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SLIDE 1

Computer Science Laboratory, SRI International

❋♦r♠❛❧ ❚❡❝❤♥✐q✉❡s ❢♦r ❆♥❛❧②③✐♥❣ ❍②❜r✐❞ ❙②st❡♠s

❆s❤✐s❤ ❚✐✇❛r✐

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SLIDE 2

❘❡♣r❡s❡♥t✐♥❣ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s

❍♦✇ t♦ s♦❧✈❡ t❤❡ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ♣r♦❜❧❡♠ S | = φ ❄ ❈r✉❝✐❛❧❧② ❞❡♣❡♥❞s ♦♥ ❤♦✇ ✇❡ r❡♣r❡s❡♥t S ❘❡❝❛❧❧ t❤❛t S = (X, F, Init) ❲❡ ♥❡❡❞ t♦ r❡♣r❡s❡♥t t❤❡ st❛t❡ s♣❛❝❡✱ t❤❡ ✐♥✐t✐❛❧ s❡t Init✱ ❛♥❞ t❤❡ s❡t ♦❢ tr❛❥❡❝t♦r✐❡s

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SLIDE 3

❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ❙t❛t❡ ❙♣❛❝❡

❙t❛t❡ s♣❛❝❡ X

  • ❙t❛t❡ s♣❛❝❡ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s t❤❡ s❡t ♦❢ ❛❧❧ ✈❛❧✉❛t✐♦♥s ♦❢ ❛ s❡t ♦❢ ✈❛r✐❛❜❧❡s
  • ❲❡ ❥✉st ♥❡❡❞ t♦ ❞❡❝❧❛r❡ ✈❛r✐❛❜❧❡s X ❛♥❞ t❤❡✐r t②♣❡s

❚❤❡ ✐♥✐t✐❛❧ st❛t❡s Init

  • ❚❤✐s ✐s ❥✉st ❛ s✉❜s❡t ♦❢ t❤❡ st❛t❡ s♣❛❝❡
  • ❈❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ❢♦r♠✉❧❛ ♦✈❡r X

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SLIDE 4

❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ❚r❛❥❡❝t♦r✐❡s

❚❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s ✐s ✉s✉❛❧❧② r❡♣r❡s❡♥t❡❞ ❜② ❧♦❝❛❧ r✉❧❡s ❚❤❛t s♣❡❝✐❢② t❤❡ ♥❡①t st❛t❡✭s✮ ❣✐✈❡♥ t❤❡ ❝✉rr❡♥t st❛t❡ ✭❛♥❞ ✐♥♣✉t✱ ✐❢ ❛♥②✮ ❚❤❡ ♥♦t✐♦♥ ♦❢ ♥❡①t ✐s ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ t✐♠❡ ❞♦♠❛✐♥ T ❲❤❡♥ T = N✱ ♥❡①t ✐s ❥✉st ✭❝✉rr❡♥t t✐♠❡ ✰ ✶ ✮

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SLIDE 5

❋r♦♠ ❝♦♥t✐♥✉♦✉s t✐♠❡ t♦ ❞✐s❝r❡t❡✲t✐♠❡

❲❤❡♥ T = R≥0✱ t❤❡r❡ ✐s ❛ ♣♦♣✉❧❛r ✇❛② t♦ ♠❛♣ ✐t t♦ T = N s → s′ ✐❢ t❤❡r❡ ❡①✐sts ❛ t′ > t s✳t✳ s′ = F(t′) ❛♥❞ s = F(t) ❋r♦♠ ♠② ❝✉rr❡♥t st❛t❡ s✱ t❤❡ s②st❡♠ ❝❛♥ ♥♦♥❞❡t❡r♠✐♥✐st✐❝❛❧❧② tr❛♥s✐t✐♦♥ ❛♥② st❛t❡ r❡❛❝❤❛❜❧❡ ✐♥ ❛♥② ❢✉t✉r❡ t✐♠❡ ✐♥st❛♥❝❡ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ dx

dt = 1 ✐s ❛ ❝❧♦❝❦✱ ✐ts ❞✐s❝r❡t❡ t✐♠❡ s❡♠❛♥t✐❝s ✐s✿

x → x′ ✐❢ x′ > x ❚❤✐s ✐s t✐♠❡ ♦❜❧✐✈✐♦✉s ■❢ ✇❡ ✇❛♥t t♦ ❦❡❡♣ tr❛❝❦ ♦❢ t✐♠❡ ❡❧❛♣s❡❞✱ ✇❡ ❝❛♥ ❛❞❞ ❛ ♥❡✇ st❛t❡ ✈❛r✐❛❜❧❡ t ❛♥❞ ❤❛✈❡ (x, t) → (x′, t′) ✐❢ x′ > x ❛♥❞ x′ − x = t′ − t

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SLIDE 6

❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ❚r❛❥❡❝t♦r✐❡s

▲❡t ✉s ❢♦❝✉s ♦♥ t✐♠❡ ❞♦♠❛✐♥ N✱ ❛♥❞ ❝♦♥s✐❞❡r ✇❛②s t♦ s♣❡❝✐❢② ❛ s✐♥❣❧❡ st❡♣ s → s′ st❛t❡ tr❛♥s✐t✐♦♥ ❚❤✐s ❝❛♥ ❜❡ ❣✐✈❡♥ ❛s ❛ ❢♦r♠✉❧❛ ♦✈❡r ❛ ♣❛✐r ♦❢ st❛t❡ ✈❛r✐❛❜❧❡s✿ φ(x, x′) ❋♦r ♦♣❡♥ s②st❡♠s✱ t❤✐s ❢♦r♠✉❧❛ ❝❛♥ ❞❡♣❡♥❞ ♦♥ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡s ❋♦r ❝♦♥t✐♥✉♦✉s t✐♠❡ s②st❡♠s✱ t❤❡ r✉❧❡ ✐s ❛ ❢♦r♠✉❧❛ ♦✈❡r (x, ˙ x)

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SLIDE 7

❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ▼♦❞✉❧❛r❧②

❲❡ ❝❛♥ r❡♣r❡s❡♥t ❞②♥❛♠✐❝❛❧ s②st❡♠s s✉❝❝✐♥❝t❧② ❜② ❞❡s❝r✐❜✐♥❣ t❤❡♠ ❛s ❛ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t✇♦ ♦r ♠♦r❡ s♠❛❧❧❡r ❞②♥❛♠✐❝❛❧ s②st❡♠s✿

input

  • utput

local state local

  • utput

input

❲❡ ♥❡❡❞ t♦ t❤✐♥❦ ❛❜♦✉t ♦♣❡♥ s②st❡♠s ✜rst

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SLIDE 8

❖♣❡♥ ❉②♥❛♠✐❝❛❧ ❙②st❡♠s

❙②st❡♠s ✇✐t❤ ✐♥♣✉ts ■♥♣✉ts ❞♦ ♥♦t ❝❤❛♥❣❡ t❤❡ st❛t❡ s♣❛❝❡ ♦❢ t❤❡ s②st❡♠ st❛t❡ s♣❛❝❡ st✐❧❧ ❞❡✜♥❡❞ ❜② t❤❡ ❧♦❝❛❧ ❛♥❞ ♦✉t♣✉t ✈❛r✐❛❜❧❡s ■♥♣✉ts ❝❛♥ ✐♥✢✉❡♥❝❡ t❤❡ ♣♦ss✐❜❧❡ tr❛❥❡❝t♦r✐❡s ❍❡♥❝❡✱ r✉❧❡ φ(s, i, s′) ❝❛♥ ❞❡♣❡♥❞ ♦♥ t❤❡ ✐♥♣✉t ✈❛r✐❛❜❧❡s i s → s′ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ✐♥♣✉t i ∈ I : φ(s, i, s′)

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SLIDE 9

❘❡♣r❡s❡♥t✐♥❣ ❙②st❡♠s✿ ▼♦❞✉❧❛r❧②

❙t❛t❡ s♣❛❝❡ ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❂ ❝r♦ss ♣r♦❞✉❝t ♦❢ t❤❡ t✇♦ st❛t❡ s♣❛❝❡s ❂ ✈❛❧✉❛t✐♦♥s ♦❢ L1 ∪ O1 ∪ L2 ∪ O2 ❚r❛❥❡❝t♦r✐❡s ♦❢ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ❂ ❄❄ ❞❡♣❡♥❞s ♦♥ t❤❡ ❝♦♠♣♦s✐t✐♦♥❛❧ ❢r❛♠❡✇♦r❦

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SLIDE 10

❚✇♦ ❈♦♠♣♦s✐t✐♦♥❛❧ ❋r❛♠❡✇♦r❦s

❙②♥❝❤r♦♥♦✉s✿

  • (s1, s2) → (s′

1, s′ 2) ✐❢ s1 → s′ 1 ❆◆❉ s2 → s′ 2

  • ❇♦t❤ s✉❜s②st❡♠s ♠❛❦❡ ❛ tr❛♥s✐t✐♦♥ s✐♠✉❧t❛♥❡♦✉s❧②

❆s②♥❝❤r♦♥♦✉s✿

  • (s1, s2) → (s′

1, s′ 2) ✐❢ s1 → s′ 1, s′ 2 = s2 ❖❘ s2 → s′ 2, s′ 1 = s1

  • ❖♥❡ ♦❢ t❤❡ t✇♦ s✉❜s②st❡♠s ♠❛❦❡s ❛ tr❛♥s✐t✐♦♥

■❢ t❤❡ t✇♦ s✉❜s②st❡♠s ❛r❡ ✇✐r❡❞ t♦ ❡❛❝❤ ♦t❤❡r✱ t❤❡ tr❛♥s✐t✐♦♥s ♦♥ t❤❡ s✉❜s②st❡♠s s❤♦✉❧❞ ❜❡ ❝♦♥s✐st❡♥t ✇✐t❤ t❤❡ ✐♥♣✉ts ✭❣❡♥❡r❛t❡❞ ❜② t❤❡ ♦t❤❡r s✉❜s②st❡♠✮

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SLIDE 11

▼♦❞❡❧✐♥❣ ✐♥ ❙❆▲

❉✐s❝r❡t❡✲t✐♠❡ ❤②❜r✐❞✲s♣❛❝❡ ❞②♥❛♠✐❝❛❧ s②st❡♠s ❝❛♥ ❜❡ ♠♦❞❡❧❡❞ ✐♥ ❙❆▲ ❊✈❡r② ❙❆▲ ▼❖❉❯▲❊ ❞❡s❝r✐❜❡s ❛ ❞②♥❛♠✐❝❛❧ ✭s✉❜✮s②st❡♠ ❝♦❧❧❛t③✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ❖❯❚P❯❚ ①✿ ◆❆❚❯❘❆▲ ■◆■❚■❆▲■❩❆❚■❖◆ ① ■◆ ④✈✿ ◆❆❚❯❘❆▲ ⑤ ❚r✉❡⑥❀ ❚❘❆◆❙■❚■❖◆ ❬ ① ▼❖❉ ✷ ❂ ✵ ✲✲❃ ①✬ ❂ ①✴✷❀ ❬❪ ❡❧s❡ ✲✲❃ ①✬ ❂ ✸✯①✰✶❀ ❪ ❊◆❉❀

✶✵

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SLIDE 12

❆s ❚✇♦ ❙✉❜s②st❡♠s

❝♦❧❧❛t③❊✈❡♥✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ■◆P❯❚ ①✿ ◆❆❚❯❘❆▲ ❖❯❚P❯❚ ②✿ ◆❆❚❯❘❆▲ ❚❘❆◆❙■❚■❖◆ ❬ ① ▼❖❉ ✷ ❂ ✵ ✲✲❃ ②✬ ❂ ①✴✷❀ ❬❪ ② ▼❖❉ ✷ ❂ ✵ ✲✲❃ ②✬ ❂ ②✴✷❀ ❪ ❊◆❉❀ ❝♦❧❧❛t③❖❞❞✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ■◆P❯❚ ②✿ ◆❆❚❯❘❆▲ ❖❯❚P❯❚ ①✿ ◆❆❚❯❘❆▲ ❚❘❆◆❙■❚■❖◆ ❬ ② ▼❖❉ ✷ ❂ ✶ ✲✲❃ ①✬ ❂ ✸✯②✰✶❀ ❪ ❊◆❉❀

✶✶

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SLIDE 13

✳✳✳ ❈♦♠♣♦s❡❞ ❚♦❣❡t❤❡r

❝♦❧❧❛t③✿ ▼❖❉❯▲❊ ❂ ❝♦❧❧❛t③❊✈❡♥ ❬❪ ❝♦❧❧❛t③❖❞❞ ❀ ■♥♣✉t ❛♥❞ ❖✉t♣✉t ♠❛t❝❤ ✉♣ ❜② ♥❛♠❡ ❤❡r❡❀ ❝♦✉❧❞ ❛♣♣❧② ♦✉t❡r r❡♥❛♠❡ ♦♣❡r❛t♦r t♦ ❛ ♠♦❞✉❧❡ ♦✳✇✳ ❆❧❧ ♠♦❞✉❧❡s ❝❛♥ ❜❡ ♣✉t ✇✐t❤✐♥ ❛ ❝♦♥t❡①t ✭✜❧❡✮ ❝♦❧❧❛t③✳s❛❧✿ ❝♦❧❧❛t③✿ ❈❖◆❚❊❳❚ ❂ ❇❊●■◆ ❝♦❧❧❛t③❊✈❡♥✿ ▼❖❉❯▲❊ ❂ ✳✳✳ ❝♦❧❧❛t③❖❞❞✿ ▼❖❉❯▲❊ ❂ ✳✳✳ ❝♦❧❧❛t③✿ ▼❖❉❯▲❊ ❂ ✳✳✳ ❊◆❉

✶✷

slide-14
SLIDE 14

❚❡♠♣♦r❛❧ Pr♦♣❡rt✐❡s ✐♥ ❙❆▲

❚❤❡ ♦♣❡♥ ♣r♦❜❧❡♠✿ ❛♠❴■❴tr✉❡✿ ❚❍❊❖❘❊▼ ❝♦❧❧❛t③ ⑤✲ ❋✭ ① ❂ ✶ ✮❀

✶✸

slide-15
SLIDE 15

▼♦❞❡❧✐♥❣ ❊①❡r❝✐s❡

❉✐s❝r❡t❡✲t✐♠❡ ❞✐s❝r❡t❡✲s♣❛❝❡ ❞②♥❛♠✐❝❛❧ s②st❡♠ ♠♦❞❡❧✐♥❣ ❛ ✈❡♥❞✐♥❣ ♠❛❝❤✐♥❡

  • ❯s❡r ❝❛♥ st❛rt ✇✐t❤ ❞♦❧❧❛r ❛♠♦✉♥t ✐♥ t❤❡ r❛♥❣❡ [0, . . . , 5]
  • ❯s❡r ❝❛♥ ❜✉② ❛ ❝❛❦❡ ❢♦r $1
  • ❯s❡r ❝❛♥ ❜✉② ❛♥ ❛♣♣❧❡ ❢♦r $0.75

❚❤❡ st❛t❡ s♣❛❝❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ✭✶✮ ❛♠♦✉♥t ♦❢ ♠♦♥❡② r❡♠❛✐♥✐♥❣✱ ✭✷✮ ♥✉♠❜❡r ♦❢ ❝❛❦❡s ❜♦✉❣❤t✱ ❛♥❞ ✭✸✮ ♥✉♠❜❡r ♦❢ ❛♣♣❧❡s ❜♦✉❣❤t

✶✹

slide-16
SLIDE 16

❙❆▲ ❊①❛♠♣❧❡✿ ▼♦❞❡❧✐♥❣

✪ ❱❡♥❞✐♥❣ ♠❛❝❤✐♥❡ ✐♥ ❙❆▲ ✈♠✿ ❈❖◆❚❊❳❚ ❂ ❇❊●■◆ ❉♦❧❧❛r❆♠♦✉♥t✿ ❚❨P❊ ❂ ❬✵✳✳✺❪❀ ◗✉❛rt❡r❆♠♦✉♥t✿ ❚❨P❊ ❂ ❬✵✳✳✷✵❪❀ ❈❛❦❡❆♠♦✉♥t✿ ❚❨P❊ ❂ ❬✵✳✳✺❪❀ ❆♣♣❧❡❆♠♦✉♥t✿ ❚❨P❊ ❂ ❬✵✳✳✶✵❪❀

✶✺

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SLIDE 17

❙❆▲ ❊①❛♠♣❧❡✿ ▼♦❞❡❧✐♥❣

♠❛❝❤✐♥❡✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ❖❯❚P❯❚ ❞✿ ❉♦❧❧❛r❆♠♦✉♥t✱ q✿ ◗✉❛rt❡r❆♠♦✉♥t✱ ❝✿ ❈❛❦❡❆♠♦✉♥t✱ ❛✿ ❆♣♣❧❡❆♠♦✉♥t ■◆■❚■❆▲■❩❆❚■❖◆ ❞ ■◆ ④✈✿ ❉♦❧❧❛r❆♠♦✉♥t ⑤ ✈ ❁❂ ✹⑥❀ q ❂ ✵❀ ❝ ❂ ✵❀ ❛ ❂ ✵❀

✶✻

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SLIDE 18

❙❆▲ ❊①❛♠♣❧❡✿ ▼♦❞❡❧✐♥❣ ❈♦♥t❞

❚❘❆◆❙■❚■❖◆ ❬ ❣❡t❴❝✿ ❞ ❃❂ ✶ ✲✲❃ ❞✬ ❂ ❞✲✶❀ ❝✬ ❂ ❝✰✶ ❬❪ ❣❡t❴❛✿ ❞ ❃❂ ✶ ✲✲❃ ❞✬ ❂ ❞✲✶❀ ❛✬ ❂ ❛✰✶❀ q✬ ❂ q✰✶ ❬❪ ❝❤❛♥❣❡✿ q ❃❂ ✹ ✲✲❃ ❞✬ ❂ ❞✰✶❀ q✬ ❂ q✲✹ ❬❪ ❡❧s❡ ✲✲❃ ❪ ❊◆❉❀

✶✼

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SLIDE 19

❙❆▲ ❊①❛♠♣❧❡✿ ▼♦❞❡❧✐♥❣ ❈♦♥t❞

♣r♦♣✿ ❚❍❊❖❘❊▼ ♠❛❝❤✐♥❡ ⑤✲ ◆❖❚ ✭ ❯✭q ❁❂ ✹✱ ❛ ❃❂ ✺✮ ✮❀ ♣r♦♣✷✿ ❚❍❊❖❘❊▼ ♠❛❝❤✐♥❡ ⑤✲

  • ✭ ◆❖❚ ✭ ❛ ❃❂ ✸ ❆◆❉ ❝ ❃❂ ✷✮ ✮❀

❊◆❉ ❊①❡r❝✐s❡✿ ■s ♣r♦♣ ✈❛❧✐❞ ♦r ❢❛❧s❡❄ ❊①❡r❝✐s❡✿ ■s ♣r♦♣✷ ✈❛❧✐❞ ♦r ❢❛❧s❡❄

✶✽

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SLIDE 20

❙❆▲ ❊①❛♠♣❧❡✿ ❆♥❛❧②s✐s

❖♥❡ ❝❛♥ ✉s❡ t❤❡ s❛❧ s②♠❜♦❧✐❝ ♠♦❞❡❧ ❝❤❡❝❦❡r t♦ ♠♦❞❡❧ ❝❤❡❝❦ t❤❡ t✇♦ ♣r♦♣❡rt✐❡s s❛❧✲s♠❝ ✈♠ ♣r♦♣ s❛❧✲s♠❝ ✈♠ ♣r♦♣✷ ❚❤❡ ✜rst ❣✐✈❡s ❛ ❝♦✉♥t❡r✲❡①❛♠♣❧❡✿ st❛rt ✇✐t❤ ✹ ❞♦❧❧❛r ❜✐❧❧s✱ ❝❤❛♥❣❡ ✇❤❡♥ ②♦✉ ❤❛✈❡ ✹ q✉❛rt❡rs ❚❤❡ s❡❝♦♥❞ s❛②s ✧♣r♦✈❡❞✧✳

✶✾

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SLIDE 21

❙❆▲ ▼♦❞❡❧✐♥❣✿ ❖t❤❡r ❆s♣❡❝ts

  • ▼♦❞✉❧❡s ❝❛♥ ❜❡ ❝♦♠♣♦s❡❞✿

▼♦❞✉❧❡ ❂ ▼♦❞✉❧❡ ❬❪ ▼♦❞✉❧❡❀ ▼♦❞✉❧❡ ❂ ▼♦❞✉❧❡ ⑤⑤ ▼♦❞✉❧❡❀

  • ▼♦❞✉❧❡s ❝❛♥ ❜❡ ♣❛r❛♠❡t❡r✐③❡❞✿ ▼♦❞✉❧❡✭✐✿ ■♥❞❡①✮ ❂ ✳✳✳
  • ❆♥❞ ❝♦♠♣♦s❡❞✿

▼♦❞✉❧❡ ❂ ✭❬❪ ✭✐✿ ■♥❞❡①✮ ▼♦❞✉❧❡❬✐❪✮ ⑤⑤ ❖❜s❡r✈❡r

  • ▼♦❞✉❧❡s ❝❛♥ ✉s❡ ✬❤❡❧♣❡r✬ ❢✉♥❝t✐♦♥s✱ ❞❛t❛t②♣❡s ❛s ❆❘❘❆❨❙ ♦❢ ❆❘❘❆❨❙ ♦❢ ✳✳✳

✷✵

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SLIDE 22

❯♥s♦✉♥❞♥❡ss❄

❞❡❛❞✿ ❈❖◆❚❊❳❚ ❂ ❇❊●■◆ ❞❡❛❞✿ ▼❖❉❯▲❊ ❂ ❇❊●■◆ ▲❖❈❆▲ ①✿◆❆❚❯❘❆▲ ■◆■❚■❆▲■❩❆❚■❖◆ ① ❂ ✵ ❀ ❚❘❆◆❙■❚■❖◆ ❬ ① ❁ ✹ ✲✲❃ ①✬ ❂ ① ✰ ✶❀ ❪ ❊◆❉ ❀ ✇r♦♥❣✿ ❚❍❊❖❘❊▼ ❞❡❛❞ ⑤✲ ●✭ ① ❁ ✸ ✮ ❀ ❊◆❉

✷✶

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SLIDE 23

❯♥s♦✉♥❞❡ss❄ ❈♦♥t✐♥✉❡❞✳✳✳

◆♦✇✱ ❧❡t ✉s tr② t♦ ♣r♦✈❡ t❤❡ ♣r♦♣❡rt✐❡s✿ s❛❧✲✐♥❢✲❜♠❝ ✲❞ ✺ ✲✐ ❞❡❛❞ ✇r♦♥❣ ❚❤✐s ✐s ♣r♦✈❡❞✦ s❛❧✲✐♥❢✲❜♠❝ ✲❞ ✸ ✲✐ ❞❡❛❞ ✇r♦♥❣ ❨♦✉ ❣❡t ❛ ❝♦✉♥t❡r✲❡①❛♠♣❧❡ ❲❤②❄

✷✷

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SLIDE 24

❙❆▲ ❆♥❛❧②s✐s✿ ❖t❤❡r ❆s♣❡❝ts

❙❛❧ ❚♦♦❧s

  • s❛❧✲s♠❝✿ s②♠❜♦❧✐❝ ♠♦❞❡❧ ❝❤❡❝❦❡r
  • s❛❧✲❜♠❝✿ ❜♦✉♥❞❡❞ ♠♦❞❡❧ ❝❤❡❝❦❡r ✭❝♦♥✈❡rts t♦ ❙❆❚✮
  • s❛❧✲✐♥❢✲❜♠❝✿ ✐♥✜♥✐t❡ ❜♦✉♥❞❡❞ ♠♦❞❡❧ ❝❤❡❝❦❡r ✭❙▼❚✮
  • s❛❧✲♣❛t❤✲✜♥❞❡r✿ ❢♦r s✐♠✉❧❛t✐♥❣
  • s❛❧✲❜♠❝ ✲✐✱ s❛❧✲✐♥❢✲❜♠❝ ✲✐✿ ♣r♦✈❡ ❜② ❦✲✐♥❞✉❝t✐♦♥
  • s❛❧✲❡♠❝✿ ❡①♣❧✐❝✐t st❛t❡ ♠♦❞❡❧ ❝❤❡❝❦❡r
  • s❛❧✲❞❡❛❞❧♦❝❦✲❝❤❡❝❦❡r✿ ❝❤❡❝❦ ❢♦r ❞❡❛❞❧♦❝❦❡❞ st❛t❡s
  • ②♦✉r s❝r✐♣t✿ s❛❧✲❛t❣✿ ❆✉t♦♠❛t❡❞ t❡st ❣❡♥❡r❛t♦r

✷✸