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■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡ ✐♥ ❉❡❝✐❞❛❜❧❡ ❋r❛❣♠❡♥ts ♦❢ ❋✐rst✲❖r❞❡r ▲♦❣✐❝

■❛♥ Pr❛tt✲❍❛rt♠❛♥♥

❙❝❤♦♦❧ ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ▼❛♥❝❤❡st❡r ❯♥✐✈❡rs✐t②✱ ❯❑ ■♥st②t✉t ■♥❢♦r♠❛t②❦✐✱ ❯♥✐✇❡rs②t❡t ❖♣♦❧s❦✐✱ P♦❧s❦❛ ❲②❞③✐❛➟ ▼❛t❡♠❛t②❦✐✱ ■♥❢♦r♠❛t②❦✐ ✐ ▼❡❝❤❛♥✐❦✐✱ ❯♥✐✇❡rs②t❡t ❲❛rs③❛✇s❦✐✱ P♦❧s❦❛ ❡♠❛✐❧✿ ✐♣r❛tt❅❝s✳♠❛♥✳❛❝✳✉❦

■❈▲❆ ✷✵✶✾ ■♥❞✐❛♥ ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣② ❉❡❧❤✐✱ ■♥❞✐❛

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❊①t❡♥s✐♦♥s ✇✐t❤ ❝♦✉♥t✐♥❣ ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ✉♥❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ❊♥ts❝❤❡✐❞✉♥❣s♣r♦❜❧❡♠✿

✭❍✐❧❜❡rt ❛♥❞ ❆❝❦❡r♠❛♥♥✱ ✶✾✷✽✮✿ ❣✐✈❡♥ ❛ ❢♦r♠✉❧❛ ♦❢ FO✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ✐t ❤❛s ❛ ♠♦❞❡❧✳

• ❚❤✐s ♣r♦❜❧❡♠ ✐s ✉♥❞❡❝✐❞❛❜❧❡ ✭❈❤✉r❝❤ ❛♥❞ ❚✉r✐♥❣✱ ✶✾✸✻✴✶✾✸✼✮✳
• ❚❤❡ ✜♥✐t❡ ❊♥ts❝❤❡✐❞✉♥❣s♣r♦❜❧❡♠✿

❣✐✈❡♥ ❛ ❢♦r♠✉❧❛ ♦❢ FO✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ✐t ❤❛s ❛ ✜♥✐t❡ ♠♦❞❡❧✳

• ❚❤✐s ♣r♦❜❧❡♠ t♦♦ ✐s ✉♥❞❡❝✐❞❛❜❧❡ ✭❚r❛❦❤t❡♥❜r♦t✱ ✶✾✺✵✮✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r FO✷✿

❣✐✈❡♥ ❛ ❢♦r♠✉❧❛ ♦❢ FO✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ✐t ❤❛s ❛ ♠♦❞❡❧✳

• ❚❤✐s ♣r♦❜❧❡♠ ✐s ✉♥❞❡❝✐❞❛❜❧❡ ✭❈❤✉r❝❤ ❛♥❞ ❚✉r✐♥❣✱ ✶✾✸✻✴✶✾✸✼✮✳
• ❚❤❡ ✜♥✐t❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r FO✷✿

❣✐✈❡♥ ❛ ❢♦r♠✉❧❛ ♦❢ FO✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ✐t ❤❛s ❛ ✜♥✐t❡ ♠♦❞❡❧✳

• ❚❤✐s ♣r♦❜❧❡♠ t♦♦ ✐s ✉♥❞❡❝✐❞❛❜❧❡ ✭❚r❛❦❤t❡♥❜r♦t✱ ✶✾✺✵✮✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(FO)✿

❣✐✈❡♥ ❛ ❢♦r♠✉❧❛ ♦❢ FO✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ✐t ❤❛s ❛ ♠♦❞❡❧✳

• ❚❤✐s ♣r♦❜❧❡♠ ✐s ✉♥❞❡❝✐❞❛❜❧❡ ✭❈❤✉r❝❤ ❛♥❞ ❚✉r✐♥❣✱ ✶✾✸✻✴✶✾✸✼✮✳
• ❋✐♥❙❛t(FO)✿ ❣✐✈❡♥ ❛ ❢♦r♠✉❧❛ ♦❢ FO✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ✐t

❤❛s ❛ ✜♥✐t❡ ♠♦❞❡❧✳

• ❚❤✐s ♣r♦❜❧❡♠ t♦♦ ✐s ✉♥❞❡❝✐❞❛❜❧❡ ✭❚r❛❦❤t❡♥❜r♦t✱ ✶✾✺✵✮✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ■t ✇❛s q✉✐❝❦❧② r❡❛❧✐③❡❞ t❤❛t✱ ❢♦r ✈❛r✐♦✉s ❢r❛❣♠❡♥ts L FO✷✱

❙❛t(L) ✐s ❞❡❝✐❞❛❜❧❡✳

• ◗✉❛♥t✐✜❡r✲♣r❡✜① ❢r❛❣♠❡♥ts
• ∃∗∀∗ ❬❡q✉❛❧✐t②✱ ♥♦ ❢✉♥❝t✐♦♥s❪ ✭❇❡r♥❛②s✲❙❝❤ö♥✜♥❦❡❧✱ ✶✾✷✽✮
• ∃∗∀∀∃∗ ❬♥♦ ❡q✉❛❧✐t② ♦r ❢✉♥❝t✐♦♥s❪ ✭●ö❞❡❧✱ ✶✾✸✸✮
• ∃∗∀∃∗ ❬❢✉♥❝t✐♦♥s✱ ♥♦ ❡q✉❛❧✐t②❪ ✭●✉r❡✈✐❝❤✱ ✶✾✼✸✮

❙❡❡ ❇ör❣❡r✱ ●rä❞❡❧ ❛♥❞ ●✉r❡✈✐❝❤✱ ✶✾✾✼✳

• ❚✇♦✲✈❛r✐❛❜❧❡ ❢r❛❣♠❡♥t ✭❙❝♦tt ✶✾✻✷✱ ▼♦rt✐♠❡r✱ ✶✾✼✺✮
• ●✉❛r❞❡❞ ❢r❛❣♠❡♥t ✭●rä❞❡❧✱ ✶✾✾✾✮
• ❆❧❧ ♦❢ t❤❡ ❛❜♦✈❡ ❢r❛❣♠❡♥ts ❤❛✈❡ t❤❡ ✜♥✐t❡ ♠♦❞❡❧ ♣r♦♣❡rt②✿ ✐❢ ❛

❢♦r♠✉❧❛ ♦❢ L ❤❛s ❛ ♠♦❞❡❧✱ t❤❡♥ ✐t ❤❛s ❛ ✜♥✐t❡ ♠♦❞❡❧ ✭❙❛t(L) = ❋✐♥❙❛t(L)✮✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ t✇♦✲✈❛r✐❛❜❧❡ ❢r❛❣♠❡♥t✱ ❞❡♥♦t❡❞ FO✷✱ ✐s t❤❡ s❡t ♦❢

✜rst✲♦r❞❡r ❢♦r♠✉❧❛s ✭❡q✉❛❧✐t② ❛❧❧♦✇❡❞✮ ✐♥✈♦❧✈✐♥❣ ♦♥❧② t❤❡ t✇♦ ✈❛r✐❛❜❧❡s x ❛♥❞ y✳

• ❋♦r ❡①❛♠♣❧❡✿

❊✈❡r② ❣r❛❞✉❛t❡ st✉❞❡♥t ✐s s✉♣❡r✈✐s❡❞ ❜② ❛ ♣r♦❢❡ss♦r ✇❤♦ t❡❛❝❤❡s s♦♠❡ ❝♦✉rs❡

∀x(❣r❛❞(x) → ∃y(s✉♣(x, y) ∧ ♣r♦❢(y) ∧ ∃x(t❡❛❝❤(y, x) ∧ ❝♦✉rs❡(x)))).

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ❢r❛❣♠❡♥t FO✷ ✭✇✐t❤ ❡q✉❛❧✐t②✮ ❤❛s ❛♥ ❡①♣♦♥❡♥t✐❛❧✲s✐③❡❞

♠♦❞❡❧ ♣r♦♣❡rt②✿ ❛♥② s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛ ✐s s❛t✐s✜❛❜❧❡ ✐♥ ❛ ♠♦❞❡❧ ♦❢ ❡①♣♦♥❡♥t✐❛❧ s✐③❡✳

• ■t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ s❤♦✇ t❤❛t t❤✐s ❜♦✉♥❞ ✐s ♦♣t✐♠❛❧✿

FO✷✲❢♦r♠✉❧❛s ❝❛♥ ❢♦r❝❡ ❡①♣♦♥❡♥t✐❛❧❧② ❧❛r❣❡ ♠♦❞❡❧s✱ ❛♥❞ ✐♥❞❡❡❞ ❡①♣♦♥❡♥t✐❛❧❧② ❧❛r❣❡ ❣r✐❞s✳

• ❲❡ ❤❛✈❡✿

❚❤❡♦r❡♠ ✭●rä❞❡❧✱ ❑♦❧❛✐t✐s✱ ❱❛r❞✐ ✾✼✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(FO✷) ❛♥❞ ❋✐♥❙❛t(FO✷) ❛r❡ ✐❞❡♥t✐❝❛❧✱ ❛♥❞ ❛r❡ ◆❊①♣❚✐♠❡✲❝♦♠♣❧❡t❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ❣✉❛r❞❡❞ ❢r❛❣♠❡♥t✱ ❞❡♥♦t❡❞ G✱ ✐s t❤❡ s✉❜s❡t ♦❢ FO✷ ✐♥

✇❤✐❝❤ ❛❧❧ q✉❛♥t✐✜❡rs ❣♦✈❡r♥✐♥❣ ❢♦r♠✉❧❛s ✇✐t❤ ♠♦r❡ t❤❛♥ ♦♥❡ ❢r❡❡ ✈❛r✐❛❜❧❡ ❤❛✈❡ ❣✉❛r❞s✿ ∀y(γ → ϕ) ∃y(γ ∧ ϕ), ✇❤❡r❡ γ ✐s ❛♥ ❛t♦♠ ❢❡❛t✉r✐♥❣ ❛❧❧ t❤❡ ❢r❡❡ ✈❛r✐❛❜❧❡s ♦❢ ϕ✳

• ❚❤❡ k✲✈❛r✐❛❜❧❡ ❣✉❛r❞❡❞ ❢r❛❣♠❡♥t✱ ❞❡♥♦t❡❞ Gk✱ ✐s t❤❡ s✉❜s❡t ♦❢

G ✐♥ ✇❤✐❝❤ ❛t ♠♦st k ✈❛r✐❛❜❧❡s ❛♣♣❡❛r✳

• ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ✐s ✐♥ G✷✿

∀x(❣r❛❞(x) → ∃y(s✉♣(x, y) ∧ ♣r♦❢(y) ∧ ∃x(t❡❛❝❤(y, x) ∧ ❝♦✉rs❡(x))))

• ❲❡ s❤❛❧❧ ♠♦st❧② ❜❡ ❝♦♥❝❡r♥❡❞ ✇✐t❤ G✷ ✐♥ t❤❡ s❡q✉❡❧✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❆♥② ✭✜♥✐t❡❧②✮ s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛ ♦❢ G✷✱

❤❛s ❛ ♠♦❞❡❧s ✇❤✐❝❤ ✐s ❛ tr❡❡ ✭✇✐t❤ ❜❛❝❦✲❧♦♦♣s✮✳

• ❋♦r ❛❧❧ k ≥ ✷✱ t❤❡ ❢r❛❣♠❡♥t Gk ❤❛s ❛ ❜♦✉♥❞❡❞ tr❡❡✲✇✐❞t❤

♣r♦♣❡rt②✳

• ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ r✉♥s ♦❢ ❛❧t❡r♥❛t✐♥❣ ♣♦❧②♥♦♠✐❛❧ s♣❛❝❡

❚✉r✐♥❣ ♠❛❝❤✐♥❡s ❝❛♥ ❜❡ ❝❛♣t✉r❡❞ ❡❛s✐❧② ❜② G✷ ❢♦r♠✉❧❛s✳

• ❲❡ ❤❛✈❡✿

❚❤❡♦r❡♠ ✭●rä❞❡❧✱ ✾✾✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(Gk) ❛♥❞ ❋✐♥❙❛t(Gk) ❛r❡ ✐❞❡♥t✐❝❛❧✱ ❛♥❞ ❛r❡ ❊①♣❚✐♠❡✲❝♦♠♣❧❡t❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❲❡ ❝❛♥ ❡❛s✐❧② s❛② ✐♥ ✭❛❧♠♦st✮ ❛❧❧ t❤❡ ❛❜♦✈❡ ❢r❛❣♠❡♥ts✿
• r ✐s r❡✢❡①✐✈❡✿ ∀x.r(x, x)❀
• r ✐s s②♠♠❡tr✐❝✿ ∀x∀y(r(x, y) → r(y, x))
• ❲❡ ❝❛♥♥♦t s❛② ✐♥ ❡✐t❤❡r FO✷ ♦r G✿
• r ✐s tr❛♥s✐t✐✈❡✿ ∀x∀y∀z(r(x, y) ∧ r(y, z) → r(x, z))❀
• r ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳
• ❖✉r q✉❡st✐♦♥✿
• ❲❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ❢r❛❣♠❡♥t✱ FO✷ ❛♥❞ G✷✱ ✇❤❡♥ ♦♥❡ ♦r

♠♦r❡ r❡❧❛t✐♦♥s ❛r❡ s♣❡❝✐✜❡❞ t♦ ❜❡ tr❛♥s✐t✐✈❡✱ ♦r t♦ ❜❡ ❡q✉✐✈❛❧❡♥❝❡s❄

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• Pr♦♣♦s✐t✐♦♥❛❧ ✭♠✉❧t✐✲✮ ♠♦❞❛❧ ❧♦❣✐❝✱ ❤❡r❡ ❞❡♥♦t❡❞ K✱ ❡①t❡♥❞s

♣r♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ✇✐t❤ ✐♥❞❡①❡❞ ♠♦❞❛❧✐t✐❡s r ❛♥❞ ♦r✱ tr❛❞✐t✐♦♥❛❧❧② ✐♥t❡r♣r❡t❡❞ ❛s ♥❡❝❡ss✐t② ❛♥❞ ♣♦ss✐❜✐❧✐t②✳

• ❇✉t ✉♥❞❡r t❤❡ r❡❧❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ♦❢ ❑r✐♣❦❡✱ K ❤❛s ❛ st❛♥❞❛r❞

tr❛♥s❧❛t✐♦♥ ✐♥t♦ ✜rst✲♦r❞❡r ❧♦❣✐❝✿ (rϕ)∗(x) = ∀y(r(x, y) → ϕ∗(y)) (♦rϕ)∗(x) = ∃y(r(x, y) ∧ ϕ∗(y))

• ❚❤❡ tr❛♥s❧❛t✐♦♥ ♦❢ ❛♥② K✲❢♦r♠✉❧❛ ✐s ❛ G✷ ❢♦r♠✉❧❛✿ ❤❡♥❝❡ K ✐s

❛ s✉❜❢r❛❣♠❡♥t ♦❢ G✷✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤✉s✱ ✇❡ ❝❛♥ r❡✲✐♥t❡r♣r❡t ❢♦r♠✉❧❛s ♦❢ K ❛s ❞❡s❝r✐♣t✐♦♥s ♦❢

❡♥t✐t✐❡s✿

• ♦rϕ✿ ✐s r❡❧❛t❡❞ ❜② r t♦ s♦♠❡ t❤✐♥❣s ✇❤✐❝❤ ❛r❡ ϕ
• rϕ✿ ✐s r❡❧❛t❡❞ ❜② r ♦♥❧② t♦ t❤✐♥❣s ✇❤✐❝❤ ❛r❡ ϕ
• ❋♦r ❡①❛♠♣❧❡✿

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

❣r❛❞ ∧ ♦s✉♣(♣r♦❢ ∧ t❡❛❝❤♠❛t❤.❝♦✉rs❡).

• K ✐s ✭♠♦r❡ ♦r ❧❡ss✮ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❞❡s❝r✐♣t✐♦♥✲❧♦❣✐❝ ALC✳
• ❲❡ s❤❛❧❧ ❝♦♥✜♥❡ ❛tt❡♥t✐♦♥ t♦ t❤❡ ❝❛s❡ ♦❢ K ✇✐t❤ ❥✉st ♦♥❡

♠♦❞❛❧✐t② ♦r✱ ❛♥❞ ✇❡ ❝❛❧❧ r t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥s❤✐♣✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚r❛❞✐t✐♦♥❛❧❧②✱ ♠♦❞❛❧ ❧♦❣✐❝✐❛♥s ❤❛✈❡ ❛s❦❡❞ ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥

t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ✐s ❛ss✉♠❡❞ t♦ s❛t✐s❢② ✈❛r✐♦✉s ❝♦♥❞✐t✐♦♥s ✭r❡✢❡①✐✈✐t②✱ s②♠♠❡tr②✱ trr❛♥s✐t✐✈✐t②✱ ✳ ✳ ✳ ✮✳

• ❉❡♥♦t❡ ❜② ❑✹ t❤❡ ❧♦❣✐❝ K ✐♥ ✇❤✐❝❤ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ✐s

r❡q✉✐r❡❞ t♦ ❜❡ tr❛♥s✐t✐✈❡ ✭✇❡ ♠✐❣❤t ❝❛❧❧ ✐t K✶T✮✳

• ❉❡♥♦t❡ ❜② ❙✺ t❤❡ ❧♦❣✐❝ K ✐♥ ✇❤✐❝❤ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ✐s

r❡q✉✐r❡❞ t♦ ❜❡ ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ✭✇❡ ♠✐❣❤t ❝❛❧❧ ✐t K✶E ✮✳

❚❤❡♦r❡♠ ✭▲❛❞♥❡r ✶✾✼✾✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(K) ❛♥❞ ❋✐♥❙❛t(K) ❛r❡ ✐❞❡♥t✐❝❛❧✱ ❛♥❞ ❛r❡ P❙♣❛❝❡✲❝♦♠♣❧❡t❡✳ ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(K✹) ❛♥❞ ❋✐♥❙❛t(K✹) ❛r❡ ✐❞❡♥t✐❝❛❧✱ ❛♥❞ ❛r❡ P❙♣❛❝❡✲❝♦♠♣❧❡t❡✳ ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(S✺) ❛♥❞ ❋✐♥❙❛t(S✺) ❛r❡ ✐❞❡♥t✐❝❛❧✱ ❛♥❞ ❛r❡ ◆P❚✐♠❡✲❝♦♠♣❧❡t❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❊①t❡♥s✐♦♥s ✇✐t❤ ❝♦✉♥t✐♥❣ ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ✉♥❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❲❡ ❞❡♥♦t❡ ❜② C✷ t❤❡ ✷✲✈❛r✐❛❜❧❡ ❢r❛❣♠❡♥t ♦❢ ✶st✲♦r❞❡r ❧♦❣✐❝

✇✐t❤ ❝♦✉♥t✐♥❣ q✉❛♥t✐✜❡rs✳ ∀x(❛rt✐st(x) → ∃≤✶✸y(❜❡❡❦❡❡♣❡r(y)∧ ∃=✻✹✼x(❝❛r♣❡♥t❡r(x) ∧ ❞❡s♣✐s❡(x, y)) ∧ ❡♥✈②(x, y))).

• ❊q✉❛❧✐t② ✐s ❞❡✜♥❛❜❧❡ ✐♥ C✷✿ ∀x(r(x, x) ∧ ∃≤✶y r(x, y))✳
• C✷ ❧❛❝❦s t❤❡ ❋▼P✿

∃x∀y¬r(y, x) ∧ ∀x∃y r(x, y) ∧ ∀x∃≤✶y r(y, x).

a✵ a✶ a✷ a✸ a✹ . . .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• P❛❝❤♦❧s❦✐ ❡t ❛❧✳ ✶✾✾✼ s❤♦✇❡❞ t❤❛t t❤❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r

C✷ ✐s ✐♥ ✷✲◆❊①♣❚✐♠❡✱ ✉s✐♥❣ ❛ t❛❜❧❡❛✉✲❧✐❦❡ ❛♣♣r♦❛❝❤✳

• ●rä❞❡❧ ❛♥❞ ❖tt♦ ✶✾✾✼ s❤♦✇❡❞ ❞❡❝✐❞❛❜✐❧✐t② ✉s✐♥❣ ✐♥t❡❣❡r ❧✐♥❡❛r

♣r♦❣r❛♠♠✐♥❣✳

• ❋✐♥✐t❡❧② s❛t✐s✜❛❜❧❡ ❢♦r♠✉❧❛s ♦❢ C✷ ❝❛♥ ❤❛✈❡ ❵❞♦✉❜❧②✲

❡①♣♦♥❡♥t✐❛❧✬ ♠♦❞❡❧s ✭●rä❞❡❧✱ ❖tt♦ ❛♥❞ ❘♦s❡♥✱ ✶✾✾✼✮✳

• ❲✐t❤ ❥✉st ❛ ❧✐tt❧❡ ♠♦r❡ ✇♦r❦✱ ✇❡ ❝❛♥ s❤♦✇

❚❤❡♦r❡♠ ✭P✲❍ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(C✷) ❛♥❞ ❋✐♥❙❛t(C✷) ❛r❡ ❜♦t❤ ◆❊①♣❚✐♠❡✲❝♦♠♣❧❡t❡✳ ❋✉rt❤❡r✱ ❛♥② ✜♥✐t❡❧② s❛t✐s✜❛❜❧❡ C✷✲❢♦r♠✉❧❛ ϕ ❤❛s ❛ ♠♦❞❡❧ ♦❢ s✐③❡ ❜♦✉♥❞❡❞ ❜② ❛ ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ♦❢ | |ϕ| |✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ❣✉❛r❞❡❞ t✇♦✲✈❛r✐❛❜❧❡ ❢r❛❣♠❡♥t ✇✐t❤ ❝♦✉♥t✐♥❣✱ GC✷ ✐s t❤❡

❢r❛❣♠❡♥t ♦❢ C✷ ✇✐t❤✿

• ✭♥♦ ✐♥❞✐✈✐❞✉❛❧ ❝♦♥st❛♥ts✮
• ♦♥❧② ❣✉❛r❞❡❞ ✭❝♦✉♥t✐♥❣✮ q✉❛♥t✐✜❝❛t✐♦♥

∀u.ϕ, ∃u.ϕ ✭ϕ ❤❛s ❛t ♠♦st ♦♥❡ ❢r❡❡ ✈❛r✐❛❜❧❡✮ ∀u(α → ψ) ∃≤Cu(α ∧ ψ), ∃≥Cu(α ∧ ψ), ∃=Cu(α ∧ ψ), ✭❱❛rs(α) = {x, y}✮

• ❊①❛♠♣❧❡ ♦❢ GC✷✲❢♦r♠✉❧❛✿

∀x(❛rt✐st(x) → ∃≤✶✼y(❞❡s♣✐s❡(x, y) ∧ ❜❡❡❦❡❡♣❡r(y)))

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• GC✷ s❤❛r❡s ♠❛♥② ♣r♦♣❡rt✐❡s ✇✐t❤ C✷
• ❋▼P ❢❛✐❧s❀
• ✜♥✐t❡ ♠♦❞❡❧s ❝❛♥ ❜❡ ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧❧② ❧❛r❣❡✳
• ❍♦✇❡✈❡r✱ t❤❡ ❝♦♠♣❧❡①✐t✐❡s ❛r❡ ❛ ❧✐tt❧❡ ❧♦✇❡r✿

❚❤❡♦r❡♠ ✭❑❛③❛❦♦✈ ✷✵✵✹✴P✲❍ ✷✵✵✻✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(C✷) ❛♥❞ ❋✐♥❙❛t(C✷) ❛r❡ ❜♦t❤ ❊①♣❚✐♠❡✲❝♦♠♣❧❡t❡✳

• ❈♦♠♣❧❡①✐t② r❡t✉r♥s t♦ ◆❊①♣❚✐♠❡✲❝♦♠♣❧❡t❡ ❛s s♦♦♥ ❛s ♦♥❡

✐♥❞✐✈✐❞✉❛❧ ❝♦♥st❛♥t ✐s ❛❞❞❡❞✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ●r❛❞❡❞ ♠♦❞❛❧ ❧♦❣✐❝✱ ❤❡r❡ ❞❡♥♦t❡❞ ●rK✱ ✐s t❤❡ ❡①t❡♥s✐♦♥ ♦❢ K

✇✐t❤ ❝♦✉♥t✐♥❣ ♠♦❞❛❧✐t✐❡s✿ ♦≤Cϕ ♦=Cϕ ♦≥Cϕ ✇❤❡r❡ C ≥ ✵✳

• ❲❡ r❡❛❞ ♦≤Cϕ ❛s ✏❚❤❡r❡ ❛r❡ ❛t ♠♦st C ❛❝❝❡ss✐❜❧❡ ❡♥t✐t✐❡s

s✉❝❤ t❤❛t ϕ✱✑ ❛♥❞ s✐♠✐❧❛r❧② ❢♦r ♦=C✱ ♦≥C✳

• ♥♦t❡ t❤❛t ♦≥✶ϕ ✐s s✐♠♣❧② ♦ϕ✱ ❛♥❞ ♦≤✵¬ϕ ✐s s✐♠♣❧② ϕ✳
• ●rK ♠❛② ❜❡ r❡❣❛r❞❡❞ ❛s ❛ s✉❜✲❢r❛❣♠❡♥t ♦❢ GC✷ ✈✐❛ t❤❡

❢♦❧❧♦✇✐♥❣ tr❛♥s❧❛t✐♦♥✿ (♦≥Cϕ)∗ =∃≥Cy(r(x, y) ∧ ϕ∗(y)) (♦≤Cϕ)∗ =∃≤Cy(r(x, y) ∧ ϕ∗(y)).

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❆❣❛✐♥✱ ✇❡ ❤❛✈❡ ❛ tr❡❡✲♠♦❞❡❧ ♣r♦♣❡rt②✱ ✇❤✐❝❤ ②✐❡❧❞s✿

❚❤❡♦r❡♠ ✭❚♦❜✐❡s ✷✵✵✵✮

❚❤❡ ♣r♦❜❧❡♠ ❙❛t(●rK) ✐s P❙♣❛❝❡✲❝♦♠♣❧❡t❡✳

• ❚❤❡ tr❡❡✲♣r♦♣❡rt② ❢❛✐❧s ✐❢ t❤❡ ❛❝❝❡ss✐❜✐❧✐t② r❡❧❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✿

A | =w✵ q✵ ∧ ♦≥✷(¬q✵ ∧ q✶ ∧ ♦≥✶(¬q✵ ∧ ¬q✶)) ∧ ♦≤✶¬q✶.

w✵ q✵ ♦≤✶¬q✶ w✶ ¬q✵✱ q✶ ♦≥✶(¬q✵ ∧ ¬q✶) w✷ ¬q✵✱ q✶ ♦≥✶(¬q✵ ∧ ¬q✶) w✸ ¬q✵✱ ¬q✶ w✹ ¬q✵✱ ¬q✶ =

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❊①t❡♥s✐♦♥s ✇✐t❤ ❝♦✉♥t✐♥❣ ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ✉♥❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ❜❧✉❡

✶

❣r❡❡♥

✶

r❡❞ r❡❞

✶

❞❛r❦❣r❡❡♥

✶

♣✐♥❦ ♣✐♥❦

✶

❣r❡❡♥

✷

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ❜❧✉❡

✶

❣r❡❡♥

✶

r❡❞ r❡❞

✶

❞❛r❦❣r❡❡♥

✶

♣✐♥❦ ♣✐♥❦

✶

❣r❡❡♥

✷

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) r❡❞

✶

❞❛r❦❣r❡❡♥

✶

♣✐♥❦ ♣✐♥❦

✶

❣r❡❡♥

✷

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) r❡❞

✶

❞❛r❦❣r❡❡♥

✶

♣✐♥❦ ♣✐♥❦

✶

❣r❡❡♥

✷

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ♣✐♥❦

✶

❣r❡❡♥

✷

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ♣✐♥❦

✶

❣r❡❡♥

✷

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ♣✐♥❦

✶

❣r❡❡♥

✷

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ♣✐♥❦

✶

❣r❡❡♥

✷

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ∀x(♣✐♥❦(x) → ∀y(r✶(x, y) → (❣r❡❡♥(y) → r✷(x, y)))) . . .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ∀x(♣✐♥❦(x) → ∀y(r✶(x, y) → (❣r❡❡♥(y) → r✷(x, y)))) . . .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ∀x(♣✐♥❦(x) → ∀y(r✶(x, y) → (❣r❡❡♥(y) → r✷(x, y)))) . . .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ∀x(♣✐♥❦(x) → ∀y(r✶(x, y) → (❣r❡❡♥(y) → r✷(x, y)))) . . .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ∀x(♣✐♥❦(x) → ∀y(r✶(x, y) → (❣r❡❡♥(y) → r✷(x, y)))) . . .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ∀x(♣✐♥❦(x) → ∀y(r✶(x, y) → (❣r❡❡♥(y) → r✷(x, y)))) . . .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷❊) ❛♥❞ ❋✐♥❙❛t(G✷✷❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✿

∃x.❜❧✉❡(x) ∀x(❜❧✉❡(x) → ∃y(r✶(x, y) ∧ ❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ r❡❞(y))) ∀x(r❡❞(x) → ∃y(r✶(x, y) ∧ ❞❛r❦❣r❡❡♥(y)) ∧ ∃y(r✶(y, x) ∧ ♣✐♥❦(y))) . . . ∀x(♣✐♥❦(x) → ∀y(r✶(x, y) → (❣r❡❡♥(y) → r✷(x, y)))) . . .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤✉s✱ ✇✐t❤ t✇♦ tr❛♥s✐t✐✈❡ r❡❧❛t✐♦♥s✱ ✇❡ ❝❛♥ ❝♦♥str✉❝t

✭♣♦t❡♥t✐❛❧❧②✮ ✐♥✜♥✐t❡ ❣r✐❞s✿

• ❇② ❝♦❧♦✉r✐♥❣ t❤❡s❡ ❣r✐❞s ✇✐t❤ ❛ ✭♥❡✇✮ s❡t ♦❢ ❝♦❧♦✉rs✱ ✇❡ ❝❛♥

❡♥❝♦❞❡ r✉♥s ♦❢ ❚✉r✐♥❣ ♠❛❝❤✐♥❡s✳

• ❆❧❧ ♦❢ t❤❡ ❢♦r♠✉❧❛s ✉s❡❞ ❛❜♦✈❡ ❧✐❡ ✐♥ G✷✷T✳
• ❚❤✉s ✇❡ ❤❛✈❡

❚❤❡♦r❡♠

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✷T) ❛♥❞ ❋✐♥❙❛t(G✷✷T) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t

✷✸

❛♥❞ ❋✐♥❙❛t

✷✸

❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❛t(G✷✸❊) ❛♥❞ ❋✐♥❙❛t(G✷✸❊) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳
• ❍❡♥❝❡

❚❤❡♦r❡♠ ✭❑✐❡r♦➠s❦✐ ❛♥❞ ❖tt♦ ✷✵✵✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(G✷✸E) ❛♥❞ ❋✐♥❙❛t(G✷✸E) ❛r❡ ✉♥❞❡❝✐❞❛❜❧❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❇② ❛ s✐♠✐❧❛r ❛r❣✉♠❡♥t✱ ❙❛t(GC✷✶❚) ❛♥❞ ❋✐♥❙❛t(GC✷✶❚) ❛r❡

✉♥❞❡❝✐❞❛❜❧❡✳

• ❙❛t(GC✷✷❊) ❛♥❞ ❋✐♥❙❛t(GC✷✷❊) ❛r❡ ❛❧s♦ ✉♥❞❡❝✐❞❛❜❧❡✳
• ❯s✐♥❣ t❤✐s ❧♦❣✐❝✱ ✇❡ ❡♥❝♦❞❡ r✉♥s ♦❢ ❝♦✉♥t❡r ♠❛❝❤✐♥❡s✿

E✶ E✶ E✷ E✷ s✱ s✵ s s s s d✶ d✶ d✶ d✷ d✷

• ✭P❛rt✐❛❧✮ ❜✐❥❡❝t✐♦♥s ❛r❡ ✉s❡❞ t♦ ✜①✴✐♥❝r❡♠❡♥t✴❞❡❝r❡♠❡♥t

❝♦✉♥t❡rs ✐♥ s✉❝❝❡ss✐✈❡ st❛t❡s✳

Ek s s ci ci rk

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❊①t❡♥s✐♦♥s ✇✐t❤ ❝♦✉♥t✐♥❣ ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ✉♥❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ▲❡t ●rKE ❜❡ ●rK ✉♥❞❡r t❤❡ st✐♣✉❧❛t✐♦♥ t❤❛t t❤❡ ❛❝❝❡ss✐❜✐❧✐t②

r❡❧❛t✐♦♥ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡✳

• ❊✛❡❝t✐✈❡❧②✱✇❡ ❛r❡ ✇✐t❤✐♥ t❤❡ ✶✲✈❛r✐❛❜❧❡ ❢r❛❣♠❡♥t ✇✐t❤ ❝♦✉♥t✐♥❣✳
• ❚❤✐s ❛❧❧♦✇s ✉s t♦ ♣r♦✈❡✿

❚❤❡♦r❡♠ ✭❑❛③❛❦♦✈✱P✲❍ ✷✵✵✾✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(●rKE) ❛♥❞ ❋✐♥❙❛t(●rKE) ❛r❡ ✐❞❡♥t✐❝❛❧ ❛♥❞ ❛r❡ ✐♥ ◆P❚✐♠❡✳

• ❇✉t ♦❜s❡r✈❡ t❤❛t ●rK ❧❛❝❦s t❤❡ ♣♦❧②✲s✐③❡❞ ♠♦❞❡❧ ♣r♦♣❡rt②

♦✈❡r s✉❝❤ ❢r❛♠❡s✿ t❤❡ ❢♦r♠✉❧❛ ♦≥✷np ❝♦♥t❛✐♥s ≈ n s②♠❜♦❧s✱ ❜✉t ✐ts s♠❛❧❧❡st ♠♦❞❡❧ ❤❛s ✷n ✇♦r❧❞s✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ▲❡t ●rKT ❜❡ ●rK ✉♥❞❡r t❤❡ st✐♣✉❧❛t✐♦♥ t❤❛t t❤❡ ❛❝❝❡ss✐❜✐❧✐t②

r❡❧❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳

• ❲✐t❤ ●rKT✱ ✇❡ ❝❛♥ ❝r❡❛t❡ tr❡❡✲❧✐❦❡ str✉❝t✉r❡s ❛s ♥♦r♠❛❧✱ ❛♥❞

sq✉❡❡③❡ t❤❡♠ t♦❣❡t❤❡r ✉s✐♥❣ ❣r❛❞❡❞ ♠♦❞❛❧✐t✐❡s✿

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤✐s ♣r♦❝❡ss r❡♣❡❛ts ✐ts❡❧❢✱ t♦ ❝r❡❛t❡ ❛ ✭r❛t❤❡r ❥✉♠❜❧❡❞✮

♣②r❛♠✐❞✲❧✐❦❡ str✉❝t✉r❡✱ ❛t t❤❡ ❜❛s❡ ♦❢ ✇❤✐❝❤ ✐s ❛ ✷n × ✷n ❣r✐❞✳

• ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐t ✐s r♦✉t✐♥❡ t♦ s❤♦✇ t❤❛t ❡①♣♦♥❡♥t✐❛❧✲s✐③❡❞

♠♦❞❡❧s ❛❧✇❛②s s✉✣❝❡ ❢♦r ●rKT✿

❚❤❡♦r❡♠ ✭❑❛③❛❦♦✈✱P✲❍ ✷✵✵✾✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(●rKT) ❛♥❞ ❋✐♥❙❛t(●rKT) ❛r❡ ✐❞❡♥t✐❝❛❧ ❛♥❞ ❛r❡ ◆❊①♣❚✐♠❡✲❝♦♠♣❧❡t❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ▲❡t FO✷✷❊❈✱ ❜❡ t❤❡ t✇♦✲✈❛r✐❛❜❧❡ ❢r❛❣♠❡♥t ✇✐t❤ t✇♦

❡q✉✐✈❛❧❡♥❝❡ ❝❧♦s✉r❡s✱

• ❆ str✉❝t✉r❡ ✐♥t❡r♣r❡t✐♥❣ r

#

✶ ✱ r

#

✷ ❛s t❤❡ ❡q✉✐✈❛❧❡♥❝❡✲❝❧♦s✉r❡s ♦❢

r✶✱ r✷ ❧♦♦❦s s♦♠❡t❤✐♥❣ ❧✐❦❡ t❤✐s✿

• ❲❡ t❤✉s ❤❛✈❡ t❤r❡❡ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥s t♦ ❞❡❛❧ ✇✐t❤✿

r

#

✶ ,

r

#

✷ ,

r

#

✶ ∩ r

#

✷ .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❆ ❜❡tt❡r ✇❛② t♦ ♣✐❝t✉r❡ ❛♥ FO✷✷❊❈✲str✉❝t✉r❡ ✐s ❛s ❛♥

❡❞❣❡✲❝♦❧♦✉r❡❞ ❜✐♣❛rt✐t❡ ❣r❛♣❤✿

• ◆♦t✐❝❡ t❤❛t t❤❡ ❡❞❣❡s ♦❢ t❤✐s ❣r❛♣❤ ❛r❡ ✐♥t❡rs❡❝t✐♦♥✲❝❧❛ss❡s
• ■♥t❡rs❡❝t✐♦♥✲❝❧❛ss❡s ♠❛② ❜❡ ❛ss✉♠❡❞ t♦ ❜❡ ❡①♣♦♥❡♥t✐❛❧❧②

❜♦✉♥❞❡❞ ✐♥ s✐③❡✳

• ❚❤❡r❡❢♦r❡✱ t❤❡② ♠❛② ❜❡ ❛ss✉♠❡❞ t♦ ❜❡ ❛ss✉♠❡❞ t♦ ❤❛✈❡ ♦♥❡ ♦❢

❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧❧② ♠❛♥② t②♣❡s✱ ♦r ❝♦❧♦✉rs✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❆ ❜❡tt❡r ✇❛② t♦ ♣✐❝t✉r❡ ❛♥ FO✷✷❊❈✲str✉❝t✉r❡ ✐s ❛s ❛♥

❡❞❣❡✲❝♦❧♦✉r❡❞ ❜✐♣❛rt✐t❡ ❣r❛♣❤✿

• ◆♦t✐❝❡ t❤❛t t❤❡ ❡❞❣❡s ♦❢ t❤✐s ❣r❛♣❤ ❛r❡ ✐♥t❡rs❡❝t✐♦♥✲❝❧❛ss❡s
• ■♥t❡rs❡❝t✐♦♥✲❝❧❛ss❡s ♠❛② ❜❡ ❛ss✉♠❡❞ t♦ ❜❡ ❡①♣♦♥❡♥t✐❛❧❧②

❜♦✉♥❞❡❞ ✐♥ s✐③❡✳

• ❚❤❡r❡❢♦r❡✱ t❤❡② ♠❛② ❜❡ ❛ss✉♠❡❞ t♦ ❜❡ ❛ss✉♠❡❞ t♦ ❤❛✈❡ ♦♥❡ ♦❢

❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧❧② ♠❛♥② t②♣❡s✱ ♦r ❝♦❧♦✉rs✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❦❡✇ ❡❞❣❡s ✭s❤❛r❡ ♥♦ ❡♥❞♣♦✐♥t✮
• ❚❤❡ ♦r❞❡r ♦❢ ❛ ♥♦❞❡ u✿ ❛ ❢✉♥❝t✐♦♥ fu ♠❛♣♣✐♥❣ ❡❛❝❤ ❝♦❧♦✉r t♦

t❤❡ ♥✉♠❜❡r ♦❢ ❡❞❣❡s t❤❡ ♥♦❞❡ ♦❢ t❤❛t ❝♦❧♦✉r ✐s ✐♥❝✐❞❡♥t ♦♥✳ u fu(❜❧✉❡) = ✷ fu(r❡❞) = ✶ fu(❣r❡❡♥) = ✵ . . .

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ◆♦✇ ❧❡t ϕ ❜❡ ❛ ❢♦r♠✉❧❛ ♦❢ FO✷✷❊❈✳
• ϕ ❝❛♥ ❜❡ ✭✜♥✐t❡❧②✮ ❡q✉✐s❛t✐s✜❛❜❧② ✇r✐tt❡♥ ✐♥ t❤❡ ❢♦r♠

∀x∀y.θ∧

m✶

• i=✶

∀x(p✶,i(x) → ∃y(r

#

✶ (x, y) ∧ r

#

✷ (x, y) ∧ θ✶,i))∧ m✷

• i=✶

∀x(p✷,i(x) → ∃y(r

#

✶ (x, y) ∧ ¬r

#

✷ (x, y) ∧ θ✷,i))∧ m✸

• i=✶

∀x(p✸,i(x) → ∃y(¬r

#

✶ (x, y) ∧ r

#

✷ (x, y) ∧ θ✸,i))∧ m✹

• i=✶

∀x(p✹,i(x) → ∃y(¬r

#

✶ (x, y) ∧ ¬r

#

✷ (x, y) ∧ θ✹,i))

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ❝♦♥❥✉♥❝ts

∀x∀y.θ ∧

m✶

• i=✶

∀x(p✶,i(x) → ∃y(r

#

✶ (x, y) ∧ r

#

✷ (x, y) ∧ θ✶,i))

s♣❡❝✐❢② t❤❡ s❡t ♦❢ ♣♦ss✐❜❧❡ ❡❞❣❡✲❝♦❧♦✉rs✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ❝♦♥❥✉♥❝ts

∀x∀y.θ ∧

m✷

• i=✶

∀x(p✷,i(x) → ∃y(r

#

✶ (x, y) ∧ ¬r

#

✷ (x, y) ∧ θ✶,i))

❝♦♥str❛✐♥ t❤❡ ♦r❞❡rs ♦❢ t❤❡ ❧❡❢t✲❤❛♥❞ ♥♦❞❡s ✭✉♣ t♦ ❛ ❝❡rt❛✐♥ ❜♦✉♥❞✮✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙②♠♠❡tr✐❝❛❧❧②✱ t❤❡ ❝♦♥❥✉♥❝ts

∀x∀y.θ ∧

m✸

• i=✶

∀x(p✸,i(x) → ∃y(¬r

#

✶ (x, y) ∧ r

#

✷ (x, y) ∧ θ✶,i))

❝♦♥str❛✐♥ t❤❡ ♦r❞❡rs ♦❢ t❤❡ ❧❡❢t✲❤❛♥❞ ♥♦❞❡s✳

• ❚❤❡ ❡①✐st❡♥t✐❛❧ ❝♦♥❥✉♥❝ts ❥♦✐♥t❧② s♣❡❝✐❢② ❡❞❣❡✲❝♦❧♦✉rs t❤❛t

♠✉st ❜❡ r❡❛❧✐③❡❞✳

• ❋✐♥❛❧❧②✱ t❤❡ ❝♦♥❥✉♥❝ts

∀x∀y.θ ∧

m✹

• i=✶

∀x(p✹,i(x) → ∃y(¬r

#

✶ (x, y) ∧ ¬r

#

✷ (x, y) ∧ θ✶,i))

✐♠♣♦s❡ ❝♦♥str❛✐♥ts ♦♥ t❤❡ ❝♦❧♦✉rs s❦❡✇ ♣❛✐rs ♦❢ ❡❞❣❡s✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡ ♣r♦❜❧❡♠ ❇●❊❙❈ ✭❜✐♣❛rt✐t❡ ❣r❛♣❤ ❡①✐st❡♥❝❡ ✇✐t❤ s❦❡✇

❝♦♥str❛✐♥ts ❛♥❞ ❝❡✐❧✐♥❣✮ ✐s ❛ s❡①t✉♣❧❡ (∆, ∆✵, M, F, G, X) ✇❤❡r❡✿

• ∆ ✐s ❛ s❡t ♦❢ ❵❝♦❧♦✉rs✬ ❛♥❞ ∆✵ ⊆ ∆❀
• M ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ✭❝❡✐❧✐♥❣✮
• F ❛♥❞ G ❛r❡ s❡ts ♦❢ ❢✉♥❝t✐♦♥s ∆ → [✵, M]
• X ✐s ❛ s❡t ♦❢ ♣❛✐rs ♦❢ ❝♦❧♦✉rs ✭s❦❡✇ ❝♦♥str❛✐♥ts✮❀
• ❚❤❡ ✐♥st❛♥❝❡ ✐s ♣♦s✐t✐✈❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❜✐♣❛rt✐t❡ ❣r❛♣❤ ✇✐t❤

❡❞❣❡s ❝♦❧♦✉r❡❞ ❜② ∆ s✉❝❤ t❤❛t

• ❆❧❧ ♦r❞❡rs ♦❢ ❧❡❢t✲❤❛♥❞ ♥♦❞❡s ✭❝❛♣♣❡❞ ❛t M✮ ❛r❡ ✐♥ F❀
• ❆❧❧ ♦r❞❡rs ♦❢ r✐❣❤t✲❤❛♥❞ ♥♦❞❡s ✭❝❛♣♣❡❞ ❛t M✮ ❛r❡ ✐♥ G❀
• ❆❧❧ ❝♦❧♦✉rs ✐♥ ∆✵ ❛r❡ r❡❛❧✐③❡❞❀
• ❚❤❡r❡ ✐s ♥♦ s❦❡✇ ♣❛✐r ✇✐t❤ ❝♦❧♦✉rs ❢r♦♠ X✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ■t ✐s ♣♦ss✐❜❧❡ t♦ s❤♦✇ t❤❡ ❢♦❧❧♦✇✐♥❣✿

❚❤❡♦r❡♠ ✭❑▼❚P✲❍✶✷✮

❇●❊❙❈ ❛♥❞ ✜♥✐t❡ ❇●❊❙❈ ❛r❡ ❜♦t❤ ◆P❚✐♠❡✲❝♦♠♣❧❡t❡✳

• ✭❚❤❡ ♣r♦♦❢ ✐s ❜② tr❛♥s❧❛t✐♦♥ t♦ ❧✐♥❡❛r ✐♥t❡❣❡r ♣r♦❣r❛♠♠✐♥❣✮✳
• ❚❤❡ ✐♥st❛♥❝❡ ♦❢ ❇●❊❙❈ ✇❡ ♦❜t❛✐♥ ✐s ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧ ✐♥ t❤❡

s✐③❡ ♦❢ ϕ✳

❚❤❡♦r❡♠ ✭❑▼❚P✲❍✶✷✮

❙❛t(FO✷✷❊❈) ❛♥❞ ❋✐♥❙❛t(FO✷✷❊❈) ❛r❡ ❜♦t❤ ✷✲◆❊①♣❚✐♠❡✲❝♦♠♣❧❡t❡✳

• ❋♦r G✷✷❊❈✱ t❤❡ s❦❡✇ ❝♦♥str❛✐♥ts ❞✐s❛♣♣❡❛r ❛♥❞ ❝♦♠♣❧❡①✐t②

❞r♦♣s t♦ ✷✲❊①♣❚✐♠❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❙❤♦✇✐♥❣ t❤❡ ❞❡❝✐❞❛❜✐❧✐t② ♦❢ C✷❊ r❡q✉✐r❡s ❛ ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤✳
• ❲❡ ❝❛♥ ❝♦♥✜♥❡ ❛tt❡♥t✐♦♥ t♦ ❢♦r♠✉❧❛s ♦❢ t❤❡ ❢♦r♠

ϕ := ∀x∀y(α ∨ x = y) ∧

• ✶≤h≤m

∀x∃=Bhy(βh(x, y) ∧ x = y), ✇❤❡r❡ α ❛♥❞ t❤❡ βh ❛r❡ q✉❛♥t✐✜❡r✲ ❛♥❞ ❡q✉❛❧✐t②✲❢r❡❡✳

• ❈❛❧❧ ❛♥ ❛t♦♠✐❝ ✷✲t②♣❡ ✇❤✐❝❤ ❡♥t❛✐❧s ❛♥② ♦❢ t❤❡s❡ βh ❛ r❛②✲t②♣❡✳
• ❊❛❝❤ ❡❧❡♠❡♥t ✐♥ ❛ str✉❝t✉r❡ ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡

♥✉♠❜❡r ♦❢ r❛②s ♦❢ ❡❛❝❤ t②♣❡s ✐t ❵❡♠✐ts✬✿

a ρ✶ ρ✶ ρ✶ ρ✷ ρ✷ ρ✹ ρ✹ ρ✹

(✸, ✷, ✵, ✸, ✵, . . . , ✵)

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤❡r❡ ✐s ❛ ❜♦✉♥❞❡❞ ♥✉♠❜❡r ♦❢ t❤❡s❡ st❛r✲t②♣❡s✿ σ✶, . . . , σN
• ❲❡ ❝❛♥ t❤✐♥❦ ♦❢ ❛ str✉❝t✉r❡ ❛s ❛ ❥✐❣❣s❛✇ ♣✉③③❧❡ ♠❛❞❡ ❜②

✜tt✐♥❣ t♦❣❡t❤❡r wk ❝♦♣✐❡s ♦❢ ❡❛❝❤ st❛r✲t②♣❡ σk✿

a

• ▼♦r❡♦✈❡r✱ ❢♦r ❛ ❣✐✈❡♥ C✷❊✲❢♦r♠✉❧❛ ϕ✱ ✇❡ ❝♦♥✜♥❡ ❛tt❡♥t✐♦♥ t♦

t❤♦s❡ st❛r✲t②♣❡s ✇❤✐❝❤ ❛r❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ϕ✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❆♥② ✜♥✐t❡ str✉❝t✉r❡ A r❡❛❧✐③✐♥❣ ❡①❛❝t❧② t❤❡ st❛r✲t②♣❡s

σ✶, . . . , σK ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛ ✈❡❝t♦r ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs (w✶, . . . , wK), ✇❤❡r❡ wk ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ✐♥ A ❤❛✈✐♥❣ st❛r✲t②♣❡ σk✳

• ❲❡ ♠✐❣❤t ❝❛❧❧ t❤✐s ✈❡❝t♦r t❤❡ ♣r♦✜❧❡ ♦❢ A✳

◗✉❡st✐♦♥✿ ❋♦r ✜①❡❞ σ✶, . . . , σK✱ ✇❤❡♥ ✐s (w✶, . . . , wK) t❤❡ ♣r♦✜❧❡ ♦❢ s♦♠❡ ✜♥✐t❡ ♠♦❞❡❧ ♦❢ ϕ❄ ❆♥s✇❡r✿ ❲❤❡♥ ✐t s❛t✐s✜❡s ❛ ❝❡rt❛✐♥ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠✳

• ❙✐♠✐❧❛r❧② ❢♦r ✭❝♦✉♥t❛❜❧②✮ ✐♥✜♥✐t❡ ♠♦❞❡❧s✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❲r✐t❡ N∗ = N ∪ {ℵ✵}✳
• ❋r♦♠ ❛♥② C✷❊✲❢♦r♠✉❧❛✱ ✇❡ ♦❜t❛✐♥ ❛♥ ✐♥t❡❣❡r ❧✐♥❡❛r

♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠ ✇❤✐❝❤ ❤❛s ❛ s♦❧✉t✐♦♥ ♦✈❡r N∗ ✭♦✈❡r N✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ϕ ❤❛s ❛ ✭✜♥✐t❡✮ ♠♦❞❡❧✳

• ❚❤✐s ❧✐♥❡❛r ✐♥t❡❣❡r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠ ❤❛s s✐♥❣❧②

❡①♣♦♥❡♥t✐❛❧❧② ♠❛♥② ❡q✉❛t✐♦♥s ✭♦r ✐♥❡q✉❛❧✐t✐❡s✮✱ ✐♥ ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧❧② ♠❛♥② ✈❛r✐❛❜❧❡s✳

• ❯s✐♥❣ ✈❛r✐♦✉s r❡s✉❧ts ❢r♦♠ t❤❡ t❤❡♦r② ♦❢ ✐♥t❡❣❡r ❧✐♥❡❛r

♣r♦❣r❛♠♠✐♥❣✱ ✇❡ ❝❛♥ ❞❡r✐✈❡✿

❚❤❡♦r❡♠ ✭P✲❍ ✷✵✶✺✮

❚❤❡ ♣r♦❜❧❡♠s ❙❛t(C✷❊) ❛♥❞ ❋✐♥❙❛t(C✷❊) ❛r❡ ❜♦t❤ ✐♥ ◆❊①♣❚✐♠❡

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

• ❚❤✐s ❧❡❛✈❡s ♦♥❧② t❤❡ ❢r❛❣♠❡♥t FO✷✶❊ t♦ ❝♦♥s✐❞❡r✳
• ❆ ♣❛♣❡r ❜② ❙③✇❛st ❛♥❞ ❚❡♥❞❡r❛ ✷✵✶✸ ❝❧❛✐♠❡❞ t❤❛t

❙❛t(FO✷✶❊) ✐s ✐♥ ✷✲◆❊①♣❚✐♠❡✳ ❍♦✇❡✈❡r✱ t❤❡ ♣r♦♦❢ ✐s ✢❛✇❡❞✱ ❛♥❞ ❞❡❝✐❞❛❜✐❧✐t② r❡♠❛✐♥s ♦♣❡♥✳

• ❋✐♥✐t❡ s❛t✐s✜❛❜✐❧✐t② ✐s ❞❡❝✐❞❛❜❧❡✿

❚❤❡♦r❡♠ ✭P✲❍ ✷✵✶✽✮

❚❤❡ ♣r♦❜❧❡♠ ❋✐♥❙❛t(FO✷✶❊) ✐s ✐♥ ✸✲◆❊①♣❚✐♠❡✳

• ❈♦♠♣❧❡①✐t② ❢❛❧❧s t♦ ✷✲◆❊①♣❚✐♠❡ ✐s t❤❡ ❞✐st✐♥❣✉✐s❤❡❞ r❡❧❛✲

t✐♦♥ ✐s r❡q✉✐r❡❞ t♦ ❜❡ ❛ ♣❛rt✐❛❧ ♦r❞❡r✳

• ❚❤❡ ❜❡st ❝✉rr❡♥t ❧♦✇❡r ❜♦✉♥❞ ✐s ✷✲❊①♣❚✐♠❡✳

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

❖✉t❧✐♥❡

■♥tr♦❞✉❝t✐♦♥ ❊①t❡♥s✐♦♥s ✇✐t❤ ❝♦✉♥t✐♥❣ ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ✉♥❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❚r❛♥s✐t✐✈✐t② ❛♥❞ ❊q✉✐✈❛❧❡♥❝❡✿ t❤❡ ❞❡❝✐❞❛❜❧❡ ❡①t❡♥s✐♦♥s ❙✉♠♠❛r②

■♥tr♦❞✉❝t✐♦♥ ❈♦✉♥t✐♥❣ ❯♥❞❡❝✐❞❛❜✐❧✐t② ❉❡❝✐❞❛❜✐❧✐t② ❙✉♠♠❛r②

✶❚ k❚ ✭≥ ✷✮ ✶❊ ✷❊ k❊ ✭≥ ✸✮ K P❙♣❛❝❡ P❙♣❛❝❡ ◆P❚✐♠❡ G✷ ❊①♣❚✐♠❡ ✷✲❊①♣❚✐♠❡ ❯♥❞❡❝ ◆❊①♣❚✐♠❡ ✷✲❊①♣❚✐♠❡ ❯♥❞❡❝ FO✷ ◆❊①♣❚✐♠❡ [✷✲❊①♣❚✐♠❡, ✸✲◆❊①♣❚✐♠❡]∗ ❯♥❞❡❝ ◆❊①♣❚✐♠❡ ✷✲◆❊①♣❚✐♠❡ ❯♥❞❡❝

• rK

P❙♣❛❝❡ ◆❊①♣❚✐♠❡ ◆P❚✐♠❡ GC✷ ❊①♣❚✐♠❡ ❯♥❞❡❝ ❯♥❞❡❝ ❊①♣❚✐♠❡ ❯♥❞❡❝ ❯♥❞❡❝ C✷ ◆❊①♣❚✐♠❡ ❯♥❞❡❝ ❯♥❞❡❝ ◆❊①♣❚✐♠❡ ❯♥❞❡❝ ❯♥❞❡❝

❋♦r ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥✱ s❡❡✿ ❊♠❛♥✉❡❧ ❑✐❡r♦➠s❦✐✱ ■❛♥ Pr❛tt✲❍❛rt♠❛♥♥ ❛♥❞ ▲✐❞✐❛ ❚❡♥❞❡r❛✿ ❆❈▼ ❙■●▲❖● ◆❡✇s ✷✷ ❏✉❧② ✷✵✶✽✱ ❱♦❧✳ ✺✱ ◆♦✳ ✸