SLIDE 1
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s
rs s - - PowerPoint PPT Presentation
rs s - - PowerPoint PPT Presentation
rs s ts rs s ts
SLIDE 2
SLIDE 3
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s
▼♦t✐✈❛t✐♦♥s
❢ ✲❛❧❣❡❜r❛s ❛r❡ ❛ ✈❡r② ✇❡❧❧ ❦♥♦✇ ❛♥❞ st✉❞✐❡❞ s✉❜❥❡❝t✱ ✇✐t❤ s❡✈❡r❛❧ ❛♥❛❧✐t✐❝s ❛♥❞ ❢✉♥❝t✐♦♥❛❧ r❡s✉❧ts ♦♥ t❤❡♠❀ ❢▼❱ ✲❛❧❣❡❜r❛s ❛s ❝♦♠♠♦♥ ❡①t❡♥t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♣t ♦❢ P▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ❘✐❡s③ ▼❱ ✲❛❧❣❡❜r❛s❀ ❇② ♠❡❛♥s ♦❢ ❛❞❥✉❝t✐♦♥ ❢▼❱ ✲❛❧❣❡❜r❛s ❣✐✈❡ ❛ ❞✐✛❡r❡♥t ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ ❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡❀ ❢♦r ❜♦t❤ P▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ❢▼❱ ✲❛❧❣❡❜r❛s ✇❡ ❛r❡ ❛❜❧❡ t♦ ❣❡t ❛ ✈❡rs✐♦♥ ♦❢ ❍❛✉s❞♦r✛ ▼♦♠❡♥t Pr♦❜❧❡♠✳ ■t ✐s ❛ ✈❡r② ❝❡♥tr❛❧ ❛♥❞ ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠ ✐♥ st❛t✐st✐❝ ❛♥❞ ♣r♦❜❛❜✐❧✐t②✳
SLIDE 4
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s
▼♦t✐✈❛t✐♦♥s
❢ ✲❛❧❣❡❜r❛s ❛r❡ ❛ ✈❡r② ✇❡❧❧ ❦♥♦✇ ❛♥❞ st✉❞✐❡❞ s✉❜❥❡❝t✱ ✇✐t❤ s❡✈❡r❛❧ ❛♥❛❧✐t✐❝s ❛♥❞ ❢✉♥❝t✐♦♥❛❧ r❡s✉❧ts ♦♥ t❤❡♠❀ ❢▼❱ ✲❛❧❣❡❜r❛s ❛s ❝♦♠♠♦♥ ❡①t❡♥t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♣t ♦❢ P▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ❘✐❡s③ ▼❱ ✲❛❧❣❡❜r❛s❀ ❇② ♠❡❛♥s ♦❢ ❛❞❥✉❝t✐♦♥ ❢▼❱ ✲❛❧❣❡❜r❛s ❣✐✈❡ ❛ ❞✐✛❡r❡♥t ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ ❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡❀ ❢♦r ❜♦t❤ P▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ❢▼❱ ✲❛❧❣❡❜r❛s ✇❡ ❛r❡ ❛❜❧❡ t♦ ❣❡t ❛ ✈❡rs✐♦♥ ♦❢ ❍❛✉s❞♦r✛ ▼♦♠❡♥t Pr♦❜❧❡♠✳ ■t ✐s ❛ ✈❡r② ❝❡♥tr❛❧ ❛♥❞ ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠ ✐♥ st❛t✐st✐❝ ❛♥❞ ♣r♦❜❛❜✐❧✐t②✳
SLIDE 5
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s
▼♦t✐✈❛t✐♦♥s
❢ ✲❛❧❣❡❜r❛s ❛r❡ ❛ ✈❡r② ✇❡❧❧ ❦♥♦✇ ❛♥❞ st✉❞✐❡❞ s✉❜❥❡❝t✱ ✇✐t❤ s❡✈❡r❛❧ ❛♥❛❧✐t✐❝s ❛♥❞ ❢✉♥❝t✐♦♥❛❧ r❡s✉❧ts ♦♥ t❤❡♠❀ ❢▼❱ ✲❛❧❣❡❜r❛s ❛s ❝♦♠♠♦♥ ❡①t❡♥t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♣t ♦❢ P▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ❘✐❡s③ ▼❱ ✲❛❧❣❡❜r❛s❀ ❇② ♠❡❛♥s ♦❢ ❛❞❥✉❝t✐♦♥ ❢▼❱ ✲❛❧❣❡❜r❛s ❣✐✈❡ ❛ ❞✐✛❡r❡♥t ♣♦✐♥t ♦❢ ✈✐❡✇ ♦♥ ❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡❀ ❢♦r ❜♦t❤ P▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ❢▼❱ ✲❛❧❣❡❜r❛s ✇❡ ❛r❡ ❛❜❧❡ t♦ ❣❡t ❛ ✈❡rs✐♦♥ ♦❢ ❍❛✉s❞♦r✛ ▼♦♠❡♥t Pr♦❜❧❡♠✳ ■t ✐s ❛ ✈❡r② ❝❡♥tr❛❧ ❛♥❞ ✐♠♣♦rt❛♥t ♣r♦❜❧❡♠ ✐♥ st❛t✐st✐❝ ❛♥❞ ♣r♦❜❛❜✐❧✐t②✳
SLIDE 6
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❙❡❝t✐♦♥ ✶ Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
SLIDE 7
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▼❱✲❛❧❣❡❜r❛s✳
■♥ ✶✾✺✽✱ ❈✳❈✳ ❈❤❛♥❣ ✐♥tr♦❞✉❝❡❞ ▼❱ ✲❛❧❣❡❜r❛s ❛s ❛❧❣❡❜r❛✐❝ ❝♦✉♥t❡r♣❛rt ♦❢ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✱ ❛♥❞ ♣r♦✈❡❞ ❈♦♠♣❧❡t❡♥❡ss ❚❤❡♦r❡♠ ✐♥ t❤❡ ❛❧❣❡❜r❛✐❝ ✇❛②✳ ❈❤❛♥❣✱ ❈✳❈✳✱ ❆❧❣❡❜r❛✐❝ ❛♥❛❧②s✐s ♦❢ ♠❛♥② ✈❛❧✉❡❞ ❧♦❣✐❝s✱ ❚r❛♥s❛❝t✐♦♥s ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✈♦❧ ✽✽ ✭✶✾✺✽✮✱ ♣♣✳ ✹✻✼✲✹✾✵✳ ❈❤❛♥❣✱ ❈✳❈✳✱ ❆ ♥❡✇ ♣r♦♦❢ ♦❢ t❤❡ ❝♦♠♣❧❡t❡♥❡ss ♦❢ t❤❡ ❾✉❦❛s✐❡✇✐❝③ ❛①✐♦♠s✱ ❚r❛♥s❛❝t✐♦♥s ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ✈♦❧ ✾✸ ✭✶✾✺✾✮✱ ♣♣✳✼✹✲✽✵✳
SLIDE 8
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▼❱✲❛❧❣❡❜r❛s
❉❡✜♥✐t✐♦♥ ❆♥ ▼❱✲❛❧❣❡❜r❛ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ st✉❝t✉r❡ ❆ ✇✐t❤ t✇♦ ♦♣❡r❛t✐♦♥ ⊕ ❛♥❞
∗
❛♥❞ ❛ ❞✐st✐♥❣✉✐s❤❡❞ ❡❧❡♠❡♥t ✵✱ t❤❛t s❛t✐s✜❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛①✐♦♠s✿ ❢♦r ❛♥② ①, ②, ③ ∈ ❆✱ ① ⊕ ② = ② ⊕ ①❀ ① ⊕ (② ⊕ ③) = (① ⊕ ②) ⊕ ③❀ ① ⊕ ✵ = ①❀ (①∗)∗ = ①❀ ① ⊕ ✵∗ = ✵∗❀ (①∗ ⊕ ②)∗ ⊕ ② = (② ∗ ⊕ ①)∗ ⊕ ①✳
SLIDE 9
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▼❱✲❛❧❣❡❜r❛s
❆ ▼❱✲❛❧❣❡❜r❛ ①, ② ∈ ❆ ① ⊙ ② = (①∗ ⊕ ② ∗)∗, ① ⊖ ② = ① ⊙ ② ∗. ❖r❞❡r ♦♥ ❆✿ ① ≤ ② ✐✛ ①∗ ⊕ ② = ✶✳ ❆ ✐s ❛ ❧❛tt✐❝❡✱ ✇✐t❤ ① ∨ ② = (① ⊙ ② ∗) ⊕ ②, ① ∧ ② = (①∗ ∨ ② ∗)∗ = ① ⊙ (①∗ ⊕ ②).
SLIDE 10
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▼❱✲❛❧❣❡❜r❛s
❆ ▼❱✲❛❧❣❡❜r❛ ①, ② ∈ ❆ ① ⊙ ② = (①∗ ⊕ ② ∗)∗, ① ⊖ ② = ① ⊙ ② ∗. ❖r❞❡r ♦♥ ❆✿ ① ≤ ② ✐✛ ①∗ ⊕ ② = ✶✳ ❆ ✐s ❛ ❧❛tt✐❝❡✱ ✇✐t❤ ① ∨ ② = (① ⊙ ② ∗) ⊕ ②, ① ∧ ② = (①∗ ∨ ② ∗)∗ = ① ⊙ (①∗ ⊕ ②).
SLIDE 11
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▼❱✲❛❧❣❡❜r❛s
❆ ▼❱✲❛❧❣❡❜r❛ ①, ② ∈ ❆ ① ⊙ ② = (①∗ ⊕ ② ∗)∗, ① ⊖ ② = ① ⊙ ② ∗. ❖r❞❡r ♦♥ ❆✿ ① ≤ ② ✐✛ ①∗ ⊕ ② = ✶✳ ❆ ✐s ❛ ❧❛tt✐❝❡✱ ✇✐t❤ ① ∨ ② = (① ⊙ ② ∗) ⊕ ②, ① ∧ ② = (①∗ ∨ ② ∗)∗ = ① ⊙ (①∗ ⊕ ②).
SLIDE 12
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝
▲❡t L ❜❡ t❤❡ ❾✉❦❛s✐❡✇✐❝③ ♣r♦♣♦s✐t✐♦♥❛❧ ❝❛❧❝✉❧✉s✳ ❉❡✜♥✐t✐♦♥ ▲❡t ϕ, ψ ∈ ❋♦r♠▲✱ ✇❡ s❛② ϕ ≡ ψ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ⊢ ϕ ↔ ψ✳ ❲❡ ❞❡✜♥❡ L = (❋♦r♠▲/ ≡, ⊕, ¬, ✵), ✇❤❡r❡ [ϕ] ⊕ [ψ] = [¬ϕ → ψ] [ϕ]∗ = [¬ϕ] ✵ = [ϕ] ✇❤❡r❡ ⊢ ¬ϕ✳ L ✐s ❛ ▼❱✲❛❧❣❡❜r❛✳
SLIDE 13
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝
▲❡t L ❜❡ t❤❡ ❾✉❦❛s✐❡✇✐❝③ ♣r♦♣♦s✐t✐♦♥❛❧ ❝❛❧❝✉❧✉s✳ ❉❡✜♥✐t✐♦♥ ▲❡t ϕ, ψ ∈ ❋♦r♠▲✱ ✇❡ s❛② ϕ ≡ ψ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ⊢ ϕ ↔ ψ✳ ❲❡ ❞❡✜♥❡ L = (❋♦r♠▲/ ≡, ⊕, ¬, ✵), ✇❤❡r❡ [ϕ] ⊕ [ψ] = [¬ϕ → ψ] [ϕ]∗ = [¬ϕ] ✵ = [ϕ] ✇❤❡r❡ ⊢ ¬ϕ✳ L ✐s ❛ ▼❱✲❛❧❣❡❜r❛✳
SLIDE 14
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝
▲❡t L ❜❡ t❤❡ ❾✉❦❛s✐❡✇✐❝③ ♣r♦♣♦s✐t✐♦♥❛❧ ❝❛❧❝✉❧✉s✳ ❉❡✜♥✐t✐♦♥ ▲❡t ϕ, ψ ∈ ❋♦r♠▲✱ ✇❡ s❛② ϕ ≡ ψ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ⊢ ϕ ↔ ψ✳ ❲❡ ❞❡✜♥❡ L = (❋♦r♠▲/ ≡, ⊕, ¬, ✵), ✇❤❡r❡ [ϕ] ⊕ [ψ] = [¬ϕ → ψ] [ϕ]∗ = [¬ϕ] ✵ = [ϕ] ✇❤❡r❡ ⊢ ¬ϕ✳ L ✐s ❛ ▼❱✲❛❧❣❡❜r❛✳
SLIDE 15
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝
▲❡t L ❜❡ t❤❡ ❾✉❦❛s✐❡✇✐❝③ ♣r♦♣♦s✐t✐♦♥❛❧ ❝❛❧❝✉❧✉s✳ ❉❡✜♥✐t✐♦♥ ▲❡t ϕ, ψ ∈ ❋♦r♠▲✱ ✇❡ s❛② ϕ ≡ ψ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ⊢ ϕ ↔ ψ✳ ❲❡ ❞❡✜♥❡ L = (❋♦r♠▲/ ≡, ⊕, ¬, ✵), ✇❤❡r❡ [ϕ] ⊕ [ψ] = [¬ϕ → ψ] [ϕ]∗ = [¬ϕ] ✵ = [ϕ] ✇❤❡r❡ ⊢ ¬ϕ✳ L ✐s ❛ ▼❱✲❛❧❣❡❜r❛✳
SLIDE 16
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▲❛tt✐❝❡✲♦r❞❡r❡❞ str✉❝t✉r❡s
(●, +, ✵) ❣r♦✉♣✱ (●, +, ✵, ≤) (●, ≤) ❧❛tt✐❝❡✱ ℓ✲❣r♦✉♣ ① ≤ ② ✐♠♣❧✐❡s ① + ③ ≤ ② + ③ (❱ , +, ✵, ≤) ❛❜❡❧✐❛♥ ℓ✲❣r♦✉♣ (❱ , +, {r|r ∈ R}, ✵, ≤) (❱ , +, {r|r ∈ R}, ✵) r❡❛❧ ✈❡❝t♦r s♣❛❝❡ ❘✐❡s③ s♣❛❝❡ ① ≤ ② ✐♠♣❧✐❡s r · ① ≤ r · ② ❢♦r r ≥ ✵ (❘, +, ✵, ≤) ❛❜❡❧✐❛♥ ℓ✲❣r♦✉♣✱ (❘, +, ·, ✵, ≤) (❘, +, ·, ✵) r✐♥❣ ℓ✲r✐♥❣ ① ≤ ② ✐♠♣❧✐❡s ① · ③ ≤ ② · ③ ❛♥❞ ③ · ① ≤ ③ · ② ❢♦r ③ ≥ ✵ (❆, +, ·, ✵, ≤) ℓ✲r✐♥❣ (❆, +, ·, {r|r ∈ R}, ✵, ≤) (❆, +, {r|r ∈ R}, ✵, ≤) ❘✐❡s③ s♣❛❝❡ ℓ✲❛❧❣❡❜r❛ r(① · ②) = (r①) · ② = ① · (r②)
SLIDE 17
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▲❛tt✐❝❡✲♦r❞❡r❡❞ str✉❝t✉r❡s
(●, +, ✵) ❣r♦✉♣✱ (●, +, ✵, ≤) (●, ≤) ❧❛tt✐❝❡✱ ℓ✲❣r♦✉♣ ① ≤ ② ✐♠♣❧✐❡s ① + ③ ≤ ② + ③ (❱ , +, ✵, ≤) ❛❜❡❧✐❛♥ ℓ✲❣r♦✉♣ (❱ , +, {r|r ∈ R}, ✵, ≤) (❱ , +, {r|r ∈ R}, ✵) r❡❛❧ ✈❡❝t♦r s♣❛❝❡ ❘✐❡s③ s♣❛❝❡ ① ≤ ② ✐♠♣❧✐❡s r · ① ≤ r · ② ❢♦r r ≥ ✵ (❘, +, ✵, ≤) ❛❜❡❧✐❛♥ ℓ✲❣r♦✉♣✱ (❘, +, ·, ✵, ≤) (❘, +, ·, ✵) r✐♥❣ ℓ✲r✐♥❣ ① ≤ ② ✐♠♣❧✐❡s ① · ③ ≤ ② · ③ ❛♥❞ ③ · ① ≤ ③ · ② ❢♦r ③ ≥ ✵ (❆, +, ·, ✵, ≤) ℓ✲r✐♥❣ (❆, +, ·, {r|r ∈ R}, ✵, ≤) (❆, +, {r|r ∈ R}, ✵, ≤) ❘✐❡s③ s♣❛❝❡ ℓ✲❛❧❣❡❜r❛ r(① · ②) = (r①) · ② = ① · (r②)
SLIDE 18
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▲❛tt✐❝❡✲♦r❞❡r❡❞ str✉❝t✉r❡s
❢ ✲r✐♥❣ ✭❢ ✲❛❧❣❡❜r❛✮ ❂ s✉❜❞✐r❡❝t ♣r♦❞✉❝t ♦❢ ❝❤❛✐♥s ❆ ❢✲r✐♥❣ ✭❢✲❛❧❣❡❜r❛✮✳ ❢♦r ❛♥② ①, ② ∈ ❆✱ ③ ∈ ❆+✱ ✐❢ ① ∧ ② = ✵ t❤❡♥ ③① ∧ ② = ①③ ∧ ② = ✵✳ ❉❡✜♥✐t✐♦♥ ❆ str♦♥❣ ✉♥✐t ❢♦r ❛♥ ℓ✲❣r♦✉♣ ● ✐s ❛♥ ❡❧❡♠❡♥t ✉ ≥ ✵ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ∈ ● t❤❡r❡ ✐s ❛♥ ✐♥t❡❣❡r ♥ ≥ ✵ ✇✐t❤ |①| ≤ ♥✉✳
SLIDE 19
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▲❛tt✐❝❡✲♦r❞❡r❡❞ str✉❝t✉r❡s
❢ ✲r✐♥❣ ✭❢ ✲❛❧❣❡❜r❛✮ ❂ s✉❜❞✐r❡❝t ♣r♦❞✉❝t ♦❢ ❝❤❛✐♥s ❆ ❢✲r✐♥❣ ✭❢✲❛❧❣❡❜r❛✮✳ ❢♦r ❛♥② ①, ② ∈ ❆✱ ③ ∈ ❆+✱ ✐❢ ① ∧ ② = ✵ t❤❡♥ ③① ∧ ② = ①③ ∧ ② = ✵✳ ❉❡✜♥✐t✐♦♥ ❆ str♦♥❣ ✉♥✐t ❢♦r ❛♥ ℓ✲❣r♦✉♣ ● ✐s ❛♥ ❡❧❡♠❡♥t ✉ ≥ ✵ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ∈ ● t❤❡r❡ ✐s ❛♥ ✐♥t❡❣❡r ♥ ≥ ✵ ✇✐t❤ |①| ≤ ♥✉✳
SLIDE 20
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▲❛tt✐❝❡✲♦r❞❡r❡❞ str✉❝t✉r❡s
❢ ✲r✐♥❣ ✭❢ ✲❛❧❣❡❜r❛✮ ❂ s✉❜❞✐r❡❝t ♣r♦❞✉❝t ♦❢ ❝❤❛✐♥s ❆ ❢✲r✐♥❣ ✭❢✲❛❧❣❡❜r❛✮✳ ❢♦r ❛♥② ①, ② ∈ ❆✱ ③ ∈ ❆+✱ ✐❢ ① ∧ ② = ✵ t❤❡♥ ③① ∧ ② = ①③ ∧ ② = ✵✳ ❉❡✜♥✐t✐♦♥ ❆ str♦♥❣ ✉♥✐t ❢♦r ❛♥ ℓ✲❣r♦✉♣ ● ✐s ❛♥ ❡❧❡♠❡♥t ✉ ≥ ✵ s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ ① ∈ ● t❤❡r❡ ✐s ❛♥ ✐♥t❡❣❡r ♥ ≥ ✵ ✇✐t❤ |①| ≤ ♥✉✳
SLIDE 21
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▼✉♥❞✐❝✐✬s ❝❛t❡❣♦r✐❛❧ ❡q✉✐✈❛❧❡♥❝❡
❚❤❡♦r❡♠ ❚❤❡ ❝❛t❡❣♦r② ♦❢ ℓ✲❣r♦✉♣s ✇✐t❤ str♦♥❣ ✉♥✐t ❛♥❞ t❤❡ ❝❛t❡❣♦r② ♦❢ ▼❱✲❛❧❣❡❜r❛s ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ▼✉♥❞✐❝✐ ❉✳✱ ■♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❆❈❋✯✲❛❧❣❡❜r❛s ✐♥ ❾✉❦❛s✐❡✇✐❝③ s❡♥t❡♥t✐❛❧ ❝❛❧❝✉❧✉s✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳ ✻✺ ✭✶✾✽✻✮ ✶✺✲✻✸✳
SLIDE 22
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
▼✉♥❞✐❝✐✬s ❝❛t❡❣♦r✐❛❧ ❡q✉✐✈❛❧❡♥❝❡
❚❤❡♦r❡♠ ❚❤❡ ❝❛t❡❣♦r② ♦❢ ℓ✲❣r♦✉♣s ✇✐t❤ str♦♥❣ ✉♥✐t ❛♥❞ t❤❡ ❝❛t❡❣♦r② ♦❢ ▼❱✲❛❧❣❡❜r❛s ❛r❡ ❡q✉✐✈❛❧❡♥t✳ ▼✉♥❞✐❝✐ ❉✳✱ ■♥t❡r♣r❡t❛t✐♦♥ ♦❢ ❆❈❋✯✲❛❧❣❡❜r❛s ✐♥ ❾✉❦❛s✐❡✇✐❝③ s❡♥t❡♥t✐❛❧ ❝❛❧❝✉❧✉s✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳ ✻✺ ✭✶✾✽✻✮ ✶✺✲✻✸✳
SLIDE 23
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
Pr♦❞✉❝t ▼❱✲❛❧❣❡❜r❛s
❉❡✜♥✐t✐♦♥ ▲❡t ❆ ❜❡ ❛♥ ▼❱✲❛❧❣❡❜r❛✱ ❢♦r ❛♥② ①, ② ∈ ❆ ① + ② ✐s ❞❡✜♥❡❞ ✐✛ ① ≤ ② ∗, ❛♥❞ ① + ② = ① ⊕ ②. ❆ ❛❞♠✐ts ♣r♦❞✉❝t✱ ✐❢ t❤❡r❡ ✐s ❛ ❜✐♥❛r② ♦♣❡r❛t✐♦♥ · s✉❝❤ t❤❛t ✭✐✮ ✐❢ ① + ② ✐s ❞❡✜♥❡❞ ✐♥ ❆✱ ❛❧s♦ ③ · ① + ③ · ② ❛♥❞ ① · ③ + ② · ③ ❛r❡ ❞❡✜♥❡❞✱ ❛♥❞ ③ · (① + ②) = ③ · ① + ③ · ②, (① + ②) · ③ = ① · ③ + ② · ③. ✭✐✐✮ (① · ②) · ③ = ① · (② · ③)✳ ❢♦r ❛♥② ①, ②, ③ ∈ ❆✳ ❆ P▼❱✲❛❧❣❡❜r❛ t❤❛t ❤❛s ✉♥✐t ❢♦r ·✱ ✐t ✐s ❝❛❧❧❡❞ ✉♥✐t❛❧✳
SLIDE 24
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
Pr♦❞✉❝t ▼❱✲❛❧❣❡❜r❛s
❉❡✜♥✐t✐♦♥ ▲❡t ❆ ❜❡ ❛♥ ▼❱✲❛❧❣❡❜r❛✱ ❢♦r ❛♥② ①, ② ∈ ❆ ① + ② ✐s ❞❡✜♥❡❞ ✐✛ ① ≤ ② ∗, ❛♥❞ ① + ② = ① ⊕ ②. ❆ ❛❞♠✐ts ♣r♦❞✉❝t✱ ✐❢ t❤❡r❡ ✐s ❛ ❜✐♥❛r② ♦♣❡r❛t✐♦♥ · s✉❝❤ t❤❛t ✭✐✮ ✐❢ ① + ② ✐s ❞❡✜♥❡❞ ✐♥ ❆✱ ❛❧s♦ ③ · ① + ③ · ② ❛♥❞ ① · ③ + ② · ③ ❛r❡ ❞❡✜♥❡❞✱ ❛♥❞ ③ · (① + ②) = ③ · ① + ③ · ②, (① + ②) · ③ = ① · ③ + ② · ③. ✭✐✐✮ (① · ②) · ③ = ① · (② · ③)✳ ❢♦r ❛♥② ①, ②, ③ ∈ ❆✳ ❆ P▼❱✲❛❧❣❡❜r❛ t❤❛t ❤❛s ✉♥✐t ❢♦r ·✱ ✐t ✐s ❝❛❧❧❡❞ ✉♥✐t❛❧✳
SLIDE 25
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❉❡✜♥✐t✐♦♥ ❆ P▼❱❢ ✲❛❧❣❡❜r❛ ✐s ❛ P▼❱ ✲❛❧❣❡❜r❛ s✉❝❤ t❤❛t ✐❢ ① ∧ ② = ✵✱ t❤❡♥ ① · ③ ∧ ② = ③ · ① ∧ ② = ✵✱ ❢♦r ❛♥② ①, ②, ③ ✐♥ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡♦r❡♠ P▼❱✲❛❧❣❡❜r❛s ❛r❡ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥t t♦ ❧❛tt✐❝❡✲♦r❞❡r❡❞ r✐♥❣s ✇✐t❤ str♦♥❣ ✉♥✐t✳ ❉✐ ◆♦❧❛ ❆✳✱ ❉✈✉r❡❝❡♥s❦✐❥ ❆✳✱ Pr♦❞✉❝t ▼❱✲❛❧❣❡❜r❛s✱ ▼✉❧t✐♣❧❡✲❱❛❧✉❡❞ ▲♦❣✐❝s ✻ ✭✷✵✵✶✮✱ ✶✾✸✲✷✶✺✳ ▼♦♥t❛❣♥❛ ❋✳✱ ❆♥ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤ t♦ Pr♦♣♦s✐t✐♦♥❛❧ ❋✉③③② ▲♦❣✐❝✱ ❏♦✉r♥❛❧ ♦❢ ▲♦❣✐❝✱ ▲❛♥❣✉❛❣❡ ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ✾ ✭✷✵✵✵✮ ♣♣ ✾✶✲✶✷✹✳
SLIDE 26
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❉❡✜♥✐t✐♦♥ ❆ P▼❱❢ ✲❛❧❣❡❜r❛ ✐s ❛ P▼❱ ✲❛❧❣❡❜r❛ s✉❝❤ t❤❛t ✐❢ ① ∧ ② = ✵✱ t❤❡♥ ① · ③ ∧ ② = ③ · ① ∧ ② = ✵✱ ❢♦r ❛♥② ①, ②, ③ ✐♥ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡♦r❡♠ P▼❱✲❛❧❣❡❜r❛s ❛r❡ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥t t♦ ❧❛tt✐❝❡✲♦r❞❡r❡❞ r✐♥❣s ✇✐t❤ str♦♥❣ ✉♥✐t✳ ❉✐ ◆♦❧❛ ❆✳✱ ❉✈✉r❡❝❡♥s❦✐❥ ❆✳✱ Pr♦❞✉❝t ▼❱✲❛❧❣❡❜r❛s✱ ▼✉❧t✐♣❧❡✲❱❛❧✉❡❞ ▲♦❣✐❝s ✻ ✭✷✵✵✶✮✱ ✶✾✸✲✷✶✺✳ ▼♦♥t❛❣♥❛ ❋✳✱ ❆♥ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤ t♦ Pr♦♣♦s✐t✐♦♥❛❧ ❋✉③③② ▲♦❣✐❝✱ ❏♦✉r♥❛❧ ♦❢ ▲♦❣✐❝✱ ▲❛♥❣✉❛❣❡ ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ✾ ✭✷✵✵✵✮ ♣♣ ✾✶✲✶✷✹✳
SLIDE 27
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❉❡✜♥✐t✐♦♥ ❆ P▼❱❢ ✲❛❧❣❡❜r❛ ✐s ❛ P▼❱ ✲❛❧❣❡❜r❛ s✉❝❤ t❤❛t ✐❢ ① ∧ ② = ✵✱ t❤❡♥ ① · ③ ∧ ② = ③ · ① ∧ ② = ✵✱ ❢♦r ❛♥② ①, ②, ③ ✐♥ t❤❡ ❛❧❣❡❜r❛✳ ❚❤❡♦r❡♠ P▼❱✲❛❧❣❡❜r❛s ❛r❡ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥t t♦ ❧❛tt✐❝❡✲♦r❞❡r❡❞ r✐♥❣s ✇✐t❤ str♦♥❣ ✉♥✐t✳ ❉✐ ◆♦❧❛ ❆✳✱ ❉✈✉r❡❝❡♥s❦✐❥ ❆✳✱ Pr♦❞✉❝t ▼❱✲❛❧❣❡❜r❛s✱ ▼✉❧t✐♣❧❡✲❱❛❧✉❡❞ ▲♦❣✐❝s ✻ ✭✷✵✵✶✮✱ ✶✾✸✲✷✶✺✳ ▼♦♥t❛❣♥❛ ❋✳✱ ❆♥ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤ t♦ Pr♦♣♦s✐t✐♦♥❛❧ ❋✉③③② ▲♦❣✐❝✱ ❏♦✉r♥❛❧ ♦❢ ▲♦❣✐❝✱ ▲❛♥❣✉❛❣❡ ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ✾ ✭✷✵✵✵✮ ♣♣ ✾✶✲✶✷✹✳
SLIDE 28
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❘✐❡s③ ▼❱✲❛❧❣❡❜r❛s
❉❡✜♥✐t✐♦♥ (❘, ⋆, ⊕, ∗, ✵) s✉❝❤ t❤❛t (❘, ⊕, ∗, ✵) ✐s ❛ ▼❱ ✲❛❧❣❡❜r❛ ❛♥❞ ⋆ : [✵, ✶] × ❘ → ❘ s❛t✐s✜❡s r ⋆ (① ⊙ ② ∗) = (r ⋆ ①) ⊙ (r ⋆ ②)∗✱ (r ⊙ q∗) ⋆ ① = (r ⋆ ①) ⊙ (q ⋆ ①)∗✱ r ⋆ (q ⋆ ①) = (rq) ⋆ ①✱ ✶ ⋆ ① = ①✳ ❢♦r ❛♥② r, q ∈ [✵, ✶] ❛♥❞ ❛♥② ①, ② ∈ ❘✿
SLIDE 29
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❘✐❡s③ ▼❱✲❛❧❣❡❜r❛s✱ ❡q✉✐✈❛❧❡♥t ❞❡✜♥✐t✐♦♥
❉❡✜♥✐t✐♦♥ [✵, ✶] st❛♥❞❛r❞ P▼❱ ✲❛❧❣❡❜r❛✱ (❘, ⋆, ⊕, ∗, ✵) s✉❝❤ t❤❛t
✭✐✮ ① + ② ❞❡✜♥❡❞ ✐♥ ❘ t❤❡♥ r ⋆ ① + r ⋆ ② ❞❡✜♥❡❞ ❛♥❞ r ⋆ (① + ②) = r ⋆ ① + r ⋆ ②, ✭✐✐✮ r + q ❞❡✜♥❡❞ ✐♥ [✵, ✶] t❤❡♥ r ⋆ ① + q ⋆ ① ✐s ❞❡✜♥❡❞ ❛♥❞ (r + q) ⋆ ① = r ⋆ ① + q ⋆ ①, ✭✐✐✐✮ (r · q) ⋆ ① = r ⋆ (q ⋆ ①). ✭✐✈✮ ✶ ⋆ ① = ①✳ ①, ② ∈ ❘✱ r, q ∈ [✵, ✶]✳
SLIDE 30
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❘✐❡s③ ▼❱✲❛❧❣❡❜r❛s✱ ❡q✉✐✈❛❧❡♥t ❞❡✜♥✐t✐♦♥
❉❡✜♥✐t✐♦♥ [✵, ✶] st❛♥❞❛r❞ P▼❱ ✲❛❧❣❡❜r❛✱ (❘, ⋆, ⊕, ∗, ✵) s✉❝❤ t❤❛t
✭✐✮ ① + ② ❞❡✜♥❡❞ ✐♥ ❘ t❤❡♥ r ⋆ ① + r ⋆ ② ❞❡✜♥❡❞ ❛♥❞ r ⋆ (① + ②) = r ⋆ ① + r ⋆ ②, ✭✐✐✮ r + q ❞❡✜♥❡❞ ✐♥ [✵, ✶] t❤❡♥ r ⋆ ① + q ⋆ ① ✐s ❞❡✜♥❡❞ ❛♥❞ (r + q) ⋆ ① = r ⋆ ① + q ⋆ ①, ✭✐✐✐✮ (r · q) ⋆ ① = r ⋆ (q ⋆ ①). ✭✐✈✮ ✶ ⋆ ① = ①✳ ①, ② ∈ ❘✱ r, q ∈ [✵, ✶]✳
SLIDE 31
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❘✐❡s③ ▼❱✲❛❧❣❡❜r❛s✱ ❡q✉✐✈❛❧❡♥t ❞❡✜♥✐t✐♦♥
❉❡✜♥✐t✐♦♥ [✵, ✶] st❛♥❞❛r❞ P▼❱ ✲❛❧❣❡❜r❛✱ (❘, ⋆, ⊕, ∗, ✵) s✉❝❤ t❤❛t
✭✐✮ ① + ② ❞❡✜♥❡❞ ✐♥ ❘ t❤❡♥ r ⋆ ① + r ⋆ ② ❞❡✜♥❡❞ ❛♥❞ r ⋆ (① + ②) = r ⋆ ① + r ⋆ ②, ✭✐✐✮ r + q ❞❡✜♥❡❞ ✐♥ [✵, ✶] t❤❡♥ r ⋆ ① + q ⋆ ① ✐s ❞❡✜♥❡❞ ❛♥❞ (r + q) ⋆ ① = r ⋆ ① + q ⋆ ①, ✭✐✐✐✮ (r · q) ⋆ ① = r ⋆ (q ⋆ ①). ✭✐✈✮ ✶ ⋆ ① = ①✳ ①, ② ∈ ❘✱ r, q ∈ [✵, ✶]✳
SLIDE 32
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❘✐❡s③ ▼❱✲❛❧❣❡❜r❛s✱ ❡q✉✐✈❛❧❡♥t ❞❡✜♥✐t✐♦♥
❉❡✜♥✐t✐♦♥ [✵, ✶] st❛♥❞❛r❞ P▼❱ ✲❛❧❣❡❜r❛✱ (❘, ⋆, ⊕, ∗, ✵) s✉❝❤ t❤❛t
✭✐✮ ① + ② ❞❡✜♥❡❞ ✐♥ ❘ t❤❡♥ r ⋆ ① + r ⋆ ② ❞❡✜♥❡❞ ❛♥❞ r ⋆ (① + ②) = r ⋆ ① + r ⋆ ②, ✭✐✐✮ r + q ❞❡✜♥❡❞ ✐♥ [✵, ✶] t❤❡♥ r ⋆ ① + q ⋆ ① ✐s ❞❡✜♥❡❞ ❛♥❞ (r + q) ⋆ ① = r ⋆ ① + q ⋆ ①, ✭✐✐✐✮ (r · q) ⋆ ① = r ⋆ (q ⋆ ①). ✭✐✈✮ ✶ ⋆ ① = ①✳ ①, ② ∈ ❘✱ r, q ∈ [✵, ✶]✳
SLIDE 33
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❚❤❡♦r❡♠ ❘✐❡s③ ▼❱✲❛❧❣❡❜r❛s ✇✐t❤ ❧✐♥❡❛r ▼❱✲❛❧❣❡❜r❛ ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥t t♦ ❘✐❡s③ ❙♣❛❝❡s ✇✐t❤ str♦♥❣ ✉♥✐t ❛♥❞ ❧✐♥❡❛r ❡❧❧✲❣r♦✉♣s ♠❛♣s✳ ❉✐ ◆♦❧❛ ❆✳✱ ▲❡✉st❡❛♥ ■✳✱ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❛♥❞ ❘✐❡s③ ❙♣❛❝❡s✱ ❙♦❢t ❈♦♠♣✳ ✱ ❛❝❝❡♣t❡❞
SLIDE 34
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s Pr❡❧✐♠✐♥❛r② ♥♦t✐♦♥s
❚❤❡♦r❡♠ ❘✐❡s③ ▼❱✲❛❧❣❡❜r❛s ✇✐t❤ ❧✐♥❡❛r ▼❱✲❛❧❣❡❜r❛ ❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥t t♦ ❘✐❡s③ ❙♣❛❝❡s ✇✐t❤ str♦♥❣ ✉♥✐t ❛♥❞ ❧✐♥❡❛r ❡❧❧✲❣r♦✉♣s ♠❛♣s✳ ❉✐ ◆♦❧❛ ❆✳✱ ▲❡✉st❡❛♥ ■✳✱ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❛♥❞ ❘✐❡s③ ❙♣❛❝❡s✱ ❙♦❢t ❈♦♠♣✳ ✱ ❛❝❝❡♣t❡❞
SLIDE 35
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙❡❝t✐♦♥ ✷ ❢▼❱✲❛❧❣❡❜r❛s
SLIDE 36
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❢▼❱✲❛❧❣❡❜r❛s✿ ❜❛s✐❝ ❞❡✜♥✐t✐♦♥s
❉❡✜♥✐t✐♦♥ ❆ P▼❱ ✲❛❧❣❡❜r❛ ❛♥❞ ❘✐❡s③ ▼❱ ✲❛❧❣❡❜r❛s✳ ❆ ✐s ❛♥ ❢▼❱ ✲❛❧❣❡❜r❛ ✐❢ ✭❢✶✮ ✐❢ ① ∧ ② = ✵ t❤❡♥ ① ∧ (③ · ②) = ① ∧ (② · ③) = ✵❀ ✭❢✷✮ α(① · ②) = (α①) · ② = ① · (α②)✳ α ∈ [✵, ✶] ❛♥❞ ❛♥② ①, ②, ③ ∈ ❆ ❚❤❡ P▼❱✲❛❧❣❡❜r❛ r❡❞✉❝t ♦❢ ❛♥ ❢▼❱✲❛❧❣❡❜r❛ ✐s ❛ P▼❱❢✲❛❧❣❡❜r❛✳
SLIDE 37
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❢▼❱✲❛❧❣❡❜r❛s✿ ❜❛s✐❝ ❞❡✜♥✐t✐♦♥s
❉❡✜♥✐t✐♦♥ ❆ P▼❱ ✲❛❧❣❡❜r❛ ❛♥❞ ❘✐❡s③ ▼❱ ✲❛❧❣❡❜r❛s✳ ❆ ✐s ❛♥ ❢▼❱ ✲❛❧❣❡❜r❛ ✐❢ ✭❢✶✮ ✐❢ ① ∧ ② = ✵ t❤❡♥ ① ∧ (③ · ②) = ① ∧ (② · ③) = ✵❀ ✭❢✷✮ α(① · ②) = (α①) · ② = ① · (α②)✳ α ∈ [✵, ✶] ❛♥❞ ❛♥② ①, ②, ③ ∈ ❆ ❚❤❡ P▼❱✲❛❧❣❡❜r❛ r❡❞✉❝t ♦❢ ❛♥ ❢▼❱✲❛❧❣❡❜r❛ ✐s ❛ P▼❱❢✲❛❧❣❡❜r❛✳
SLIDE 38
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❊q✉❛t✐♦♥❛❧ ❝❤❛r❛❝t❡r✐③❛t✐♦♥
❚❤❡♦r❡♠ ❆ ✐s ❛ ❢▼❱✲❛❧❣❡❜r❛ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t s❛t✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ (▼✶) α(① ⊙ ② ∗) = (α①) ⊙ (α②)∗ (▼✷) (α ⊙ β∗)① = (α①) ⊙ (β)①)∗ (▼✸) α(β①) = (α · β)① (▼✹) ✶① = ① (P✶❛) ③ · (① ⊙ (① ∧ ②)∗) = (③ · ①) ⊙ (③ · (① ∧ ②))∗ (P✶❜) (① ⊙ (① ∧ ②)∗) · ③ = (① · ③) ⊙ ((① ∧ ②) · ③)∗ (P✷) ① · (② · ③) = (① · ②) · ③ (❋✶❛) (③ · (① ⊙ ② ∗)) ∧ (② ⊙ ①∗) = ✵ (❋✶❜) ((① ⊙ ② ∗) · ③) ∧ (② ⊙ ①∗) = ✵ (❋✷) α(① · ②) = (α①) · ② = ① · (α②).
SLIDE 39
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❈❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡
❢▼❱❛❧❣✱ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ ❢▼❱✲❛❧❣❡❜r❛s ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠s ❛r❡ ▼❱ ✲❛❧❣❡❜r❛s ❤♦♠♦♠♦r♣❤✐s♠s t❤❛t ♣r❡s❡r✈❡ ❜♦t❤ ✐♥t❡r♥❛❧ ❛♥❞ ❡①t❡r♥❛❧ ♣r♦❞✉❝t✳ ❢❛❧❣✱ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ ❢✲❛❧❣❡❜r❛s ✇✐t❤ str♦♥❣ ✉♥✐t ✉ s✉❝❤ t❤❛t ✉ · ✉ ≤ ✉ ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠s ❛r❡ ❧✐♥❡❛r ℓ✉✲r✐♥❣ ❤♦♠♦♠♦r♣❤✐s♠s✱ t❤❛t ✐s ℓ✉✲r✐♥❣s ❤♦♠♦♠♦r♣❤✐s♠s t❤❛t ♣r❡s❡r✈❡ t❤❡ ❡①t❡r♥❛❧ ♣r♦❞✉❝t✳ ❲❡ ✇✐❧❧ ❝❛❧❧ Γ❢ t❤❡ ❢✉♥❝t♦r ❢r♦♠ ❢❛❧❣ t♦ ❢▼❱❛❧❣ t❤❛t ❡①t❡♥❞ ▼✉♥❞✐❝✐✬s ❢✉♥❝t♦r Γ✳
SLIDE 40
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❈❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡
❢▼❱❛❧❣✱ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ ❢▼❱✲❛❧❣❡❜r❛s ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠s ❛r❡ ▼❱ ✲❛❧❣❡❜r❛s ❤♦♠♦♠♦r♣❤✐s♠s t❤❛t ♣r❡s❡r✈❡ ❜♦t❤ ✐♥t❡r♥❛❧ ❛♥❞ ❡①t❡r♥❛❧ ♣r♦❞✉❝t✳ ❢❛❧❣✱ ❝❛t❡❣♦r② ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ ❢✲❛❧❣❡❜r❛s ✇✐t❤ str♦♥❣ ✉♥✐t ✉ s✉❝❤ t❤❛t ✉ · ✉ ≤ ✉ ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠s ❛r❡ ❧✐♥❡❛r ℓ✉✲r✐♥❣ ❤♦♠♦♠♦r♣❤✐s♠s✱ t❤❛t ✐s ℓ✉✲r✐♥❣s ❤♦♠♦♠♦r♣❤✐s♠s t❤❛t ♣r❡s❡r✈❡ t❤❡ ❡①t❡r♥❛❧ ♣r♦❞✉❝t✳ ❲❡ ✇✐❧❧ ❝❛❧❧ Γ❢ t❤❡ ❢✉♥❝t♦r ❢r♦♠ ❢❛❧❣ t♦ ❢▼❱❛❧❣ t❤❛t ❡①t❡♥❞ ▼✉♥❞✐❝✐✬s ❢✉♥❝t♦r Γ✳
SLIDE 41
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❈❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡
❚❤❡♦r❡♠ ❚❤❡ ❢✉♥❝t♦r Γ❢ ❡st❛❜❧✐s❤ ❛ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝❛t❡❣♦r② ❢❛❧❣ ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ ❢✲❛❧❣❡❜r❛s ✇✐t❤ str♦♥❣ ✉♥✐t ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠s ❛r❡ ℓ✉✲r✐♥❣s ❤♦♠♦♠♦r♣❤✐s♠s ♣r❡s❡r✈✐♥❣ t❤❡ ❡①t❡r♥❛❧ ♣r♦❞✉❝t✱ ❛♥❞ t❤❡ ❝❛t❡❣♦r② ❢▼❱❛❧❣ ✇❤♦s❡ ♦❜❥❡❝ts ❛r❡ ❢▼❱✲❛❧❣❡❜r❛ ❛♥❞ ✇❤♦s❡ ♠♦r♣❤✐s♠s ❛r❡ ▼❱✲❤♦♠♦♠♦r♣❤✐s♠s ♣r❡s❡r✈✐♥❣ ❜♦t❤ ♣r♦❞✉❝ts✳
SLIDE 42
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
■❞❡❛❧s ❛♥❞ ❘❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡♦r❡♠
❉❡✜♥✐t✐♦♥ ▲❡t ■ ❜❡ ❛ s✉❜s❡t ♦❢ ❛♥ ❢▼❱✲❛❧❣❡❜r❛ ❆✳ ❲❡ ✇✐❧❧ ❝❛❧❧ ■ ❢ ✲✐❞❡❛❧ ✐❢✿ ✭■✶✮ ■ ✐s ❛♥ ▼❱ ✲✐❞❡❛❧❀ ✭■✷✮ ❢♦r ❛♥② ① ∈ ❆✱ ② ∈ ■ ✇❡ ❤❛✈❡ ① · ② ∈ ■ ❛♥❞ ② · ① ∈ ■❀ ✭■✸✮ ❢♦r ❛♥② α ∈ [✵, ✶] ❛♥❞ ❛♥② ① ∈ ■✱ α① ∈ ■✳ ❚❤❡♦r❡♠ ❆♥② ❢▼❱✲❛❧❣❡❜r❛ ❆ ✐s s✉❜❞✐r❡❝t ♣r♦❞✉❝t ♦❢ t♦t❛❧❧② ♦r❞❡r❡❞ ❢▼❱✲❛❧❣❡❜r❛s✳
SLIDE 43
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
■❞❡❛❧s ❛♥❞ ❘❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡♦r❡♠
❉❡✜♥✐t✐♦♥ ▲❡t ■ ❜❡ ❛ s✉❜s❡t ♦❢ ❛♥ ❢▼❱✲❛❧❣❡❜r❛ ❆✳ ❲❡ ✇✐❧❧ ❝❛❧❧ ■ ❢ ✲✐❞❡❛❧ ✐❢✿ ✭■✶✮ ■ ✐s ❛♥ ▼❱ ✲✐❞❡❛❧❀ ✭■✷✮ ❢♦r ❛♥② ① ∈ ❆✱ ② ∈ ■ ✇❡ ❤❛✈❡ ① · ② ∈ ■ ❛♥❞ ② · ① ∈ ■❀ ✭■✸✮ ❢♦r ❛♥② α ∈ [✵, ✶] ❛♥❞ ❛♥② ① ∈ ■✱ α① ∈ ■✳ ❚❤❡♦r❡♠ ❆♥② ❢▼❱✲❛❧❣❡❜r❛ ❆ ✐s s✉❜❞✐r❡❝t ♣r♦❞✉❝t ♦❢ t♦t❛❧❧② ♦r❞❡r❡❞ ❢▼❱✲❛❧❣❡❜r❛s✳
SLIDE 44
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❙❡♠✐♣r✐♠❡
❉❡✜♥✐t✐♦♥ ✭✐✮ ❆♥ ❢ ✲❛❧❣❡❜r❛ ❱ ✐s ❝❛❧❧❡❞ s❡♠✐♣r✐♠❡ ✐❢ t❤❡ ♦♥❧② ♥✐❧♣♦t❡♥t ❡❧❡♠❡♥t ✐s ✵✳ ❚❤❛t ✐s✱ ✐❢ ① · ① = ✵✱ t❤❡♥ ① = ✵ ❢♦r ❛♥② ① ∈ ❱ ✳ ✭✐✐✮ ❆♥ ❢▼❱ ✲❛❧❣❡❜r❛ ❆ ✐s ❝❛❧❧❡❞ s❡♠✐♣r✐♠❡ ✐❢ t❤❡ ♦♥❧② ♥✐❧♣♦t❡♥t ❡❧❡♠❡♥t ✐s ✵✳ ❚❤❡② ❛r❡ r❡❧❛t❡❞ t♦ ▼♦♥t❛❣♥❛✬s P▼❱ +✳ ▼♦♥t❛❣♥❛ ❋✳✱ ❙✉❜r❡❞✉❝ts ♦❢ ▼❱✲❛❧❣❡❜r❛s ✇✐t❤ ♣r♦❞✉❝t ❛♥❞ ♣r♦❞✉❝t r❡s✐❞✉❛t✐♦♥✱ ❆❧❣❡❜r❛ ❯♥✐✈❡rs❛❧✐s ✺✸ ✭✷✵✵✺✮ ♣♣ ✶✵✾✲✶✸✼✳
SLIDE 45
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❙❡♠✐♣r✐♠❡
❉❡✜♥✐t✐♦♥ ✭✐✮ ❆♥ ❢ ✲❛❧❣❡❜r❛ ❱ ✐s ❝❛❧❧❡❞ s❡♠✐♣r✐♠❡ ✐❢ t❤❡ ♦♥❧② ♥✐❧♣♦t❡♥t ❡❧❡♠❡♥t ✐s ✵✳ ❚❤❛t ✐s✱ ✐❢ ① · ① = ✵✱ t❤❡♥ ① = ✵ ❢♦r ❛♥② ① ∈ ❱ ✳ ✭✐✐✮ ❆♥ ❢▼❱ ✲❛❧❣❡❜r❛ ❆ ✐s ❝❛❧❧❡❞ s❡♠✐♣r✐♠❡ ✐❢ t❤❡ ♦♥❧② ♥✐❧♣♦t❡♥t ❡❧❡♠❡♥t ✐s ✵✳ ❚❤❡② ❛r❡ r❡❧❛t❡❞ t♦ ▼♦♥t❛❣♥❛✬s P▼❱ +✳ ▼♦♥t❛❣♥❛ ❋✳✱ ❙✉❜r❡❞✉❝ts ♦❢ ▼❱✲❛❧❣❡❜r❛s ✇✐t❤ ♣r♦❞✉❝t ❛♥❞ ♣r♦❞✉❝t r❡s✐❞✉❛t✐♦♥✱ ❆❧❣❡❜r❛ ❯♥✐✈❡rs❛❧✐s ✺✸ ✭✷✵✵✺✮ ♣♣ ✶✵✾✲✶✸✼✳
SLIDE 46
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❙❡♠✐♣r✐♠❡
❉❡✜♥✐t✐♦♥ ❇② ❢▼❱ + ✇❡ ✇✐❧❧ ❞❡♥♦t❡ t❤❡ ❝❧❛ss ♦❢ ✉♥✐t❛❧✱ ❝♦♠♠✉t❛t✐✈❡ ❛♥❞ s❡♠✐♣r✐♠❡ ❢▼❱ ✲❛❧❣❡❜r❛s✳ Pr♦♣♦s✐t✐♦♥ ❆ ❢▼❱✲❛❧❣❡❜r❛ ❆ ✐s s❡♠✐♣r✐♠❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✲❛❧❣❡❜r❛ ❱ ❛r✐s✐♥❣ ❢r♦♠ t❤❡ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡ ✐s s❡♠✐♣r✐♠❡✳
SLIDE 47
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❙❡♠✐♣r✐♠❡
❉❡✜♥✐t✐♦♥ ❇② ❢▼❱ + ✇❡ ✇✐❧❧ ❞❡♥♦t❡ t❤❡ ❝❧❛ss ♦❢ ✉♥✐t❛❧✱ ❝♦♠♠✉t❛t✐✈❡ ❛♥❞ s❡♠✐♣r✐♠❡ ❢▼❱ ✲❛❧❣❡❜r❛s✳ Pr♦♣♦s✐t✐♦♥ ❆ ❢▼❱✲❛❧❣❡❜r❛ ❆ ✐s s❡♠✐♣r✐♠❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✲❛❧❣❡❜r❛ ❱ ❛r✐s✐♥❣ ❢r♦♠ t❤❡ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡ ✐s s❡♠✐♣r✐♠❡✳
SLIDE 48
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❙❡♠✐♣r✐♠❡
Pr♦♣♦s✐t✐♦♥ ❆♥② ❢▼❱ +✲❛❧❣❡❜r❛ ✐s s✉❜❞✐r❡❝t ♣r♦❞✉❝t ♦❢ t♦t❛❧❧② ♦r❞❡r❡❞ ❢▼❱ +✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ❚❤❡ ❝❧❛ss ♦❢ ❢▼❱ +✲❛❧❣❡❜r❛s ✐s t❤❡ q✉❛s✐✲✈❛r✐❡t② ❣❡♥❡r❛t❡❞ ❜② [✵, ✶]✳
SLIDE 49
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❙❡♠✐♣r✐♠❡
Pr♦♣♦s✐t✐♦♥ ❆♥② ❢▼❱ +✲❛❧❣❡❜r❛ ✐s s✉❜❞✐r❡❝t ♣r♦❞✉❝t ♦❢ t♦t❛❧❧② ♦r❞❡r❡❞ ❢▼❱ +✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ❚❤❡ ❝❧❛ss ♦❢ ❢▼❱ +✲❛❧❣❡❜r❛s ✐s t❤❡ q✉❛s✐✲✈❛r✐❡t② ❣❡♥❡r❛t❡❞ ❜② [✵, ✶]✳
SLIDE 50
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❋♦r♠❛❧❧② r❡❛❧
❉❡✜♥✐t✐♦♥ ❆ ❢▼❱ ✲❛❧❣❡❜r❛ ✭P▼❱ ✲❛❧❣❡❜r❛✮ ✐s ❢♦r♠❛❧❧② r❡❛❧ ✐❢ ✐t ❜❡❧♦♥❣s t♦ ❍❙P([✵, ✶])✳ ❲❡ ❞❡♥♦t❡ ❜② FR t❤❡ ❝❧❛ss ♦❢ ❢♦r♠❛❧❧② r❡❛❧ ❢▼❱ ✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ❋♦r ❛♥② ❢♦r♠❛❧❧② r❡❛❧ ❢▼❱✲❛❧❣❡❜r❛ ❆ t❤❡r❡ ❡①✐sts ❛♥ ✉❧tr❛♣♦✇❡r ♦❢
∗[✵, ✶]
♦❢ [✵, ✶] s✉❝❤ t❤❛t ❆ ❡♠❜❡❞❞s ✐♥ ( ∗[✵, ✶])■✱ ❢♦r s♦♠❡ s❡t ■✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢✳ ■t ✐s ❥✉st ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✹✳✷ ♦❢ t❤❡ ♣❛♣❡r ❋❧❛♠✐♥✐♦ ❚✳✱ ❇✐❛♥❝❤✐ ▼✳✱ ❆ ♥♦t❡ ❢♦r s❛t✉r❛t❡❞ ♠♦❞❡❧s ❢♦r ♠❛♥② ✈❛❧✉❡❞ ❧♦❣✐❝✱ ▼❛t❤❡♠❛t✐❝❛ ❙❧♦✈❛❝❛✱ s✉❜♠✐tt❡❞✳
SLIDE 51
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❋♦r♠❛❧❧② r❡❛❧
❉❡✜♥✐t✐♦♥ ❆ ❢▼❱ ✲❛❧❣❡❜r❛ ✭P▼❱ ✲❛❧❣❡❜r❛✮ ✐s ❢♦r♠❛❧❧② r❡❛❧ ✐❢ ✐t ❜❡❧♦♥❣s t♦ ❍❙P([✵, ✶])✳ ❲❡ ❞❡♥♦t❡ ❜② FR t❤❡ ❝❧❛ss ♦❢ ❢♦r♠❛❧❧② r❡❛❧ ❢▼❱ ✲❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ❋♦r ❛♥② ❢♦r♠❛❧❧② r❡❛❧ ❢▼❱✲❛❧❣❡❜r❛ ❆ t❤❡r❡ ❡①✐sts ❛♥ ✉❧tr❛♣♦✇❡r ♦❢
∗[✵, ✶]
♦❢ [✵, ✶] s✉❝❤ t❤❛t ❆ ❡♠❜❡❞❞s ✐♥ ( ∗[✵, ✶])■✱ ❢♦r s♦♠❡ s❡t ■✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢✳ ■t ✐s ❥✉st ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✹✳✷ ♦❢ t❤❡ ♣❛♣❡r ❋❧❛♠✐♥✐♦ ❚✳✱ ❇✐❛♥❝❤✐ ▼✳✱ ❆ ♥♦t❡ ❢♦r s❛t✉r❛t❡❞ ♠♦❞❡❧s ❢♦r ♠❛♥② ✈❛❧✉❡❞ ❧♦❣✐❝✱ ▼❛t❤❡♠❛t✐❝❛ ❙❧♦✈❛❝❛✱ s✉❜♠✐tt❡❞✳
SLIDE 52
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❋♦r♠❛❧❧② ❘❡❛❧
◆♦t ❛♥② ✉♥✐t❛❧ ❛♥❞ ❝♦♠♠✉t❛t✐✈❡ ❢▼❱✲❛❧❣❡❜r❛s ✐s ❢♦r♠❛❧❧② r❡❛❧ ❊①❛♠♣❧❡ ■t ❢♦❧❧♦✇s ❢r♦♠ ❊①❛♠♣❧❡ ✸✳✶✹ ✐♥ ❍♦r❝í❦ ❘✳✱ ❈✐♥t✉❧❛ P✳✱ Pr♦❞✉❝t ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✱ ❆r❝❤✐✈❡ ❢♦r ▼❛t❤❡♠❛t✐❝❛❧ ▲♦❣✐❝✱ ✹✸✭✹✮ ✹✼✼✲✺✵✸ ✭✷✵✵✹✮✳
SLIDE 53
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s✿ ❋♦r♠❛❧❧② ❘❡❛❧
◆♦t ❛♥② ✉♥✐t❛❧ ❛♥❞ ❝♦♠♠✉t❛t✐✈❡ ❢▼❱✲❛❧❣❡❜r❛s ✐s ❢♦r♠❛❧❧② r❡❛❧ ❊①❛♠♣❧❡ ■t ❢♦❧❧♦✇s ❢r♦♠ ❊①❛♠♣❧❡ ✸✳✶✹ ✐♥ ❍♦r❝í❦ ❘✳✱ ❈✐♥t✉❧❛ P✳✱ Pr♦❞✉❝t ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✱ ❆r❝❤✐✈❡ ❢♦r ▼❛t❤❡♠❛t✐❝❛❧ ▲♦❣✐❝✱ ✹✸✭✹✮ ✹✼✼✲✺✵✸ ✭✷✵✵✹✮✳
SLIDE 54
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❢▼❱✲❛❧❣❡❜r❛s
❘❡❢❡r❡♥❝❡s
❇✐r❦❤♦✛ ●✳✱ P✐❡r❝❡ ❘✳❙✳✱ ▲❛tt✐❝❡✲♦r❞❡r❡❞ r✐♥❣s ❆♥✳ ❆❝❛❞✳ ❇r❛s✐❧✳ ❈✐❡♥❝✳ ✷✽ ✭✶✾✺✻✮ ♣♣✳ ✹✶✲✻✾✳ ▼✳❍❡♥r✐❦s❡♥✱ ❏✳ ■s❜❡❧❧✱ ▲❛tt✐❝❡✲♦r❞❡r❡❞ r✐♥❣s ❛♥❞ ❢✉♥❝t✐♦♥ r✐♥❣s P❛❝✐✜❝ ❏✳ ▼❛t❤✳ ✶✷ ✭✶✾✻✷✮✱ ✺✸✸✲✺✻✺✳ ❏✳▼❛❞❞❡♥✱ ❍❡♥r✐❦s❡♥ ❛♥❞ ■s❜❡❧❧ ♦♥ ❢✲r✐♥❣s ❚♦♣♦❧♦❣② ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s ✶✺✽ ✭✷✵✶✶✮✱ ✶✼✻✽✲✶✼✼✸✳ ❩❛♥❡❡♥ ❆✳❈✳✱ ❘✐❡s③ ❙♣❛❝❡ ■■✱ ◆♦rt❤ ❍♦❧❧❛♥❞✱ ❆♠st❡r❞❛♠ ✶✾✽✸✳ ▲❛♣❡♥t❛ ❙✳ ▲❡✉st❡❛♥ ■✳✱ ❯♥✐t ✐♥t❡r✈❛❧s ✐♥ ❢✲❛❧❣❡❜r❛s✱ ❞r❛❢t✳
SLIDE 55
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❙❡❝t✐♦♥ ✸ P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
SLIDE 56
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
✲κ ❝❛r❞✐♥❛❧ ♥✉♠❜❡r❀ ✲α < κ✱ ❞❡✜♥❡ πκ
α : ❆κ → ❆✱ πκ α(❛✶, . . . , ❛α, . . .) = ❛α.
✲❙ ❜❡ ❛ s✉❜r✐♥❣ ♦❢ R L❙ ✐s t❤❡ ❛❧♣❤❛❜❡t {⊕,∗ , ·, ✵, } ∪ {δr | r ∈ [✵, ✶] ∩ ❙}✱ δr ✐s ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ t❤❛t ✐s ✐♥t❡r♣r❡t❡❞ ❜② ① → r① ❢♦r ❛♥② r ∈ [✵, ✶] ∩ ❙✳ ✲❛ t❡r♠ ♦✈❡r t❤❡ s❡t ♦❢ ✈❛r✐❛❜❧❡s {❳α}α<κ ✐s ❛ ✜♥✐t❡ str✐♥❣ ♦❢ ❡❧❡♠❡♥t ♦✈❡r t❤❡ ❛❧♣❤❛❜❡t L❙✳ ❚❡r♠❆
♥ (❙) = {t❡r♠s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ L❙}
SLIDE 57
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
✲κ ❝❛r❞✐♥❛❧ ♥✉♠❜❡r❀ ✲α < κ✱ ❞❡✜♥❡ πκ
α : ❆κ → ❆✱ πκ α(❛✶, . . . , ❛α, . . .) = ❛α.
✲❙ ❜❡ ❛ s✉❜r✐♥❣ ♦❢ R L❙ ✐s t❤❡ ❛❧♣❤❛❜❡t {⊕,∗ , ·, ✵, } ∪ {δr | r ∈ [✵, ✶] ∩ ❙}✱ δr ✐s ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ t❤❛t ✐s ✐♥t❡r♣r❡t❡❞ ❜② ① → r① ❢♦r ❛♥② r ∈ [✵, ✶] ∩ ❙✳ ✲❛ t❡r♠ ♦✈❡r t❤❡ s❡t ♦❢ ✈❛r✐❛❜❧❡s {❳α}α<κ ✐s ❛ ✜♥✐t❡ str✐♥❣ ♦❢ ❡❧❡♠❡♥t ♦✈❡r t❤❡ ❛❧♣❤❛❜❡t L❙✳ ❚❡r♠❆
♥ (❙) = {t❡r♠s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ L❙}
SLIDE 58
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
✲κ ❝❛r❞✐♥❛❧ ♥✉♠❜❡r❀ ✲α < κ✱ ❞❡✜♥❡ πκ
α : ❆κ → ❆✱ πκ α(❛✶, . . . , ❛α, . . .) = ❛α.
✲❙ ❜❡ ❛ s✉❜r✐♥❣ ♦❢ R L❙ ✐s t❤❡ ❛❧♣❤❛❜❡t {⊕,∗ , ·, ✵, } ∪ {δr | r ∈ [✵, ✶] ∩ ❙}✱ δr ✐s ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ t❤❛t ✐s ✐♥t❡r♣r❡t❡❞ ❜② ① → r① ❢♦r ❛♥② r ∈ [✵, ✶] ∩ ❙✳ ✲❛ t❡r♠ ♦✈❡r t❤❡ s❡t ♦❢ ✈❛r✐❛❜❧❡s {❳α}α<κ ✐s ❛ ✜♥✐t❡ str✐♥❣ ♦❢ ❡❧❡♠❡♥t ♦✈❡r t❤❡ ❛❧♣❤❛❜❡t L❙✳ ❚❡r♠❆
♥ (❙) = {t❡r♠s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ L❙}
SLIDE 59
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
✲κ ❝❛r❞✐♥❛❧ ♥✉♠❜❡r❀ ✲α < κ✱ ❞❡✜♥❡ πκ
α : ❆κ → ❆✱ πκ α(❛✶, . . . , ❛α, . . .) = ❛α.
✲❙ ❜❡ ❛ s✉❜r✐♥❣ ♦❢ R L❙ ✐s t❤❡ ❛❧♣❤❛❜❡t {⊕,∗ , ·, ✵, } ∪ {δr | r ∈ [✵, ✶] ∩ ❙}✱ δr ✐s ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ t❤❛t ✐s ✐♥t❡r♣r❡t❡❞ ❜② ① → r① ❢♦r ❛♥② r ∈ [✵, ✶] ∩ ❙✳ ✲❛ t❡r♠ ♦✈❡r t❤❡ s❡t ♦❢ ✈❛r✐❛❜❧❡s {❳α}α<κ ✐s ❛ ✜♥✐t❡ str✐♥❣ ♦❢ ❡❧❡♠❡♥t ♦✈❡r t❤❡ ❛❧♣❤❛❜❡t L❙✳ ❚❡r♠❆
♥ (❙) = {t❡r♠s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ L❙}
SLIDE 60
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
✲κ ❝❛r❞✐♥❛❧ ♥✉♠❜❡r❀ ✲α < κ✱ ❞❡✜♥❡ πκ
α : ❆κ → ❆✱ πκ α(❛✶, . . . , ❛α, . . .) = ❛α.
✲❙ ❜❡ ❛ s✉❜r✐♥❣ ♦❢ R L❙ ✐s t❤❡ ❛❧♣❤❛❜❡t {⊕,∗ , ·, ✵, } ∪ {δr | r ∈ [✵, ✶] ∩ ❙}✱ δr ✐s ❛ ✉♥❛r② ♦♣❡r❛t✐♦♥ t❤❛t ✐s ✐♥t❡r♣r❡t❡❞ ❜② ① → r① ❢♦r ❛♥② r ∈ [✵, ✶] ∩ ❙✳ ✲❛ t❡r♠ ♦✈❡r t❤❡ s❡t ♦❢ ✈❛r✐❛❜❧❡s {❳α}α<κ ✐s ❛ ✜♥✐t❡ str✐♥❣ ♦❢ ❡❧❡♠❡♥t ♦✈❡r t❤❡ ❛❧♣❤❛❜❡t L❙✳ ❚❡r♠❆
♥ (❙) = {t❡r♠s ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ L❙}
SLIDE 61
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
❉❡✜♥✐t✐♦♥ t ∈ ❚❡r♠❆
♥ (❙)✱ ❛♥❞ ❆ ❛ ❢▼❱✲❛❧❣❡❜r❛✳ ❚❤❡ t❡r♠ ❢✉♥❝t✐♦♥ ˜
t : ❆♥ → ❆ ♦❢ t ✐s ❞❡✜♥❡❞ ❜② ✭✐✮ ❋♦r ❛♥② ♠ ≤ ♥✱ ❳♠ = π♥
♠❀
✭✐✐✮ ✵ ✐s t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❡q✉❛❧ t♦ ✵ ♦♥ ❆♥❀ ✭✐✐✐✮ t∗ = ( t)∗❀ ✭✐✈✮ t✶ ⊕ t✷ = t✶ ⊕ t✷❀ ✭✈✮ δrt = r t❀ ✭✈✐✮ t✶ · t✷ = t✶ · t✷✳ ❋❚ ❆
♥ (❙) = {
t : ❆♥ → ❆ | t ∈ ❚❡r♠❆
♥ (❙) ❛♥❞
t ✐s t❤❡ t❡r♠ ❢✉♥❝t✐♦♥ ♦❢ t}
SLIDE 62
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
❉❡✜♥✐t✐♦♥ t ∈ ❚❡r♠❆
♥ (❙)✱ ❛♥❞ ❆ ❛ ❢▼❱✲❛❧❣❡❜r❛✳ ❚❤❡ t❡r♠ ❢✉♥❝t✐♦♥ ˜
t : ❆♥ → ❆ ♦❢ t ✐s ❞❡✜♥❡❞ ❜② ✭✐✮ ❋♦r ❛♥② ♠ ≤ ♥✱ ❳♠ = π♥
♠❀
✭✐✐✮ ✵ ✐s t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❡q✉❛❧ t♦ ✵ ♦♥ ❆♥❀ ✭✐✐✐✮ t∗ = ( t)∗❀ ✭✐✈✮ t✶ ⊕ t✷ = t✶ ⊕ t✷❀ ✭✈✮ δrt = r t❀ ✭✈✐✮ t✶ · t✷ = t✶ · t✷✳ ❋❚ ❆
♥ (❙) = {
t : ❆♥ → ❆ | t ∈ ❚❡r♠❆
♥ (❙) ❛♥❞
t ✐s t❤❡ t❡r♠ ❢✉♥❝t✐♦♥ ♦❢ t}
SLIDE 63
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
❉❡✜♥✐t✐♦♥ t ∈ ❚❡r♠❆
♥ (❙)✱ ❛♥❞ ❆ ❛ ❢▼❱✲❛❧❣❡❜r❛✳ ❚❤❡ t❡r♠ ❢✉♥❝t✐♦♥ ˜
t : ❆♥ → ❆ ♦❢ t ✐s ❞❡✜♥❡❞ ❜② ✭✐✮ ❋♦r ❛♥② ♠ ≤ ♥✱ ❳♠ = π♥
♠❀
✭✐✐✮ ✵ ✐s t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ❡q✉❛❧ t♦ ✵ ♦♥ ❆♥❀ ✭✐✐✐✮ t∗ = ( t)∗❀ ✭✐✈✮ t✶ ⊕ t✷ = t✶ ⊕ t✷❀ ✭✈✮ δrt = r t❀ ✭✈✐✮ t✶ · t✷ = t✶ · t✷✳ ❋❚ ❆
♥ (❙) = {
t : ❆♥ → ❆ | t ∈ ❚❡r♠❆
♥ (❙) ❛♥❞
t ✐s t❤❡ t❡r♠ ❢✉♥❝t✐♦♥ ♦❢ t}
SLIDE 64
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
✲✐❢ ❆ = [✵, ✶]✱ t❤❡♥ ❋❚ [✵,✶]
♥
(❙) ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② ❋❚♥(❙)❀ ✲❢r❡❡ ❢▼❱✲❛❧❣❡❜r❛ ✐♥ FR ❡①✐st ❛♥❞ ✐t ✐s ❣✐✈❡♥ ❜② ❋❘♥ = { t | t ∈ ❚❡r♠♥, t : [✵, ✶]♥ → [✵, ✶] ✐s t❤❡ t❡r♠ ❢✉♥❝t✐♦♥ ♦❢ t}❀ ✲❋❚♥(R) = ❋❘♥✳
SLIDE 65
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
✲✐❢ ❆ = [✵, ✶]✱ t❤❡♥ ❋❚ [✵,✶]
♥
(❙) ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② ❋❚♥(❙)❀ ✲❢r❡❡ ❢▼❱✲❛❧❣❡❜r❛ ✐♥ FR ❡①✐st ❛♥❞ ✐t ✐s ❣✐✈❡♥ ❜② ❋❘♥ = { t | t ∈ ❚❡r♠♥, t : [✵, ✶]♥ → [✵, ✶] ✐s t❤❡ t❡r♠ ❢✉♥❝t✐♦♥ ♦❢ t}❀ ✲❋❚♥(R) = ❋❘♥✳
SLIDE 66
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❚❡r♠s ❛♥❞ t❡r♠ ❢✉♥❝t✐♦♥s
✲✐❢ ❆ = [✵, ✶]✱ t❤❡♥ ❋❚ [✵,✶]
♥
(❙) ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❜② ❋❚♥(❙)❀ ✲❢r❡❡ ❢▼❱✲❛❧❣❡❜r❛ ✐♥ FR ❡①✐st ❛♥❞ ✐t ✐s ❣✐✈❡♥ ❜② ❋❘♥ = { t | t ∈ ❚❡r♠♥, t : [✵, ✶]♥ → [✵, ✶] ✐s t❤❡ t❡r♠ ❢✉♥❝t✐♦♥ ♦❢ t}❀ ✲❋❚♥(R) = ❋❘♥✳
SLIDE 67
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s
❉❡✜♥✐t✐♦♥ ❆ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥ ✐♥ ♥ ✈❛r✐❛❜❧❡s ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❙ ✭P❲P♥✭❙✮✲❢✉♥❝t✐♦♥✱ s❤♦rt❧②✮ ✐s ❢ : R♥ → R s✉❝❤ t❤❛t✿ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦❧②♥♦♠✐❛❧s ❢✶✱ . . .✱ ❢❦ ∈ ❙[①✶, . . . , ①♥] s✉❝❤ t❤❛t ❢♦r ❛♥② (❛✶, . . . , ❛♥) ∈ [✵, ✶]♥ t❤❡r❡ ✐s ✐ ∈ {✶, . . . , ❦} ✇✐t❤ ❢ (❛✶, . . . , ❛♥) = ❢✐(❛✶, . . . , ❛♥)✳ ❲❡ s❛② t❤❛t ❢✶✱ . . .✱ ❢❦ ❛r❡ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ❢ ✳
SLIDE 68
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s
❉❡✜♥✐t✐♦♥ ❆ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥ ✐♥ ♥ ✈❛r✐❛❜❧❡s ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❙ ✭P❲P♥✭❙✮✲❢✉♥❝t✐♦♥✱ s❤♦rt❧②✮ ✐s ❢ : R♥ → R s✉❝❤ t❤❛t✿ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦❧②♥♦♠✐❛❧s ❢✶✱ . . .✱ ❢❦ ∈ ❙[①✶, . . . , ①♥] s✉❝❤ t❤❛t ❢♦r ❛♥② (❛✶, . . . , ❛♥) ∈ [✵, ✶]♥ t❤❡r❡ ✐s ✐ ∈ {✶, . . . , ❦} ✇✐t❤ ❢ (❛✶, . . . , ❛♥) = ❢✐(❛✶, . . . , ❛♥)✳ ❲❡ s❛② t❤❛t ❢✶✱ . . .✱ ❢❦ ❛r❡ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ❢ ✳
SLIDE 69
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s
❉❡✜♥✐t✐♦♥ ❆ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥ ✐♥ ♥ ✈❛r✐❛❜❧❡s ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❙ ✭P❲P♥✭❙✮✲❢✉♥❝t✐♦♥✱ s❤♦rt❧②✮ ✐s ❢ : R♥ → R s✉❝❤ t❤❛t✿ t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ♣♦❧②♥♦♠✐❛❧s ❢✶✱ . . .✱ ❢❦ ∈ ❙[①✶, . . . , ①♥] s✉❝❤ t❤❛t ❢♦r ❛♥② (❛✶, . . . , ❛♥) ∈ [✵, ✶]♥ t❤❡r❡ ✐s ✐ ∈ {✶, . . . , ❦} ✇✐t❤ ❢ (❛✶, . . . , ❛♥) = ❢✐(❛✶, . . . , ❛♥)✳ ❲❡ s❛② t❤❛t ❢✶✱ . . .✱ ❢❦ ❛r❡ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ ❢ ✳
SLIDE 70
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
◆♦t❛t✐♦♥s
P❋♥(❙) = {❢ : [✵, ✶]♥ → [✵, ✶] | ❢ ✐s ❛ ❝♦♥t✳ P❲P♥✭❙✮✲❢✉♥❝t✐♦♥ ❞❡❢✳ ♦♥ t❤❡ ♥✲❝✉❜❡} P❋♥(❙)r = {❢ |[✵,✶]♥|❢ : R♥ → [✵, ✶] ✐s ❛ ❝♦♥t✐♥✉♦✉s P❲P♥✭❙✮✲❢✉♥❝t✐♦♥} ❙■❉♥(❙) = {❣ : [✵, ✶]♥ → [✵, ✶] | ❣ ∈ P❋♥(❙), ❣ =
✐∈■
- ❥∈❏ ❣✐❥} ✇❤❡r❡
❣✐❥ ❛r❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❙[①✶, . . . , ①♥].
SLIDE 71
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
◆♦t❛t✐♦♥s
P❋♥(❙) = {❢ : [✵, ✶]♥ → [✵, ✶] | ❢ ✐s ❛ ❝♦♥t✳ P❲P♥✭❙✮✲❢✉♥❝t✐♦♥ ❞❡❢✳ ♦♥ t❤❡ ♥✲❝✉❜❡} P❋♥(❙)r = {❢ |[✵,✶]♥|❢ : R♥ → [✵, ✶] ✐s ❛ ❝♦♥t✐♥✉♦✉s P❲P♥✭❙✮✲❢✉♥❝t✐♦♥} ❙■❉♥(❙) = {❣ : [✵, ✶]♥ → [✵, ✶] | ❣ ∈ P❋♥(❙), ❣ =
✐∈■
- ❥∈❏ ❣✐❥} ✇❤❡r❡
❣✐❥ ❛r❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❙[①✶, . . . , ①♥].
SLIDE 72
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
◆♦t❛t✐♦♥s
P❋♥(❙) = {❢ : [✵, ✶]♥ → [✵, ✶] | ❢ ✐s ❛ ❝♦♥t✳ P❲P♥✭❙✮✲❢✉♥❝t✐♦♥ ❞❡❢✳ ♦♥ t❤❡ ♥✲❝✉❜❡} P❋♥(❙)r = {❢ |[✵,✶]♥|❢ : R♥ → [✵, ✶] ✐s ❛ ❝♦♥t✐♥✉♦✉s P❲P♥✭❙✮✲❢✉♥❝t✐♦♥} ❙■❉♥(❙) = {❣ : [✵, ✶]♥ → [✵, ✶] | ❣ ∈ P❋♥(❙), ❣ =
✐∈■
- ❥∈❏ ❣✐❥} ✇❤❡r❡
❣✐❥ ❛r❡ ♣♦❧②♥♦♠✐❛❧s ✐♥ ❙[①✶, . . . , ①♥].
SLIDE 73
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❆ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ❢♦r ❋❘♥
Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ ❋❚♥(❙) ❛r❡ ❝♦♥t✐♥✉♦✉s ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ t❤❡ ♥✲❝✉❜❡✱ ✐✳❡✳ ❋❚♥(❙) ⊆ P❋♥(❙)✳ ❉❡✜♥✐t✐♦♥ ̺ : R → [✵, ✶]✱ ̺(①) = ① ∧ ✶ ∨ ✵✱ ❢♦r ❛♥② ① ∈ R✳
SLIDE 74
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❆ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ❢♦r ❋❘♥
Pr♦♣♦s✐t✐♦♥ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ ❋❚♥(❙) ❛r❡ ❝♦♥t✐♥✉♦✉s ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ t❤❡ ♥✲❝✉❜❡✱ ✐✳❡✳ ❋❚♥(❙) ⊆ P❋♥(❙)✳ ❉❡✜♥✐t✐♦♥ ̺ : R → [✵, ✶]✱ ̺(①) = ① ∧ ✶ ∨ ✵✱ ❢♦r ❛♥② ① ∈ R✳
SLIDE 75
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❆ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ❢♦r ❋❘♥
Pr♦♣♦s✐t✐♦♥ ▲❡t ❙ ❜❡ ❛ s✉❜r✐♥❣ ♦❢ R✳ ✭❛✮❋♦r ❛♥② ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥ ♣ : [✵, ✶]♥ → R ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❙✱ t❤❡r❡ ❡①✐sts ❛ t❡r♠ t ∈ ❚❡r♠♥(❙) s✉❝❤ t❤❛t ̺ ◦ ♣ = t ❛♥❞ t ∈ ❋❚♥(❙)✳ ✭❜✮❋♦r ❛♥② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❣ ∈ ❙■❉♥(❙) t❤❡r❡ ❡①✐sts ❛ t❡r♠ t ∈ ❚❡r♠♥(❙) s✉❝❤ t❤❛t ❣ = t✳
SLIDE 76
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❆ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ❢♦r ❋❘♥
❈♦r♦❧❧❛r② ✭✶✮ ❙■❉♥(❙) ⊆ ❋❚♥(❙) ⊆ P❋♥(❙) ✭✷✮ ❙■❉♥(❙) ⊆ P❋♥(❙)r ⊆ P❋♥(❙)✳
SLIDE 77
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❆ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ❢♦r ❋❘♥
❚❤❡♦r❡♠ ❋♦r ♥ ≤ ✷✱ P❋♥(R)r = P❋♥(R) = ❋❘♥ = ❙■❉♥(R)✳ ■♥ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡ ❢▼❱ ✲❛❧❣❡❜r❛ ❋❘♥ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❝♦♥t✐♥✉♦✉s ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ t❤❡ ♥✲❝✉❜❡✱ ✐✳❡ ❛♥② ❝♦♥t✐♥✉♦✉s ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ t❤❡ ♥✲❝✉❜❡ ✐s ❛ t❡r♠ ❢✉♥❝t✐♦♥ ❢r♦♠ ❋❘♥✳
❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡ ✐s ♣r♦✈❡❞ ❢♦r ♥ < ✸ ✐♥ ▼❛❤é ▲✳✱ ❖♥ t❤❡ ❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡✱ ❘♦❝❦② ▼✳ ❏✳ ✶✹✭✹✮ ✭✶✾✽✹✮ ✾✽✸✲✾✽✺ ❛ P❲P ❢✉♥❝t✐♦♥ ♦♥ [✵, ✶]✷ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ R✷ ❜② ❆✳ ❋✐s❝❤❡r✱ ▼✳ ▼❛rs❤❛❧❧✱ ❊①t❡♥❞✐♥❣ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ✐♥ t✇♦ ✈❛r✐❛❜❧❡s✱ ❆♥♥❛❧❡s ❞❡ ❧❛ ❋❛❝✉❧t❡ ❞❡s ❙❝✐❡♥❝❡s ❚♦✉❧♦✉s❡✱ ✷✷ ✭✷✵✶✸✮ ✷✺✸✲✷✻✽✳
SLIDE 78
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❆ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ❢♦r ❋❘♥
❚❤❡♦r❡♠ ❋♦r ♥ ≤ ✷✱ P❋♥(R)r = P❋♥(R) = ❋❘♥ = ❙■❉♥(R)✳ ■♥ ❝♦♥s❡q✉❡♥❝❡✱ t❤❡ ❢▼❱ ✲❛❧❣❡❜r❛ ❋❘♥ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❝♦♥t✐♥✉♦✉s ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ t❤❡ ♥✲❝✉❜❡✱ ✐✳❡ ❛♥② ❝♦♥t✐♥✉♦✉s ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ t❤❡ ♥✲❝✉❜❡ ✐s ❛ t❡r♠ ❢✉♥❝t✐♦♥ ❢r♦♠ ❋❘♥✳
❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡ ✐s ♣r♦✈❡❞ ❢♦r ♥ < ✸ ✐♥ ▼❛❤é ▲✳✱ ❖♥ t❤❡ ❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡✱ ❘♦❝❦② ▼✳ ❏✳ ✶✹✭✹✮ ✭✶✾✽✹✮ ✾✽✸✲✾✽✺ ❛ P❲P ❢✉♥❝t✐♦♥ ♦♥ [✵, ✶]✷ ❝❛♥ ❜❡ ❡①t❡♥❞❡❞ t♦ R✷ ❜② ❆✳ ❋✐s❝❤❡r✱ ▼✳ ▼❛rs❤❛❧❧✱ ❊①t❡♥❞✐♥❣ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ✐♥ t✇♦ ✈❛r✐❛❜❧❡s✱ ❆♥♥❛❧❡s ❞❡ ❧❛ ❋❛❝✉❧t❡ ❞❡s ❙❝✐❡♥❝❡s ❚♦✉❧♦✉s❡✱ ✷✷ ✭✷✵✶✸✮ ✷✺✸✲✷✻✽✳
SLIDE 79
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
- ✐✈❡♥ ❛ ✐♥t❡r✈❛❧ ■ ⊆ R✱ t❤❡ ♥t❤✲♠♦♠❡♥t ♦❢ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ ♦♥ ■
✐s ❞❡✜♥❡❞ ❛s
- ■ ①♥❞µ✳ ▲❡t {♠❦}❦≥✵ ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ♥✉♠❜❡rs✱ t❤❡
▼♦♠❡♥t Pr♦❜❧❡♠s ♦♥ ■ ❝♦♥s✐st❡s ♦♥ ✜♥❞✐♥❣ ♦✉t t❤❡ ❝♦♥❞✐t✐♦♥ ♦♥ {♠❦}❦≥✵ ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ ♦♥ ■ s✉❝❤ t❤❛t ♠❦ ✐s t❤❡ ❦t❤ ♠♦♠❡♥t ♦❢ µ✳ ❲❤❡♥ ■ = [✵, ✶] ✇❡ ❣❡t t❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ♣r♦❜❧❡♠ ❍❛✉s❞♦r✛ ❋✳✱ ❙✉♠♠❛t✐♦♥♠❡t❤♦❞❡♥ ✉♥❞ ▼♦♠❡♥t❢♦❧❣❡♥ ■✱ ▼❛t❤✳ ❩✳ ✾ ✭✶✾✷✶✮✱ ✼✹ ✲✶✵✾✳ ❍❛✉s❞♦r✛ ❋✳✱ ❙✉♠♠❛t✐♦♥♠❡t❤♦❞❡♥ ✉♥❞ ▼♦♠❡♥t❢♦❧❣❡♥ ■■✱ ▼❛t❤✳ ❩✳ ✾ ✭✶✾✷✶✮✱ ✷✽✵✲✷✾✾✳
SLIDE 80
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
- ✐✈❡♥ ❛ ✐♥t❡r✈❛❧ ■ ⊆ R✱ t❤❡ ♥t❤✲♠♦♠❡♥t ♦❢ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ ♦♥ ■
✐s ❞❡✜♥❡❞ ❛s
- ■ ①♥❞µ✳ ▲❡t {♠❦}❦≥✵ ❜❡ ❛ s❡q✉❡♥❝❡ ♦❢ r❡❛❧ ♥✉♠❜❡rs✱ t❤❡
▼♦♠❡♥t Pr♦❜❧❡♠s ♦♥ ■ ❝♦♥s✐st❡s ♦♥ ✜♥❞✐♥❣ ♦✉t t❤❡ ❝♦♥❞✐t✐♦♥ ♦♥ {♠❦}❦≥✵ ❢♦r ✇❤✐❝❤ t❤❡r❡ ❡①✐sts ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ ♦♥ ■ s✉❝❤ t❤❛t ♠❦ ✐s t❤❡ ❦t❤ ♠♦♠❡♥t ♦❢ µ✳ ❲❤❡♥ ■ = [✵, ✶] ✇❡ ❣❡t t❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ♣r♦❜❧❡♠ ❍❛✉s❞♦r✛ ❋✳✱ ❙✉♠♠❛t✐♦♥♠❡t❤♦❞❡♥ ✉♥❞ ▼♦♠❡♥t❢♦❧❣❡♥ ■✱ ▼❛t❤✳ ❩✳ ✾ ✭✶✾✷✶✮✱ ✼✹ ✲✶✵✾✳ ❍❛✉s❞♦r✛ ❋✳✱ ❙✉♠♠❛t✐♦♥♠❡t❤♦❞❡♥ ✉♥❞ ▼♦♠❡♥t❢♦❧❣❡♥ ■■✱ ▼❛t❤✳ ❩✳ ✾ ✭✶✾✷✶✮✱ ✷✽✵✲✷✾✾✳
SLIDE 81
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
❉❡✜♥✐t✐♦♥ ❆ st❛t❡ ❢♦r ❛ ▼❱ ✲❛❧❣❡❜r❛ ❆ ✐s ❛ ♠❛♣ s : ❆ → [✵, ✶] s✉❝❤ t❤❛t ❢♦r ❛♥② ①, ② ∈ ❆ ✇✐t❤ ① ⊙ ② = ✵✱ s(① ⊕ ②) = s(①) + s(②) ❛♥❞ s(✶) = ✶✳ ▼✉♥❞✐❝✐ ❉✳✱ ❆✈❡r❛❣✐♥❣ t❤❡ ❚r✉t❤✲❱❛❧✉❡ ✐♥ ❾✉❦❛s✐❡✇✐❝③ ▲♦❣✐❝✱ ❙t✉❞✐❛ ▲♦❣✐❝❛ ✺✺ ✭✶✾✾✺✮✱ ✶✶✸✲✶✷✼✳ ❉❡✜♥✐t✐♦♥ ❆ st❛t❡ ❢♦r ❛ ❢▼❱ ✲❛❧❣❡❜r❛ ✐s ❛ st❛t❡ ❢♦r ✐ts ▼❱ ✲❛❧❣❡❜r❛ r❡❞✉❝t✳
SLIDE 82
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
❉❡✜♥✐t✐♦♥ ❆ st❛t❡ ❢♦r ❛ ▼❱ ✲❛❧❣❡❜r❛ ❆ ✐s ❛ ♠❛♣ s : ❆ → [✵, ✶] s✉❝❤ t❤❛t ❢♦r ❛♥② ①, ② ∈ ❆ ✇✐t❤ ① ⊙ ② = ✵✱ s(① ⊕ ②) = s(①) + s(②) ❛♥❞ s(✶) = ✶✳ ▼✉♥❞✐❝✐ ❉✳✱ ❆✈❡r❛❣✐♥❣ t❤❡ ❚r✉t❤✲❱❛❧✉❡ ✐♥ ❾✉❦❛s✐❡✇✐❝③ ▲♦❣✐❝✱ ❙t✉❞✐❛ ▲♦❣✐❝❛ ✺✺ ✭✶✾✾✺✮✱ ✶✶✸✲✶✷✼✳ ❉❡✜♥✐t✐♦♥ ❆ st❛t❡ ❢♦r ❛ ❢▼❱ ✲❛❧❣❡❜r❛ ✐s ❛ st❛t❡ ❢♦r ✐ts ▼❱ ✲❛❧❣❡❜r❛ r❡❞✉❝t✳
SLIDE 83
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
❋♦r ❛♥② ❦ ≥ ✶✱ ♣❦ : [✵, ✶] → [✵, ✶] ✐s ♣❦(①) = ①❦ ❢♦r ❛♥② ① ∈ [✵, ✶]✳ ♣✵(①) = ✶ ❢♦r ❛♥② ① ∈ [✵, ✶]✳ ◆♦t❡ t❤❛t ♣❦ ∈ ❋❘✶ ❢♦r ❛♥② ❦ ≥ ✵✳ {♠❦|❦ ≥ ✵} ⊆ [✵, ✶]✳ ∆✵♠❦ = ♠❦✱ ∆r♠❦ = ∆r−✶♠❦+✶ − ∆r−✶♠❦ ❢♦r ❛♥② r✱ ❦ ≥ ✵✳ ❚❤❡ s❡q✉❡♥❝❡ {♠❦}❦ s❛t✐s✜❡s t❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ❝♦♥❞✐t✐♦♥ ✐❢ ♠✵ = ✶ ❛♥❞ (−✶)r∆r♠❦ ≥ ✵ ❢♦r ❛♥② r✱ ❦ ≥ ✵✳ ❈([✵, ✶]) = Γ(❈([✵, ✶], R), ✶)✱ ❈ ❜❡ ❛♥② s❡♠✐s✐♠♣❧❡ P▼❱ ✲s✉❜❛❧❣❡❜r❛ ✭✉♥✐t❛❧ ❛♥❞ ❝♦♠♠✉t❛t✐✈❡✮ ♦❢ ❈([✵, ✶]) s✉❝❤ t❤❛t ♣✶ ∈ ❈✳
SLIDE 84
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
❋♦r ❛♥② ❦ ≥ ✶✱ ♣❦ : [✵, ✶] → [✵, ✶] ✐s ♣❦(①) = ①❦ ❢♦r ❛♥② ① ∈ [✵, ✶]✳ ♣✵(①) = ✶ ❢♦r ❛♥② ① ∈ [✵, ✶]✳ ◆♦t❡ t❤❛t ♣❦ ∈ ❋❘✶ ❢♦r ❛♥② ❦ ≥ ✵✳ {♠❦|❦ ≥ ✵} ⊆ [✵, ✶]✳ ∆✵♠❦ = ♠❦✱ ∆r♠❦ = ∆r−✶♠❦+✶ − ∆r−✶♠❦ ❢♦r ❛♥② r✱ ❦ ≥ ✵✳ ❚❤❡ s❡q✉❡♥❝❡ {♠❦}❦ s❛t✐s✜❡s t❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ❝♦♥❞✐t✐♦♥ ✐❢ ♠✵ = ✶ ❛♥❞ (−✶)r∆r♠❦ ≥ ✵ ❢♦r ❛♥② r✱ ❦ ≥ ✵✳ ❈([✵, ✶]) = Γ(❈([✵, ✶], R), ✶)✱ ❈ ❜❡ ❛♥② s❡♠✐s✐♠♣❧❡ P▼❱ ✲s✉❜❛❧❣❡❜r❛ ✭✉♥✐t❛❧ ❛♥❞ ❝♦♠♠✉t❛t✐✈❡✮ ♦❢ ❈([✵, ✶]) s✉❝❤ t❤❛t ♣✶ ∈ ❈✳
SLIDE 85
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
❋♦r ❛♥② ❦ ≥ ✶✱ ♣❦ : [✵, ✶] → [✵, ✶] ✐s ♣❦(①) = ①❦ ❢♦r ❛♥② ① ∈ [✵, ✶]✳ ♣✵(①) = ✶ ❢♦r ❛♥② ① ∈ [✵, ✶]✳ ◆♦t❡ t❤❛t ♣❦ ∈ ❋❘✶ ❢♦r ❛♥② ❦ ≥ ✵✳ {♠❦|❦ ≥ ✵} ⊆ [✵, ✶]✳ ∆✵♠❦ = ♠❦✱ ∆r♠❦ = ∆r−✶♠❦+✶ − ∆r−✶♠❦ ❢♦r ❛♥② r✱ ❦ ≥ ✵✳ ❚❤❡ s❡q✉❡♥❝❡ {♠❦}❦ s❛t✐s✜❡s t❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ❝♦♥❞✐t✐♦♥ ✐❢ ♠✵ = ✶ ❛♥❞ (−✶)r∆r♠❦ ≥ ✵ ❢♦r ❛♥② r✱ ❦ ≥ ✵✳ ❈([✵, ✶]) = Γ(❈([✵, ✶], R), ✶)✱ ❈ ❜❡ ❛♥② s❡♠✐s✐♠♣❧❡ P▼❱ ✲s✉❜❛❧❣❡❜r❛ ✭✉♥✐t❛❧ ❛♥❞ ❝♦♠♠✉t❛t✐✈❡✮ ♦❢ ❈([✵, ✶]) s✉❝❤ t❤❛t ♣✶ ∈ ❈✳
SLIDE 86
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐sts ❛ st❛t❡ s : ❈ → [✵, ✶] s✉❝❤ t❤❛t s(♣❦) = ♠❦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡q✉❡♥❝❡ {♠❦} s❛t✐s✜❡s t❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ❝♦♥❞✐t✐♦♥✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢✳ ❇② ❑r♦✉♣❛✲P❛♥t✐ r❛♣r❡s❡♥t❛t✐♦♥ ❢♦r st❛t❡s ✇❡ ❣❡t s(❢ ) = ✶
✵ ❢❞µ✳ ❚❤❡♥ ✐t
❢♦❧❧♦✇s ❜② ❝❛❧❝✉❧❛t✐♦♥s✳ ❖♥ t❤❡ ♦t❤❡r ❞✐r❡❝t✐♦♥ ✐s ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢
▼✐r❛♥❞❛ ❊✳✱ ❞❡ ❈♦♦♠❛♥ ●✳✱ ◗✉❛❡❣❤❡❜❡✉r ❊✳✱ ❚❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ♣r♦❜❧❡♠ ✉♥❞❡r ✜♥✐t❡ ❛❞❞✐t✐✈✐t②✱ ❏♦✉r♥❛❧ ♦❢ ❚❤❡♦r❡t✐❝❛❧ Pr♦❜❛❜✐❧✐t② ✷✵✭✸✮ ✷✵✵✼ ♣♣ ✻✻✸✲✻✾✸✳
SLIDE 87
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
❚❤❡♦r❡♠ ❚❤❡r❡ ❡①✐sts ❛ st❛t❡ s : ❈ → [✵, ✶] s✉❝❤ t❤❛t s(♣❦) = ♠❦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡q✉❡♥❝❡ {♠❦} s❛t✐s✜❡s t❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ❝♦♥❞✐t✐♦♥✳ ❖✉t❧✐♥❡ ♦❢ t❤❡ ♣r♦♦❢✳ ❇② ❑r♦✉♣❛✲P❛♥t✐ r❛♣r❡s❡♥t❛t✐♦♥ ❢♦r st❛t❡s ✇❡ ❣❡t s(❢ ) = ✶
✵ ❢❞µ✳ ❚❤❡♥ ✐t
❢♦❧❧♦✇s ❜② ❝❛❧❝✉❧❛t✐♦♥s✳ ❖♥ t❤❡ ♦t❤❡r ❞✐r❡❝t✐♦♥ ✐s ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢
▼✐r❛♥❞❛ ❊✳✱ ❞❡ ❈♦♦♠❛♥ ●✳✱ ◗✉❛❡❣❤❡❜❡✉r ❊✳✱ ❚❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ♣r♦❜❧❡♠ ✉♥❞❡r ✜♥✐t❡ ❛❞❞✐t✐✈✐t②✱ ❏♦✉r♥❛❧ ♦❢ ❚❤❡♦r❡t✐❝❛❧ Pr♦❜❛❜✐❧✐t② ✷✵✭✸✮ ✷✵✵✼ ♣♣ ✻✻✸✲✻✾✸✳
SLIDE 88
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
▼♦♠❡♥t Pr♦❜❧❡♠
❈♦r♦❧❧❛r② ❚❤❡r❡ ❡①✐sts ❛ st❛t❡ s : ❋❘ → [✵, ✶] s✉❝❤ t❤❛t s(♣❦) = ♠❦ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ s❡q✉❡♥❝❡ {♠❦} s❛t✐s✜❡s t❤❡ ❍❛✉s❞♦r✛ ♠♦♠❡♥t ❝♦♥❞✐t✐♦♥✳
SLIDE 89
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❈♦♥❝❧✉s✐♦♥s
❞❡✜♥✐t✐♦♥ ❛♥❞ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❢♦r ❢▼❱ ✲❛❧❣❡❜r❛s❀ ❞❡s❝r✐♣t✐♦♥ ♦❢ s♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s❀ ❢♦r ♥ ≤ ✷ ❛ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❢r❡❡ ❢♦r♠❛❧❧② r❡❛❧ ❢▼❱ ✲❛❧❣❡❜r❛ ✇✐t❤ t✇♦ ❣❡♥❡r❛t♦rs✱ r❡❧②✐♥❣ ♦♥ ❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡❀ ❍❛✉s❞♦r✛ ▼♦♠❡♥t ♣r♦❜❧❡♠ ✐♥ t❤❡ ▼❱ ✲❛❧❣❡❜r❛✐❝ s❡tt✐♥❣✳ ❋✉t✉r❡ ❞❡✈❡❧♦♣♠❡♥ts✿ r❡❧❛t✐♦♥ ✇✐t❤ ✜♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱ ✲❛❧❣❡❜r❛s✱ st✉❞② ♦❢ ♦rt❤♦♠♦r♣❤✐s♠s ❢♦r ❢▼❱ ✲❛❧❣❡❜r❛s✱ st✉❞② ♦❢ t❤❡ s♣❛❝❡ ♦❢ ♠✐♥✐♠❛❧ ♣r✐♠❡ ✐❞❡❛❧s✳✳✳
SLIDE 90
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❈♦♥❝❧✉s✐♦♥s
❞❡✜♥✐t✐♦♥ ❛♥❞ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❢♦r ❢▼❱ ✲❛❧❣❡❜r❛s❀ ❞❡s❝r✐♣t✐♦♥ ♦❢ s♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s❀ ❢♦r ♥ ≤ ✷ ❛ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❢r❡❡ ❢♦r♠❛❧❧② r❡❛❧ ❢▼❱ ✲❛❧❣❡❜r❛ ✇✐t❤ t✇♦ ❣❡♥❡r❛t♦rs✱ r❡❧②✐♥❣ ♦♥ ❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡❀ ❍❛✉s❞♦r✛ ▼♦♠❡♥t ♣r♦❜❧❡♠ ✐♥ t❤❡ ▼❱ ✲❛❧❣❡❜r❛✐❝ s❡tt✐♥❣✳ ❋✉t✉r❡ ❞❡✈❡❧♦♣♠❡♥ts✿ r❡❧❛t✐♦♥ ✇✐t❤ ✜♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱ ✲❛❧❣❡❜r❛s✱ st✉❞② ♦❢ ♦rt❤♦♠♦r♣❤✐s♠s ❢♦r ❢▼❱ ✲❛❧❣❡❜r❛s✱ st✉❞② ♦❢ t❤❡ s♣❛❝❡ ♦❢ ♠✐♥✐♠❛❧ ♣r✐♠❡ ✐❞❡❛❧s✳✳✳
SLIDE 91
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❈♦♥❝❧✉s✐♦♥s
❞❡✜♥✐t✐♦♥ ❛♥❞ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❢♦r ❢▼❱ ✲❛❧❣❡❜r❛s❀ ❞❡s❝r✐♣t✐♦♥ ♦❢ s♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s❀ ❢♦r ♥ ≤ ✷ ❛ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❢r❡❡ ❢♦r♠❛❧❧② r❡❛❧ ❢▼❱ ✲❛❧❣❡❜r❛ ✇✐t❤ t✇♦ ❣❡♥❡r❛t♦rs✱ r❡❧②✐♥❣ ♦♥ ❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡❀ ❍❛✉s❞♦r✛ ▼♦♠❡♥t ♣r♦❜❧❡♠ ✐♥ t❤❡ ▼❱ ✲❛❧❣❡❜r❛✐❝ s❡tt✐♥❣✳ ❋✉t✉r❡ ❞❡✈❡❧♦♣♠❡♥ts✿ r❡❧❛t✐♦♥ ✇✐t❤ ✜♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱ ✲❛❧❣❡❜r❛s✱ st✉❞② ♦❢ ♦rt❤♦♠♦r♣❤✐s♠s ❢♦r ❢▼❱ ✲❛❧❣❡❜r❛s✱ st✉❞② ♦❢ t❤❡ s♣❛❝❡ ♦❢ ♠✐♥✐♠❛❧ ♣r✐♠❡ ✐❞❡❛❧s✳✳✳
SLIDE 92
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠
❈♦♥❝❧✉s✐♦♥s
❞❡✜♥✐t✐♦♥ ❛♥❞ ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡ ❢♦r ❢▼❱ ✲❛❧❣❡❜r❛s❀ ❞❡s❝r✐♣t✐♦♥ ♦❢ s♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❢▼❱ ✲❛❧❣❡❜r❛s❀ ❢♦r ♥ ≤ ✷ ❛ ❞✐✛❡r❡♥t ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❢r❡❡ ❢♦r♠❛❧❧② r❡❛❧ ❢▼❱ ✲❛❧❣❡❜r❛ ✇✐t❤ t✇♦ ❣❡♥❡r❛t♦rs✱ r❡❧②✐♥❣ ♦♥ ❇✐r❦❤♦✛✲P✐❡r❝❡ ❝♦♥❥❡❝t✉r❡❀ ❍❛✉s❞♦r✛ ▼♦♠❡♥t ♣r♦❜❧❡♠ ✐♥ t❤❡ ▼❱ ✲❛❧❣❡❜r❛✐❝ s❡tt✐♥❣✳ ❋✉t✉r❡ ❞❡✈❡❧♦♣♠❡♥ts✿ r❡❧❛t✐♦♥ ✇✐t❤ ✜♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱ ✲❛❧❣❡❜r❛s✱ st✉❞② ♦❢ ♦rt❤♦♠♦r♣❤✐s♠s ❢♦r ❢▼❱ ✲❛❧❣❡❜r❛s✱ st✉❞② ♦❢ t❤❡ s♣❛❝❡ ♦❢ ♠✐♥✐♠❛❧ ♣r✐♠❡ ✐❞❡❛❧s✳✳✳
SLIDE 93
❢▼❱ ✲❛❧❣❡❜r❛s ❛♥❞ ♣✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s P✐❡❝❡✇✐s❡ ♣♦❧②♥♦♠✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠♦♠❡♥t ♣r♦❜❧❡♠