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❋♦✉♥❞❛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 Q1 ♠♦r❡ ❡①♣r❡ss✐✈❡ t❤❛♥ Q2❄ ⇒ ❊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 AC0 NP✲❝♦♠♣❧❡t❡ ❋❖ AC0 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❡ ❉✐s❥✉♥✳ ❉❛t❛❧♦❣ Πp

2✲❝♦♠♣❧❡t❡

co✲NEXPTIMENP✲❝♦♠♣❧❡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 AC0 NP✲❝♦♠♣❧❡t❡ ❋❖ AC0 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❡ ❉✐s❥✉♥✳ ❉❛t❛❧♦❣ Πp

2✲❝♦♠♣❧❡t❡

co✲NEXPTIMENP✲❝♦♠♣❧❡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 ❧❛♥❣✉❛❣❡✱ Din ✐♥♣✉t ❞❛t❛❜❛s❡ ❛♥❞ A ❛ ❣r♦✉♥❞ ❛t♦♠✳ ❞❛t❛ ❝♦♠♣❧❡①✐t② ▲❡t P ❜❡ ✜①❡❞ ■♥st❛♥❝❡✳ Din ❛♥❞ A✳ ◗✉❡st✐♦♥✳ ❉♦❡s Din ∪ P | = A ❤♦❧❞❄ ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t② ✭❛✳❦✳❛✳ ❡①♣r❡ss✐♦♥ ❝♦♠♣❧❡①✐t②✮ ▲❡t Din ❜❡ ✜①❡❞✳ ■♥st❛♥❝❡✳ P ❛♥❞ A✳ ◗✉❡st✐♦♥✳ ❉♦❡s Din ∪ P | = A ❤♦❧❞❄ ❝♦♠❜✐♥❡❞ ❝♦♠♣❧❡①✐t② ■♥st❛♥❝❡✳ P✱ Din ❛♥❞ A✳ ◗✉❡st✐♦♥✳ ❉♦❡s Din ∪ P | = A ❤♦❧❞❄

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✾✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❈♦♠♣❧❡①✐t② ❝❧❛ss❡s

L ⊆ NL ⊆ P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ❚❤❡s❡ ❛r❡ t❤❡ ❝❧❛ss❡s ♦❢ ♣r♦❜❧❡♠s ✇❤✐❝❤ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ❧♦❣❛r✐t❤♠✐❝ s♣❛❝❡ ✭L✮✱ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❧♦❣❛r✐t❤♠✐❝ s♣❛❝❡ ✭NL✮✱ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭P✮✱ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭NP✮✱ ♣♦❧②♥♦♠✐❛❧ s♣❛❝❡ ✭PSPACE✮✱ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ✭EXPTIME✮✱ ❛♥❞ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ✭NEXPTIME✮✳ ✇❡ s❤❛❧❧ ❡♥❝♦✉♥t❡r ✐♥ t❤✐s ❝♦✉rs❡✿ P, NP, PSPACE, EXPTIME

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✵✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❈♦♠♣❧❡①✐t② ❝❧❛ss❡s ✕ ❝♦ Pr♦❜❧❡♠s

❆♥② ❝♦♠♣❧❡①✐t② ❝❧❛ss C ❤❛s ✐ts ❝♦♠♣❧❡♠❡♥t❛r② ❝❧❛ss ❞❡♥♦t❡❞ ❜② co✲C✳ ❋♦r ❡✈❡r② ❧❛♥❣✉❛❣❡ L ⊆ Σ∗✱ ❧❡t L ❞❡♥♦t❡ ✐ts ❝♦♠♣❧❡♠❡♥t✱ ✐✳❡✳ t❤❡ s❡t Σ∗ \ L✳ ❚❤❡♥ co✲C ✐s {L | L ∈ C}✳ ❊✈❡r② ❞❡t❡r♠✐♥✐st✐❝ ❝♦♠♣❧❡①✐t② ❝❧❛ss ✐s ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t✱ ❜❡❝❛✉s❡ ♦♥❡ ❝❛♥ s✐♠♣❧② ❛❞❞ ❛ ❧❛st st❡♣ t♦ t❤❡ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ r❡✈❡rs❡s t❤❡ ❛♥s✇❡r✳ ✭❝♦✲P❄✮

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✶✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❈♦♠♣❧❡①✐t② ❝❧❛ss❡s ✕ ❘❡❞✉❝t✐♦♥s

▲♦❣s♣❛❝❡ ❘❡❞✉❝t✐♦♥

• ▲❡t L1 ❛♥❞ L2 ❜❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ✭❧❛♥❣✉❛❣❡s ♦✈❡r s♦♠❡ ❛❧♣❤❛❜❡t Σ✮✳
• R : Σ∗ → Σ∗ ❜❡ ❛ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ❧♦❣❛r✐t❤♠✐❝ s♣❛❝❡
• ❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt② ❤♦❧❞s✿ ❢♦r ❡✈❡r② x ∈ Σ∗✱ x ∈ L1 ✐✛ R(x) ∈ L2✳
• ❚❤❡♥ R ✐s ❝❛❧❧❡❞ ❛ ❧♦❣❛r✐t❤♠✐❝✲s♣❛❝❡ r❡❞✉❝t✐♦♥ ❢r♦♠ L1 t♦ L2 ❛♥❞ ✇❡ s❛②

t❤❛t L1 ✐s r❡❞✉❝✐❜❧❡ t♦ L2✳

❍❛r❞♥❡ss✱ ❈♦♠♣❧❡t❡♥❡ss ▲❡t C ❜❡ ❛ s❡t ♦❢ ❧❛♥❣✉❛❣❡s✳ ❆ ❧❛♥❣✉❛❣❡ L ✐s ❝❛❧❧❡❞ C✲❤❛r❞ ✐❢ ❛♥② ❧❛♥❣✉❛❣❡ L′ ✐♥ C ✐s r❡❞✉❝✐❜❧❡ t♦ L✳ ■❢ L ✐s C✲❤❛r❞ ❛♥❞ L ∈ C t❤❡♥ L ✐s ❝❛❧❧❡❞ ❝♦♠♣❧❡t❡ ❢♦r C ♦r s✐♠♣❧② C✲❝♦♠♣❧❡t❡✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✷✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❚✉r✐♥❣ ♠❛❝❤✐♥❡s

❆ ❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✭❉❚▼✮ ✐s ❞❡✜♥❡❞ ❛s ❛ q✉❛❞r✉♣❧❡ (S, Σ, δ, s0) S ✐s ❛ ✜♥✐t❡ s❡t ♦❢ st❛t❡s✱ Σ ✐s ❛ ✜♥✐t❡ ❛❧♣❤❛❜❡t ♦❢ s②♠❜♦❧s✱ ✇❤✐❝❤ ❝♦♥t❛✐♥s ❛ s♣❡❝✐❛❧ s②♠❜♦❧ ✥ ❝❛❧❧❡❞ t❤❡ ❜❧❛♥❦✳ δ ✐s ❛ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✱ ❛♥❞ s0 ∈ S ✐s t❤❡ ✐♥✐t✐❛❧ st❛t❡✳ ❚❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ δ ✐s ❛ ♠❛♣ δ : S × Σ → (S ∪ {②❡s, ♥♦}) × Σ × {✲✶✱ ✵✱ ✰✶}, ✇❤❡r❡ ②❡s✱ ❛♥❞ ♥♦ ❞❡♥♦t❡ t✇♦ ❛❞❞✐t✐♦♥❛❧ st❛t❡s ♥♦t ♦❝❝✉rr✐♥❣ ✐♥ S✱ ❛♥❞ ✲✶✱ ✵✱ ✰✶ ❞❡♥♦t❡ ♠♦t✐♦♥ ❞✐r❡❝t✐♦♥s✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✸✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❚✉r✐♥❣ ♠❛❝❤✐♥❡s

❉❚▼ q✉❛❞r✉♣❧❡✿ (Σ, S, δ, s0) ❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ δ(s, σ) = (s′, σ′, d). ❚❤❡ t❛♣❡ ♦❢ t❤❡ ❚▼ ⊲ a b . . . b a a ✥ ✥ ✥ ✥ . . . s1

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❚✉r✐♥❣ ♠❛❝❤✐♥❡s

❉❚▼ q✉❛❞r✉♣❧❡✿ (Σ, S, δ, s0) ❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ δ(s, σ) = (s′, σ′, d). ❚❤❡ t❛♣❡ ♦❢ t❤❡ ❚▼ ⊲ a b . . . b a a ✥ ✥ ✥ ✥ . . . s0

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❚✉r✐♥❣ ♠❛❝❤✐♥❡s

❉❚▼ q✉❛❞r✉♣❧❡✿ (Σ, S, δ, s0) ❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ δ(s, σ) = (s′, σ′, d). ❚❤❡ t❛♣❡ ♦❢ t❤❡ ❚▼ ⊲ a b . . . b a a ✥ ✥ ✥ ✥ . . . s0 c0 c1 c|I|−1

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❚✉r✐♥❣ ♠❛❝❤✐♥❡s

❉❚▼ q✉❛❞r✉♣❧❡✿ (Σ, S, δ, s0) ❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ δ(s, σ) = (s′, σ′, d). ❚❤❡ t❛♣❡ ♦❢ t❤❡ ❚▼ ⊲ a b . . . b a a ✥ ✥ ✥ ✥ . . . s ❚r❛♥s✐t✐♦♥ ❋✉♥❝t✐♦♥ ❡①❛♠♣❧❡✿ δ(s, a) = (s′, b, −1)

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❚✉r✐♥❣ ♠❛❝❤✐♥❡s

❉❚▼ q✉❛❞r✉♣❧❡✿ (Σ, S, δ, s0) ❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ δ(s, σ) = (s′, σ′, d). ❚❤❡ t❛♣❡ ♦❢ t❤❡ ❚▼ ⊲ a b . . . b b a ✥ ✥ ✥ ✥ . . . s ❚r❛♥s✐t✐♦♥ ❋✉♥❝t✐♦♥ ❡①❛♠♣❧❡✿ δ(s, a) = (s′, b, −1)

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❚✉r✐♥❣ ♠❛❝❤✐♥❡s

❉❚▼ q✉❛❞r✉♣❧❡✿ (Σ, S, δ, s0) ❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ δ(s, σ) = (s′, σ′, d). ❚❤❡ t❛♣❡ ♦❢ t❤❡ ❚▼ ⊲ a b . . . b b a ✥ ✥ ✥ ✥ . . . s′ ❚r❛♥s✐t✐♦♥ ❋✉♥❝t✐♦♥ ❡①❛♠♣❧❡✿ δ(s, a) = (s′, b, −1)

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❚✉r✐♥❣ ♠❛❝❤✐♥❡s

❉❚▼ q✉❛❞r✉♣❧❡✿ (Σ, S, δ, s0) ❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ δ(s, σ) = (s′, σ′, d). ❚❤❡ t❛♣❡ ♦❢ t❤❡ ❚▼ ⊲ a b . . . b b a a a b b . . . ②❡s ❆❝❝❡♣t✦ T ❤❛❧ts✱ ✇❤❡♥ ❛♥② ♦❢ t❤❡ st❛t❡s ②❡s ♦r ♥♦ ✐s r❡❛❝❤❡❞

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

❚✉r✐♥❣ ♠❛❝❤✐♥❡s

❉❚▼ q✉❛❞r✉♣❧❡✿ (Σ, S, δ, s0) ❚r❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥✿ δ(s, σ) = (s′, σ′, d). ❚❤❡ t❛♣❡ ♦❢ t❤❡ ❚▼ ⊲ a b . . . b b a a a b b . . . ♥♦ ❘❡❥❡❝t✦ T ❤❛❧ts✱ ✇❤❡♥ ❛♥② ♦❢ t❤❡ st❛t❡s ②❡s ♦r ♥♦ ✐s r❡❛❝❤❡❞

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

◆❉❚▼

❆ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✭◆❉❚▼✮ ✐s ❞❡✜♥❡❞ ❛s ❛ q✉❛❞r✉♣❧❡ (S, Σ, ∆, s0) S, Σ, s0 ❛r❡ t❤❡ s❛♠❡ ❛s ❉❚▼ ∆ ✐s ♥♦ ❧♦♥❣❡r ❛ ❢✉♥❝t✐♦♥✱ ❜✉t ❛ r❡❧❛t✐♦♥✿ ∆ ⊆ (S × Σ) × (S ∪ {②❡s, ♥♦}) × Σ × {✲✶✱ ✵✱ ✰✶}. ❆ t✉♣❧❡ ✇✐t❤ s ❛♥❞ σ✳ ■❢ t❤❡ ♥✉♠❜❡r ♦❢ s✉❝❤ t✉♣❧❡s ✐s ❣r❡❛t❡r t❤❛♥ ♦♥❡✱ t❤❡ ◆❉❚▼ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝❛❧❧② ❝❤♦♦s❡s ❛♥② ♦❢ t❤❡♠ ❛♥❞ ♦♣❡r❛t❡s ❛❝❝♦r❞✐♥❣❧②✳ ❯♥❧✐❦❡ t❤❡ ❝❛s❡ ♦❢ ❛ ❉❚▼✱ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛❝❝❡♣t❛♥❝❡ ❛♥❞ r❡❥❡❝t✐♦♥ ❜② ❛ ◆❉❚▼ ✐s ❛s②♠♠❡tr✐❝✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✺✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

◆♦♥❞❡t❡r♠✐♥✐st✐❝ ❈♦♠♣✉t❛t✐♦♥ ✭❆❝❝❡♣t✮

• ②❡s

♥♦

• ♥♦

♥♦

• ②❡s

②❡s

• ♥♦

♥♦

• ♥♦

②❡s

• ♥♦

♥♦

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✻✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

◆♦♥❞❡t❡r♠✐♥✐st✐❝ ❈♦♠♣✉t❛t✐♦♥ ✭❆❝❝❡♣t✮

• ②❡s

♥♦

• ♥♦

♥♦

• ②❡s

②❡s

• ♥♦

♥♦

• ♥♦

②❡s

• ♥♦

♥♦ ❆❝❝❡♣t✦

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✻✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

◆♦♥❞❡t❡r♠✐♥✐st✐❝ ❈♦♠♣✉t❛t✐♦♥ ✭❆❝❝❡♣t✮

• ②❡s

♥♦

• ♥♦

♥♦

• ②❡s

②❡s

• ♥♦

♥♦

• ♥♦

②❡s

• ♥♦

♥♦ ❆❝❝❡♣t✦

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✻✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

◆♦♥❞❡t❡r♠✐♥✐st✐❝ ❈♦♠♣✉t❛t✐♦♥ ✭❘❡❥❡❝t✐♦♥✮

• ♥♦

♥♦

• ♥♦

♥♦

• ♥♦

♥♦

• ♥♦

♥♦

• ♥♦

♥♦

• ♥♦

♥♦

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✼✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✶ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s ❛♥❞ ❘❡❞✉❝t✐♦♥s

◆♦♥❞❡t❡r♠✐♥✐st✐❝ ❈♦♠♣✉t❛t✐♦♥ ✭❘❡❥❡❝t✐♦♥✮

• ♥♦

♥♦

• ♥♦

♥♦

• ♥♦

♥♦

• ♥♦

♥♦

• ♥♦

♥♦

• ♥♦

♥♦ ❘❡❥❡❝t✦

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✼✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✷ ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣

❖✉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 ✼✳✷ ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣

✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣

❚♦❞❛② ✇❡ s❤❛❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ ◗✉❡r② ❉❛t❛ ❈♦♠♣❧❡①✐t② Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t② ❈♦♥❥✉♥❝t✐✈❡ q✉❡r② AC0 NP✲❝♦♠♣❧❡t❡ ❋❖ AC0 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❡ ❉✐s❥✉♥✳ ❉❛t❛❧♦❣ Πp

2✲❝♦♠♣❧❡t❡

co✲NEXPTIMENP✲❝♦♠♣❧❡t❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✶✾✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✷ ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣

Pr♦♣♦s✐t✐♦♥❛❧ ▲P

❚❤❡♦r❡♠

Pr♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ✐s P✲❝♦♠♣❧❡t❡✳ Pr♦♦❢✿ ✭▼❡♠❜❡rs❤✐♣✮ ❚❤❡ s❡♠❛♥t✐❝s ♦❢ ❛ ❣✐✈❡♥ ♣r♦❣r❛♠ P ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s t❤❡ ❧❡❛st ✜①♣♦✐♥t ♦❢ t❤❡ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦♣❡r❛t♦r TP ❚❤✐s ❧❡❛st ✜①♣♦✐♥t lfp(TP ) ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❡✈❡♥ ✐❢ t❤❡ ✏♥❛✐✈❡✑ ❡✈❛❧✉❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✐s ❛♣♣❧✐❡❞✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ✭✐✳❡✳ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ T

P ✮ ✐s ❜♦✉♥❞❡❞ ❜② t❤❡

♥✉♠❜❡r ♦❢ r✉❧❡s ♣❧✉s ✶✳ ❊❛❝❤ ✐t❡r❛t✐♦♥ st❡♣ ✐s ❝❧❡❛r❧② ❢❡❛s✐❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✵✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✷ ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣

Pr♦♣♦s✐t✐♦♥❛❧ ▲P P✲❤❛r❞♥❡ss Pr♦♦❢

Pr♦♦❢✿ ✭❍❛r❞♥❡ss✮ ❊♥❝♦❞✐♥❣ ♦❢ ❛ ❛ ❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✭❉❚▼✮ T✳ ●✐✈❡♥ ❛ ❉❚▼ T✱ ❛♥ ✐♥♣✉t str✐♥❣ I ❛♥❞ ❛ ♥✉♠❜❡r ♦❢ st❡♣s N✱ ✇❤❡r❡ N ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ |I|✱ ❝♦♥str✉❝t ✐♥ ❧♦❣s♣❛❝❡ ❛ ♣r♦❣r❛♠ P = P(T, I, N)✳ ❆♥ ❛t♦♠ A s✉❝❤ ❛s P | = A ✐✛ T ❛❝❝❡♣ts I ✐♥ N st❡♣s✳ ❚❤❡ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ δ ♦❢ ❛ ❉❚▼ ✇✐t❤ ❛ s✐♥❣❧❡ t❛♣❡ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ t❛❜❧❡ ✇❤♦s❡ r♦✇s ❛r❡ t✉♣❧❡s t = s, σ, s′, σ′, d✳ ❙✉❝❤ ❛ t✉♣❧❡ t ❡①♣r❡ss❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❢✲t❤❡♥✲r✉❧❡✿

✐❢ ❛t s♦♠❡ t✐♠❡ ✐♥st❛♥t τ t❤❡ ❉❚▼ ✐s ✐♥ st❛t❡ s✱ t❤❡ ❝✉rs♦r ♣♦✐♥ts t♦ ❝❡❧❧ ♥✉♠❜❡r π✱ ❛♥❞ t❤✐s ❝❡❧❧ ❝♦♥t❛✐♥s s②♠❜♦❧ σ t❤❡♥ ❛t ✐♥st❛♥t τ + 1 t❤❡ ❉❚▼ ✐s ✐♥ st❛t❡ s′✱ ❝❡❧❧ ♥✉♠❜❡r π ❝♦♥t❛✐♥s s②♠❜♦❧ σ′✱ ❛♥❞ t❤❡ ❝✉rs♦r ♣♦✐♥ts t♦ ❝❡❧❧ ♥✉♠❜❡r π + d✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✶✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✷ ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣

Pr♦♣♦s✐t✐♦♥❛❧ ▲P P✲❤❛r❞♥❡ss✿ t❤❡ ❛t♦♠s

❚❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ ❛t♦♠s ✐♥ P(T, I, N)✳ ✭t❤❡r❡ ❛r❡ ♠❛♥②✱ ❜✉t ♦♥❧② ♣♦❧②♥♦♠✐❛❧❧② ♠❛♥②✳✳✳✮ s②♠❜♦❧α[τ, π] ❢♦r 0 ≤ τ ≤ N✱ 0 ≤ π ≤ N ❛♥❞ α ∈ Σ✳ ■♥t✉✐t✐✈❡ ♠❡❛♥✐♥❣✿ ❛t ✐♥st❛♥t τ ♦❢ t❤❡ ❝♦♠♣✉t❛t✐♦♥✱ ❝❡❧❧ ♥✉♠❜❡r π ❝♦♥t❛✐♥s s②♠❜♦❧ α✳ ❝✉rs♦r[τ, π] ❢♦r 0 ≤ τ ≤ N ❛♥❞ 0 ≤ π ≤ N✳ ■♥t✉✐t✐✈❡ ♠❡❛♥✐♥❣✿ ❛t ✐♥st❛♥t τ✱ t❤❡ ❝✉rs♦r ♣♦✐♥ts t♦ ❝❡❧❧ ♥✉♠❜❡r π✳ st❛t❡s[τ] ❢♦r 0 ≤ τ ≤ N ❛♥❞ s ∈ S✳ ■♥t✉✐t✐✈❡ ♠❡❛♥✐♥❣✿ ❛t ✐♥st❛♥t τ✱ t❤❡ ❉❚▼ T ✐s ✐♥ st❛t❡ s✳ ❛❝❝❡♣t ■♥t✉✐t✐✈❡ ♠❡❛♥✐♥❣✿ T ❤❛s r❡❛❝❤❡❞ st❛t❡ ②❡s✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✷✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✷ ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣

Pr♦♣♦s✐t✐♦♥❛❧ ▲P P✲❤❛r❞♥❡ss✿ t❤❡ r✉❧❡s

✐♥✐t✐❛❧✐③❛t✐♦♥ ❢❛❝ts✿ ✐♥ P(T, I, N)✿ s②♠❜♦❧σ[0, π] ← ❢♦r 0 ≤ π < |I|✱ ✇❤❡r❡ Iπ = σ s②♠❜♦❧✥[0, π] ← ❢♦r |I| ≤ π ≤ N ❝✉rs♦r[0, 0] ← st❛t❡s0[0] ← ❚❤❡ t❛♣❡ ♦❢ t❤❡ ❚▼ ⊲ a b . . . b a a ✥ ✥ ✥ ✥ . . . s0 c0 c1 c|I|−1

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✸✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✷ ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣

Pr♦♣♦s✐t✐♦♥❛❧ ▲P P✲❤❛r❞♥❡ss✿ t❤❡ r✉❧❡s

tr❛♥s✐t✐♦♥ r✉❧❡s✿ ❢♦r ❡❛❝❤ ❡♥tr② s, σ, s′, σ′, d✱ 0 ≤ τ < N✱ 0 ≤ π < N✱ ❛♥❞ 0 ≤ π + d✳ s②♠❜♦❧σ′[τ + 1, π] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] ❝✉rs♦r[τ + 1, π + d] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] st❛t❡s′[τ + 1] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] ✐♥❡rt✐❛ r✉❧❡s✿ ✇❤❡r❡ 0 ≤ τ < N✱ 0 ≤ π < π′ ≤ N s②♠❜♦❧σ[τ + 1, π] ← s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π′] s②♠❜♦❧σ[τ + 1, π′] ← s②♠❜♦❧σ[τ, π′], ❝✉rs♦r[τ, π] ❛❝❝❡♣t r✉❧❡s✿ ❢♦r 0 ≤ τ ≤ N ❛❝❝❡♣t ← st❛t❡②❡s[τ]

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✷ ✼✳✷ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣

Pr♦♣♦s✐t✐♦♥❛❧ ▲P P✲❤❛r❞♥❡ss

❚❤❡ ❡♥❝♦❞✐♥❣ ♣r❡❝✐s❡❧② s✐♠✉❧❛t❡s t❤❡ ❜❡❤❛✈✐♦✉r ♠❛❝❤✐♥❡ T ♦♥ ✐♥♣✉t I ✉♣ t♦ N st❡♣s✳ ✭❚❤✐s ❝❛♥ ❜❡ ❢♦r♠❛❧❧② s❤♦✇♥ ❜② ✐♥❞✉❝t✐♦♥ ♦♥ t❤❡ t✐♠❡ st❡♣s✳✮ P(T, I, N) | = ❛❝❝❡♣t ✐✛ t❤❡ ❉❚▼ T ❛❝❝❡♣ts t❤❡ ✐♥♣✉t str✐♥❣ I ✇✐t❤✐♥ N st❡♣s✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ✐s ❢❡❛s✐❜❧❡ ✐♥ ❧♦❣s♣❛❝❡ ❍♦r♥ ❝❧❛✉s❡ ✐♥❢❡r❡♥❝❡ ✐s P✲❝♦♠♣❧❡t❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✺✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❖✉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 ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❚♦❞❛② ✇❡ s❤❛❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ ◗✉❡r② ❉❛t❛ ❈♦♠♣❧❡①✐t② Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t② ❈♦♥❥✉♥❝t✐✈❡ q✉❡r② AC0 NP✲❝♦♠♣❧❡t❡ ❋❖ AC0 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❡ ❉✐s❥✉♥✳ ❉❛t❛❧♦❣ Πp

2✲❝♦♠♣❧❡t❡

co✲NEXPTIMENP✲❝♦♠♣❧❡t❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✼✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❈♦♠♣❧❡①✐t② ♦❢ ❉❛t❛❧♦❣ Pr♦❣r❛♠s ✕ ❉❛t❛ ❝♦♠♣❧❡①✐t②

❚❤❡♦r❡♠

❉❛t❛❧♦❣ ✐s ❞❛t❛ ❝♦♠♣❧❡t❡ ❢♦r P✳ Pr♦♦❢✿ ✭▼❡♠❜❡rs❤✐♣✮ ❊✛❡❝t✐✈❡ r❡❞✉❝t✐♦♥ t♦ Pr♦♣♦s✐t✐♦♥❛❧ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ✐s ♣♦ss✐❜❧❡✳ ●✐✈❡♥ P✱ D✱ A✿

• ❡♥❡r❛t❡ ground(P, D)

❉❡❝✐❞❡ ✇❤❡t❤❡r ground(P, D) | = A

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✽✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

• r♦✉♥❞✐♥❣ ♦❢ ❉❛t❛❧♦❣ ❘✉❧❡s

▲❡t UD ❜❡ t❤❡ ✉♥✐✈❡rs❡ ♦❢ D ✭✉s✉❛❧❧② t❤❡ ❛❝t✐✈❡ ✉♥✐✈❡rs❡ ✭❞♦♠❛✐♥✮✱ ✐✳❡✳✱ t❤❡ s❡t ♦❢ ❛❧❧ ❞♦♠❛✐♥ ❡❧❡♠❡♥ts ♣r❡s❡♥t ✐♥ D✮✳ ❚❤❡ ❣r♦✉♥❞✐♥❣ ♦❢ ❛ r✉❧❡ r✱ ❞❡♥♦t❡❞ ground(r, D)✱ ✐s t❤❡ s❡t ♦❢ ❛❧❧ r✉❧❡s ♦❜t❛✐♥❡❞ ❢r♦♠ r ❜② ❛❧❧ ♣♦ss✐❜❧❡ ✉♥✐❢♦r♠ s✉❜st✐t✉t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ♦❢ UD ❢♦r t❤❡ ✈❛r✐❛❜❧❡s ✐♥ r✳ ❋♦r ❛♥② ❞❛t❛❧♦❣ ♣r♦❣r❛♠ P ❛♥❞ ❞❛t❛❜❛s❡ D✱ ground(P, D) =

• r∈P

ground(r, D).

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✷✾✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

• r♦✉♥❞✐♥❣ ❡①❛♠♣❧❡

P ❛♥❞ D✿

parent(X, Y ) ← father(X, Y ) parent(X, Y ) ← mother(X, Y ). ancestor(X, Y ) ← parent(X, Y ). ancestor(X, Y ) ← parent(X, Z), ancestor(Z, Y ). father(john, mary). father(joe, kurt). mother(mary, joe).mother(tina, kurt).

ground(P, D)✿

parent(john, john) ← father(john, john) parent(john, john) ← father(john, marry) . . . parent(john, john) ← mother(john, john) parent(john, marry) ← mother(john, marry) . . . ancestor(john, john) ← parent(john, john) . . .

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✵✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

• r♦✉♥❞✐♥❣ ❝♦♠♣❧❡①✐t②
• ✐✈❡♥ P, D✱ t❤❡ ♥✉♠❜❡r ♦❢ r✉❧❡s ✐♥ ground(P, D) ✐s ❜♦✉♥❞❡❞ ❜②

|P| ∗ #consts(D)vmax vmax(≥ 1) ✐s t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥t ✈❛r✐❛❜❧❡s ✐♥ ❛♥② r✉❧❡ r ∈ P #consts(D) = |UD| ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥st❛♥ts ✐♥ D ✭❛ss✳✿ |UD| > 0✮✳ ground(P ∪ D) ❝❛♥ ❜❡ ❡①♣♦♥❡♥t✐❛❧ ✐♥ t❤❡ s✐③❡ ♦❢ P✳ ground(P ∪ D) ✐s ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ s✐③❡ ♦❢ D✳ ❍❡♥❝❡✱ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ✐s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❢♦r t❤❡ ❞❛t❛ ❝♦♠♣❧❡①✐t②✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✶✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❉❛t❛❧♦❣ ❞❛t❛ ❝♦♠♣❧❡①✐t②✿ ❤❛r❞♥❡ss

Pr♦♦❢✿ ❍❛r❞♥❡ss ❚❤❡ P✲❤❛r❞♥❡ss ❝❛♥ ❜❡ s❤♦✇♥ ❜② ✇r✐t✐♥❣ ❛ s✐♠♣❧❡ ❞❛t❛❧♦❣ ♠❡t❛✲✐♥t❡r♣r❡t❡r ❢♦r ♣r♦♣♦s✐t✐♦♥❛❧ ▲P(k)✱ ✇❤❡r❡ k ✐s ❛ ❝♦♥st❛♥t✳ ❘❡♣r❡s❡♥t r✉❧❡s A0 ← A1, . . . , Ai✱ ✇❤❡r❡ 0 ≤ i ≤ k✱ ❜② t✉♣❧❡s A0, . . . , Ai ✐♥ ❛♥ (i + 1)✲❛r② r❡❧❛t✐♦♥ Ri ♦♥ t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ ❛t♦♠s✳ ❚❤❡♥✱ ❛ ♣r♦❣r❛♠ P ✐♥ ▲P(k) ✇❤✐❝❤ ✐s st♦r❡❞ t❤✐s ✇❛② ✐♥ ❛ ❞❛t❛❜❛s❡ D(P) ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ❜② ❛ ✜①❡❞ ❞❛t❛❧♦❣ ♣r♦❣r❛♠ PMI(k) ✇❤✐❝❤ ❝♦♥t❛✐♥s ❢♦r ❡❛❝❤ r❡❧❛t✐♦♥ Ri✱ 0 ≤ i ≤ k✱ ❛ r✉❧❡ T(X0) ← T(X1), . . . , T(Xi), Ri(X0, . . . , Xi). T(x) ✐♥t✉✐t✐✈❡❧② ♠❡❛♥s t❤❛t ❛t♦♠ x ✐s tr✉❡✳ ❚❤❡♥✱ P | = A ❥✉st ✐❢ PMI ∪ P(D) | = T(A)✳ P✲❤❛r❞♥❡ss ♦❢ t❤❡ ❞❛t❛ ❝♦♠♣❧❡①✐t② ♦❢ ❞❛t❛❧♦❣ ✐s t❤❡♥ ✐♠♠❡❞✐❛t❡❧② ♦❜t❛✐♥❡❞✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✷✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t② ❉❛t❛❧♦❣

❚❤❡♦r❡♠

❉❛t❛❧♦❣ ✐s ♣r♦❣r❛♠ ❝♦♠♣❧❡t❡ ❢♦r EXPTIME✳ ▼❡♠❜❡rs❤✐♣✳ ●r♦✉♥❞✐♥❣ P ♦♥ D ❧❡❛❞s t♦ ❛ ♣r♦♣♦s✐t✐♦♥❛❧ ♣r♦❣r❛♠ grounding(P, D) ✇❤♦s❡ s✐③❡ ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ ✜①❡❞ ✐♥♣✉t ❞❛t❛❜❛s❡ D✳ ❍❡♥❝❡✱ t❤❡ ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t② ✐s ✐♥ EXPTIME✳ ❍❛r❞♥❡ss✳

• ❆❞❛♣t t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ ♣r♦❣r❛♠ P(T, I, N) ❞❡❝✐❞✐♥❣ ❛❝❝❡♣t❛♥❝❡ ♦❢ ✐♥♣✉t I

❢♦r T ✇✐t❤✐♥ N st❡♣s✱ ✇❤❡r❡ N = 2m✱ m = nk(n = |I|) t♦ ❛ ❞❛t❛❧♦❣ ♣r♦❣r❛♠ Pdat(T, I, N)

• ◆♦t❡✿ ❲❡ ❝❛♥ ♥♦t s✐♠♣❧② ❣❡♥❡r❛t❡ P(T, I, N)✱ s✐♥❝❡ t❤✐s ♣r♦❣r❛♠ ✐s

❡①♣♦♥❡♥t✐❛❧❧② ❧❛r❣❡ ✭❛♥❞ t❤✉s t❤❡ r❡❞✉❝t✐♦♥ ✇♦✉❧❞ ♥♦t ❜❡ ♣♦❧②♥♦♠✐❛❧✦✮

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✸✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❉❛t❛❧♦❣ Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t②✿ ❍❛r❞♥❡ss

▼❛✐♥ ✐❞❡❛s ❢♦r ❧✐❢t✐♥❣ P(T, I, N) t♦ Pdat(T, I, N)✿ ✉s❡ t❤❡ ♣r❡❞✐❝❛t❡s s②♠❜♦❧σ(X, Y )✱ ❝✉rs♦r(X, Y ) ❛♥❞ st❛t❡s(X) ✐♥st❡❛❞ ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ ❧❡tt❡rs s②♠❜♦❧σ[X, Y ]✱ ❝✉rs♦r[X, Y ] ❛♥❞ st❛t❡s[X] r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ t✐♠❡ ♣♦✐♥ts τ ❛♥❞ t❛♣❡ ♣♦s✐t✐♦♥s π ❢r♦♠ 0 t♦ N − 1 ❛r❡ ❡♥❝♦❞❡❞ ✐♥ ❜✐♥❛r②✱ ✐✳❡✳ ❜② m✲❛r② t✉♣❧❡s tτ ❂ c1, . . . , cm✱ ci ∈ {0, 1}✱ i = 1, . . . , m✱ s✉❝❤ t❤❛t 0 = 0, . . . , 0✱ 1 = 0, . . . , 1✱ N − 1 = 1, . . . , 1✳ ❚❤❡ ❢✉♥❝t✐♦♥s τ + 1 ❛♥❞ π + d ❛r❡ r❡❛❧✐③❡❞ ❜② ♠❡❛♥s ♦❢ t❤❡ s✉❝❝❡ss♦r ❙✉❝❝m ❢r♦♠ ❛ ❧✐♥❡❛r ♦r❞❡r ≤m ♦♥ U m✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❉❛t❛❧♦❣ Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t②✿ ❍❛r❞♥❡ss

❚❤❡ ♣r❡❞✐❝❛t❡s ❙✉❝❝m✱ ❋✐rstm✱ ❛♥❞ ▲❛stm ❛r❡ ♣r♦✈✐❞❡❞✳ ❚❤❡ ✐♥✐t✐❛❧✐③❛t✐♦♥ ❢❛❝ts s②♠❜♦❧σ[0, π] ❛r❡ r❡❛❞✐❧② tr❛♥s❧❛t❡❞ ✐♥t♦ t❤❡ ❞❛t❛❧♦❣ r✉❧❡s s②♠❜♦❧σ(X, t) ← ❋✐rstm(X), ✇❤❡r❡ t r❡♣r❡s❡♥ts t❤❡ ♣♦s✐t✐♦♥ π✱ ❙✐♠✐❧❛r❧② t❤❡ ❢❛❝ts ❝✉rs♦r[0, 0] ❛♥❞ st❛t❡s0[0]✳ ■♥✐t✐❛❧✐③❛t✐♦♥ ❢❛❝ts s②♠❜♦❧✥[0, π]✱ ✇❤❡r❡ |I| ≤ π ≤ N✱ ❛r❡ tr❛♥s❧❛t❡❞ t♦ t❤❡ r✉❧❡ s②♠❜♦❧✥(X, Y) ← ❋✐rstm(X), ≤m(t, Y) ✇❤❡r❡ t r❡♣r❡s❡♥ts t❤❡ ♥✉♠❜❡r |I|✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✺✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❉❛t❛❧♦❣ Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t②✿ ❍❛r❞♥❡ss

❚r❛♥s✐t✐♦♥ ❛♥❞ ✐♥❡rt✐❛ r✉❧❡s✿ ❢♦r r❡❛❧✐③✐♥❣ τ + 1 ❛♥❞ π + d✱ ✉s❡ ✐♥ t❤❡ ❜♦❞② ❛t♦♠s ❙✉❝❝m(X, X′)✳ ❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❝❧❛✉s❡ s②♠❜♦❧σ′[τ + 1, π] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] ✐s tr❛♥s❧❛t❡❞ ✐♥t♦ s②♠❜♦❧σ′(X′, Y) ← st❛t❡s(X), s②♠❜♦❧σ(X, Y), ❝✉rs♦r(X, Y), ❙✉❝❝m(X, X′). ❚❤❡ tr❛♥s❧❛t✐♦♥ ♦❢ t❤❡ ❛❝❝❡♣t r✉❧❡s ✐s str❛✐❣❤t❢♦r✇❛r❞✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✻✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❉❡✜♥✐♥❣ ❙✉❝❝m(X, X′) ❛♥❞ ≤m

❚❤❡ ❣r♦✉♥❞ ❢❛❝ts ❙✉❝❝1(0, 1)✱ ❋✐rst1(0)✱ ❛♥❞ ▲❛st1(1) ❛r❡ ♣r♦✈✐❞❡❞✳ ❋♦r ❛♥ ✐♥❞✉❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ s✉♣♣♦s❡ ❙✉❝❝i(X, Y)✱ ❋✐rsti(X)✱ ❛♥❞ ▲❛sti(X) t❡❧❧ t❤❡ s✉❝❝❡ss♦r✱ t❤❡ ✜rst✱ ❛♥❞ t❤❡ ❧❛st ❡❧❡♠❡♥t ❢r♦♠ ❛ ❧✐♥❡❛r ♦r❞❡r ≤i ♦♥ U i✱ ✇❤❡r❡ X ❛♥❞ Y ❤❛✈❡ ❛r✐t② i✳ ❚❤❡♥✱ ✉s❡ r✉❧❡s ❙✉❝❝i+1(Z, X, Z, Y) ← ❙✉❝❝i(X, Y) ❙✉❝❝i+1(Z, X, Z′, Y) ← ❙✉❝❝1(Z, Z′), ▲❛sti(X), ❋✐rsti(Y) ❋✐rsti+1(Z, X) ← ❋✐rst1(Z), ❋✐rsti(X) ▲❛sti+1(Z, X) ← ▲❛st1(Z), ▲❛sti(X)

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✼✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❉❡✜♥✐♥❣ ❙✉❝❝m(X, X′) ❛♥❞ ≤m

❚❤❡ ❣r♦✉♥❞ ❢❛❝ts ❙✉❝❝1(0, 1)✱ ❋✐rst1(0)✱ ❛♥❞ ▲❛st1(1) ❛r❡ ♣r♦✈✐❞❡❞✳ ❋♦r ❛♥ ✐♥❞✉❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ s✉♣♣♦s❡ ❙✉❝❝i(X, Y)✱ ❋✐rsti(X)✱ ❛♥❞ ▲❛sti(X) t❡❧❧ t❤❡ s✉❝❝❡ss♦r✱ t❤❡ ✜rst✱ ❛♥❞ t❤❡ ❧❛st ❡❧❡♠❡♥t ❢r♦♠ ❛ ❧✐♥❡❛r ♦r❞❡r ≤i ♦♥ U i✱ ✇❤❡r❡ X ❛♥❞ Y ❤❛✈❡ ❛r✐t② i✳ ❚❤❡♥✱ ✉s❡ r✉❧❡s ❙✉❝❝i+1(0, X, 0, Y) ← ❙✉❝❝i(X, Y) ❙✉❝❝i+1(1, X, 1, Y) ← ❙✉❝❝i(X, Y) ❙✉❝❝i+1(0, X, 1, Y) ← ▲❛sti(X), ❋✐rsti(Y) ❋✐rsti+1(0, X) ← ❋✐rsti(X) ▲❛sti+1(1, X) ← ▲❛sti(X)

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✼✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❉❡✜♥✐♥❣ ❙✉❝❝m(X, X′) ❛♥❞ ≤m

❚❤❡ ❣r♦✉♥❞ ❢❛❝ts ❙✉❝❝1(0, 1)✱ ❋✐rst1(0)✱ ❛♥❞ ▲❛st1(1) ❛r❡ ♣r♦✈✐❞❡❞✳ ❋♦r ❛♥ ✐♥❞✉❝t✐✈❡ ❞❡✜♥✐t✐♦♥✱ s✉♣♣♦s❡ ❙✉❝❝i(X, Y)✱ ❋✐rsti(X)✱ ❛♥❞ ▲❛sti(X) t❡❧❧ t❤❡ s✉❝❝❡ss♦r✱ t❤❡ ✜rst✱ ❛♥❞ t❤❡ ❧❛st ❡❧❡♠❡♥t ❢r♦♠ ❛ ❧✐♥❡❛r ♦r❞❡r ≤i ♦♥ U i✱ ✇❤❡r❡ X ❛♥❞ Y ❤❛✈❡ ❛r✐t② i✳ ❚❤❡♥✱ ✉s❡ r✉❧❡s ❙✉❝❝i+1(0, X, 0, Y) ← ❙✉❝❝i(X, Y) ❙✉❝❝i+1(1, X, 1, Y) ← ❙✉❝❝i(X, Y) ❙✉❝❝i+1(0, X, 1, Y) ← ▲❛sti(X), ❋✐rsti(Y) ❋✐rsti+1(0, X) ← ❋✐rsti(X) ▲❛sti+1(1, X) ← ▲❛sti(X) ❚❤❡ ♦r❞❡r ≤m ✐s ❡❛s✐❧② ❞❡✜♥❡❞ ❢r♦♠ ❙✉❝❝m ❜② t✇♦ ❝❧❛✉s❡s ≤m(X, X) ← ≤m(X, Y) ← ❙✉❝❝m(X, Z), ≤m (Z, Y)

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✼✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❉❛t❛❧♦❣ Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t② ❈♦♥❝❧✉s✐♦♥

▲❡t Pdat(T, I, N) ❞❡♥♦t❡ t❤❡ ❞❛t❛❧♦❣ ♣r♦❣r❛♠ ✇✐t❤ ❡♠♣t② edb ❞❡s❝r✐❜❡❞ ❢♦r T✱ I✱ ❛♥❞ N = 2m✱ m = nk ✭✇❤❡r❡ n ❂ |I|✮ Pdat(T, I, N) ✐s ❝♦♥str✉❝t✐❜❧❡ ❢r♦♠ T ❛♥❞ I ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✭✐♥ ❢❛❝t✱ ❝❛r❡❢✉❧ ❛♥❛❧②s✐s s❤♦✇s ❢❡❛s✐❜✐❧✐t② ✐♥ ❧♦❣❛r✐t❤♠✐❝ s♣❛❝❡✮✳ Pdat(T, I, N) ❤❛s ❛❝❝❡♣t ✐♥ ✐ts ❧❡❛st ♠♦❞❡❧ ⇔ T ❛❝❝❡♣ts ✐♥♣✉t I ✇✐t❤✐♥ N st❡♣s✳ ❚❤✉s✱ t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❛♥② ❧❛♥❣✉❛❣❡ ✐♥ ❊❳P❚■▼❊ ✐s r❡❞✉❝✐❜❧❡ t♦ ❞❡❝✐❞✐♥❣ P | = A ❢♦r ❞❛t❛❧♦❣ ♣r♦❣r❛♠ P ❛♥❞ ❢❛❝t A✳ ❈♦♥s❡q✉❡♥t❧②✱ ❞❡❝✐❞✐♥❣ P | = A ❢♦r ❛ ❣✐✈❡♥ ❞❛t❛❧♦❣ ♣r♦❣r❛♠ P ❛♥❞ ❢❛❝t A ✐s ❊❳P❚■▼❊✲❤❛r❞✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✽✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✸ ✼✳✸ ❉❛t❛❧♦❣ ❈♦♠♣❧❡①✐t②

❈♦♠♣❧❡①✐t② ♦❢ ❉❛t❛❧♦❣ ✇✐t❤ ❙tr❛t✐✜❡❞ ◆❡❣❛t✐♦♥

❚❤❡♦r❡♠

❙tr❛t✐✜❡❞ ♣r♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ✇✐t❤ ♥❡❣❛t✐♦♥ ✐s P✲❝♦♠♣❧❡t❡✳ ❙tr❛t✐✜❡❞ ❞❛t❛❧♦❣ ✇✐t❤ ♥❡❣❛t✐♦♥ ✐s ❞❛t❛ ❝♦♠♣❧❡t❡ ❢♦r P ❛♥❞ ♣r♦❣r❛♠ ❝♦♠♣❧❡t❡ ❢♦r EXPTIME✳ str❛t✐✜❡❞ P ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❞✐s❥♦✐♥t s❡ts S1, . . . , Sn s✳t✳ t❤❡ s❡♠❛♥t✐❝s ♦❢ P ✐s ❝♦♠♣✉t❡❞ ❜② s✉❝❝❡ss✐✈❡❧② ❝♦♠♣✉t✐♥❣ ✜①♣♦✐♥ts ♦❢ t❤❡ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦♣❡r❛t♦rs T

S1✱ ✳ ✳ ✳ ✱ T Sn✳

▲❡t I0 ❜❡ t❤❡ ✐♥✐t✐❛❧ ✐♥st❛♥❝❡ ♦✈❡r t❤❡ ❡①t❡♥s✐♦♥❛❧ ♣r❡❞✐❝❛t❡ s②♠❜♦❧s ♦❢ P ❛♥❞ ❧❡t Ii ✭✇✐t❤ 1 ≤ i ≤ n✮ ❜❡ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ I1 := Tω

S1(I0), I2 := Tω S2(I1), . . . , In := Tω Sn(In−1)

❚❤❡♥ t❤❡ s❡♠❛♥t✐❝s ♦❢ ♣r♦❣r❛♠ P ✐s ❣✐✈❡♥ t❤r♦✉❣❤ t❤❡ s❡t In✳ ■♥ t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ ❝❛s❡✱ In ✐s ❝❧❡❛r❧② ♣♦❧②♥♦♠✐❛❧❧② ❝♦♠♣✉t❛❜❧❡✳ ❍❡♥❝❡✱ str❛t✐✜❡❞ ♥❡❣❛t✐♦♥ ❞♦❡s ♥♦t ✐♥❝r❡❛s❡ t❤❡ ❝♦♠♣❧❡①✐t②✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✸✾✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❖✉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 ❙t❛❜❧❡ ▼♦❞❡❧

✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❚♦❞❛② ✇❡ s❤❛❧❧ ❝♦♥❝❡♥tr❛t❡ ♦♥ ◗✉❡r② ❉❛t❛ ❈♦♠♣❧❡①✐t② Pr♦❣r❛♠ ❈♦♠♣❧❡①✐t② ❈♦♥❥✉♥❝t✐✈❡ q✉❡r② AC0 NP✲❝♦♠♣❧❡t❡ ❋❖ AC0 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❡ ❉✐s❥✉♥✳ ❉❛t❛❧♦❣ Πp

2✲❝♦♠♣❧❡t❡

co✲NEXPTIMENP✲❝♦♠♣❧❡t❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✶✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❘❡❝❛❧❧ ❙t❛❜❧❡ ▼♦❞❡❧ ❙❡♠❛♥t✐❝s

▲❡t S ❜❡ ❛ ✭♣♦ss✐❜❧② ✐♥✜♥✐t❡✮ s❡t ♦❢ ❣r♦✉♥❞ ♥♦r♠❛❧ ❝❧❛✉s❡s✱ ✐✳❡✳✱ ♦❢ ❢♦r♠✉❧❛s ♦❢ t❤❡ ❢♦r♠ A ← L1 ∧ . . . ∧ Ln ✇❤❡r❡ n ≥ 0 ❛♥❞ A ✐s ❛ ❣r♦✉♥❞ ❛t♦♠ ❛♥❞ t❤❡ Li ❢♦r 1 ≤ i ≤ n ❛r❡ ❣r♦✉♥❞ ❧✐t❡r❛❧s✳

• ❡❧❢♦♥❞✲▲✐❢s❝❤✐t③ ❚r❛♥s❢♦r♠❛t✐♦♥

▲❡t B ⊆ HB✳ ❚❤❡ ●❡❧❢♦♥❞✲▲✐❢s❝❤✐t③ tr❛♥s❢♦r♠ GLB(S) ♦❢ S ✇✐t❤ r❡s♣❡❝t t♦ B ✐s ♦❜t❛✐♥❡❞ ❢r♦♠ S ❛s ❢♦❧❧♦✇s✿

✶ r❡♠♦✈❡ ❡❛❝❤ ❝❧❛✉s❡ ✇❤♦s❡ ❛♥t❡❝❡❞❡♥t ❝♦♥t❛✐♥s ❛ ❧✐t❡r❛❧ ¬A ✇✐t❤ A ∈ B✳ ✷ r❡♠♦✈❡ ❢r♦♠ t❤❡ ❛♥t❡❝❡❞❡♥ts ♦❢ t❤❡ r❡♠❛✐♥✐♥❣ ❝❧❛✉s❡s ❛❧❧ ♥❡❣❛t✐✈❡ ❧✐t❡r❛❧s✳

❙t❛❜❧❡ ▼♦❞❡❧

❆♥ ❍❡r❜r❛♥❞ ✐♥t❡r♣r❡t❛t✐♦♥ HI (B) ✐s ❛ st❛❜❧❡ ♠♦❞❡❧ ♦❢ S ✐✛ ✐t ✐s t❤❡ ✉♥✐q✉❡ ♠✐♥✐♠❛❧ ❍❡r❜r❛♥❞ ♠♦❞❡❧ ♦❢ GLB(S)✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✷✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❈♦♠♣❧❡①✐t② Pr♦♣✳ ▲P ❙t❛❜❧❡ ♠♦❞❡❧

❚❤❡♦r❡♠

• ✐✈❡♥ ❛ ♣r♦♣♦s✐t✐♦♥❛❧ ♥♦r♠❛❧ ❧♦❣✐❝ ♣r♦❣r❛♠ P✱ ❞❡❝✐❞✐♥❣ ✇❤❡t❤❡r P ❤❛s ❛ st❛❜❧❡

♠♦❞❡❧ ✐s NP✲❝♦♠♣❧❡t❡✳ ▼❡♠❜❡rs❤✐♣✳ ❈❧❡❛r❧②✱ P I ✐s ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❝♦♠♣✉t❛❜❧❡ ❢r♦♠ P ❛♥❞ I✳ ❍❡♥❝❡✱ ❛ st❛❜❧❡ ♠♦❞❡❧ M ♦❢ P ❝❛♥ ❜❡ ❣✉❡ss❡❞ ❛♥❞ ❝❤❡❝❦❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✸✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❙t❛❜❧❡ ▼♦❞❡❧ Pr♦♣✳ ▲P ✲ ❍❛r❞♥❡ss

Pr♦♦❢ ❤❛r❞♥❡ss ❊♥❝♦❞✐♥❣ ♦❢ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✭◆❉❚▼✮ T✳

• ●✐✈❡♥ ❛ ◆❉❚▼ T✱ ❛♥ ✐♥♣✉t str✐♥❣ I ❛♥❞ ❛ ♥✉♠❜❡r ♦❢ st❡♣s N✱ ✇❤❡r❡ N ✐s ❛

♣♦❧②♥♦♠✐❛❧ ♦❢ |I|✱ ❝♦♥str✉❝t ✐♥ ❧♦❣s♣❛❝❡ ❛ ♣r♦❣r❛♠ P = P(T, I, N)✳

• P ❤❛s ❛ st❛❜❧❡ ♠♦❞❡❧ ✐✛ T ❛❝❝❡♣ts I ✐♥ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝❛❧❧② N st❡♣s✳

▼✉❝❤ s✐♠✐❧❛r t♦ t❤❡ ❡♥❝♦❞✐♥❣ ♦❢ ❉❚▼ ✇✐t❤ ♣r♦♣♦s✐t✐♦♥❛❧ ▲P✳ ▼♦❞✐✜❝❛t✐♦♥ ♦♥ ❞❡t❡r♠✐♥✐st✐❝ ♣r♦♣❡rt②✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❙t❛❜❧❡ ▼♦❞❡❧ Pr♦♣✳ ▲P ✲ ❍❛r❞♥❡ss

❊①❛♠♣❧❡✿ s, σ, s1, σ′

1, d1✱ s, σ, s2, σ′ 2, d2

❚r❛♥s✐t✐♦♥ r✉❧❡s 0 ≤ τ < N✱ 0 ≤ π < N✱ ❛♥❞ 0 ≤ π + d✳ s②♠❜♦❧σ′

1[τ + 1, π]

← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] ❝✉rs♦r[τ + 1, π + d1] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] st❛t❡s1[τ + 1] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] s②♠❜♦❧σ′

2[τ + 1, π]

← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] ❝✉rs♦r[τ + 1, π + d2] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] st❛t❡s2[τ + 1] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π] ❲❤❛t ✐s ✇r♦♥❣ ❤❡r❡❄ ❊♥❢♦r❝❡♠❡♥t ✈✐♦❧❛t❡❞✿ ❆t ❛♥② t✐♠❡ ✐♥st❛♥❝❡ τ✱ t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ ❝✉rs♦r❀ ❡❛❝❤ ❝❡❧❧ ♦❢ t❤❡ t❛♣❡ ❝♦♥t❛✐♥s ❡①❛❝t❧② ♦♥❡ ❡❧❡♠❡♥t❀ ✐♥ ❡①❛❝t❧② ♦♥❡ st❛t❡✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✺✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❙t❛❜❧❡ ▼♦❞❡❧ Pr♦♣✳ ▲P ✲ ❍❛r❞♥❡ss

❋♦r ❡❛❝❤ st❛t❡ s ❛♥❞ s②♠❜♦❧ σ✱ ✐♥tr♦❞✉❝❡ ❛t♦♠s Bs,σ,1[τ]✱✳ ✳ ✳ ✱ Bs,σ,k[τ] ❢♦r ❛❧❧ 1 ≤ τ < N ❛♥❞ ❢♦r ❛❧❧ tr❛♥s✐t✐♦♥s s, σ, si, σ′

i, di✱ ✇❤❡r❡ 1 ≤ i ≤ k✳

❆❞❞ Bs,σ,i[τ] ✐♥ t❤❡ ❜♦❞✐❡s ♦❢ t❤❡ tr❛♥s✐t✐♦♥ r✉❧❡s ❢♦r s, σ, si, σ′

i, di✳

❆❞❞ t❤❡ r✉❧❡ Bs,σ,i[τ] ← ¬Bs,σ,1[τ], . . . , ¬Bs,σ,i−1[τ], ¬Bs,σ,i+1[τ], . . . , ¬Bs,σ,k[τ]. ■♥t✉✐t✐✈❡❧②✱ t❤❡s❡ r✉❧❡s ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝❛❧❧② s❡❧❡❝t ♣r❡❝✐s❡❧② ♦♥❡ ♦❢ t❤❡ ♣♦ss✐❜❧❡ tr❛♥s✐t✐♦♥s ❢♦r s ❛♥❞ σ ❛t t✐♠❡ ✐♥st❛♥t τ✱ ✇❤♦s❡ tr❛♥s✐t✐♦♥ r✉❧❡s ❛r❡ ❡♥❛❜❧❡❞ ✈✐❛ Bs,σ,i[τ]✳ ❋✐♥❛❧❧②✱ ❛❞❞ ❛ r✉❧❡ ❛❝❝❡♣t ← ¬❛❝❝❡♣t. ■t ❡♥s✉r❡s t❤❛t ❛❝❝❡♣t ✐s tr✉❡ ✐♥ ❡✈❡r② st❛❜❧❡ ♠♦❞❡❧✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✻✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❙t❛❜❧❡ ▼♦❞❡❧ Pr♦♣✳ ▲P ✲ ❍❛r❞♥❡ss

❊①❛♠♣❧❡✿ s, σ, s1, σ′

1, d1✱ s, σ, s2, σ′ 2, d2

s②♠❜♦❧σ′

1[τ + 1, π]

← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π], Bs,σ,1[τ] ❝✉rs♦r[τ + 1, π + d1] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π], Bs,σ,1[τ] st❛t❡s1[τ + 1] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π], Bs,σ,1[τ] s②♠❜♦❧σ′

2[τ + 1, π]

← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π], Bs,σ,2[τ] ❝✉rs♦r[τ + 1, π + d2] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π], Bs,σ,2[τ] st❛t❡s2[τ + 1] ← st❛t❡s[τ], s②♠❜♦❧σ[τ, π], ❝✉rs♦r[τ, π], Bs,σ,2[τ] Bs,σ,1[τ] ← ¬Bs,σ,2[τ] Bs,σ,2[τ] ← ¬Bs,σ,1[τ] ❖♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ❛t♦♠ ❢r♦♠ Bs,σ,1[τ] ❛♥❞ Bs,σ,2[τ] ✐s tr✉❡✳ ❲❤✐❝❤ ♦♥❡❄ ◆♦♥✲❞❡t❡r♠✐♥✐st✐❝

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✼✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❙t❛❜❧❡ ▼♦❞❡❧ Pr♦♣✳ ▲P ✲ ❍❛r❞♥❡ss

Pr♦♦❢✳ ❆ss✉♠❡ t❤❡r❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ❝❤♦✐❝❡s ❧❡❛❞✐♥❣ t♦ t❤❡ st❛t❡ ②❡s✳ ▲❡t I ❜❡ t❤❡ s❡t ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ ❛t♦♠s ❛❧♦♥❣ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♣❛t❤ r❡❛❝❤✐♥❣ t❤❡ st❛t❡ ❛❝❝❡♣t✳ ❚❤❡♥ ❛❝❝❡♣t ∈ I ❞✉❡ t♦ t❤❡ r✉❧❡✿ ❛❝❝❡♣t ← st❛t❡②❡s[τ] ❈❧❡❛r❧② I ✐s ❛ st❛❜❧❡ ♠♦❞❡❧ ♦❢ P✳ ❆ss✉♠❡ t❤❡r❡ ❡①✐sts ♥♦ s❡q✉❡♥❝❡ ♦❢ ❝❤♦✐❝❡s ❧❡❛❞✐♥❣ t♦ t❤❡ st❛t❡ ②❡s ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ tr❡❡✳ ❙✉♣♣♦s❡ I ✐s ❛ st❛❜❧❡ ♠♦❞❡❧ ♦❢ P ❛♥❞ ❛❝❝❡♣t ∈ I✳ ❇② ♠✐♥✐♠❛❧✐t② ♦❢ I ❢♦r P I✱ ✐t ❢♦❧❧♦✇s t❤❛t st❛t❡②❡s[τ] ∈ I ❢♦r s♦♠❡ τ❀ ♠♦r❡♦✈❡r✱ t❤✐s ♠❡❛♥s t❤❛t ❛ s❡q✉❡♥❝❡ ♦❢ ❝❤♦✐❝❡s ❧❡❛❞s t♦ ②❡s✳ ❈♦♥tr❛❞✐❝t✐♦♥✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✽✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❋✉rt❤❡r ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

❚❤❡♦r❡♠

Pr♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ✇✐t❤ ♥❡❣❛t✐♦♥ ✉♥❞❡r ✇❡❧❧✲❢♦✉♥❞❡❞ s❡♠❛♥t✐❝s ✐s P✲❝♦♠♣❧❡t❡✳ ❉❛t❛❧♦❣ ✇✐t❤ ♥❡❣❛t✐♦♥ ✉♥❞❡r ✇❡❧❧✲❢♦✉♥❞❡❞ s❡♠❛♥t✐❝s ✐s ❞❛t❛ ❝♦♠♣❧❡t❡ ❢♦r P ❛♥❞ ♣r♦❣r❛♠ ❝♦♠♣❧❡t❡ ❢♦r EXPTIME✳

❚❤❡♦r❡♠

Pr♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ✇✐t❤ ♥❡❣❛t✐♦♥ ✉♥❞❡r ✐♥✢❛t✐♦♥❛r② s❡♠❛♥t✐❝s ✐s P✲❝♦♠♣❧❡t❡✳ ❉❛t❛❧♦❣ ✇✐t❤ ♥❡❣❛t✐♦♥ ✉♥❞❡r ✐♥✢❛t✐♦♥❛r② s❡♠❛♥t✐❝s ✐s ❞❛t❛ ❝♦♠♣❧❡t❡ ❢♦r P ❛♥❞ ♣r♦❣r❛♠ ❝♦♠♣❧❡t❡ ❢♦r EXPTIME✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✹✾✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❋✉rt❤❡r ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

❚❤❡♦r❡♠

Pr♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ✇✐t❤ ♥❡❣❛t✐♦♥ ✉♥❞❡r st❛❜❧❡ ♠♦❞❡❧ s❡♠❛♥t✐❝s ✐s co✲NP✲❝♦♠♣❧❡t❡✳ ❉❛t❛❧♦❣ ✇✐t❤ ♥❡❣❛t✐♦♥ ✉♥❞❡r st❛❜❧❡ ♠♦❞❡❧ s❡♠❛♥t✐❝s ✐s ❞❛t❛ ❝♦♠♣❧❡t❡ ❢♦r co✲NP ❛♥❞ ♣r♦❣r❛♠ ❝♦♠♣❧❡t❡ ❢♦r co✲NEXPTIME✳ ◆♦t❡ t❤❛t t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ❤❡r❡ ✐s ✇❤❡t❤❡r ❛♥ ❛t♦♠ ✐s tr✉❡ ✐♥ ❛❧❧ st❛❜❧❡ ♠♦❞❡❧s✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✵✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❋✉rt❤❡r ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

❚❤❡♦r❡♠

❚❤❡ ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t② ♦❢ ❝♦♥❥✉♥❝t✐✈❡ q✉❡r✐❡s ✐s NP✲❝♦♠♣❧❡t❡✳

❚❤❡♦r❡♠

❋✐rst✲♦r❞❡r q✉❡r✐❡s ❛r❡ ♣r♦❣r❛♠✲❝♦♠♣❧❡t❡ ❢♦r PSPACE✳ ❚❤❡✐r ❞❛t❛ ❝♦♠♣❧❡①✐t② ✐s ✐♥ t❤❡ ❝❧❛ss AC0✱ ✇❤✐❝❤ ❝♦♥t❛✐♥s t❤❡ ❧❛♥❣✉❛❣❡s r❡❝♦❣♥✐③❡❞ ❜② ✉♥❜♦✉♥❞❡❞ ❢❛♥✲✐♥ ❝✐r❝✉✐ts ♦❢ ♣♦❧②♥♦♠✐❛❧ s✐③❡ ❛♥❞ ❝♦♥st❛♥t ❞❡♣t❤✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✶✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✹ ✼✳✹ ❈♦♠♣❧❡①✐t② ❙t❛❜❧❡ ▼♦❞❡❧

❋✉rt❤❡r ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

❚❤❡♦r❡♠

▲♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ✐s r✳❡✳✲❝♦♠♣❧❡t❡✳

❚❤❡♦r❡♠

◆♦♥r❡❝✉rs✐✈❡ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ✐s NEXPTIME✲❝♦♠♣❧❡t❡✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✷✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❖✉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♣r❡ss✐✈❡ P♦✇❡r

✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❆ q✉❡r② q ❞❡✜♥❡s ❛ ♠❛♣♣✐♥❣ Mq t❤❛t ❛ss✐❣♥s t♦ ❡❛❝❤ s✉✐t❛❜❧❡ ✐♥♣✉t ❞❛t❛❜❛s❡ Din ✭♦✈❡r ❛ ✜①❡❞ ✐♥♣✉t s❝❤❡♠❛✮ ❛ r❡s✉❧t ❞❛t❛❜❛s❡ Dout = Mq(Din) ✭♦✈❡r ❛ ✜①❡❞ ♦✉t♣✉t s❝❤❡♠❛✮ ❋♦r♠❛❧❧②✱ t❤❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ♦❢ ❛ q✉❡r② ❧❛♥❣✉❛❣❡ Q ✐s t❤❡ s❡t ♦❢ ♠❛♣♣✐♥❣s Mq ❢♦r ❛❧❧ q✉❡r✐❡s q ❡①♣r❡ss✐❜❧❡ ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ Q ❜② s♦♠❡ q✉❡r② ❡①♣r❡ss✐♦♥ ✭♣r♦❣r❛♠✮ E ❘❡s❡❛r❝❤ t❛s❦s ❝♦♥❝❡r♥✐♥❣ ❡①♣r❡ss✐✈❡ ♣♦✇❡r✿

• ❈♦♠♣❛r✐♥❣ t✇♦ q✉❡r② ❧❛♥❣✉❛❣❡s Q1 ❛♥❞ Q2 ✐♥ t❤❡✐r r❡❧❛t✐✈❡ ❡①♣r❡ss✐✈❡

♣♦✇❡r ✭❡✳❣✳ ❋❖ ✈s✳ ❙◗▲ ✈s✳ ❉❛t❛❧♦❣✮✳ ❚❤✐s ✐s ✐♠♣♦rt❛♥t ❢♦r ❞❡s✐❣♥✐♥❣ ❛♥❞ ❛♥❛❧②s✐♥❣ ❛ q✉❡r② ❧❛♥❣✉❛❣❡✳

• ❉❡t❡r♠✐♥✐♥❣ t❤❡ ❛❜s♦❧✉t❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ♦❢ ❛ q✉❡r② ❧❛♥❣✉❛❣❡✱ ❡✳❣✳ ♣r♦✈✐♥❣

t❤❛t ❛ ❣✐✈❡♥ q✉❡r② ❧❛♥❣✉❛❣❡ Q ✐s ❛❜❧❡ t♦ ❡①♣r❡ss ❡①❛❝t❧② ❛❧❧ q✉❡r✐❡s ✇❤♦s❡ ❡✈❛❧✉❛t✐♦♥ ❝♦♠♣❧❡①✐t② ✐s ✐♥ ❛ ❝♦♠♣❧❡①✐t② ❝❧❛ss C✳

❲❡ s❛② Q ❝❛♣t✉r❡s C ❛♥❞ ✇r✐t❡ s✐♠♣❧② Q = C✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✹✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❊①♣r❡ss✐✈❡ P♦✇❡r

❚❤❡r❡ ✐s ❛ s✉❜st❛♥t✐❛❧ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ s❤♦✇✐♥❣ t❤❛t t❤❡ q✉❡r② ❡✈❛❧✉❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❛ ❝❡rt❛✐♥ q✉❡r② ❧❛♥❣✉❛❣❡ Q ✐s C✲❝♦♠♣❧❡t❡ ❛♥❞ s❤♦✇✐♥❣ t❤❛t Q ❝❛♣t✉r❡s C✳ ■❢ t❤❡ ❡✈❛❧✉❛t✐♦♥ ♣r♦❜❧❡♠ ❢♦r Q ✐s C✲❝♦♠♣❧❡t❡✱ t❤❡♥ ❛t ❧❡❛st ♦♥❡ C✲❤❛r❞ q✉❡r② ✐s ❡①♣r❡ss✐❜❧❡ ✐♥ Q✳ ■❢ Q ❝❛♣t✉r❡s C✱ t❤❡♥ Q ❡①♣r❡ss❡s ❛❧❧ q✉❡r✐❡s ❡✈❛❧✉❛❜❧❡ ✐♥ C ✭✐♥❝❧✉❞✐♥❣✱ ♦❢ ❝♦✉rs❡✱ ❛❧❧ C✲❤❛r❞ q✉❡r✐❡s✮✳ ❊①❛♠♣❧❡✿ ❊✈❛❧✉❛t✐♥❣ ❉❛t❛❧♦❣ ✐s P ❤❛r❞ ✭❞❛t❛ ❝♦♠♣❧❡①✐t②✮✱ ❜✉t ♣♦s✐t✐✈❡ ❉❛t❛❧♦❣ ❝❛♥ ♦♥❧② ❡①♣r❡ss ♠♦♥♦t♦♥❡ ♣r♦♣❡rt✐❡s✱ ❤♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ♦❢ ❝♦✉rs❡ ♣r♦❜❧❡♠s ✐♥ P ✇❤✐❝❤ ❛r❡ ♥♦♥✲♠♦♥♦t♦♥✐❝✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✺✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❊①♣r❡ss✐✈❡ P♦✇❡r✿ ❖r❞❡r❡❞ ❙tr✉❝t✉r❡s

❚♦ ♣r♦✈❡ t❤❛t ❛ q✉❡r② ❧❛♥❣✉❛❣❡ Q ❝❛♣t✉r❡s ❛ ♠❛❝❤✐♥❡✲❜❛s❡❞ ❝♦♠♣❧❡①✐t② ❝❧❛ss C✱ ♦♥❡ ✉s✉❛❧❧② s❤♦✇s t❤❛t ❡❛❝❤ C✲♠❛❝❤✐♥❡ ✇✐t❤ ✭❡♥❝♦❞✐♥❣s ♦❢✮ ✜♥✐t❡ str✉❝t✉r❡s ❛s ✐♥♣✉ts t❤❛t ❝♦♠♣✉t❡s ❛ ❣❡♥❡r✐❝ q✉❡r② ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛♥ ❡①♣r❡ss✐♦♥ ✐♥ ❧❛♥❣✉❛❣❡ Q✳ ❆ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✇♦r❦s ♦♥ ❛ str✐♥❣ ❡♥❝♦❞✐♥❣ ♦❢ t❤❡ ✐♥♣✉t ❞❛t❛❜❛s❡ D✳ ❙✉❝❤ ❛♥ ❡♥❝♦❞✐♥❣ ♣r♦✈✐❞❡s ❛♥ ✐♠♣❧✐❝✐t ❧✐♥❡❛r ♦r❞❡r ♦♥ D✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ♦♥ ❛❧❧ ❡❧❡♠❡♥ts ♦❢ t❤❡ ✉♥✐✈❡rs❡ UD ❚❤❡r❡❢♦r❡✱ ♦♥❡ ♦❢t❡♥ ❛ss✉♠❡s t❤❛t ❛ ❧✐♥❡❛r ♦r❞❡r✐♥❣ ♦❢ t❤❡ ✉♥✐✈❡rs❡ ❡❧❡♠❡♥ts ✐s ♣r❡❞❡✜♥❡❞ ❲❡ ❝♦♥s✐❞❡r ❤❡r❡ ♦r❞❡r❡❞ ❞❛t❛❜❛s❡s ✇❤♦s❡ s❝❤❡♠❛s ❝♦♥t❛✐♥ s♣❡❝✐❛❧ r❡❧❛t✐♦♥ s②♠❜♦❧s ❙✉❝❝✱ ❋✐rst✱ ❛♥❞ ▲❛st

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✻✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❊①♣r❡ss✐✈❡ P♦✇❡r✿ ❉❛t❛❧♦❣

❞❛t❛❧♦❣+✿ ❞❛t❛❧♦❣ ✇✐t❤ ✐♥♣✉t ♥❡❣❛t✐♦♥✳

❚❤❡♦r❡♠

❞❛t❛❧♦❣+ P✳ ❙❤♦✇ t❤❛t t❤❡r❡ ❡①✐sts ♥♦ ❞❛t❛❧♦❣+ ♣r♦❣r❛♠ P t❤❛t ❝❛♥ t❡❧❧ ✇❤❡t❤❡r t❤❡ ✉♥✐✈❡rs❡ U ♦❢ t❤❡ ✐♥♣✉t ❞❛t❛❜❛s❡ ❤❛s ❛♥ ❡✈❡♥ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts✳

❚❤❡♦r❡♠

❖♥ ♦r❞❡r❡❞ ❞❛t❛❜❛s❡s✱ ❞❛t❛❧♦❣+ ❝❛♣t✉r❡s P✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✼✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❊①♣r❡ss✐✈❡ P♦✇❡r✿ ▼♦r❡ ❘❡s✉❧ts

❚❤❡♦r❡♠

◆♦♥r❡❝✉rs✐✈❡ r❛♥❣❡✲r❡str✐❝t❡❞ ❞❛t❛❧♦❣ ✇✐t❤ ♥❡❣❛t✐♦♥ ❂ r❡❧❛t✐♦♥❛❧ ❛❧❣❡❜r❛ ❂ ❞♦♠❛✐♥✲✐♥❞❡♣❡♥❞❡♥t r❡❧❛t✐♦♥❛❧ ❝❛❧❝✉❧✉s ❂ ✜rst✲♦r❞❡r ❧♦❣✐❝ ✭✇✐t❤♦✉t ❢✉♥❝t✐♦♥ s②♠❜♦❧s✮✳

❚❤❡♦r❡♠

❖♥ ♦r❞❡r❡❞ ❞❛t❛❜❛s❡s✱ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡r② ❧❛♥❣✉❛❣❡s ❝❛♣t✉r❡ P✿ str❛t✐✜❡❞ ❞❛t❛❧♦❣✱ ❞❛t❛❧♦❣ ✉♥❞❡r ✇❡❧❧✲❢♦✉♥❞❡❞ s❡♠❛♥t✐❝s✱ ❞❛t❛❧♦❣ ✉♥❞❡r ✐♥✢❛t✐♦♥❛r② s❡♠❛♥t✐❝s✳

❚❤❡♦r❡♠

❉❛t❛❧♦❣ ✉♥❞❡r st❛❜❧❡ ♠♦❞❡❧ s❡♠❛♥t✐❝s ❝❛♣t✉r❡s co✲NP✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✽✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❋r♦♠ ❊①♣r❡ss✐✈❡ P♦✇❡r t♦ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

■❢ q✉❡r② ❧❛♥❣✉❛❣❡ L ❝❛♣t✉r❡s ❝♦♠♣❧❡①✐t② ❝❧❛ss C✱ ❛♥❞ C ❤❛s ❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s✱ t❤❡♥ L ❤❛s C✲❝♦♠♣❧❡t❡ ❞❛t❛ ❝♦♠♣❧❡①✐t②✳ ❯♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s✱ r❡s✉❧ts ♦♥ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ❛❧❧♦✇ ♦♥❡ t♦ ❞❡r✐✈❡ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❛❜♦✉t ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t②✳ ❚❤✐s ✐s ❧✐♥❦❡❞ t♦ s✉❝❝✐♥❝t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣r♦❜❧❡♠ ✐♥♣✉ts

r❡♣r❡s❡♥t ✐♥♣✉ts ❜② ❇♦♦❧❡❛♥ ❝✐r❝✉✐ts ✇✐t❤ ✐♥♣✉t ❜✐ts ✇❤❡r❡

✏❈♦♠♣❧❡①✐t② ❯♣❣r❛❞✐♥❣✑ r❡s✉❧ts ✭r♦♦t❡❞ ✐♥ P❛♣❛❞✐♠✐tr✐♦✉✴❨❛♥♥❛❦❛❦✐s✱ ✶✾✽✺✮

❝♦♥❝❧✉❞❡ ❢r♦♠ ❝♦♠♣❧❡t❡♥❡ss ♦❢ ♣r♦❜❧❡♠ ❢♦r ❝❧❛ss ✉♥❞❡r s♣❡❝✐✜❝ r❡❞✉❝t✐♦♥s✱ t❤❡ ❤❛r❞♥❡ss ♦❢ ✐ts ✏s✉❝❝✐♥❝t✑ ✈❛r✐❛♥t ❢♦r ❛♥ ✏❡①♣♦♥❡♥t✐❛❧❧②✑ ❤❛r❞❡r ❝❧❛ss

❊①❛♠♣❧❡✿ ❙✉❝❝✐♥❝t ✸✲❈♦❧♦r❛❜✐❧✐t② ✐s ✲❝♦♠♣❧❡t❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✾✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❋r♦♠ ❊①♣r❡ss✐✈❡ P♦✇❡r t♦ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

■❢ q✉❡r② ❧❛♥❣✉❛❣❡ L ❝❛♣t✉r❡s ❝♦♠♣❧❡①✐t② ❝❧❛ss C✱ ❛♥❞ C ❤❛s ❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s✱ t❤❡♥ L ❤❛s C✲❝♦♠♣❧❡t❡ ❞❛t❛ ❝♦♠♣❧❡①✐t②✳ ❯♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s✱ r❡s✉❧ts ♦♥ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ❛❧❧♦✇ ♦♥❡ t♦ ❞❡r✐✈❡ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❛❜♦✉t ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t②✳ ❚❤✐s ✐s ❧✐♥❦❡❞ t♦ s✉❝❝✐♥❝t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣r♦❜❧❡♠ ✐♥♣✉ts

r❡♣r❡s❡♥t ✐♥♣✉ts ❜② ❇♦♦❧❡❛♥ ❝✐r❝✉✐ts ✇✐t❤ ✐♥♣✉t ❜✐ts ✇❤❡r❡

✏❈♦♠♣❧❡①✐t② ❯♣❣r❛❞✐♥❣✑ r❡s✉❧ts ✭r♦♦t❡❞ ✐♥ P❛♣❛❞✐♠✐tr✐♦✉✴❨❛♥♥❛❦❛❦✐s✱ ✶✾✽✺✮

❝♦♥❝❧✉❞❡ ❢r♦♠ ❝♦♠♣❧❡t❡♥❡ss ♦❢ ♣r♦❜❧❡♠ ❢♦r ❝❧❛ss ✉♥❞❡r s♣❡❝✐✜❝ r❡❞✉❝t✐♦♥s✱ t❤❡ ❤❛r❞♥❡ss ♦❢ ✐ts ✏s✉❝❝✐♥❝t✑ ✈❛r✐❛♥t ❢♦r ❛♥ ✏❡①♣♦♥❡♥t✐❛❧❧②✑ ❤❛r❞❡r ❝❧❛ss

❊①❛♠♣❧❡✿ ❙✉❝❝✐♥❝t ✸✲❈♦❧♦r❛❜✐❧✐t② ✐s ✲❝♦♠♣❧❡t❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✾✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❋r♦♠ ❊①♣r❡ss✐✈❡ P♦✇❡r t♦ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

■❢ q✉❡r② ❧❛♥❣✉❛❣❡ L ❝❛♣t✉r❡s ❝♦♠♣❧❡①✐t② ❝❧❛ss C✱ ❛♥❞ C ❤❛s ❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s✱ t❤❡♥ L ❤❛s C✲❝♦♠♣❧❡t❡ ❞❛t❛ ❝♦♠♣❧❡①✐t②✳ ❯♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s✱ r❡s✉❧ts ♦♥ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ❛❧❧♦✇ ♦♥❡ t♦ ❞❡r✐✈❡ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❛❜♦✉t ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t②✳ ❚❤✐s ✐s ❧✐♥❦❡❞ t♦ s✉❝❝✐♥❝t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣r♦❜❧❡♠ ✐♥♣✉ts

• r❡♣r❡s❡♥t ✐♥♣✉ts w ∈ {0, 1}∗ ❜② ❇♦♦❧❡❛♥ ❝✐r❝✉✐ts Cw ✇✐t❤ n = log |w| ✐♥♣✉t

❜✐ts

• Cw(b1, . . . , bn) = 1 ⇔ wi = 1 ✇❤❡r❡ bin(i) = (b1, . . . , bn)

✏❈♦♠♣❧❡①✐t② ❯♣❣r❛❞✐♥❣✑ r❡s✉❧ts ✭r♦♦t❡❞ ✐♥ P❛♣❛❞✐♠✐tr✐♦✉✴❨❛♥♥❛❦❛❦✐s✱ ✶✾✽✺✮

❝♦♥❝❧✉❞❡ ❢r♦♠ ❝♦♠♣❧❡t❡♥❡ss ♦❢ ♣r♦❜❧❡♠ ❢♦r ❝❧❛ss ✉♥❞❡r s♣❡❝✐✜❝ r❡❞✉❝t✐♦♥s✱ t❤❡ ❤❛r❞♥❡ss ♦❢ ✐ts ✏s✉❝❝✐♥❝t✑ ✈❛r✐❛♥t ❢♦r ❛♥ ✏❡①♣♦♥❡♥t✐❛❧❧②✑ ❤❛r❞❡r ❝❧❛ss

❊①❛♠♣❧❡✿ ❙✉❝❝✐♥❝t ✸✲❈♦❧♦r❛❜✐❧✐t② ✐s ✲❝♦♠♣❧❡t❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✾✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❋r♦♠ ❊①♣r❡ss✐✈❡ P♦✇❡r t♦ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts

■❢ q✉❡r② ❧❛♥❣✉❛❣❡ L ❝❛♣t✉r❡s ❝♦♠♣❧❡①✐t② ❝❧❛ss C✱ ❛♥❞ C ❤❛s ❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s✱ t❤❡♥ L ❤❛s C✲❝♦♠♣❧❡t❡ ❞❛t❛ ❝♦♠♣❧❡①✐t②✳ ❯♥❞❡r ❝❡rt❛✐♥ ❝♦♥❞✐t✐♦♥s✱ r❡s✉❧ts ♦♥ ❡①♣r❡ss✐✈❡ ♣♦✇❡r ❛❧❧♦✇ ♦♥❡ t♦ ❞❡r✐✈❡ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ❛❜♦✉t ♣r♦❣r❛♠ ❝♦♠♣❧❡①✐t②✳ ❚❤✐s ✐s ❧✐♥❦❡❞ t♦ s✉❝❝✐♥❝t r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ♣r♦❜❧❡♠ ✐♥♣✉ts

• r❡♣r❡s❡♥t ✐♥♣✉ts w ∈ {0, 1}∗ ❜② ❇♦♦❧❡❛♥ ❝✐r❝✉✐ts Cw ✇✐t❤ n = log |w| ✐♥♣✉t

❜✐ts

• Cw(b1, . . . , bn) = 1 ⇔ wi = 1 ✇❤❡r❡ bin(i) = (b1, . . . , bn)

✏❈♦♠♣❧❡①✐t② ❯♣❣r❛❞✐♥❣✑ r❡s✉❧ts ✭r♦♦t❡❞ ✐♥ P❛♣❛❞✐♠✐tr✐♦✉✴❨❛♥♥❛❦❛❦✐s✱ ✶✾✽✺✮

• ❝♦♥❝❧✉❞❡ ❢r♦♠ ❝♦♠♣❧❡t❡♥❡ss ♦❢ ♣r♦❜❧❡♠ Π ❢♦r ❝❧❛ss C1 ✉♥❞❡r s♣❡❝✐✜❝

r❡❞✉❝t✐♦♥s✱ t❤❡ ❤❛r❞♥❡ss ♦❢ ✐ts ✏s✉❝❝✐♥❝t✑ ✈❛r✐❛♥t sΠ ❢♦r ❛♥ ✏❡①♣♦♥❡♥t✐❛❧❧②✑ ❤❛r❞❡r ❝❧❛ss C2

❊①❛♠♣❧❡✿ ❙✉❝❝✐♥❝t ✸✲❈♦❧♦r❛❜✐❧✐t② ✐s NEXPTIME✲❝♦♠♣❧❡t❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✺✾✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❋r♦♠ ❊①♣r❡ss✐✈❡ P♦✇❡r t♦ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts✱ ❝♦♥t✬❞

■❢ L ✐s ❝❛♣❛❜❧❡ ♦❢ s✐♠✉❧❛t✐♥❣ ❣❡♥❡r✐❝ ❇♦♦❧❡❛♥ ❝✐r❝✉✐ts ✭q✉❡r② QBC✮✱ t❤❡♥ ❛ ❣✐✈❡♥ s✉❝❝✐♥❝t ✐♥♣✉t Cw ❝❛♥ ❜❡ ✏✉♥❝♦♠♣r❡ss❡❞✑ t♦ w ❊①❛♠♣❧❡✿ ❉❛t❛❧♦❣ ✭✇✐t❤ ✐♥♣✉t ♥❡❣❛t✐♦♥✮ ■❢ ❝❛♣t✉r❡s ✱ ❛❧❧ ❞❛t❛❜❛s❡ q✉❡r✐❡s ✭t❤✉s ❛❧s♦ ❡✈❡r② q✉❡r② ❤❛r❞ ✉♥❞❡r t❤❡ s♣❡❝✐✜❝ r❡❞✉❝t✐♦♥s✮ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✭ ✮ ❇② ❝♦♠♣♦s✐♥❣ ❛♥❞ ✭❛ss✉♠✐♥❣ ❝❡rt❛✐♥ ♠♦❞✉❧❛r✐t② ♦❢ ✮✱ s♦♠❡ ✲❤❛r❞ q✉❡r② ✐s ❡①♣r❡ss❡❞✳ ❊①❛♠♣❧❡✿ ❉❛t❛❧♦❣ ✉♥❞❡r st❛❜❧❡ s❡♠❛♥t✐❝s ✐s ✲❝♦♠♣❧❡t❡ ❚♦ t❤✐s ❡♥❞✱ s♦♠❡ ❛❜str❛❝t ♣r♦♣❡rt✐❡s ♦❢ ❛r❡ r❡q✉✐r❡❞ ▼♦r❡ ❞❡t❛✐❧s✿

❊❴✱ ●♦tt❧♦❜✱ ▼❛♥♥✐❧❛✿ ❉✐s❥✉♥❝t✐✈❡ ❉❛t❛❧♦❣✱ ❆❈▼ ❚❖❉❙ ✷✷✿✸✻✹✲✹✶✼✱ ✶✾✾✼

• ✳ ●♦tt❧♦❜✱ ◆✳ ▲❡♦♥❡✱ ❍✳❱❡✐t❤✱ ❙✉❝❝✐♥❝t♥❡ss ❛s ❛ ❙♦✉r❝❡ ♦❢ ❈♦♠♣❧❡①✐t② ✐♥

▲♦❣✐❝❛❧ ❋♦r♠❛❧✐s♠s✱ ❆P❆▲ ✾✼✿✷✸✶✕✷✻✵✱ ✶✾✾✾✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✻✵✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❋r♦♠ ❊①♣r❡ss✐✈❡ P♦✇❡r t♦ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts✱ ❝♦♥t✬❞

■❢ L ✐s ❝❛♣❛❜❧❡ ♦❢ s✐♠✉❧❛t✐♥❣ ❣❡♥❡r✐❝ ❇♦♦❧❡❛♥ ❝✐r❝✉✐ts ✭q✉❡r② QBC✮✱ t❤❡♥ ❛ ❣✐✈❡♥ s✉❝❝✐♥❝t ✐♥♣✉t Cw ❝❛♥ ❜❡ ✏✉♥❝♦♠♣r❡ss❡❞✑ t♦ w ❊①❛♠♣❧❡✿ ❉❛t❛❧♦❣ ✭✇✐t❤ ✐♥♣✉t ♥❡❣❛t✐♦♥✮ ■❢ L ❝❛♣t✉r❡s C1✱ ❛❧❧ ❞❛t❛❜❛s❡ q✉❡r✐❡s ✭t❤✉s ❛❧s♦ ❡✈❡r② q✉❡r② f ❤❛r❞ ✉♥❞❡r t❤❡ s♣❡❝✐✜❝ r❡❞✉❝t✐♦♥s✮ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✭Qf✮ ❇② ❝♦♠♣♦s✐♥❣ ❛♥❞ ✭❛ss✉♠✐♥❣ ❝❡rt❛✐♥ ♠♦❞✉❧❛r✐t② ♦❢ ✮✱ s♦♠❡ ✲❤❛r❞ q✉❡r② ✐s ❡①♣r❡ss❡❞✳ ❊①❛♠♣❧❡✿ ❉❛t❛❧♦❣ ✉♥❞❡r st❛❜❧❡ s❡♠❛♥t✐❝s ✐s ✲❝♦♠♣❧❡t❡ ❚♦ t❤✐s ❡♥❞✱ s♦♠❡ ❛❜str❛❝t ♣r♦♣❡rt✐❡s ♦❢ ❛r❡ r❡q✉✐r❡❞ ▼♦r❡ ❞❡t❛✐❧s✿

❊❴✱ ●♦tt❧♦❜✱ ▼❛♥♥✐❧❛✿ ❉✐s❥✉♥❝t✐✈❡ ❉❛t❛❧♦❣✱ ❆❈▼ ❚❖❉❙ ✷✷✿✸✻✹✲✹✶✼✱ ✶✾✾✼

• ✳ ●♦tt❧♦❜✱ ◆✳ ▲❡♦♥❡✱ ❍✳❱❡✐t❤✱ ❙✉❝❝✐♥❝t♥❡ss ❛s ❛ ❙♦✉r❝❡ ♦❢ ❈♦♠♣❧❡①✐t② ✐♥

▲♦❣✐❝❛❧ ❋♦r♠❛❧✐s♠s✱ ❆P❆▲ ✾✼✿✷✸✶✕✷✻✵✱ ✶✾✾✾✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✻✵✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❋r♦♠ ❊①♣r❡ss✐✈❡ P♦✇❡r t♦ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts✱ ❝♦♥t✬❞

■❢ L ✐s ❝❛♣❛❜❧❡ ♦❢ s✐♠✉❧❛t✐♥❣ ❣❡♥❡r✐❝ ❇♦♦❧❡❛♥ ❝✐r❝✉✐ts ✭q✉❡r② QBC✮✱ t❤❡♥ ❛ ❣✐✈❡♥ s✉❝❝✐♥❝t ✐♥♣✉t Cw ❝❛♥ ❜❡ ✏✉♥❝♦♠♣r❡ss❡❞✑ t♦ w ❊①❛♠♣❧❡✿ ❉❛t❛❧♦❣ ✭✇✐t❤ ✐♥♣✉t ♥❡❣❛t✐♦♥✮ ■❢ L ❝❛♣t✉r❡s C1✱ ❛❧❧ ❞❛t❛❜❛s❡ q✉❡r✐❡s ✭t❤✉s ❛❧s♦ ❡✈❡r② q✉❡r② f ❤❛r❞ ✉♥❞❡r t❤❡ s♣❡❝✐✜❝ r❡❞✉❝t✐♦♥s✮ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✭Qf✮ ❇② ❝♦♠♣♦s✐♥❣ QBC ❛♥❞ Qf ✭❛ss✉♠✐♥❣ ❝❡rt❛✐♥ ♠♦❞✉❧❛r✐t② ♦❢ L✮✱ s♦♠❡ C2✲❤❛r❞ q✉❡r② ✐s ❡①♣r❡ss❡❞✳ ❊①❛♠♣❧❡✿ ❉❛t❛❧♦❣ ✉♥❞❡r st❛❜❧❡ s❡♠❛♥t✐❝s ✐s NEXPTIME✲❝♦♠♣❧❡t❡ ❚♦ t❤✐s ❡♥❞✱ s♦♠❡ ❛❜str❛❝t ♣r♦♣❡rt✐❡s ♦❢ ❛r❡ r❡q✉✐r❡❞ ▼♦r❡ ❞❡t❛✐❧s✿

❊❴✱ ●♦tt❧♦❜✱ ▼❛♥♥✐❧❛✿ ❉✐s❥✉♥❝t✐✈❡ ❉❛t❛❧♦❣✱ ❆❈▼ ❚❖❉❙ ✷✷✿✸✻✹✲✹✶✼✱ ✶✾✾✼

• ✳ ●♦tt❧♦❜✱ ◆✳ ▲❡♦♥❡✱ ❍✳❱❡✐t❤✱ ❙✉❝❝✐♥❝t♥❡ss ❛s ❛ ❙♦✉r❝❡ ♦❢ ❈♦♠♣❧❡①✐t② ✐♥

▲♦❣✐❝❛❧ ❋♦r♠❛❧✐s♠s✱ ❆P❆▲ ✾✼✿✷✸✶✕✷✻✵✱ ✶✾✾✾✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✻✵✴✻✵

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❈♦♠♣❧❡①✐t② ❛♥❞ ❊①♣r❡ss✐✈❡ P♦✇❡r ✼✳✺ ✼✳✺ ❊①♣r❡ss✐✈❡ P♦✇❡r

❋r♦♠ ❊①♣r❡ss✐✈❡ P♦✇❡r t♦ ❈♦♠♣❧❡①✐t② ❘❡s✉❧ts✱ ❝♦♥t✬❞

■❢ L ✐s ❝❛♣❛❜❧❡ ♦❢ s✐♠✉❧❛t✐♥❣ ❣❡♥❡r✐❝ ❇♦♦❧❡❛♥ ❝✐r❝✉✐ts ✭q✉❡r② QBC✮✱ t❤❡♥ ❛ ❣✐✈❡♥ s✉❝❝✐♥❝t ✐♥♣✉t Cw ❝❛♥ ❜❡ ✏✉♥❝♦♠♣r❡ss❡❞✑ t♦ w ❊①❛♠♣❧❡✿ ❉❛t❛❧♦❣ ✭✇✐t❤ ✐♥♣✉t ♥❡❣❛t✐♦♥✮ ■❢ L ❝❛♣t✉r❡s C1✱ ❛❧❧ ❞❛t❛❜❛s❡ q✉❡r✐❡s ✭t❤✉s ❛❧s♦ ❡✈❡r② q✉❡r② f ❤❛r❞ ✉♥❞❡r t❤❡ s♣❡❝✐✜❝ r❡❞✉❝t✐♦♥s✮ ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✭Qf✮ ❇② ❝♦♠♣♦s✐♥❣ QBC ❛♥❞ Qf ✭❛ss✉♠✐♥❣ ❝❡rt❛✐♥ ♠♦❞✉❧❛r✐t② ♦❢ L✮✱ s♦♠❡ C2✲❤❛r❞ q✉❡r② ✐s ❡①♣r❡ss❡❞✳ ❊①❛♠♣❧❡✿ ❉❛t❛❧♦❣ ✉♥❞❡r st❛❜❧❡ s❡♠❛♥t✐❝s ✐s NEXPTIME✲❝♦♠♣❧❡t❡ ❚♦ t❤✐s ❡♥❞✱ s♦♠❡ ❛❜str❛❝t ♣r♦♣❡rt✐❡s ♦❢ L ❛r❡ r❡q✉✐r❡❞ ▼♦r❡ ❞❡t❛✐❧s✿

• ❊❴✱ ●♦tt❧♦❜✱ ▼❛♥♥✐❧❛✿ ❉✐s❥✉♥❝t✐✈❡ ❉❛t❛❧♦❣✱ ❆❈▼ ❚❖❉❙ ✷✷✿✸✻✹✲✹✶✼✱ ✶✾✾✼
• ●✳ ●♦tt❧♦❜✱ ◆✳ ▲❡♦♥❡✱ ❍✳❱❡✐t❤✱ ❙✉❝❝✐♥❝t♥❡ss ❛s ❛ ❙♦✉r❝❡ ♦❢ ❈♦♠♣❧❡①✐t② ✐♥

▲♦❣✐❝❛❧ ❋♦r♠❛❧✐s♠s✱ ❆P❆▲ ✾✼✿✷✸✶✕✷✻✵✱ ✶✾✾✾✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷ ✻✵✴✻✵