❝♦♥t❛✐♥s ❛ ❝♦♠♣❧❡♠❡♥t❛r② ♣❛✐r ✐s ❛ t❛✉t♦❧♦❣② ❊①❛♠♣❧❡ ❈ = P(①) ∨ ◗(②, ❢ (②)) ∨ ¬❘(③) ∨ ¬P(①) ✐s ❛ t❛✉t♦❧♦❣② ❛s {P(①), ¬P(①)} ✐s ❛ ❝♦♠♣❧❡♠❡♥t❛r② ♣❛✐r

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SLIDE 32

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ■

❊①❛♠♣❧❡

¬P, ¬◗ ¬❘ ❘ ◗ P ❘ ¬❘ ¬P ❘ ¬❘ ¬❘ ❘ ◗ ❘ ¬◗, ¬❘ ◗ ¬◗ P Z X Y

❋✐❣✉r❡✿ ❙❡♠❛♥t✐❝ tr❡❡ ❢♦r t❤❡ ❛t♦♠ s❡t ❆ = P, ◗, ❘

■(X) = {◗, P} ■(Y) = {¬❘, ¬P, ◗} ■(Z) = {¬❘, ¬P, ¬◗} ◆♦t❡ t❤❛t ❢♦r t❤❡ r♦♦t ♥♦❞❡ ✇❡ ❤❛✈❡ ◗✶ = {P} ❛♥❞ ◗✷ = ◗ ❛♥❞ ◗✸ = {¬P, ¬◗} t❤❡r❡❢♦r❡ ◗✶ ∨ ◗✷ ∨ ◗✸ ✐s ❛ ✈❛❧✐❞ ❢♦r♠✉❧❛✳

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SLIDE 33

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡ ■■

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ s❡t ♦❢ ❝❧❛✉s❡s ❙ = {P(①), P(❛)}✳ ❚❤❡ ❛t♦♠ s❡t ❢♦r t❤✐s s❡t ♦❢ ❝❧❛✉s❡s ✐s ❆ = P(❛)

¬P(❛) P(❛)

❋✐❣✉r❡✿ ❙❡♠❛♥t✐❝ tr❡❡ t❤❡ ❛t♦♠ s❡t ❆ = P(❛)

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SLIDE 34

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❈♦♠♣❧❡t❡ ❙❡♠❛♥t✐❝ tr❡❡✿ ❉❡✜♥✐t✐♦♥

❉❡✜♥✐t✐♦♥ ❈♦♠♣❧❡t❡ ❙❡♠❛♥t✐❝ ❚r❡❡ ●✐✈❡♥ ❛♥ ❛t♦♠ s❡t ❆ = ❆✶, · · · , ❆❦, · · · ❆ s❡♠❛♥t✐❝ tr❡❡ ✐s ❝♦♠♣❧❡t❡ ✐✛ ❢♦r ❡✈❡r② ❧❡❛❢ ♥♦❞❡ ◆✱ ■(◆) ❝♦♥t❛✐♥s ❆✐ ♦r ¬❆✐ ❢♦r ✐ = ✶, ✷, · · · ◆♦t❡ ❆❧❧ ♣r❡✈✐♦✉s s❡♠❛♥t✐❝ tr❡❡s ✇❡r❡ ❝♦♠♣❧❡t❡

slide-35
SLIDE 35

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❛♠♣❧❡

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ s❡t ♦❢ ❝❧✉s❡s ❙ = P(❢ (①))✱ t❤❡ ❍❡r❜r❛♥❞ ❇❛s❡ ❢♦r ❙ ✐s ❆ = {P(❛), P(❢ (❛)), · · · } ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❙❡♠❛♥t✐❝ ❚r❡❡ r❡♣r❡s❡♥ts ❙ ❛♥❞ ✐s ♥♦t ❝♦♠♣❧❡t❡

¬P(❛) P(❛) ¬P(❢ (❛)) P(❢ (❛)) P(❢ (❛)) ¬P(❢ (❛))

❋✐❣✉r❡✿ ◆♦t ❈♦♠♣❧❡t❡ ❙❡♠❛♥t✐❝ tr❡❡

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SLIDE 36

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❖❜s❡r✈❛t✐♦♥s ♦♥ ❙❡♠❛♥t✐❝ ❚r❡❡

  • ✐✈❡♥ ❛ s❡♠❛♥t✐❝ tr❡❡ ❚ r❡♣r❡s❡♥t✐♥❣ ❛ s❡t ♦❢ ❝❧❛✉s❡ ❙ ❢♦r

❡❛❝❤ ♥♦❞❡ ◆✱ ■(◆) ✐s ❛ s✉❜s❡t ♦❢ ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ❢♦r ❙ ■(◆) ✐s t❤❡r❡❢♦r❡ ❛ ♣❛rt✐❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❙

  • ✐✈❡♥ ❙✱ ✐❢ ❆ ✐♥✜♥✐t❡ t❤❡♥ ❛♥② ❝♦♠♣❧❡t❡ s❡♠❛♥t✐❝ tr❡❡ ❚

❢♦r ❙ ✐s ✐♥✜♥✐t❡

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SLIDE 37

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❖❜s❡r✈❛t✐♦♥s ♦♥ ❙❡♠❛♥t✐❝ ❚r❡❡s ❛♥❞ ❙❛t✐s✜❛❜✐❧✐t②

❲❡ ❝❛♥ ✉s❡ s❡♠❛♥t✐❝ tr❡❡s t♦ ❝❤❡❝❦ s❛t✐s✜❛❜✐❧✐t② ♦❢ ❙

  • ✐✈❡♥ ❛ s❡t ♦❢ ❝❧❛✉s❡ ❙ ❛♥② ❝♦♠♣❧❡t❡ s❡♠❛♥t✐❝ tr❡❡ ❢♦r ❙

❝♦♥t❛✐♥s ❛❧❧ ♣♦ss✐❜❧❡ ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ ❙✳ ❲❤❡♥ ❡①♣❛♥❞✐♥❣ t❤❡ s❡♠❛♥t✐❝ tr❡❡✱ ✇❡ ❝❛♥ st♦♣ ❡①♣❛♥❞✐♥❣ ❛s s♦♦♥ ❛s ❛ ♣❛rt✐❛❧ ✐♥t❡r♣r❡t❛t✐♦♥ ❢❛❧s✐✜❡s ❙✳ ■❢ ■(◆) ❢❛❧s✐✜❡s ❙ ✇❡ ❝❛♥ st♦♣ ❛t ♥♦❞❡ ◆✳

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SLIDE 38

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❉❡✜♥✐t✐♦♥✿ ❋❛✐❧✉r❡ ◆♦❞❡

❉❡✜♥✐t✐♦♥ ✭❋❛✐❧✉r❡ ◆♦❞❡✮

  • ✐✈❡♥ ❛ s❡t ♦❢ ❝❧❛✉s❡s ❙ ❛♥❞ ❛ s❡♠❛♥t✐❝ tr❡❡ ❢♦r ❙✱ ❛ ♥♦❞❡ ◆ ✐s

❝❛❧❧❡❞ ❛ ❢❛✐❧✉r❡ ♥♦❞❡ ✐✛ ■(◆) ❢❛❧s✐✜❡s s♦♠❡ ❣r♦✉♥❞ ✐♥st❛♥❝❡s ♦❢ ❛ ❝❧❛✉s❡ ✐♥ ❙✱ ❜✉t ■(◆′) ❞♦❡s ♥♦t ❢❛❧s✐❢② ❛♥② ❣r♦✉♥❞ ✐♥st❛♥❝❡ ♦❢ ❛ ❝❧❛✉s❡ ✐♥ ❙ ❢♦r ❡✈❡r② ❛♥❝❡st♦r ◆′ ♦❢ ◆✳ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❝❧❛✉s❡ ❙ = {P ∨ ◗, ◗} ❜✉✐❧❞ ❛ s❡♠❛♥t✐❝ tr❡❡ ❛♥❞ ❝❤❡❝❦ ✇❤✐❝❤ ♥♦❞❡ ✐s ❛ ❢❛✐❧✉r❡ ♥♦❞❡✳

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SLIDE 39

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❉❡✜♥✐t✐♦♥✿ ❈❧♦s❡❞ ❚r❡❡

❉❡✜♥✐t✐♦♥ ✭❈❧♦s❡❞ ❙❡♠❛♥t✐❝ ❚r❡❡✮ ❆ s❡♠❛♥t✐❝ tr❡❡ ❚ ✐s s❛✐❞ t♦ ❜❡ ❝❧♦s❡❞ ✐✛ ❡✈❡r② ❜r❛♥❝❤ ♦❢ ❚ t❡r♠✐♥❛t❡s ❛t ❛ ❢❛✐❧✉r❡ ♥♦❞❡✳ ❉❡✜♥✐t✐♦♥ ✭■♥❢❡r❡♥❝❡ ◆♦❞❡✮ ❆ ♥♦❞❡ ◆ ♦❢ ❛ ❝❧♦s❡❞ s❡♠❛♥t✐❝ tr❡❡ ✐s ❝❛❧❧❡❞ ❛♥ ✐♥❢❡r❡♥❝❡ ♥♦❞❡ ✐❢ ❛❧❧ ✐ts ✐♠♠❡❞✐❛t❡ ❞❡s❝❡♥❞❛♥t ♥♦❞❡s ❛r❡ ❢❛✐❧✉r❡ ♥♦❞❡s✳

slide-40
SLIDE 40

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❈❧♦s❡❞ ❙❡♠❛♥t✐❝ ❚r❡❡✿ ❊①❛♠♣❧❡ ■

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦r♠✉❧❛ ❙ = {P, ◗ ∨ ❘, ¬P ∨ ¬◗, ¬P ∨ ¬❘} ❆ = {P, ◗, ❘}

X X X X

❘ ¬❘ ¬◗ ◗ P ¬P

¬P ∨ ¬❘ ¬P ∨ ¬◗ ◗ ∨ ❘ P

❋✐❣✉r❡✿ ❈❧♦s❡❞ ❙❡♠❛♥t✐❝ tr❡❡

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SLIDE 41

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❈❧♦s❡❞ ❙❡♠❛♥t✐❝ ❚r❡❡✿ ❊①❛♠♣❧❡ ■■

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦r♠✉❧❛ ❙ = {¬P(①) ∨ ◗(①), P(❛), ¬◗(③)}✱ ❍ = {❛} ❆ = {P(❛), ◗(❛)}

P ❛ P ❛ ◗ ❛ ◗ ❛

◗ ③ ❛ P ① ❛ ◗ ① ❛ P ❛

❋✐❣✉r❡✿ ❈❧♦s❡❞ ❙❡♠❛♥t✐❝ tr❡❡

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SLIDE 42

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❈❧♦s❡❞ ❙❡♠❛♥t✐❝ ❚r❡❡✿ ❊①❛♠♣❧❡ ■■

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦r♠✉❧❛ ❙ = {¬P(①) ∨ ◗(①), P(❛), ¬◗(③)}✱ ❍ = {❛} ❆ = {P(❛), ◗(❛)}

X X X

P(❛) ¬P(❛) ¬◗(❛) ◗(❛)

¬◗(③ = ❛) ¬P(① = ❛) ∨ ◗(① = ❛) P(❛)

❋✐❣✉r❡✿ ❈❧♦s❡❞ ❙❡♠❛♥t✐❝ tr❡❡

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SLIDE 43

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❈❧♦s❡❞ ❙❡♠❛♥t✐❝ ❚r❡❡✿ ❊①❛♠♣❧❡ ■■■

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦r♠✉❧❛ ❙ = {¬P(①) ∨ ◗(①), P(❢ (❛)), ¬◗(③)}

❍ = {❛, ❢ (❛), ❢ (❢ (❛)), · · · } ❆ = {P(❛), ◗(❛), P(❢ (❛)), ◗(❢ (❛)), · · · }

P ❛ P ❛ ◗ ❛ ◗ ❛ P ❢ ❛ P ❢ ❛ ◗ ❢ ❛ ◗ ❢ ❛ ◗ ❛ ◗ ❛ P ① ❛ ◗ ① ❛ ◗ ③ ❛ ◗ ③ ❢ ❛ P ① ❢ ❛ ◗ ① ❢ ❛ P ❢ ❛ ◗ ③ ❛

❋✐❣✉r❡✿ ❈❧♦s❡❞ ❙❡♠❛♥t✐❝ tr❡❡

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SLIDE 44

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❈❧♦s❡❞ ❙❡♠❛♥t✐❝ ❚r❡❡✿ ❊①❛♠♣❧❡ ■■■

❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦r♠✉❧❛ ❙ = {¬P(①) ∨ ◗(①), P(❢ (❛)), ¬◗(③)}

❍ = {❛, ❢ (❛), ❢ (❢ (❛)), · · · } ❆ = {P(❛), ◗(❛), P(❢ (❛)), ◗(❢ (❛)), · · · }

X X X X X X

P(❛) ¬P(❛) ◗(❛) ¬◗(❛) P(❢ (❛)) ¬P(❢ (❛)) ◗(❢ (❛)) ¬◗(❢ (❛)) ¬◗(❛) ◗(❛) ¬P(① = ❛) ∨ ◗(① = ❛) ¬◗(③ = ❛) ¬◗(③ = ❢ (❛)) ¬P(① = ❢ (❛)) ∨ ◗(① = ❢ (❛)) P(❢ (❛)) ¬◗(③ = ❛)

❋✐❣✉r❡✿ ❈❧♦s❡❞ ❙❡♠❛♥t✐❝ tr❡❡

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SLIDE 45

❍✲ ■♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ❍✲ ❙❛t✐s✜❛❜✐❧✐t② ❙❡♠❛♥t✐❝ ❚r❡❡s

❊①❡r❝✐s❡✿ ❙❡♠❛♥t✐❝ ❚r❡❡

❊①❡r❝✐s❡

✶ ❙ = {P, ¬P ∨ ◗, ¬◗} ●✐✈❡ ❛ ❝❧♦s❡❞ ❙❡♠❛♥t✐❝ ❚r❡❡ ♦❢ ❙

❬❈❤❛♥❣✲▲❡❡ ❊① ✶✶✱ ♣❛❣❡ ✻✽❪

✷ ❙ = {P(①), ¬P(①) ∨ ◗(①, ❛), ¬◗(②, ❛)} ❬❈❤❛♥❣✲▲❡❡ ❊① ✶✷✱

♣❛❣❡ ✻✽❪

  • ✐✈❡ t❤❡ ❛t♦♠ s❡t ♦❢ ❙
  • ✐✈❡ ❛ ❝♦♠♣❧❡t❡ ❙❡♠❛♥t✐❝ ❚r❡❡ ♦❢ ❙
  • ✐✈✇ ❛ ❝❧♦s❡❞ ❙❡♠❛♥t✐❝ ❚r❡❡ ♦❢ ❙