❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❛t❛ ❛♥❞ ❑♥♦✇❧❡❞❣❡ ❙②st❡♠s ❊P❈▲ ❇❛s✐❝ ❚r❛✐♥✐♥❣ ❈❛♠♣ ✷✵✶✷ P❛rt ❙✐① ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ■♥st✐t✉t ❢ür ■♥❢♦r♠❛t✐♦♥ss②st❡♠❡ ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ❲✐❡♥ ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✴✻✵
❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ❖✉t❧✐♥❡ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t② ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✴✻✵
❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s ❖✉t❧✐♥❡ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t② ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✴✻✵
❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s ❚❤❡ ❙t♦r② ❙♦ ❢❛r ◗✉❡r② ❧❛♥❣✉❛❣❡s ✇✐t❤ t❤❡ ❢♦r♠ ♦❢ ❧♦❣✐❝s ❙②♥t❛①✱ ❞❡❝❧❛r❛t✐✈❡ ❛♥❞ ♦♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ❍♦✇ ♠✉❝❤ r❡s♦✉r❝❡ ✭t✐♠❡✱ s♣❛❝❡✮ ❞♦ ✇❡ ♥❡❡❞ ❢♦r t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡s❡ s❡♠❛♥t✐❝s❄ ⇒ ❈♦♠♣❧❡①✐t② ❲❤❛t ❦✐♥❞ ♦❢ ♣r♦♣❡rt✐❡s ❝❛♥ ❛ ❣✐✈❡♥ q✉❡r② ❧❛♥❣✉❛❣❡ ❡①♣r❡ss❄ ■s Q 1 ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ Q 2 ❄ ⇒ ❊①♣r❡ss✐✈❡ ♣♦✇❡r ❆ ❞r❡❛♠ q✉❡r② ❧❛♥❣✉❛❣❡ s❤♦✉❧❞ ❤❛✈❡✿ ❧♦✇❡r ❝♦♠♣❧❡①✐t②✱ ❛♥❞ ♠♦r❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✴✻✵
❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s ❚❤❡ ❘❡s✉❧ts ❖✈❡r✈✐❡✇ ◗✉❡r② ❉❛t❛ ❈♦♠♣❧❡①✐t② Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t② ❈♦♥❥✉♥❝t✐✈❡ q✉❡r② AC 0 NP ✲❝♦♠♣❧❡t❡ ❋❖ AC 0 P❙P❆❈❊✲❝♦♠♣❧❡t❡ Pr♦♣✳ ▲P P✲❝♦♠♣❧❡t❡ ❉❛t❛❧♦❣ P✲❝♦♠♣❧❡t❡ ❊❳P❚■▼❊✲❝♦♠♣❧❡t❡ ❙tr❛t✐✜❡❞ ❉❛t❛❧♦❣ P✲❝♦♠♣❧❡t❡ ❊❳P❚■▼❊✲❝♦♠♣❧❡t❡ ❉❛t❛❧♦❣✭❲❋▼✮ P✲❝♦♠♣❧❡t❡ ❊❳P❚■▼❊✲❝♦♠♣❧❡t❡ ❉❛t❛❧♦❣✭■◆❋✮ P✲❝♦♠♣❧❡t❡ ❊❳P❚■▼❊✲❝♦♠♣❧❡t❡ ❉❛t❛❧♦❣✭❙t❛❜❧❡ ▼♦❞❡❧✮ co ✲ NP ✲❝♦♠♣❧❡t❡ co ✲ NEXPTIME ✲❝♦♠♣❧❡t❡ Π p co ✲ NEXPTIME NP ✲❝♦♠♣❧❡t❡ ❉✐s❥✉♥✳ ❉❛t❛❧♦❣ 2 ✲❝♦♠♣❧❡t❡ ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✴✻✵
❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s ❚❤❡ ❘❡s✉❧ts ❖✈❡r✈✐❡✇ ❚♦❞❛② ✇❡ s❤❛❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ ◗✉❡r② ❉❛t❛ ❈♦♠♣❧❡①✐t② Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t② ❈♦♥❥✉♥❝t✐✈❡ q✉❡r② AC 0 NP ✲❝♦♠♣❧❡t❡ ❋❖ AC 0 P❙P❆❈❊✲❝♦♠♣❧❡t❡ Pr♦♣✳ ▲P P✲❝♦♠♣❧❡t❡ ❉❛t❛❧♦❣ P✲❝♦♠♣❧❡t❡ ❊❳P❚■▼❊✲❝♦♠♣❧❡t❡ ❙tr❛t✐✜❡❞ ❉❛t❛❧♦❣ P✲❝♦♠♣❧❡t❡ ❊❳P❚■▼❊✲❝♦♠♣❧❡t❡ ❉❛t❛❧♦❣✭❲❋▼✮ P✲❝♦♠♣❧❡t❡ ❊❳P❚■▼❊✲❝♦♠♣❧❡t❡ ❉❛t❛❧♦❣✭■◆❋✮ P✲❝♦♠♣❧❡t❡ ❊❳P❚■▼❊✲❝♦♠♣❧❡t❡ ❉❛t❛❧♦❣✭❙t❛❜❧❡ ▼♦❞❡❧✮ ❝♦✲◆P✲❝♦♠♣❧❡t❡ ❝♦✲◆❊❳P❚■▼❊✲❝♦♠♣❧❡t❡ Π p co ✲ NEXPTIME NP ✲❝♦♠♣❧❡t❡ ❉✐s❥✉♥✳ ❉❛t❛❧♦❣ 2 ✲❝♦♠♣❧❡t❡ ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✻✴✻✵
❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s ●♦❛❧ ♦❢ t❤✐s ▲❡❝t✉r❡ ❘❡❝❛❧❧ ❜❛s✐❝ ❝♦♥❝❡♣t ♦❢ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ❘❡❝❛❧❧ ♣r♦❜❧❡♠ r❡❞✉❝t✐♦♥ ✭❧♦❣s♣❛❝❡ r❡❞✉❝t✐♦♥✮✱ ❝♦♠♣❧❡t❡♥❡ss✱ ❉❡✜♥❡ ❞❛t❛ ❝♦♠♣❧❡①✐t② ❛♥❞ ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t② ●❡t ❛ t❛st❡ ♦❢ t❤❡ ❤❛r❞♥❡ss ♣r♦♦❢s ♦❢ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ✈✐❛ ✭♥✐❝❡✮ ❡♥❝♦❞✐♥❣ ♦❢ ❛ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ▲❡❛r♥ ❜❛s✐❝s ❛❜♦✉t ❡①♣r❡ss✐✈❡ ♣♦✇❡r ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✼✴✻✵
❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s ❉❡❝✐s✐♦♥ Pr♦❜❧❡♠s Pr♦❜❧❡♠s ✇❤❡r❡ t❤❡ ❛♥s✇❡r ✐s ✏②❡s✑ ♦r ✏♥♦✑ ❋♦r♠❛❧❧②✱ • ❆ ❧❛♥❣✉❛❣❡ L ♦✈❡r s♦♠❡ ❛❧♣❤❛❜❡t Σ ✳ • ❆♥ ✐♥st❛♥❝❡ ✐s ❣✐✈❡♥ ❛s ❛ ✇♦r❞ x ∈ Σ ∗ ✳ • ◗✉❡st✐♦♥✿ ✇❤❡t❤❡r x ∈ L ❤♦❧❞s ❚❤❡ r❡s♦✉r❝❡s ✭✐✳❡✳✱ ❡✐t❤❡r t✐♠❡ ♦r s♣❛❝❡✮ r❡q✉✐r❡❞ ✐♥ t❤❡ ✇♦rst ❝❛s❡ t♦ ✜♥❞ t❤❡ ❝♦rr❡❝t ❛♥s✇❡r ❢♦r ❛♥② ✐♥st❛♥❝❡ x ♦❢ ❛ ♣r♦❜❧❡♠ L ✐s r❡❢❡rr❡❞ t♦ ❛s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♣r♦❜❧❡♠ L ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✽✴✻✵
❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s ❈♦♠♣❧❡①✐t✐❡s ▲❡t P ❜❡ ❛ ♣r♦❣r❛♠ ✇✐t❤ s♦♠❡ q✉❡r② ❧❛♥❣✉❛❣❡✱ D in ✐♥♣✉t ❞❛t❛❜❛s❡ ❛♥❞ A ❛ ❣r♦✉♥❞ ❛t♦♠✳ ❞❛t❛ ❝♦♠♣❧❡①✐t② ▲❡t P ❜❡ ✜①❡❞ ■♥st❛♥❝❡✳ D in ❛♥❞ A ✳ ◗✉❡st✐♦♥✳ ❉♦❡s D in ∪ P | = A ❤♦❧❞❄ ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t② ✭❛✳❦✳❛✳ ❡①♣r❡ss✐♦♥ ❝♦♠♣❧❡①✐t② ✮ ▲❡t D in ❜❡ ✜①❡❞ ✳ ■♥st❛♥❝❡✳ P ❛♥❞ A ✳ ◗✉❡st✐♦♥✳ ❉♦❡s D in ∪ P | = A ❤♦❧❞❄ ❝♦♠❜✐♥❡❞ ❝♦♠♣❧❡①✐t② ■♥st❛♥❝❡✳ P ✱ D in ❛♥❞ A ✳ ◗✉❡st✐♦♥✳ ❉♦❡s D in ∪ P | = A ❤♦❧❞❄ ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✾✴✻✵
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