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❯♥✐❢♦r♠ ❊✈❛❧✉❛t✐♦♥ ♦❢ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s

❚❤♦♠❛s ❊✐t❡r ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❚✳ ❑r❡♥♥✇❛❧❧♥❡r✱ P✳ ❙❝❤♥❡✐❞❡r✱ ●✳ ❳✐❛♦

■♥st✐t✉t ❢ür ■♥❢♦r♠❛t✐♦♥s②st❡♠❡✱ ❚❯ ❲✐❡♥ ❡✐t❡r❅❦r✳t✉✇✐❡♥✳❛❝✳❛t ❊P❈▲ ❚r❛✐♥✐♥❣ ❈❛♠♣✱ ❉r❡s❞❡♥✱ ❉❡❝❡♠❜❡r ✷✶✱ ✷✵✶✷

❆✉str✐❛♥ ❙❝✐❡♥❝❡ ❋✉♥❞ ✭❋❲❋✮ ❣r❛♥t P✷✵✽✹✵✱ P✷✵✽✹✶ ■❈❚ ❖♥t♦r✉❧❡ ✭❋P✼ ✷✸✶✽✼✺✮ ✶✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✶✳ ■♥tr♦❞✉❝t✐♦♥

❇❛❝❦❣r♦✉♥❞✿ ❙❡♠❛♥t✐❝ ❲❡❜ ✭❲✸❈✮

❘❉❋ ✭❘❡s♦✉r❝❡ ❉❡s❝r✐♣t✐♦♥ ❋r❛♠❡✇♦r❦✮ ✐s t❤❡ ❞❛t❛ ♠♦❞❡❧ ❘❉❋❙ ✭❙❝❤❡♠❛✮ ❡♥r✐❝❤❡s ❘❉❋ ❜② s✐♠♣❧❡ t❛①♦♥♦♠✐❡s ❛♥❞ ❤✐❡r❛r❝❤✐❡s ▼♦r❡ ❡①♣r❡ss✐✈❡✿ ❖❲▲ ✭❲❡❜ ❖♥t♦❧♦❣② ▲❛♥❣✉❛❣❡✮ ✭✷✵✵✹❀ ✷✵✵✾✮

• str♦♥❣❧② ❜✉✐❧❞s ♦♥ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝s

❘✉❧❡ ❧❛♥❣✉❛❣❡s✿ ❘✉❧❡ ■♥t❡r❝❤❛♥❣❡ ❋♦r♠❛t ✭❘■❋✮ ✭✷✵✶✵✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✶✳ ■♥tr♦❞✉❝t✐♦♥

❈♦♠❜✐♥✐♥❣ ❘✉❧❡s ❛♥❞ ❖♥t♦❧♦❣✐❡s

▼❛❥♦r ■ss✉❡✿ ❝♦♠❜✐♥✐♥❣ r✉❧❡s ❛♥❞ ♦♥t♦❧♦❣✐❡s ✭❧♦❣✐❝ ❢r❛♠❡✇♦r❦✮ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ❛♥❞ ♦♥t♦❧♦❣② ❢♦r♠❛❧✐s♠s ❧✐❦❡ ❘❉❋✴s✱ ❖❲▲ r❡s♣✳ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝s ❤❛✈❡ r❡❧❛t❡❞ ②❡t ❞✐✛❡r❡♥t ✉♥❞❡r❧②✐♥❣ s❡tt✐♥❣s ❆t t❤❡ ❤❡❛rt✱ t❤❡ ❞✐✛❡r❡♥❝❡ ✐s ❜❡t✇❡❡♥ ▲P ❛♥❞ ❈❧❛ss✐❝❛❧ ❧♦❣✐❝ ❚❤✐s ♠❛❦❡s ❝♦♠❜✐♥❛t✐♦♥ ♥♦♥✲tr✐✈✐❛❧ ▼❛✐♥ ❉✐✛❡r❡♥❝❡s✿

• ❈❧♦s❡❞ ✈s✳ ❖♣❡♥ ❲♦r❧❞ ❆ss✉♠♣t✐♦♥ ✭❈❲❆ ✈s✳ ❖❲❆✮
• ◆❡❣❛t✐♦♥ ❛s ❢❛✐❧✉r❡✱ str♦♥❣ ♥❡❣❛t✐♦♥ ✈s✳ ❝❧❛ss✐❝❛❧ ♥❡❣❛t✐♦♥
• ❯♥✐q✉❡ ♥❛♠❡s ❛ss✉♠♣t✐♦♥ ✭❯◆❆✮✱ tr❡❛t♠❡♥t ♦❢ ❡q✉❛❧✐t②

s✉♣♣❧✐❡r ❜r❛♥❝❤ ❛❞❞r❡ss ❇❛rr✐❧❧❛ ❘♦♠❛ P✐❛③③❛ ❊s♣❛❣♥❛ ✶ ❉❡❈❡❝❝♦ ▼✐❧❛♥♦ ❱✐❛ ❈❛❞♦r♥♦ ✷ ❇❛r✐❧❧❛ ❘♦♠❛ ❱✐❛ ❙❛❧❛r✐❛ ✶✵

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✶✳ ■♥tr♦❞✉❝t✐♦♥

❙♦♠❡ ❆♣♣r♦❛❝❤❡s

❍②❜r✐❞ ❦♥♦✇❧❡❞❣❡ ❜❛s❡✿ KB = (L, P)

• L ✐s ❛♥ ♦♥t♦❧♦❣②

Father ≡ Man ⊓ ∃hasChild.Human

• P ✐s t❤❡ r✉❧❡s ♣❛rt ✭♣r♦❣r❛♠✮

rich(X) ← famous(X), not scientist(X)

Pr♦♣♦s❛❧s✿

• ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝ Pr♦❣r❛♠s ❬●r♦s♦❢ ❡t ❛❧✳✱ ✷✵✵✸❪
• ❉▲✲s❛❢❡ r✉❧❡s ❬▼♦t✐❦ ❡t ❛❧✳✱ ✷✵✵✺❪
• r✲❤②❜r✐❞ ❑❇s ❬❘♦s❛t✐✱ ✷✵✵✺❪
• ▼❑◆❋ ❑❇s ❬▼♦t✐❦ ❛♥❞ ❘♦s❛t✐✱ ✷✵✵✼❪
• ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝ ❘✉❧❡s ❬❑röt③s❝❤ ❡t ❛❧✳✱ ✷✵✵✽❛❪
• ❊▲P ❬❑röt③s❝❤ ❡t ❛❧✳✱ ✷✵✵✽❜❪
• DL+log ❬❘♦s❛t✐✱ ✷✵✵✻❪
• ❙❲❘▲ ❬❍♦rr♦❝❦s ❡t ❛❧✳✱ ✷✵✵✹❪
• ❞❧✲♣r♦❣r❛♠s ❬❊❴❡t ❛❧✳✱ ✷✵✵✽❜❪

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✶✳ ■♥tr♦❞✉❝t✐♦♥

❞❧✲Pr♦❣r❛♠s

❆♥ ❡①t❡♥s✐♦♥ ♦❢ ❛♥s✇❡r s❡t ♣r♦❣r❛♠s ✇✐t❤ q✉❡r✐❡s t♦ ❉▲ ❦♥♦✇❧❡❞❣❡ ❜❛s❡s ✭❑❇s✮ ✭t❤r♦✉❣❤ ❞❧✲❛t♦♠s✮ ❬❊❴❡t ❛❧✳✱ ✷✵✵✽❜❪ ❞❧✲❛t♦♠s ❛❧❧♦✇ t♦ q✉❡r② ❛ ❉▲ ❦♥♦✇❧❡❞❣❡ ❜❛s❡ ❞✐✛❡r❡♥t❧② ❜✐❞✐r❡❝t✐♦♥❛❧ ✢♦✇ ♦❢ ✐♥❢♦r♠❛t✐♦♥✱ ✇✐t❤ ❝❧❡❛♥ t❡❝❤♥✐❝❛❧ s❡♣❛r❛t✐♦♥ ♦❢ ❉▲ ❡♥❣✐♥❡ ❛♥❞ ❆❙P s♦❧✈❡r ✭✏❧♦♦s❡ ❝♦✉♣❧✐♥❣✑✮ DL Engine ASP Solver

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❯s❡ ❞❧✲♣r♦❣r❛♠s ❛s ✏❣❧✉❡✑ ❢♦r ❝♦♠❜✐♥✐♥❣ ✐♥❢❡r❡♥❝❡s ♦♥ ❛ ❉▲ ❑❇✳ ❙②st❡♠ Pr♦t♦t②♣❡s

• ◆▲P✲❉▲ ❤tt♣✿✴✴✇✇✇✳❦r✳t✉✇✐❡♥✳❛❝✳❛t✴r❡s❡❛r❝❤✴s②st❡♠s✴s❡♠✇❡❜❧♣✴
• ★❋✲▲♦❣✐❝ ♣r♦❣r❛♠s ✭❖♥t♦♣r✐s❡✱ ❡①t❡♥s✐♦♥ t♦ ❋✲❧♦❣✐❝ ♣r♦❣r❛♠s✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✶✳ ■♥tr♦❞✉❝t✐♦♥

▲♦♦s❡ ❈♦✉♣❧✐♥❣ ✲ ❋❡❛t✉r❡s

❆❞✈❛♥t❛❣❡✿

• ❈❧❡❛♥ s❡♠❛♥t✐❝s✱ ❝❛♥ ✉s❡ ❧❡❣❛❝② s②st❡♠s
• ❋❛✐r❧② ❡❛s② t♦ ✐♥❝♦r♣♦r❛t❡ ❢✉rt❤❡r ❦♥♦✇❧❡❞❣❡ ❢♦r♠❛ts

✭❡✳❣✳ ❘❉❋✮

• Pr✐✈❛❝②✱ ✐♥❢♦r♠❛t✐♦♥ ❤✐❞✐♥❣

❘✉❧❡s ❖♥t♦❧♦❣② ❞❧✲❛t♦♠ ✶ ❞❧✲❛t♦♠ ✷ ❘✉❧❡ ❘❡❛s♦♥❡r ❖♥t♦❧♦❣② ❘❡❛s♦♥❡r ❍②❜r✐❞ ❘❡❛s♦♥❡r

❉r❛✇❜❛❝❦✿ ✐♠♣❡❞❛♥❝❡ ♠✐s♠❛t❝❤✱ ♣❡r❢♦r♠❛♥❝❡

• ❊✈❛❧✉❛t✐♦♥ ♦❢ ❞❧✲♣r♦❣r❛♠ ♥❡❡❞s ♠✉❧t✐♣❧❡ ❝❛❧❧s ♦❢ ❛

❉▲✲r❡❛s♦♥❡r

• ❈❛❧❧s ❛r❡ ❡①♣❡♥s✐✈❡

♦♣t✐♠✐③❛t✐♦♥s ✭❝❛❝❤✐♥❣✱ ♣r✉♥✐♥❣ ✳✳✳✮

• ■♥ s♦♠❡ ❝❛s❡✱ ❡①♣♦♥❡♥t✐❛❧❧② ♠❛♥② ❝❛❧❧s ♠✐❣❤t ❜❡

✉♥❛✈♦✐❞❛❜❧❡

• ❊✈❡♥ ♣♦❧②♥♦♠✐❛❧❧② ♠❛♥② ❝❛❧❧s ♠✐❣❤t ❜❡ t♦♦ ❝♦st❧②

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✻✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✶✳ ■♥tr♦❞✉❝t✐♦♥

❯♥✐❢♦r♠ ❊✈❛❧✉❛t✐♦♥

❈♦♥✈❡rt t❤❡ ❡✈❛❧✉❛t✐♦♥ ♣r♦❜❧❡♠ ✐♥t♦ ♦♥❡ ❢♦r ❛ s✐♥❣❧❡ r❡❛s♦♥✐♥❣ ❡♥❣✐♥❡

L✲❢♦r♠✉❧❛s ▲♦❣✐❝ L ❘❡❛s♦♥❡r

❚❤✐s ♠❡❛♥s t♦ tr❛♥s❢♦r♠ ❛ ❞❧✲♣r♦❣r❛♠ ✐♥t♦ ❛♥ ✭❡q✉✐✈❛❧❡♥t✮ ❦♥♦✇❧❡❞❣❡ ❜❛s❡ ✐♥ ♦♥❡ ❢♦r♠❛❧✐s♠ L ❢♦r ❡✈❛❧✉❛t✐♦♥ ✭✉♥✐❢♦r♠ ❡✈❛❧✉❛t✐♦♥✮ ◆♦t❡✿ ✉♥✐❢♦r♠ ❡✈❛❧✉❛t✐♦♥ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t✐❣❤t ✐♥t❡❣r❛t✐♦♥ ♦❢ ❑❇s ✐♥ ❛ s✐♥❣❧❡ ✉♥✐❢②✐♥❣ ❧♦❣✐❝

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✼✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✶✳ ■♥tr♦❞✉❝t✐♦♥

■ss✉❡s

❚❤✐s ✐❞❡❛ ♦❢ ❛ ✏✉♥✐❢♦r♠ ❡✈❛❧✉❛t✐♦♥✑ ❛♣♣r♦❛❝❤ r❛✐s❡s s❡✈❡r❛❧ ✐ss✉❡s✿ ✶✮ ❈♦st ♦❢ ❛ tr❛♥s❢♦r♠❛t✐♦♥ ❘❡❞✉❝t✐♦♥ ♦❢ ❝♦♥❥✉♥❝t✐✈❡ q✉❡r✐❡s ♦✈❡r ❉▲✲▲✐t❡ ♦♥t♦❧♦❣✐❡s t♦

• ✜rst✲♦r❞❡r ✭❋❖✮ ▲♦❣✐❝ ❬❈❛❧✈❛♥❡s❡ ❡t ❛❧✳✱ ✷✵✵✼❪
• ♥♦♥✲r❡❝✉rs✐✈❡ Datalog ❬●♦tt❧♦❜ ❛♥❞ ❙❝❤✇❡♥t✐❝❦✱ ✷✵✶✶❪

❘❡❞✉❝t✐♦♥ ♦❢ SHIQ t♦ ❞✐s❥✳ ❞❛t❛❧♦❣ ❬❍✉st❛❞t ❡t ❛❧✳✱ ✷✵✵✼❪✳ ✷✮ ❊①✐st❡♥❝❡ ♦❢ ❛ tr❛♥s❢♦r♠❛t✐♦♥ ✭♣♦ss✐❜❧② ✉♥❞❡r ❝♦♥str❛✐♥ts✮ ❊♠❜❡❞❞✐♥❣ ♦❢ ❛ ❢♦r♠❛❧✐s♠ ✐♥t♦ ❛♥♦t❤❡r Pr♦♣❡rt✐❡s ✭❡✳❣✳ ♠♦❞✉❧❛r✐t② ❬❏❛♥❤✉♥❡♥✱ ✶✾✾✾❪✮ ❊♠❜❡❞❞✐♥❣ ♦❢ ❞❧✲♣r♦❣r❛♠s ❡✳❣✳ ✐♥t♦ ❆❊▲ ❬❞❡ ❇r✉✐❥♥ ❡t ❛❧✳✱ ✷✵✵✽❪✱ ❊q✉✐❧✐❜r✐✉♠ ▲♦❣✐❝ ❬❋✐♥❦ ❛♥❞ P❡❛r❝❡✱ ✷✵✶✵❪✱ ▼❑◆❋ ❬▼♦t✐❦ ❛♥❞ ❘♦s❛t✐✱ ✷✵✶✵❪✱ ❉❡❢❛✉❧t ▲♦❣✐❝ ❬❲❛♥❣ ❡t ❛❧✳✱ ✷✵✶✶❪ ✸✮ ❈♦♠♣❧❡①✐t② ♦❢ t❤❡ t❛r❣❡t ❢♦r♠❛❧✐s♠ ✭❞❛t❛ ❝♦♠♣❧❡①✐t②✮ ✹✮ ❋❡❛s✐❜✐❧✐t② ♦❢ tr❛♥s❢♦r♠❛t✐♦♥s ❢♦r ♣r❛❝t✐❝❛❧ ❝♦♥❝❡r♥s

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✽✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✶✳ ■♥tr♦❞✉❝t✐♦♥

❘♦❛❞♠❛♣

❯♥✐❢♦r♠ ❡✈❛❧✉❛t✐♦♥ ♦❢ ✈❛r✐♦✉s ❢r❛❣♠❡♥ts ♦❢ ❞❧✲♣r♦❣r❛♠s ❤❛s ❜❡❡♥ ❝♦♥s✐❞❡r❡❞ ❛t t❤❡ ❑❇❙ ❣r♦✉♣ ♦❢ ❚❯ ❲✐❡♥❀ ❘❡✈✐❡✇ s♦♠❡ ♦❢ t❤✐s ✇♦r❦ ✇✐t❤ ❛ ❢♦❝✉s ♦♥ ✐t❡♠s ✶ ❛♥❞ ✹ ❢r♦♠ ❛❜♦✈❡✱ ❛♥❞ r❡♣♦rts s♦♠❡ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛✳ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❞❧✲Pr♦❣r❛♠s ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✺✳ ❈♦♥❝❧✉s✐♦♥

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✾✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✶ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝ ❖♥t♦❧♦❣✐❡s

❖♥t♦❧♦❣✐❡s

❑♥♦✇❧❡❞❣❡ ❛❜♦✉t ❝♦♥❝❡♣ts✱ ✐♥❞✐✈✐❞✉❛❧s✱ t❤❡✐r ♣r♦♣❡rt✐❡s ❛♥❞ r❡❧❛t✐♦♥s❤✐♣s ❲✸❈ st❛♥❞❛r❞ ✭✵✹✴✷✵✵✹✮✿ ❲❡❜ ❖♥t♦❧♦❣② ▲❛♥❣✉❛❣❡ ✭❖❲▲✮ ❚❤r❡❡ ✐♥❝r❡❛s✐♥❣❧② ❡①♣r❡ss✐✈❡ s✉❜❧❛♥❣✉❛❣❡s

• ❖❲▲ ▲✐t❡✿ ❈♦♥❝❡♣t ❤✐❡r❛r❝❤✐❡s✱

s✐♠♣❧❡ ❝♦♥str❛✐♥t ❢❡❛t✉r❡s✳ ✭ ⇋ SHIF(D)✮

• ❖❲▲ ❉▲ ✿ ❇❛s✐❝❛❧❧②✱ ❉❆▼▲✰❖■▲✳

✭ ⇋ SHOIN(D)✮

• ❖❲▲ ❋✉❧❧✿ ❆❧❧♦✇ ❡✳❣✳ t♦ tr❡❛t ❝❧❛ss❡s ❛s ✐♥❞✐✈✐❞✉❛❧s✳

❖❲▲✷ ✭✷✵✵✾✮✿ tr❛❝t❛❜❧❡ ♣r♦✜❧❡s ❖❲▲✷ ❊▲✱ ❖❲▲✷ ◗▲✱ ❖❲▲✷ ❘▲ ❖❲▲ s②♥t❛① ✐s ❜❛s❡❞ ♦♥ ❘❉❋

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✵✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✶ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝ ❖♥t♦❧♦❣✐❡s

❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝s ✭❉▲s✮

❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝s ♦✛❡r ♠♦r❡ ❡①♣r❡ss✐✈✐t② t❤❛♥ ❘❉❋✴❙✦

❚❤❡ ✈♦❝❛❜✉❧❛r② ♦❢ ❜❛s✐❝ ❉▲s ❝♦♠♣r✐s❡s✿

• ❈♦♥❝❡♣ts

✭❡✳❣✳✱ ❲✐♥❡✱ ❲❤✐t❡❲✐♥❡✮

• ❘♦❧❡s

✭❡✳❣✳✱ ❤❛s▼❛❦❡r✱ ♠❛❞❡❋r♦♠●r❛♣❡✮

• ■♥❞✐✈✐❞✉❛❧s

✭❡✳❣✳✱ ❙❡❧❛❦s■❝❡❲✐♥❡✱ ❚❛②❧♦rP♦rt✮ ❙t❛t❡♠❡♥ts r❡❧❛t❡ ✐♥❞✐✈✐❞✉❛❧s ❛♥❞ t❤❡✐r ♣r♦♣❡rt✐❡s ✉s✐♥❣

• ❧♦❣✐❝❛❧ ❝♦♥♥❡❝t✐✈❡s ✭⊓✱ ⊔✱ ¬✱ ⊑✱ ❡t❝✮✱ ❛♥❞
• q✉❛♥t✐✜❡rs ✭∃✱ ∀✱ ≤k✱ ≥k✱ ❡t❝✮

❆ ❉▲ ❦♥♦✇❧❡❞❣❡ L = (T , A) ❜❛s❡ ✉s✉❛❧❧② ❝♦♠♣r✐s❡s

• ❛ ❚❇♦① T ✭t❡r♠✐♥♦❧♦❣②✱ ❝♦♥❝❡♣t✉❛❧✐③❛t✐♦♥✮✱ ❛♥❞
• ❛♥ ❆❇♦① A ✭❛ss❡rt✐♦♥s✱ ❡①t❡♥s✐♦♥❛❧ ❦♥♦✇❧❡❞❣❡✮

❉▲s ❛r❡ t❛✐❧♦r❡❞ ❢♦r ❞❡❝✐❞❛❜❧❡ r❡❛s♦♥✐♥❣ ✭❦❡② t❛s❦✿ s❛t✐s✜❛❜✐❧✐t②✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✶✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✶ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝ ❖♥t♦❧♦❣✐❡s

❊①❛♠♣❧❡✿ ❚❤❡ ❲✐♥❡ ❖♥t♦❧♦❣②

❆✈❛✐❧❛❜❧❡ ❛t ❤tt♣✿✴✴✇✇✇✳✇✸✳♦r❣✴❚❘✴♦✇❧✲❣✉✐❞❡✴✇✐♥❡✳r❞❢

• wl:Thing

Wine Region Red Wine White Wine locatedIn WineDescriptor WineTaste WineFlavor hasFlavor

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✷✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✶ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝ ❖♥t♦❧♦❣✐❡s

❊①❛♠♣❧❡✿ ❚❤❡ ❲✐♥❡ ❖♥t♦❧♦❣② ✴✷

❙♦♠❡ ❛①✐♦♠s ❢r♦♠ t❤❡ ❚❇♦① Wine ⊑ PotableLiquid ⊓ =1hasMaker ⊓ ∀hasMaker.Winery; ∃hasColor−.Wine ⊑ {”White”, ”Rose”, ”Red”}; WhiteWine ≡ Wine ⊓ ∀hasColor.{”White”}.

• ❆ ✇✐♥❡ ✐s ❛ ♣♦t❛❜❧❡ ❧✐q✉✐❞✱ ❤❛✈✐♥❣ ❡①❛❝t❧② ♦♥❡ ♠❛❦❡r✱ ✇❤♦ ✐s ❛

♠❡♠❜❡r ♦❢ t❤❡ ❝❧❛ss ✏Winery✑✳

• ❲✐♥❡s ❤❛✈❡ ❝♦❧♦rs ✏White✑✱ ✏Rose✑✱ ♦r ✏Red✑✳
• ❆ WhiteWine ✐s ❛ ✇✐♥❡ ✇✐t❤ ❡①❝❧✉s✐✈❡ ❝♦❧♦r ✏White✑✳

❚❤❡ ❆❇♦① ❝♦♥t❛✐♥s✱ ❡✳❣✳✱

WhiteWine(”StGenevieveTexasWhite”)✱ hasMaker(”TaylorPort”, ”Taylor”)

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✸✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✶ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝ ❖♥t♦❧♦❣✐❡s

❋♦r♠❛❧ ❖❲▲ ✴ ❉▲ ❙❡♠❛♥t✐❝s

❚❤❡ s❡♠❛♥t✐❝s ♦❢ ❝♦r❡ ❉▲s ✐s ❣✐✈❡♥ ❜② ❛ ♠❛♣♣✐♥❣ t♦ ✜rst✲♦r❞❡r ❧♦❣✐❝ ■♥ ♠❛♥② ❉▲s✱ ❜❛s✐❝ r❡❛s♦♥✐♥❣ t❛s❦s ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ ❝♦r❡ ❉▲s ■♥ ❡ss❡♥❝❡✱ ❉▲s ❛r❡ ❋❖ ❧♦❣✐❝ ✐♥ ❞✐s❣✉✐s❡

❖❲▲ ♣r♦♣❡rt② ❛①✐♦♠s ❛s ❘❉❋ ❚r✐♣❧❡s ❉▲ s②♥t❛① ❋❖▲ s❤♦rt r❡♣r❡s❡♥t❛t✐♦♥ P r❞❢s✿❞♦♠❛✐♥ C ⊤ ⊑ ∀P −.C ∀x, y.P (x, y) ⊃ C(x) P r❞❢s✿r❛♥❣❡ C ⊤ ⊑ ∀P.C ∀x, y.P (x, y) ⊃ C(y) P ♦✇❧✿✐♥✈❡rs❡❖❢ P0 P ≡ P − ∀x, y.P (x, y) ≡ P0(y, x) P r❞❢✿t②♣❡ ♦✇❧✿❙②♠♠❡tr✐❝Pr♦♣❡rt② P ≡ P − ∀x, y.P (x, y) ≡ P (y, x) P r❞❢✿t②♣❡ ♦✇❧✿❋✉♥❝t✐♦♥❛❧Pr♦♣❡rt② ⊤ ⊑ 1P ∀x, y1, y2.P (x, y1)∧P (x, y2) ⊃ y1=y2 P r❞❢✿t②♣❡ ♦✇❧✿❚r❛♥s✐t✐✈❡Pr♦♣❡rt② P + ⊑ P ∀x, y, z.P (x, y) ∧ P (y, z) ⊃ P (x, z) ❖❲▲ ❝♦♠♣❧❡① ❝❧❛ss ❞❡s❝r✐♣t✐♦♥s ❉▲ s②♥t❛① ❋❖▲ s❤♦rt r❡♣r❡s❡♥t❛t✐♦♥ ♦✇❧✿❚❤✐♥❣ ⊤ x = x ♦✇❧✿◆♦t❤✐♥❣ ⊥ ¬x = x ♦✇❧✿✐♥t❡rs❡❝t✐♦♥❖❢ ✭C1 ✳ ✳ ✳ Cn✮ C1⊓. . .⊓Cn Ci(x) ♦✇❧✿✉♥✐♦♥❖❢ ✭C1 ✳ ✳ ✳ Cn✮ C1⊔. . .⊔Cn Ci(x) ♦✇❧✿❝♦♠♣❧❡♠❡♥t❖❢ ✭C✮ ¬C ¬C(x) ♦✇❧✿♦♥❡❖❢ ✭o1 ✳ ✳ ✳ on✮ {o1 . . . on} x = oi ♦✇❧✿r❡str✐❝t✐♦♥ ✭P ♦✇❧✿s♦♠❡❱❛❧✉❡s❋r♦♠ ✭C✮✮ ∃P.C ∃y.P (x, y) ∧ C(y) ♦✇❧✿r❡str✐❝t✐♦♥ ✭P ♦✇❧✿❛❧❧❱❛❧✉❡s❋r♦♠ ✭C✮✮ ∀P.C ∀y.P (x, y) ⊃ C(y) ♦✇❧✿r❡str✐❝t✐♦♥ ✭P ♦✇❧✿✈❛❧✉❡ ✭o✮✮ ∃P.{o} P (x, o) ♦✇❧✿r❡str✐❝t✐♦♥ ✭P ♦✇❧✿♠✐♥❈❛r❞✐♥❛❧✐t② ✭n✮✮ n P ∃n

i=1yi. n j=1 P (x, yj) ∧ i=j yi=yj

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✹✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✷ ❉▲ ❛t♦♠s

❞❧✲❛t♦♠s

❇❛s✐❝ ■❞❡❛✿ ◗✉❡r② t❤❡ ❉▲ ❑❇ L ✉s✐♥❣ t❤❡ q✉❡r② ✐♥t❡r❢❛❝❡ ♦❢ t❤❡ ❉▲ ❡♥❣✐♥❡ ◗✉❡r② Q ♠❛② ❜❡ ❝♦♥❝❡♣t✴r♦❧❡ ✐♥st❛♥❝❡ C(X)✴R(X, Y )❀ s✉❜s✉♠♣t✐♦♥ t❡st C ⊑ D❀ ❡t❝ ✭r❡❝❡♥t ❡①t❡♥s✐♦♥✿ ❝♦♥❥✉♥❝t✐✈❡ q✉❡r✐❡s✮ ■♠♣♦rt❛♥t✿ P♦ss✐❜❧❡ t♦ ♠♦❞✐❢② t❤❡ ❡①t❡♥s✐♦♥❛❧ ♣❛rt ✭❆❇♦①✮ ♦❢ L✱ ❜② ❛❞❞✐♥❣ ♣♦s✐t✐✈❡ ✭⊎✮ ♦r ♥❡❣❛t✐✈❡ ✭− ∪✮ ❛ss❡rt✐♦♥s✱ ❜❡❢♦r❡ q✉❡r②✐♥❣ Q ❡✈❛❧✉❛t❡s t♦ tr✉❡ ✐✛ t❤❡ ♠♦❞✐✜❡❞ L ♣r♦✈❡s Q✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✺✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✷ ❉▲ ❛t♦♠s

❞❧✲❛t♦♠s✿ ❊①❛♠♣❧❡s

❲✐♥❡ ♦♥t♦❧♦❣②

• wl:Thing

Wine Region Red Wine White Wine locatedIn WineDescriptor WineTaste WineFlavor hasFlavor

DL[Wine](“ChiantiClassico”) DL[Wine](X) DL[DryWine ⊎ dry; Wine](W) ❛❞❞ ❛❧❧ ❛ss❡rt✐♦♥s DryWine(c) t♦ L✱ s✉❝❤ t❤❛t dry(c) ❤♦❧❞s✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✻✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✷ ❉▲ ❛t♦♠s

❞❧✲❆t♦♠s✿ ❙②♥t❛①

❆ ❞❧✲❛t♦♠ ❤❛s t❤❡ ❢♦r♠

DL[S1op1p1, . . . , Smopm pm; Q](t) , m ≥ 0, ✇❤❡r❡ ❡❛❝❤ Si ✐s ❡✐t❤❡r ❛ ❝♦♥❝❡♣t ♦r ❛ r♦❧❡

• pi ∈ {⊎, −

∪}✱ pi ✐s ❛ ✉♥❛r② r❡s♣✳ ❜✐♥❛r② ♣r❡❞✐❝❛t❡ ✭✐♥♣✉t ♣r❡❞✐❝❛t❡✮✱ Q(t) ✐s ❛ ❉▲ q✉❡r② ✭t ❝♦♥t❛✐♥s ✈❛r✐❛❜❧❡s ❛♥❞✴♦r ❝♦♥st❛♥ts✮✳ ■♥t✉✐t✐✈❡❧②✿

• pi = ⊎ ✐♥❝r❡❛s❡s Si ❜② pi✳
• pi = −

∪ ✐♥❝r❡❛s❡s ¬Si ❜② pi✳ ❙❤♦rt❤❛♥❞✿ λ = S1op1p1, . . . , Smopmpm

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✼✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✸ ❉▲ ◗✉❡r✐❡s

❞❧✲◗✉❡r✐❡s

❆ ❉▲ q✉❡r② Q(t) ✐s ♦♥❡ ♦❢

✭❛✮ ❛ ❝♦♥❝❡♣t ✐♥❝❧✉s✐♦♥ ❛①✐♦♠ C ⊑ D✱ ♦r ✐ts ♥❡❣❛t✐♦♥ ¬(C ⊑ D)✱ ✭❜✮ C(t) ♦r ¬C(t)✱ ❢♦r ❛ ❝♦♥❝❡♣t C ❛♥❞ t❡r♠ t✱ ♦r ✭❝✮ R(t1, t2) ♦r ¬R(t1, t2)✱ ❢♦r ❛ r♦❧❡ R ❛♥❞ t❡r♠s t1✱ t2✳ ❘❡♠❛r❦s✿ ❋✉rt❤❡r ✏✉♣❞❛t❡ ♦♣❡r❛t♦rs✑ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ✭ − ∩✮ ❋✉rt❤❡r q✉❡r✐❡s ❛r❡ ❝♦♥❝❡✐✈❛❜❧❡ ✭❡✳❣✳✱ ✉♥✐♦♥ ♦❢ ❝♦♥❥✉♥❝t✐✈❡ q✉❡r✐❡s ❬❊❴❡t ❛❧✳✱ ✷✵✵✽❛❪✮ ❚❤❡ q✉❡r✐❡s ❛❜♦✈❡ ❛r❡ st❛♥❞❛r❞ q✉❡r✐❡s✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✽✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✹ ❉▲ Pr♦❣r❛♠s

❞❧✲Pr♦❣r❛♠s

❞❧✲♣r♦❣r❛♠s ❛r❡ ❤②❜r✐❞ ❑❇s ✇✐t❤ ❞❧✲❛t♦♠s ✐♥ r✉❧❡s

❉❡✜♥✐t✐♦♥ ✭❉▲✲Pr♦❣r❛♠✮

❆ ❞❧✲♣r♦❣r❛♠ ✐s ❛ ♣❛✐r KB = (L, P) ✇❤❡r❡ L ✐s ❛ ❉▲ ❦♥♦✇❧❡❞❣❡ ❜❛s❡ P ❝♦♥s✐sts ♦❢ ❞❧✲r✉❧❡s a ← b1, . . . , bk, not bk+1, . . . , not bm, m > 0✱ ✇❤❡r❡

• a ✐s ❛♥ ❛t♦♠✱
• b1, . . . , bm✱ m ≥ 0✱ ❛r❡ ❛t♦♠s ♦r ❞❧✲❛t♦♠s ✭♥♦ ❢✉♥❝t✐♦♥ s②♠❜♦❧s✮✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✶✾✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✺ ❆♥s✇❡r ❙❡ts

❙❡♠❛♥t✐❝s

HBΦ

P ✿ ❙❡t ♦❢ ❛❧❧ ❣r♦✉♥❞ ✭❝❧❛ss✐❝❛❧✮ ❛t♦♠s ✇✐t❤ ♣r❡❞✐❝❛t❡ s②♠❜♦❧

✐♥ P ❛♥❞ ❝♦♥st❛♥ts C ❢r♦♠ ✜♥✐t❡ r❡❧❛t✐♦♥❛❧ ❛❧♣❤❛❜❡t Φ✳ ❈♦♥st❛♥ts C✿ t❤♦s❡ ✐♥ P ❛♥❞ ✭❛❧❧✮ ✐♥❞✐✈✐❞✉❛❧s ✐♥ t❤❡ ❆❇♦① ♦❢ L✳ ❍❡r❜r❛♥❞ ✐♥t❡r♣r❡t❛t✐♦♥✿ s✉❜s❡t I ⊆ HBΦ

P

❉❡✜♥✐t✐♦♥ ✭❙❛t✐s❢❛❝t✐♦♥✮

• I s❛t✐s✜❡s ❛ ❝❧❛ss✐❝❛❧ ❣r♦✉♥❞ ❛t♦♠ a ✭I |

=L a✮✱ ✐✛ a ∈ I❀

• I s❛t✐s✜❡s ❛ ❣r♦✉♥❞ ❞❧✲❛t♦♠ α = DL[λ; Q](c) ✭I |

=L α✮ ✐✛ L ∪ A1(I) ∪ · · · ∪ Am(I) | = Q(c)✱ ✇❤❡r❡

Ai(I) = {Si(e) | pi(e) ∈ I}✱ ❢♦r opi = ⊎❀ Ai(I) = {¬Si(e) | pi(e) ∈ I}✱ ❢♦r opi = − ∪✳

I ✐s ❛ ♠♦❞❡❧ ♦❢ KB = (L, P) ✐❢ ✐t s❛t✐s✜❡s ❡❛❝❤ r✉❧❡ ✐♥ grnd(P) ❛s ✉s✉❛❧

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✵✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✺ ❆♥s✇❡r ❙❡ts

❊①❛♠♣❧❡s

❙✉♣♣♦s❡ L | = Wine(“TaylorPort”)✱ ❛♥❞ I ❝♦♥t❛✐♥s wineBottle(“TaylorPort”) ❚❤❡♥ I | =L DL[“Wine”](“TaylorPort”) ❛♥❞ I | =L wineBottle(“TaylorPort”) ← DL[“Wine”](“TaylorPort”) ❙✉♣♣♦s❡ I = {white(“siw”), not❴dry(“siw”)}✳ ❚❤❡♥ I | =L DL[“WhiteWine” ⊎ white, “DryWine”− ∪not❴dry; “Wine”](“siw”)

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✶✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✺ ❆♥s✇❡r ❙❡ts

❆♥s✇❡r ❙❡ts

❯s❡ ❛ r❡❞✉❝t KBI ❛❦✐♥ t♦ t❤❡ ●❡❧❢♦♥❞✲▲✐❢s❝❤✐t③ r❡❞✉❝t P I ■♥ ❜✉✐❧❞✐♥❣ KBI✱ tr❡❛t ❞❧✲❛t♦♠s ❧✐❦❡ ♦r❞✐♥❛r② ❛t♦♠s✿ ❉❡✜♥✐t✐♦♥ ✭❘❡❞✉❝t KBI ♦❢ KB = (L, P)✮ KBI ❝♦♥t❛✐♥s ❛❧❧ r✉❧❡s ♦❜t❛✐♥❡❞ ❢r♦♠ grnd(P) ❜② r❡♠♦✈✐♥❣ ✭✐✮ ❛❧❧ r✉❧❡ ✐♥st❛♥❝❡s a ← b1, . . . , bk, not bk+1, . . . , not bm s✉❝❤ t❤❛t I | =L bj ❢♦r s♦♠❡ bj✱ k < j ≤ m✱ ❛♥❞ ✭✐✐✮ ❛❧❧ ❧✐t❡r❛❧s not bj ❢r♦♠ t❤❡ r❡♠❛✐♥✐♥❣ r✉❧❡s✳ ❉❡✜♥✐t✐♦♥ ✭❆♥s✇❡r s❡t✮ I ✐s ❛ ✭str♦♥❣✮ ❛♥s✇❡r s❡t ♦❢ KB ✐✛ I ✐s t❤❡ ❧❡❛st ♠♦❞❡❧ ♦❢ KBI✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✷✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✺ ❆♥s✇❡r ❙❡ts

❊①❛♠♣❧❡✿ ❘❡✈✐❡✇❡r s❡❧❡❝t✐♦♥ ❬❊❴❡t ❛❧✳✱ ✷✵✵✽❜❪ ✭❛❞❛♣t❡❞✮

paper(p1); kw(p1, ”Semantic❴Web”); ✭✶✮ paper(p2); kw(p2, ”Bioinformatics”); kw(p2, ASP); ✭✷✮ kw(P, K2) ← kw(P, K1), DL[hasMember](S, K1), DL[hasMember](S, K2); ✭✸✮ paperArea(P, A) ← DL[keywords ⊎ kw; inArea](P, A); ✭✹✮ cand❴rev(X, P) ← paperArea(P, A), DL[CandidateReviewer](X), DL[expert](X, A); ✭✺✮ assign(X, P) ← cand❴rev(X, P), not unassign(X, P); ✭✻✮ unassign(Y, P) ← cand❴rev(Y, P), assign(X, P), X = Y ; ✭✼✮ has❴rev(P) ← assign(X, P); ✭✽✮ error(P) ← paper(P), not has❴rev(P). ✭✾✮

❉❡t❡r♠✐♥❡ ♣❛♣❡r ❛r❡❛ ✇✐t❤ ❡♥❤❛♥❝❡❞ ❦❡②✇♦r❞ ✐♥❢♦ ✭❦❡② ✇♦r❞ ❝❧✉st❡rs✮ ✭✸✮✱ ✭✹✮ ❯s❡ ♦♥t♦❧♦❣② t♦ ❞❡t❡r♠✐♥❡ ❝❛♥❞✐❞❛t❡ r❡✈✐❡✇❡rs ✭✺✮ ✭✻✮✕✭✾✮ ✐s ❛ ♣❧❛✐♥ ❆❙P s❡❧❡❝t✐♦♥ ♣r♦❣r❛♠ ✭❝❤♦♦s❡ ♦♥❡ cand❴rev ♣❡r ♣❛♣❡r✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✸✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✺ ❆♥s✇❡r ❙❡ts

❘❡✈✐❡✇❡r s❡❧❡❝t✐♦♥ ✭❝t❞✳✮

❆♥s✇❡r s❡ts ♦❢ KB ❞❡♣❡♥❞ ♦♥ t❤❡ ✐♥st❛♥❝❡s ♦❢ hasMember✱ keywords✱ inArea✱ expert CandidateReviewer ❙✉♣♣♦s❡ ✐♥ L expert✭jim✱”A1”✮✱ expert✭tim✱”A1”✮✱ expert✭sue✱”A2”✮

ReviewerCandidate✭jim✮✱ ReviewerCandidate✭tim✮✱ ReviewerCandidate✭sue✱”LP”✮✱ hasMember(c1, ”ASP”)✱ hasMember(c1, ”LP”) ❛r❡ tr✉❡ ✭♥❛♠❡❞ ❝❧✉st❡rs✮

❋✉rt❤❡r✱ t❤❛t inArea✭p1✱”A1”✮ ✐s tr✉❡ ❛♥❞ inArea✭p2✱”A2”✮ ✐s tr✉❡ ❛❢t❡r ❛ss❡rt✐♥❣ keywords✭p2✱”LP”✮✳ M =        (1), (2), kw(p2, ”LP”), paperArea(p1, ”A1”), paperArea(p2, ”A2”), cand❴rev(p1, jim), cand❴rev(p1, tim), cand❴rev(p2, sue), assign(jim, p1), unassign(tim, p1), assign(sue, p2), has❴rev(p1), has❴rev(p2)        ✐s ❛♥ ❛♥s✇❡r s❡t ♦❢ ❑ ❇✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✹✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✺ ❆♥s✇❡r ❙❡ts

❊①❛♠♣❧❡✿ ❘❡✈✐❡✇❡r s❡❧❡❝t✐♦♥ ✭❝t❞✳✮ ✴✷

M ❂ ④ (1), (2)✱ kw(p2, ”LP”)✱ paperArea(p1, ”A1”)✱ paperArea(p2, ”A2”)✱

cand❴rev(p1, jim)✱ cand❴rev(p1, tim)✱ cand❴rev(p2, sue)✱ assign(jim, p1)✱ unassign(tim, p1)✱ assign(sue, p2)✱ has❴rev(p1)✱ has❴rev(p2)

⑥ P❛rt ✵✿ ❋❛❝ts P❛rt ✶✿ kw, paperArea✱ ✭LP✱ ASP ✐♥ s❛♠❡ ❝❧✉st❡r✮ P❛rt ✷ cand❴rev P❛rt ✸✿ ❝❤♦✐❝❡ ❢♦r assign❀ has❴rev❀ r❡❞✉❝t sP M ✭r❡❧❡✈❛♥t ♣❛rt✮ ◆♦t❡✿ s❡❝♦♥❞ ❛♥s✇❡r s❡t ✐s M ❂ {. . . unassign(jim, p1)✱assign(tim, p1)✳ ✳ ✳ ⑥

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✺✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✺ ❆♥s✇❡r ❙❡ts

❈♦♠♣✉t❛t✐♦♥❛❧ ❈♦♠♣❧❡①✐t②

❉❡❝✐❞✐♥❣ str♦♥❣ ❛♥s✇❡r s❡t ❡①✐st❡♥❝❡ ❢♦r ❞❧✲♣r♦❣r❛♠s ✭❝♦♠♣❧❡t❡♥❡ss r❡s✉❧ts✮

KB = (L, P) ♥♦ ❞❧✲❛t♦♠s L ✐♥ SHIF(D) L ✐♥ SHOIN(D) ♣♦s✐t✐✈❡ ExpTime ExpTime NExpTime str❛t✐✜❡❞ ExpTime ExpTime PNExpTime ❣❡♥❡r❛❧ NExpTime NExpTime NPNExpTime

◆♦t❡✿ ❙❛t✐s✜❛❜✐❧✐t② ✐♥ SHIF(D) / SHOIN(D) ✐s ExpTime✲✴NExpTime✲❝♦♠♣❧❡t❡✳ ❑❡② ♦❜s❡r✈❛t✐♦♥✿ ❚❤❡ ♥✉♠❜❡r ♦❢ ❣r♦✉♥❞ ❞❧✲❛t♦♠s ✐s ♣♦❧②♥♦♠✐❛❧ NPNExpTime ❂ PNExpTime ✐s ❧❡ss ♣♦✇❡r❢✉❧ t❤❛♥ ❆♥s✇❡r ❙❡ts ❢♦r ❞✐s❥✉♥❝t✐✈❡ ♣r♦❣r❛♠s ✭≡ NExpTimeNP✮ ❙❛♠❡ ❝♦♠♣❧❡①✐t② ❛s ❢♦r ♥♦ ❞❧✲❛t♦♠s✱ ✐❢ L ✐s ❢r♦♠ ❛ ♣♦❧②♦♥♠✐❛❧ DL ❝❧❛ss ✭❡✳❣✳✱ ❖❲▲ ✷ Pr♦✜❧❡s ❘▲✱ ❊▲✱ ◗▲✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✻✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✻ ❆♣♣❧✐❝❛t✐♦♥s

❆♣♣❧✐❝❛t✐♦♥s

❞❧✲♣r♦❣r❛♠s ❢❛❝✐❧✐t❛t❡ s♦♠❡ ❛❞✈❛♥❝❡❞ r❡❛s♦♥✐♥❣ t❛s❦s ❉❡❢❛✉❧t ❘❡❛s♦♥✐♥❣ P♦♦❧❡✲st②❧❡ ❛♥❞ ❘❡✐t❡r✲st②❧❡ ❉❡❢❛✉❧t ▲♦❣✐❝ ♦✈❡r ❉▲ ❦♥♦✇❧❡❞❣❡ ❜❛s❡s ✭❢♦r r❡str✐❝t❡❞ ❢r❛❣♠❡♥ts✱ t♦ t❤❡ ❡✛❡❝t ♦❢ ❚❡r♠✐♥♦❧♦❣✐❝❛❧ ❉❡❢❛✉❧t ▲♦❣✐❝ ❬❇❛❛❞❡r ❛♥❞ ❍♦❧❧✉♥❞❡r✱ ✶✾✾✺❪✮✳ ❋r♦♥t✲❡♥❞ ❬❉❛♦✲❚r❛♥ ❡t ❛❧✳✱ ✷✵✵✾❪ ❈❧♦s❡❞ ❲♦r❧❞ ❘❡❛s♦♥✐♥❣ ❊♠✉❧❛t❡ ❈❲❆ ❛♥❞ ❊①t❡♥❞❡❞ ❈❲❆ ✭❊❈❲❆✮ ♦♥ t♦♣ ♦❢ ❛ ❉▲ ❑❇✳ ▼✐♥✐♠❛❧ ▼♦❞❡❧ ❘❡❛s♦♥✐♥❣ ❙✐♥❣❧❡ ♦✉t ✏♠✐♥✐♠❛❧✑ ♠♦❞❡❧s ♦❢ ❛ ❉▲ ❑❇

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✼✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✷✳ ◆♦♥♠♦♥♦t♦♥✐❝ ❉▲✲Pr♦❣r❛♠s ✷✳✼ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s

❲❡❧❧✲❢♦✉♥❞❡❞ s❡♠❛♥t✐❝s

▲✐❢t ✇❡❧❧✲❢♦✉♥❞❡❞ s❡♠❛♥t✐❝s ❢♦r ♦r❞✐♥❛r② t♦ ❞❧✲♣r♦❣r❛♠s ❬❊❴❡t ❛❧✳✱ ✷✵✶✶❪ ▲❡t ❢♦r KB ❛♥❞ I ❜❡ γKB(I) = LM(KBI) t❤❡ ❧❡❛st ♠♦❞❡❧ ♦❢ t❤❡ r❡❞✉❝t KBI✳ γKB ✐s ❛♥t✐✲♠♦♥♦t♦♥❡✱ t❤✉s γ2

KB ✐s ✐s ♠♦♥♦t♦♥❡ ❛♥❞ ❤❛s ❛ ❧❡❛st

✜①♣♦✐♥t lfp(γ2

KB)✳

❲❡❧❧✲❢♦✉♥❞❡❞ ❛t♦♠s ♦❢ KB = (L, P)

• WFS(KB) = lfp(γ2

KB) ✐s t❤❡ s❡t ♦❢ ✇❡❧❧✲❢♦✉♥❞❡❞ ❛t♦♠s ♦❢ KB❀

• ❋♦r ❡✈❡r② ❣r♦✉♥❞ ❛t♦♠ a✱

KB | =wf a ❞❡♥♦t❡s t❤❛t a ∈ WFS(KB)✱ KB | =wf ¬a ❞❡♥♦t❡s t❤❛t a / ∈ γKB(WFS(KB))✳

❲❡❧❧✲❢♦✉♥❞❡❞ ❛♥❞ ❛♥s✇❡r s❡t s❡♠❛♥t✐❝s r❡❧❛t❡ s✐♠✐❧❛r❧② ❛s ❢♦r ♦r❞✐♥❛r② ♣r♦❣r❛♠s

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✽✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t②

❋❖✲❘❡✇r✐t❛❜✐❧✐t②

❇❛s✐❝ ■❞❡❛✿

• ❚r❛♥s❢♦r♠ ❛ ❞❧✲♣r♦❣r❛♠ KB = (L, P) ✐♥t♦ ❛♥ ❙◗▲ ❡①♣r❡ss✐♦♥ S(KB)

♦✈❡r t❤❡ ✈♦❝❛❜✉❧❛r② ♦❢ L

• ❉❡s✐r❡❞ ♣r♦♣❡rt②✿ S(KB) ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❝♦♥❝r❡t❡ ❆❇♦① ♦❢ L

♠❛♥❛❣❡♠❡♥t s②st❡♠s ✭❉❇▼❙✮

• ❚♦ ❡✈❛❧✉❛t❡ S(KB)✱ ✇❡ ❝❛♥ ✉s❡ ❡✣❝✐❡♥t r❡❧❛t✐♦♥❛❧ ❞❛t❛❜❛s❡

❚❤❡ ❢❛♠✐❧② ♦❢ ❉▲✲▲✐t❡ ❉▲s s❛t✐s✜❡s t❤❡ ✭❛♥❛❧♦❣✮ ♣r♦♣❡rt② ✭❝❛❧❧❡❞ ❋❖✲r❡❞✉❝✐❜✐❧✐t②✮ ❢♦r ❝♦♥❥✉♥❝t✐✈❡ q✉❡r✐❡s ❋♦r ❞❧✲♣r♦❣r❛♠s✱ ✇❡ ♥❡❡❞ r❡str✐❝t✐♦♥s ♦♥ t❤❡ r✉❧❡s ❛♥❞ t❤❡ ♦♥t♦❧♦❣②

❉❡✜♥✐t✐♦♥

❆ ❞❧✲♣r♦❣r❛♠ KB = (L, P)✱ L = (T , A)✱ ✐s ❋❖✲r❡✇r✐t❛❜❧❡✱ ✐❢ KB | = p( c) ❢♦r ❛t♦♠ p( c)✱ ✐s ❡①♣r❡ss✐❜❧❡ ❜② ❛ ❋❖ ❢♦r♠✉❧❛ φ( x) ♦✈❡r t❤❡ r❡❧❛t✐♦♥❛❧ s❝❤❡♠❛ ✐♥❞✉❝❡❞ ❜② t❤❡ ✈♦❝❛❜✉❧❛r② ♦❢ L✱ s✉❝❤ t❤❛t KB | = p( c) ✐✛ A | = φ( c)✱ ✇❤❡r❡ φ ♦♥❧② ❞❡♣❡♥❞s ♦♥ p✱ P ❛♥❞ T ✱ ❜✉t ♥♦t ♦♥ A✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✷✾✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✶ ❆❝②❝❧✐❝ ❞❧✲♣r♦❣r❛♠s

❆❝②❝❧✐❝ ❞❧✲♣r♦❣r❛♠s

❚♦ ❡♥s✉r❡ ❋❖✲r❡✇r✐t❛❜✐❧✐t②✱ ❜❛♥ ✐♥tr✐♥s✐❝ r❡❝✉rs✐♦♥ ❢r♦♠ KB = (L, P) ❚❤✐s ✐s ❡♥s✉r❡❞ ❜② ❛❝②❝❧✐❝✐t②✿ P ✐s ❛❝②❝❧✐❝✱ ✐❢ s♦♠❡ ♠❛♣♣✐♥❣ K: Preds(P) → {0, . . . , n} ❡①✐sts s✉❝❤ t❤❛t ❢♦r ❡✈❡r② r✉❧❡ a ← b1, . . . , bk, not bk+1, . . . , not bm, ✐♥ P✱ ❛♥❞ ❡✈❡r② p, q ∈ Preds(P) ✇❤❡r❡ p ♦❝❝✉rs ✐♥ a ❛♥❞ q ♦❝❝✉rs ✐♥ s♦♠❡ bi✱ ✐t ❤♦❧❞s K(p) > K(q)✳

❊①❛♠♣❧❡

KB = (L, P) ✇❤❡r❡ L = {C ⊑ D} ❛♥❞ P = p(a); p(b); q(c); s(X) ← DL[C ⊎ p; D](X), not DL[C ⊎ q, C − ∪p; D](X)

• .

◆♦t❡✿ ❡✈❡r② ❛❝②❝❧✐❝ KB ❤❛s WFS(KB) ❛s ✐ts ✉♥✐q✉❡ ❛♥s✇❡r s❡t✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✵✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✷ ❋❖✲❘❡✇r✐t❛❜❧❡ ❞❧✲❛t♦♠s

❋❖✲❘❡✇r✐t❛❜❧❡ ❞❧✲❛t♦♠s

❋♦r ❋❖✲r❡✇r✐t❛❜✐❧✐t② ♦❢ KB = (L, P)✱ L = (T , A) ❡❛❝❤ ❞❧✲q✉❡r② Q(x) ✐♥ P ♠✉st ❜❡ ❋❖✲r❡✇r✐t❛❜❧❡✱ ✐✳❡✳✱ s♦♠❡ ❋❖✲❢♦r♠✉❧❛ φQ(x) ♦♥ L✬s ✈♦❝❛❜✉❧❛r② ❡①✐sts✱ s✉❝❤ t❤❛t L | = Q(c) ✐✛ A | = φQ(c)✱ ❢♦r ❡❛❝❤ c φQ(x) ♠✉st ❞❡♣❡♥❞ ♦♥❧② ♦♥ T ✱ ❜✉t ♥♦t ♦♥ A

❊①❛♠♣❧❡ ✭❝♦♥t✬❞✮

❞❧✲q✉❡r② Q = D(X) ♦✈❡r L = {C ⊑ D} ✐s tr❛♥s❧❛t❡❞ t♦ φQ(x) = C(x) ∨ D(x) ❋♦r ❞❧✲❛t♦♠ DL[λ, Q](x)✱ ❛❧s♦ ✉♣❞❛t❡s Si opi pi ♠✉st ❜❡ r❡s♣❡❝t❡❞

❊①❛♠♣❧❡ ✭❝♦♥t✬❞✮

❚❤❡ ❞❧✲❛t♦♠ DL[C ⊎ p; D](X) ✐s tr❛♥s❧❛t❡❞ ✐♥t♦ δ1(x) = (C(x) ∨ p(x)) ∨ D(x)✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✶✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✷ ❋❖✲❘❡✇r✐t❛❜❧❡ ❞❧✲❛t♦♠s

❋❖✲r❡✇r✐t❛❜❧❡ ❞❧✲❛t♦♠s ✭❝♦♥t✬❞✮

✐❢ opi = − ∪ ♦❝❝✉rs ✐♥ P✱ ❛✈♦✐❞ tr❛♥s❧❛t✐♥❣ Si − ∪pi t♦ Si(x) ∨ ¬pi(x) ❆ss✉♠❡ L ✐s ♦✈❡r ❛ ❉▲ t❤❛t ✐s ✭✐✮ ❈❲❆✲s❛t✐s✜❛❜❧❡ ✭✐✳❡✳✱ ❢♦r ❡✈❡r② ❉▲ ❑❇ L′✱ t❤❡ ❉▲ ❑❇ ❈❲❆✭L′✮ ❂ L′ ∪ {¬α | α ∈ AΣ, L′ | = α} ✐s s❛t✐s✜❛❜❧❡✱ ✇❤❡r❡ AΣ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ❛❧❧ ♠❡♠❜❡rs❤✐♣ ❛ss❡rt✐♦♥s ✐♥ t❤❡ ✉♥❞❡r❧②✐♥❣ ✈♦❝❛❜✉❧❛r② Σ✱ ✱ ❛♥❞ ✭✐✐✮ ❛❧❧♦✇s ❢♦r ❋❖✲r❡✇r✐t❛❜❧❡ ❝♦♥❝❡♣t ❛♥❞ r♦❧❡ ♠❡♠❜❡rs❤✐♣s✳

❊①❛♠♣❧❡ ✭❝♦♥t✬❞✮

❚❤❡ ❞❧✲❛t♦♠ DL[C ⊎ q; C − ∪p; D](X) ✐s tr❛♥s❧❛t❡❞ ✐♥t♦ δ2(x) = (C(x) ∨ q(x)) ∨ D(x) ∨ ∃y((C(y) ∨ q(y)) ∧ p(y))

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✷✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✸ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ❘❡s✉❧t

❋❖✲❘❡✇r✐t❛❜✐❧✐t② ❘❡s✉❧t

❚❤❡♦r❡♠ ✭❬❊❴❡t ❛❧✳✱ ✷✵✶✶❪✮

▲❡t KB = (L, P) ❜❡ ❛❝②❝❧✐❝✱ ❛♥❞ p( c) ❛♥ ❛t♦♠✱ s✉❝❤ t❤❛t ✭✶✮ ❡✈❡r② ❞❧✲q✉❡r② ✐♥ P ✐s ❋❖✲r❡✇r✐t❛❜❧❡✱ ❛♥❞ ✭✷✮ ✐❢ − ∪ ♦❝❝✉rs ✐♥ P✱ t❤❡♥ L ✐s ❞❡✜♥❡❞ ♦✈❡r ❛ ❉▲ t❤❛t ✭✷❛✮ ✐s ❈❲❆✲s❛t✐s✜❛❜❧❡✱ ❛♥❞ ✭✷❜✮ ❛❧❧♦✇s ❢♦r ❋❖✲r❡✇r✐t❛❜❧❡ ❝♦♥❝❡♣t ❛♥❞ r♦❧❡ ♠❡♠❜❡rs❤✐♣s✳ ❚❤❡♥✱ KB | =wf p( c) ✐s ❋❖✲r❡✇r✐t❛❜❧❡✳ ❈♦♥str✉❝t✐✈❡ ♣r♦♦❢✿ ✭❛✮ ❡✈❡r② ❞❧✲❛t♦♠ δ ✐s ❡①♣r❡ss✐❜❧❡ ❛s ❋❖ ❢♦r♠✉❧❛ ♦✈❡r t❤❡ ❆❇♦① ♦❢ L❀ ✭❜✮ ❡✈❡r② ♣r❡❞✐❝❛t❡ ♦❢ r❛♥❦ ✵ ✐s ❡❛s✐❧② ❋❖✲❡①♣r❡ss✐❜❧❡ ♦✈❡r t❤❡ ❢❛❝ts ♦❢ P❀ ✭❝✮ ❡✈❡r② ♦t❤❡r ♣r❡❞✐❝❛t❡ pI ✐s ❡①♣r❡ss✐❜❧❡ ❜② r✉❧❡ ♠❡r❣✐♥❣ ✭❝❢✳ ❈❧❛r❦ ❈♦♠♣❧❡t✐♦♥✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✸✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✸ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ❘❡s✉❧t

❊①❛♠♣❧❡ ✭❝♦♥t✬❞✮

❚❤❡ r✉❧❡ ❢♦r ♣r❡❞✐❝❛t❡ s ✐s tr❛♥s❧❛t❡❞ ✐♥t♦ φs(x) = (δ1(x) ∧ ¬δ2(x)) ❚❤❡♥ KB | =wf s(o) ✐✛ F | = φs(o)✱ ❢♦r ❛♥② ❝♦♥st❛♥t o✳ ❘❡♠❛r❦✿ ❚❤❡ ❉▲✲▲✐t❡ ❢❛♠✐❧② ✐s ❈❲❆✲s❛t✐s✜❛❜❧❡ ❚❤❡r❡✱ ❞❧✲q✉❡r✐❡s C(X)✱ R(X, Y ) ❛r❡ ✐♠♠❡❞✐❛t❡❧② ❋❖✲r❡✇r✐t❛❜❧❡ ❖t❤❡r ❞❧✲q✉❡r✐❡s ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ s✉❝❤ q✉❡r✐❡s ✭✐♥tr♦❞✉❝✐♥❣ ❢r❡s❤ ✐♥❞✈✐✉❛❧s✮✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✹✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✹ ■♠♣❧❡♠❡♥t❛t✐♦♥

■♠♣❧❡♠❡♥t❛t✐♦♥

▼❖❘ ✭▼❡r❣❡❘✉❧❡❖♥t♦❧♦❣②✮ ❬❙❝❤♥❡✐❞❡r✱ ✷✵✶✵❪✿ ❡①♣❡r✐♠❡♥t❛❧ ♣r♦t♦t②♣❡ ❊✈❛❧✉❛t❡s ❝♦♥❥✉♥❝t✐✈❡ q✉❡r✐❡s CQ ♦✈❡r ❛♥ ❛❝②❝❧✐❝ ❞❧✲♣r♦❣r❛♠ KB = (L, P) ✉s✐♥❣ ❛♥ ❘❉❇▼❙ ✭P♦st❣r❡❙◗▲ ✽✳✹✮ ▼❛✐♥ ♠♦❞✉❧❡s✿

• ❉❛t❛❧♦❣✲t♦✲❙◗▲ r❡✇r✐t❡r✿

♣✉ts t❤❡ ❢❛❝ts ♦❢ P ❛♥❞ t❤❡ ❆❇♦① ♦❢ L ✐♥ t❤❡ ❉❇ ❛♥❞ r❡✇r✐t❡s t❤❡ r✉❧❡s ♦❢ P ✐♥t♦ ❝❛s❝❛❞✐♥❣ VIEWS ✭♥♦t ♠❛t❡r✐❛❧✐③❡❞✮

• ❉▲✲▲✐t❡ ♣❧✉❣✐♥✿

tr❛♥s❢♦r♠s ❞❧✲❛t♦♠s✱ ✉s✐♥❣ t❤❡ ♣❡r❢❡❝t r❡✇r✐t✐♥❣ ♦❢ ❛ q✉❡r② ❛♥❞ ❛ ❚❇♦① T ❢r♦♠ t❤❡ ❛❧❣♦r✐t❤♠ PerfectRef ❬❈❛❧✈❛♥❡s❡ ❡t ❛❧✳✱ ✷✵✵✼❪

• ❖✇❧❣r❡s ✭❛❞❛♣t❡❞✮✿

❝♦♥str✉❝t t❤❡ ♣❡r❢❡❝t r❡✇r✐t✐♥❣ ✭♥♦ ❡①❡❝✉t✐♦♥✮

❘❡❛❧✐③❡ ❤②♣♦t❤❡t✐❝❛❧ ✉♣❞❛t❡s Si ⊎ pi ✐♥ ❞❧✲❛t♦♠s ❜② ✈✐❡✇s ❖t❤❡r ♣❧✉❣✐♥s t❤❛♥ ❉▲✲❧✐t❡ ❛r❡ ♣♦ss✐❜❧❡ ✭❛❝❝❡ss ♦t❤❡r ❉▲s✱ ❡✈❡♥ ♦t❤❡r ❢♦r♠❛❧✐s♠s✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✺✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✺ ❊①♣❡r✐♠❡♥ts

❊①♣❡r✐♠❡♥ts

❇❡♥❝❤♠❛r❦s ✭n . . . ♥✉♠❜❡r ♦❢ ❢❛❝ts ✴ ❛ss❡rt✐♦♥s✮ ❘❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ s❡ts ♦❢ ❢❛❝ts ✭Rn✮

• ❆❧❧♦✇✐♥❣ ❛ ❤✐❣❤ s❡❧❡❝t✐✈✐t② ❛♠♦♥❣ t❤❡ ❥♦✐♥ ❛ttr✐❜✉t❡s✳

❆ s✐♠♣❧✐✜❡❞ ✈❡rs✐♦♥ ♦❢ ❉❇♣❡❞✐❛ ✭Dn✮

• ❞✐✛❡r❡♥t s❡ts ♦❢ ❜♦♦❦s✱ ♣❡r✐♦❞✐❝❛❧s✱ ♣✉❜❧✐❝❛t✐♦♥s✱ ✐♥❝❧✳ ❛ s✐♥❣❧❡ r♦❧❡✳

▲❡❤✐❣❤ ❯♥✐✈❡rs✐t② ❇❡♥❝❤♠❛r❦ ✭▲❯❇▼✮ ❬●✉♦ ❡t ❛❧✳✱ ✷✵✵✺❪ ✭Un✮

• ✐s ♥♦t ✐♥ ❉▲✲▲✐t❡R ✿ ❝❤❛♥❣❡❞ ≈✶✵✪ ❚❇♦① ❛①✐♦♠s ✭❡✳❣✳✱ ❞r♦♣ r♦❧❡

tr❛♥s✐t✐✈✐t②✱ ✇❡❛❦❡♥ B ≡ C1 ⊓ C2 t♦ B ⊑ C1 ❛♥❞ B ⊑ C2✮✳

• ✉s❡❞ t❤❡ ▲❯❇▼ ✐♥st❛♥❝❡ ❣❡♥❡r❛t♦r ✭U100k ❤❛s ≈✶✷❦ ✐♥❞✐✈✐❞✉❛❧s✮✳

◗✉❡r✐❡s

◆❛♠❡ ❉❡s❝r✐♣t✐♦♥ ❉❛t❛ ❘❡❢❡r❡♥❝❡ FO1 ❚r❡❡ ♦❢ ❜✐♥❛r② ❥♦✐♥s ✭✇✐t❤ ♥❡❣❛t✐♦♥✮ ❘❛♥❞♦♠ ❬❙❝❤♥❡✐❞❡r✱ ✷✵✶✵✱ ❊①✳ ✺✳✷✳✶❪ FO2 ❙❡❧❡❝t ❛ r❛♥❣❡ ♦❢ t❤❡ ❑❇ ✉♣♦♥ ❡①t❡♥s✐♦♥ ✇✐t❤ ❜♦♦❦s ❢r♦♠ ❛♥ ❡①t❡r♥❛❧ s♦✉r❝❡ ❉❇♣❡❞✐❛ ❬❙❝❤♥❡✐❞❡r✱ ✷✵✶✵✱ ❊① ✺✳✸✳✸❪ FO3 ❙❡❡❦ st✉❞❡♥ts t❛❦✐♥❣ ❝♦✉rs❡s ♦❢ ❢❛❝✉❧t② ❛❞✈✐✲ s♦rs ✇❤♦ ❛r❡ ♥♦t ❢✉❧❧ ♣r♦❢❡ss♦rs ▲❯❇▼ ❬❙❝❤♥❡✐❞❡r✱ ✷✵✶✵✱ ❊① ✺✳✹✳✷❪ ❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✻✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✺ ❊①♣❡r✐♠❡♥ts

❊①♣❡r✐♠❡♥ts ✭❝♦♥t✬❞✮

❙②st❡♠s ▼❖❘ ❉▲❱ ❬▲❡♦♥❡ ❡t ❛❧✳✱ ✷✵✵✻❪ ✷✵✶✵✲✶✵✲✶✹ ❞❧✈❤❡①❬❉▲❪✿ ❞❧✈❤❡① ❬❊❴❡t ❛❧✳✱ ✷✵✵✻❪ ✰ ❘❛❝❡rPr♦ ❉▲✲♣❧✉❣✐♥✿

• ✉s❛❣❡ ♦❢ ❆❧❧❡❣r♦●r❛♣❤ ❧✐❜r❛r②✿ ❘❛❝❡rPr♦ ✶✳✾✳✷ ❝❛♥ ❤❛♥❞❧❡ ♦♥❧② ❧✐♠✐t❡❞

✐♥st❛♥❝❡ s✐③❡

❉▲❱❉❇❬❚❡rr❛❝✐♥❛ ❡t ❛❧✳✱ ✷✵✵✽❪ P❧❛t❢♦r♠ ✫ ▼❡❛s✉r❡♠❡♥t ♦♣❡♥❙❯❙❊ ✶✶✳✶ ✭①✽✻❴✻✹✮ s❡r✈❡r✱ ■♥t❡❧ ❳❡♦♥ ❈P❯ ❊✺✹✺✵ ✸✳✵✵●❍③✱ ✶✺✳✼ ●❇ ❘❆▼✳ P♦st❣r❡❙◗▲ ✽✳✹ ❞❛t❛❜❛s❡ ✐t❡♠ ❛✈❡r❛❣❡ ♦❢ ✜✈❡ r✉♥s ✭t✐♠❡♦✉t ♦❢ ✻❤✱ ♠❡♠♦✉t ♦❢ ✶✹✳✼ ●❇✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✼✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✺ ❊①♣❡r✐♠❡♥ts

❘❡s✉❧ts

r✉♥t✐♠❡ ✐♥ s❡❝s❀ ✏✖✑ ❂ ♦✉t ♦❢ ♠❡♠♦r②

FO1 FO2 FO3 ■♥st ▼❖❘ ❉▲❱❉❇ ❉▲❱ R10k <1 <1 ✶ R100k ✶ ✶ ✶✵✺ R250k ✸ ✹ ✾✼✼ R500k ✺ ✾ ✷✱✼✾✺ R1M ✶✶ ✶✾ ✶✶✱✹✹✻ ■♥st ▼❖❘ ❉▲❱❉❇ ❞❧✈❤❡①❬❉▲❪ D10k ✶ <1 ✼ D100k ✹ ✻ ✖ D250k ✾ ✷✺ ✖ D500k ✶✽ ✺✵ ✖ D1M ✹✷ ✶✹✺ ✖ ■♥st ▼❖❘ ❞❧✈❤❡①❬❉▲❪ U10k ✶ ✸✻ U100k ✹ ✶✶✼ U250k ✶✶ ✖ U500k ✷✵ ✖ U1M ✹✹ ✖

▲✐♥❡❛r r✉♥t✐♠❡ ❜❡❤❛✈✐♦r ♦❢ ▼❖❘ ❢♦r ❛❧❧ ❜❡♥❝❤♠❛r❦s ❋♦r FO1✱ ▼❖❘ ❛♥❞ ❉▲❱❉❇ s❝❛❧❡ s✐♠✐❧❛r❧② ✭❉▲❱❉❇♠❛t❡r✐❛❧✐③❡s ✈✐❡✇s✮ ❚❤❡ r✉❧❡ r❡✇r✐t✐♥❣ ♦❢ ❉▲❱❉❇ s❡❡♠s t♦ ❜❡ ❡✛❡❝t✐✈❡ ❚❡♠♣♦r❛r② ✉♣❞❛t❡ ♦❢ ❉▲ ❑❇ ✐♥ FO2 ✭✻✵ ✐♥❞✐✈✐❞✉❛❧s✮ ❞✐❞ ♥♦t ❤✐t ♠✉❝❤ ♦♥ ▼❖❘✬s ♣❡r❢♦r♠❛♥❝❡ ▼♦r❡ ❞❡t❛✐❧s✿

❤tt♣✿✴✴✇✇✇✳❦r✳t✉✇✐❡♥✳❛❝✳❛t✴r❡s❡❛r❝❤✴s②st❡♠s✴❞r❡✇✴❡①♣❡r✐♠❡♥ts✳❤t♠❧

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✽✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✻ ▲✐♠✐t❡❞ ❘❡❝✉rs✐♦♥

▲✐♠✐t❡❞ ❘❡❝✉rs✐♦♥

❙◗▲✿✶✾✾✾ st❛♥❞❛r❞✿ ❧✐♠✐t❡❞ ❢♦r♠ ♦❢ ❧✐♥❡❛r r❡❝✉rs✐♦♥ ✐♥ q✉❡r✐❡s

p(X, Y ) ← a(X, Y ). p(X, Y ) ← a(X, Z), p(Z, Y ). ✐♥ ❙◗▲✿

WITH RECURSIVE p AS (SELECT ∗ FROM a UNION SELECT a.1 p.2 FROM a, p WHERE a.2 = p.1) SELECT ∗ FROM p

❆❧❧♦✇ ❞❧✲♣r♦❣r❛♠s t❤❛t ✭✶✮ ❤❛✈❡ str❛t✐✜❡❞ ♥❡❣❛t✐♦♥ ✭♥♦ ♣r❡❞✐❝❛t❡ ❞❡♣❡♥❞s ♥❡❣❛t✐✈❡❧② ♦♥ ✐ts❡❧❢✮ ❛♥❞ ✭✷✮ ♥♦ ❝②❝❧❡s t❤r♦✉❣❤ ❞❧✲❛t♦♠s ❘❡♠❛r❦✿ ❍✐❡r❛r❝❤✐❝ ✐♥♣✉t t♦ ❞❧✲❛t♦♠s st✐❧❧ ♣♦ss✐❜❧❡ ❇❡♥❝❤♠❛r❦✿ ▲❯❇▼ ♦♥t♦❧♦❣② ❛♥❞ ❛ ❧✐♥❡❛r r❡❝✉rs✐✈❡ ❞❧✲♣r♦❣r❛♠ ❈❛❧❝✉❧❛t❡ t❤❡ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ♦❢ t❤❡ s♣❛rs❡✱ tr❡❡✲❧✐❦❡ str✉❝t✉r❡ ♦❢ t❤❡ subOrganization r♦❧❡ ✭✐✳❡✳✱ ❤✐❡r❛r❝❤② ♦❢ t❤❡ ▲❯❇▼ ✉♥✐✈❡rs✐t②✮ ❜② r✉❧❡s✱ ❢❡❡❞ ✐t t♦ t❤❡ ♦♥t♦❧♦❣②

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✸✾✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✸✳ ❋❖✲❘❡✇r✐t❛❜✐❧✐t② ✸✳✻ ▲✐♠✐t❡❞ ❘❡❝✉rs✐♦♥

❊①♣❡r✐♠❡♥t

❘❡s✉❧ts ✭r✉♥t✐♠❡ ✐♥ s❡❝s❀ s❛♠❡ ♣❧❛t❢♦r♠✮

■♥st❛♥❝❡ ▼❖❘ ❞❧✈❤❡①❬❉▲❪ ✶✵❦ ✶ ✸✺ ✶✵✵❦ ✶ ✶✵✽ ✷✺✵❦ ✷ ✖ ✺✵✵❦ ✹ ✖ ✶▼ ✶✶ ✖

❋♦r Un✱ q✉❛❞r❛t✐❝ tr❡♥❞ ❢♦r ▼❖❘✱ ❞❧✈❤❡①❬❉▲❪ r✉♥s ♦✉t ♦❢ ♠❡♠♦r② ❖❜s❡r✈❛t✐♦♥✿ r❡❝✉rs✐✈❡ q✉❡r✐❡s ❛r❡ ♥♦t ✇❡❧❧ s✉♣♣♦rt❡❞ ❜② ❘❉❇▼❙

• P♦st❣r❡s ✐t❡r❛t❡s ❥♦✐♥s ✇✐t❤♦✉t ❝②❝❧❡
• ♦♥ ❝②❝❧✐❝ ❞❛t❛✱ q✉❡r✐❡s ♠❛② ♥♦t t❡r♠✐♥❛t❡

✭⇒ ❜♦✉♥❞ ✐t❡r❛t✐♦♥s ❜② LIMIT ♣❛r❛♠❡t❡r✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✵✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t②

❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t②

❋❖✲r❡✇r✐t❛❜✐❧✐t② ❡①❝❧✉❞❡s r❡❝✉rs✐♦♥ ◗✉❡r② ❛♥s✇❡r✐♥❣ ✐s ♥♦t ❋❖✲r❡✇r✐t❛❜❧❡ ✐♥ ♠♦r❡ ❡①♣r❡ss✐✈❡ ❉▲s

• ❡✳❣✳✱ ✐♥ EL✱ SHIQ

❇✉t✱ ✐t ♠❛② ❜❡ ❡①♣r❡ss✐❜❧❡ ✐♥ ❉❛t❛❧♦❣ ❘❡❝❛❧❧✿

❚❤❡♦r❡♠ ✭❱❛r❞✐✱ ■♠♠❡r♠❛♥✮

❉❛t❛❧♦❣+ ✭❉❛t❛❧♦❣ ✇✐t❤ ✐♥♣✉t ♥❡❣❛t✐♦♥✮ ❝❛♣t✉r❡s P ♦♥ ♦r❞❡r❡❞ ❞❛t❛❜❛s❡s ✭✐✳❡✳✱ ✐♥ ♣r❡s❡♥❝❡ ♦❢ ❛ s✉❝❝❡ss♦r r❡❧❛t✐♦♥✮✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✶✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✶ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜❧❡ ❉▲s

❉❛t❛❧♦❣✲❘❡✇r✐t❛❜❧❡ ❉▲s

❉❡✜♥✐t✐♦♥

❆ ❉▲ DL ✐s Datalog✲r❡✇r✐t❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ tr❛♥s❢♦r♠❛t✐♦♥ ΦDL ❢r♦♠ DL ❑❇s t♦ Datalog ♣r♦❣r❛♠s s✉❝❤ t❤❛t✱ ❢♦r ❛♥② DL ❑❇ L✱ ✭✐✮ L | = Q(o) ✐✛ ΦDL(L) | = Q(o) ❢♦r ❛♥② ❝♦♥❝❡♣t ♦r r♦❧❡ ♥❛♠❡ Q ❢r♦♠ L✱ ❛♥❞ ✐♥❞✐✈✐❞✉❛❧s o ❢r♦♠ L❀ ✭✐✐✮ ΦDL ✐s ♠♦❞✉❧❛r✱ ✐✳❡✳✱ ❢♦r L = T , A ✇❤❡r❡ T ✐s ❛ ❚❇♦① ❛♥❞ A ❛♥ ❆❇♦①✱ ΦDL(L) = ΦDL(T ) ∪ A❀ P♦❧②♥♦♠✐❛❧ ❘❡✇r✐t❛❜✐❧✐t②✿ ❆ Datalog✲r❡✇r✐t❛❜❧❡ ❉▲ DL ✐s ♣♦❧②♥♦♠✐❛❧ r❡✇r✐t❛❜❧❡✱ ✐❢ ΦDL(L) ✐s ❝♦♠♣✉t❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✷✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✶ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜❧❡ ❉▲s

❊①❛♠♣❧❡ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜❧❡ ❉▲s

LDL+ ❬❍❡②♠❛♥s ❡t ❛❧✳✱ ✷✵✶✵❪✿ ❛ ❧✐❣❤t✇❡✐❣❤t ♦♥t♦❧♦❣② ❧❛♥❣✉❛❣❡

• ❊①t❡♥❞s ✐♥ ❡ss❡♥❝❡ ❖❲▲ ✷ ❘▲ ✇✐t❤ s✐♥❣❧❡t♦♥ ♥♦♠✐♥❛❧s✱ r♦❧❡

❝♦♥❥✉♥❝t✐♦♥s✱ ❛♥❞ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡✳

• ❆ tr❛♥s❢♦r♠❛t✐♦♥ ΦLDL+ ♦❢ LDL+ ✐♥t♦ Datalog ✐s str❛✐❣❤t❢♦r✇❛r❞❀

❍♦r♥✲SHIQ ❬❖rt✐③ ❡t ❛❧✳✱ ✷✵✶✵❪ ❛♥❞ r❡❧❛t✐✈❡s ❬●♦tt❧♦❜ ❛♥❞ ❙❝❤✇❡♥t✐❝❦✱ ✷✵✶✶❪ SROEL(⊓, ×) ❬❑röt③s❝❤✱ ✷✵✶✶❪✿ ❛ s✉♣❡rs❡t ♦❢ ❖❲▲ ✷ ❊▲ ❬▼♦t✐❦ ❡t ❛❧✳✱ ✷✵✵✽❪

• ❞✐sr❡❣❛r❞✐♥❣ ❞❛t❛t②♣❡s
• ❛❞❞✐♥❣ ❝♦♥❝❡♣t ♣r♦❞✉❝t✐♦♥ ✭C × D ⊑ T✱ R ⊑ C × D✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✸✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✷ ❞❧✲♣r♦❣r❛♠ ❚r❛♥s❢♦r♠❛t✐♦♥

❞❧✲♣r♦❣r❛♠ ❚r❛♥s❢♦r♠❛t✐♦♥

▲❡t ΛP = {λ | DL[λ; Q] ♦❝❝✉rs ✐♥ P} ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ✐♥♣✉t ❧✐sts ♦❢ ❞❧✲❛t♦♠s ❛♣♣❡❛r✐♥❣ ✐♥ P✳ Ψ(KB) := ΦDL(LΛP ) ∪ P ord ∪ ρ(ΛP ) ∪ TP ✇❤❡r❡ LΛP =

λ∈ΛP Lλ✱ ✇❤❡r❡ Lλ ✐s L ✇✐t❤ ❛❧❧ ❝♦♥❝❡♣t ❛♥❞ r♦❧❡ ♥❛♠❡s

s✉❜s❝r✐♣t❡❞ ✇✐t❤ λ ✭✐♥t✉✐t✐✈❡❧②✱ ❝r❡❛t❡ ✐♥❞✐✈✐❞✉❛❧ ❝♦♣✐❡s✮ ρ(ΛP ) ❝♦♥s✐sts ♦❢ r✉❧❡s Siλ(Xi) ← pi(Xi)✱ ❢♦r ❛❧❧ λ ∈ ΛP ✇❤❡r❡ λ = S1 ⊎ p1, . . . , Sm ⊎ pm ❛♥❞ 1 ≤ i ≤ m ✭✐♥t✉✐t✐✈❡❧②✱ ❛❞❞ t❤❡ ❡①t❡♥s✐♦♥ ♦❢ pi t♦ Si✮ P ord ✐s P ✇✐t❤ ❡❛❝❤ DL[λ; Q](t) r❡♣❧❛❝❡❞ ❜② ❛ ♥❡✇ ❛t♦♠ Qλ(t) TP = {⊤(a), top2(a, b) | a, b ❢r♦♠ HUP }

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✹✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✷ ❞❧✲♣r♦❣r❛♠ ❚r❛♥s❢♦r♠❛t✐♦♥

❚r❛♥s❢♦r♠❛t✐♦♥ ✭❝♦♥t✬❞✮

❊①❛♠♣❧❡

KB = (L, P) L = {C ⊑ D} P = p(a); p(b); q(c); s(X) ← DL[C ⊎ p; D](X), not DL[C ⊎ q; D](X)

• .

❚❤❡♥ ΛP = {λ1 = C ⊎ p, λ2 = C ⊎ q}✱ Φ(LΛP ) = {Dλ1(X) ← Cλ1(X); Dλ2(X) ← Cλ2(X)}✱ ρ(ΛP ) = {Cλ1(X) ← p(X); Cλ2(X) ← q(X)}✳ P ord = { p(a); p(b); q(c); s(X) ← Dλ1(X), not Dλ2(X) }✳ TP = {⊤(o) | o ∈ {a, b, c}} ∪ {⊤2(o1, o2) | {o1, o2} ⊂ {a, b, c}}✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✺✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✷ ❞❧✲♣r♦❣r❛♠ ❚r❛♥s❢♦r♠❛t✐♦♥

❈♦rr❡❝t♥❡ss

❚❤❡♦r❡♠

▲❡t KB = (L, P) ❜❡ ❛ ❞❧✲♣r♦❣r❛♠ ♦✈❡r ❛ Datalog✲r❡✇r✐t❛❜❧❡ ❉▲✳ ❚❤❡♥ ✭✶✮ ❢♦r ❡✈❡r② a ∈ HBP ✱ KB | =wf a ✐✛ Ψ(KB) | =wf a❀ ✭✷✮ t❤❡ ❛♥s✇❡r s❡ts ♦❢ KB ❝♦rr❡s♣♦♥❞ ✶✲✶ t♦ t❤❡ ❛♥s✇❡r s❡ts ♦❢ Ψ(KB)✱ s✉❝❤ t❤❛t ✭✐✮ ❡✈❡r② ❛♥s✇❡r s❡t ♦❢ KB ✐s ❡①♣❡♥❞❛❜❧❡ t♦ ❛♥ ❛♥s✇❡r s❡t ♦❢ Ψ(KB)❀ ❛♥❞ ✭✐✐✮ ❢♦r ❡✈❡r② ❛♥s✇❡r s❡t J ♦❢ Ψ(KB)✱ ✐ts r❡str✐❝t✐♦♥ I = J |HBP t♦ HBP ✐s ❛♥ ❛♥s✇❡r s❡t ♦❢ KB✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✻✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✸ SROEL(⊓, ×)

❉❛t❛❧♦❣✲r❡✇r✐t❛❜✐❧✐t② ♦❢ SROEL(⊓, ×)

isa(X, X) ← nom(X) ✭✶✵✮ self(X, V ) ← nom(X), triple(X, V, X) ✭✶✶✮ isa(X, Z) ← top(Z), isa(X, Z′) ✭✶✷✮ isa(X, Y ) ← bot(Z), isa(U, Z), isa(X, Z′), cls(Y ) ✭✶✸✮ isa(X, Z) ← subClass(Y, Z), isa(X, Y ) ✭✶✹✮ isa(X, Z) ← subConj(Y1, Y2, Z), isa(X, Y1), isa(X, Y2) ✭✶✺✮ isa(X, Z) ← subEx(V, Y, Z), triple(X, V, X′), isa(X′, Y ) ✭✶✻✮ isa(X, Z) ← subEx(V, Y, Z), self(X, V ), isa(X, Y ) ✭✶✼✮ triple(X, V, X′) ← supEx(Y, V, Z, X′), isa(X, Y ) ✭✶✽✮ isa(X′, Z) ← supEx(Y, V, Z, X′), isa(X, Y ) ✭✶✾✮ isa(X, Z) ← subSelf(V, Z), self(X, V ) ✭✷✵✮ self(X, V ) ← supSelf(Y, V ), isa(X, Y ) ✭✷✶✮ triple(X, W, X′) ← subRole(V, W ), triple(X, V, X′) ✭✷✷✮ self(X, W ) ← subRole(V, W ), self(X, V ) ✭✷✸✮ triple(X, W, X′′) ← subRChain(U, V, W ), triple(X, U, X′), triple(X′, V, X′′) ✭✷✹✮ triple(X, W, X′) ← subRChain(U, V, W ), self(X, U), triple(X, V, X′) ✭✷✺✮ triple(X, W, X′) ← subRChain(U, V, W ), triple(X, U, X′), self(X′, V ) ✭✷✻✮ triple(X, W, X) ← subRChain(U, V, W ), self(X, U), self(X, V ) ✭✷✼✮ triple(X, W, X′) ← subRConj(V1, V2, W ), triple(X, V1, X′), triple(X, V2, X′) ✭✷✽✮ self(X, W ) ← subRConj(V1, V2, W ), self(X, V1), self(X, V2) ✭✷✾✮ isa(Y, Z) ← isa(X, Y ), nom(Y ), isa(X, Z) ✭✸✵✮ isa(X, Z) ← isa(X, Y ), nom(Y ), isa(Y, Z) ✭✸✶✮ triple(Z, U, Y ) ← isa(X, Y ), nom(Y ), triple(Z, U, X). ✭✸✷✮

Pr♦♦❢ s②st❡♠ Pinst ❢♦r SROEL(⊓, ×) ✐♥ ❉❛t❛❧♦❣ ❬❑röt③s❝❤✱ ✷✵✶✶❪ ❘❡✐❢② ❝♦♥❝❡♣t ✴ r♦❧❡ ♥❛♠❡s ❢♦r ✉♥✐❢♦r♠✐t② isa(X, Y ) C(a) ❀ isa(a, C)✱ R(a, b) ❀ triple(a, R, b)✱ a ∈ NI ❀ nom(a) r❡✐❢② ❛①✐♦♠s✱ ❡✳❣✳✱ A ⊑ C ❀ subClass(A, C)✱ A ⊓ B ⊑ C ❀ subConj(A, B, C)

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✼✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✸ SROEL(⊓, ×)

❉❛t❛❧♦❣✲r❡✇r✐t❛❜✐❧✐t② ♦❢ SROEL(⊓, ×)

❯s❡ ❛ r❡✐✜❡❞ ❡♥❝♦❞✐♥❣ ♦❢ L✿ Iinst(L) = {Iinst(α) | α ∈ L} ∪ {Iinst(s) | s ∈ NI ∪ NC ∪ NR}

❊①❛♠♣❧❡

L1 = {A(a), A ⊑ ∃R.B, B ⊑ C, ∃R.C ⊑ D} ✐s tr❛♥s❧❛t❡❞ t♦

Iinst(L) =

• isa(a, A), supEx(A, R, B, eA⊑∃R.B), subClass(B, C),

subEx(R, C, D), nom(a), cls(A), cls(B), cls(C), cls(D), rol(R)

• .

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✽✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✸ SROEL(⊓, ×)

❆♥s✇❡r✐♥❣ ❉▲✲◗✉❡r✐❡s

Pinst ∪ Iinst(L) ❝❛♥ ❜❡ r❡❛❞✐❧② ✉s❡❞ t♦ ❛♥s✇❡r q✉❡r✐❡s L | = C(a) ❋♦r ♥❡❣❛t✐✈❡ ✐♥st❛♥❝❡ q✉❡r✐❡s✱ ✉s❡ s✐♠♣❧❡ r❡❞✉❝t✐♦♥✿ L | = ¬C(a) ⇔ L ∪ {C(a)} ✐s ✉♥s❛t✐s✜❛❜❧❡ ⇔ L ∪ {C(a)} | = ⊥(o) ❢♦r ❛♥② ✐♥❞✐✈✐❞✉❛❧ o✳ ❋♦r ✐♥st❛♥❝❡ r❡tr✐❡✈❛❧ ♦❢ ¬C(X)✱ ✉s❡ ❛♥ ❵✐♥❞❡①❡❞ ✈❛r✐❛♥t✬ ♦❢ Pinst ❢♦r ❡❛❝❤ ❝❛s❡ ✭L ∪ {C(a)})✿

• isa(X, Y ) ❀ isa❴n(X, Y,′ C′, a)
• s✐♠✐❧❛r❧② triple(X, Y, Z) ❀ triple❴n(X, Y, Z,′ C′, a)
• ♠♦❞✐❢② Pinst t♦ P ¬

inst✱

• ✉s❡ r✉❧❡s s♣❡❝✐❛❧ P ¬ t♦ t❡st L |

= ¬C(a) ✈✐❛ ❞❡r✐✈✐♥❣ isnota(a, C) ❢r♦♠ isa❴n(·, ⊥,′ C′, a)

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✹✾✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✸ SROEL(⊓, ×)

❆♥s✇❡r✐♥❣ ❉▲✲◗✉❡r✐❡s ✴✷

❯s❡ ❞❛t❛❧♦❣ tr❛♥s❢♦r♠❛t✐♦♥s ΦEL(L) = Pinst ∪ Iinst(L) Φ¬

EL(L) = P ¬ inst ∪ Iinst(L) ∪ P ¬

❚❤❡♦r❡♠

❋♦r ❡✈❡r② EL ♦♥t♦❧♦❣② L✱ ✭✐✮ L | = C(a) ✐✛ ΦEL(L) | = isa(a, C)❀ ✭✐✐✮ L | = ¬C(a) ✐✛ Φ¬

EL(L) |

= isnota(a, C)✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✵✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✹ ■♠♣❧❡♠❡♥t❛t✐♦♥ ❛♥❞ ❊①♣❡r✐♠❡♥ts

■♠♣❧❡♠❡♥t❛t✐♦♥

❉❘❡❲ ♣r♦t♦t②♣❡✿ ✉♥✐❢♦r♠ ❡✈❛❧✉❛t✐♦♥ ✐♥ Datalog¬ ❤tt♣✿✴✴✇✇✇✳❦r✳t✉✇✐❡♥✳❛❝✳❛t✴r❡s❡❛r❝❤✴s②st❡♠s✴❞r❡✇✴ ❲r✐tt❡♥ ✐♥ ❏❛✈❛ ✭∼ ✶✹❦ ❧✐♥❡s✮ ❉❛t❛❧♦❣ ❘❡❛s♦♥❡r✿ ❉▲❱ ♦r ❝❧✐♥❣♦ ❖♥t♦❧♦❣② P❛rs❡r✿ ♦✇❧✲❛♣✐ ❉▲ ❘❡❛s♦♥❡r✿ ❝♦♥❥✉♥❝t✐✈❡ q✉❡r② ❢♦r Datalog✲r❡✇r✐t❛❜❧❡ ♦♥t♦❧♦❣✐❡s ✭❉▲✲s❛❢❡♥❡ss✮ ❉▲✲♣r♦❣r❛♠ ❘❡❛s♦♥❡r✿

• ❝♦♠♣✉t❡ ✇❡❧❧✲❢♦✉♥❞❡❞ ♠♦❞❡❧ ❢♦r ❞❧✲♣r♦❣r❛♠s ♦✈❡r Datalog✲r❡✇r✐t❛❜❧❡

♦♥t♦❧♦❣✐❡s

• ❝❛♥ ❜❡ ✉s❡❞ ❢♦r LDL+ ❛♥❞ EL ♦♥t♦❧♦❣✐❡s

❉❘❡❲ ✈✵✳✸ ❢❡❛t✉r❡s✿ s✉♣♣♦rt ♦❢✿ ❖❲▲ ✷ ❘▲✱ ❖❲▲ ✷ ❊▲❀ ❛♥s✇❡r s❡t s❡♠❛♥t✐❝s

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✶✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✺ ❊①♣❡r✐♠❡♥t ✶

❊①♣❡r✐♠❡♥t ✶✿ ■♥st❛♥❝❡ ❘❡tr✐❡✈❛❧ ✇✐t❤ ▲❛r❣❡ EL ❚❇♦①❡s

❇❡♥❝❤♠❛r❦ ❊❧✲●❛❧❡♥✶ ✇✐t❤ ✷✸✱✶✹✶ ❝♦♥❝❡♣ts ❛♥❞ ✾✺✵ r♦❧❡s ✐♥ t❤❡ ❚❇♦① ❆ EL ✈❛r✐❛♥t ♦❢ ●❛❧❡♥✷ ❛ ♣r♦♠✐♥❡♥t✱ ❧❛r❣❡ ❜✐♦♠❡❞✐❝❛❧ ♦♥t♦❧♦❣② ❈r❡❛t❡❞ ♦♥t♦❧♦❣② ✐♥st❛♥❝❡s G1✲G4 ✇✐t❤ ✜①❡❞ ❚❇♦① ❛♥❞ ✐♥❝r❡❛s✐♥❣ ❆❇♦①❡s ✇✐t❤ 10∗i ❛ss❡rt✐♦♥s✱ ❡❛❝❤ ✉s✐♥❣ ≈ ✶✵ ❝♦♥❝❡♣ts ❛♥❞ r♦❧❡s✳ ❋♦✉r ✐♥st❛♥❝❡ r❡tr✐❡✈❛❧ q✉❡r✐❡s ♦✈❡r Gi

q1(X) = Substance(X) q3(X) = MaleAdult(X) q2(X) = Animal(X) q4(X) = Human(X)

P❧❛t❢♦r♠ ❯❜✉♥t✉ ▲✐♥✉① ✶✶✳✶✵ s②st❡♠ ♦♥ ❛♥ ❆▼❉ ❖♣t❡r♦♥ ▼❛❣♥②✲❈♦✉rs ✻✶✼✻ ❙❊ ✷✳✸●❍③ s②st❡♠ ✇✐t❤ ✷✹ ❝♦r❡s ❛♥❞ ✶✷✽●❇ ❘❆▼

✶❤tt♣✿✴✴❝♦♥❞♦r✲r❡❛s♦♥❡r✳❣♦♦❣❧❡❝♦❞❡✳❝♦♠✴❢✐❧❡s✴❊▲✲●❆▲❊◆✳♦✇❧ ✷❤tt♣✿✴✴✇✇✇✳♦♣❡♥❣❛❧❡♥✳♦r❣✴ ❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✷✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✺ ❊①♣❡r✐♠❡♥t ✶

❘❡s✉❧ts ✭❘✉♥t✐♠❡ ✐♥ s❡❝s✮

❖♥t♦❧♦❣② ❉❘❡❲ ❍❡r♠✐t ◗✉❡r② ❬❉▲❱❪ ❬❝❧✐♥❣♦❪ G1 q1 ✷✳✵ ✶✳✸ ✽✳✶ q2 ✷✳✵ ✶✳✸ ✽✳✷ q3 ✷✳✵ ✶✳✸ ✽✳✵ q4 ✷✳✵ ✶✳✹ ✽✳✶ G2 q1 ✷✳✵ ✶✳✸ ✽✳✾ q2 ✷✳✵ ✶✳✹ ✽✳✾ q3 ✷✳✶ ✶✳✹ ✽✳✼ q4 ✷✳✵ ✶✳✸ ✾✳✵ ❖♥t♦❧♦❣② ❉❘❡❲ ❍❡r♠✐t ◗✉❡r② ❬❉▲❱❪ ❬❝❧✐♥❣♦❪ G3 q1 ✷✳✵ ✶✳✸ ✾✳✺ q2 ✷✳✶ ✶✳✹ ✾✳✺ q3 ✷✳✵ ✶✳✹ ✾✳✼ q4 ✷✳✶ ✶✳✹ ✾✳✺ G4 q1 ✷✳✶ ✶✳✹ ✶✵✳✸ q2 ✷✳✶ ✶✳✹ ✶✵✳✷ q3 ✷✳✶ ✶✳✹ ✶✵✳✷ q4 ✷✳✶ ✶✳✹ ✶✵✳✷

❉❘❡❲ ✇✐t❤ ❝❧✐♥❣♦ ✸✳✵✳✸ ❬●❡❜s❡r ❡t ❛❧✳✱ ✷✵✶✶❪ ❛♥❞ ❉▲❱ ✷✵✶✵✲✶✵✲✶✹ ❬▲❡♦♥❡ ❡t ❛❧✳✱ ✷✵✵✻❪✳ ❍❡r♠✐❚✶✳✸✳✺ ❬▼♦t✐❦ ❡t ❛❧✳✱ ✷✵✵✾❪ ❛♥❞ P❡❧❧❡t ✷✳✸✳✵ ❬❙✐r✐♥ ❡t ❛❧✳✱ ✷✵✵✼❪✳ ❉❘❡❲ ✐s s✉♣❡r✐♦r t♦ ❍❡r♠✐❚ ❛♥❞ P❡❧❧❡t P❡❧❧❡t ❛❧✇❛②s t✐♠❡❞ ♦✉t ✭✶ ❤♦✉r✮ ❡✈❡♥ ✇✐t❤ s♠❛❧❧ ❆❇♦①❡s

◆♦t❡✿ ❈❇ ❬❑❛③❛❦♦✈✱ ✷✵✵✾❪ ❛♥❞ ❊▲❑ ❬❑❛③❛❦♦✈ ❡t ❛❧✳✱ ✷✵✶✶❪ ❞♦ ❢❛st ❝❧❛ss✐✜❝❛t✐♦♥✱ ❜✉t ❝❛♥✬t ❜❡ ✉s❡❞ ❢♦r ✐♥st❛♥❝❡ r❡tr✐❡✈❛❧ ♦✉t ♦❢ t❤❡ ❜♦①✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✸✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✻ ❊①♣❡r✐♠❡♥t ✷

❊①♣❡r✐♠❡♥t ✷✿ ❉❡❢❛✉❧t ❘❡❛s♦♥✐♥❣ ✇✐t❤ EL✲P♦❧✐❝✐❡s

T =        Staff ⊑ User, Blacklisted ⊑ Staff , Deny ⊓ Grant ⊑ ⊥, UserRequest ≡ ∃hasAction.Action ⊓ ∃hasSubject.User ⊓ ∃hasTarget.Project, StaffRequest ≡ ∃hasAction.Action ⊓ ∃hasSubject.Staff ⊓ ∃hasTarget.Project, BlacklistedStaffRequest ≡ StaffRequest ⊓ ∃hasSubject.Blacklisted        A = {StaffRequest(joe), Blacklisted(jim), . . . } D =    UserRequest(X) : Deny(X)/Deny(X), StaffRequest(X) : ¬BlacklistedStaffRequest(X)/Grant(X), BlacklistedStaffRequest(X) : ⊤/Deny(X)   

❆❝❝❡ss ❝♦♥tr♦❧ ♣♦❧✐❝②✱ ❛s ✐♥ ❬❇♦♥❛tt✐ ❡t ❛❧✳✱ ✷✵✶✶❪✱ ❝♦✉❝❤❡❞ ✐♥ t❡r♠✐♥♦❧♦❣✐❝❛❧ ❞❡❢❛✉❧t ❧♦❣✐❝ ❬❇❛❛❞❡r ❛♥❞ ❍♦❧❧✉♥❞❡r✱ ✶✾✾✺❪ ❑♥♦✇❧❡❞❣❡ ❜❛s❡ ∆ = (L, D) ✇✐t❤ ♦♥t♦❧♦❣② L = (T , A) ❛♥❞ ❉❡❢❛✉❧t r✉❧❡s D✿

• ✉s❡rs ♥♦r♠❛❧❧② ❛r❡ ❞❡♥✐❡❞ ❛❝❝❡ss t♦ ✜❧❡s
• st❛✛ ✐s ♥♦r♠❛❧❧② ❣r❛♥t❡❞ ❛❝❝❡ss t♦ ✜❧❡s
• ❜❧❛❝❦❧✐st❡❞ st❛✛ ❛r❡ ❞❡♥✐❡❞ ❛♥② ❛❝❝❡ss

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✹✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✹✳ ❉❛t❛❧♦❣✲❘❡✇r✐t❛❜✐❧✐t② ✹✳✻ ❊①♣❡r✐♠❡♥t ✷

❇❡♥❝❤♠❛r❦ ✫ ❘❡s✉❧ts

❑❇ ❉❘❡❲ ❚②♣✐♥❣ ❬❉▲❱❪ ❬❝❧✐♥❣♦❪ ∆1 ✺ ✶✳✶ ✵✳✽ ✺✵ ✷✳✹ ✶✳✸ ✶✵✵ ✻✳✵ ✸✳✵ ∆5 ✺ ✻✳✻ ✹✳✹ ✺✵ ✽✳✸ ✺✳✵ ✶✵✵ ✶✷✳✷ ✼✳✹ ∆10 ✺ ✶✸✳✾ ✾✳✹ ✺✵ ✶✺✳✼ ✶✵✳✶ ✶✵✵ ✷✵✳✺ ✶✸✳✸ ∆25 ✺ ✸✺✳✽ ✷✻✳✵ ✺✵ ✹✵✳✵ ✷✻✳✹ ✶✵✵ ✹✸✳✼ ✸✷✳✼

❖♥t♦❧♦❣② ✐♥st❛♥❝❡s Li✱ i ∈ {1, 5, 10, 25}✱ t❤❛t ❤❛✈❡ ❛ ✜①❡❞ ❚❇♦① ❛♥❞ ✐♥❝r❡❛s✐♥❣ ❆❜♦①❡s ✇✐t❤ i∗1000 ✐♥st❛♥❝❡s ♦❢ ✉s❡r r❡q✉❡sts✳ ❛s❦ ✇❤❡t❤❡r ❛ s❡t ♦❢ k ♣❛rt✐❝✉❧❛r ✐♥❞✐✈✐❞✉❛❧s✱ ❞❡s✐❣♥❛t❡❞ ❜② ❝♦♥❝❡♣ts Qk✱ k ∈ {5, 50, 100}✱ ❛r❡ ❣r❛♥t❡❞ ❛❝❝❡ss✳ ❉❘❡❲ s❝❛❧❡s s✉❜❧✐♥❡❛r❧②✱ ♦♥ t♦♣ ♦❢ ❜♦t❤ ❉▲❱ ❛♥❞ ❝❧✐♥❣♦✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✺✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✺✳ ❉✐s❝✉ss✐♦♥ ✫ ❈♦♥❝❧✉s✐♦♥ ✺✳✶ ❉✐s❝✉ss✐♦♥

❉✐s❝✉ss✐♦♥

❋❖✲❘❡✇r✐t❛❜✐❧✐t② ❘❡❧❛①❡❞ ♥♦t✐♦♥ ♦❢ ❋❖✲r❡✇r✐t❛❜✐❧✐t②

• ❈♦♠❜✐♥❡❞ ❛♣♣r♦❛❝❤ ❬❑♦♥t❝❤❛❦♦✈ ❡t ❛❧✳✱ ✷✵✶✵❪✿ ♣❡r♠✐t ♠♦❞✐✜❝❛t✐♦♥s

♦❢ t❤❡ ❆❇♦① ✭❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ❚❇♦① ❜✉t ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ q✉❡r②✮

• ❋♦r ❞❧✲❛t♦♠s✱ ♣r❡❝✐s❡ ❝♦♥t❡♥ts ♦❢ ✉♣❞❛t❡❞ ❆❇♦① ✉♥❦♥♦✇♥
• ❙t✐❧❧ ♣♦ss✐❜❧❡✿ ❛✉①✐❧✐❛r② r❡❧❛t✐♦♥s ✭❡✳❣✳✱ ❛ ❧✐♥❡❛r ♦r❞❡r✐♥❣ ♦❢ t❤❡

✐♥❞✐✈✐❞✉❛❧s✱ ♦r ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s✮✳

❘❡❝✉rs✐✈❡ ❙◗▲✿ r♦♦♠ ❢♦r ✐♠♣r♦✈❡♠❡♥t ✭❝❢✳ ❬❚❡rr❛❝✐♥❛ ❡t ❛❧✳✱ ✷✵✵✽❪✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✻✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✺✳ ❉✐s❝✉ss✐♦♥ ✫ ❈♦♥❝❧✉s✐♦♥ ✺✳✶ ❉✐s❝✉ss✐♦♥

❉✐s❝✉ss✐♦♥ ✭❝♦♥t✬❞✮

Datalog¬✲❘❡✇r✐t❛❜✐❧✐t② ❘❡❧❛①❡❞ ♥♦t✐♦♥s ♦❢ Datalog✲r❡✇r✐t❛❜✐❧✐t② ✭❛❧❧♦✇ ❢♦r ❛✉①✐❧✐❛r② r❡❧❛t✐♦♥s✮✳ Datalog¬✲r❡✇r✐t❛❜✐❧✐t② ♦❢ ❞❧✲❛t♦♠s✿

• Pr♦❣r❛♠ ΦDL(L) ❝♦✉❧❞ ❤❛✈❡ ♠✉❧t✐♣❧❡ ❛♥s✇❡r s❡ts ✭♦r ♥♦♥❡✮✳
• P❧✉❣❣✐♥❣ ✐♥ ΦDL(Lλ) ❢♦r s♦♠❡ ❞❧✲❛t♦♠ DL[λ, Q](

t) ♠❛② ❧❡❛❞ t♦ ✉♥✇❛♥t❡❞ ❡✛❡❝ts ✭❡✳❣✳✱ ❛❞❞✐t✐♦♥❛❧ ❛♥s✇❡r s❡ts✮✳ ⇒ ✉s❡ s②♥t❛❝t✐❝ r❡str✐❝t✐♦♥s ✭❡✳❣✳✱ ❛❝②❝❧✐❝✐t②✱ ❞❧✲❛t♦♠s ❛r❡ ♥♦t ✐♥✈♦❧✈❡❞ ✐♥ ❝②❝❧❡s✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✼✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✺✳ ❉✐s❝✉ss✐♦♥ ✫ ❈♦♥❝❧✉s✐♦♥ ✺✳✷ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❚❤❡ ✉♥✐❢♦r♠ ❡✈❛❧✉❛t✐♦♥ ❛♣♣r♦❛❝❤ ✐s ❛ ✢❡①✐❜❧❡ ❢r❛♠❡✇♦r❦ ❋✉rt❤❡r ✉s❛❣❡✿ ❊♠♣❧♦② ▼♦❞✉❧❛r ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ t♦ ❡✈❛❧✉❛t❡ ❞❛t❛❧♦❣✲r❡✇r✐t❛❜❧❡ ❞❧✲♣r♦❣r❛♠s ✭❚❉✲▼▲P✮ ❚❤❡ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❢♦r s✐♠♣❧❡ ♣r♦t♦t②♣❡s ✭▼❖❘✱ ❉❘❡❲✱ ❚❉✲▼▲P✮ ❛r❡ ❡♥❝♦✉r❛❣✐♥❣ ▼♦r❡ ❞❡t❛✐❧s✿

❤tt♣✿✴✴✇✇✇✳❦r✳t✉✇✐❡♥✳❛❝✳❛t✴r❡s❡❛r❝❤✴s②st❡♠s✴❞r❡✇✴❡①♣❡r✐♠❡♥ts✳❤t♠❧

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✽✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✺✳ ❉✐s❝✉ss✐♦♥ ✫ ❈♦♥❝❧✉s✐♦♥ ✺✳✷ ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥ ✭❝♦♥t✬❞✮

■ss✉❡s✿ ❖t❤❡r ❢♦r♠❛❧✐s♠s ✇✐t❤ ✭❡♠❡r❣✐♥❣✮ r❡❛s♦♥✐♥❣ ❡♥❣✐♥❡s ♠❛② ❜❡ ❝♦♥s✐❞❡r❡❞

• ❋❖✭·✮ ▲♦❣✐❝ ❬❉❡♥❡❝❦❡r ❛♥❞ ❚❡r♥♦✈s❦❛✱ ✷✵✵✽❪
• ❋✲❧♦❣✐❝ ❬❑✐❢❡r ❡t ❛❧✳✱ ✶✾✾✺❪
• Datalog ± ❬❈❛❧ì ❡t ❛❧✳✱ ✷✵✶✵❪ ✭s✉❜s✉♠❡s ❉▲✲▲✐t❡✮
• ✳✳✳

■♠♣r♦✈❡❞ ♣r♦t♦t②♣❡s ✴ s②st❡♠s ❆❧t❡r♥❛t✐✈❡ ❡♥❝♦❞✐♥❣s ✴ r❡❞✉❝t✐♦♥s ❖♣t✐♠✐③❛t✐♦♥ ♠❡t❤♦❞s ❛♥❞ t❡❝❤♥✐q✉❡s

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✺✾✴✻✹

❘❡❢❡r❡♥❝❡s ■

❋r❛♥③ ❇❛❛❞❡r ❛♥❞ ❇❡r♥❤❛r❞ ❍♦❧❧✉♥❞❡r✳ ❊♠❜❡❞❞✐♥❣ ❞❡❢❛✉❧ts ✐♥t♦ t❡r♠✐♥♦❧♦❣✐❝❛❧ ❦♥♦✇❧❡❞❣❡ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r♠❛❧✐s♠s✳ ❏✳ ❆✉t♦♠✳ ❘❡❛s♦♥✐♥❣✱ ✶✹✭✶✮✿✶✹✾✕✶✽✵✱ ✶✾✾✺✳ P✐❡r♦ ❆✳ ❇♦♥❛tt✐✱ ▼❛r❝♦ ❋❛❡❧❧❛✱ ❛♥❞ ▲✉✐❣✐ ❙❛✉r♦✳ ❆❞❞✐♥❣ ❞❡❢❛✉❧t ❛ttr✐❜✉t❡s t♦ ❊▲✰✰✳ ■♥ ❲♦❧❢r❛♠ ❇✉r❣❛r❞ ❛♥❞ ❉❛♥ ❘♦t❤✱ ❡❞✐t♦rs✱ ❆❆❆■✳ ❆❆❆■ Pr❡ss✱ ✷✵✶✶✳ ❆♥❞r❡❛ ❈❛❧ì✱ ●❡♦r❣ ●♦tt❧♦❜✱ ❚❤♦♠❛s ▲✉❦❛s✐❡✇✐❝③✱ ❇r✉♥♦ ▼❛r♥❡tt❡✱ ❛♥❞ ❆♥❞r❡❛s P✐❡r✐s✳ ❉❛t❛❧♦❣✰✴✲✿ ❆ ❢❛♠✐❧② ♦❢ ❧♦❣✐❝❛❧ ❦♥♦✇❧❡❞❣❡ r❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ q✉❡r② ❧❛♥❣✉❛❣❡s ❢♦r ♥❡✇ ❛♣♣❧✐❝❛t✐♦♥s✳ ■♥ ▲■❈❙✱ ♣❛❣❡s ✷✷✽✕✷✹✷✳ ■❊❊❊ ❈♦♠♣✉t❡r ❙♦❝✐❡t②✱ ✷✵✶✵✳

❘❡❢❡r❡♥❝❡s ■■

❉✳ ❈❛❧✈❛♥❡s❡✱ ●✳ ❞❡ ●✐❛❝♦♠♦✱ ❉✳ ▲❡♠❜♦✱ ▼✳ ▲❡♥③❡r✐♥✐✱ ❛♥❞ ❘✐❝❝❛r❞♦ ❘♦s❛t✐✳ ❚r❛❝t❛❜❧❡ r❡❛s♦♥✐♥❣ ❛♥❞ ❡✣❝✐❡♥t q✉❡r② ❛♥s✇❡r✐♥❣ ✐♥ ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝s✿ ❚❤❡ ❉▲✲▲✐t❡ ❢❛♠✐❧②✳ ❏❆❘✱ ✸✾✭✸✮✿✸✽✺✕✹✷✾✱ ✷✵✵✼✳ ▼✐♥❤ ❉❛♦✲❚r❛♥✱ ❚❤♦♠❛s ❊✐t❡r✱ ❛♥❞ ❚❤♦♠❛s ❑r❡♥♥✇❛❧❧♥❡r✳ ❘❡❛❧✐③✐♥❣ ❞❡❢❛✉❧t ❧♦❣✐❝ ♦✈❡r ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝ ❦♥♦✇❧❡❞❣❡ ❜❛s❡s✳ ■♥ ❈✳ ❙♦ss❛✐ ❛♥❞ ●✳ ❈❤❡♠❡❧❧♦✱ ❡❞✐t♦rs✱ Pr♦❝✳ ✶✵t❤ ❊✉r♦♣❡❛♥ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙②♠❜♦❧✐❝ ❛♥❞ ◗✉❛♥t✐t❛t✐✈❡ ❆♣♣r♦❛❝❤❡s t♦ ❘❡❛s♦♥✐♥❣ ✇✐t❤ ❯♥❝❡rt❛✐♥t②✱ ✭❊❈❙◗❆❘❯ ✷✵✵✾✮✱ ✈♦❧✉♠❡ ✺✺✾✵ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✻✵✷✕✻✶✸✳ ❙♣r✐♥❣❡r✱ ✷✵✵✾✳

❘❡❢❡r❡♥❝❡s ■■■

❏♦s ❞❡ ❇r✉✐❥♥✱ ❚❤♦♠❛s ❊✐t❡r✱ ❛♥❞ ❍❛♥s ❚♦♠♣✐ts✳ ❊♠❜❡❞❞✐♥❣ ❛♣♣r♦❛❝❤❡s t♦ ❝♦♠❜✐♥✐♥❣ r✉❧❡s ❛♥❞ ♦♥t♦❧♦❣✐❡s ✐♥t♦ ❛✉t♦❡♣✐st❡♠✐❝ ❧♦❣✐❝✳ ■♥ Pr♦❝❡❡❞✐♥❣s ✶✶t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ Pr✐♥❝✐♣❧❡s ♦❢ ❑♥♦✇❧❡❞❣❡ ❘❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ❘❡❛s♦♥✐♥❣ ✭❑❘ ✷✵✵✽✮✱ ❙❡♣t❡♠❜❡r ✶✻✲✶✾✱ ✷✵✵✽✱ ❙✐❞♥❡②✱ ❆✉str❛❧✐❛✱ ♣❛❣❡s ✹✽✺✕✹✾✺✱ ✷✵✵✽✳ ▼❛r❝ ❉❡♥❡❝❦❡r ❛♥❞ ❊✉❣❡♥✐❛ ❚❡r♥♦✈s❦❛✳ ❆ ❧♦❣✐❝ ♦❢ ♥♦♥✲♠♦♥♦t♦♥❡ ✐♥❞✉❝t✐✈❡ ❞❡✜♥✐t✐♦♥s✳ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲♦❣✐❝✱ ✾✱ ■ss✉❡ ✷✱ ✷✵✵✽✳ ❆rt✐❝❧❡ ✶✹✱ ✺✷ ♣♣✳ ❚✳ ❊✐t❡r✱ ●✳ ■❛♥♥✐✱ ❘✳ ❙❝❤✐♥❞❧❛✉❡r✱ ❛♥❞ ❍✳ ❚♦♠♣✐ts✳ ❊✛❡❝t✐✈❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❞❡❝❧❛r❛t✐✈❡ r✉❧❡s ✇✐t❤ ❡①t❡r♥❛❧ ❡✈❛❧✉❛t✐♦♥s ❢♦r s❡♠❛♥t✐❝✲✇❡❜ r❡❛s♦♥✐♥❣✳ ■♥ ❊❙❲❈✱ ✈♦❧✉♠❡ ✹✵✶✶ ♦❢ ▲◆❈❙✱ ♣❛❣❡s ✷✼✸✕✷✽✼✳ ❙♣r✐♥❣❡r✱ ✷✵✵✻✳

❘❡❢❡r❡♥❝❡s ■❱

❚❤♦♠❛s ❊✐t❡r✱ ●✐♦✈❛♠❜❛tt✐st❛ ■❛♥♥✐✱ ❚❤♦♠❛s ❑r❡♥♥✇❛❧❧♥❡r✱ ❛♥❞ ❘♦♠❛♥ ❙❝❤✐♥❞❧❛✉❡r✳ ❊①♣❧♦✐t✐♥❣ ❝♦♥❥✉♥❝t✐✈❡ q✉❡r✐❡s ✐♥ ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝ ♣r♦❣r❛♠s✳ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❆rt✐✜❝✐❛❧ ■♥t❡❧❧✐❣❡♥❝❡✱ ✺✸✭✶✲✹✮✿✶✶✺✕✶✺✷✱ ✷✵✵✽✳ ❞♦✐✿✶✵✳✶✵✵✼✴s✶✵✹✼✷✲✵✵✾✲✾✶✶✶✲✸✳ ❚❤♦♠❛s ❊✐t❡r✱ ●✐♦✈❛♠❜❛tt✐st❛ ■❛♥♥✐✱ ❚❤♦♠❛s ▲✉❦❛s✐❡✇✐❝③✱ ❘♦♠❛♥ ❙❝❤✐♥❞❧❛✉❡r✱ ❛♥❞ ❍❛♥s ❚♦♠♣✐ts✳ ❈♦♠❜✐♥✐♥❣ ❆♥s✇❡r ❙❡t Pr♦❣r❛♠♠✐♥❣ ✇✐t❤ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝s ❢♦r t❤❡ ❙❡♠❛♥t✐❝ ❲❡❜✳ ❆rt✐✜❝✐❛❧ ■♥t❡❧❧✐❣❡♥❝❡✱ ✶✼✷✭✶✷✲✶✸✮✿✶✹✾✺✕✶✺✸✾✱ ✷✵✵✽✳ ❞♦✐✿✶✵✳✶✵✶✻✴❥✳❛rt✐♥t✳✷✵✵✽✳✵✹✳✵✵✷✳ Pr❡❧✐♠✐♥❛r② ✈❡rs✐♦♥ ❛✈❛✐❧❛❜❧❡ ❛s ❚❡❝❤✳❘❡♣✳ ■◆❋❙❨❙ ❘❘✲✶✽✹✸✲✵✼✲✵✹✱ ■♥st✐t✉t❡ ♦❢ ■♥❢♦r♠❛t✐♦♥ ❙②st❡♠s✱ ❚❯ ❱✐❡♥♥❛✱ ❏❛♥✉❛r② ✷✵✵✼✳

❘❡❢❡r❡♥❝❡s ❱

❚✳ ❊✐t❡r✱ ●✳ ■❛♥♥✐✱ ❚✳ ▲✉❦❛s✐❡✇✐❝③✱ ❛♥❞ ❘✳ ❙❝❤✐♥❞❧❛✉❡r✳ ❲❡❧❧✲❢♦✉♥❞❡❞ s❡♠❛♥t✐❝s ❢♦r ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝ ♣r♦❣r❛♠s ✐♥ t❤❡ ❙❡♠❛♥t✐❝ ❲❡❜✳ ❆❈▼ ❚r❛♥s✳ ❈♦♠♣✉t✳ ▲♦❣✳✱ ✶✷✭✷✮✿✶✶✱ ✷✵✶✶✳ ▼✐❝❤❛❡❧ ❋✐♥❦ ❛♥❞ ❉❛✈✐❞ P❡❛r❝❡✳ ❆ ❧♦❣✐❝❛❧ s❡♠❛♥t✐❝s ❢♦r ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝ ♣r♦❣r❛♠s✳ ■♥ ❚♦♠✐ ❏❛♥❤✉♥❡♥ ❛♥❞ ■❧❦❦❛ ◆✐❡♠❡❧ä✱ ❡❞✐t♦rs✱ ❏❊▲■❆✱ ✈♦❧✉♠❡ ✻✸✹✶ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✶✺✻✕✶✻✽✳ ❙♣r✐♥❣❡r✱ ✷✵✶✵✳ ▼❛rt✐♥ ●❡❜s❡r✱ ❇❡♥❥❛♠✐♥ ❑❛✉❢♠❛♥♥✱ ❘♦❧❛♥❞ ❑❛♠✐♥s❦✐✱ ▼❛① ❖str♦✇s❦✐✱ ❚♦rst❡♥ ❙❝❤❛✉❜✱ ❛♥❞ ▼❛r✐✉s ❙❝❤♥❡✐❞❡r✳ P♦t❛ss❝♦✿ ❚❤❡ P♦ts❞❛♠ ❆♥s✇❡r ❙❡t ❙♦❧✈✐♥❣ ❈♦❧❧❡❝t✐♦♥✳ ❆■ ❈♦♠♠✉♥✳✱ ✷✹✭✷✮✿✶✵✼✕✶✷✹✱ ✷✵✶✶✳

❘❡❢❡r❡♥❝❡s ❱■

• ❡♦r❣ ●♦tt❧♦❜ ❛♥❞ ❚❤♦♠❛s ❙❝❤✇❡♥t✐❝❦✳

❘❡✇r✐t✐♥❣ ♦♥t♦❧♦❣✐❝❛❧ q✉❡r✐❡s ✐♥t♦ s♠❛❧❧ ♥♦♥r❡❝✉rs✐✈❡ ❞❛t❛❧♦❣ ♣r♦❣r❛♠s✳ ■♥ ❘✐❝❝❛r❞♦ ❘♦s❛t✐✱ ❙❡❜❛st✐❛♥ ❘✉❞♦❧♣❤✱ ❛♥❞ ▼✐❝❤❛❡❧ ❩❛❦❤❛r②❛s❝❤❡✈✱ ❡❞✐t♦rs✱ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝s✱ ✈♦❧✉♠❡ ✼✹✺ ♦❢ ❈❊❯❘ ❲♦r❦s❤♦♣ Pr♦❝❡❡❞✐♥❣s✳ ❈❊❯❘✲❲❙✳♦r❣✱ ✷✵✶✶✳ ❇✳ ◆✳ ●r♦s♦❢✱ ■✳ ❍♦rr♦❝❦s✱ ❘✳ ❱♦❧③✱ ❛♥❞ ❙✳ ❉❡❝❦❡r✳ ❉❡s❝r✐♣t✐♦♥ ❧♦❣✐❝ ♣r♦❣r❛♠s✿ ❈♦♠❜✐♥✐♥❣ ❧♦❣✐❝ ♣r♦❣r❛♠s ✇✐t❤ ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝✳ ■♥ Pr♦❝✳ ❲❲❲ ✷✵✵✸✱ ♣❛❣❡s ✹✽✕✺✼✳ ❆❈▼✱ ✷✵✵✸✳ ❨✉❛♥❜♦ ●✉♦✱ ❩❤❡♥❣①✐❛♥❣ P❛♥✱ ❛♥❞ ❏❡✛ ❍❡✢✐♥✳ ▲✉❜♠✿ ❆ ❜❡♥❝❤♠❛r❦ ❢♦r ♦✇❧ ❦♥♦✇❧❡❞❣❡ ❜❛s❡ s②st❡♠s✳ ❲❡❜ ❙❡♠❛♥t✐❝s✱ ✸✭✷✲✸✮✿✶✺✽ ✕ ✶✽✷✱ ✷✵✵✺✳

❘❡❢❡r❡♥❝❡s ❱■■

❙✳ ❍❡②♠❛♥s✱ ❚✳ ❊✐t❡r✱ ❛♥❞ ●✳ ❳✐❛♦✳ ❚r❛❝t❛❜❧❡ r❡❛s♦♥✐♥❣ ✇✐t❤ ❉▲✲♣r♦❣r❛♠s ♦✈❡r ❞❛t❛❧♦❣✲r❡✇r✐t❛❜❧❡ ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝s✳ ■♥ Pr♦❝✳ ✶✾t❤ ❊✉r♦♣❡❛♥ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❆rt✐✜❝✐❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ✭❊❈❆■✮✳ ■❖❙ Pr❡ss✱ ✷✵✶✵✳ ❆❝❝❡♣t❡❞ ❢♦r P✉❜❧✐❝❛t✐♦♥✳ ■✳ ❍♦rr♦❝❦s✱ P✳❋✳ P❛t❡❧✲❙❝❤♥❡✐❞❡r✱ ❍✳ ❇♦❧❡②✱ ❙✳ ❚❛❜❡t✱ ❇✳ ●r♦s♦❢✱ ❛♥❞ ▼✳ ❉❡❛♥✳ ❙❲❘▲✿ ❆ s❡♠❛♥t✐❝ ✇❡❜ r✉❧❡ ❧❛♥❣✉❛❣❡ ❝♦♠❜✐♥✐♥❣ ❖❲▲ ❛♥❞ ❘✉❧❡▼▲✳ ❲✸❈ ▼❡♠❜❡r ❙✉❜♠✐ss✐♦♥✱ ❲♦r❧❞ ❲✐❞❡ ❲❡❜ ❈♦♥s♦rt✐✉♠✱ ✷✵✵✹✳ ❯❧❧r✐❝❤ ❍✉st❛❞t✱ ❇♦r✐s ▼♦t✐❦✱ ❛♥❞ ❯❧r✐❦❡ ❙❛tt❧❡r✳ ❘❡❛s♦♥✐♥❣ ✐♥ ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝s ❜② ❛ r❡❞✉❝t✐♦♥ t♦ ❞✐s❥✉♥❝t✐✈❡ ❞❛t❛❧♦❣✳ ❏✳ ❆✉t♦♠✳ ❘❡❛s♦♥✐♥❣✱ ✸✾✭✸✮✿✸✺✶✕✸✽✹✱ ✷✵✵✼✳

❘❡❢❡r❡♥❝❡s ❱■■■

❚♦♠✐ ❏❛♥❤✉♥❡♥✳ ❖♥ t❤❡ ✐♥t❡rtr❛♥s❧❛t❛❜✐❧✐t② ♦❢ ♥♦♥✲♠♦♥♦t♦♥✐❝ ❧♦❣✐❝s✳ ❆♥♥✳ ▼❛t❤✳ ❆rt✐❢✳ ■♥t❡❧❧✳✱ ✷✼✭✶✲✹✮✿✼✾✕✶✷✽✱ ✶✾✾✾✳ ❨❡✈❣❡♥② ❑❛③❛❦♦✈✱ ▼❛r❦✉s ❑röt③s❝❤✱ ❛♥❞ ❋r❛♥t✐s❡❦ ❙✐♠❛♥❝✐❦✳ ❈♦♥❝✉rr❡♥t ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❡❧ ♦♥t♦❧♦❣✐❡s✳ ■♥ ■♥t❡r♥❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝ ❲❡❜ ❈♦♥❢❡r❡♥❝❡ ✭✶✮✱ ✈♦❧✉♠❡ ✼✵✸✶ ♦❢ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡s ✸✵✺✕✸✷✵✳ ❙♣r✐♥❣❡r✱ ✷✵✶✶✳ ❨✳ ❑❛③❛❦♦✈✳ ❈♦♥s❡q✉❡♥❝❡✲❞r✐✈❡♥ r❡❛s♦♥✐♥❣ ❢♦r ❍♦r♥ SHIQ ♦♥t♦❧♦❣✐❡s✳ ■♥ ❈r❛✐❣ ❇♦✉t✐❧✐❡r✱ ❡❞✐t♦r✱ ■❏❈❆■✱ ♣❛❣❡s ✷✵✹✵✕✷✵✹✺✱ ✷✵✵✾✳ ▼✐❝❤❛❡❧ ❑✐❢❡r✱ ●❡♦r❣ ▲❛✉s❡♥✱ ❛♥❞ ❏❛♠❡s ❲✉✳ ▲♦❣✐❝❛❧ ❢♦✉♥❞❛t✐♦♥s ♦❢ ♦❜❥❡❝t✲♦r✐❡♥t❡❞ ❛♥❞ ❢r❛♠❡✲❜❛s❡❞ ❧❛♥❣✉❛❣❡s✳ ❏✳ ❆❈▼✱ ✹✷✭✹✮✿✼✹✶✕✽✹✸✱ ✶✾✾✺✳

❘❡❢❡r❡♥❝❡s ■❳

❘♦♠❛♥ ❑♦♥t❝❤❛❦♦✈✱ ❈❛rst❡♥ ▲✉t③✱ ❉❛✈✐❞ ❚♦♠❛♥✱ ❋r❛♥❦ ❲♦❧t❡r✱ ❛♥❞ ▼✐❝❤❛❡❧ ❩❛❦❤❛r②❛s❝❤❡✈✳ ❚❤❡ ❝♦♠❜✐♥❡❞ ❛♣♣r♦❛❝❤ t♦ q✉❡r② ❛♥s✇❡r✐♥❣ ✐♥ ❞❧✲❧✐t❡✳ ■♥ ❑❘✱ ✷✵✶✵✳ ▼✳ ❑röt③s❝❤✱ ❙✳ ❘✉❞♦❧♣❤✱ ❛♥❞ P✳ ❍✐t③❧❡r✳ ❉❡s❝r✐♣t✐♦♥ ❧♦❣✐❝ r✉❧❡s✳ ■♥ Pr♦❝✳ ❊❈❆■✱ ♣❛❣❡s ✽✵✕✽✹✳ ■❖❙ Pr❡ss✱ ✷✵✵✽✳ ▼✳ ❑röt③s❝❤✱ ❙✳ ❘✉❞♦❧♣❤✱ ❛♥❞ P✳ ❍✐t③❧❡r✳ ❊▲P✿ ❚r❛❝t❛❜❧❡ r✉❧❡s ❢♦r ❖❲▲ ✷✳ ■♥ Pr♦❝✳ ■❙❲❈ ✷✵✵✽✱ ♣❛❣❡s ✻✹✾✕✻✻✹✱ ✷✵✵✽✳

❘❡❢❡r❡♥❝❡s ❳

▼❛r❦✉s ❑röt③s❝❤✳ ❊✣❝✐❡♥t r✉❧❡✲❜❛s❡❞ ✐♥❢❡r❡♥❝✐♥❣ ❢♦r ❖❲▲ ❊▲✳ ■♥ ❚♦❜② ❲❛❧s❤✱ ❡❞✐t♦r✱ ■❏❈❆■ ✷✵✶✶✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✷✷♥❞ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✐♥t ❈♦♥❢❡r❡♥❝❡ ♦♥ ❆rt✐✜❝✐❛❧ ■♥t❡❧❧✐❣❡♥❝❡✱ ❇❛r❝❡❧♦♥❛✱ ❈❛t❛❧♦♥✐❛✱ ❙♣❛✐♥✱ ❏✉❧② ✶✻✲✷✷✱ ✷✵✶✶✱ ♣❛❣❡s ✷✻✻✽✕✷✻✼✸✳ ■❏❈❆■✴❆❆❆■✱ ✷✵✶✶✳ ◆✳ ▲❡♦♥❡✱ ●✳ P❢❡✐❢❡r✱ ❲✳ ❋❛❜❡r✱ ❚✳ ❊✐t❡r✱ ●✳ ●♦tt❧♦❜✱ ❙✳ P❡rr✐✱ ❛♥❞ ❋✳ ❙❝❛r❝❡❧❧♦✳ ❚❤❡ ❉▲❱ ❙②st❡♠ ❢♦r ❑♥♦✇❧❡❞❣❡ ❘❡♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ❘❡❛s♦♥✐♥❣✳ ❆❈▼ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ▲♦❣✐❝ ✭❚❖❈▲✮✱ ✼✭✸✮✱ ❏✉❧② ✷✵✵✻✳ ❇✳ ▼♦t✐❦ ❛♥❞ ❘✳ ❘♦s❛t✐✳ ❆ ❢❛✐t❤❢✉❧ ✐♥t❡❣r❛t✐♦♥ ♦❢ ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝s ✇✐t❤ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣✳ ■♥ Pr♦❝✳ ■❏❈❆■✱ ♣❛❣❡s ✹✼✼✕✹✽✷✱ ✷✵✵✼✳

❘❡❢❡r❡♥❝❡s ❳■

❇✳ ▼♦t✐❦ ❛♥❞ ❘✳ ❘♦s❛t✐✳ ❘❡❝♦♥❝✐❧✐♥❣ ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝s ❛♥❞ ❘✉❧❡s✳ ❏♦✉r♥❛❧ ♦❢ t❤❡ ❆❈▼✱ ✺✼✭✺✮✿✶✕✻✷✱ ✷✵✶✵✳ ❇✳ ▼♦t✐❦✱ ❯✳ ❙❛tt❧❡r✱ ❛♥❞ ❘✳ ❙t✉❞❡r✳ ◗✉❡r② ❛♥s✇❡r✐♥❣ ❢♦r ❖❲▲✲❉▲ ✇✐t❤ r✉❧❡s✳ ❏♦✉r♥❛❧ ♦❢ ❲❡❜ ❙❡♠❛♥t✐❝s✱ ✸✭✶✮✿✹✶✕✻✵✱ ❏✉❧② ✷✵✵✺✳ ❇♦r✐s ▼♦t✐❦✱ ❇❡r♥❛r❞♦ ❈✉❡♥❝❛ ●r❛✉✱ ■❛♥ ❍♦rr♦❝❦s✱ ❩❤❡ ❲✉✱ ❆❝❤✐❧❧❡ ❋♦❦♦✉❡✱ ❛♥❞ ❈❛rst❡♥ ▲✉t③✳ ❖✇❧ ✷ ✇❡❜ ♦♥t♦❧♦❣② ❧❛♥❣✉❛❣❡✿ ❖✇❧ ✷ ✇❡❜ ♦♥t♦❧♦❣② ❧❛♥❣✉❛❣❡✿ Pr♦✜❧❡s✳ ❲✸❈ ❲♦r❦✐♥❣ ❉r❛❢t✱ ❲♦r❧❞ ❲✐❞❡ ❲❡❜ ❈♦♥s♦rt✐✉♠✱ ❤tt♣✿✴✴✇✇✇✳✇✸✳♦r❣✴❚❘✴✷✵✵✽✴❲❉✲♦✇❧✷✲♣r♦✜❧❡s✲✷✵✵✽✶✵✵✽✴✱ ✷✵✵✽✳

❘❡❢❡r❡♥❝❡s ❳■■

❇♦r✐s ▼♦t✐❦✱ ❘♦❜ ❙❤❡❛r❡r✱ ❛♥❞ ■❛♥ ❍♦rr♦❝❦s✳ ❍②♣❡rt❛❜❧❡❛✉ ❘❡❛s♦♥✐♥❣ ❢♦r ❉❡s❝r✐♣t✐♦♥ ▲♦❣✐❝s✳ ❏♦✉r♥❛❧ ♦❢ ❆rt✐✜❝✐❛❧ ■♥t❡❧❧✐❣❡♥❝❡ ❘❡s❡❛r❝❤✱ ✸✻✿✶✻✺✕✷✷✽✱ ✷✵✵✾✳ ▼❛❣❞❛❧❡♥❛ ❖rt✐③✱ ❙❡❜❛st✐❛♥ ❘✉❞♦❧♣❤✱ ❛♥❞ ▼❛♥t❛s ❙✐♠❦✉s✳ ❲♦rst✲❝❛s❡ ♦♣t✐♠❛❧ r❡❛s♦♥✐♥❣ ❢♦r t❤❡ ❤♦r♥✲❞❧ ❢r❛❣♠❡♥ts ♦❢ ♦✇❧ ✶ ❛♥❞ ✷✳ ■♥ ❋❛♥❣③❤❡♥ ▲✐♥✱ ❯❧r✐❦❡ ❙❛tt❧❡r✱ ❛♥❞ ▼✐r♦s❧❛✇ ❚r✉s③❝③②♥s❦✐✱ ❡❞✐t♦rs✱ ❑❘✳ ❆❆❆■ Pr❡ss✱ ✷✵✶✵✳ ❘✳ ❘♦s❛t✐✳ ❖♥ t❤❡ ❞❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❝♦♠♣❧❡①✐t② ♦❢ ✐♥t❡❣r❛t✐♥❣ ♦♥t♦❧♦❣✐❡s ❛♥❞ r✉❧❡s✳ ❏♦✉r♥❛❧ ♦❢ ❲❡❜ ❙❡♠❛♥t✐❝s✱ ✸✭✶✮✿✹✶✕✻✵✱ ✷✵✵✺✳ ❘✳ ❘♦s❛t✐✳ ❉▲✰❧♦❣✿ ❚✐❣❤t ✐♥t❡❣r❛t✐♦♥ ♦❢ ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝s ❛♥❞ ❞✐s❥✉♥❝t✐✈❡ ❞❛t❛❧♦❣✳ ■♥ Pr♦❝✳ ❑❘✱ ♣❛❣❡s ✻✽✕✼✽✱ ✷✵✵✻✳

❘❡❢❡r❡♥❝❡s ❳■■■

P✳ ❙❝❤♥❡✐❞❡r✳ ❊✈❛❧✉❛t✐♦♥ ♦❢ ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝ ♣r♦❣r❛♠s ✉s✐♥❣ ❛♥ ❘❉❇▼❙✳ ▼❛st❡r✬s t❤❡s✐s✱ ❱✐❡♥♥❛ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✱ ❉❡❝❡♠❜❡r ✷✵✶✵✳ ❊✳ ❙✐r✐♥✱ ❇✳ P❛rs✐❛✱ ❇✳ ❈✉❡♥❝❛ ●r❛✉✱ ❆✳ ❑❛❧②❛♥♣✉r✱ ❛♥❞ ❨✳ ❑❛t③✳ P❡❧❧❡t✿ ❆ ♣r❛❝t✐❝❛❧ ❖❲▲✲❉▲ r❡❛s♦♥❡r✳ ❏✳ ❲❡❜ ❙❡♠✳✱ ✺✭✷✮✿✺✶✕✺✸✱ ✷✵✵✼✳

• ✐♦r❣✐♦ ❚❡rr❛❝✐♥❛✱ ◆✐❝♦❧❛ ▲❡♦♥❡✱ ❱✐♥❝❡♥③✐♥♦ ▲✐♦✱ ❛♥❞ ❈❧❛✉❞✐♦ P❛♥❡tt❛✳

❊①♣❡r✐♠❡♥t✐♥❣ ✇✐t❤ r❡❝✉rs✐✈❡ q✉❡r✐❡s ✐♥ ❞❛t❛❜❛s❡ ❛♥❞ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ s②st❡♠s✳ ❚P▲P✱ ✽✭✷✮✿✶✷✾✕✶✻✺✱ ✷✵✵✽✳ ❨✐s♦♥❣ ❲❛♥❣✱ ❏✐❛✲❍✉❛✐ ❨♦✉✱ ▲✐✲❨❛♥ ❨✉❛♥✱ ❨✐✲❉♦♥❣ ❙❤❡♥✱ ❛♥❞ ❚❤♦♠❛s ❊✐t❡r✳ ❊♠❜❡❞❞✐♥❣ ❞❡s❝r✐♣t✐♦♥ ❧♦❣✐❝ ♣r♦❣r❛♠s ✐♥t♦ ❞❡❢❛✉❧t ❧♦❣✐❝✳ ❈♦❘❘✱ ❛❜s✴✶✶✶✶✳✶✹✽✻✱ ✷✵✶✶✳

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✼✳ ❆♣♣❡♥❞✐①✿ ▼♦❞✉❧❛r Datalog¬

❆♣♣❡♥❞✐①✿ ▼♦❞✉❧❛r Datalog¬

❉❡❝❧❛r❛t✐♦♥✿ ✐♥t ❢✉♥✭✐♥t✱ ✐♥t✮❀ ❉❡✜♥✐t✐♦♥✿ ✐♥t ❢✉♥✭✐♥t ①✱ ✐♥t ②✮ ④ ✳ ✳ ✳ r❡t✉r♥ · · · ❀ ⑥ ❯s❡✿ ✐♥t ③ ❂ ❢✉♥✭✶✱✷✮❀

■♠♣❡r❛t✐✈❡ Pr♦❣r❛♠♠✐♥❣ ✭❈✲st②❧❡✮

❉❡❝❧❛r❛t✐♦♥✿ ❢✉♥ ✿✿ ■♥t ✲❃ ■♥t ✲❃ ■♥t ❉❡✜♥✐t✐♦♥✿ ❢✉♥ ① ② ❂ · · · ❯s❡✿ ❧❡t ③ ❂ ❢✉♥ ✶ ✷ ✐♥ · · ·

❋✉♥❝t✐♦♥❛❧ Pr♦❣r❛♠♠✐♥❣ ✭❍❛s❦❡❧❧✲st②❧❡✮

❉❡❝❧❛r❛t✐♦♥✿ m = (fun[p, q], R)

• fun ✐s ❛ ♠♦❞✉❧❡ ♥❛♠❡
• p, q ❛r❡ ♣r❡❞✐❝❛t❡ ♥❛♠❡s
• R ✐s ❛ s❡t ♦❢ r✉❧❡s

❉❡✜♥✐t✐♦♥✿ R = {even(X) ← · · · ; . . . } ❯s❡✿ z(X) ← fun[r, s].even(X)

▼♦❞✉❧❛r ◆♦♥♠♦♥♦t♦♥✐❝ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ✭▼▲P✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✻✵✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✼✳ ❆♣♣❡♥❞✐①✿ ▼♦❞✉❧❛r Datalog¬ ✼✳✶ ▼▲P ❘❡❞✉❝t✐♦♥

❯s✐♥❣ ▼▲Ps

■❞❡❛✿ ❋♦r ✉♥✐❢♦r♠ ❡✈❛❧✉❛t✐♦♥✱ ✉s❡ ✭❞❛t❛❧♦❣✮ ♠♦❞✉❧❡s t♦ ❡♠✉❧❛t❡ ❉▲✲❛t♦♠s ❊①♣r❡ss ❉▲✲❛t♦♠ DL[λ; Q](X)✱ λ = S1 ⊎ p1, . . . , Sm ⊎ pm✱ ❜② ❛ ❝❛❧❧ ❛t♦♠ Pλ[p1, . . . , pm].Q(X) ▼♦❞✉❧❡ m = (Pλ[q1, . . . , qm], R) ❤❛s R ❡♥❝♦❞✐♥❣ I | = DL[λ; Q](X)

❊①❛♠♣❧❡

KB = (L, P)✱ ✇❤❡r❡ L = {C ⊑ D} ❛♥❞

• P =

p(a); p(b); q(c); s(X) ← DL[C ⊎ p; D](X), not DL[C ⊎ q; D](X)

• .

▼▲P P = (m1, mDL)✱ ✇❤❡r❡

• m1 = (P1[], R1) ✇❤❡r❡ R1 =

p(a), p(b), q(c), s(X) ← PDL[p].D(X), not PDL[q].D(X)

• ✱
• mDL = (PDL[C], RDL) ✇❤❡r❡ RDL = TP ∪ {D(X) ← C(X)}✳

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✻✶✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✼✳ ❆♣♣❡♥❞✐①✿ ▼♦❞✉❧❛r Datalog¬ ✼✳✷ ❊①♣❡r✐♠❡♥ts

❊①♣❡r✐♠❡♥ts

▼❛✐♥ ◗✉❡st✐♦♥

❚r❛❞❡♦✛ ❜❡t✇❡❡♥ str✉❝t✉r❡❞ ♣r♦❣r❛♠♠✐♥❣ ✭♠♦❞✉❧❡s✮ ❛♥❞ ❝♦❞❡ r❡♣❡t✐t✐♦♥ ❯s❡ t❤❡ ❚❉✲▼▲P s♦❧✈❡r ✭❤❛♥❞❧❡s ❛ ❢r❛❣♠❡♥t ♦❢ ▼▲Ps✮

• ◆♦t❡✿ ▼▲P ✐s ✈❡r② ❡①♣r❡ss✐✈❡ ✭2 − NExpTime✲❝♦♠♣❧❡t❡✮

❇❡♥❝❤♠❛r❦s ❊①♣❡r✐♠❡♥ts ♦♥ ❞❧✲♣r♦❣r❛♠s KB = (Ui, Pj)✱ ✇❤❡r❡

• Ui ✐s ❛♥ EL ✈❡rs✐♦♥ ♦❢ t❤❡ ▲❯❇▼ ♦♥t♦❧♦❣②✱ ✇✐t❤ i ✉♥✐✈❡rs✐t✐❡s

s❦✐♣ ✷ ❚❇♦① ❛①✐♦♠s ✇✐t❤ ✐♥✈❡rs❡ r♦❧❡s s❦✐♣ ✷✽✺✼ ✭✸✸✶✺✹✮ ❆❇♦① ❛①✐♦♠s ✇✐t❤ ❞❛t❛t②♣❡ ✈✐♦❧❛t✐♦♥s ✐♥ U1 ✭U15✮ U ❤❛s ✽✻ ❚❇♦① ❛①✐♦♠s ✉s✐♥❣ ✹✸ ❝♦♥❝❡♣ts ❛♥❞ ✷✺ r♦❧❡s❀ U1 ✭U15✮ ❤❛s ✺✼✸✽ ✭✻✼✻✾✶✮ ❆❇♦① ❛①✐♦♠s✱ ✶✺✺✺ ✭✶✼✶✼✹✮ ✐♥❞✐✈✐❞✉❛❧s

Pj ✭❛❝②❝❧✐❝✮ ✐s ❛ ✈❛r✐❛♥t ♦❢ t❤❡ ▲❯❇▼ q✉❡r② Qi✸

• P0✕P4✿ ✷✕✺ ❞❧✲❛t♦♠s✱ ♥♦ ✐♥♣✉t ❧✐st
• P5✕P9✿ ✷✕✾ ❞❧✲❛t♦♠s✱ ❡❛❝❤ ✇✐t❤ ❞✐st✐♥❝t ✐♥♣✉t ❧✐st

✸❤tt♣✿✴✴s✇❛t✳❝s❡✳❧❡❤✐❣❤✳❡❞✉✴♣r♦❥❡❝ts✴❧✉❜♠✴q✉❡r②✳❤t♠ ❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✻✷✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✼✳ ❆♣♣❡♥❞✐①✿ ▼♦❞✉❧❛r Datalog¬ ✼✳✷ ❊①♣❡r✐♠❡♥ts

❘❡s✉❧ts ✭❘✉♥t✐♠❡ ✐♥ s❡❝s❀ ❍❛r❞✇❛r❡ ❛s ❢♦r Datalog¬✮

❖♥t♦❧♦❣② U1 Pr♦❣r❛♠ ❉❘❡❲ ❚❉✲▼▲P ❬❝❧✐♥❣♦❪ ❬❉▲❱❪ ❬❝❧✐♥❣♦❪ ❬❉▲❱❪ P0 ✵✳✸✶ ✵✳✹✺ ✶✳✾✽ ✷✳✽✽ P1 ✵✳✸✷ ✵✳✹✹ ✶✳✻✾ ✷✳✹✼ P2 ✵✳✸✷ ✵✳✹✹ ✷✳✻✸ ✸✳✽✷ P3 ✵✳✸✶ ✵✳✹✸ ✶✳✻✻ ✷✳✹✷ P4 ✵✳✸✷ ✵✳✹✺ ✷✳✹✺ ✸✳✻✸ P5 ✵✳✻✶ ✵✳✽✻ ✶✳✻✻ ✷✳✹✻ P6 ✶✳✼✾ ✷✳✼✻ ✺✳✻✺ ✽✳✹✶ P7 ✷✳✼✵ ✹✳✸✵ ✹✳✽✼ ✼✳✽✹ P8 ✷✳✼✻ ✹✳✷✻ ✾✳✼✵ ✶✹✳✶✷ P9 ✷✳✼✸ ✹✳✸✶ ✽✳✵✹ ✶✶✳✻✵ ❖♥t♦❧♦❣② U15 Pr♦❣r❛♠ ❉❘❡❲ ❚❉✲▼▲P ❬❝❧✐♥❣♦❪ ❬❉▲❱❪ ❬❝❧✐♥❣♦❪ ❬❉▲❱❪ P0 ✻✳✹✾ ✶✵✳✷✼ ✸✵✳✹✸ ✹✷✳✺✸ P1 ✹✳✵✵ ✻✳✷✼ ✷✶✳✷✷ ✸✵✳✶✷ P2 ✸✳✾✺ ✻✳✵✽ ✸✷✳✻✺ ✹✺✳✷✹ P3 ✸✳✾✽ ✻✳✶✸ ✷✵✳✾✹ ✸✵✳✸✸ P4 ✹✳✶✺ ✻✳✹✸ ✷✽✳✶✾ ✸✾✳✾✸ P5 ✼✳✾✼ ✶✷✳✻✻ ✷✶✳✺✹ ✸✵✳✽✼ P6 ✷✸✳✺✷ ✹✵✳✺✻ ✼✷✳✽✻ ✶✵✸✳✼✻ P7 ✸✻✳✸✸ ✻✹✳✵✺ ✶✶✺✳✵✸ ✶✻✷✳✷✶ P8 ✸✻✳✺✽ ✻✶✳✼✶ ✶✷✽✳✵✶ ✶✽✶✳✹✶ P9 ✸✺✳✷✻ ✻✷✳✶✾ ✶✵✽✳✸✽ ✶✹✺✳✵✹

❉❘❡❲ ♦✉t♣❡r❢♦r♠s ❚❉✲▼▲P ✐♥ ❛❧❧ t❡sts✱ ✇✐t❤✐♥ ❛ ❝♦♥st❛♥t ❢❛❝t♦r✳ ❉❘❡❲✬s ❧❡❛❞ s❤r✐♥❦s ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ❞❧✲❛t♦♠s ✐♥ P5✕P9✳ ■♥t✉✐t✐♦♥✿

• ❉❘❡❲ ❝r❡❛t❡s ❝♦♣✐❡s ♦❢ s✉❜♣r♦❣r❛♠s ❢♦r ❉▲✲❛t♦♠s ✉♣❢r♦♥t✱ ✇❤✐❧❡ ✐♥

❚❉✲▼▲P ❝♦♣✐❡s ❛r❡ ❝r❡❛t❡❞ ❜② t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥

• ❈✉rr❡♥t ❚❉✲▼▲P ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❤❛s ♦✈❡r❤❡❛❞ ❢♦r ✐♥st❛♥t✐❛t✐♥❣ ♠♦❞✉❧❡s

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

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✻✸✴✻✹

❯♥✐❢♦r♠ ❉▲✲Pr♦❣r❛♠ ❊✈❛❧✉❛t✐♦♥ ✼✳ ❆♣♣❡♥❞✐①✿ ▼♦❞✉❧❛r Datalog¬ ✼✳✷ ❊①♣❡r✐♠❡♥ts

❉✐s❝✉ss✐♦♥

❊①♣❡r✐♠❡♥t✿ t❤❡ ▼▲P ❡♥❝♦❞✐♥❣ ❧❛❣s ✐♥ t♦t❛❧ r✉♥t✐♠❡ ❜❡❤✐♥❞ t❤❡ ❛❞ ❤♦❝ ✐♥❧✐♥❡❞ ❛♣♣r♦❛❝❤✱ ❜✉t ✐t s❝❛❧❡s ❛t ❛ s❧♦✇❡r ❣r♦✇t❤ r❛t❡ ❋♦r ❧❛r❣❡ ❛♠♦✉♥ts ♦❢ ❞❛t❛✱ t❤❡ ❣❛♣ ♠❛② ❣❡t ❝❧♦s❡❞ ■ss✉❡✿ ❞❡s✐❣♥ ♦❢ ♠♦❞✉❧❡s ❢♦r ❞❧✲❛t♦♠s

• ❋❡✇✱ ❣❡♥❡r❛❧ ♠♦❞✉❧❡s ✭❛ ✉♥✐✈❡rs❛❧ ♦♥❡ ❢♦r ❛❧❧ ❞❧✲❛t♦♠s✱ ❝❢✳ ❉▲✲♣❧✉❣✐♥

✐♥ ❞❧✈❤❡①✮

• ▼❛♥② s♣❡❝✐❛❧✐③❡❞ ♠♦❞✉❧❡s ✭❡✳❣✳ ♦♥❡ ♠♦❞✉❧❡ ♣❡r ❞❧✲❛t♦♠✮

❚✳ ❊✐t❡r✴❚❯ ❲✐❡♥ ❊P❈▲ ✷✶✳✶✷✳✷✵✶✷ ✻✹✴✻✹