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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✱ Rb ◮ I ✕ ✐♥❤✐❜✐t♦rs✱ Ib ◮ P ✕ ♣r♦❞✉❝ts✱ Pb

✹✴✸✸

❊①❛♠♣❧❡

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

✺✴✸✸

❊♥✈✐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}) ✐♥✐t✐❛❧ st❛t❡✿ {1, 2} ❡♥✈✐r♦♥♠❡♥t ✭❝♦♥t❡①t✮✿ 2{4} = {∅, {4}} a = ({1, 4}, {2}, {1, 2}) b = ({2}, {4}, {1, 3, 4}) c = ({1, 3}, {2}, {1, 2}) d = ({3}, {2}, {1}) {1, 2} st❛rt {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♦❞✉❝❡❞✿ AΨG(¬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 | =b e ∼ e′ ✐✛ b(e) ∼ b(e′) b | =b ¬a ✐✛ b | =b a b | =b a ∨ a′ ✐✛ b | =b a ♦r b | =b a′

✶✷✴✸✸

❘❡❛❝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❡❞✿ ra✱ ia✱ ❛♥❞ pa

✶✸✴✸✸

❘❡❛❝t✐♦♥ s②st❡♠s ✇✐t❤ ❝♦♥❝❡♥tr❛t✐♦♥s✿ ❡♥❛❜❧❡♠❡♥t

❆ ❝✲r❡❛❝t✐♦♥ a ∈ A ✐s ❡♥❛❜❧❡❞ ❜② t ∈ B(S)✱ ❞❡♥♦t❡❞ ena(t)✱ ✐❢ ra t ❛♥❞ t(e) < ia(e)✱ ❢♦r ❡✈❡r② e ∈ carr(ia) resa(t) ✕ t❤❡ r❡s✉❧t ♦❢ a ♦♥ t✿ ◮ resa(t) = pa ✐❢ ena(t) ◮ resa(t) = ∅S ♦t❤❡r✇✐s❡ resA(t) = ✦{resa(t) | a ∈ A} = ✦{pa | a ∈ A ❛♥❞ ena(t)}✳ ✦(B)(x) = max({b(x) | b ∈ B}) ❢♦r ♥♦♥✲❡♠♣t② B ⊆ B(S)✱ b b′ ✐❢ b(x) b′(x) ❢♦r ❡✈❡r② x ∈ X

✶✹✴✸✸

❈♦♥t❡①t✲r❡str✐❝t❡❞ rs❝

❈♦♥t❡①t ❛✉t♦♠❛t♦♥ ♦✈❡r t❤❡ s❡t B(S)✿ ca = (Q, q0, R)✱ ✇❤❡r❡✿ ◮ Q ✕ ✜♥✐t❡ s❡t ♦❢ st❛t❡s ◮ q0 ∈ Q ✕ t❤❡ ✐♥✐t✐❛❧ st❛t❡ ◮ R ⊆ Q × B(S) × Q ✕ tr❛♥s✐t✐♦♥ r❡❧❛t✐♦♥ crrsc = (rsc, ca) ✖ ❝♦♥t❡①t✲r❡str✐❝t❡❞ rs❝✿ ◮ rsc = (S, A) ✕ r❡❛❝t✐♦♥ s②st❡♠ ✇✐t❤ ❞✐s❝r❡t❡ ❝♦♥❝❡♥tr❛t✐♦♥s ◮ ca = (Q, q0, R) ✕ ❝♦♥t❡①t ❛✉t♦♠❛t♦♥ ♦✈❡r B(S)

✶✺✴✸✸

■♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss❡s ♦❢ ❝rrs❝

π = (ζ, γ, δ) ✕ ✭n✲st❡♣✮ ✐♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss ✐♥ crrsc i = 0 i = 1 i = 2 i = n − 1 i = n z0 z1 z2 · · · zn−1 zn c0 c1 c2 · · · cn−1 cn d0 d1 d2 · · · dn−1 dn ζ = (z0, z1, . . . , zn) z0, z1, . . . , zn ∈ Q ✇✐t❤ z0 = q0

✶✻✴✸✸

■♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss❡s ♦❢ ❝rrs❝

π = (ζ, γ, δ) ✕ ✭n✲st❡♣✮ ✐♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss ✐♥ crrsc i = 0 i = 1 i = 2 i = n − 1 i = n z0 z1 z2 · · · zn−1 zn c0 c1 c2 · · · cn−1 cn d0 d1 d2 · · · dn−1 dn γ = (c0, c1, . . . , cn) c0, c1, . . . , cn ∈ B(S)

✶✻✴✸✸

■♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss❡s ♦❢ ❝rrs❝

π = (ζ, γ, δ) ✕ ✭n✲st❡♣✮ ✐♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss ✐♥ crrsc i = 0 i = 1 i = 2 i = n − 1 i = n z0 z1 z2 · · · zn−1 zn c0 c1 c2 · · · cn−1 cn d0 d1 d2 · · · dn−1 dn δ = (d0, d1, . . . , dn) d0, d1, . . . , dn ∈ B(S)

✶✻✴✸✸

■♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss❡s ♦❢ ❝rrs❝

π = (ζ, γ, δ) ✕ ✭n✲st❡♣✮ ✐♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss ✐♥ crrsc i = 0 i = 1 i = 2 i = n − 1 i = n z0 z1 z2 · · · zn−1 zn c0 c1 c2 · · · cn−1 cn d0 d1 d2 · · · dn−1 dn c0 c1 cn−1 (zi, ci, zi+1) ∈ R✱ ❢♦r ❡✈❡r② i ∈ {0, . . . , n − 1}

✶✻✴✸✸

■♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss❡s ♦❢ ❝rrs❝

π = (ζ, γ, δ) ✕ ✭n✲st❡♣✮ ✐♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss ✐♥ crrsc i = 0 i = 1 i = 2 i = n − 1 i = n z0 z1 z2 · · · zn−1 zn c0 c1 c2 · · · cn−1 cn d0 d1 d2 · · · dn−1 dn c0 c1 cn−1 d0 = ∅B(S)✱ di = resA(✦{di−1, ci−1})✱ ❢♦r ❡✈❡r② i ∈ {1, . . . , n}

✶✻✴✸✸

■♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss❡s ♦❢ ❝rrs❝

π = (ζ, γ, δ) ✕ ✭n✲st❡♣✮ ✐♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss ✐♥ crrsc i = 0 i = 1 i = 2 i = n − 1 i = n z0 z1 z2 · · · zn−1 zn c0 c1 c2 · · · cn−1 cn d0 d1 d2 · · · dn−1 dn c0 c1 cn−1 st❛t❡ s❡q✉❡♥❝❡ ♦❢ π✿ (w0, . . . , wn) = (✦{c0, d0}, . . . , ✦{cn, dn})

✶✻✴✸✸

▲❚▲ ❢♦r ❘❙ ✕ rs▲❚▲

❚❤❡ s②♥t❛① ♦❢ rs▲❚▲ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♠♠❛r✿ φ ::= a | φ ∧ φ | φ ∨ φ | Xaφ | φUaφ | φRaφ ✇❤❡r❡ a ∈ BE(S) σ | = Xaφ φ a

✶✼✴✸✸

▲❚▲ ❢♦r ❘❙ ✕ rs▲❚▲

❚❤❡ s②♥t❛① ♦❢ rs▲❚▲ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♠♠❛r✿ φ ::= a | φ ∧ φ | φ ∨ φ | Xaφ | φUaφ | φRaφ ✇❤❡r❡ a ∈ BE(S) σ | = φUaψ φ φ ψ a a

✶✼✴✸✸

▲❚▲ ❢♦r ❘❙ ✕ rs▲❚▲

❚❤❡ s②♥t❛① ♦❢ rs▲❚▲ ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♠♠❛r✿ φ ::= a | φ ∧ φ | φ ∨ φ | Xaφ | φUaφ | φRaφ ✇❤❡r❡ a ∈ BE(S) σ | = φRaψ ψ ψ ψ φ a a

✶✼✴✸✸

❘❡❛❝t✐♦♥ ▼✐♥✐♥❣

◮ ❘❡❛❝t✐♦♥s ♠❛② ❜❡ ❞❡✜♥❡❞ ♣❛rt✐❛❧❧② ❡✳❣✳✱ ♠✐ss✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✐♥❤✐❜✐t♦rs✱ r❡❛❝t❛♥ts✱ ❡t❝✳

P❛rt✐❛❧❧② ❞❡✜♥❡❞ r❡❛❝t✐♦♥ s②st❡♠s✿ ♣❛r❛♠❡tr✐❝ r❡❛❝t✐♦♥ s②st❡♠s

◮ ❋r♦♠ ❡①♣❡r✐♠❡♥ts ✇❡ ♦❜t❛✐♥ ♦❜s❡r✈❛t✐♦♥s ✇❤✐❝❤ ❤❡❧♣ ✜❧❧ ✐♥ t❤❡ ♠✐ss✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t r❡❛❝t✐♦♥s

rs▲❚▲ ✐s ✉s❡❞ t♦ ❡①♣r❡ss t❤❡s❡ ♦❜s❡r✈❛t✐♦♥s

◮ ❆ss✉♠♣t✐♦♥✿ ❡①♣❡r✐♠❡♥ts r❡s✉❧t ✐♥ ❡①✐st❡♥t✐❛❧ ♦❜s❡r✈❛t✐♦♥s

rs▲❚▲ ✐♥t❡r♣r❡t❡❞ ❡①✐st❡♥t✐❛❧❧②

✶✽✴✸✸

❘❡❛❝t✐♦♥ ▼✐♥✐♥❣

◮ ❘❡❛❝t✐♦♥s ♠❛② ❜❡ ❞❡✜♥❡❞ ♣❛rt✐❛❧❧② ❡✳❣✳✱ ♠✐ss✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✐♥❤✐❜✐t♦rs✱ r❡❛❝t❛♥ts✱ ❡t❝✳

◮ P❛rt✐❛❧❧② ❞❡✜♥❡❞ r❡❛❝t✐♦♥ s②st❡♠s✿ ♣❛r❛♠❡tr✐❝ r❡❛❝t✐♦♥ s②st❡♠s

◮ ❋r♦♠ ❡①♣❡r✐♠❡♥ts ✇❡ ♦❜t❛✐♥ ♦❜s❡r✈❛t✐♦♥s ✇❤✐❝❤ ❤❡❧♣ ✜❧❧ ✐♥ t❤❡ ♠✐ss✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t r❡❛❝t✐♦♥s

◮ rs▲❚▲ ✐s ✉s❡❞ t♦ ❡①♣r❡ss t❤❡s❡ ♦❜s❡r✈❛t✐♦♥s

◮ ❆ss✉♠♣t✐♦♥✿ ❡①♣❡r✐♠❡♥ts r❡s✉❧t ✐♥ ❡①✐st❡♥t✐❛❧ ♦❜s❡r✈❛t✐♦♥s

◮ rs▲❚▲ ✐♥t❡r♣r❡t❡❞ ❡①✐st❡♥t✐❛❧❧②

✶✽✴✸✸

P❛r❛♠❡tr✐❝ r❡❛❝t✐♦♥ s②st❡♠s ✭✇✐t❤ ❞✐s❝r❡t❡ ❝♦♥❝❡♥tr❛t✐♦♥s✮

P❛r❛♠❡tr✐❝ r❡❛❝t✐♦♥ s②st❡♠✿ prs = (S, P, A)✱ ✇❤❡r❡✿ ◮ S ✕ ❜❛❝❦❣r♦✉♥❞ s❡t ◮ P ✕ s❡t ♦❢ ♣❛r❛♠❡t❡rs ◮ A ✕ s❡t ♦❢ ♣❛r❛♠❡tr✐❝ r❡❛❝t✐♦♥s✱ A = ∅ S✱ P✱ A ❛r❡ ✜♥✐t❡ ▲❡t a = (r, i, p) ∈ A✿ r, i, p ∈ B(S) ∪ P ◮ r✱ i✱ p ✖ ❞❡♥♦t❡❞ ❜② ra✱ ia✱ ❛♥❞ pa ◮ r❡❛❝t❛♥ts✱ ✐♥❤✐❜✐t♦rs✱ ❛♥❞ ♣r♦❞✉❝ts ♦❢ ♣❛r❛♠❡tr✐❝ r❡❛❝t✐♦♥ a ❊①❛♠♣❧❡✿ ▲❡t λ1, λ2 ∈ P ◮ P❛r❛♠❡tr✐❝ r❡❛❝t✐♦♥s✿ ({x, y}, λ1, {z})✱ (λ1, {x}, λ2)

✶✾✴✸✸

P❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥s

P❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ ♦❢ prs✿ v : P → B(S) ◮ ✇❡ ✇r✐t❡ b←v ❢♦r v(b) ◮ PVprs ✕ ❛❧❧ t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥s ❢♦r prs P❛r❛♠❡t❡r s✉❜st✐t✉t✐♦♥s ◮ P❛r❛♠❡t❡rs ❛r❡ s✉❜st✐t✉t❡❞ ❛❝❝♦r❞✐♥❣ t♦ v ∈ PVprs ◮ X←v def = {(a←v

r

, a←v

i

, a←v

p

) | a ∈ X} ❢♦r X ⊆ A ◮ prs←v def = (S, A←v) v ∈ PVprs ✐s ❛ ✈❛❧✐❞ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ ✐❢ prs←v ②✐❡❧❞s ❛♥ rs❝

✷✵✴✸✸

❈♦♥t❡①t✲r❡str✐❝t❡❞ P❘❙

❈♦♥t❡①t✲r❡str✐❝t❡❞ ♣❛r❛♠❡tr✐❝ r❡❛❝t✐♦♥ s②st❡♠ ✭❝r♣rs✮✿ crprs = (prs, ca) ✇❤❡r❡✿ ◮ prs = (S, P, A) ◮ ca = (Q, q0, R) ✕ ❝♦♥t❡①t ❛✉t♦♠❛t♦♥ ♦✈❡r B(S) ❋♦r v ∈ PVprs ✇❡ ❞❡✜♥❡✿ crprs←v = (prs←v, ca)

✷✶✴✸✸

P❛r❛♠❡t❡r ❝♦♥str❛✐♥ts

c ∈ PC(prs)✿ c ::= true | λ[e] ∼ c | λ[e] ∼ λ[e] | ¬c | c ∨ c, ✇❤❡r❡✿ λ ∈ P e ∈ S c ∈ I N ∼ ∈ {<, , =, , >} ▲❡t v ∈ PVprs ◮ c ❤♦❧❞s ✐♥ v ✐s ❞❡♥♦t❡❞ v | =p c✿ v | =p true ❢♦r ❡✈❡r② v v | =p λ[e] ∼ c ✐❢ λ←v(e) ∼ c v | =p λ1[e1] ∼ λ2[e2] ✐❢ λ←v

1

(e1) ∼ λ←v

2

(e2) v | =p ¬c ✐❢ v | =p c v | =p c1 ∨ c2 ✐❢ v | =p c1 ♦r v | =p c2

✷✷✴✸✸

❈♦♥str❛✐♥❡❞ P❘❙

❈♦♥str❛✐♥❡❞ ♣❛r❛♠❡tr✐❝ r❡❛❝t✐♦♥ s②st❡♠✿ cprs = (S, P, A, c) ✇❤❡r❡✿ ◮ prs = (S, P, A) ◮ c ∈ PC(prs) ❈♦♥t❡①t✲r❡str✐❝t❡❞ ❝♣rs✿ cr✲cprs = (cprs, ca) ✇❤❡r❡✿ ◮ cprs = (S, P, A, c) ◮ ca = (Q, q0, R) ✕ ❝♦♥t❡①t ❛✉t♦♠❛t♦♥ ♦✈❡r B(S) cr✲cprs←v = (cprs←v, ca)

✷✸✴✸✸

P❛r❛♠❡t❡r s②♥t❤❡s✐s

◮ cr✲cprs = (cprs, ca) ◮ F = {φ1, . . . , φn} ✕ rs▲❚▲ ❢♦r♠✉❧❛❡ ◮ c ✕ ♣❛r❛♠❡t❡r ❝♦♥str❛✐♥t ❈❛❧❝✉❧❛t❡ ❛ ✈❛❧✐❞ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ v ♦❢ cr✲cprs s✉❝❤ t❤❛t✿ (M(cr✲cprs←v) | =∃ φ1) ∧ · · · ∧ (M(cr✲cprs←v) | =∃ φn) ❚❤❡♦r❡♠✳ ❚❤❡ ♣r♦❜❧❡♠ ✇❤❡t❤❡r t❤❡r❡ ✐s ❛ ✈❛❧✐❞ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ ✐s P❙P❆❈❊✲❝♦♠♣❧❡t❡✳ ■♥❝r❡♠❡♥t❛❧ ❛♣♣r♦❛❝❤✿ ❑❡❡♣ ✐♥❝r❡❛s✐♥❣ k 0 ✉♥t✐❧ ❛ ✈❛❧✐❞ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ ✐s ❢♦✉♥❞✿ (M(cr✲cprs←v) | =k

∃ φ1) ∧ · · · ∧ (M(cr✲cprs←v) |

=k

∃ φn)

✷✹✴✸✸

❊♥❝♦❞✐♥❣ ♦❢ ♣❛r❛♠❡t❡r s②♥t❤❡s✐s ✐♥t♦ ❙▼❚

fps =  

φf∈F

Pathsk

f ∧ Loopsk f ∧ |

[φf] |k   ∧ PC(ppar) ✶✳ ❚❡st s❛t✐s✜❛❜✐❧✐t② ♦❢ fps ✷✳ ❲❤❡♥ fps ✐s ❙❆❚ → ❡①tr❛❝t ✈❛❧✉❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ✸✳ ❲❤❡♥ fps ✐s ❯◆❙❆❚ → ♥♦ ✈❛❧✐❞ ✈❛❧✉❛t✐♦♥ ❡①✐sts

✷✺✴✸✸

❊①♣❡r✐♠❡♥t❛❧ ❡✈❛❧✉❛t✐♦♥

◮ ■♥❝r❡♠❡♥t❛❧ ❛♣♣r♦❛❝❤✿ ✉♥r♦❧❧✐♥❣ ♦❢ ✐♥t❡r❛❝t✐✈❡ ♣r♦❝❡ss❡s ◮ ❚✇♦ ✐♠♣❧❡♠❡♥t❛t✐♦♥s✿

◮ P❛r❛♠❡tr✐❝✿

◮ ✇✐t❤ ❙▼❚ ❡♥❝♦❞✐♥❣ ❛❧❧♦✇✐♥❣ ❢♦r ♣❛r❛♠❡t❡r s②♥t❤❡s✐s

◮ ◆♦♥✲♣❛r❛♠❡tr✐❝ ✕ ✉s✐♥❣ ❞✐✛❡r❡♥t ❙▼❚ ❡♥❝♦❞✐♥❣

✭♦♣t✐♠✐s❡❞ ❢♦r ♥♦♥✲♣❛r❛♠❡tr✐❝ ✈❡r✐✜❝❛t✐♦♥✮

◮ ❯s✐♥❣ P②t❤♦♥ ❛♥❞ ❩✸ ❙▼❚✲s♦❧✈❡r ✭✹✳✺✳✵✮

✷✻✴✸✸

▼✉t❡①

◮ n 2 ♣r♦❝❡ss❡s ◮ ❝♦♠♣❡t✐♥❣ ❢♦r ❡①❝❧✉s✐✈❡ ❛❝❝❡ss t♦ ❝r✐t✐❝❛❧ s❡❝t✐♦♥ ◮ ❇❛❝❦❣r♦✉♥❞ s❡t✿ S = n

i=1 Si✿

◮ i✲t❤ ♣r♦❝❡ss✿ Si = {outi, reqi, ini, acti, lock, done, s} ◮ lock✱ done✱ s ✕ s❤❛r❡❞ ❛♠♦♥❣st ❛❧❧ t❤❡ ♣r♦❝❡ss❡s

◮ ❘❡❛❝t✐♦♥s✿ A = n

i=1 Ai ∪

• ({lock}, {done}, {lock})
• ◮ Ai ✐s t❤❡ s❡t ♦❢ r❡❛❝t✐♦♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ i✲t❤ ♣r♦❝❡ss

◮ ❈♦♥t❡①t ❛✉t♦♠❛t♦♥ ♣r♦✈✐❞❡s✿

◮ t❤❡ ✐♥✐t✐❛❧ ❝♦♥t❡①t s❡t ◮ ❝♦♥t❡①t s❡ts ✕ ❛t ♠♦st t✇♦ ❛❝t✐✈❡ ♣r♦❝❡ss❡s ❛❧❧♦✇❡❞

✷✼✴✸✸

▼✉t❡①

◮ ❆ss✉♠♣t✐♦♥✿ ♦♣❡♥ s②st❡♠ ◮ n✲t❤ ♣r♦❝❡ss✿ ❛❞❞✐t✐♦♥❛❧ ✭♠❛❧✐❝✐♦✉s✮ r❡❛❝t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡rs✿ P = {λr, λi, λp} cr✲cprsM = ((S, P, A ∪ {(λr, λi, λp)}, c), ca) ◮ c = (λp[inn] = 0) ∧

λ∈P,e∈S\Sn(λ[e] = 0) ✕ ❛❞❞✐t✐♦♥❛❧

r❡❛❝t✐♦♥✿

◮ ♣r♦❞✉❝❡s ♦♥❧② ❡♥t✐t✐❡s r❡❧❛t❡❞ t♦ t❤❡ n✲t❤ ♣r♦❝❡ss ◮ ❝❛♥♥♦t ♣r♦❞✉❝❡ inn ✭t♦ ❛✈♦✐❞ tr✐✈✐❛❧ s♦❧✉t✐♦♥s✮

❙②♥t❤❡s✐s✿ ♣❛r❛♠❡t❡r ✈❛❧✉❛t✐♦♥ v ♦❢ cr✲cprsM✿ ◮ φ = F(in1 ∧ inn) ✕ ✈✐♦❧❛t✐♦♥ ♦❢ ♠✉t✉❛❧ ❡①❝❧✉s✐♦♥ M(cr✲cprs←v

M ) |

=∃ φ

✷✽✴✸✸

❘❡s✉❧ts✿ t✐♠❡

10 20 30 40 50 200 400 600 800 1,000 n t✐♠❡ ✭✐♥ s❡❝♦♥❞s✮ cr✲cprs crrscnp

λ←v

r

= {outn}✱ λ←v

i

= {s}✱ ❛♥❞ λ←v

p

= {reqn, done}

✷✾✴✸✸

❘❡s✉❧ts✿ t✐♠❡

10 20 30 40 50 200 400 600 800 1,000 n t✐♠❡ ✭✐♥ s❡❝♦♥❞s✮ cr✲cprs crrscnp

cr✲cprs ✕ ♣❛r❛♠❡tr✐❝ ✐♠♣❧❡♠❡♥t❛t✐♦♥

✷✾✴✸✸

❘❡s✉❧ts✿ t✐♠❡

10 20 30 40 50 200 400 600 800 1,000 n t✐♠❡ ✭✐♥ s❡❝♦♥❞s✮ cr✲cprs crrscnp

crrscnp ✕ ♥♦♥✲♣❛r❛♠❡tr✐❝ ✭❤❛r❞✲❝♦❞❡❞ ✈❛❧✉❛t✐♦♥✮

✷✾✴✸✸

❘❡s✉❧ts✿ ♠❡♠♦r②

10 20 30 40 50 100 200 300 400 n ♠❡♠♦r② ✭✐♥ ▼❇✮ cr✲cprs crrscnp

✸✵✴✸✸

❘❡❛❝t✐♦♥ ❙②st❡♠s ▼♦❞❡❧ ❈❤❡❝❦✐♥❣ ❚♦♦❧❦✐t

✓ rs❈❚▲✿ ◮ ❇✐♥❛r② ❉❡❝✐s✐♦♥ ❉✐❛❣r❛♠s ✉s❡❞ ❢♦r st♦r✐♥❣ ❛♥❞ ♣❡r❢♦r♠✐♥❣ ♦♣❡r❛t✐♦♥s ♦♥ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ◮ ❯s❡s ❇❉❉✲❜❛s❡❞ ❜♦✉♥❞❡❞ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ❢♦r ❡✣❝✐❡♥t ✈❡r✐✜❝❛t✐♦♥ ♦❢ ❡①✐st❡♥t✐❛❧ ❢♦r♠✉❧❛❡ ✓ rs▲❚▲✿ ◮ ❇❛s❡❞ ♦♥ tr❛♥s❧❛t✐♦♥ t♦ t❤❡ ❙❆❚ ♣r♦❜❧❡♠ ✭❙▼❚✮ ◮ ❊①✐st❡♥t✐❛❧ ✈❡r✐✜❝❛t✐♦♥ ✓ ❘❡❛❝t✐♦♥ ♠✐♥✐♥❣ ❢♦r rs▲❚▲✿ ◮ ❖❜s❡r✈❛t✐♦♥s ❡①♣r❡ss❡❞ ✐♥ rs▲❚▲ ◮ ❯s❡s ❙▼❚ ❢♦r ❇▼❈✲❜❛s❡❞ ♣❛r❛♠❡t❡r s②♥t❤❡s✐s

✸✶✴✸✸

❘❡❛❝t✐♦♥ ❙②st❡♠s ▼♦❞❡❧ ❈❤❡❝❦✐♥❣ ❚♦♦❧❦✐t

❋♦r♠❛❧✐s♠ rs❈❚▲ rs▲❚▲ rs ✉♠❝✴❜♠❝ ❜♠❝ rs❝ ✗ ❜♠❝ ♣rs ✗ ❜♠❝

✸✷✴✸✸

❈♦♥❝❧✉s✐♦♥s

◮ ❙②♥t❤❡s✐s ♠❡t❤♦❞ ❢♦r ♣❛rt✐❛❧❧② ❞❡✜♥❡❞ r❡❛❝t✐♦♥ s②st❡♠s ✭❘❙✮ ◮ Pr♦♣❡rt✐❡s s♣❡❝✐✜❡❞ ✉s✐♥❣ ❧✐♥❡❛r✲t❡♠♣♦r❛❧ ❧♦❣✐❝ ❢♦r ❘❙ ◮ ❉❡♠♦♥str❛t❡❞ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ❛tt❛❝❦ s②♥t❤❡s✐s ❋✉rt❤❡r ✇♦r❦✿ ◮ ❚❛❝❦❧❡ ✉♥✐✈❡rs❛❧ ♦❜s❡r✈❛t✐♦♥s ◮ ❖♣t✐♠✐s❛t✐♦♥ ♦❢ ❙▼❚✲❡♥❝♦❞✐♥❣ ❤tt♣✿✴✴r❡❛❝t✐♦♥s②st❡♠s✳♦r❣

❚❤❛♥❦ ②♦✉✦

✸✸✴✸✸

❈♦♥❝❧✉s✐♦♥s

◮ ❙②♥t❤❡s✐s ♠❡t❤♦❞ ❢♦r ♣❛rt✐❛❧❧② ❞❡✜♥❡❞ r❡❛❝t✐♦♥ s②st❡♠s ✭❘❙✮ ◮ Pr♦♣❡rt✐❡s s♣❡❝✐✜❡❞ ✉s✐♥❣ ❧✐♥❡❛r✲t❡♠♣♦r❛❧ ❧♦❣✐❝ ❢♦r ❘❙ ◮ ❉❡♠♦♥str❛t❡❞ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ❛tt❛❝❦ s②♥t❤❡s✐s ❋✉rt❤❡r ✇♦r❦✿ ◮ ❚❛❝❦❧❡ ✉♥✐✈❡rs❛❧ ♦❜s❡r✈❛t✐♦♥s ◮ ❖♣t✐♠✐s❛t✐♦♥ ♦❢ ❙▼❚✲❡♥❝♦❞✐♥❣ ❤tt♣✿✴✴r❡❛❝t✐♦♥s②st❡♠s✳♦r❣

❚❤❛♥❦ ②♦✉✦

✸✸✴✸✸

❈♦♥❝❧✉s✐♦♥s

◮ ❙②♥t❤❡s✐s ♠❡t❤♦❞ ❢♦r ♣❛rt✐❛❧❧② ❞❡✜♥❡❞ r❡❛❝t✐♦♥ s②st❡♠s ✭❘❙✮ ◮ Pr♦♣❡rt✐❡s s♣❡❝✐✜❡❞ ✉s✐♥❣ ❧✐♥❡❛r✲t❡♠♣♦r❛❧ ❧♦❣✐❝ ❢♦r ❘❙ ◮ ❉❡♠♦♥str❛t❡❞ ❛♣♣❧✐❝❛t✐♦♥ ✐♥ ❛tt❛❝❦ s②♥t❤❡s✐s ❋✉rt❤❡r ✇♦r❦✿ ◮ ❚❛❝❦❧❡ ✉♥✐✈❡rs❛❧ ♦❜s❡r✈❛t✐♦♥s ◮ ❖♣t✐♠✐s❛t✐♦♥ ♦❢ ❙▼❚✲❡♥❝♦❞✐♥❣ ❤tt♣✿✴✴r❡❛❝t✐♦♥s②st❡♠s✳♦r❣

❚❤❛♥❦ ②♦✉✦

✸✸✴✸✸