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❙▼❚✲❜❛s❡❞ ❇♦✉♥❞❡❞ ▼♦❞❡❧ ❈❤❡❝❦✐♥❣ ❢♦r P❛r❛♠❡tr✐❝ ❘❡❛❝t✐♦♥ ❙②st❡♠s ❲♦❥❝✐❡❝❤ P❡♥❝③❡❦ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❆rt✉r ▼➛s❦✐ ■♥st✐t✉t❡ ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ P♦❧✐s❤ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ❯♥✐✈❡rs✐t② ♦❢ ◆❛t✉r❛❧ ❙❝✐❡♥❝❡s ❛♥❞ ❍✉♠❛♥✐t✐❡s ❙②♥❈♦P✱ ❆♣r✐❧ ✼✱ ✷✵✶✾✱ Pr❛❣✉❡ ✶✴✸✸

❘❡❧❛t❡❞ ❲♦r❦ ✶✳ ▼♦❞❡❧ ❝❤❡❝❦✐♥❣ t❡♠♣♦r❛❧ ♣r♦♣❡rt✐❡s ♦❢ r❡❛❝t✐♦♥ s②st❡♠s ■♥❢♦r♠❛t✐♦♥ ❙❝✐❡♥❝❡s ✸✶✸✱ ✷✵✶✺❀ ❆✳ ▼➛s❦✐✱ ❲✳ P❡♥❝③❡❦✱ ●✳ ❘♦③❡♥❜❡r❣ ✷✳ ❈♦♠♣❧❡①✐t② ♦❢ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ❢♦r r❡❛❝t✐♦♥ s②st❡♠s ✱ ❚❈❙ ✻✷✸✱ ✷✵✶✻ ❙✳ ❆③✐♠✐✱ ❈✳ ●r❛t✐❡✱ ❙✳ ■✈❛♥♦✈✱ ▲✳ ▼❛♥③♦♥✐✱ ■✳ P❡tr❡✱ ❆✳ ❊✳ P♦rr❡❝❛ ✸✳ ❱❡r✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r✲❚✐♠❡ ❚❡♠♣♦r❛❧ Pr♦♣❡rt✐❡s ❢♦r ❘❡❛❝t✐♦♥ ❙②st❡♠s ✇✐t❤ ❉✐s❝r❡t❡ ❈♦♥❝❡♥tr❛t✐♦♥s ❋✉♥❞❛♠❡♥t❛ ■♥❢♦r♠❛t✐❝❛❡✱ ✷✵✶✼❀ ❆✳ ▼➛s❦✐✱ ▼✳ ❑♦✉t♥②✱ ❲✳ P❡♥❝③❡❦ ✹✳ ❘❡❛❝t✐♦♥ ▼✐♥✐♥❣ ❢♦r ❘❡❛❝t✐♦♥ ❙②st❡♠s ❯❈◆❈✱ ✷✵✶✽❀ ❆✳ ▼➛s❦✐✱ ▼✳ ❑♦✉t♥②✱ ❲✳ P❡♥❝③❡❦ ✷✴✸✸

❖✉t❧✐♥❡ ❘❡❛❝t✐♦♥ s②st❡♠s ▼♦❞❡❧ ❝❤❡❝❦✐♥❣ ❢♦r rs❈❚▲ ♦✈❡r ❘❙ ❘❡❛❝t✐♦♥ s②st❡♠s ✇✐t❤ ❞✐s❝r❡t❡ ❝♦♥❝❡♥tr❛t✐♦♥s P❛r❛♠❡tr✐❝ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ❢♦r rs▲❚▲ ♦✈❡r ❘❙❈ ❊①♣❡r✐♠❡♥t❛❧ ❡✈❛❧✉❛t✐♦♥ ✸✴✸✸

❘❡❛❝t✐♦♥ s②st❡♠s ❆ r❡❛❝t✐♦♥ s②st❡♠ ✐s ❛ ♣❛✐r rs = ( S , A ) ✱ ✇❤❡r❡✿ ◮ S ✕ ✜♥✐t❡ ❜❛❝❦❣r♦✉♥❞ s❡t ❡♥t✐t✐❡s✴♠♦❧❡❝✉❧❡s ◮ A ✕ s❡t ♦❢ r❡❛❝t✐♦♥s ♦✈❡r S ❊❛❝❤ r❡❛❝t✐♦♥ ✐♥ A ✐s ❛ tr✐♣❧❡ b = ( R , I , P ) s✉❝❤ t❤❛t R ✱ I ✱ P ❛r❡ ♥♦♥❡♠♣t② s✉❜s❡ts ♦❢ S ✇✐t❤ R ∩ I = ∅ ✳ ◮ R ✕ r❡❛❝t❛♥ts✱ R b ◮ I ✕ ✐♥❤✐❜✐t♦rs✱ I b ◮ P ✕ ♣r♦❞✉❝ts✱ P b ✹✴✸✸

❊①❛♠♣❧❡ ( S , A ) = ( { 1, 2, 3, 4 } , { a , b , c , d } ) = ( { 1, 4 } , { 2 } , { 1, 2 } ) = ( { 2 } , { 4 } , { 1, 3, 4 } ) a b c = ( { 1, 3 } , { 2 } , { 1, 2 } ) d = ( { 3 } , { 2 } , { 1 } ) ■♥ st❛t❡ { 1, 3, 4 } ✿ ◮ a ✱ c ✱ d ✕ ❡♥❛❜❧❡❞ r❡❛❝t✐♦♥s st❛rt { 1, 3, 4 } ■♥❞✐✈✐❞✉❛❧ r❡s✉❧ts ❢♦r t❤❡ r❡❛❝t✐♦♥s✿ ◮ a − → { 1, 2 } ◮ b − → ∅ ◮ c − → { 1, 2 } { 1, 2 } ◮ d − → { 1 } ❘❡s✉❧t st❛t❡✿ { 1, 2 } ✺✴✸✸

❊♥✈✐r♦♥♠❡♥t ◮ ❊①❡❝✉t✐♦♥ ♦❢ r❡❛❝t✐♦♥ s②st❡♠s ❞❡♣❡♥❞s ♦♥ t❤❡✐r ❡♥✈✐r♦♥♠❡♥t ◮ ❊♥✈✐r♦♥♠❡♥t ✐s ❞❡✜♥❡❞ ✐♥ r❡❛❝t✐♦♥ s②st❡♠s ❛s ❝♦♥t❡①t ◮ ❈♦♥t❡①t ✕ s❡q✉❡♥❝❡ ♦❢ s❡ts ♦❢ ❡♥t✐t✐❡s ◮ ❙✉♣♣❧✐❡❞ ❛t ❡❛❝❤ st❡♣ ♦❢ ❡①❡❝✉t✐♦♥ ◮ ❆✛❡❝ts r❡❛❝t✐♦♥s ❡♥❛❜❧❡♠❡♥t ✿ st❛t❡s ❛r❡ ❡①t❡♥❞❡❞ ✇✐t❤ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♥t❡①t ✻✴✸✸

❊①❛♠♣❧❡ ( S , A ) = ( { 1, 2, 3, 4 } , { a , b , c , d } ) ❡♥✈✐r♦♥♠❡♥t ✭❝♦♥t❡①t✮✿ 2 { 4 } = { ∅ , { 4 }} ✐♥✐t✐❛❧ st❛t❡ ✿ { 1, 2 } = ( { 1, 4 } , { 2 } , { 1, 2 } ) = ( { 2 } , { 4 } , { 1, 3, 4 } ) a b c = ( { 1, 3 } , { 2 } , { 1, 2 } ) d = ( { 3 } , { 2 } , { 1 } ) ∅ st❛rt { 1, 2 } { 1, 3, 4 } ∅ { 4 } { 4 } { 4 } ∅ ∅ ✼✴✸✸

▼♦❞❡❧ ❝❤❡❝❦✐♥❣ ❢♦r rs❈❚▲ ❬▼P❘✶✺❪ ■♥♣✉t✿ ◮ ■♥✐t✐❛❧✐s❡❞ ❝♦♥t❡①t r❡str✐❝t❡❞ r❡❛❝t✐♦♥ s②st❡♠✿ icrrs ◮ rs❈❚▲ ❢♦r♠✉❧❛ φ ✭rs❈❚▲ ✲ ❈❚▲ ✇✐t❤ ♣❛t❤ s❡❧❡❝t✐♦♥ ❜② r❡❢❡rr✐♥❣ t♦ ❝♦♥t❡①ts✮ ❉❡❝✐s✐♦♥ ♣r♦❜❧❡♠✿ ❄ M φ | = ♠♦❞❡❧ ❢♦r ✐❝rrs ❢♦r♠✉❧❛ ❚❤❡♦r❡♠✳ ❚❤❡ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ♣r♦❜❧❡♠ ❢♦r rs❈❚▲ ✐s P❙P❆❈❊✲❝♦♠♣❧❡t❡✳ ▼♦❞❡❧ ❝❤❡❝❦✐♥❣ ❛❧❣♦r✐t❤♠ ✐s ❜❛s❡❞ ♦♥ ❇❉❉s✳ ✽✴✸✸

❊①❛♠♣❧❡✳ ●❡♥❡ r❡❣✉❧❛t♦r② ♥❡t✇♦r❦ ❚❤r❡❡ ✭❛❜str❛❝t✮ ❣❡♥❡s x ✱ y ✱ z ❡①♣r❡ss✐♥❣ ♣r♦t❡✐♥s X ✱ Y ✱ Z ✱ r❡s♣❡❝t✐✈❡❧②✱ ♣r♦t❡✐♥ U ✱ ❛♥❞ ♣r♦t❡✐♥ ❝♦♠♣❧❡① Q ❢♦r♠❡❞ ❜② X ❛♥❞ U ✳ ❚❤❡ ❡①♣r❡ss✐♦♥ ♦❢ X ❜② x ✐s ✐♥❤✐❜✐t❡❞ ❜② Y ❛♥❞ Z ✱ t❤❡ ❡①♣r❡ss✐♦♥ ♦❢ Z ❜② z ✐s ✐♥❤✐❜✐t❡❞ ❜② X ✱ ❛♥❞ ❡①♣r❡ss✐♦♥ ♦❢ Y ❜② y ✐s ✐♥❤✐❜✐t❡❞ ❜② t❤❡ ♣r♦t❡✐♥ ❝♦♠♣❧❡① Q ✳ ✾✴✸✸

❊①❛♠♣❧❡✳ ●❡♥❡ r❡❣✉❧❛t♦r② ♥❡t✇♦r❦✿ ♣r♦♣❡rt✐❡s ✶✳ ■t ✐s ♣♦ss✐❜❧❡ t❤❛t t❤❡ ♣r♦t❡✐♥ Q ✇✐❧❧ ♥❡✈❡r ❜❡ ♣r♦❞✉❝❡❞✿ EG ( ¬ Q ) . ✷✳ ■❢ ✇❡ ❞♦ ♥♦t s✉♣♣❧② U ✐♥ t❤❡ ❝♦♥t❡①t✱ t❤❡♥ Q ✇✐❧❧ ♥❡✈❡r ❜❡ ♣r♦❞✉❝❡❞✿ A Ψ G ( ¬ Q ) , ✇❤❡r❡ Ψ = { α ⊆ E | U �∈ α } = { ∅ } . ✶✵✴✸✸

▲✐♥❡❛r✲❚✐♠❡ ❚❡♠♣♦r❛❧ Pr♦♣❡rt✐❡s ♦❢ ❘❙ ❱❡r✐✜❝❛t✐♦♥ ♦❢ ▲✐♥❡❛r✲❚✐♠❡ ❚❡♠♣♦r❛❧ Pr♦♣❡rt✐❡s ❢♦r ❘❡❛❝t✐♦♥ ❙②st❡♠s ✇✐t❤ ❉✐s❝r❡t❡ ❈♦♥❝❡♥tr❛t✐♦♥s ❋✉♥❞❛♠❡♥t❛ ■♥❢♦r♠❛t✐❝❛❡✱ ✷✵✶✼❀ ❆✳ ▼➛s❦✐✱ ▼✳ ❑♦✉t♥②✱ ❲✳ P❡♥❝③❡❦ ✶✶✴✸✸

▼✉❧t✐s❡ts ♦✈❡r S : B ( S ) ◮ s �→ i ✕ ♠✉❧t✐♣❧✐❝✐t② ♦❢ s ❡✳❣✳ { s �→ 2, x �→ 3, y } ▼✉❧t✐s❡t ❡①♣r❡ss✐♦♥s ✿ a ∈ BE ( S ) a ::= true | e ∼ c | e ∼ e | ¬ a | a ∨ a ✇❤❡r❡✿ ◮ ∼ ∈ { < , � , = , � , > } ◮ e ∈ S ◮ c ∈ I N ❚❤❡♥ b | = b a ♠❡❛♥s t❤❛t a ❤♦❧❞s ❢♦r b ∈ B ( S ) ✿ b | = b true ❢♦r ❡✈❡r② b ∈ B ( S ) b | = b e ∼ c ✐✛ b ( e ) ∼ c = b e ∼ e ′ b ( e ) ∼ b ( e ′ ) b | ✐✛ b | = b ¬ a ✐✛ b � | = b a = b a ∨ a ′ = b a ′ b | ✐✛ b | = b a ♦r b | ✶✷✴✸✸

❘❡❛❝t✐♦♥ s②st❡♠s ✇✐t❤ ❝♦♥❝❡♥tr❛t✐♦♥s✿ ❞❡✜♥✐t✐♦♥ rsc = ( S , A ) ✕ r❡❛❝t✐♦♥ s②st❡♠ ✇✐t❤ ✭❞✐s❝r❡t❡✮ ❝♦♥❝❡♥tr❛t✐♦♥s ✿ ◮ S ✕ ✜♥✐t❡ ❜❛❝❦❣r♦✉♥❞ s❡t ◮ A ✕ ♥♦♥❡♠♣t② ✜♥✐t❡ s❡t ♦❢ ❝✲r❡❛❝t✐♦♥s ♦✈❡r S B ( S ) ✕ s❡t ♦❢ ❛❧❧ ❜❛❣s ♦✈❡r S ❀ a = ( r , i , p ) ∈ A ✕ ❝✲r❡❛❝t✐♦♥ ◮ r , i , p ∈ B ( S ) ✇✐t❤ r ( e ) < i ( e ) ✱ ❢♦r ❡✈❡r② e ∈ carr ( i ) ✭ carr ( b ) = { s ∈ S | b ( s ) > 0 } ✮ ◮ r ✱ i ✱ p ✕ r❡❛❝t❛♥t✱ ✐♥❤✐❜✐t♦r✱ ❛♥❞ ♣r♦❞✉❝t ❝♦♥❝❡♥tr❛t✐♦♥ ❧❡✈❡❧s ◮ ❞❡♥♦t❡❞✿ r a ✱ i a ✱ ❛♥❞ p a ✶✸✴✸✸

❘❡❛❝t✐♦♥ s②st❡♠s ✇✐t❤ ❝♦♥❝❡♥tr❛t✐♦♥s✿ ❡♥❛❜❧❡♠❡♥t ❆ ❝✲r❡❛❝t✐♦♥ a ∈ A ✐s ❡♥❛❜❧❡❞ ❜② t ∈ B ( S ) ✱ ❞❡♥♦t❡❞ en a ( t ) ✱ ✐❢ r a � t ❛♥❞ t ( e ) < i a ( e ) ✱ ❢♦r ❡✈❡r② e ∈ carr ( i a ) res a ( t ) ✕ t❤❡ r❡s✉❧t ♦❢ a ♦♥ t ✿ ◮ res a ( t ) = p a ✐❢ en a ( t ) ◮ res a ( t ) = ∅ S ♦t❤❡r✇✐s❡ res A ( t ) = ✦ { res a ( t ) | a ∈ A } = ✦ { p a | a ∈ A ❛♥❞ en a ( t ) } ✳ ✦ ( B )( x ) = max ( { b ( x ) | b ∈ B } ) ❢♦r ♥♦♥✲❡♠♣t② B ⊆ B ( S ) ✱ b � b ′ ✐❢ b ( x ) � b ′ ( x ) ❢♦r ❡✈❡r② x ∈ X ✶✹✴✸✸