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t Pt Prts r Prsrr - - PowerPoint PPT Presentation

Feb 07, 2024 184 likes •952 views

t Pt Prts r Prsrr rtt r sr s st

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SLIDE 1

❙✉❝❝✐♥❝t P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s ❢♦r Pr❡s❜✉r❣❡r ❆r✐t❤♠❡t✐❝

▼✐❝❤❛❡❧ ❇❧♦♥❞✐♥✱ ❏❛✈✐❡r ❊s♣❛r③❛✱ ❇❧❛✐s❡ ●❡♥❡st✱ ▼❛rt✐♥ ❍❡❧❢r✐❝❤ ❛♥❞ ❙t❡❢❛♥ ❏❛❛①

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SLIDE 2

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

❋♦r♠❛❧ ♠♦❞❡❧ ♦❢ ❞✐str✐❜✉t❡❞ ❝♦♠♣✉t❛t✐♦♥ ❜② ❝♦❧❧❡❝t✐♦♥s ♦❢ ✐❞❡♥t✐❝❛❧✱ ✜♥✐t❡✲st❛t❡✱ ❛♥❞ ♠♦❜✐❧❡ ❛❣❡♥ts ❧✐❦❡ ❛❞✲❤♦❝ ♥❡t✇♦r❦s ♦❢ ♠♦❜✐❧❡ s❡♥s♦rs ♣❡♦♣❧❡ ✐♥ s♦❝✐❛❧ ♥❡t✇♦r❦s ✏s♦✉♣s✑ ♦❢ ♠♦❧❡❝✉❧❡s

✭❈❤❡♠✐❝❛❧ ❘❡❛❝t✐♦♥ ◆❡t✇♦r❦s✮

✳ ✳ ✳

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SLIDE 3

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

❋♦r♠❛❧ ♠♦❞❡❧ ♦❢ ❞✐str✐❜✉t❡❞ ❝♦♠♣✉t❛t✐♦♥ ❜② ❝♦❧❧❡❝t✐♦♥s ♦❢ ✐❞❡♥t✐❝❛❧✱ ✜♥✐t❡✲st❛t❡✱ ❛♥❞ ♠♦❜✐❧❡ ❛❣❡♥ts ❧✐❦❡ ❛❞✲❤♦❝ ♥❡t✇♦r❦s ♦❢ ♠♦❜✐❧❡ s❡♥s♦rs ♣❡♦♣❧❡ ✐♥ s♦❝✐❛❧ ♥❡t✇♦r❦s ✏s♦✉♣s✑ ♦❢ ♠♦❧❡❝✉❧❡s

✭❈❤❡♠✐❝❛❧ ❘❡❛❝t✐♦♥ ◆❡t✇♦r❦s✮

✳ ✳ ✳

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SLIDE 4

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

❋♦r♠❛❧ ♠♦❞❡❧ ♦❢ ❞✐str✐❜✉t❡❞ ❝♦♠♣✉t❛t✐♦♥ ❜② ❝♦❧❧❡❝t✐♦♥s ♦❢ ✐❞❡♥t✐❝❛❧✱ ✜♥✐t❡✲st❛t❡✱ ❛♥❞ ♠♦❜✐❧❡ ❛❣❡♥ts ❧✐❦❡ ❛❞✲❤♦❝ ♥❡t✇♦r❦s ♦❢ ♠♦❜✐❧❡ s❡♥s♦rs ♣❡♦♣❧❡ ✐♥ s♦❝✐❛❧ ♥❡t✇♦r❦s ✏s♦✉♣s✑ ♦❢ ♠♦❧❡❝✉❧❡s

✭❈❤❡♠✐❝❛❧ ❘❡❛❝t✐♦♥ ◆❡t✇♦r❦s✮

✳ ✳ ✳

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SLIDE 5

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

❋♦r♠❛❧ ♠♦❞❡❧ ♦❢ ❞✐str✐❜✉t❡❞ ❝♦♠♣✉t❛t✐♦♥ ❜② ❝♦❧❧❡❝t✐♦♥s ♦❢ ✐❞❡♥t✐❝❛❧✱ ✜♥✐t❡✲st❛t❡✱ ❛♥❞ ♠♦❜✐❧❡ ❛❣❡♥ts ❧✐❦❡ ❛❞✲❤♦❝ ♥❡t✇♦r❦s ♦❢ ♠♦❜✐❧❡ s❡♥s♦rs ♣❡♦♣❧❡ ✐♥ s♦❝✐❛❧ ♥❡t✇♦r❦s ✏s♦✉♣s✑ ♦❢ ♠♦❧❡❝✉❧❡s

✭❈❤❡♠✐❝❛❧ ❘❡❛❝t✐♦♥ ◆❡t✇♦r❦s✮

✳ ✳ ✳

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SLIDE 6

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

❋♦r♠❛❧ ♠♦❞❡❧ ♦❢ ❞✐str✐❜✉t❡❞ ❝♦♠♣✉t❛t✐♦♥ ❜② ❝♦❧❧❡❝t✐♦♥s ♦❢ ✐❞❡♥t✐❝❛❧✱ ✜♥✐t❡✲st❛t❡✱ ❛♥❞ ♠♦❜✐❧❡ ❛❣❡♥ts ❧✐❦❡ ❛❞✲❤♦❝ ♥❡t✇♦r❦s ♦❢ ♠♦❜✐❧❡ s❡♥s♦rs ♣❡♦♣❧❡ ✐♥ s♦❝✐❛❧ ♥❡t✇♦r❦s ✏s♦✉♣s✑ ♦❢ ♠♦❧❡❝✉❧❡s

✭❈❤❡♠✐❝❛❧ ❘❡❛❝t✐♦♥ ◆❡t✇♦r❦s✮

✳ ✳ ✳and birds!

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SLIDE 7

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡ ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s ❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s ❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡ ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥

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SLIDE 8

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s

❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s ❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡ ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥

slide-9
SLIDE 9

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s

❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s ❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡ ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥

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SLIDE 10

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s

❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s ❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡ ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥

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SLIDE 11

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s

❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s ❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡ ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥

slide-12
SLIDE 12

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s
  • ❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s

❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡ ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥

slide-13
SLIDE 13

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s
  • ❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s

❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡ ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥

slide-14
SLIDE 14

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s
  • ❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s
  • ❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡

❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥

slide-15
SLIDE 15

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s
  • ❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s
  • ❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡
  • ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥
slide-16
SLIDE 16

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s
  • ❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s
  • ❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡
  • ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥
slide-17
SLIDE 17

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

  • ❛♥♦♥②♠♦✉s ♠♦❜✐❧❡ ❛❣❡♥ts ✇✐t❤ ✈❡r② ❢❡✇ r❡s♦✉r❝❡s
  • ❛❣❡♥ts ❝❤❛♥❣❡ st❛t❡s ✈✐❛ r❛♥❞♦♠ ♣❛✐r✇✐s❡ ✐♥t❡r❛❝t✐♦♥s
  • ❡❛❝❤ ❛❣❡♥t ❤❛s ♦♣✐♥✐♦♥ tr✉❡✴❢❛❧s❡
  • ❝♦♠♣✉t❡s ❜② st❛❜✐❧✐③✐♥❣ ❛❣❡♥ts t♦ s♦♠❡ ♦♣✐♥✐♦♥
slide-18
SLIDE 18

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡ ♦❢ ✵ ✶ ✷ ✸ ✹ ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥ st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵ ✵ ✐❢ ✹ ✹ ✹ ✐❢ ✹

slide-19
SLIDE 19

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵ ✵ ✐❢ ✹ ✹ ✹ ✐❢ ✹

✶ ✶ ✶ ✵ ✶

slide-20
SLIDE 20

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵ ✵ ✐❢ ✹ ✹ ✹ ✐❢ ✹

✶ ✶ ✶ ✵ ✶

slide-21
SLIDE 21

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✶ ✶ ✶ ✵ ✶

slide-22
SLIDE 22

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✷ ✶ ✶ ✵ ✵

slide-23
SLIDE 23

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✷ ✶ ✶ ✵ ✵

slide-24
SLIDE 24

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✷ ✶ ✶ ✵ ✵

slide-25
SLIDE 25

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✷ ✶ ✶ ✵ ✵

slide-26
SLIDE 26

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✷ ✷ ✵ ✵ ✵

slide-27
SLIDE 27

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✷ ✷ ✵ ✵ ✵

slide-28
SLIDE 28

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✹ ✹ ✵ ✵ ✵

slide-29
SLIDE 29

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✹ ✹ ✵ ✵ ✵

slide-30
SLIDE 30

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✹ ✹ ✵ ✹ ✵

slide-31
SLIDE 31

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✹ ✹ ✵ ✹ ✵

slide-32
SLIDE 32

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✹ ✹ ✵ ✹ ✹

slide-33
SLIDE 33

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✹ ✹ ✵ ✹ ✹

slide-34
SLIDE 34

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✹ ✹ ✹ ✹ ✹

slide-35
SLIDE 35

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least 4 sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, ✷, ✸, ✹}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < ✹

  • (m, n) → (✹, ✹)

✐❢ m + n ≥ ✹

✹ ✹ ✹ ✹ ✹

slide-36
SLIDE 36

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❆♥ ❊①❛♠♣❧❡

Are there at least c sick birds?

  • ❊❛❝❤ ❜✐r❞ ✐s ✐♥ ❛ st❛t❡

♦❢ {✵, ✶, . . . , c}

  • ■♥✐t✐❛❧❧②✱ s✐❝❦ ❜✐r❞s ✐♥

st❛t❡ ✶✱ ❤❡❛❧t❤② ❜✐r❞s ✐♥ st❛t❡ ✵

  • (m, n) → (m + n, ✵)

✐❢ m + n < c

  • (m, n) → (c, c)

✐❢ m + n ≥ c

✹ ✹ ✹ ✹

slide-37
SLIDE 37

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❋♦r♠❛❧ ▼♦❞❡❧

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

✵ ✶ ✷ ✸ ✹

  • ❙t❛t❡s✿

✜♥✐t❡ s❡t Q

  • ❖♣✐♥✐♦♥s✿

O : Q → {✵, ✶}

  • ■♥✐t✐❛❧ st❛t❡s✿

I ⊆ Q

  • ❚r❛♥s✐t✐♦♥s✿

T ⊆ Q✷ × Q✷ ❈♦♥✜❣✉r❛t✐♦♥s✿ ■♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✿

slide-38
SLIDE 38

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❋♦r♠❛❧ ▼♦❞❡❧

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

✵ ✶ ✷ ✸ ✹

  • ❙t❛t❡s✿

✜♥✐t❡ s❡t Q

  • ❖♣✐♥✐♦♥s✿

O : Q → {✵, ✶}

  • ■♥✐t✐❛❧ st❛t❡s✿

I ⊆ Q

  • ❚r❛♥s✐t✐♦♥s✿

T ⊆ Q✷ × Q✷ ❈♦♥✜❣✉r❛t✐♦♥s✿ ■♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✿

slide-39
SLIDE 39

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❋♦r♠❛❧ ▼♦❞❡❧

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

✵ ✶ ✷ ✸ ✹

  • ❙t❛t❡s✿

✜♥✐t❡ s❡t Q

  • ❖♣✐♥✐♦♥s✿

O : Q → {✵, ✶}

  • ■♥✐t✐❛❧ st❛t❡s✿

I ⊆ Q

  • ❚r❛♥s✐t✐♦♥s✿

T ⊆ Q✷ × Q✷ ❈♦♥✜❣✉r❛t✐♦♥s✿ ■♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✿

slide-40
SLIDE 40

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❋♦r♠❛❧ ▼♦❞❡❧

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

✵ ✶ ✷ ✸ ✹

✶ ✷ ✸ ✵

  • ❙t❛t❡s✿

✜♥✐t❡ s❡t Q

  • ❖♣✐♥✐♦♥s✿

O : Q → {✵, ✶}

  • ■♥✐t✐❛❧ st❛t❡s✿

I ⊆ Q

  • ❚r❛♥s✐t✐♦♥s✿

T ⊆ Q✷ × Q✷ ❈♦♥✜❣✉r❛t✐♦♥s✿ ■♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✿

slide-41
SLIDE 41

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❋♦r♠❛❧ ▼♦❞❡❧

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

✵ ✶ ✷ ✸ ✹ ✭

✵

✷

✶

✺

✷

✶

✸

✸

✹

✶ ✮

  • ❙t❛t❡s✿

✜♥✐t❡ s❡t Q

  • ❖♣✐♥✐♦♥s✿

O : Q → {✵, ✶}

  • ■♥✐t✐❛❧ st❛t❡s✿

I ⊆ Q

  • ❚r❛♥s✐t✐♦♥s✿

T ⊆ Q✷ × Q✷

  • ❈♦♥✜❣✉r❛t✐♦♥s✿

Q → N ■♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✿

slide-42
SLIDE 42

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❋♦r♠❛❧ ▼♦❞❡❧

❆♥❣❧✉✐♥ ❡t ❛❧✳ P❖❉❈✬✵✹

✵ ✶ ✷ ✸ ✹ ✭

✵

✶

✶

✹

✷

✵

✸

✵

✹

✵ ✮

  • ❙t❛t❡s✿

✜♥✐t❡ s❡t Q

  • ❖♣✐♥✐♦♥s✿

O : Q → {✵, ✶}

  • ■♥✐t✐❛❧ st❛t❡s✿

I ⊆ Q

  • ❚r❛♥s✐t✐♦♥s✿

T ⊆ Q✷ × Q✷

  • ❈♦♥✜❣✉r❛t✐♦♥s✿

Q → N

  • ■♥✐t✐❛❧ ❝♦♥✜❣✉r❛t✐♦♥s✿

I → N

slide-43
SLIDE 43

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❈♦♠♣✉t✐♥❣ Pr❡❞✐❝❛t❡s

❯♥❞❡r❧②✐♥❣ ▼❛r❦♦✈ ❝❤❛✐♥ ❢♦r (✶, ✹, ✵, ✵, ✵)✿

✭♣❛✐rs ♦❢ ❛❣❡♥ts ❛r❡ ♣✐❝❦❡❞ ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✮

✭✶✱✹✱✵✱✵✱✵✮ ✭✷✱✷✱✶✱✵✱✵✮ ✭✸✱✵✱✷✱✵✱✵✮ ✭✸✱✶✱✵✱✶✱✵✮ ✭✸✱✵✱✵✱✵✱✷✮ ✭✷✱✵✱✵✱✵✱✸✮ ✭✶✱✵✱✵✱✵✱✹✮ ✭✵✱✵✱✵✱✵✱✺✮

✻ ✶✵ ✶ ✶✵ ✶ ✶✵ ✻ ✶✵ ✻ ✶✵ ✹ ✶✵ ✶✵ ✶✵ ✷ ✶✵ ✶ ✶✵ ✹ ✶✵ ✼ ✶✵ ✾ ✶✵ ✾ ✶✵ ✹ ✶✵ ✹ ✶✵ ✻ ✶✵

slide-44
SLIDE 44

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❈♦♠♣✉t✐♥❣ Pr❡❞✐❝❛t❡s

Pr♦t♦❝♦❧ ❝♦♠♣✉t❡s ϕ: ■♥✐t❈ → {✵, ✶}✿ ❢♦r ❡✈❡r② C ∈ ■♥✐t❈ ✱ t❤❡ r✉♥s st❛rt✐♥❣ ❛t C r❡❛❝❤ st❛❜❧❡ ❝♦♥s❡♥s✉s ϕ(C) ✇✐t❤ ♣r♦❜❛❜✐❧✐t② ✶✳

❈✵

✵ ✵

❈✶

✶ ✶

❈✷

✶

✳ ✳ ✳

Pr♦t♦❝♦❧ ❝♦♠♣✉t❡s ϕ(C✵) = ✵, ϕ(C✶) = ✶, ϕ(C✷) = ✶, . . .

slide-45
SLIDE 45

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❊①♣r❡ss✐✈❡ P♦✇❡r

❆♥❣❧✉✐♥ ❡t ❛❧✳ ✭❉✐st✳ ❈♦♠♣✳✬✵✼✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s ❝♦♠♣✉t❡ ♣r❡❝✐s❡❧② t❤❡ ♣r❡❞✐❝❛t❡s ❞❡✜♥❛❜❧❡ ✐♥ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝✱ ✐✳❡✳ ❋❖✭N✱ +✱ <✮

Pr♦♦❢✿ PPs ❝♦♠♣✉t❡ ❛❧❧ Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡s

❙✐♥❝❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ❤❛s q✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥✱ ✐t s✉✣❝❡s t♦ ❡①❤✐❜✐t PPs ❢♦r✿ t❤r❡s❤♦❧❞ ❛♥❞ ♠♦❞✉❧♦ ♣r❡❞✐❝❛t❡s

✶ ✶ ✶ ✶

t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦t♦❝♦❧s

slide-46
SLIDE 46

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❊①♣r❡ss✐✈❡ P♦✇❡r

❆♥❣❧✉✐♥ ❡t ❛❧✳ ✭❉✐st✳ ❈♦♠♣✳✬✵✼✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s ❝♦♠♣✉t❡ ♣r❡❝✐s❡❧② t❤❡ ♣r❡❞✐❝❛t❡s ❞❡✜♥❛❜❧❡ ✐♥ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝✱ ✐✳❡✳ ❋❖✭N✱ +✱ <✮

y > ✷x ∧ ∃i ∈ N : x = ✸i + ✷

Pr♦♦❢✿ PPs ❝♦♠♣✉t❡ ❛❧❧ Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡s

❙✐♥❝❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ❤❛s q✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥✱ ✐t s✉✣❝❡s t♦ ❡①❤✐❜✐t PPs ❢♦r✿ t❤r❡s❤♦❧❞ ❛♥❞ ♠♦❞✉❧♦ ♣r❡❞✐❝❛t❡s

✶ ✶ ✶ ✶

t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦t♦❝♦❧s

slide-47
SLIDE 47

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❊①♣r❡ss✐✈❡ P♦✇❡r

❆♥❣❧✉✐♥ ❡t ❛❧✳ ✭❉✐st✳ ❈♦♠♣✳✬✵✼✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s ❝♦♠♣✉t❡ ♣r❡❝✐s❡❧② t❤❡ ♣r❡❞✐❝❛t❡s ❞❡✜♥❛❜❧❡ ✐♥ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝✱ ✐✳❡✳ ❋❖✭N✱ +✱ <✮

Pr♦♦❢✿ PPs ❝♦♠♣✉t❡ ❛❧❧ Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡s

❙✐♥❝❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ❤❛s q✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥✱ ✐t s✉✣❝❡s t♦ ❡①❤✐❜✐t PPs ❢♦r✿ t❤r❡s❤♦❧❞ ❛♥❞ ♠♦❞✉❧♦ ♣r❡❞✐❝❛t❡s

✶ ✶ ✶ ✶

t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦t♦❝♦❧s

slide-48
SLIDE 48

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❊①♣r❡ss✐✈❡ P♦✇❡r

❆♥❣❧✉✐♥ ❡t ❛❧✳ ✭❉✐st✳ ❈♦♠♣✳✬✵✼✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s ❝♦♠♣✉t❡ ♣r❡❝✐s❡❧② t❤❡ ♣r❡❞✐❝❛t❡s ❞❡✜♥❛❜❧❡ ✐♥ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝✱ ✐✳❡✳ ❋❖✭N✱ +✱ <✮

Pr♦♦❢✿ PPs ❝♦♠♣✉t❡ ❛❧❧ Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡s

❙✐♥❝❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ❤❛s q✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥✱ ✐t s✉✣❝❡s t♦ ❡①❤✐❜✐t PPs ❢♦r✿

  • t❤r❡s❤♦❧❞ ❛♥❞ ♠♦❞✉❧♦ ♣r❡❞✐❝❛t❡s

a✶x✶ + · · · + anxn ≤ b a✶x✶ + · · · + anxm ≡c b t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦t♦❝♦❧s

slide-49
SLIDE 49

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❊①♣r❡ss✐✈❡ P♦✇❡r

❆♥❣❧✉✐♥ ❡t ❛❧✳ ✭❉✐st✳ ❈♦♠♣✳✬✵✼✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s ❝♦♠♣✉t❡ ♣r❡❝✐s❡❧② t❤❡ ♣r❡❞✐❝❛t❡s ❞❡✜♥❛❜❧❡ ✐♥ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝✱ ✐✳❡✳ ❋❖✭N✱ +✱ <✮

Pr♦♦❢✿ PPs ❝♦♠♣✉t❡ ❛❧❧ Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡s

❙✐♥❝❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ❤❛s q✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥✱ ✐t s✉✣❝❡s t♦ ❡①❤✐❜✐t PPs ❢♦r✿

  • t❤r❡s❤♦❧❞ ❛♥❞ ♠♦❞✉❧♦ ♣r❡❞✐❝❛t❡s

a✶x✶ + · · · + anxn ≤ b a✶x✶ + · · · + anxm ≡c b

  • t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦t♦❝♦❧s
slide-50
SLIDE 50

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❊①♣r❡ss✐✈❡ P♦✇❡r

❆♥❣❧✉✐♥ ❡t ❛❧✳ ✭❉✐st✳ ❈♦♠♣✳✬✵✼✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧s ❝♦♠♣✉t❡ ♣r❡❝✐s❡❧② t❤❡ ♣r❡❞✐❝❛t❡s ❞❡✜♥❛❜❧❡ ✐♥ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝✱ ✐✳❡✳ ❋❖✭N✱ +✱ <✮

Pr♦♦❢✿ PPs ❝♦♠♣✉t❡ ❛❧❧ Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡s

❙✐♥❝❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ❤❛s q✉❛♥t✐✜❡r ❡❧✐♠✐♥❛t✐♦♥✱ ✐t s✉✣❝❡s t♦ ❡①❤✐❜✐t PPs ❢♦r✿

  • t❤r❡s❤♦❧❞ ❛♥❞ ♠♦❞✉❧♦ ♣r❡❞✐❝❛t❡s

a✶x✶ + · · · + anxn ≤ b a✶x✶ + · · · + anxm ≡c b

  • t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ❛♥❞ ♥❡❣❛t✐♦♥ ♦❢ ♣r♦t♦❝♦❧s
  • synthesis procedure!
slide-51
SLIDE 51

❙②♥t❤❡s✐s

❜② ❆♥❣❧✉✐♥ ❡t ❛❧✳

Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡ ϕ ✶✳ ❊❧✐♠✐♥❛t❡ q✉❛♥t✐✜❡rs ✷✳ ●❡♥❡r❛t❡ ❜❛s❡ ♣r♦t♦❝♦❧s✿

r❡♠❛✐♥❞❡r✿

✶

t❤r❡s❤♦❧❞✿

✶

❛❝❝✉♠✉❧❛t❡ ✈❛❧✉❡ ✉♣ t♦ s♦♠❡ ✭❧✐❦❡ ✐♥ ✏s✐❝❦ ❜✐r❞s✑ ❡①❛♠♣❧❡✮

✵

✶

✳✳✳

✸✳ ❈♦♠❜✐♥❡ ❜❛s❡ ♣r♦t♦❝♦❧s✿

♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥ ✭r❡♠❡♠❜❡r ❜♦t❤ st❛t❡s✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ❢♦r

✷

✸ ✷

✶ ✷ ✸ ✳✳✳

✷ ✶ ✷ ✸

✳✳✳

✸ ✷

✶

✶

✶

✷

✳✳✳

✷

✶

✷

✷

✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳

✷

✸ ✷

slide-52
SLIDE 52

❙②♥t❤❡s✐s

❜② ❆♥❣❧✉✐♥ ❡t ❛❧✳

Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡ ϕ ✶✳ ❊❧✐♠✐♥❛t❡ q✉❛♥t✐✜❡rs ✷✳ ●❡♥❡r❛t❡ ❜❛s❡ ♣r♦t♦❝♦❧s✿

r❡♠❛✐♥❞❡r✿

✶

t❤r❡s❤♦❧❞✿

✶

❛❝❝✉♠✉❧❛t❡ ✈❛❧✉❡ ✉♣ t♦ s♦♠❡ ✭❧✐❦❡ ✐♥ ✏s✐❝❦ ❜✐r❞s✑ ❡①❛♠♣❧❡✮

✵

✶

✳✳✳

✸✳ ❈♦♠❜✐♥❡ ❜❛s❡ ♣r♦t♦❝♦❧s✿

♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥ ✭r❡♠❡♠❜❡r ❜♦t❤ st❛t❡s✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ❢♦r

ϕ ↓ (✷x > y) ∧ (x ≡✸ ✷)

✶ ✷ ✸ ✳✳✳

✷ ✶ ✷ ✸

✳✳✳

✸ ✷

✶

✶

✶

✷

✳✳✳

✷

✶

✷

✷

✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳

✷

✸ ✷

slide-53
SLIDE 53

❙②♥t❤❡s✐s

❜② ❆♥❣❧✉✐♥ ❡t ❛❧✳

Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡ ϕ ✶✳ ❊❧✐♠✐♥❛t❡ q✉❛♥t✐✜❡rs ✷✳ ●❡♥❡r❛t❡ ❜❛s❡ ♣r♦t♦❝♦❧s✿

  • r❡♠❛✐♥❞❡r✿ k

i=✶ aixi ≡m b

  • t❤r❡s❤♦❧❞✿ k

i=✶ aixi ≤ b

→ ❛❝❝✉♠✉❧❛t❡ ✈❛❧✉❡ ✉♣ t♦ s♦♠❡ c ✭❧✐❦❡ ✐♥ ✏s✐❝❦ ❜✐r❞s✑ ❡①❛♠♣❧❡✮

✵

✶

✳✳✳

c

✸✳ ❈♦♠❜✐♥❡ ❜❛s❡ ♣r♦t♦❝♦❧s✿

♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥ ✭r❡♠❡♠❜❡r ❜♦t❤ st❛t❡s✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ❢♦r

ϕ ↓ (✷x > y) ∧ (x ≡✸ ✷) ↓

✶ ✷ ✸ ✳✳✳

✷x > y ✶ ✷ ✸

✳✳✳

x ≡✸ ✷ ✶

✶

✶

✷

✳✳✳

✷

✶

✷

✷

✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳

✷

✸ ✷

slide-54
SLIDE 54

❙②♥t❤❡s✐s

❜② ❆♥❣❧✉✐♥ ❡t ❛❧✳

Pr❡s❜✉r❣❡r ♣r❡❞✐❝❛t❡ ϕ ✶✳ ❊❧✐♠✐♥❛t❡ q✉❛♥t✐✜❡rs ✷✳ ●❡♥❡r❛t❡ ❜❛s❡ ♣r♦t♦❝♦❧s✿

  • r❡♠❛✐♥❞❡r✿ k

i=✶ aixi ≡m b

  • t❤r❡s❤♦❧❞✿ k

i=✶ aixi ≤ b

→ ❛❝❝✉♠✉❧❛t❡ ✈❛❧✉❡ ✉♣ t♦ s♦♠❡ c ✭❧✐❦❡ ✐♥ ✏s✐❝❦ ❜✐r❞s✑ ❡①❛♠♣❧❡✮

✵

✶

✳✳✳

c

✸✳ ❈♦♠❜✐♥❡ ❜❛s❡ ♣r♦t♦❝♦❧s✿

→ ♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥ ✭r❡♠❡♠❜❡r ❜♦t❤ st❛t❡s✮

P♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ❢♦r ϕ

ϕ ↓ (✷x > y) ∧ (x ≡✸ ✷) ↓

✶ ✷ ✸ ✳✳✳

✷x > y ✶ ✷ ✸

✳✳✳

x ≡✸ ✷

×

↓

✶

✶

✶

✷

✳✳✳

✷

✶

✷

✷

✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳

(✷x > y) ∧ (x ≡✸ ✷)

slide-55
SLIDE 55

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❙✐③❡

How large are the resulting protocols?

❙✐③❡ ♦❢ ♣♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧✿ ♥✉♠❜❡r ♦❢ st❛t❡s ❙✐③❡ ♦❢ ✭q✉❛♥t✐✜❡r✲❢r❡❡✮ ♣r❡❞✐❝❛t❡ ✿ ❧❡♥❣t❤ ♦❢ ❜✐♥❛r② ❡♥❝♦❞✐♥❣ ❡✳❣✳ ✸✵✵✵ ✹✷ ❤❛s s✐③❡ ✶✵

❚❤❡♦r❡♠ ✭❆♥❣❧✉✐♥ ❡t ❛❧✳✮

  • ✐✈❡♥ ❛ ❢♦r♠✉❧❛

♦❢ q✉❛♥t✐✜❡r✲❢r❡❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ✇✐t❤ r❡♠❛✐♥❞❡r ♣r❡❞✐❝❛t❡s✱ t❤❡r❡ ✐s ❛ ♣r♦❝❡❞✉r❡ t❤❛t ❝♦♥str✉❝ts ❛ ♣♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ✇✐t❤ ✷ st❛t❡s t❤❛t ❝♦♠♣✉t❡s ✳

slide-56
SLIDE 56

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❙✐③❡

How large are the resulting protocols?

❙✐③❡ ♦❢ ♣♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧✿ ♥✉♠❜❡r ♦❢ st❛t❡s ❙✐③❡ |ϕ| ♦❢ ✭q✉❛♥t✐✜❡r✲❢r❡❡✮ ♣r❡❞✐❝❛t❡ ϕ✿ ❧❡♥❣t❤ ♦❢ ❜✐♥❛r② ❡♥❝♦❞✐♥❣ ❡✳❣✳ ✸✵✵✵x > y + ✹✷ ❤❛s s✐③❡ ✶✵

❚❤❡♦r❡♠ ✭❆♥❣❧✉✐♥ ❡t ❛❧✳✮

  • ✐✈❡♥ ❛ ❢♦r♠✉❧❛

♦❢ q✉❛♥t✐✜❡r✲❢r❡❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ✇✐t❤ r❡♠❛✐♥❞❡r ♣r❡❞✐❝❛t❡s✱ t❤❡r❡ ✐s ❛ ♣r♦❝❡❞✉r❡ t❤❛t ❝♦♥str✉❝ts ❛ ♣♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ✇✐t❤ ✷ st❛t❡s t❤❛t ❝♦♠♣✉t❡s ✳

slide-57
SLIDE 57

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❙✐③❡

How large are the resulting protocols?

❙✐③❡ ♦❢ ♣♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧✿ ♥✉♠❜❡r ♦❢ st❛t❡s ❙✐③❡ |ϕ| ♦❢ ✭q✉❛♥t✐✜❡r✲❢r❡❡✮ ♣r❡❞✐❝❛t❡ ϕ✿ ❧❡♥❣t❤ ♦❢ ❜✐♥❛r② ❡♥❝♦❞✐♥❣ ❡✳❣✳ ✸✵✵✵x > y + ✹✷ ❤❛s s✐③❡ ✶✵

❚❤❡♦r❡♠ ✭❆♥❣❧✉✐♥ ❡t ❛❧✳✮

  • ✐✈❡♥ ❛ ❢♦r♠✉❧❛ ϕ ♦❢ q✉❛♥t✐✜❡r✲❢r❡❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝

✇✐t❤ r❡♠❛✐♥❞❡r ♣r❡❞✐❝❛t❡s✱ t❤❡r❡ ✐s ❛ ♣r♦❝❡❞✉r❡ t❤❛t ❝♦♥str✉❝ts ❛ ♣♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ✇✐t❤ ✷O(|ϕ|) st❛t❡s t❤❛t ❝♦♠♣✉t❡s ϕ✳

slide-58
SLIDE 58

❙②♥t❤❡s✐s✿ ❊①♣♦♥❡♥t✐❛❧ ❇❧♦✇✉♣

❜② ❆♥❣❧✉✐♥ ❡t ❛❧✳

■ss✉❡ ❊①♣❧❛♥❛t✐♦♥ ❊①❛♠♣❧❡ ✫ ❙✐③❡

❱❛❧✉❡ ❘❡♣r❡s❡♥t❛t✐♦♥

✭❇❛s❡ ❈♦♥str✉❝t✐♦♥✮ ✉♥❛r② ✐♥ s✐♥❣❧❡ ❛❣❡♥t ✭st❛t❡ ❢♦r ❡✈❡r② ✈❛❧✉❡✮

✵ ✶

❙t❛t❡s✿

❇♦♦❧❡❛♥ ❈♦♠❜✐♥❛t✐♦♥

♦❢ P✶ ✇✐t❤ n✶ st❛t❡s ❛♥❞ P✷ ✇✐t❤ n✷ st❛t❡s ❛❣❡♥ts ♠✉❧t✐t❛s❦ ✭♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥✮

✹ ✶ ✹ ✶

✶ ✷ ✶ ✷

❙t❛t❡s✿

✶ ✷

slide-59
SLIDE 59

❙②♥t❤❡s✐s✿ ❊①♣♦♥❡♥t✐❛❧ ❇❧♦✇✉♣

❜② ❆♥❣❧✉✐♥ ❡t ❛❧✳

■ss✉❡ ❊①♣❧❛♥❛t✐♦♥ ❊①❛♠♣❧❡ ✫ ❙✐③❡

❱❛❧✉❡ ❘❡♣r❡s❡♥t❛t✐♦♥

✭❇❛s❡ ❈♦♥str✉❝t✐♦♥✮ ✉♥❛r② ✐♥ s✐♥❣❧❡ ❛❣❡♥t ✭st❛t❡ ❢♦r ❡✈❡r② ✈❛❧✉❡✮

✵ ✶

· · ·

c

ϕ = x ≥ c |ϕ| = log c ❙t❛t❡s✿ c

  • EXP(|ϕ|)

❇♦♦❧❡❛♥ ❈♦♠❜✐♥❛t✐♦♥

♦❢ P✶ ✇✐t❤ n✶ st❛t❡s ❛♥❞ P✷ ✇✐t❤ n✷ st❛t❡s ❛❣❡♥ts ♠✉❧t✐t❛s❦ ✭♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥✮

✹ ✶ ✹ ✶

✶ ✷ ✶ ✷

❙t❛t❡s✿

✶ ✷

slide-60
SLIDE 60

❙②♥t❤❡s✐s✿ ❊①♣♦♥❡♥t✐❛❧ ❇❧♦✇✉♣

❜② ❆♥❣❧✉✐♥ ❡t ❛❧✳

■ss✉❡ ❊①♣❧❛♥❛t✐♦♥ ❊①❛♠♣❧❡ ✫ ❙✐③❡

❱❛❧✉❡ ❘❡♣r❡s❡♥t❛t✐♦♥

✭❇❛s❡ ❈♦♥str✉❝t✐♦♥✮ ✉♥❛r② ✐♥ s✐♥❣❧❡ ❛❣❡♥t ✭st❛t❡ ❢♦r ❡✈❡r② ✈❛❧✉❡✮

✵ ✶

· · ·

c

ϕ = x ≥ c |ϕ| = log c ❙t❛t❡s✿ c

  • EXP(|ϕ|)

❇♦♦❧❡❛♥ ❈♦♠❜✐♥❛t✐♦♥

♦❢ P✶ ✇✐t❤ n✶ st❛t❡s ❛♥❞ P✷ ✇✐t❤ n✷ st❛t❡s ❛❣❡♥ts ♠✉❧t✐t❛s❦ ✭♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥✮

✹ ✶ ✹ ✶

ϕ = ϕ✶ ∧ ϕ✷ |ϕ| = |ϕ✶| + |ϕ✷| ❙t❛t❡s✿ n✶ · n✷

  • EXP(|ϕ|)
slide-61
SLIDE 61

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❙✉❝❝✐♥❝t♥❡ss

Is there a succinct protocol for every predicate?

▼❛✐♥ ❚❤❡♦r❡♠ ✭❙❚❆❈❙✷✵✮

  • ✐✈❡♥ ❛ ❢♦r♠✉❧❛

♦❢ q✉❛♥t✐✜❡r✲❢r❡❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ✇✐t❤ r❡♠❛✐♥❞❡r ♣r❡❞✐❝❛t❡s✱ t❤❡r❡ ✐s ❛ ♣r♦❝❡❞✉r❡ t❤❛t ❝♦♥str✉❝ts ❛ ♣♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ✇✐t❤ st❛t❡s t❤❛t ❝♦♠♣✉t❡s ✳

slide-62
SLIDE 62

P♦♣✉❧❛t✐♦♥ Pr♦t♦❝♦❧s✿ ❙✉❝❝✐♥❝t♥❡ss

Is there a succinct protocol for every predicate?

▼❛✐♥ ❚❤❡♦r❡♠ ✭❙❚❆❈❙✷✵✮

  • ✐✈❡♥ ❛ ❢♦r♠✉❧❛ ϕ ♦❢ q✉❛♥t✐✜❡r✲❢r❡❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝

✇✐t❤ r❡♠❛✐♥❞❡r ♣r❡❞✐❝❛t❡s✱ t❤❡r❡ ✐s ❛ ♣r♦❝❡❞✉r❡ t❤❛t ❝♦♥str✉❝ts ❛ ♣♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ✇✐t❤ O(POLY (|ϕ|)) st❛t❡s t❤❛t ❝♦♠♣✉t❡s ϕ✳

  • Yes!
slide-63
SLIDE 63

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♠♣❛r✐s♦♥

❆♥❣❧✉✐♥ ❡t ❛❧✳ ❖✉r ❙♦❧✉t✐♦♥ ❱❛❧✉❡ ❘❡♣r❡s❡♥t❛t✐♦♥

✭❇❛s❡ ❈♦♥str✉❝t✐♦♥✮ ✉♥❛r② ✐♥ s✐♥❣❧❡ ❛❣❡♥t ✭st❛t❡ ❢♦r ❡✈❡r② ✈❛❧✉❡✮

✼

→ EXP(|ϕ|) ❜✐♥❛r② ❛♥❞ ❞✐str✐❜✉t❡❞ ✭st❛t❡ ❢♦r ♣♦✇❡rs ♦❢ ✷✮

✹ ✷ ✶

❇♦♦❧❡❛♥ ❈♦♠❜✐♥❛t✐♦♥

♦❢ P✶ ✇✐t❤ n✶ st❛t❡s ❛♥❞ P✷ ✇✐t❤ n✷ st❛t❡s ❛❣❡♥ts ♠✉❧t✐t❛s❦ ✭♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥✮

✹ ✶ ✹ ✶

→ n✶ · n✷ st❛t❡s → EXP(|ϕ|) ❛❣❡♥ts ❞✐✈✐❞❡ ✇♦r❦ ✭s♣❡❝✐❛❧✐③❛t✐♦♥✮

✷ ✽

✶ ✷ st❛t❡s

slide-64
SLIDE 64

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♠♣❛r✐s♦♥

❆♥❣❧✉✐♥ ❡t ❛❧✳ ❖✉r ❙♦❧✉t✐♦♥ ❱❛❧✉❡ ❘❡♣r❡s❡♥t❛t✐♦♥

✭❇❛s❡ ❈♦♥str✉❝t✐♦♥✮ ✉♥❛r② ✐♥ s✐♥❣❧❡ ❛❣❡♥t ✭st❛t❡ ❢♦r ❡✈❡r② ✈❛❧✉❡✮

✼

→ EXP(|ϕ|) ❜✐♥❛r② ❛♥❞ ❞✐str✐❜✉t❡❞ ✭st❛t❡ ❢♦r ♣♦✇❡rs ♦❢ ✷✮

✹ ✷ ✶

→ POLY (|ϕ|)

❇♦♦❧❡❛♥ ❈♦♠❜✐♥❛t✐♦♥

♦❢ P✶ ✇✐t❤ n✶ st❛t❡s ❛♥❞ P✷ ✇✐t❤ n✷ st❛t❡s ❛❣❡♥ts ♠✉❧t✐t❛s❦ ✭♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥✮

✹ ✶ ✹ ✶

→ n✶ · n✷ st❛t❡s → EXP(|ϕ|) ❛❣❡♥ts ❞✐✈✐❞❡ ✇♦r❦ ✭s♣❡❝✐❛❧✐③❛t✐♦♥✮

✷ ✽

✶ ✷ st❛t❡s

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SLIDE 65

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♠♣❛r✐s♦♥

❆♥❣❧✉✐♥ ❡t ❛❧✳ ❖✉r ❙♦❧✉t✐♦♥ ❱❛❧✉❡ ❘❡♣r❡s❡♥t❛t✐♦♥

✭❇❛s❡ ❈♦♥str✉❝t✐♦♥✮ ✉♥❛r② ✐♥ s✐♥❣❧❡ ❛❣❡♥t ✭st❛t❡ ❢♦r ❡✈❡r② ✈❛❧✉❡✮

✼

→ EXP(|ϕ|) ❜✐♥❛r② ❛♥❞ ❞✐str✐❜✉t❡❞ ✭st❛t❡ ❢♦r ♣♦✇❡rs ♦❢ ✷✮

✹ ✷ ✶

→ POLY (|ϕ|)

❇♦♦❧❡❛♥ ❈♦♠❜✐♥❛t✐♦♥

♦❢ P✶ ✇✐t❤ n✶ st❛t❡s ❛♥❞ P✷ ✇✐t❤ n✷ st❛t❡s ❛❣❡♥ts ♠✉❧t✐t❛s❦ ✭♣r♦❞✉❝t ❝♦♥str✉❝t✐♦♥✮

✹ ✶ ✹ ✶

→ n✶ · n✷ st❛t❡s → EXP(|ϕ|) ❛❣❡♥ts ❞✐✈✐❞❡ ✇♦r❦ ✭s♣❡❝✐❛❧✐③❛t✐♦♥✮

✷ ✽

→ n✶ + n✷ st❛t❡s → POLY (|ϕ|)

slide-66
SLIDE 66

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♥✈❡rt✐♥❣ t♦ ❇✐♥❛r② ❈❛s❡ st✉❞②✿ ■♥✐t✐❛❧ ✈❛❧✉❡s ✫ ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥

✸ ✸✷

healthy sick very sick

✵ ✶ ✷ ✹ ✽ ✶✻ ✸✷ ✼ ✸✷

slide-67
SLIDE 67

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♥✈❡rt✐♥❣ t♦ ❇✐♥❛r② ❈❛s❡ st✉❞②✿ ■♥✐t✐❛❧ ✈❛❧✉❡s ✫ ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥

ϕ = +✸· ≥ ✸✷

healthy sick very sick

✵ ✶ ✷ ✹ ✽ ✶✻ ✸✷ ✼ ✸✷

slide-68
SLIDE 68

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♥✈❡rt✐♥❣ t♦ ❇✐♥❛r② ❈❛s❡ st✉❞②✿ ■♥✐t✐❛❧ ✈❛❧✉❡s ✫ ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥

ϕ = +✸· ≥ ✸✷

healthy sick very sick

✵ ✶ ✷ ✹ ✽ ✶✻ ✸✷ ✼ ✸✷

slide-69
SLIDE 69

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♥✈❡rt✐♥❣ t♦ ❇✐♥❛r② ❈❛s❡ st✉❞②✿ ■♥✐t✐❛❧ ✈❛❧✉❡s ✫ ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥

ϕ = +✸· ≥ ✸✷

healthy sick very sick

✵ ✶ ✷ ✹ ✽ ✶✻ ✸✷ ✼ ✸✷

slide-70
SLIDE 70

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♥✈❡rt✐♥❣ t♦ ❇✐♥❛r② ❈❛s❡ st✉❞②✿ ■♥✐t✐❛❧ ✈❛❧✉❡s ✫ ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥

ϕ = +✸· ≥ ✸✷

healthy sick very sick

✵ ✶ ✷ ✹ ✽ ✶✻ ✸✷

? ? ?

✼ ✸✷

slide-71
SLIDE 71

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♥✈❡rt✐♥❣ t♦ ❇✐♥❛r② ❈❛s❡ st✉❞②✿ ■♥✐t✐❛❧ ✈❛❧✉❡s ✫ ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥

ϕ = +✸· ≥ ✸✷

healthy sick very sick helper

✵ ✶ ✷ ✹ ✽ ✶✻ ✸✷ ✼ ✸✷

slide-72
SLIDE 72

❙✉❝❝✐♥❝t♥❡ss✿ ❈♦♥✈❡rt✐♥❣ t♦ ❇✐♥❛r② ❈❛s❡ st✉❞②✿ ■♥✐t✐❛❧ ✈❛❧✉❡s ✫ ❜✐♥❛r② r❡♣r❡s❡♥t❛t✐♦♥

ϕ = +✸· ≥ ✸✷

healthy sick very sick helper

✵ ✶ ✷ ✹ ✽ ✶✻ ✸✷ ✼ ✸✷

slide-73
SLIDE 73

❙✉❝❝✐♥❝t♥❡ss✿ ❍❡❧♣❡rs ❍❡❧♣❡rs✿ ❛✉①✐❧✐❛r② ❛❣❡♥ts ✇✐t❤ ✐❞❡♥t✐t② ✭✐✳❡✳ ✏r❡s❡r✈♦✐r ♦❢ ❜❛❝❦✉♣ ❛❣❡♥ts✑✮

✸ ✷ ✶

❯s❡❢✉❧ ❢♦r✿ ❝♦♥✈❡rt✐♥❣ ✈❛❧✉❡s t♦ ❜✐♥❛r② ❞✐str✐❜✉t✐♥❣ ✇♦r❦ ❤❛♥❞❧✐♥❣ t❤❡ ♦✉t♣✉t ❍❡❧♣❡r r❡♠♦✈❛❧ ✭✇✐t❤♦✉t ❡①♣♦♥❡♥t✐❛❧ ❜❧♦✇✉♣✮✿ ❋♦r ✏❧❛r❣❡✑ ✐♥♣✉ts✿ ❡♥♦✉❣❤ ❛❣❡♥ts ❛✈❛✐❧❛❜❧❡ s✐♠✉❧❛t❡ ❤❡❧♣❡rs ❋♦r ✏s♠❛❧❧✑ ✐♥♣✉ts✿ ❧✐♠✐t❡❞ ♣♦ss✐❜✐❧✐t✐❡s ❝♦♠♣✉t❡ r❡s✉❧t ❞✐r❡❝t❧②

slide-74
SLIDE 74

❙✉❝❝✐♥❝t♥❡ss✿ ❍❡❧♣❡rs ❍❡❧♣❡rs✿ ❛✉①✐❧✐❛r② ❛❣❡♥ts ✇✐t❤ ✐❞❡♥t✐t② ✭✐✳❡✳ ✏r❡s❡r✈♦✐r ♦❢ ❜❛❝❦✉♣ ❛❣❡♥ts✑✮

✸ ✷ ✶

❯s❡❢✉❧ ❢♦r✿

  • ❝♦♥✈❡rt✐♥❣ ✈❛❧✉❡s t♦ ❜✐♥❛r②
  • ❞✐str✐❜✉t✐♥❣ ✇♦r❦
  • ❤❛♥❞❧✐♥❣ t❤❡ ♦✉t♣✉t

❍❡❧♣❡r r❡♠♦✈❛❧ ✭✇✐t❤♦✉t ❡①♣♦♥❡♥t✐❛❧ ❜❧♦✇✉♣✮✿ ❋♦r ✏❧❛r❣❡✑ ✐♥♣✉ts✿ ❡♥♦✉❣❤ ❛❣❡♥ts ❛✈❛✐❧❛❜❧❡ s✐♠✉❧❛t❡ ❤❡❧♣❡rs ❋♦r ✏s♠❛❧❧✑ ✐♥♣✉ts✿ ❧✐♠✐t❡❞ ♣♦ss✐❜✐❧✐t✐❡s ❝♦♠♣✉t❡ r❡s✉❧t ❞✐r❡❝t❧②

slide-75
SLIDE 75

❙✉❝❝✐♥❝t♥❡ss✿ ❍❡❧♣❡rs ❍❡❧♣❡rs✿ ❛✉①✐❧✐❛r② ❛❣❡♥ts ✇✐t❤ ✐❞❡♥t✐t② ✭✐✳❡✳ ✏r❡s❡r✈♦✐r ♦❢ ❜❛❝❦✉♣ ❛❣❡♥ts✑✮

✸ ✷ ✶

❯s❡❢✉❧ ❢♦r✿

  • ❝♦♥✈❡rt✐♥❣ ✈❛❧✉❡s t♦ ❜✐♥❛r②
  • ❞✐str✐❜✉t✐♥❣ ✇♦r❦
  • ❤❛♥❞❧✐♥❣ t❤❡ ♦✉t♣✉t

❍❡❧♣❡r r❡♠♦✈❛❧ ✭✇✐t❤♦✉t ❡①♣♦♥❡♥t✐❛❧ ❜❧♦✇✉♣✮✿

  • ❋♦r ✏❧❛r❣❡✑ ✐♥♣✉ts✿ ❡♥♦✉❣❤ ❛❣❡♥ts ❛✈❛✐❧❛❜❧❡

→ s✐♠✉❧❛t❡ ❤❡❧♣❡rs

  • ❋♦r ✏s♠❛❧❧✑ ✐♥♣✉ts✿ ❧✐♠✐t❡❞ ♣♦ss✐❜✐❧✐t✐❡s

→ ❝♦♠♣✉t❡ r❡s✉❧t ❞✐r❡❝t❧②

slide-76
SLIDE 76

▼❛✐♥ ❚❤❡♦r❡♠ ✭❙❚❆❈❙✷✵✮

  • ✐✈❡♥ ❛ ❢♦r♠✉❧❛ ϕ ♦❢ q✉❛♥t✐✜❡r✲❢r❡❡ Pr❡s❜✉r❣❡r ❛r✐t❤♠❡t✐❝ ✇✐t❤

r❡♠❛✐♥❞❡r ♣r❡❞✐❝❛t❡s✱ t❤❡r❡ ✐s ❛ ♣r♦❝❡❞✉r❡ t❤❛t ❝♦♥str✉❝ts ❛ ♣♦♣✉❧❛t✐♦♥ ♣r♦t♦❝♦❧ ✇✐t❤ O(POLY (|ϕ|)) st❛t❡s t❤❛t ❝♦♠♣✉t❡s ϕ✳

THANK YOU!