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❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s ❆①❡❧ P♦❧❧❡r❡s ❯♥✐✈❡rs✐❞❛❞ ❘❡② ❏✉❛♥ ❈❛r❧♦s✱ ▼❛❞r✐❞ ❏♦✐♥t ✇♦r❦ ✇✐t❤✿ ❏♦s ❞❡ ❇r✉✐❥♥ ✱ ❉❊❘■ ■♥♥s❜r✉❝❦✱ ❉❛✈✐❞ P❡❛r❝❡ ✱ ❯♥✐✈✳ ❘❡② ❏✉❛♥ ❈❛r❧♦s✱ ❆❣✉stí♥ ❱❛❧✈❡r❞❡ ✱ ❯♥✐✈✳ ❞❡ ▼á❧❛❣❛ ❘✉❧❡▼▲ ❲♦r❦s❤♦♣ ✷✵✵✻ ✶ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

❖✉t❧✐♥❡ ✷ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

▼♦t✐✈❛t✐♦♥ ✕ ❍②❜r✐❞ ❑♥♦✇❧❡❞❣❡ ❇❛s❡s ❈♦♠❜✐♥❡ r✉❧❡s ✇✐t❤ ♥❡❣❛t✐♦♥ ❛s ❢❛✐❧✉r❡ ✇✐t❤ ❝❧❛ss✐❝❛❧ t❤❡♦r✐❡s ✿ • ❍②r✐❞ ❑❇ ❛♣♣r♦❛❝❤❡s r❡❧② ♦♥ ✭✈❛r✐❛♥ts ♦❢✮ t❤❡ ❆♥s✇❡r ❙❡t ❙❡♠❛♥t✐❝s✳ ❬❘♦s❛t✐✱✷✵✵✺✴✷✵✵✺❜✴✷✵✵✻✱ ❍❡②♠❛♥s✱ ❡t ❛❧✳ ✷✵✵✻❪ • ❆❧❧ ❣✐✈❡ ❛ ♠♦❞✉❧❛r ❞❡✜♥✐t✐♦♥ ♦❢ ♠♦❞❡❧s ❜② ♣r♦❥❡❝t✐♦♥✰r❡❞✉❝t✳ • ❉r✐✈❡♥ ❜② ❞❡❝✐❞❛❜✐❧✐t② ❝♦♥❝❡r♥s✿ ❉❡✜♥❡❞ ❢♦r s②♥t❛❝t✐❝❛❧❧② ❧✐♠✐t❡❞ ♣r♦❣r❛♠s✴❋❖▲ t❤❡♦r✐❡s ❲❡ ♠✐❣❤t ❤❛✈❡ s♦♠❡ ♦t❤❡r ◗✉❡st✐♦♥s✿ ❄ • ❈❛♥ ✇❡ ❣❡♥❡r❛❧✐③❡ t❤❡s❡ ❝♦♠❜✐♥❛t✐♦♥s ✐♥ ❛ ✭♥♦♥✲❝❧❛ss✐❝❛❧✮ ❧♦❣✐❝✱ ✐✳❡✳ ✇✐t❤ ❛ ♥♦♥✲♠♦❞✉❧❛r ♠♦❞❡❧ ❞❡✜♥✐t✐♦♥❄ • ❉♦❡s t❤✐s ♣r♦✈✐❞❡ ✉s ✇✐t❤ ♥♦t✐♦♥s ♦❢ ❡q✉✐✈❛❧❡♥❝❡ ❝♦♠♠♦♥❧② ✉s❡❞ ✭str♦♥❣ ❡q✉✐✈❛❧❡♥❝❡✱ ✉♥✐❢♦r♠ ❡q✉✐✈❛❧❡♥❝❡✱ ❡t❝✳✮❄ ✸ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

❍②❜r✐❞ ❑♥♦✇❧❡❞❣❡ ❇❛s❡s ✕ ●❡♥❡r❛❧✐③❡❞ ❉❡✜♥✐t✐♦♥ K = ( T , P ) ❤②❜r✐❞ ❦♥♦✇❧❡❞❣❡ ❜❛s❡ ✿ • ❝❧❛ss✐❝❛❧ ✜rst✲♦r❞❡r t❤❡♦r② T ♦✈❡r ❢✉♥❝t✐♦♥✲❢r❡❡ ❧❛♥❣✉❛❣❡ L T = � C, P T � • ❛ ❧♦❣✐❝ ♣r♦❣r❛♠ P ♦✈❡r ❢✉♥❝t✐♦♥✲❢r❡❡ ❧❛♥❣✉❛❣❡ L = � C, P T ∪ P P � ✱ ✐✳❡✳ ❛ s❡t ♦❢ r✉❧❡s✿ a 1 ∨ a 2 ∨ . . . ∨ a k ∨¬ a k +1 ∨ . . . ∨¬ a l ← b 1 , . . . , b m , ¬ b m +1 , . . . , ¬ b n ✇❤❡r❡ P T ∩ P P = ∅ ◆♦t❡✿ • T ❛♥❞ P t❛❧❦ ❛❜♦✉t t❤❡ s❛♠❡ ❝♦♥st❛♥ts✱ ❛♥❞ • ❛❧❧♦✇❡❞ ♣r❡❞✐❝❛t❡ s②♠❜♦❧s ✐♥ P ❛r❡ ❛ s✉♣❡rs❡t ♦❢ t❤❡ ♣r❡❞✐❝❛t❡ s②♠❜♦❧s ✐♥ L T ✳ ✹ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

❍②❜r✐❞ ❑♥♦✇❧❡❞❣❡ ❇❛s❡s ✕ Pr♦❥❡❝t✐♦♥✿ ❖✈❡r❛❧❧ ✐❞❡❛ ❢♦r ❛ ♥♦♥♠♦♥♦t♦♥✐❝ s❡♠❛♥t✐❝s✿ ✏❡✈❛❧✉❛t❡✑ P ✇rt ❛ ❝❧❛ss✐❝❛❧ ♠♦❞❡❧ ♦❢ t❤❡ t❤❡♦r② ❛♥❞ t❤❡♥ ❝♦♠♣✉t❡ st❛❜❧❡ ♠♦❞❡❧s✳ ▲❡t P ❜❡ ❛ ❣r♦✉♥❞ ♣r♦❣r❛♠ ❛♥ I = � U, I � ❛♥ L ✲str✉❝t✉r❡✱ ✇✐t❤ U = ( D, σ ) ✳ Π( P , I ) ✱ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ P ✇rt I ✱ ♦❜t❛✐♥❡❞ ❜② ✶ ❞❡❧❡t✐♥❣ ❡❛❝❤ r✉❧❡ ✇✐t❤ ❤❡❛❞ ❧✐t❡r❛❧ p ( t ) ✭♦r ¬ p ( t ) ✮ ♦✈❡r At D ( C, P T ) s✉❝❤ t❤❛t p ( σ ( t )) ∈ I ✭♦r p ( σ ( t )) �∈ I ✮ ✷ ❞❡❧❡t✐♥❣ ❡❛❝❤ r✉❧❡ ✇✐t❤ ❜♦❞② ❧✐t❡r❛❧ p ( t ) ✭♦r ¬ p ( t ) ✮ ♦✈❡r At D ( C, P T ) s✉❝❤ t❤❛t p ( σ ( t )) �∈ I ✭♦r p ( σ ( t )) ∈ I ✮❀ ❛♥❞ ❞❡❧❡t✐♥❣ ♦❝❝✉rr❡♥❝❡s ♦❢ ❧✐t❡r❛❧s ❢r♦♠ L T ❢r♦♠ r❡♠❛✐♥✐♥❣ r✉❧❡s✳ ✺ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

❍②❜r✐❞ ❑♥♦✇❧❡❞❣❡ ❇❛s❡s ✕ ◆▼✲♠♦❞❡❧✿ ❖✈❡r❛❧❧ ✐❞❡❛ ❢♦r ❛ ♥♦♥♠♦♥♦t♦♥✐❝ s❡♠❛♥t✐❝s✿ ✏❡✈❛❧✉❛t❡✑ P ✇rt ❛ ❝❧❛ss✐❝❛❧ ♠♦❞❡❧ ♦❢ t❤❡ t❤❡♦r② ❛♥❞ t❤❡♥ ❝♦♠♣✉t❡ st❛❜❧❡ ♠♦❞❡❧s✳ ▲❡t K = ( T , P ) ❜❡ ❛ ❤②❜r✐❞ ❦♥♦✇❧❡❞❣❡ ❜❛s❡✳ ❆♥ ◆▼✲♠♦❞❡❧ M = � U, I � ♦❢ ❛ ❤②❜r✐❞ ❦♥♦✇❧❡❞❣❡ ❜❛s❡ K ✐s ❛ ✜rst✲♦r❞❡r L ✲str✉❝t✉r❡ s✉❝❤ t❤❛t ✶ M| L T ✐s ❛ ♠♦❞❡❧ ♦❢ T ❛♥❞ ✷ M| L P ✐s ❛ st❛❜❧❡ ♠♦❞❡❧ s❡t ♦❢ Π( gr U ( P ) , M| L T ) ✱ ✐✳❡✳ M| L P ✐s ❛ ♠✐♥✐♠❛❧ ❍❡r❜r❛♥❞ ▼♦❞❡❧ ♦❢ t❤❡ r❡❞✉❝t Π( gr U ( P ) , M ) M| LP ✱ ♦❜t❛✐♥❡❞ ❜② t❛❦✐♥❣ ❛❧❧ r✉❧❡s✿ • s✉❝❤ t❤❛t M| L P | = a i ❢♦r ♥❡❣❛t✐✈❡ ❤❡❛❞ ❛t♦♠s a i ❛♥❞ • I �| = b j ❢♦r ❛❧❧ ♥❡❣❛t✐✈❡ ❜♦❞② ❛t♦♠s b j ✳ ✻ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

❊①❛♠♣❧❡ ✕ ❛ s♠❛❧❧ ❍②❜r✐❞ ❑❇✿ ▲❡t K = ( T , P ) ✇✐t❤ T ✿ ❊❛❝❤ ❢♦❛❢✿P❡rs♦♥ ✐s ❛ ❢♦❛❢✿❆❣❡♥t✿ ∀ x.PERSON ( x ) → AGENT ( x ) AGENT ( David ) P ✿ ❙♦♠❡ ♥♦♥♠♦♥♦t♦♥✐❝ r✉❧❡ ♦♥ t♦♣ PERSON ( x ) ← pcmember ( x, LPNMR ) , AGENT ( x ) , ¬ machine ( x ) pcmember ( David, LPNMR ) ■s ❉❛✈✐❞ ❛ P❊❘❙❖◆❄ ✼ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

❊①❛♠♣❧❡ ✲ ◆▼✲♠♦❞❡❧s✿ ❈❧❛ss✐❝❛❧ ♠♦❞❡❧s ♦❢ T ✿ ∀ x.PERSON ( x ) → AGENT ( x ) AGENT ( David ) M 1 | L T = { AGENT ( David ) , ¬ PERSON ( David ) , ¬ AGENT ( LPNMR ) , . . . } M 2 | L T = { AGENT ( David ) , PERSON ( David ) , ¬ AGENT ( LPNMR ) , . . . } gr U ( P ) PERSON ( David ) ← pcmember ( David, LPNMR ) , AGENT ( David ) , ¬ machine ( David ) PERSON ( LPNMR ) ← pcmember ( LPNMR, LPNMR ) , AGENT ( LPNMR ) , ¬ machine ( LPNMR ) pcmember ( David, LPNMR ) ■s ❉❛✈✐❞ ❛ P❊❘❙❖◆❄ ✽ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

❊①❛♠♣❧❡ ✲ ◆▼✲♠♦❞❡❧s✿ ❈❧❛ss✐❝❛❧ ♠♦❞❡❧s ♦❢ T ✿ ∀ x.PERSON ( x ) → AGENT ( x ) AGENT ( David ) M 1 | L T = { AGENT ( David ) , ¬ PERSON ( David ) , ¬ AGENT ( LPNMR ) , . . . } M 2 | L T = { AGENT ( David ) , PERSON ( David ) , ¬ AGENT ( LPNMR ) , . . . } Π( gr U ( P ) , M 1 | L T ) ← pcmember ( David, LPNMR ) , ¬ machine ( David ) . pcmember ( David, LPNMR ) ◆♦ st❛❜❧❡ ♠♦❞❡❧s✦ ■s ❉❛✈✐❞ ❛ P❊❘❙❖◆❄ ✾ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

❊①❛♠♣❧❡ ✲ ◆▼✲♠♦❞❡❧s✿ ❈❧❛ss✐❝❛❧ ♠♦❞❡❧s ♦❢ T ✿ ∀ x.PERSON ( x ) → AGENT ( x ) AGENT ( David ) M 1 | L T = { AGENT ( David ) , ¬ PERSON ( David ) , ¬ AGENT ( LPNMR ) , . . . } M 2 | L T = { AGENT ( David ) , PERSON ( David ) , ¬ AGENT ( LPNMR ) , . . . } Π( gr U ( P ) , M 2 | L T ) pcmember ( David, LPNMR ) ❖♥❡ st❛❜❧❡ ♠♦❞❡❧✳✳✳ ■s ❉❛✈✐❞ ❛ P❊❘❙❖◆❄ ❨❡s✦ PERSON ( David ) ✐♥ ❛❧❧ ◆▼✲♠♦❞❡❧s✱ ✐✳❡✳ K | = NM PERSON ( David ) ✶✵ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s

❊q✉✐❧✐❜r✐✉♠ ▲♦❣✐❝ • ❊q✉✐❧✐❜r✐✉♠ ❧♦❣✐❝ ✭P❡❛r❝❡✱ ✶✾✾✼✮ ❣❡♥❡r❛❧✐s❡s st❛❜❧❡ ♠♦❞❡❧ s❡♠❛♥t✐❝s ❛♥❞ ❛♥s✇❡r s❡t s❡♠❛♥t✐❝s ❢♦r ❧♦❣✐❝ ♣r♦❣r❛♠s t♦ ❛r❜✐tr❛r② ♣r♦♣♦s✐t✐♦♥❛❧ t❤❡♦r✐❡s✳ • ■t ✐s ❛ ♥♦♥♠♦♥♦t♦♥✐❝ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❧♦❣✐❝ ♦❢ ❍❡r❡✲❛♥❞✲t❤❡r❡ ✇✐t❤ str♦♥❣ ♥❡❣❛t✐♦♥✳ • ▼♦❞❡❧ t❤❡♦r② ❜❛s❡❞ ♦♥ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❢♦r ✐♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝ • ❲❡ ♥❡❡❞ ❛ ✜rst✲♦r❞❡r ✈❡rst✐♦♥ ❤❡r❡✳✳✳ ✶✶ ❆①❡❧ P♦❧❧❡r❡s ❆ ▲♦❣✐❝ ❢♦r ❍②❜r✐❞ ❘✉❧❡s