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rss t t - - PowerPoint PPT Presentation

May 11, 2023 409 likes •1.1k views

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SLIDE 1

❍♦♠♦♠♦r♣❤✐s♠s t♦ ♠♦❞❡❧ ♥❛✈✐❣❛t✐♦♥❛❧ q✉❡r✐❡s ♦❢ ❣r❛♣❤ ❞❛t❛❜❛s❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ✭▲■❋❖✱ ❯♥✐✈❡rs✐té ❞✬❖r❧é❛♥s✱ ❋r❛♥❝❡✮ ❥♦✐♥t ✇♦r❦ ✇✐t❤✿ ▲❛✉r❡♥t ❇❡❛✉❞♦✉✱ ❋❧♦r❡♥t ▼❛❞❡❧❛✐♥❡✱ ▲❤♦✉❛r✐ ◆♦✉r✐♥❡✱ ●❛ét❛♥ ❘✐❝❤❛r❞ ❆♣r✐❧ ✷✵✶✾

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SLIDE 2

❘❡❧❛t✐♦♥❛❧ ❞❛t❛❜❛s❡s

❉❛t❛❜❛s❡ ✭❉❇✮ ✿ st♦r❛❣❡ s②st❡♠ ✰ ♦r❣❛♥✐s❛t✐♦♥ ♦❢ ♥✉♠❡r✐❝❛❧ ❞❛t❛✳ ❘❡❧❛t✐♦♥❛❧ ❞❛t❛❜❛s❡ ✭❘❉❇✮ ✿ ♠♦❞❡❧ ❞❡✜♥❡❞ ❜② ❈♦❞❞✱ ✶✾✼✵✳ ❙❡t ♦❢ t❛❜❧❡s ✇✐t❤ ❦❡②s ✐❞❡♥t✐❢②✐♥❣ ❡❛❝❤ ❞❛t❛ t✉♣❧❡✳ ◗✉❡r② ✿ ✐♥t❡rr♦❣❛t✐♦♥ ♦❢ t❤❡ ❉❇ t♦ ♦❜t❛✐♥ s♦♠❡ ✐♥❢♦r♠❛t✐♦♥ ❚②♣✐❝❛❧ q✉❡r② ❧❛♥❣❛❣❡ ✿ ❙◗▲✱ ❙tr✉❝t✉r❡❞ ◗✉❡r② ▲❛♥❣✉❛❣❡ ✭✏s❡q✉❡❧✑✮ ❇♦②❝❡✲❈❤❛♠❜❡r❧❛✐♥✱ ■❇▼✱ ✶✾✼✾ ❡① ✿ ❙❊▲❊❈❚ ❆✉t❤♦r■❉ ❋❘❖▼ ❇♦♦❦s ❲❍❊❘❊ ❉❛t❡ ❃ ✶✾✼✵ ❀

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷ ✴ ✷✹

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SLIDE 3
  • r❛♣❤ ❞❛t❛❜❛s❡s
  • r❛♣❤ ❞❛t❛❜❛s❡ ✭●❉❇✮ ✿ ❛❧❧ r❡❧❛t✐♦♥s ❛r❡ ❜✐♥❛r②✳

❉❛t❛ ✐s st♦r❡❞ ❛s ❛ ❣r❛♣❤ ✭✏❞❛t❛ ❣r❛♣❤✑✮✳ ❋❛st✲❞❡✈❡❧♦♣✐♥❣ t❡❝❤♥♦❧♦❣② → ◆❡♦✹❏✱ ❖r✐❡♥t❉❇✱ ●r❛♣❤❉❇✳✳✳ ◗✉❡r② ❧❛♥❣✉❛❣❡s ✿ ❈②♣❤❡r✱ ❙P❆❘◗▲✱ ❳P❛t❤✳✳✳ P❛rt ♦❢ t❤❡ r❡❝❡♥t tr❡♥❞ ❢♦r ❣r❛♣❤✲❜❛s❡❞ ■❚ s②st❡♠s → ●♦♦❣❧❡ ✭Pr❡❣❡❧✮✱ ❋❛❝❡❜♦♦❦ ✭●r❛♣❤ ❆P■✮✱ ❚✇✐tt❡r ✭❈❛ss♦✈❛r②✮✳✳✳

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✸ ✴ ✷✹

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SLIDE 4

❙t❛♥❞❛r❞s ❛r❡ ❜❡✐♥❣ ✇r✐tt❡♥

❈♦♥❢❡r❡♥❝❡ ❆❈▼ ❙■●▼❖❉✬✶✽ ✿ ✐♥❞✉str② ❛♥❞ ❛❝❛❞❡♠✐❝s tr② t♦ ❞❡s✐❣♥ ❢✉t✉r❡ st❛♥❞❛r❞s ❢♦r ✏●r❛♣❤ ◗✉❡r② ▲❛♥❣✉❛❣❡s✑

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✹ ✴ ✷✹

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SLIDE 5

❈②♣❤❡r ❜② ◆❡♦✹❏

❈②♣❤❡r ✭✷✵✶✶✮ ✰ ❖♣❡♥ ❈②♣❤❡r ✭✷✵✶✺✮ ♦♥ Pr♦♣❡rt② ❣r❛♣❤s ❊①❛♠♣❧❡s ♦❢ ❈②♣❤❡r q✉❡r✐❡s ✿ ▼❆❚❈❍✭ ✿P❡rs♦♥ {♥❛♠❡ ✿✬❏❡♥♥✐❢❡r✬}✮✲❬ ✿❲❖❘❑❙❴❋❖❘❪✲❃✭❝ ✿❈♦♠♣❛♥②✮ ❘❊❚❯❘◆ ❝ ▼❆❚❈❍ ♣❂✭❛ ✿P❡rs♦♥✮✲❬✯✶✳✳✸❪✲✭❜ ✿P❡rs♦♥✮ ❲❍❊❘❊ ❛✳♥❛♠❡=✬❆❧✐❝❡✬ ❆◆❉ ❜✳♥❛♠❡=✬❇♦❜✬ ❘❊❚❯❘◆ ♣ ❆❧s♦ ✿ s❤♦rt❡st ♣❛t❤s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✺ ✴ ✷✹

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SLIDE 6
  • r❛♣❤ ❞❛t❛❜❛s❡s ✿ s✐♠♣❧❡r t❤❛♥ ❝❧❛ss✐❝ ❘❉❇ ❄
  • r❛♣❤s ✿ ❜♦t❤ s✐♠♣❧❡ ❛♥❞ ❡①♣r❡ss✐✈❡

✭❡①❛♠♣❧❡ ❢r♦♠ ◆❡♦✹❏✮

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✻ ✴ ✷✹

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SLIDE 7
  • r❛♣❤ ❞❛t❛❜❛s❡s ✲ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✼ ✴ ✷✹

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SLIDE 8
  • r❛♣❤ ❞❛t❛❜❛s❡s ✲ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✼ ✴ ✷✹

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SLIDE 9
  • r❛♣❤ ❞❛t❛❜❛s❡s ✲ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✼ ✴ ✷✹

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SLIDE 10
  • r❛♣❤ ❞❛t❛❜❛s❡s ✲ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✼ ✴ ✷✹

slide-11
SLIDE 11
  • r❛♣❤ ❞❛t❛❜❛s❡s ✲ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✼ ✴ ✷✹

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SLIDE 12
  • r❛♣❤ ❞❛t❛❜❛s❡s ✲ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✼ ✴ ✷✹

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SLIDE 13
  • r❛♣❤ ❞❛t❛❜❛s❡s ✲ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✼ ✴ ✷✹

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SLIDE 14
  • r❛♣❤ ❞❛t❛❜❛s❡s ✲ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✼ ✴ ✷✹

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SLIDE 15
  • r❛♣❤ ❞❛t❛❜❛s❡s ✲ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✼ ✴ ✷✹

slide-16
SLIDE 16
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

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SLIDE 17
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

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SLIDE 18
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍

1 1 2 2 2 3 3 4 5

❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

1 2 3 5 4

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-19
SLIDE 19
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

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SLIDE 20
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-21
SLIDE 21
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-22
SLIDE 22
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-23
SLIDE 23
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-24
SLIDE 24
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-25
SLIDE 25
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-26
SLIDE 26
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-27
SLIDE 27
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-28
SLIDE 28
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-29
SLIDE 29
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-30
SLIDE 30
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-31
SLIDE 31
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-32
SLIDE 32
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-33
SLIDE 33
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❚❛r❣❡t ❣r❛♣❤ ✿ ❍ = ❈✺

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-34
SLIDE 34
  • r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠s

▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❞❥❛❝❡♥❝②✱ ❛r❝ ❧❛❜❡❧s ❛♥❞ ♦r✐❡♥t❛✲ t✐♦♥ ✿ ①② ∈ ❊(●) = ⇒ ❤(①)❤(②) ∈ ❊(❍) ◆♦t❛t✐♦♥ ✿ ● → ❍✳ ❉❡✜♥✐t✐♦♥ ✲ ●r❛♣❤ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ● t♦ ❍ ❈♦♠♣❧❡t❡ ❣r❛♣❤ ❑✻

  • → ❑❦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ χ(●) ≤ ❦

❘❡♠❛r❦✿ ❍♦♠♦♠♦r♣❤✐s♠s ❣❡♥❡r❛❧✐③❡ ♣r♦♣❡r ❝♦❧♦✉r✐♥❣s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✽ ✴ ✷✹

slide-35
SLIDE 35

❈♦r❡s

  • ❈♦r❡ ♦❢ ● ✿ ♠✐♥✐♠❛❧ s✉❜❣r❛♣❤ ❍ ✇✐t❤ ● → ❍
  • ● ✐s ❛ ❝♦r❡ ✐❢ ❝♦r❡(●) = ●

❉❡✜♥✐t✐♦♥ ✲ ❈♦r❡ ❚❤❡ ❝♦r❡ ♦❢ ❛ ❣r❛♣❤ ✐s ✉♥✐q✉❡ ✭✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✮ Pr♦♣♦s✐t✐♦♥ ❊①❛♠♣❧❡s ✿ t❤❡ ❝♦r❡ ♦❢ ❛♥② ♥♦♥tr✐✈✐❛❧ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✐s ❑✷ ❝♦♠♣❧❡t❡ ❣r❛♣❤s ❛♥❞ ♦❞❞ ❝②❝❧❡s ❛r❡ ❝♦r❡s

  • ❍ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❝♦r❡ ●

❝♦r❡ ❍ Pr♦♣♦s✐t✐♦♥

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✾ ✴ ✷✹

slide-36
SLIDE 36

❈♦r❡s

  • ❈♦r❡ ♦❢ ● ✿ ♠✐♥✐♠❛❧ s✉❜❣r❛♣❤ ❍ ✇✐t❤ ● → ❍
  • ● ✐s ❛ ❝♦r❡ ✐❢ ❝♦r❡(●) = ●

❉❡✜♥✐t✐♦♥ ✲ ❈♦r❡ ❚❤❡ ❝♦r❡ ♦❢ ❛ ❣r❛♣❤ ✐s ✉♥✐q✉❡ ✭✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✮ Pr♦♣♦s✐t✐♦♥ ❊①❛♠♣❧❡s ✿ • t❤❡ ❝♦r❡ ♦❢ ❛♥② ♥♦♥tr✐✈✐❛❧ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✐s ❑✷

  • ❝♦♠♣❧❡t❡ ❣r❛♣❤s ❛♥❞ ♦❞❞ ❝②❝❧❡s ❛r❡ ❝♦r❡s
  • ❍ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❝♦r❡ ●

❝♦r❡ ❍ Pr♦♣♦s✐t✐♦♥

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✾ ✴ ✷✹

slide-37
SLIDE 37

❈♦r❡s

  • ❈♦r❡ ♦❢ ● ✿ ♠✐♥✐♠❛❧ s✉❜❣r❛♣❤ ❍ ✇✐t❤ ● → ❍
  • ● ✐s ❛ ❝♦r❡ ✐❢ ❝♦r❡(●) = ●

❉❡✜♥✐t✐♦♥ ✲ ❈♦r❡ ❚❤❡ ❝♦r❡ ♦❢ ❛ ❣r❛♣❤ ✐s ✉♥✐q✉❡ ✭✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✮ Pr♦♣♦s✐t✐♦♥ ❊①❛♠♣❧❡s ✿ • t❤❡ ❝♦r❡ ♦❢ ❛♥② ♥♦♥tr✐✈✐❛❧ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✐s ❑✷

  • ❝♦♠♣❧❡t❡ ❣r❛♣❤s ❛♥❞ ♦❞❞ ❝②❝❧❡s ❛r❡ ❝♦r❡s
  • → ❍ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❝♦r❡(●) → ❝♦r❡(❍)

Pr♦♣♦s✐t✐♦♥

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✾ ✴ ✷✹

slide-38
SLIDE 38

❈♦r❡s

  • ❈♦r❡ ♦❢ ● ✿ ♠✐♥✐♠❛❧ s✉❜❣r❛♣❤ ❍ ✇✐t❤ ● → ❍
  • ● ✐s ❛ ❝♦r❡ ✐❢ ❝♦r❡(●) = ●

❉❡✜♥✐t✐♦♥ ✲ ❈♦r❡ ❚❤❡ ❝♦r❡ ♦❢ ❛ ❣r❛♣❤ ✐s ✉♥✐q✉❡ ✭✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✮ Pr♦♣♦s✐t✐♦♥ ❊①❛♠♣❧❡s ✿ • t❤❡ ❝♦r❡ ♦❢ ❛♥② ♥♦♥tr✐✈✐❛❧ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✐s ❑✷

  • ❝♦♠♣❧❡t❡ ❣r❛♣❤s ❛♥❞ ♦❞❞ ❝②❝❧❡s ❛r❡ ❝♦r❡s
  • → ❍ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❝♦r❡(●) → ❝♦r❡(❍)

Pr♦♣♦s✐t✐♦♥

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✾ ✴ ✷✹

slide-39
SLIDE 39

❚❤❡ ❤♦♠♦♠♦r♣❤✐s♠ ♦r❞❡r

❉❡✜♥❡❞ ❜② ● ❍ ✐✛ ● → ❍ ✭✐❢ r❡str✐❝t❡❞ t♦ ❝♦r❡s ✿ ♣❛rt✐❛❧ ♦r❞❡r✮✳ ❉❡✜♥✐t✐♦♥ ✲ ❍♦♠♦♠♦r♣❤✐s♠ q✉❛s✐✲♦r❞❡r

...

✐s ✿

  • r❡✢❡①✐✈❡
  • tr❛♥s✐t✐✈❡
  • ❛♥t✐s②♠♠❡tr✐❝ ✭♦♥ ❝♦r❡s✮

❚❤❡ ♦r❞❡r ✐s ✿

  • ❞❡♥s❡
  • ✉♥✐✈❡rs❛❧
  • ❢r❛❝t❛❧

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✵ ✴ ✷✹

slide-40
SLIDE 40

❍✲❈❖▲❖❯❘■◆● ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠

■◆P❯❚ ✿ ❛ ❣r❛♣❤ ●✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s ● ❤❛✈❡ ❛ ❤♦♠♦♠♦r♣❤✐s♠ t♦ ❍ ❄ ❍✲❈❖▲❖❯❘■◆● ❍✲❈❖▲❖❯❘■◆● ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ✐❢ ❍ ✐s ❜✐♣❛rt✐t❡ ♦r ❤❛s ❛ ❧♦♦♣✳ ❖t❤❡r✇✐s❡✱ ✐t ✐s ◆P✲❝♦♠♣❧❡t❡✳ ❚❤❡♦r❡♠ ✭❍❡❧❧✲◆❡➨❡t➦✐❧✱ ✶✾✾✵✮ ◆♦t❡ ✿ ❢♦r ❞✐r❡❝t❡❞ ❣r❛♣❤s✱ s✉❝❤ ♥✐❝❡ ❞✐❝❤♦t♦♠② ✐s ✉♥❦♥♦✇♥ ✭❛♥❞ ♣r♦❜❛❜❧② ❞♦❡s ♥♦t ❡①✐st✮ ✦

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✶ ✴ ✷✹

slide-41
SLIDE 41

❈♦♥str❛✐♥t ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❧❡♠ ✭❈❙P✮

❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ❙ = (❳,❱ ) ✿ ❞♦♠❛✐♥ ❳ ✰ r❡❧❛t✐♦♥s ❘✶,...,❘❦ ♦❢ ❛r✐t② ❛✶,...,❛❦ ✭❘✐ ⊆ ❳ ❛✐ ✮✳ ❊① ✿ ❣r❛♣❤s✱ ❞✐❣r❛♣❤s✱ ❦✲❙❆❚ ❇♦♦❧❡❛♥ ❢♦r♠✉❧❛❡✳✳✳ ■◆P❯❚ ✿ t✇♦ r❡❧❛t✐♦♥❛❧ str✉❝t✉r❡s ❙ ❡t ❚✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s ❙ ❤❛✈❡ ❛ ❤♦♠♦♠♦r♣❤✐s♠ t♦ ❚ ❄ ❈❙P ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✷ ✴ ✷✹

slide-42
SLIDE 42

❈♦♥str❛✐♥t ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❧❡♠ ✭❈❙P✮

❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ❙ = (❳,❱ ) ✿ ❞♦♠❛✐♥ ❳ ✰ r❡❧❛t✐♦♥s ❘✶,...,❘❦ ♦❢ ❛r✐t② ❛✶,...,❛❦ ✭❘✐ ⊆ ❳ ❛✐ ✮✳ ❊① ✿ ❣r❛♣❤s✱ ❞✐❣r❛♣❤s✱ ❦✲❙❆❚ ❇♦♦❧❡❛♥ ❢♦r♠✉❧❛❡✳✳✳ ■◆P❯❚ ✿ ❛ r❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ❙✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s ❙ ❤❛✈❡ ❛ ❤♦♠♦♠♦r♣❤✐s♠ t♦ ❚ ❄ ✭◆♦♥✲✉♥✐❢♦r♠✮ ❈❙P ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❛ ✜①❡❞ t❡♠♣❧❛t❡ ❚

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✷ ✴ ✷✹

slide-43
SLIDE 43

❈♦♥str❛✐♥t ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❧❡♠ ✭❈❙P✮

❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ❙ = (❳,❱ ) ✿ ❞♦♠❛✐♥ ❳ ✰ r❡❧❛t✐♦♥s ❘✶,...,❘❦ ♦❢ ❛r✐t② ❛✶,...,❛❦ ✭❘✐ ⊆ ❳ ❛✐ ✮✳ ❊① ✿ ❣r❛♣❤s✱ ❞✐❣r❛♣❤s✱ ❦✲❙❆❚ ❇♦♦❧❡❛♥ ❢♦r♠✉❧❛❡✳✳✳ ■◆P❯❚ ✿ ❛ r❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ❙✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s ❙ ❤❛✈❡ ❛ ❤♦♠♦♠♦r♣❤✐s♠ t♦ ❚ ❄ ✭◆♦♥✲✉♥✐❢♦r♠✮ ❈❙P ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❛ ✜①❡❞ t❡♠♣❧❛t❡ ❚ ❋♦r ❡✈❡r② r❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ❚✱ ❚✲❈❙P ✐s ❡✐t❤❡r ◆P✲❝♦♠♣❧❡t❡ ♦r ♣♦❧②♥♦♠✐❛❧✲ t✐♠❡ ✭✐✳❡✳ ♥♦t ◆P✲✐♥t❡r♠❡❞✐❛t❡✮✳ ❈♦♥❥❡❝t✉r❡ ✭❋❡❞❡r✲❱❛r❞✐✱ ✶✾✾✽ ✲ ❉✐❝❤♦t♦♠② ❈♦♥❥❡❝t✉r❡✮

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✷ ✴ ✷✹

slide-44
SLIDE 44

❈♦♥str❛✐♥t ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❧❡♠ ✭❈❙P✮

❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ❙ = (❳,❱ ) ✿ ❞♦♠❛✐♥ ❳ ✰ r❡❧❛t✐♦♥s ❘✶,...,❘❦ ♦❢ ❛r✐t② ❛✶,...,❛❦ ✭❘✐ ⊆ ❳ ❛✐ ✮✳ ❊① ✿ ❣r❛♣❤s✱ ❞✐❣r❛♣❤s✱ ❦✲❙❆❚ ❇♦♦❧❡❛♥ ❢♦r♠✉❧❛❡✳✳✳ ■◆P❯❚ ✿ ❛ r❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ❙✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s ❙ ❤❛✈❡ ❛ ❤♦♠♦♠♦r♣❤✐s♠ t♦ ❚ ❄ ✭◆♦♥✲✉♥✐❢♦r♠✮ ❈❙P ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❛ ✜①❡❞ t❡♠♣❧❛t❡ ❚ ❋♦r ❡✈❡r② r❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ❚✱ ❚✲❈❙P ✐s ❡✐t❤❡r ◆P✲❝♦♠♣❧❡t❡ ♦r ♣♦❧②♥♦♠✐❛❧✲ t✐♠❡ ✭✐✳❡✳ ♥♦t ◆P✲✐♥t❡r♠❡❞✐❛t❡✮✳ ❈♦♥❥❡❝t✉r❡ ✭❋❡❞❡r✲❱❛r❞✐✱ ✶✾✾✽ ✲ ❉✐❝❤♦t♦♠② ❈♦♥❥❡❝t✉r❡✮ ❙✐❣❣❡rs ♣♦❧②♠♦r♣❤✐s♠ ✿ ❤♦♠♦♠♦r♣❤✐s♠ ❢ : ❚ ✹ → ❚ ✇✐t❤ ❢ (❛,r,❡,❛) = ❢ (r,❛,r,❡) ❢♦r ❛❧❧ ❛,❡,r ∈ ❚✳ ■❢ ❚ ❤❛s ❛ ❙✐❣❣❡rs ♣♦❧②♠♦r♣❤✐s♠✱ ❚✲❈❙P ✐s ♣♦❧②♥♦♠✐❛❧✳ ❖t❤❡r✇✐s❡✱ ◆P✲❤❛r❞✳ ❚❤❡♦r❡♠ ✭❇✉❧❛t♦✈ ✷✵✶✼ ✰ ❩❤✉❦ ✷✵✶✼✮ Pr♦✈❡❞ ❛❢t❡r ✷✵ ②❡❛rs ♦❢ ✐♥t❡♥s✐✈❡ ✇♦r❦s ✉s✐♥❣ ❛❧❣❡❜r❛✐❝ t❡❝❤♥✐q✉❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✷ ✴ ✷✹

slide-45
SLIDE 45

❇❛❝❦ t♦ ♦✉r q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✸ ✴ ✷✹

slide-46
SLIDE 46

❉❛t❛❜❛s❡s ✲ s♦♠❡ ❛❧❣♦r✐t❤♠✐❝ t❛s❦s

■◆P❯❚ ✿ ❛ ❞❛t❛❜❛s❡ ❇ ❛♥❞ ❛ q✉❡r② ◗✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ♠❛t❝❤ ♦❢ ◗ ✐♥ ❇ ❄ ◗✉❡r② ❡✈❛❧✉❛t✐♦♥ ♣r♦❜❧❡♠ ■◆P❯❚ ✿ ❛ ❞❛t❛❜❛s❡ ❇ ❛♥❞ ❛ q✉❡r② ◗✳ ❚❆❙❑ ✿ ▲✐st ❛❧❧ ♠❛t❝❤❡s ♦❢ ◗ ✐♥ ❇✳ ❊♥✉♠❡r❛t✐✈❡ q✉❡r② ❡✈❛❧✉❛t✐♦♥ ♣r♦❜❧❡♠

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✹ ✴ ✷✹

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SLIDE 47

❉❛t❛❜❛s❡s ✲ s♦♠❡ ❛❧❣♦r✐t❤♠✐❝ t❛s❦s

■◆P❯❚ ✿ ❛ ❞❛t❛❜❛s❡ ❇ ❛♥❞ ❛ q✉❡r② ◗✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ♠❛t❝❤ ♦❢ ◗ ✐♥ ❇ ❄ ◗✉❡r② ❡✈❛❧✉❛t✐♦♥ ♣r♦❜❧❡♠ ■◆P❯❚ ✿ ❛ ❞❛t❛❜❛s❡ ❇ ❛♥❞ ❛ q✉❡r② ◗✳ ❚❆❙❑ ✿ ▲✐st ❛❧❧ ♠❛t❝❤❡s ♦❢ ◗ ✐♥ ❇✳ ❊♥✉♠❡r❛t✐✈❡ q✉❡r② ❡✈❛❧✉❛t✐♦♥ ♣r♦❜❧❡♠

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✹ ✴ ✷✹

slide-48
SLIDE 48

❆❧❣♦r✐t❤♠✐❝ t❛s❦s ✲ q✉❡r✐❡s

■◆P❯❚ ✿ ❛ q✉❡r② ◗✳ ◗❯❊❙❚■❖◆ ✿ ■s ◗ ♠✐♥✐♠❛❧ ✭✐s ❛ ❝♦r❡✮ ❄ ◗✉❡r② ♠✐♥✐♠✐③❛t✐♦♥ ◗✶ ❛♥❞ ◗✷ ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ ❢♦r ❡❛❝❤ ❞❛t❛❜❛s❡ ❇✱ ◗✶ ♠❛t❝❤❡s ❇ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ◗✷ ♠❛t❝❤❡s ❇✳ ■◆P❯❚ ✿ t✇♦ q✉❡r✐❡s ◗✶✱ ◗✷✳ ◗❯❊❙❚■❖◆ ✿ ❆r❡ ◗✶ ❛♥❞ ◗✷ ❡q✉✐✈❛❧❡♥t ❄ ◗✉❡r② ❡q✉✐✈❛❧❡♥❝❡ ◗✶ ✐s ❝♦♥t❛✐♥❡❞✴✐♥❝❧✉❞❡❞ ✐♥ ◗✷ ✐❢ ❢♦r ❡❛❝❤ ❞❛t❛❜❛s❡ ❇✱ ❛♥② ♠❛t❝❤ ♦❢ ◗✶ ✐♥ ❇ ✐s ❛ ♠❛t❝❤ ♦❢ ◗✷ ✐♥ ❇✳ ■◆P❯❚ ✿ t✇♦ q✉❡r✐❡s ◗✶✱ ◗✷✳ ◗❯❊❙❚■❖◆ ✿ ■s ◗✶ ❝♦♥t❛✐♥❡❞ ✐♥ ◗✷ ❄ ◗✉❡r② ❝♦♥t❛✐♥♠❡♥t

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✺ ✴ ✷✹

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SLIDE 49

❆❧❣♦r✐t❤♠✐❝ t❛s❦s ✲ q✉❡r✐❡s

■◆P❯❚ ✿ ❛ q✉❡r② ◗✳ ◗❯❊❙❚■❖◆ ✿ ■s ◗ ♠✐♥✐♠❛❧ ✭✐s ❛ ❝♦r❡✮ ❄ ◗✉❡r② ♠✐♥✐♠✐③❛t✐♦♥ ◗✶ ❛♥❞ ◗✷ ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ ❢♦r ❡❛❝❤ ❞❛t❛❜❛s❡ ❇✱ ◗✶ ♠❛t❝❤❡s ❇ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ◗✷ ♠❛t❝❤❡s ❇✳ ■◆P❯❚ ✿ t✇♦ q✉❡r✐❡s ◗✶✱ ◗✷✳ ◗❯❊❙❚■❖◆ ✿ ❆r❡ ◗✶ ❛♥❞ ◗✷ ❡q✉✐✈❛❧❡♥t ❄ ◗✉❡r② ❡q✉✐✈❛❧❡♥❝❡ ◗✶ ✐s ❝♦♥t❛✐♥❡❞✴✐♥❝❧✉❞❡❞ ✐♥ ◗✷ ✐❢ ❢♦r ❡❛❝❤ ❞❛t❛❜❛s❡ ❇✱ ❛♥② ♠❛t❝❤ ♦❢ ◗✶ ✐♥ ❇ ✐s ❛ ♠❛t❝❤ ♦❢ ◗✷ ✐♥ ❇✳ ■◆P❯❚ ✿ t✇♦ q✉❡r✐❡s ◗✶✱ ◗✷✳ ◗❯❊❙❚■❖◆ ✿ ■s ◗✶ ❝♦♥t❛✐♥❡❞ ✐♥ ◗✷ ❄ ◗✉❡r② ❝♦♥t❛✐♥♠❡♥t

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✺ ✴ ✷✹

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SLIDE 50

❆❧❣♦r✐t❤♠✐❝ t❛s❦s ✲ q✉❡r✐❡s

■◆P❯❚ ✿ ❛ q✉❡r② ◗✳ ◗❯❊❙❚■❖◆ ✿ ■s ◗ ♠✐♥✐♠❛❧ ✭✐s ❛ ❝♦r❡✮ ❄ ◗✉❡r② ♠✐♥✐♠✐③❛t✐♦♥ ◗✶ ❛♥❞ ◗✷ ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ ❢♦r ❡❛❝❤ ❞❛t❛❜❛s❡ ❇✱ ◗✶ ♠❛t❝❤❡s ❇ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ◗✷ ♠❛t❝❤❡s ❇✳ ■◆P❯❚ ✿ t✇♦ q✉❡r✐❡s ◗✶✱ ◗✷✳ ◗❯❊❙❚■❖◆ ✿ ❆r❡ ◗✶ ❛♥❞ ◗✷ ❡q✉✐✈❛❧❡♥t ❄ ◗✉❡r② ❡q✉✐✈❛❧❡♥❝❡ ◗✶ ✐s ❝♦♥t❛✐♥❡❞✴✐♥❝❧✉❞❡❞ ✐♥ ◗✷ ✐❢ ❢♦r ❡❛❝❤ ❞❛t❛❜❛s❡ ❇✱ ❛♥② ♠❛t❝❤ ♦❢ ◗✶ ✐♥ ❇ ✐s ❛ ♠❛t❝❤ ♦❢ ◗✷ ✐♥ ❇✳ ■◆P❯❚ ✿ t✇♦ q✉❡r✐❡s ◗✶✱ ◗✷✳ ◗❯❊❙❚■❖◆ ✿ ■s ◗✶ ❝♦♥t❛✐♥❡❞ ✐♥ ◗✷ ❄ ◗✉❡r② ❝♦♥t❛✐♥♠❡♥t

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✺ ✴ ✷✹

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SLIDE 51

◆❛✈✐❣❛t✐♦♥❛❧ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✻ ✴ ✷✹

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SLIDE 52

◆❛✈✐❣❛t✐♦♥❛❧ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✻ ✴ ✷✹

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SLIDE 53

◆❛✈✐❣❛t✐♦♥❛❧ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✻ ✴ ✷✹

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SLIDE 54

◆❛✈✐❣❛t✐♦♥❛❧ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✻ ✴ ✷✹

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SLIDE 55

◆❛✈✐❣❛t✐♦♥❛❧ q✉❡r✐❡s

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✻ ✴ ✷✹

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SLIDE 56

◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s

  • r❛♣❤ ❞❛t❛❜❛s❡ ✿ ❆r❝✲❧❛❜❡❧❡❞ ❞✐r❡❝t❡❞ ❣r❛♣❤✱ ❧❛❜❡❧s ❛r❡ ❢r♦♠ ❛ ✜①❡❞ ❛❧♣❤❛❜❡t

Σ = {❛,❜,❝,...} ◗✉❡r② ❣r❛♣❤ ✿ ❉✐r❡❝t❡❞ ❣r❛♣❤ ✇✐t❤ ❧❛❜❡❧ ❊(①,②) ❢♦r ❡❛❝❤ ❛r❝ (①,②)✳ ▲❛❜❡❧s ❛r❡ s❡ts ♦❢ ✇♦r❞s ♦✈❡r Σ✳ ▼❛♣♣✐♥❣ ❤ : ❱ (●) → ❱ (❍) t❤❛t ♣r❡s❡r✈❡s ❧❛❜❡❧s ✿ (①,②) ∈ ❆(◗) = ⇒ t❤❡r❡ ❡①✐sts ❛ ❞✐r❡❝t❡❞ ✇❛❧❦ ❢r♦♠ ❤(①) t♦ ❤(②) ✐♥ ❇ ✇❤♦s❡ ❛ss♦❝✐❛t❡❞ ✇♦r❞ ✐s ✐♥ ❊(①,②)✳ ❉❡✜♥✐t✐♦♥ ✲ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ◗ t♦ ❇

x y

E(x,y)={aab,aaab} b a a a a b b b a h(y) h(x) h b

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✼ ✴ ✷✹

slide-57
SLIDE 57

❘❡❣✉❧❛r ♣❛t❤ q✉❡r✐❡s ✭❘P◗✮

❚❤❡ s❡ts ♦❢ ✇♦r❞s ❧❛❜❡❧✐♥❣ t❤❡ ❛r❝s ❛r❡ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s✳ ❘❡❣✉❧❛r ❧❛♥❣✉❛❣❡ ✿ s❡t ♦❢ ✇♦r❞s ♦✈❡r ❛❧♣❤❛❜❡t Σ t❤❛t ✐s ❝❧♦s❡❞ ✉♥❞❡r ✉♥✐♦♥ ✰✱ ❝♦♥❝❛t❡♥❛t✐♦♥✱ ❛♥❞ ❑❧❡❡♥❡✲st❛r ∗✳ ❊①❛♠♣❧❡ ✿ ❛(❛∗❜❛)∗❝(❞❜)∗ ❈♦rr❡s♣♦♥❞ t♦ ✇♦r❞ s❡ts r❡❝♦❣♥✐③❡❞ ❜② ✜♥✐t❡ ❛✉t♦♠❛t❛ ✿

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✽ ✴ ✷✹

slide-58
SLIDE 58

❈❘P◗ ❡✈❛❧✉❛t✐♦♥

❘P◗ ✿ ♦♥❡ ❛r❝ ❝♦♥❥✉♥❝t✐✈❡ ❘P◗ ✭❈❘P◗✮ ✿ r❡❣✉❧❛r ❣r❛♣❤ ♣❛tt❡r♥ ✭✏❣r❛♣❤ ♦❢ ❘P◗s✑✮ ❇ ✿ ❛ ✜①❡❞ ❞❛t❛❜❛s❡ ■◆P❯❚ ✿ ❛ r❡❣✉❧❛r ❣r❛♣❤ ♣❛tt❡r♥ ◗✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ♠❛t❝❤ ✭♥✲❤♦♠♦♠♦r♣❤✐s♠✮ ♦❢ ◗ ✐♥ ❇ ❄ ❈❘P◗ ❡✈❛❧✉❛t✐♦♥ ❢♦r ❇ → ❇❛r❝❡❧ó✲❘♦♠❡r♦✲❱❛r❞✐ ✭▲■❈❙✬✶✼✮ ✿ ◆P✲❝♦♠♣❧❡t❡✴♣♦❧②♥♦♠✐❛❧ ❞✐❝❤♦t♦♠② ✉s✐♥❣ t❤❡ ❈❙P ❞✐❝❤♦t♦♠② t❤❡♦r❡♠ ✭❇✉❧❛t♦✈✬✶✼ ✰ ❩❤✉❦✬✶✼✮

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✶✾ ✴ ✷✹

slide-59
SLIDE 59

❈❘P◗ ❝♦♥t❛✐♥♠❡♥t

❍♦✇ t♦ ♠♦❞❡❧ ♥❛✈✐❣❛t✐♦♥❛❧ q✉❡r② ❝♦♥t❛✐♥♠❡♥t ✉s✐♥❣ ❤♦♠♦♠♦r♣❤✐s♠s ❄ ◗✉❡r② ❣r❛♣❤s ✿ ❉✐r❡❝t❡❞ ❣r❛♣❤s ✇✐t❤ ❧❛❜❡❧ ❊(①,②) ❢♦r ❡❛❝❤ ❛r❝ (①,②)✳ ▲❛❜❡❧s ❛r❡ s❡ts ♦❢ ✇♦r❞s ♦✈❡r Σ✳ ❊①❛♠♣❧❡ ✿ {❛,❛❛,❛❜,❜❛,❜❝❛❜} ▼❛♣♣✐♥❣ ❤ : ❱ (◗✶) → ❱ (◗✷) s✉❝❤ t❤❛t ✿ (①,②) ∈ ❆(◗✶) = ⇒ t❤❡r❡ ❡①✐sts ❛ ❞✐r❡❝t❡❞ ✇❛❧❦ ❢r♦♠ ❤(①) t♦ ❤(②) ✐♥ ◗✷ ✇❤♦s❡ ❛ss♦❝✐❛t❡❞ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ s❡ts ♦❢ ✇♦r❞s ✐s ✐♥ ❊(①,②)✳ ❉❡✜♥✐t✐♦♥ ✲ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ t✇♦ ❈❘P◗s ◗✶ ❛♥❞ ◗✷

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✵ ✴ ✷✹

slide-60
SLIDE 60

❈❘P◗ ❝♦♥t❛✐♥♠❡♥t

■◆P❯❚ ✿ t✇♦ ❈❘P◗s ◗✶ ❛♥❞ ◗✷✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ◗✶ t♦ ◗✷ ❄ ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ ❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ ❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ✐s ✐♥ ❊❳P❚■▼❊✱ ❜✉t P❙P❆❈❊✲❤❛r❞✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ▼❛❞❡❧❛✐♥❡✱ ◆♦✉r✐♥❡✱ ❘✐❝❤❛r❞✱ ✷✵✶✾✮ ❤❛r❞♥❡ss ✿ tr✐✈✐❛❧ r❡❞✉❝t✐♦♥ ❢r♦♠ ❘❡❣✉❧❛r ▲❛♥❣✉❛❣❡ ■♥❝❧✉s✐♦♥✳ ▲ ♥ ✿ ♥✲tr✉♥❝❛t✐♦♥ ♦❢ ▲ ✭✇♦r❞s ♦❢ ▲ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st ♥✮ ❆ ❇✶ ❇❦ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s r❡❝♦❣♥✐③❡❞ ❜② ❛✉t♦♠❛t❛ ✇✐t❤ ♥❆ ♥✶ ♥❦ st❛t❡s✳ ❚❤❡♥✱ ▲ ❇✶ ▲ ❇❦ ▲ ❆ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ▲ ❇✶

♥❆♥✶

▲ ❇❦

♥❆♥❦

▲ ❆ ✳ ▲❡♠♠❛

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✶ ✴ ✷✹

slide-61
SLIDE 61

❈❘P◗ ❝♦♥t❛✐♥♠❡♥t

■◆P❯❚ ✿ t✇♦ ❈❘P◗s ◗✶ ❛♥❞ ◗✷✳ ◗❯❊❙❚■❖◆ ✿ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ◗✶ t♦ ◗✷ ❄ ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ ❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ ❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ✐s ✐♥ ❊❳P❚■▼❊✱ ❜✉t P❙P❆❈❊✲❤❛r❞✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ▼❛❞❡❧❛✐♥❡✱ ◆♦✉r✐♥❡✱ ❘✐❝❤❛r❞✱ ✷✵✶✾✮ ❤❛r❞♥❡ss ✿ tr✐✈✐❛❧ r❡❞✉❝t✐♦♥ ❢r♦♠ ❘❡❣✉❧❛r ▲❛♥❣✉❛❣❡ ■♥❝❧✉s✐♦♥✳ ▲|♥ ✿ ♥✲tr✉♥❝❛t✐♦♥ ♦❢ ▲ ✭✇♦r❞s ♦❢ ▲ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st ♥✮ ❆,❇✶,...,❇❦ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s r❡❝♦❣♥✐③❡❞ ❜② ❛✉t♦♠❛t❛ ✇✐t❤ ♥❆,♥✶,...,♥❦ st❛t❡s✳ ❚❤❡♥✱ ▲(❇✶)·...·▲(❇❦) ⊆ ▲(❆) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ▲(❇✶)|♥❆♥✶ ·...·▲(❇❦)|♥❆♥❦ ⊆ ▲(❆)✳ ▲❡♠♠❛

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✶ ✴ ✷✹

slide-62
SLIDE 62

❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ ❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ✐s ✐♥ ❊❳P❚■▼❊✱ ❜✉t P❙P❆❈❊✲❤❛r❞✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ▼❛❞❡❧❛✐♥❡✱ ◆♦✉r✐♥❡✱ ❘✐❝❤❛r❞✱ ✷✵✶✾✮ ❘❡♠❛r❦ ✿ ●❡♥❡r❛❧ ❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ✐s ❊❳P❙P❆❈❊✲❝♦♠♣❧❡t❡ ✭❋❧♦r❡s❝✉ ❡t ❛❧✳ P❖❉❙✬✾✽✮

❛+ ❜ P ❛ ◗

❋✐❣ ✿ ◗ → P ❜✉t ◗ ❝♦♥t❛✐♥❡❞ ✐♥ P

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✷ ✴ ✷✹

slide-63
SLIDE 63

❈❘P◗ ❝♦♥t❛✐♥♠❡♥t

❙♣❡❝✐❛❧ ❝❛s❡ ♦❢ ✉♥❛r② ❛❧♣❤❛❜❡t ✿ Σ = {❛} ❛♥❞ ✇❛❧❦s ♦❢ t②♣❡ ✏❛✑ ♦r ✏❛+✑ ✭❛+ = {❛,❛❛,❛❛❛,❛❛❛❛,...}✮ ▼♦t✐✈❛t✐♦♥ ✿ ❳P❛t❤ ✭❳▼▲ P❛t❤ ▲❛♥❣✉❛❣❡✮✱ ❙P❆❘◗▲ ✭✮ ❳P❛t❤ ♦♣❡r❛t♦rs ✏✑ ✭❝❤✐❧❞ ♥♦❞❡✮ ❛♥❞ ✏✑ ✭❞❡s❝❡♥❞❛♥ts ♦r s❡❧❢✮ ♦❜❥❡❝ts♣r✐❝❡ ✿ r❡t✉r♥s ❛❧❧ ♣r✐❝❡s t❤❛t ❛r❡ ❜❡❧♦✇ t❤❡ ✏✐t❡♠s✑

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✸ ✴ ✷✹

slide-64
SLIDE 64

❈❘P◗ ❝♦♥t❛✐♥♠❡♥t

❙♣❡❝✐❛❧ ❝❛s❡ ♦❢ ✉♥❛r② ❛❧♣❤❛❜❡t ✿ Σ = {❛} ❛♥❞ ✇❛❧❦s ♦❢ t②♣❡ ✏❛✑ ♦r ✏❛+✑ ✭❛+ = {❛,❛❛,❛❛❛,❛❛❛❛,...}✮ ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ {❛,❛+}✲❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ❤❛s ❛ ♣♦❧②♥♦♠✐❛❧✴◆P✲ ❝♦♠♣❧❡t❡ ❞✐❝❤♦t♦♠②✳ Pr♦♣♦s✐t✐♦♥

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✸ ✴ ✷✹

slide-65
SLIDE 65

❈❘P◗ ❝♦♥t❛✐♥♠❡♥t

❙♣❡❝✐❛❧ ❝❛s❡ ♦❢ ✉♥❛r② ❛❧♣❤❛❜❡t ✿ Σ = {❛} ❛♥❞ ✇❛❧❦s ♦❢ t②♣❡ ✏❛✑ ♦r ✏❛+✑ ✭❛+ = {❛,❛❛,❛❛❛,❛❛❛❛,...}✮ ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ {❛,❛+}✲❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ❢♦r ✉♥❞✐r❡❝t❡❞ ❈❘P◗s ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ✐❢ t❤❡ ❝♦r❡ ♦❢ t❤❡ t❛r❣❡t ❤❛s ❛t ♠♦st ♦♥❡ ❡❞❣❡✱ ◆P✲❝♦♠♣❧❡t❡ ♦t❤❡r✇✐s❡✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ▼❛❞❡❧❛✐♥❡✱ ◆♦✉r✐♥❡✱ ❘✐❝❤❛r❞✱ ✷✵✶✾✮

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✸ ✴ ✷✹

slide-66
SLIDE 66

❈❘P◗ ❝♦♥t❛✐♥♠❡♥t

❙♣❡❝✐❛❧ ❝❛s❡ ♦❢ ✉♥❛r② ❛❧♣❤❛❜❡t ✿ Σ = {❛} ❛♥❞ ✇❛❧❦s ♦❢ t②♣❡ ✏❛✑ ♦r ✏❛+✑ ✭❛+ = {❛,❛❛,❛❛❛,❛❛❛❛,...}✮ ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ {❛,❛+}✲❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ✐❢ t❤❡ t❛r✲ ❣❡t ✐s ❛ ❞✐r❡❝t❡❞ ♣❛t❤✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ▼❛❞❡❧❛✐♥❡✱ ◆♦✉r✐♥❡✱ ❘✐❝❤❛r❞✱ ✷✵✶✾✮

a a a h a a a a+ a a a a a+ a+ a+ a+ a+ ❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✸ ✴ ✷✹

slide-67
SLIDE 67

❈❘P◗ ❝♦♥t❛✐♥♠❡♥t

❙♣❡❝✐❛❧ ❝❛s❡ ♦❢ ✉♥❛r② ❛❧♣❤❛❜❡t ✿ Σ = {❛} ❛♥❞ ✇❛❧❦s ♦❢ t②♣❡ ✏❛✑ ♦r ✏❛+✑ ✭❛+ = {❛,❛❛,❛❛❛,❛❛❛❛,...}✮ ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ {❛,❛+}✲❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ✐❢ t❤❡ t❛r✲ ❣❡t ✐s ❛ ❞✐r❡❝t❡❞ ♣❛t❤✳ ❚❤❡♦r❡♠ ✭❇❡❛✉❞♦✉✱ ❋✳✱ ▼❛❞❡❧❛✐♥❡✱ ◆♦✉r✐♥❡✱ ❘✐❝❤❛r❞✱ ✷✵✶✾✮ Pr♦♦❢✳ ■❢ ♦♥❧② ❛✬s ✿ ♣❛r❛❧❧❡❧ s❝❤❡❞✉❧✐♥❣ ✇✐t❤ r❡❧❛t✐✈❡ ❞❡❛❞❧✐♥❡s✳

  • ❡♥❡r❛❧❧② ✿ ♠❛❥♦r✐t② ✭♠❡❞✐❛♥✮ ♣♦❧②♠♦r♣❤✐s♠✳

t❡r♥❛r② ♠❛❥♦r✐t② ♣♦❧②♠♦r♣❤✐s♠ ♦❢ ❚ ✿ ❤♦♠♦♠♦r♣❤✐s♠ ❤ : ❚ ✸ → ❚ ✇✐t❤ ❤(①,①,②) = ❤(①,②,①) = ❤(②,①,①) = ① ■❢ ❚ ❤❛s ❛ t❡r♥❛r② ♠❛❥♦r✐t② ♣♦❧②♠♦r♣❤✐s♠✱ t❤❡♥ ❈❙P✭❚✮ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❝✉❜✐❝ t✐♠❡ ❜② ♣❛t❤✲❝♦♥s✐st❡♥❝②✳ ❚❤❡♦r❡♠ ✭❋❡❞❡r✲❱❛r❞✐✮

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✸ ✴ ✷✹

slide-68
SLIDE 68

❈♦♥❝❧✉s✐♦♥

■♥t❡r❢❛❝❡ ♦❢ r✐❝❤ r❡s❡❛r❝❤ ❛r❡❛s ✿ ❉❛t❛❜❛s❡s ✲ ●r❛♣❤ t❤❡♦r②✴❛❧❣♦r✐t❤♠s ✲ ❈❙P ✲ ▲❛♥❣✉❛❣❡✴❛✉t♦♠❛t❛ t❤❡♦r② ❙♦♠❡ s❡❧❡❝t❡❞ ♣r♦❜❧❡♠s ✿

  • ■s ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ ❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ❊❳P❚■▼❊✲❤❛r❞ ♦r ✐♥

P❙P❆❈❊ ❄

  • ■s ❍♦♠♦♠♦r♣❤✐s♠✲❜❛s❡❞ {❛,❛+}✲❈❘P◗ ❝♦♥t❛✐♥♠❡♥t ◆P✲❝♦♠♣❧❡t❡ ♦♥

r♦♦t❡❞ ❞✐r❡❝t❡❞ tr❡❡s ❄

  • ▼❛♥② s♣❡❝✐❛❧ ❝❛s❡s ❛r❡ ✐♥t❡r❡st✐♥❣ ✦

❊①❛♠♣❧❡s ✿ {❛,❛+} ❀ {❛,❛∗} ❀ {❛❦,❦ ∈ N} ❀ ❯♥❛r② r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s ✐♥ ❣❡♥❡r❛❧✳✳✳

❋❧♦r❡♥t ❋♦✉❝❛✉❞ ◆❛✈✐❣❛t✐♦♥❛❧ ❤♦♠♦♠♦r♣❤✐s♠s ❢♦r ❣r❛♣❤ ❞❛t❛❜❛s❡s ✷✹ ✴ ✷✹