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❋❖❘▼❆❚❙ ✷✵✶✻ ❲❡❞♥❡s❞❛② ✷✹t❤ ♦❢ ❆✉❣✉st✱ ✷✵✶✻ ◗✉é❜❡❝✱ ◗✉é❜❡❝

❖♥ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

➱t✐❡♥♥❡ ❆♥❞ré1,2✱ ❉✐❞✐❡r ▲✐♠❡2✱ ❖❧✐✈✐❡r ❍✳ ❘♦✉①2

1 ▲■P◆✱ ❯♥✐✈❡rs✐té P❛r✐s ✶✸✱ ❙♦r❜♦♥♥❡ P❛r✐s ❈✐té✱ ❈◆❘❙✱ ❋r❛♥❝❡ 2 ➱❝♦❧❡ ❈❡♥tr❛❧❡ ❞❡ ◆❛♥t❡s✱ ■❘❈❈②◆✱ ❈◆❘❙✱ ❯▼❘ ✻✺✾✼✱ ❋r❛♥❝❡ ➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶ ✴ ✸✺

■♥tr♦❞✉❝t✐♦♥

❈♦♥t❡①t✿ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

❚✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

y = ❞❡❧❛② x := 0 x < ♣❡r✐♦❞

❆ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ ✐s ✉♥r❡❛❝❤❛❜❧❡ ❆ ♣r♦♣❡rt② t♦ ❜❡ s❛t✐s✜❡❞ ◗✉❡st✐♦♥✿ ❨❡s ✐❢✳ ✳ ✳ ◆♦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷ ✴ ✸✺

■♥tr♦❞✉❝t✐♦♥

❈♦♥t❡①t✿ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

❚✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

y = ❞❡❧❛② x := 0 x < ♣❡r✐♦❞

❆ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ ?

| =

✐s ✉♥r❡❛❝❤❛❜❧❡ ❆ ♣r♦♣❡rt② t♦ ❜❡ s❛t✐s✜❡❞ ◗✉❡st✐♦♥✿ ❞♦❡s t❤❡ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ s❛t✐s❢② t❤❡ ♣r♦♣❡rt②❄ ❨❡s ✐❢✳ ✳ ✳ ◆♦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷ ✴ ✸✺

■♥tr♦❞✉❝t✐♦♥

❈♦♥t❡①t✿ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

❚✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

y = ❞❡❧❛② x := 0 x < ♣❡r✐♦❞

❆ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ ?

| =

✐s ✉♥r❡❛❝❤❛❜❧❡ ❆ ♣r♦♣❡rt② t♦ ❜❡ s❛t✐s✜❡❞ ◗✉❡st✐♦♥✿ ❞♦❡s t❤❡ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ s❛t✐s❢② t❤❡ ♣r♦♣❡rt②❄ ❨❡s ✐❢✳ ✳ ✳ ◆♦ ❈♦✉♥t❡r❡①❛♠♣❧❡

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷ ✴ ✸✺

■♥tr♦❞✉❝t✐♦♥

❈♦♥t❡①t✿ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

❚✐♠❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣

y = ❞❡❧❛② x := 0 x < ♣❡r✐♦❞

❆ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ ?

| =

✐s ✉♥r❡❛❝❤❛❜❧❡ ❆ ♣r♦♣❡rt② t♦ ❜❡ s❛t✐s✜❡❞ ◗✉❡st✐♦♥✿ ❢♦r ✇❤❛t ✈❛❧✉❡s ♦❢ t❤❡ ♣❛r❛♠❡t❡rs ❞♦❡s t❤❡ ♠♦❞❡❧ ♦❢ t❤❡ s②st❡♠ s❛t✐s❢② t❤❡ ♣r♦♣❡rt②❄ ❨❡s ✐❢✳ ✳ ✳ ◆♦ 2❞❡❧❛② > ♣❡r✐♦❞ ∧ ♣❡r✐♦❞ < 20.46

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷ ✴ ✸✺

❖✉t❧✐♥❡

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

▼♦t✐✈❛t✐♦♥

✸

❉❡✜♥✐t✐♦♥s

✹

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss

✻

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

▼♦t✐✈❛t✐♦♥

✸

❉❡✜♥✐t✐♦♥s

✹

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss

✻

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✹ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s✮ ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✺ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✺ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t X ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✺ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t X ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

y ≤ 5 y ≤ 8 ♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✺ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t X ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

y ≤ 5 y ≤ 8 ♣r❡ss❄ y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ y = 8 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✺ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭❚❆✮

❋✐♥✐t❡ st❛t❡ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s ❛♥❞ ❛❝t✐♦♥s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t X ♦❢ ❝❧♦❝❦s ❬❆❧✉r ❛♥❞ ❉✐❧❧✱ ✶✾✾✹✱ ❍❡♥③✐♥❣❡r ❡t ❛❧✳✱ ✶✾✾✹❪

❘❡❛❧✲✈❛❧✉❡❞ ✈❛r✐❛❜❧❡s ❡✈♦❧✈✐♥❣ ❧✐♥❡❛r❧② ❛t t❤❡ s❛♠❡ r❛t❡

❋❡❛t✉r❡s

▲♦❝❛t✐♦♥ ✐♥✈❛r✐❛♥t✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ st❛② ❛t ❛ ❧♦❝❛t✐♦♥ ❚r❛♥s✐t✐♦♥ ❣✉❛r❞✿ ❝♦♥str❛✐♥t t♦ ❜❡ ✈❡r✐✜❡❞ t♦ ❡♥❛❜❧❡ ❛ tr❛♥s✐t✐♦♥ ❈❧♦❝❦ r❡s❡t✿ s♦♠❡ ♦❢ t❤❡ ❝❧♦❝❦s ❝❛♥ ❜❡ s❡t t♦ 0 ❛t ❡❛❝❤ tr❛♥s✐t✐♦♥

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✺ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❈♦♥❝r❡t❡ s❡♠❛♥t✐❝s ♦❢ t✐♠❡❞ ❛✉t♦♠❛t❛

❈♦♥❝r❡t❡ st❛t❡ ♦❢ ❛ ❚❆✿ ♣❛✐r (l, w)✱ ✇❤❡r❡

l ✐s ❛ ❧♦❝❛t✐♦♥✱ w ✐s ❛ ✈❛❧✉❛t✐♦♥ ♦❢ ❡❛❝❤ ❝❧♦❝❦

❈♦♥❝r❡t❡ r✉♥✿ ❛❧t❡r♥❛t✐♥❣ s❡q✉❡♥❝❡ ♦❢ ❝♦♥❝r❡t❡ st❛t❡s ❛♥❞ ❛❝t✐♦♥s ♦r t✐♠❡ ❡❧❛♣s❡

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✻ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r ❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 15.4

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 15.4 ♣r❡ss❄

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 15.4 ♣r❡ss❄ 5

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 15.4 ♣r❡ss❄ 5 ❝✉♣✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y ♣r❡ss❄

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 ♣r❡ss❄ 1.5

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 ♣r❡ss❄ 1.5 ♣r❡ss❄

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 0.8 5 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄ 0.8

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 0.8 5 0.8 5 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄ 0.8 ❝✉♣✦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 0.8 5 0.8 5 3.8 8 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄ 0.8 ❝✉♣✦ 3

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❊①❛♠♣❧❡s ♦❢ ❝♦♥❝r❡t❡ r✉♥s

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

P♦ss✐❜❧❡ ❝♦♥❝r❡t❡ r✉♥s ❢♦r t❤❡ ❝♦✛❡❡ ♠❛❝❤✐♥❡

❈♦✛❡❡ ✇✐t❤ ♥♦ s✉❣❛r

x y 15.4 15.4 5 5 5 5 8 8 8 8 15.4 ♣r❡ss❄ 5 ❝✉♣✦ 3 ❝♦✛❡❡✦

❈♦✛❡❡ ✇✐t❤ ✷ ❞♦s❡s ♦❢ s✉❣❛r

x y 1.5 1.5 1.5 2.7 4.2 4.2 0.8 5 0.8 5 3.8 8 3.8 8 ♣r❡ss❄ 1.5 ♣r❡ss❄ 2.7 ♣r❡ss❄ 0.8 ❝✉♣✦ 3 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✼ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

P❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭P❚❆✮

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s✱ ❛❝t✐♦♥s ❛♥❞ ❝❧♦❝❦s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t ♦❢ ♣❛r❛♠❡t❡rs

❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸❪

❯♥❦♥♦✇♥ ❝♦♥st❛♥ts ✉s❡❞ ✐♥ ❣✉❛r❞s ❛♥❞ ✐♥✈❛r✐❛♥ts

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✽ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

P❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭P❚❆✮

❚✐♠❡❞ ❛✉t♦♠❛t♦♥ ✭s❡ts ♦❢ ❧♦❝❛t✐♦♥s✱ ❛❝t✐♦♥s ❛♥❞ ❝❧♦❝❦s✮ ❛✉❣♠❡♥t❡❞ ✇✐t❤ ❛ s❡t P ♦❢ ♣❛r❛♠❡t❡rs

❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸❪

❯♥❦♥♦✇♥ ❝♦♥st❛♥ts ✉s❡❞ ✐♥ ❣✉❛r❞s ❛♥❞ ✐♥✈❛r✐❛♥ts

y ≤ p2 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = p2 ❝✉♣✦ x ≥ p1 ♣r❡ss❄ x := 0 y = p3 ❝♦✛❡❡✦

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✽ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❱❛❧✉❛t✐♦♥ ♦❢ ❛ P❚❆

• ✐✈❡♥ ❛ P❚❆ A ❛♥❞ ❛ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ v✱ ✇❡ ❞❡♥♦t❡ ❜② v(A)

t❤❡ ✭♥♦♥✲♣❛r❛♠❡tr✐❝✮ t✐♠❡❞ ❛✉t♦♠❛t♦♥ ✇❤❡r❡ ❛❧❧ ♣❛r❛♠❡t❡rs ❛r❡ ✈❛❧✉❛t❡❞ ❜② v

♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦ ♣r❡ss❄ ❝✉♣✦ ♣r❡ss❄ ❝♦✛❡❡✦

✇✐t❤

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✾ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

❱❛❧✉❛t✐♦♥ ♦❢ ❛ P❚❆

• ✐✈❡♥ ❛ P❚❆ A ❛♥❞ ❛ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ v✱ ✇❡ ❞❡♥♦t❡ ❜② v(A)

t❤❡ ✭♥♦♥✲♣❛r❛♠❡tr✐❝✮ t✐♠❡❞ ❛✉t♦♠❛t♦♥ ✇❤❡r❡ ❛❧❧ ♣❛r❛♠❡t❡rs ❛r❡ ✈❛❧✉❛t❡❞ ❜② v

v       

y ≤ p2 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = p2 ❝✉♣✦ x ≥ p1 ♣r❡ss❄ x := 0 y = p3 ❝♦✛❡❡✦

       =

y ≤ 5 y ≤ 8 ♣r❡ss❄ x := 0 y := 0 y = 5 ❝✉♣✦ x ≥ 1 ♣r❡ss❄ x := 0 y = 8 ❝♦✛❡❡✦

✇✐t❤ v :    p1 → 1 p2 → 5 p3 → 8

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✾ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

■♥t❡❣❡rs ♦r r❛t✐♦♥❛❧s❄

■♥ P❚❆s✱ ❜♦t❤ t❤❡ ❝❧♦❝❦s ❛♥❞ t❤❡ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❡✐t❤❡r ✐♥t❡❣❡r✲✈❛❧✉❡❞ ♦r r❛t✐♦♥❛❧✲✈❛❧✉❡❞✳ ❚❤✐s ❣✐✈❡s t❤r❡❡ ♣♦ss✐❜✐❧✐t✐❡s✿ ❈❧♦❝❦s P❛r❛♠❡t❡rs ❉✐s❝r❡t❡ t✐♠❡ N N ❉❡♥s❡ t✐♠❡ R+ N ❉❡♥s❡ t✐♠❡ R+ Q+ ✳ ✳ ✳ ❛♥❞ t❤✐s ❤❛s ❛♥ ✐♠♣❛❝t ♦♥ ❞❡❝✐❞❛❜✐❧✐t②

❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸✱ ▼✐❧❧❡r✱ ✷✵✵✵✱ ❇❡♥❡➨ ❡t ❛❧✳✱ ✷✵✶✺❪

❍❡r❡✿ ✇❡ ❝♦♥s✐❞❡r ✭♠❛✐♥❧②✮ ❞❡♥s❡ t✐♠❡ ✇✐t❤ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ♣❛r❛♠❡t❡rs

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✵ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

■♥t❡❣❡rs ♦r r❛t✐♦♥❛❧s❄

■♥ P❚❆s✱ ❜♦t❤ t❤❡ ❝❧♦❝❦s ❛♥❞ t❤❡ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❡✐t❤❡r ✐♥t❡❣❡r✲✈❛❧✉❡❞ ♦r r❛t✐♦♥❛❧✲✈❛❧✉❡❞✳ ❚❤✐s ❣✐✈❡s t❤r❡❡ ♣♦ss✐❜✐❧✐t✐❡s✿ ❈❧♦❝❦s P❛r❛♠❡t❡rs ❉✐s❝r❡t❡ t✐♠❡ N N ❉❡♥s❡ t✐♠❡ R+ N ❉❡♥s❡ t✐♠❡ R+ Q+ ✳ ✳ ✳ ❛♥❞ t❤✐s ❤❛s ❛♥ ✐♠♣❛❝t ♦♥ ❞❡❝✐❞❛❜✐❧✐t②

❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸✱ ▼✐❧❧❡r✱ ✷✵✵✵✱ ❇❡♥❡➨ ❡t ❛❧✳✱ ✷✵✶✺❪

❍❡r❡✿ ✇❡ ❝♦♥s✐❞❡r ✭♠❛✐♥❧②✮ ❞❡♥s❡ t✐♠❡ ✇✐t❤ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ♣❛r❛♠❡t❡rs

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✵ ✴ ✸✺

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛ P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

■♥t❡❣❡rs ♦r r❛t✐♦♥❛❧s❄

■♥ P❚❆s✱ ❜♦t❤ t❤❡ ❝❧♦❝❦s ❛♥❞ t❤❡ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❡✐t❤❡r ✐♥t❡❣❡r✲✈❛❧✉❡❞ ♦r r❛t✐♦♥❛❧✲✈❛❧✉❡❞✳ ❚❤✐s ❣✐✈❡s t❤r❡❡ ♣♦ss✐❜✐❧✐t✐❡s✿ ❈❧♦❝❦s P❛r❛♠❡t❡rs ❉✐s❝r❡t❡ t✐♠❡ N N ❉❡♥s❡ t✐♠❡ R+ N ❉❡♥s❡ t✐♠❡ R+ Q+ ✳ ✳ ✳ ❛♥❞ t❤✐s ❤❛s ❛♥ ✐♠♣❛❝t ♦♥ ❞❡❝✐❞❛❜✐❧✐t②

❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸✱ ▼✐❧❧❡r✱ ✷✵✵✵✱ ❇❡♥❡➨ ❡t ❛❧✳✱ ✷✵✶✺❪

❍❡r❡✿ ✇❡ ❝♦♥s✐❞❡r ✭♠❛✐♥❧②✮ ❞❡♥s❡ t✐♠❡ ✇✐t❤ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ♣❛r❛♠❡t❡rs

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✵ ✴ ✸✺

▼♦t✐✈❛t✐♦♥

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

▼♦t✐✈❛t✐♦♥

✸

❉❡✜♥✐t✐♦♥s

✹

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss

✻

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✶ ✴ ✸✺

▼♦t✐✈❛t✐♦♥

❚❤❡ q✉❡st✐♦♥ ♦❢ t❤❡ s②♥t❛①

❆❧♠♦st ❡❛❝❤ ✇♦r❦ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❞❡✜♥❡s ✐ts ♦✇♥ s②♥t❛①✿ ❣✉❛r❞s ✐♥✈❛r✐❛♥ts ❛❝❝❡♣t✐♥❣ ❧♦❝❛t✐♦♥s ♦r ♥♦t ❆r❡ t❤❡s❡ ❞❡✜♥✐t✐♦♥s ❡q✉✐✈❛❧❡♥t❧② ❡①♣r❡ss✐✈❡❄ ■♥t❡❣❡r ✈s✳ r❛t✐♦♥❛❧ ✈❛❧✉❡❞ ♣❛r❛♠❡t❡rs ❋♦r ❛ r❛t✐♦♥❛❧ ✈❛❧✉❡❞ P❚❆✱ ❝❛♥ ✇❡ ✜♥❞ ❛♥ ✐♥t❡❣❡r✲✈❛❧✉❡❞ P❚❆ ✇✐t❤ t❤❡ s❛♠❡ ❡①♣r❡ss✐✈❡♥❡ss❄

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✷ ✴ ✸✺

▼♦t✐✈❛t✐♦♥

❚❤❡ q✉❡st✐♦♥ ♦❢ t❤❡ s②♥t❛①

❆❧♠♦st ❡❛❝❤ ✇♦r❦ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡ ❞❡✜♥❡s ✐ts ♦✇♥ s②♥t❛①✿ ❣✉❛r❞s ✐♥✈❛r✐❛♥ts ❛❝❝❡♣t✐♥❣ ❧♦❝❛t✐♦♥s ♦r ♥♦t ❆r❡ t❤❡s❡ ❞❡✜♥✐t✐♦♥s ❡q✉✐✈❛❧❡♥t❧② ❡①♣r❡ss✐✈❡❄ ■♥t❡❣❡r ✈s✳ r❛t✐♦♥❛❧ ✈❛❧✉❡❞ ♣❛r❛♠❡t❡rs ❋♦r ❛ r❛t✐♦♥❛❧ ✈❛❧✉❡❞ P❚❆✱ ❝❛♥ ✇❡ ✜♥❞ ❛♥ ✐♥t❡❣❡r✲✈❛❧✉❡❞ P❚❆ ✇✐t❤ t❤❡ s❛♠❡ ❡①♣r❡ss✐✈❡♥❡ss❄

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✷ ✴ ✸✺

▼♦t✐✈❛t✐♦♥

❙✉❜❝❧❛ss❡s ♦❢ P❚❆s

❙❡✈❡r❛❧ s✉❜❝❧❛ss❡s ✇❡r❡ ❞❡✜♥❡❞ ❢♦r P❚❆s✱ ❡✳ ❣✳✱✿ ▲♦✇❡r✲❜♦✉♥❞ ✉♣♣❡r✲❜♦✉♥❞ P❚❆s

❬❍✉♥❡ ❡t ❛❧✳✱ ✷✵✵✷✱ ❇♦③③❡❧❧✐ ❛♥❞ ▲❛ ❚♦rr❡✱ ✷✵✵✾❪

❇♦✉♥❞❡❞ P❚❆s ✭✐✳ ❡✳✱ ✇✐t❤ ❛ ❜♦✉♥❞❡❞ ♣❛r❛♠❡t❡r ❞♦♠❛✐♥✮ ❆r❡ t❤❡s❡ s✉❜❝❧❛ss❡s ❧❡ss ❡①♣r❡ss✐✈❡ t❤❛♥ P❚❆s❄ ❆♥❞ ❛r❡ P❚❆s ❛♥② ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ ❚❆s✳ ✳ ✳ ❄

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✸ ✴ ✸✺

▼♦t✐✈❛t✐♦♥

❙✉❜❝❧❛ss❡s ♦❢ P❚❆s

❙❡✈❡r❛❧ s✉❜❝❧❛ss❡s ✇❡r❡ ❞❡✜♥❡❞ ❢♦r P❚❆s✱ ❡✳ ❣✳✱✿ ▲♦✇❡r✲❜♦✉♥❞ ✉♣♣❡r✲❜♦✉♥❞ P❚❆s

❬❍✉♥❡ ❡t ❛❧✳✱ ✷✵✵✷✱ ❇♦③③❡❧❧✐ ❛♥❞ ▲❛ ❚♦rr❡✱ ✷✵✵✾❪

❇♦✉♥❞❡❞ P❚❆s ✭✐✳ ❡✳✱ ✇✐t❤ ❛ ❜♦✉♥❞❡❞ ♣❛r❛♠❡t❡r ❞♦♠❛✐♥✮ ❆r❡ t❤❡s❡ s✉❜❝❧❛ss❡s ❧❡ss ❡①♣r❡ss✐✈❡ t❤❛♥ P❚❆s❄ ❆♥❞ ❛r❡ P❚❆s ❛♥② ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ ❚❆s✳ ✳ ✳ ❄

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✸ ✴ ✸✺

▼♦t✐✈❛t✐♦♥

❚♦✇❛r❞s ❛♥ ❡①♣r❡ss✐✈❡♥❡ss ❢♦r P❚❆s

Pr♦❜❧❡♠

❲❤❛t ✐s t❤❡ ❡①♣r❡ss✐✈❡♥❡ss ♦❢ ❛ P❚❆❄

• ♦❛❧

Pr♦♣♦s❡ ❛ ❞❡✜♥✐t✐♦♥ ♦❢ ❡①♣r❡ss✐✈❡♥❡ss ❢♦r P❚❆s✱ ❛♥❞ ❝♦♠♣❛r❡ t❤❡ s②♥t❛① ✴ s✉❜❝❧❛ss❡s ♦❢ P❚❆s✳

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✹ ✴ ✸✺

▼♦t✐✈❛t✐♦♥

❚♦✇❛r❞s ❛♥ ❡①♣r❡ss✐✈❡♥❡ss ❢♦r P❚❆s

Pr♦❜❧❡♠

❲❤❛t ✐s t❤❡ ❡①♣r❡ss✐✈❡♥❡ss ♦❢ ❛ P❚❆❄

• ♦❛❧

Pr♦♣♦s❡ ❛ ❞❡✜♥✐t✐♦♥ ♦❢ ❡①♣r❡ss✐✈❡♥❡ss ❢♦r P❚❆s✱ ❛♥❞ ❝♦♠♣❛r❡ t❤❡ s②♥t❛① ✴ s✉❜❝❧❛ss❡s ♦❢ P❚❆s✳

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✹ ✴ ✸✺

❉❡✜♥✐t✐♦♥s

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

▼♦t✐✈❛t✐♦♥

✸

❉❡✜♥✐t✐♦♥s

✹

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss

✻

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✺ ✴ ✸✺

❉❡✜♥✐t✐♦♥s P❚❆s ✇✐t❤ ❤✐❞❞❡♥ ♣❛r❛♠❡t❡rs

❉❡✜♥✐t✐♦♥ ✭P❚❆ ✇✐t❤ ❤✐❞❞❡♥ ♣❛r❛♠❡t❡rs✮

❆ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t♦♥ ✇✐t❤ ❤✐❞❞❡♥ ♣❛r❛♠❡t❡rs ✭❤❡r❡❛❢t❡r ❤P❚❆✮ A ✐s ❛ t✉♣❧❡ (Σ, L, l0, F, X, P, I, E)✱ ✇❤❡r❡✿

✶ Σ ✐s ❛ ✜♥✐t❡ s❡t ♦❢ ❛❝t✐♦♥s✱ ✷ L ✐s ❛ ✜♥✐t❡ s❡t ♦❢ ❧♦❝❛t✐♦♥s✱ ✸ l0 ∈ L ✐s t❤❡ ✐♥✐t✐❛❧ ❧♦❝❛t✐♦♥✱ ✹ F ⊆ L ✐s ❛ s❡t ♦❢ ❛❝❝❡♣t✐♥❣ ❧♦❝❛t✐♦♥s✱ ✺ X ✐s ❛ ✜♥✐t❡ s❡t ♦❢ ❝❧♦❝❦s✱ ✻ P = Pv ⊎ Pv ✐s ❛ ✜♥✐t❡ s❡t ♦❢ ♣❛r❛♠❡t❡rs ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❤✐❞❞❡♥

♣❛r❛♠❡t❡rs Pv ❛♥❞ ✈✐s✐❜❧❡ ♣❛r❛♠❡t❡rs Pv✱

✼ I ✐s t❤❡ ✐♥✈❛r✐❛♥t✱ ❛ss✐❣♥✐♥❣ t♦ ❡✈❡r② l ∈ L ❛ ❣✉❛r❞ I(l)✱ ✽ E ✐s ❛ ✜♥✐t❡ s❡t ♦❢ ❡❞❣❡s e = (l, g, a, R, l′) ✇❤❡r❡ l, l′ ∈ L ❛r❡ t❤❡

s♦✉r❝❡ ❛♥❞ t❛r❣❡t ❧♦❝❛t✐♦♥s✱ a ∈ Σ ∪ {ǫ} ✭ǫ✿ s✐❧❡♥t ❛❝t✐♦♥✮✱ R ⊆ X ✐s ❛ s❡t ♦❢ ❝❧♦❝❦s t♦ ❜❡ r❡s❡t✱ ❛♥❞ g ✐s ❛ ❣✉❛r❞✳

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✻ ✴ ✸✺

❉❡✜♥✐t✐♦♥s P❚❆s ✇✐t❤ ❤✐❞❞❡♥ ♣❛r❛♠❡t❡rs

▲♦✇❡r✲❜♦✉♥❞ ✴ ❯♣♣❡r✲❜♦✉♥❞ P❚❆s

❉❡✜♥✐t✐♦♥ ✭❤▲✴❯✲P❚❆ ❬❍✉♥❡ ❡t ❛❧✳✱ ✷✵✵✷✱ ❇♦③③❡❧❧✐ ❛♥❞ ▲❛ ❚♦rr❡✱ ✷✵✵✾❪✮

❆♥ ❤▲✴❯✲P❚❆ ✐s ❛♥ ❤P❚❆ ✇❤❡r❡ t❤❡ s❡t ♦❢ ♣❛r❛♠❡t❡rs ✐s ♣❛rt✐t✐♦♥❡❞ ✐♥t♦✿

✶ ❛ s❡t ♦❢ ❧♦✇❡r✲❜♦✉♥❞ ♣❛r❛♠❡t❡rs P− ✇❤❡r❡ ❡❛❝❤ ♣❛r❛♠❡t❡r ❝❛♥

♦♥❧② ❜❡ ❝♦♠♣❛r❡❞ t♦ ❛ ❝❧♦❝❦ ❛s ❛ ❧♦✇❡r ❜♦✉♥❞ ✭✏p ≤ x✑✮ ❛♥❞✱

✷ ❛ s❡t ♦❢ ✉♣♣❡r✲❜♦✉♥❞ ♣❛r❛♠❡t❡rs P+ ✭✏x ≤ p✑✮✳

▲✴❯✲P❚❆s ✭✇✐t❤♦✉t ❤✐❞❞❡♥ ♣❛r❛♠❡t❡rs✮ ❜❡♥❡✜t ❢r♦♠ s♦♠❡ ❞❡❝✐❞❛❜✐❧✐t② ♣r♦♣❡rt✐❡s✱ ✐♥ ❝♦♥tr❛st t♦ ❣❡♥❡r❛❧ P❚❆s

❬❍✉♥❡ ❡t ❛❧✳✱ ✷✵✵✷✱ ❇♦③③❡❧❧✐ ❛♥❞ ▲❛ ❚♦rr❡✱ ✷✵✵✾❪

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✼ ✴ ✸✺

❉❡✜♥✐t✐♦♥s ❚✇♦ ❞❡✜♥✐t✐♦♥s ♦❢ ❡①♣r❡ss✐✈❡♥❡ss

❆ ✜rst ❞❡✜♥✐t✐♦♥ ♦❢ ❡①♣r❡ss✐✈❡♥❡ss

❉❡✜♥✐t✐♦♥ ✭✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡ ♦❢ ❛♥ ❤P❚❆✮

• ✐✈❡♥ ❛♥ ❤P❚❆ A✱ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡ ❯▲(A) ♦❢ A ✐s
• v∈V(P)
• w | w ✐s ❛♥ ✉♥t✐♠❡❞ ✇♦r❞ ❛❝❝❡♣t❡❞ ❜② v(A)
• ❯♥t✐♠❡❞ ✇♦r❞ ❛❝❝❡♣t❡❞ ❜② ❛ ❚❆✿ s❡q✉❡♥❝❡ ♦❢ ❛❝t✐♦♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ✜♥✐t❡

✭t✐♠❡❞✮ r✉♥ ❡♥❞✐♥❣ ✐♥ ❛♥ ❛❝❝❡♣t✐♥❣ ❧♦❝❛t✐♦♥

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✽ ✴ ✸✺

❉❡✜♥✐t✐♦♥s ❚✇♦ ❞❡✜♥✐t✐♦♥s ♦❢ ❡①♣r❡ss✐✈❡♥❡ss

❆ s❡❝♦♥❞ ❞❡✜♥✐t✐♦♥ ♦❢ ❡①♣r❡ss✐✈❡♥❡ss

❉❡✜♥✐t✐♦♥ ✭❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡ ♦❢ ❛♥ ❤P❚❆✮

• ✐✈❡♥ ❛♥ ❤P❚❆ A✱ t❤❡ ❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡ ❈❯▲(A) ♦❢ A ✐s
• v∈V(Pv)
• (w, v) | ∃v ∈ V(Pv) s✳t✳ w ✐s ❛♥ ✉♥t✐♠❡❞ ✇♦r❞ ♦❢ v(v(A))
• ❘❡♠❛r❦s✿

❖♥❧② t❤❡ ✈✐s✐❜❧❡ ♣❛r❛♠❡t❡rs ❛r❡ ❝♦♥s✐❞❡r❡❞ ❈♦♥str❛✐♥❡❞ ❜❡❝❛✉s❡ ❛♥ ❛❧t❡r♥❛t✐✈❡ ✇❛② t♦ r❡♣r❡s❡♥t ❈❯▲(A) ✐s ❛ s❡t ♦❢ ♣❛✐rs (w, K) ✭✇❤❡r❡ K ❞❡♥♦t❡s ❛❧❧ ✈❛❧✉❛t✐♦♥s ❛❝❝❡♣t✐♥❣ w✮ ❖♥❧② P❚❆s ✇✐t❤ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ ✈✐s✐❜❧❡ ♣❛r❛♠❡t❡rs ❝❛♥ ❜❡ ❝♦♠♣❛r❡❞ ✉s✐♥❣ ❈❯▲

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✶✾ ✴ ✸✺

❉❡✜♥✐t✐♦♥s ❚✇♦ ❞❡✜♥✐t✐♦♥s ♦❢ ❡①♣r❡ss✐✈❡♥❡ss

❊①❛♠♣❧❡

① ≤ 1 ① = 1 ❛ ① := 0 ① = 1 ∧ ① ≤ ♣✷ ❜ ① := 0 ① = ♣✶ ❛

❯▲(A) = {a} ∪ {ban | n ∈ N} ✭❛❧s♦ ✇r✐tt❡♥ ❯▲(A) = a + ba∗✮ ❈❯▲

✭❛❧s♦ ✇r✐tt❡♥ ❈❯▲ ✱ ✇✐t❤ ✮

◆♦t❡✿ ❜♦t❤ t❤❡ ♣❛r❛♠❡t❡r ❛♥❞ t❤❡ ❢❛❝t t❤❛t ♠✉st ❜❡ ❛t ❧❡❛st ✶ t♦ ❣♦ t♦ ❛r❡ ❤✐❞❞❡♥✳

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✵ ✴ ✸✺

❉❡✜♥✐t✐♦♥s ❚✇♦ ❞❡✜♥✐t✐♦♥s ♦❢ ❡①♣r❡ss✐✈❡♥❡ss

❊①❛♠♣❧❡

① ≤ 1 ① = 1 ❛ ① := 0 ① = 1 ∧ ① ≤ ♣✷ ❜ ① := 0 ① = ♣✶ ❛

❯▲(A) = {a} ∪ {ban | n ∈ N} ✭❛❧s♦ ✇r✐tt❡♥ ❯▲(A) = a + ba∗✮ ❈❯▲(A) =

• (a, p1 = i) | 0 ≤ i ≤ 1

(ban, p1 = i) | i ∈ N, n ∈ N

• ✭❛❧s♦ ✇r✐tt❡♥ ❈❯▲(A) =
• (a, p1 ≤ 1), (ba∗, p1 ≥ 0)
• ✱ ✇✐t❤ p1 ∈ N✮

◆♦t❡✿ ❜♦t❤ t❤❡ ♣❛r❛♠❡t❡r p2 ❛♥❞ t❤❡ ❢❛❝t t❤❛t p2 ♠✉st ❜❡ ❛t ❧❡❛st ✶ t♦ ❣♦ t♦ ❛r❡ ❤✐❞❞❡♥✳

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✵ ✴ ✸✺

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

▼♦t✐✈❛t✐♦♥

✸

❉❡✜♥✐t✐♦♥s

✹

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss

✻

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✶ ✴ ✸✺

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

Pr♦♣♦s✐t✐♦♥ ✭❘❛t✐♦♥❛❧s ❛♥❞ ✐♥t❡❣❡rs ❛r❡ ❡q✉✐✈❛❧❡♥t ❢♦r ❯▲✮

P❚❆s ✇✐t❤ r❛t✐♦♥❛❧ ♣❛r❛♠❡t❡rs ❛♥❞ P❚❆s ✇✐t❤ ✉♥❜♦✉♥❞❡❞ ✐♥t❡❣❡r ♣❛r❛♠❡t❡rs ❛r❡ ❡q✉✐✈❛❧❡♥t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳

Pr♦♦❢ ✐❞❡❛

❆♥② ✐♥t❡❣❡r✲✈❛❧✉❡❞ P❚❆ ❝❛♥ ❜❡ s✐♠✉❧❛t❡❞ ❜② ❛ r❛t✐♦♥❛❧✲✈❛❧✉❡❞ P❚❆ ✉s✐♥❣ ❛♣♣r♦♣r✐❛t❡ ❣❛❞❣❡ts ❆ r❛t✐♦♥❛❧✲✈❛❧✉❡❞ P❚❆ ❝❛♥ ❜❡ s✐♠✉❧❛t❡❞ ❜② ❛♥ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ❜② s❝❛❧✐♥❣ ✉♣ ❛❧❧ ❡①♣r❡ss✐♦♥s✿ s✐♥❝❡ ✇❡ ❞♦ ♥♦t ❦♥♦✇ ❜② ❤♦✇ ♠✉❝❤ ✇❡ ♥❡❡❞ t♦ s❝❛❧❡✱ ✇❡ ❛❞❞✳ ✳ ✳ ❛ ❢r❡s❤ ♣❛r❛♠❡t❡r✳ ❋♦r ❡①❛♠♣❧❡ ❜❡❝♦♠❡s ✳

❆ s✐♠✐❧❛r r❡s✉❧t ✐s s❤♦✇♥ ❢♦r ▲✴❯✲P❚❆s ❬s❡❡ ♣❛♣❡r❪

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✷ ✴ ✸✺

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

Pr♦♣♦s✐t✐♦♥ ✭❘❛t✐♦♥❛❧s ❛♥❞ ✐♥t❡❣❡rs ❛r❡ ❡q✉✐✈❛❧❡♥t ❢♦r ❯▲✮

P❚❆s ✇✐t❤ r❛t✐♦♥❛❧ ♣❛r❛♠❡t❡rs ❛♥❞ P❚❆s ✇✐t❤ ✉♥❜♦✉♥❞❡❞ ✐♥t❡❣❡r ♣❛r❛♠❡t❡rs ❛r❡ ❡q✉✐✈❛❧❡♥t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳

Pr♦♦❢ ✐❞❡❛

❆♥② ✐♥t❡❣❡r✲✈❛❧✉❡❞ P❚❆ ❝❛♥ ❜❡ s✐♠✉❧❛t❡❞ ❜② ❛ r❛t✐♦♥❛❧✲✈❛❧✉❡❞ P❚❆ ✉s✐♥❣ ❛♣♣r♦♣r✐❛t❡ ❣❛❞❣❡ts ❆ r❛t✐♦♥❛❧✲✈❛❧✉❡❞ P❚❆ ❝❛♥ ❜❡ s✐♠✉❧❛t❡❞ ❜② ❛♥ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ❜② s❝❛❧✐♥❣ ✉♣ ❛❧❧ ❡①♣r❡ss✐♦♥s✿ s✐♥❝❡ ✇❡ ❞♦ ♥♦t ❦♥♦✇ ❜② ❤♦✇ ♠✉❝❤ ✇❡ ♥❡❡❞ t♦ s❝❛❧❡✱ ✇❡ ❛❞❞✳ ✳ ✳ ❛ ❢r❡s❤ ♣❛r❛♠❡t❡r✳ ❋♦r ❡①❛♠♣❧❡ x ≤ 3p1 + 2p2 + 7 ❜❡❝♦♠❡s x ≤ 3p1 + 2p2 + 7p✳

❆ s✐♠✐❧❛r r❡s✉❧t ✐s s❤♦✇♥ ❢♦r ▲✴❯✲P❚❆s ❬s❡❡ ♣❛♣❡r❪

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✷ ✴ ✸✺

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

Pr♦♣♦s✐t✐♦♥ ✭❘❛t✐♦♥❛❧s ❛♥❞ ✐♥t❡❣❡rs ❛r❡ ❡q✉✐✈❛❧❡♥t ❢♦r ❯▲✮

P❚❆s ✇✐t❤ r❛t✐♦♥❛❧ ♣❛r❛♠❡t❡rs ❛♥❞ P❚❆s ✇✐t❤ ✉♥❜♦✉♥❞❡❞ ✐♥t❡❣❡r ♣❛r❛♠❡t❡rs ❛r❡ ❡q✉✐✈❛❧❡♥t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳

Pr♦♦❢ ✐❞❡❛

❆♥② ✐♥t❡❣❡r✲✈❛❧✉❡❞ P❚❆ ❝❛♥ ❜❡ s✐♠✉❧❛t❡❞ ❜② ❛ r❛t✐♦♥❛❧✲✈❛❧✉❡❞ P❚❆ ✉s✐♥❣ ❛♣♣r♦♣r✐❛t❡ ❣❛❞❣❡ts ❆ r❛t✐♦♥❛❧✲✈❛❧✉❡❞ P❚❆ ❝❛♥ ❜❡ s✐♠✉❧❛t❡❞ ❜② ❛♥ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ❜② s❝❛❧✐♥❣ ✉♣ ❛❧❧ ❡①♣r❡ss✐♦♥s✿ s✐♥❝❡ ✇❡ ❞♦ ♥♦t ❦♥♦✇ ❜② ❤♦✇ ♠✉❝❤ ✇❡ ♥❡❡❞ t♦ s❝❛❧❡✱ ✇❡ ❛❞❞✳ ✳ ✳ ❛ ❢r❡s❤ ♣❛r❛♠❡t❡r✳ ❋♦r ❡①❛♠♣❧❡ x ≤ 3p1 + 2p2 + 7 ❜❡❝♦♠❡s x ≤ 3p1 + 2p2 + 7p✳

❆ s✐♠✐❧❛r r❡s✉❧t ✐s s❤♦✇♥ ❢♦r ▲✴❯✲P❚❆s ❬s❡❡ ♣❛♣❡r❪

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✷ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

▼♦t✐✈❛t✐♦♥

✸

❉❡✜♥✐t✐♦♥s

✹

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss

✻

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✸ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss P❚❆s ✐♥ t❤❡ ❈❤♦♠s❦② ❤✐❡r❛r❝❤②

• ❡♥❡r❛❧ P❚❆s✿ t②♣❡ ✵

▲❡♠♠❛

❚✉r✐♥❣✲r❡❝♦❣♥✐③❛❜❧❡ ❧❛♥❣✉❛❣❡s ❛r❡ ❛❧s♦ r❡❝♦❣♥✐③❛❜❧❡ ❜② P❚❆s✳

Pr♦♦❢ ✐❞❡❛

❛❧❧ ♣r♦♦❢s ♦❢ ✉♥❞❡❝✐❞❛❜✐❧✐② ❢♦r P❚❆s r❡❞✉❝❡ ❢r♦♠ t❤❡ ❤❛❧t✐♥❣ ♣r♦❜❧❡♠ ♦❢ ❛ ✷✲❝♦✉♥t❡r ♠❛❝❤✐♥❡✳ ❡✳ ❣✳✱ ❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸❪ ✷✲❝♦✉♥t❡r ♠❛❝❤✐♥❡s ❛r❡ ❚✉r✐♥❣ ❡q✉✐✈❛❧❡♥t

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✹ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss P❚❆s ✐♥ t❤❡ ❈❤♦♠s❦② ❤✐❡r❛r❝❤②

• ❡♥❡r❛❧ P❚❆s✿ t②♣❡ ✵

▲❡♠♠❛

❚✉r✐♥❣✲r❡❝♦❣♥✐③❛❜❧❡ ❧❛♥❣✉❛❣❡s ❛r❡ ❛❧s♦ r❡❝♦❣♥✐③❛❜❧❡ ❜② P❚❆s✳

Pr♦♦❢ ✐❞❡❛

❛❧❧ ♣r♦♦❢s ♦❢ ✉♥❞❡❝✐❞❛❜✐❧✐② ❢♦r P❚❆s r❡❞✉❝❡ ❢r♦♠ t❤❡ ❤❛❧t✐♥❣ ♣r♦❜❧❡♠ ♦❢ ❛ ✷✲❝♦✉♥t❡r ♠❛❝❤✐♥❡✳ ❡✳ ❣✳✱ ❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸❪ ✷✲❝♦✉♥t❡r ♠❛❝❤✐♥❡s ❛r❡ ❚✉r✐♥❣ ❡q✉✐✈❛❧❡♥t

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✹ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss P❚❆s ✐♥ t❤❡ ❈❤♦♠s❦② ❤✐❡r❛r❝❤②

❙✐♥❣❧❡✲❝❧♦❝❦ P❚❆s✿ t②♣❡ ✸

▲❡♠♠❛

❚❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡ r❡❝♦❣♥✐③❡❞ ❜② ❛ P❚❆ ✇✐t❤ ❛ s✐♥❣❧❡ ❝❧♦❝❦ ✭❛♥❞ ❛r❜✐tr❛r✐❧② ♠❛♥② r❛t✐♦♥❛❧✲✈❛❧✉❡❞ ♣❛r❛♠❡t❡rs✮ ✐s r❡❣✉❧❛r✳

Pr♦♦❢ ✐❞❡❛

❋r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ ♣❛r❛♠❡tr✐❝ ③♦♥❡ ❣r❛♣❤ ♦❢ ❛ P❚❆ ✇✐t❤ ❛ s✐♥❣❧❡ ✭♥❡❝❡ss❛r✐❧② ♣❛r❛♠❡tr✐❝✮ ❝❧♦❝❦ ❛♥❞ ❛r❜✐tr❛r✐❧② ♠❛♥② ♣❛r❛♠❡t❡rs ✐s ✜♥✐t❡✳ ❬❆✳ ❛♥❞ ▼❛r❦❡②✱ ✷✵✶✺✱ ❚❤❡♦r❡♠ ✷✵❪

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✺ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss P❚❆s ✐♥ t❤❡ ❈❤♦♠s❦② ❤✐❡r❛r❝❤②

❙✐♥❣❧❡✲❝❧♦❝❦ P❚❆s✿ t②♣❡ ✸

▲❡♠♠❛

❚❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡ r❡❝♦❣♥✐③❡❞ ❜② ❛ P❚❆ ✇✐t❤ ❛ s✐♥❣❧❡ ❝❧♦❝❦ ✭❛♥❞ ❛r❜✐tr❛r✐❧② ♠❛♥② r❛t✐♦♥❛❧✲✈❛❧✉❡❞ ♣❛r❛♠❡t❡rs✮ ✐s r❡❣✉❧❛r✳

Pr♦♦❢ ✐❞❡❛

❋r♦♠ t❤❡ ❢❛❝t t❤❛t t❤❡ ♣❛r❛♠❡tr✐❝ ③♦♥❡ ❣r❛♣❤ ♦❢ ❛ P❚❆ ✇✐t❤ ❛ s✐♥❣❧❡ ✭♥❡❝❡ss❛r✐❧② ♣❛r❛♠❡tr✐❝✮ ❝❧♦❝❦ ❛♥❞ ❛r❜✐tr❛r✐❧② ♠❛♥② ♣❛r❛♠❡t❡rs ✐s ✜♥✐t❡✳ ❬❆✳ ❛♥❞ ▼❛r❦❡②✱ ✷✵✶✺✱ ❚❤❡♦r❡♠ ✷✵❪

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✺ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss P❚❆s ✐♥ t❤❡ ❈❤♦♠s❦② ❤✐❡r❛r❝❤②

❚✇♦ ❝❧♦❝❦s✿ ❛t ❧❡❛st t②♣❡ ✶

❚❤❡♦r❡♠

P❚❆s ✇✐t❤ ✶ ♣❛r❛♠❡tr✐❝ ❝❧♦❝❦✱ ✶ ♥♦♥✲♣❛r❛♠❡tr✐❝ ❝❧♦❝❦ ❛♥❞ ✶ ♣❛r❛♠❡t❡r ❝❛♥ r❡❝♦❣♥✐③❡ ❧❛♥❣✉❛❣❡s t❤❛t ❛r❡ ❝♦♥t❡①t✲s❡♥s✐t✐✈❡✳

Pr♦♦❢ ✐❞❡❛

❚❤❡ ❢♦❧❧♦✇✐♥❣ P❚❆ r❡❝♦❣♥✐③❡s ①✶ ①✷ ♣ ①✶ ①✷ ♣ ①✶ ①✷ ♣ ①✶ ❛ ①✶ ①✶ ①✷ ♣ ❛ ①✶ ①✷ ①✶ ❜ ①✶ ①✶ ①✷ ♣ ❜ ①✶ ①✷ ①✶ ❝ ①✶ ①✶ ①✷ ♣ ❝

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✻ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss P❚❆s ✐♥ t❤❡ ❈❤♦♠s❦② ❤✐❡r❛r❝❤②

❚✇♦ ❝❧♦❝❦s✿ ❛t ❧❡❛st t②♣❡ ✶

❚❤❡♦r❡♠

P❚❆s ✇✐t❤ ✶ ♣❛r❛♠❡tr✐❝ ❝❧♦❝❦✱ ✶ ♥♦♥✲♣❛r❛♠❡tr✐❝ ❝❧♦❝❦ ❛♥❞ ✶ ♣❛r❛♠❡t❡r ❝❛♥ r❡❝♦❣♥✐③❡ ❧❛♥❣✉❛❣❡s t❤❛t ❛r❡ ❝♦♥t❡①t✲s❡♥s✐t✐✈❡✳

Pr♦♦❢ ✐❞❡❛

❚❤❡ ❢♦❧❧♦✇✐♥❣ P❚❆ r❡❝♦❣♥✐③❡s anbncn ①✶ ≤ 1 ∧ ①✷ ≤ ♣ ①✶ ≤ 1 ∧ ①✷ ≤ ♣ ①✶ ≤ 1 ∧ ①✷ ≤ ♣ ①✶ = 1 ❛ ①✶ := 0 ①✶ = 1 ∧ ①✷ = ♣ ❛ ①✶, ①✷ := 0 ①✶ = 1 ❜ ①✶ := 0 ①✶ = 1 ∧ ①✷ = ♣ ❜ ①✶, ①✷ := 0 ①✶ = 1 ❝ ①✶ := 0 ①✶ = 1 ∧ ①✷ = ♣ ❝

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✻ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❖✉t❧✐♥❡

✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss P❚❆s ✐♥ t❤❡ ❈❤♦♠s❦② ❤✐❡r❛r❝❤② ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡ ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✼ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❚❆s ❂ ▲✴❯✲P❚❆s

Pr♦♣♦s✐t✐♦♥

▼♦st s✉❜❝❧❛ss❡s ♦❢ P❚❆s ❛r❡ ♥♦t ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ t✐♠❡❞ ❛✉t♦♠❛t❛ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳

Pr♦♦❢ ✐❞❡❛

▲✴❯✲P❚❆s✿ ✉♥✐♦♥ ♦✈❡r ❛❧❧ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❚❆ r❡♣❧❛❝✐♥❣ ❧♦✇❡r✲❜♦✉♥❞ ♣❛r❛♠❡t❡rs ✇✐t❤ ✵ ❛♥❞ ✉♣♣❡r✲❜♦✉♥❞ ♣❛r❛♠❡t❡r ✇✐t❤ ❛ s✉✣❝✐❡♥t❧② ❧❛r❣❡ ❝♦♥st❛♥t ❬❇♦③③❡❧❧✐ ❛♥❞ ▲❛ ❚♦rr❡✱ ✷✵✵✾❪ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ❜♦✉♥❞❡❞ P❚❆s✿ ❢r♦♠ t❤❡ ❝❧♦s✉r❡ ♦❢ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✽ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❚❆s ❂ ▲✴❯✲P❚❆s

Pr♦♣♦s✐t✐♦♥

▼♦st s✉❜❝❧❛ss❡s ♦❢ P❚❆s ❛r❡ ♥♦t ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ t✐♠❡❞ ❛✉t♦♠❛t❛ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳

Pr♦♦❢ ✐❞❡❛

▲✴❯✲P❚❆s✿ ✉♥✐♦♥ ♦✈❡r ❛❧❧ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❚❆ r❡♣❧❛❝✐♥❣ ❧♦✇❡r✲❜♦✉♥❞ ♣❛r❛♠❡t❡rs ✇✐t❤ ✵ ❛♥❞ ✉♣♣❡r✲❜♦✉♥❞ ♣❛r❛♠❡t❡r ✇✐t❤ ❛ s✉✣❝✐❡♥t❧② ❧❛r❣❡ ❝♦♥st❛♥t ❬❇♦③③❡❧❧✐ ❛♥❞ ▲❛ ❚♦rr❡✱ ✷✵✵✾❪ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ❜♦✉♥❞❡❞ P❚❆s✿ ❢r♦♠ t❤❡ ❝❧♦s✉r❡ ♦❢ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✽ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❚❆s ❂ ▲✴❯✲P❚❆s ❂ ❜♦✉♥❞❡❞ P❚❆s

Pr♦♣♦s✐t✐♦♥

▼♦st s✉❜❝❧❛ss❡s ♦❢ P❚❆s ❛r❡ ♥♦t ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ t✐♠❡❞ ❛✉t♦♠❛t❛ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳

Pr♦♦❢ ✐❞❡❛

▲✴❯✲P❚❆s✿ ✉♥✐♦♥ ♦✈❡r ❛❧❧ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❚❆ r❡♣❧❛❝✐♥❣ ❧♦✇❡r✲❜♦✉♥❞ ♣❛r❛♠❡t❡rs ✇✐t❤ ✵ ❛♥❞ ✉♣♣❡r✲❜♦✉♥❞ ♣❛r❛♠❡t❡r ✇✐t❤ ❛ s✉✣❝✐❡♥t❧② ❧❛r❣❡ ❝♦♥st❛♥t ❬❇♦③③❡❧❧✐ ❛♥❞ ▲❛ ❚♦rr❡✱ ✷✵✵✾❪ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ❜♦✉♥❞❡❞ P❚❆s✿ ❢r♦♠ t❤❡ ❝❧♦s✉r❡ ♦❢ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✽ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❚❆s ❁ P❚❆s

Pr♦♣♦s✐t✐♦♥

P❚❆s ❛r❡ str✐❝t❧② ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ ❚❆s ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳

Pr♦♦❢ ✐❞❡❛

❋r♦♠ t❤❡ r❡s✉❧ts ♦♥ t❤❡ ❤✐❡r❛r❝❤② ♦❢ ❈❤♦♠s❦②✳

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✷✾ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

P❚❆s ❂ ❢❝♣✲P❚❆s ❂ ❤P❚❆s

Pr♦♣♦s✐t✐♦♥

❋✉❧❧② ♣❛r❛♠❡tr✐❝ ❝♦♥str❛✐♥ts P❚❆s ✭❢♣❝✲P❚❆s✮✱ ✐✳ ❡✳✱ ❛❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❧✐♥❡❛r t❡r♠s ✭3p1 − p2 > 5p3✮ ✐♥ ❣✉❛r❞s✱ ❛r❡ ♥♦t ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ P❚❆s ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳ ❤P❚❆s ❛r❡ ❡q✉❛❧❧② ❡①♣r❡ss✐✈❡ ✇✐t❤ P❚❆s ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳

Pr♦♦❢ ✐❞❡❛

✶

❇② tr❛♥s❧❛t✐♦♥ t♦ ❛♥ ❡q✉✐✈❛❧❡♥t P❚❆ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✱ t❤❛t ❢♦❧❧♦✇s t❤❡ ♠♦st r❡str✐❝t✐✈❡ s②♥t❛① ✭t❤❛t ♦❢ ❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸❪✮

✷

❇② ✏❞❡✲❤✐❞✐♥❣✑ ♣❛r❛♠❡t❡rs

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✵ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

P❚❆s ❂ ❢❝♣✲P❚❆s ❂ ❤P❚❆s

Pr♦♣♦s✐t✐♦♥

❋✉❧❧② ♣❛r❛♠❡tr✐❝ ❝♦♥str❛✐♥ts P❚❆s ✭❢♣❝✲P❚❆s✮✱ ✐✳ ❡✳✱ ❛❧❧♦✇✐♥❣ ♣❛r❛♠❡tr✐❝ ❧✐♥❡❛r t❡r♠s ✭3p1 − p2 > 5p3✮ ✐♥ ❣✉❛r❞s✱ ❛r❡ ♥♦t ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ P❚❆s ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳ ❤P❚❆s ❛r❡ ❡q✉❛❧❧② ❡①♣r❡ss✐✈❡ ✇✐t❤ P❚❆s ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✳

Pr♦♦❢ ✐❞❡❛

✶

❇② tr❛♥s❧❛t✐♦♥ t♦ ❛♥ ❡q✉✐✈❛❧❡♥t P❚❆ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✱ t❤❛t ❢♦❧❧♦✇s t❤❡ ♠♦st r❡str✐❝t✐✈❡ s②♥t❛① ✭t❤❛t ♦❢ ❬❆❧✉r ❡t ❛❧✳✱ ✶✾✾✸❪✮

✷

❇② ✏❞❡✲❤✐❞✐♥❣✑ ♣❛r❛♠❡t❡rs

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✵ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❖✉t❧✐♥❡

✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss P❚❆s ✐♥ t❤❡ ❈❤♦♠s❦② ❤✐❡r❛r❝❤② ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡ ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✶ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❊①♣r❡ss✐✈❡♥❡ss ✇✳r✳t✳ t❤❡ ❈❯▲

▲✴❯ P❚❆s

▼❛✐♥ r❡s✉❧ts✿

✭♠♦r❡ ✐♥ ♣❛♣❡r✮

▲✴❯✲P❚❆s ❁ P❚❆s ❤▲✴❯✲P❚❆s ❂ ▲✴❯✲P❚❆s ❜▲✴❯✲P❚❆ ✐♥❝♦♠♣✳ ✇✐t❤ ▲✴❯ ❜♦✉♥❞❡❞ P❚❆s ❁ P❚❆s ❤P❚❆s ❃ P❚❆s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✷ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❊①♣r❡ss✐✈❡♥❡ss ✇✳r✳t✳ t❤❡ ❈❯▲

▲✴❯ ❂ ❤▲✴❯✲P❚❆s P❚❆s

▼❛✐♥ r❡s✉❧ts✿

✭♠♦r❡ ✐♥ ♣❛♣❡r✮

▲✴❯✲P❚❆s ❁ P❚❆s ❤▲✴❯✲P❚❆s ❂ ▲✴❯✲P❚❆s ❜▲✴❯✲P❚❆ ✐♥❝♦♠♣✳ ✇✐t❤ ▲✴❯ ❜♦✉♥❞❡❞ P❚❆s ❁ P❚❆s ❤P❚❆s ❃ P❚❆s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✷ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❊①♣r❡ss✐✈❡♥❡ss ✇✳r✳t✳ t❤❡ ❈❯▲

❜♦✉♥❞❡❞ ▲✴❯ ▲✴❯ ❂ ❤▲✴❯✲P❚❆s P❚❆s

▼❛✐♥ r❡s✉❧ts✿

✭♠♦r❡ ✐♥ ♣❛♣❡r✮

▲✴❯✲P❚❆s ❁ P❚❆s ❤▲✴❯✲P❚❆s ❂ ▲✴❯✲P❚❆s ❜▲✴❯✲P❚❆ ✐♥❝♦♠♣✳ ✇✐t❤ ▲✴❯ ❜♦✉♥❞❡❞ P❚❆s ❁ P❚❆s ❤P❚❆s ❃ P❚❆s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✷ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❊①♣r❡ss✐✈❡♥❡ss ✇✳r✳t✳ t❤❡ ❈❯▲

❜♦✉♥❞❡❞ ▲✴❯ ▲✴❯ ❂ ❤▲✴❯✲P❚❆s ❜♦✉♥❞❡❞ P❚❆s P❚❆s

▼❛✐♥ r❡s✉❧ts✿

✭♠♦r❡ ✐♥ ♣❛♣❡r✮

▲✴❯✲P❚❆s ❁ P❚❆s ❤▲✴❯✲P❚❆s ❂ ▲✴❯✲P❚❆s ❜▲✴❯✲P❚❆ ✐♥❝♦♠♣✳ ✇✐t❤ ▲✴❯ ❜♦✉♥❞❡❞ P❚❆s ❁ P❚❆s ❤P❚❆s ❃ P❚❆s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✷ ✴ ✸✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss ❈♦♠♣❛r✐s♦♥ ✇✳r✳t✳ t❤❡ ❝♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡

❊①♣r❡ss✐✈❡♥❡ss ✇✳r✳t✳ t❤❡ ❈❯▲

❜♦✉♥❞❡❞ ▲✴❯ ▲✴❯ ❂ ❤▲✴❯✲P❚❆s ❜♦✉♥❞❡❞ P❚❆s P❚❆s ❤P❚❆s

▼❛✐♥ r❡s✉❧ts✿

✭♠♦r❡ ✐♥ ♣❛♣❡r✮

▲✴❯✲P❚❆s ❁ P❚❆s ❤▲✴❯✲P❚❆s ❂ ▲✴❯✲P❚❆s ❜▲✴❯✲P❚❆ ✐♥❝♦♠♣✳ ✇✐t❤ ▲✴❯ ❜♦✉♥❞❡❞ P❚❆s ❁ P❚❆s ❤P❚❆s ❃ P❚❆s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✷ ✴ ✸✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

❖✉t❧✐♥❡

✶

P❛r❛♠❡tr✐❝ ❚✐♠❡❞ ❆✉t♦♠❛t❛

✷

▼♦t✐✈❛t✐♦♥

✸

❉❡✜♥✐t✐♦♥s

✹

■♥t❡❣❡rs ✈s✳ ❘❛t✐♦♥❛❧s

✺

❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❊①♣r❡ss✐✈❡♥❡ss

✻

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✸ ✴ ✸✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s ❙✉♠♠❛r②

❙✉♠♠❛r②

❋✐rst ❛tt❡♠♣t t♦ ❞❡✜♥❡ t❤❡ ❡①♣r❡ss✐✈❡♥❡ss ♦❢ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t❛ ❛♥❞ s✉❜❝❧❛ss❡s

■♥❝❧✉❞✐♥❣ ❜② ❤✐❞✐♥❣ ♣❛r❛♠❡t❡rs

❯♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✿ ♠♦st ❢♦r♠❛❧✐s♠s ❛r❡ ♥♦t ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ t✐♠❡❞ ❛✉t♦♠❛t❛ ❈♦♥str❛✐♥❡❞ ✉♥t✐♠❡❞ ❧❛♥❣✉❛❣❡✿ ♣r♦♣♦s❡❞ ❛ ✜rst ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ s✉❜❝❧❛ss❡s ♦❢ P❚❆s

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✹ ✴ ✸✺

❈♦♥❝❧✉s✐♦♥ ❛♥❞ P❡rs♣❡❝t✐✈❡s ❙✉♠♠❛r②

P❡rs♣❡❝t✐✈❡s

❈♦♥s✐❞❡r t❤❡ t✐♠❡❞ ❧❛♥❣✉❛❣❡ ❛♥❞ ❝♦♥str❛✐♥❡❞ t✐♠❡❞ ❧❛♥❣✉❛❣❡ ❛s ✇❡❧❧ ✭❇✉t ✐♥ ❢❛❝t✱ r❡s✉❧ts ❞♦ ♥♦t s❡❡♠ t♦ ❞✐✛❡r ❢r♦♠ t❤❡✐r ✉♥t✐♠❡❞ ❝♦✉♥t❡r♣❛rts✮ ❚❤❡ ❡①♣r❡ss✐✈❡♥❡ss ♦❢ t✇♦✲❝❧♦❝❦ P❚❆s r❡♠❛✐♥s ♦♣❡♥ ❆♥❞ ❝❧♦s❡ t♦ ✇❡❧❧✲❦♥♦✇♥ ♦♣❡♥ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s P♦✇❡r ♦❢ s✐❧❡♥t tr❛♥s✐t✐♦♥s ✐♥ ♦✉r ❞❡✜♥✐t✐♦♥s❄ ❘❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❣✉❛r❞ s②♥t❛① ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❝❧♦❝❦s ❛♥❞ ♣❛r❛♠❡t❡rs ■s ✉s✐♥❣ ❛ s②♥t❛① ✏x ∼ p + c✑ ❡q✉✐✈❛❧❡♥t t♦ ❛ s②♥t❛① ✏x ∼ p | c✑ ❛t t❤❡ ❝♦st ♦❢ ♦♥❡ ❡①tr❛ ❝❧♦❝❦ ✐♥ t❤❡ P❚❆❄

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✺ ✴ ✸✺

❇✐❜❧✐♦❣r❛♣❤②

❇✐❜❧✐♦❣r❛♣❤②

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✻ ✴ ✸✺

❇✐❜❧✐♦❣r❛♣❤②

❘❡❢❡r❡♥❝❡s ■

❆❧✉r✱ ❘✳ ❛♥❞ ❉✐❧❧✱ ❉✳ ▲✳ ✭✶✾✾✹✮✳ ❆ t❤❡♦r② ♦❢ t✐♠❡❞ ❛✉t♦♠❛t❛✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✶✷✻✭✷✮✿✶✽✸✕✷✸✺✳ ❆❧✉r✱ ❘✳✱ ❍❡♥③✐♥❣❡r✱ ❚✳ ❆✳✱ ❛♥❞ ❱❛r❞✐✱ ▼✳ ❨✳ ✭✶✾✾✸✮✳ P❛r❛♠❡tr✐❝ r❡❛❧✲t✐♠❡ r❡❛s♦♥✐♥❣✳ ■♥ ❙❚❖❈✱ ♣❛❣❡s ✺✾✷✕✻✵✶✳ ❆❈▼✳ ❆♥❞ré✱ ➱✳ ❛♥❞ ▼❛r❦❡②✱ ◆✳ ✭✷✵✶✺✮✳ ▲❛♥❣✉❛❣❡ ♣r❡s❡r✈❛t✐♦♥ ♣r♦❜❧❡♠s ✐♥ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t❛✳ ■♥ ❋❖❘▼❆❚❙✱ ✈♦❧✉♠❡ ✾✷✻✽ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✷✼✕✹✸✳ ❙♣r✐♥❣❡r✳ ❇❡♥❡➨✱ ◆✳✱ ❇❡③❞➙❦✱ P✳✱ ▲❛rs❡♥✱ ❑✳ ●✳✱ ❛♥❞ ❙r❜❛✱ ❏✳ ✭✷✵✶✺✮✳ ▲❛♥❣✉❛❣❡ ❡♠♣t✐♥❡ss ♦❢ ❝♦♥t✐♥✉♦✉s✲t✐♠❡ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t❛✳ ■♥ ■❈❆▲P✱ P❛rt ■■✱ ✈♦❧✉♠❡ ✾✶✸✺ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✻✾✕✽✶✳ ❙♣r✐♥❣❡r✳ ❇♦③③❡❧❧✐✱ ▲✳ ❛♥❞ ▲❛ ❚♦rr❡✱ ❙✳ ✭✷✵✵✾✮✳ ❉❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ❢♦r ❧♦✇❡r✴✉♣♣❡r ❜♦✉♥❞ ♣❛r❛♠❡tr✐❝ t✐♠❡❞ ❛✉t♦♠❛t❛✳ ❋♦r♠❛❧ ▼❡t❤♦❞s ✐♥ ❙②st❡♠ ❉❡s✐❣♥✱ ✸✺✭✷✮✿✶✷✶✕✶✺✶✳

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✼ ✴ ✸✺

❇✐❜❧✐♦❣r❛♣❤②

❘❡❢❡r❡♥❝❡s ■■

❍❡♥③✐♥❣❡r✱ ❚✳ ❆✳✱ ◆✐❝♦❧❧✐♥✱ ❳✳✱ ❙✐❢❛❦✐s✱ ❏✳✱ ❛♥❞ ❨♦✈✐♥❡✱ ❙✳ ✭✶✾✾✹✮✳ ❙②♠❜♦❧✐❝ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ❢♦r r❡❛❧✲t✐♠❡ s②st❡♠s✳ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ✶✶✶✭✷✮✿✶✾✸✕✷✹✹✳ ❍✉♥❡✱ ❚✳✱ ❘♦♠✐❥♥✱ ❏✳✱ ❙t♦❡❧✐♥❣❛✱ ▼✳✱ ❛♥❞ ❱❛❛♥❞r❛❣❡r✱ ❋✳ ❲✳ ✭✷✵✵✷✮✳ ▲✐♥❡❛r ♣❛r❛♠❡tr✐❝ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ♦❢ t✐♠❡❞ ❛✉t♦♠❛t❛✳ ❏♦✉r♥❛❧ ♦❢ ▲♦❣✐❝ ❛♥❞ ❆❧❣❡❜r❛✐❝ Pr♦❣r❛♠♠✐♥❣✱ ✺✷✲✺✸✿✶✽✸✕✷✷✵✳ ▼✐❧❧❡r✱ ❏✳ ❙✳ ✭✷✵✵✵✮✳ ❉❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❢♦r t✐♠❡❞ ❛✉t♦♠❛t❛ ❛♥❞ s❡♠✐✲❧✐♥❡❛r ❤②❜r✐❞ ❛✉t♦♠❛t❛✳ ■♥ ❍❙❈❈✱ ✈♦❧✉♠❡ ✶✼✾✵ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✷✾✻✕✸✵✾✳ ❙♣r✐♥❣❡r✳

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✽ ✴ ✸✺

▲✐❝❡♥s✐♥❣

▲✐❝❡♥s✐♥❣

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✸✾ ✴ ✸✺

▲✐❝❡♥s✐♥❣

❙♦✉r❝❡ ♦❢ t❤❡ ❣r❛♣❤✐❝s ✉s❡❞ ■

❚✐t❧❡✿ ❙♠✐❧❡② ❣r❡❡♥ ❛❧✐❡♥ ❜✐❣ ❡②❡s ✭❛❛❛❤✮ ❆✉t❤♦r✿ ▲❛❞②♦❢❍❛ts ❙♦✉r❝❡✿ ❤tt♣s✿✴✴❝♦♠♠♦♥s✳✇✐❦✐♠❡❞✐❛✳♦r❣✴✇✐❦✐✴❋✐❧❡✿❙♠✐❧❡②❴❣r❡❡♥❴❛❧✐❡♥❴❜✐❣❴❡②❡s✳s✈❣ ▲✐❝❡♥s❡✿ ♣✉❜❧✐❝ ❞♦♠❛✐♥ ❚✐t❧❡✿ ❙♠✐❧❡② ❣r❡❡♥ ❛❧✐❡♥ ❜✐❣ ❡②❡s ✭❝r②✮ ❆✉t❤♦r✿ ▲❛❞②♦❢❍❛ts ❙♦✉r❝❡✿ ❤tt♣s✿✴✴❝♦♠♠♦♥s✳✇✐❦✐♠❡❞✐❛✳♦r❣✴✇✐❦✐✴❋✐❧❡✿❙♠✐❧❡②❴❣r❡❡♥❴❛❧✐❡♥❴❜✐❣❴❡②❡s✳s✈❣ ▲✐❝❡♥s❡✿ ♣✉❜❧✐❝ ❞♦♠❛✐♥

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✹✵ ✴ ✸✺

▲✐❝❡♥s✐♥❣

▲✐❝❡♥s❡ ♦❢ t❤✐s ❞♦❝✉♠❡♥t

❚❤✐s ♣r❡s❡♥t❛t✐♦♥ ❝❛♥ ❜❡ ♣✉❜❧✐s❤❡❞✱ r❡✉s❡❞ ❛♥❞ ♠♦❞✐✜❡❞ ✉♥❞❡r t❤❡ t❡r♠s ♦❢ t❤❡ ❧✐❝❡♥s❡ ❈r❡❛t✐✈❡ ❈♦♠♠♦♥s ❆ttr✐❜✉t✐♦♥✲❙❤❛r❡❆❧✐❦❡ ✹✳✵ ❯♥♣♦rt❡❞ ✭❈❈ ❇❨✲❙❆ ✹✳✵✮

✭▲

❆

❚ ❊❳ s♦✉r❝❡ ❛✈❛✐❧❛❜❧❡ ♦♥ ❞❡♠❛♥❞✮

❆✉t❤♦r✿ ➱t✐❡♥♥❡ ❆♥❞ré ❤tt♣s✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✲s❛✴✹✳✵✴

➱✳ ❆♥❞ré ✭P❛r✐s ✶✸ ✴ ◆❛♥t❡s✮ ❊①♣r❡ss✐✈❡♥❡ss ♦❢ P❚❆s ✷✹t❤ ❆✉❣✉st ✷✵✶✻ ✹✶ ✴ ✸✺