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❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❛t❛ ❛♥❞ ❑♥♦✇❧❡❞❣❡ ❙②st❡♠s

❊P❈▲ ❇❛s✐❝ ❚r❛✐♥✐♥❣ ❈❛♠♣ ✷✵✶✷ P❛rt ❋✐✈❡ ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r

■♥st✐t✉t ❢ür ■♥❢♦r♠❛t✐♦♥ss②st❡♠❡ ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ❲✐❡♥

❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙

❖✉t❧✐♥❡

✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥ ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠ ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥ ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥ ✻✳✺ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥ ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s

❊✈❛❧✉❛t✐♦♥ ❙tr❛t❡❣✐❡s

❚❤❡r❡ ❛r❡ t✇♦ ❜❛s✐❝ ❡✈❛❧✉❛t✐♦♥ str❛t❡❣✐❡s ♦❢ r✉❧❡ ❜❛s❡s✿

✶ ❋♦r✇❛r❞ ❈❤❛✐♥✐♥❣✿ ■♥ t❤❡ s♣✐r✐t ♦❢ ▼♦❞✉s P♦♥❡♥s✿

ϕ, ϕ ⇒ ψ ψ ❆♣♣❧② t❤❡ r✉❧❡s t♦ ❝♦♥❝❧✉❞❡ ♥❡✇ ❢❛❝ts ✭❝❢✳ ♦♣❡r❛t♦r T

S✮✳

⇒ ❛ ❜♦tt♦♠✲✉♣ ❡✈❛❧✉❛t✐♦♥ ♦❢ r✉❧❡s✱ ❢r♦♠ ❢❛❝ts t♦ t❤❡ ❞❡s✐r❡❞ ❝♦♥❝❧✉s✐♦♥✳

✷ ❇❛❝❦✇❛r❞ ❈❤❛✐♥✐♥❣✿ ■♥ t❤❡ s♣✐r✐t ♦❢ ❆❜❞✉❝t✐✈❡ ❘❡❛s♦♥✐♥❣✿

ψ, ϕ ⇒ ψ ϕ ❘❡❞✉❝❡ ♣r♦✈✐♥❣ ψ ✈✐❛ ❛ r✉❧❡ ✇✐t❤ ❝♦♥s❡q✉❡♥t ψ t♦ ♣r♦✈✐♥❣ ✐ts ❛♥t❡❝❡❞❡♥t ϕ✳ ⇒ ❛ t♦♣✲❞♦✇♥ ❡✈❛❧✉❛t✐♦♥ ♦❢ r✉❧❡s✱ ❢r♦♠ ❛ ❞❡s✐r❡❞ ❝♦♥❝❧✉s✐♦♥ ✭❣♦❛❧✮ t♦✇❛r❞s t❤❡ ❢❛❝ts✳

✸ ▼✐①❡❞ ❢♦r♠s ♦❢ ❡✈❛❧✉❛t✐♦♥ ❡①✐st ✭r❡❛❧✐③✐♥❣ ❛ ❜✐❞✐r❡❝t✐♦♥❛❧ s❡❛r❝❤✮✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

❖✉t❧✐♥❡

✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥ ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠ ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥ ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥ ✻✳✺ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥ ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

❘❡❝❛❧❧

❉❛t❛❧♦❣✿ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣ ◆♦ ❢✉♥❝t✐♦♥s s②♠❜♦❧s✱ ♦♥❧② ❝♦♥st❛♥ts❀ ♥♦ ♥❡❣❛t✐♦♥ P❛rt✐t✐♦♥✐♥❣ ♦❢ t❤❡ ♣r❡❞✐❝❛t❡ s②♠❜♦❧s ♦❢ ❛ ♣r♦❣r❛♠ P✱ ❝❛❧❧❡❞ t❤❡ s❝❤❡♠❛ ♦❢ P✱ ✐♥t♦

• t❤❡ s❡t ext(P) ♦❢ ❡①t❡♥s✐♦♥❛❧ ♣r❡❞✐❝❛t❡s✱ ❛♥❞
• t❤❡ s❡t int(P) ♦❢ ✐♥t❡♥s✐♦♥❛❧ ♣r❡❞✐❝❛t❡s✳

❊①t❡♥s✐♦♥❛❧ ♣r❡❞✐❝❛t❡s ❝❛♥ ♥♦t ♦❝❝✉r ✐♥ r✉❧❡ ❤❡❛❞s✳ ❇② ❞❡❢❛✉❧t✱ ❛❧❧ ♣r❡❞✐❝❛t❡s ♦❝❝✉rr✐♥❣ ♦♥❧② ✐♥ r✉❧❡ ❤❡❛❞s ❛r❡ ❛ss✉♠❡❞ t♦ ❜❡ ❡①t❡♥s✐♦♥❛❧✳ ❯s✉❛❧❧②✱ ❛❧❧ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ❝♦♥s❡q✉❡♥t ♦❢ ❛ ❝❧❛✉s❡ ❛❧s♦ ♦❝❝✉r ✐♥ t❤❡ ❛♥t❡❝❡❞❡♥t ✭r❛♥❣❡✲r❡str✐❝t✐♦♥✱ s❛❢❡t②✮✳ ❙❡♠❛♥t✐❝❛❧❧②✱ ❛ ❢❛❝t✲❢r❡❡ ❉❛t❛❧♦❣ ♣r♦❣r❛♠ P s♣❡❝✐✜❡s ❛ ♠❛♣♣✐♥❣ µP ❢r♦♠ t❤❡ ❍❡r❜r❛♥❞ ✐♥t❡r♣r❡t❛t✐♦♥s ext(P) t♦ t❤❡ ❍❡r❜r❛♥❞ ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ int(P)✱ ❣✐✈❡♥ ❜② µP (I) = HI (lfp(T

P ∪I|ext(P )))|int(P ).

✭IP reds ✳ ✳ ✳ r❡str✐❝t✐♦♥ ♦❢ I t♦ Preds✮✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✺✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

❊①❛♠♣❧❡

Pr♦❣r❛♠ P ✭✐♥❝❧✉❞✐♥❣ ❡①t❡♥s✐♦♥❛❧ ❢❛❝ts✮✿

❢❡❡❞s❴♠✐❧❦✭❜❡tt② ✮✳ ❧❛②s❴❡❣❣s✭❜❡tt② ✮✳ ❤❛s❴s♣✐♥❡s✭❜❡tt② ✮✳ ♠♦♥♦tr❡♠❡✭❳✮← ❧❛②s❴❡❣❣s✭❳✮✱ ❢❡❡❞s❴♠✐❧❦✭❳✮✳ ❡❝❤✐❞♥❛✭❳✮← ♠♦♥♦tr❡♠❡✭❳✮✱ ❤❛s❴s♣✐♥❡s✭❳✮✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

❊①❛♠♣❧❡

Pr♦❣r❛♠ P ✭✐♥❝❧✉❞✐♥❣ ❡①t❡♥s✐♦♥❛❧ ❢❛❝ts✮✿

❢❡❡❞s❴♠✐❧❦✭❜❡tt② ✮✳ ❧❛②s❴❡❣❣s✭❜❡tt② ✮✳ ❤❛s❴s♣✐♥❡s✭❜❡tt② ✮✳ ♠♦♥♦tr❡♠❡✭❳✮← ❧❛②s❴❡❣❣s✭❳✮✱ ❢❡❡❞s❴♠✐❧❦✭❳✮✳ ❡❝❤✐❞♥❛✭❳✮← ♠♦♥♦tr❡♠❡✭❳✮✱ ❤❛s❴s♣✐♥❡s✭❳✮✳

❙❝❤❡♠❛ ♦❢ P✿ ext(P) int(P)

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

❊①❛♠♣❧❡

Pr♦❣r❛♠ P ✭✐♥❝❧✉❞✐♥❣ ❡①t❡♥s✐♦♥❛❧ ❢❛❝ts✮✿

❢❡❡❞s❴♠✐❧❦✭❜❡tt② ✮✳ ❧❛②s❴❡❣❣s✭❜❡tt② ✮✳ ❤❛s❴s♣✐♥❡s✭❜❡tt② ✮✳ ♠♦♥♦tr❡♠❡✭❳✮← ❧❛②s❴❡❣❣s✭❳✮✱ ❢❡❡❞s❴♠✐❧❦✭❳✮✳ ❡❝❤✐❞♥❛✭❳✮← ♠♦♥♦tr❡♠❡✭❳✮✱ ❤❛s❴s♣✐♥❡s✭❳✮✳

❙❝❤❡♠❛ ♦❢ P✿ { ❢❡❡❞s❴♠✐❧❦✱ ❧❛②s❴❡❣❣s✱ ❤❛s❴s♣✐♥❡s✱ ♠♦♥♦tr❡♠❡✱ ❡❝❤✐❞♥❛ } ❈❛❧❝✉❧❛t✐♦♥ ♦❢ lfp(T

P ) ✭b = betty✮✿

T

P ↑ 1 = {b}feeds, {b}lays, {b}spines, {}monotreme, {}echidna

T

P ↑ 2 = {b}feeds, {b}lays, {b}spines, {b}monotreme, {}echidna

T

P ↑ 3 = {b}feeds, {b}lays, {b}spines, {b}monotreme, {b}echidna

= lfp(T

P )

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

❙tr❛✐❣❤t ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ♦♣❡r❛t♦r T

P ✿ I0 ✿❂ ∅ I1 ✿❂ ❣r♦✉♥❞❴❢❛❝ts ✭P ✮ ✐ ✿❂ ✶ ✇ ❤ ✐ ❧ ❡ Ii = Ii−1 ❞♦ ✐ ✿❂ ✐ ✰ ✶ Ii ✿❂ Ii−1 ✇ ❤ ✐ ❧ ❡ ✭❘ ❂ ❘✉❧❡s ✳ ♥❡①t ✭ ✮ ✮ ■ ♥ s t s ✿❂ ✐ ♥ s t ❛ ♥ t ✐ ❛ t ✐ ♦ ♥ s ✭❘✱ Ii−1 ✮ ✇ ❤ ✐ ❧ ❡ ✭ ✐ ♥ s t ❂ ■ ♥ s t s ✳ ♥❡①t ✭ ✮ ✮ Ii ✿❂ Ii ∪ ❤❡❛❞✭✐♥st✮ r ❡ t ✉ r ♥ Ii ✐♥st❛♥t✐❛t✐♦♥s ✭❘✱ I✮✿ ❛❧❧ ✐♥st❛♥❝❡s r ♦❢ r✉❧❡s ✐♥ ❘ s✳t✳ ❜♦❞②✭r✮ ✐s s❛t✐s✜❡❞ ❜② I✳

❉✐s❛❞✈❛♥t❛❣❡

❘❡✜r✐♥❣ ♦❢ r✉❧❡s ✭❡✳❣✳✱ ❛❧❧ ❢❛❝ts ❛r❡ r❡✲♦❜t❛✐♥❡❞ ✐♥ ❡❛❝❤ st❡♣❀ ♠♦♥♦tr❡♠❡✭❜❡tt②✮ ❛❣❛✐♥ ✐♥ ❙t❡♣ ✸✮✳ ■❞❡❛✿ ♦♥❧② ❝♦♥s✐❞❡r r✉❧❡s ✇❤✐❝❤ ✐♥✈♦❧✈❡ ♥❡✇❧② ❞❡r✐✈❡❞ ❛t♦♠s✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✼✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥

✐♥❝r❡♠❡♥t❛❧ ❢♦r✇❛r❞ ❝❤❛✐♥✐♥❣✿

❑♥♦✇♥❋❛❝ts ✿❂ ∅ ■♥❦ ✿❂ ④ ❋❛❝t ⑤ ✭ ❋❛❝t ← t r ✉ ❡ ✮ ∈ P ⑥ ✇ ❤ ✐ ❧ ❡ ✭ ■♥❦ = ∅✮ ■ ♥ s t s ✿❂ ✐ ♥ s t ❛ ♥ t ✐ ❛ t ✐ ♦ ♥ s ✭❘✱ ❑♥♦✇♥❋❛❝ts ✱ ■♥❦ ✮ ❑♥♦✇♥❋❛❝ts ✿❂ ❑♥♦✇♥❋❛❝ts ∪ ■♥❦ ■♥❦ ✿❂ ❤❡❛❞s ✭ ■ ♥ s t s ✮ r ❡ t ✉ r ♥ ❑♥♦✇♥❋❛❝ts ✐♥st❛♥t✐❛t✐♦♥s ✭❘✱ ❑♥♦✇♥❋❛❝ts✱ ■♥❦✮✿ ❛❧❧ ✐♥st❛♥❝❡s r ♦❢ r✉❧❡s ✐♥ ❘ s✳t✳ ❜♦❞②✭r✮ ✐s s❛t✐s✜❡❞

❜② ❑♥♦✇♥❋❛❝ts ∪ ■♥❦ ✉s✐♥❣ s♦♠❡ ❢❛❝t ❢r♦♠ ■♥❦✳ ❋✉rt❤❡r ✐♠♣r♦✈❡♠❡♥ts✿ ❡✳❣✳✱

• ✉s❡ ♦♥❧② r✉❧❡ ✐♥st❛♥❝❡s ✇✐t❤ ❤❡❛❞ ♥♦t ✐♥ ❑♥♦✇♥❋❛❝ts ∪ ■♥❦
• st♦r❡ ♣❛rt✐❛❧❧② ✐♥st❛♥t✐❛t❡❞ r✉❧❡s ✭✐♥❝r❡♠❡♥t❛❧ s❛t✐s❢❛❝t✐♦♥ ♦❢ t❤❡ ❜♦❞②✮
• ✐♥ ❛❞❞✐t✐♦♥✱ s❤❛r❡ ❝♦♠♠♦♥ ❜♦❞② ♣❛rts ❜❡t✇❡❡♥ r✉❧❡s ✭❀ ❘❊❚❊ ❆❧❣♦r✐t❤♠✮

❖t❤❡r ✈✐❡✇✿ ♠❛♣ ❉❛t❛❧♦❣ t♦ ❘❡❧❛t✐♦♥❛❧ ❆❧❣❡❜r❛

• ❙❡❛r❝❤ s♦❧✉t✐♦♥ ❢♦r s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s✱ ✉s✐♥❣ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s ✭❡✳❣✳✱
• ❛✉ÿ✲❙❡✐❞❡❧ ✐t❡r❛t✐♦♥ ✭s❡❡ ❬❈❡r✐✱ ●♦tt❧♦❜✱ ❚❛♥❝❛ ✶✾✾✵❪✮

❊①t❡♥s✐✈❡ tr❡❛t♠❡♥t✿ ❬❆❜✐t❡❜♦✉❧ ❡t ❛❧✳✱ ✶✾✾✺❪

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✽✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠

❖✉t❧✐♥❡

✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥ ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠ ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥ ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥ ✻✳✺ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥ ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✾✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠

❘❊❚❊ ❆❧❣♦r✐t❤♠

❇② ❈❤❛r❧❡s ❋♦r❣② ✭✶✾✾✵✮✱ ❢♦r ❢♦r✇❛r❞ ❝❤❛✐♥✐♥❣ ✭♣r♦❞✉❝t✐♦♥✮ s②st❡♠s ❙t♦r❛❣❡ ♦❢ ♣❛rt✐❛❧❧② ✐♥st❛♥t✐❛t❡❞ r✉❧❡s ❙❤❛r✐♥❣ ♦❢ ✐♥st❛♥t✐❛t❡❞ ❧✐t❡r❛❧s ❛♠♦♥❣ s✐♠✐❧❛r r✉❧❡s ❙❡✈❡r❛❧ ♦♣t✐♠✐③❛t✐♦♥s✱ ✐♥❞✉str✐❛❧ ✉s❡ ✭❈❧✐♣s✱ ❉r♦♦❧s✱ ❏❘✉❧❡s✱ ✳✳✳✮ ❇❛s✐❝ ❛♣♣r♦❛❝❤✿ ❯s❡

• ♣r♦❞✉❝t✐♦♥ ♠❡♠♦r② P▼ ✭r✉❧❡ st♦r❡✮ ❛♥❞
• ✇♦r❦✐♥❣ ♠❡♠♦r② ❲▼ ✭❝✉rr❡♥t ❢❛❝ts✮

❉✐✛❡r❡♥t ❦✐♥❞s ♦❢ ♥♦❞❡s✿

• ❛❧♣❤❛✲♥♦❞❡✿ r❡♣r❡s❡♥ts ❛ s✐♥❣❧❡ ❛t♦♠✐❝ ❝♦♥❞✐t✐♦♥ ✐♥ r✉❧❡ ❜♦❞✐❡s ✭❛❝r♦ss

r✉❧❡s✮❀ ✐t ❝♦♥t❛✐♥s ❛❧❧ ❲▼ ❡❧❡♠❡♥ts t❤❛t ♠❛❦❡ ✐t tr✉❡❀

• ❜❡t❛✲♥♦❞❡✿ r❡♣r❡s❡♥ts ❛ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❛❧♣❤❛✲♥♦❞❡s❀ ✐t ❝♦♥t❛✐♥s t✉♣❧❡s ♦❢

❲▼ ❡❧❡♠❡♥ts s❛t✐s❢②✐♥❣ t❤❡♠✳

• ❥♦✐♥✲♥♦❞❡✿ ❢♦r ❝♦♠♣✉t❛t✐♦♥❛❧ ♣✉r♣♦s❡s ✭❝♦♠❜✐♥✐♥❣ ❛❧♣❤❛ ❛♥❞✴♦r ❜❡t❛ ♥♦❞❡s✮
• ♣r♦❞✉❝t✐♦♥✲♥♦❞❡✿ ♦♥❡ ♣❡r r✉❧❡✱ ❤♦❧❞✐♥❣ ❛❧❧ t✉♣❧❡s ♦❢ ❲▼ ❡❧❡♠❡♥ts t❤❛t

s❛t✐s❢② ✐ts ❜♦❞②✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✵✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠

❊①❛♠♣❧❡

Working memory: w1: anna feeds milk w2: anna lays eggs w3: anna married_to pierre w4: pierre is poisonous w5: betty feeds milk w6: betty lays eggs w7: betty has spines w8: tux has wings w9: tux lays eggs Production memory: p1: X lays eggs, X has wings ==> X is_a bird p2: X feeds milk ==> X is_a mammal p3: X feeds milk, X lays eggs ==> X is_a monotreme p4: X feeds milk, X lays eggs, X has_spines ==> X is_a echidna p5: X feeds milk, X lays eggs, X married_to Y, Y is poisonous ==> X is_a platypus, Y is_a platypus X lays eggs X feeds milk w2, w6, w9 w1, w5 w1^w2, w5^w6 (w1^w2), (w5^w6) X has spines w7 X has_husband Y w3 X has wings w8 Y is poisonous w4 w8^w9 (X lays eggs), (X feeds milk), (X married_to Y) (Y is poisonous) (X lays eggs), (X feeds milk), (X married_to Y) (X lays eggs), (X feeds milk), (X has spines) X is_a monotreme X is_a platypus, Y is_a platypus (X lays eggs), (X feeds milk), X is_a echidna (X lays eggs), (X lays eggs), (X has wings) X is_a bird X is_a mammal dummy top node w2, w6, w9 w5^w6^w7 w1^w2^w3 w1^w2^w3^w4 join on X join on X join on X join on X join on Y w5^w6^w7 w8^w9 w1, w5 w1^w2^w3^w4 ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✶✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❖✉t❧✐♥❡

✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥ ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠ ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥ ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥ ✻✳✺ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥ ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❙▲❉ ❘❡s♦❧✉t✐♦♥✿ Pr✐♥❝✐♣❧❡s

❣♦❛❧ ❞r✐✈❡♥ ❡✈❛❧✉❛t✐♦♥ ♦❢ ❧♦❣✐❝ ♣r♦❣r❛♠s ✭❜❛❝❦✇❛r❞ ❝❤❛✐♥✐♥❣✮ t♦ s❤♦✇ t❤❛t P | = ϕ✱ s❤♦✇ t❤❛t P ∪ {¬ϕ} ✐s ✉♥s❛t✐s✜❛❜❧❡ ✉s❡s ✉♥✐✜❝❛t✐♦♥ ❛♥❞ r❡s♦❧✉t✐♦♥✿ ❜❛s✐❝❛❧❧②✱ ϕ1 ∨ ψ, ¬ψ ∨ ϕ2 ϕ1 ∨ ϕ2 r❡❝❛❧❧ t❤❛t ϕ ← ψ ✐s ❡q✉✐✈❛❧❡♥t t♦ ϕ ∨ ¬ψ ❙▲❉ r❡s♦❧✉t✐♦♥✿ ❙❡❧❡❝t❡❞ ▲✐t❡r❛❧ ❉❡✜♥✐t❡ ❈❧❛✉s❡ r❡s♦❧✉t✐♦♥ ✇✐t❤ ❜❛❝❦tr❛❝❦✐♥❣ ✐s ✉s❡❞ ❛s ❝♦♥tr♦❧ ♠❡❝❤❛♥✐s♠ ✐♥ Pr♦❧♦❣

❖❜s❡r✈❡

❆ ❣♦❛❧ ← a1, . . . , an ✐s ❛ s②♥t❛❝t✐❝❛❧ ✈❛r✐❛♥t ♦❢ t❤❡ ❢✳♦✳ s❡♥t❡♥❝❡ ∀x1 · · · ∀xm(⊥ ← a1 ∧ . . . ∧ an) ✇❤❡r❡ x1, . . . , xm ❛r❡ ❛❧❧ ✈❛r✐❛❜❧❡s ♦❝❝✉rr✐♥❣ ✐♥ a1, . . . an✳ ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ¬∃x1 · · · ∃xm(a1 ∧ . . . ∧ an)✳ P | = ∃x1 · · · ∃xm(a1 ∧ . . . ∧ an) ✐✛ P ∪ {← a1, . . . , an} ✐s ✉♥s❛t✐s✜❛❜❧❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✸✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❙▲❉ ❘❡s♦❧✈❡♥t✮

▲❡t C ❜❡ t❤❡ ❝❧❛✉s❡ b ← b1, . . . , bk ❛♥❞ G ❛ ❣♦❛❧ ← a1, . . . , am, . . . , an s✉❝❤ t❤❛t G ❛♥❞ C s❤❛r❡ ♥♦ ✈❛r✐❛❜❧❡s ✭♦t❤❡r✇✐s❡✱ r❡♥❛♠❡ ✈❛r✐❛❜❧❡s ✐♥ C✮✳ ❚❤❡♥ G′ ✐s ❛♥ ❙▲❉ r❡s♦❧✈❡♥t ♦❢ G ❛♥❞ C ✉s✐♥❣ ϑ✱ ✐❢ G′ ✐s t❤❡ ❣♦❛❧ ← (a1, . . . am−1, b1, . . . bk, am+1, . . . an)ϑ ✇❤❡r❡ ϑ ✐s t❤❡ ♠❣✉ ♦❢ am ❛♥❞ b✳

❉❡✜♥✐t✐♦♥ ✭❙▲❉ ❉❡r✐✈❛t✐♦♥✮

❆♥ ❙▲❉ ❞❡r✐✈❛t✐♦♥ ♦❢ P ∪ {G} ❝♦♥s✐sts ♦❢ ❛ s❡q✉❡♥❝❡ G0, G1, . . . ♦❢ ❣♦❛❧s ✇❤❡r❡ G = G0✱ ❛ s❡q✉❡♥❝❡ C1, C2, . . . ♦❢ ✈❛r✐❛♥ts ♦❢ ♣r♦❣r❛♠ ❝❧❛✉s❡s ♦❢ P✱ ❛♥❞ ❛ s❡q✉❡♥❝❡ ϑ1, ϑ2, . . . ♦❢ ♠❣✉✬s s✉❝❤ t❤❛t Gi+1 ✐s ❛ r❡s♦❧✈❡♥t ❢r♦♠ Gi ❛♥❞ Ci+1 ✉s✐♥❣ ϑi+1✳ ❆♥ ❙▲❉✲r❡❢✉t❛t✐♦♥ ✐s ❛ ✜♥✐t❡ ❙▲❉✲❞❡r✐✈❛t✐♦♥ ✇❤♦s❡ ❧❛st ❣♦❛❧ ✐s ❡♠♣t②✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✹✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❉❡✜♥✐t✐♦♥ ✭❙▲❉ ❚r❡❡✮

❆♥ ❙▲❉ tr❡❡ T ✇✳r✳t✳ ❛ ♣r♦❣r❛♠ P ❛♥❞ ❛ ❣♦❛❧ G ✐s ❛ ❧❛❜❡❧❡❞ tr❡❡ ✇❤❡r❡ ❡✈❡r② ♥♦❞❡ ♦❢ T ✐s ❛ ❣♦❛❧✱ t❤❡ r♦♦t ♦❢ T ✐s G✱ ❛♥❞ ✐❢ G ✐s ❛ ♥♦❞❡ ✐♥ T t❤❡♥ G ❤❛s ❛ ❝❤✐❧❞ G′ ❝♦♥♥❡❝t❡❞ t♦ G ❜② ❛♥ ❡❞❣❡ ❧❛❜❡❧❡❞ (C, ϑ) ✐✛ G′ ✐s ❛♥ ❙▲❉✲r❡s♦❧✈❡♥t ♦❢ G ❛♥❞ C ✉s✐♥❣ ϑ✳

❉❡✜♥✐t✐♦♥ ✭❈♦♠♣✉t❡❞ ❆♥s✇❡r✮

• ✐✈❡♥ ❛ ❞❡✜♥✐t❡ ♣r♦❣r❛♠ P ❛♥❞ ❛ ❞❡✜♥✐t❡ ❣♦❛❧ G✱ ❛ ❝♦♠♣✉t❡❞ ❛♥s✇❡r ϑ ❢♦r

P ∪ {G} ✐s t❤❡ s✉❜st✐t✉t✐♦♥ ♦❜t❛✐♥❡❞ ❜② r❡str✐❝t✐♥❣ t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ s❡q✉❡♥❝❡ ♦❢ ♠❣✉✬s ϑ1, . . . ϑn ✉s❡❞ ✐♥ s♦♠❡ ❙▲❉✲r❡❢✉t❛t✐♦♥ ♦❢ P ∪ {G} t♦ t❤❡ ✈❛r✐❛❜❧❡s ♦❝❝✉rr✐♥❣ ✐♥ G✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✺✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:- t(1,A)

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:- t(1,A) :- e(1,A) 1, {X/1, Y/A}

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:- t(1,A) :- e(1,A) 1, {X/1, Y/A} :- 3, {A/2}

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:- t(1,A) :- e(1,A) 1, {X/1, Y/A} :- 3, {A/2} :- t(1,Y),e(Y,A) 2,{X/1,Z/A}

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:- t(1,A) :- e(1,A) 1, {X/1, Y/A} :- 3, {A/2} :- t(1,Y),e(Y,A) 2,{X/1,Z/A} :- e(1,Y),e(Y,A) 1,{X/1,Y'/Y}

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:- t(1,A) :- e(1,A) 1, {X/1, Y/A} :- 3, {A/2} :- t(1,Y),e(Y,A) 2,{X/1,Z/A} :- e(1,Y),e(Y,A) 1,{X/1,Y'/Y} :- e(2,A) 3,{Y/2}

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❈♦♠♣✉t❛t✐♦♥ r✉❧❡s

■♥ ❡❛❝❤ r❡s♦❧✉t✐♦♥ st❡♣✱ t❤❡ s❡❧❡❝t❡❞ ❧✐t❡r❛❧ ❛♥❞ t❤❡ ❝❧❛✉s❡ C ❛r❡ ❝❤♦s❡♥ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝❛❧❧②✳

❉❡✜♥✐t✐♦♥ ✭❈♦♠♣✉t❛t✐♦♥ ❘✉❧❡✮

❆ ❝♦♠♣✉t❛t✐♦♥ r✉❧❡ ✐s ❛♥② ❢✉♥❝t✐♦♥ t❤❛t ❛ss♦❝✐❛t❡s ✇✐t❤ ❡❛❝❤ ❣♦❛❧ ♦♥❡ ♦❢ ✐ts ❛t♦♠s✳

Pr♦♣♦s✐t✐♦♥ ✭■♥❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❈♦♠♣✉t❛t✐♦♥ ❘✉❧❡✮

▲❡t P ❜❡ ❛ ❞❡✜♥✐t❡ ♣r♦❣r❛♠ ❛♥❞ G ❜❡ ❛ ❞❡✜♥✐t❡ ❣♦❛❧✳ ❙✉♣♣♦s❡ t❤❡r❡ ✐s ❛♥ ❙▲❉✲r❡❢✉t❛t✐♦♥ ♦❢ P ∪ {G} ✇✐t❤ ❝♦♠♣✉t❡❞ ❛♥s✇❡r ϑ✳ ❚❤❡♥✱ ❢♦r ❡✈❡r② ❝♦♠♣✉t❛t✐♦♥ r✉❧❡ R✱ t❤❡r❡ ❡①✐sts ❛♥ ❙▲❉✲r❡❢✉t❛t✐♦♥ ♦❢ P ∪ {G} ✉s✐♥❣ t❤❡ ❛t♦♠ s❡❧❡❝t❡❞ ❜② R ❛s s❡❧❡❝t❡❞ ❛t♦♠ ✐♥ ❡❛❝❤ st❡♣ ✇✐t❤ ❝♦♠♣✉t❡❞ ❛♥s✇❡r ϑ′ s✉❝❤ t❤❛t Gϑ ✐s ❛ ✈❛r✐❛♥t ♦❢ Gϑ′✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✼✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

▲❡t ❛ ❝♦rr❡❝t ❛♥s✇❡r ❢♦r ❛ ♣r♦❣r❛♠ P ❛♥❞ ❣♦❛❧ G ❜❡ ❛♥② s✉❜st✐t✉t✐♦♥ ϑ s✉❝❤ t❤❛t P | = Gϑ✳

Pr♦♣♦s✐t✐♦♥ ✭❙♦✉♥❞♥❡ss ❛♥❞ ❈♦♠♣❧❡t❡♥❡ss ♦❢ ▲♦❣✐❝ Pr♦❣r❛♠♠✐♥❣✮

▲❡t P ❜❡ ❛ ♣r♦❣r❛♠ ❛♥❞ ❧❡t Q ❜❡ ❛ q✉❡r②✳ ❚❤❡♥ ❡✈❡r② ❝♦♠♣✉t❡❞ ❛♥s✇❡r ♦❢ P ❛♥❞ G ✐s ❛ ❝♦rr❡❝t ❛♥s✇❡r✱ ❛♥❞ ❢♦r ❡✈❡r② ❝♦rr❡❝t ❛♥s✇❡r σ ♦❢ P ❛♥❞ G t❤❡r❡ ❡①✐sts ❛ ❝♦♠♣✉t❡❞ ❛♥s✇❡r ϑ s✉❝❤ t❤❛t ϑ ✐s ♠♦r❡ ❣❡♥❡r❛❧ t❤❛t σ✳

❉❡✜♥✐t✐♦♥ ✭❙▲❉ Pr♦❝❡❞✉r❡✮

❆♥ ❙▲❉✲♣r♦❝❡❞✉r❡ ✐s ❛♥② ❞❡t❡r♠✐♥✐st✐❝ ❙▲❉✲r❡s♦❧✉t✐♦♥ ❛❧❣♦r✐t❤♠ ❝♦♥str❛✐♥❡❞ ❜② ❛ ❝♦♠♣✉t❛t✐♦♥ r✉❧❡ ❛♥❞ ❛♥ ♦r❞❡r ❢♦r ✈✐s✐t✐♥❣ t❤❡ ✜♥✐t❡ ❜r❛♥❝❤❡s ♦❢ ❛♥ ❙▲❉✲tr❡❡ ✭s❡❛r❝❤ str❛t❡❣②✮✳ ❚❤❡ ❝♦♠♣❧❡t❡♥❡ss ♦❢ ❛ ❙▲❉ ♣r♦❝❡❞✉r❡ ❞❡♣❡♥❞s ♦♥ t❤❡ s❡❛r❝❤ str❛t❡❣②✳ ❚♦ ❜❡ ❝♦♠♣❧❡t❡✱ ❡❛❝❤ ❧❡❛❢ ♦❢ ❛ ✭✜♥✐t❡✮ ❜r❛♥❝❤ ♠✉st ❜❡ ✈✐s✐t❡❞ ❛❢t❡r ✜♥✐t❡❧② ♠❛♥② st❡♣s ✭❢❛✐r♥❡ss✮✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✽✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡ ✭❝♦♥t✬❞✮

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:- t(1,A) :- e(1,A) 1, {X/1, Y/A} :- 3, {A/2} :- t(1,Y),e(Y,A) 2,{X/1,Z/A} :- e(1,Y),e(Y,A) 1,{X/1,Y'/Y} :- e(2,A) 3,{Y/2} :- 4,{A/1}

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✾✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡ ✭❝♦♥t✬❞✮

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:- t(1,A) :- e(1,A) 1, {X/1, Y/A} :- 3, {A/2} :- t(1,Y),e(Y,A) 2,{X/1,Z/A} :- e(1,Y),e(Y,A) 1,{X/1,Y'/Y} :- e(2,A) 3,{Y/2} :- 4,{A/1} :- t(1,Y'), e(Y',Y), e(Y,A) 2',{X'/1,Z'/Y} ... ....

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✾✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡ ✭❝♦♥t✬❞✮

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:- t(1,A) :- e(1,A) 1, {X/1, Y/A} :- 3, {A/2} :- t(1,Y),e(Y,A) 2,{X/1,Z/A} :- e(1,Y),e(Y,A) 1,{X/1,Y'/Y} :- e(2,A) 3,{Y/2} :- 4,{A/1} :- t(1,Y'), e(Y',Y), e(Y,A) 2',{X'/1,Z'/Y} ... .... Problem: Non-termination: t(1,2),t(1,1),t(1,2),t(1,1),...

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✶✾✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✷✿ t✭❳✱❩✮←t✭❳✱❨✮✱❡✭❨✱❩✮✳ ✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:-t(1,A) :-e(1,A) 1, {X/1, Y/A} :- 3, {A/2} :-t(1,Y),e(Y,A) 2,{X/1,Z/A} :-e(1,Y),e(Y,A) 1,{X/1,Y/Y} :-e(2,A) 3,{Y/2} :- 4,{A/1} :-t(1,Y'), e(Y',Y), e(Y,A) 2',{X'/1,Z'/Y} ... .... Non-Termination! and no solutions

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✵✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❩✮←❡✭❨✱❩✮✱t✭❳✱❨✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳

:-t(1,A) :-e(1,A) 1, {X/1, Y/A} :- 3, {A/2} :-e(Y,A),t(1,Y) 2,{X/1,Z/A} :-t(1,1) 3,{Y/1,A/2} :-t(1,2) 2',{Y'/2,A/1} ... 2 :-e(1,1) 1 false :-e(Y',1),t(1,Y') 4,{Y'/2} t(1,2) 2',{X'/1,Z'/1} :-e(1,2) 1 :- 1 ... 2 Non-termination due to circular data: Solution A=1 is not found

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✶✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❖✉t❧✐♥❡

✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥ ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠ ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥ ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥ ✻✳✺ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥ ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❖▲❉❚ ❘❡s♦❧✉t✐♦♥

◆♦♥✲t❡r♠✐♥❛t✐♦♥ ♦❢ ❙▲❉ r❡s♦❧✉t✐♦♥ ❞✉❡ t♦ ✐♥✜♥✐t❡ ❜r❛♥❝❤❡s ■♥✜♥✐t❡ ❜r❛♥❝❤❡s B = [G0, G1, . . .] ❞✉❡ t♦

✶ ✈❛r✐❛♥ts ♦❢ t❤❡ s❛♠❡ ❣♦❛❧ ♦♥ t❤❡ ✐♥✜♥✐t❡ ❜r❛♥❝❤✱ ✐✳❡✳✱ ✐♥ s♦♠❡ s✉❜s❡q✉❡♥❝❡

[Gi0, Gi1, . . .]✱ ❢♦r ❛❧❧ j, k ∈ N✱ Gij ❛♥❞ Gik ❝♦♥t❛✐♥ ❛♥ ❡q✉❛❧ ❛t♦♠ ✭✉♣ t♦ r❡♥❛♠✐♥❣ ♦❢ ✈❛r✐❛❜❧❡s✮ ♦r

✷ s✉❜s✉♠✐♥❣ ❣♦❛❧s ♦♥ t❤❡ ✐♥✜♥✐t❡ ❜r❛♥❝❤✱ ✐✳❡✳ ✐♥ s♦♠❡ s✉❜s❡q✉❡♥❝❡

[Gi0, Gi1, . . .]✱ ❢♦r ❛❧❧ j ∈ N✱ Gij ❝♦♥t❛✐♥s ❛♥ ❛t♦♠ ✇❤✐❝❤ ✐s ❛ r❡❛❧ ✐♥st❛♥❝❡ ♦❢ ❛♥ ❛t♦♠ ✐♥ Gij−1✳

■❞❡❛s

❆✈♦✐❞ r❡♣❡❛t❡❞ ❡✈❛❧✉❛t✐♦♥ ♦❢ ❛ s✉❜❣♦❛❧ ♦♥ t❤❡ s❛♠❡ ❝♦♠♣✉t❛t✐♦♥ ♣❛t❤ t❤r♦✉❣❤ t❛❜❧✐♥❣ ♦r ♠❡♠♦r✐③❛t✐♦♥✱ s✐♠✐❧❛r ❛s ✐♥ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣✳ ❙✐❞❡ ❡✛❡❝t✿ ♥♦ r❡♣❡❛t❡❞ ❡✈❛❧✉❛t✐♦♥s ♦❢ s✉❜❣♦❛❧s ❛t ❛❧❧✳ ❯s❡ ❞❡s✐❣♥❛t❡❞ t❛❜❧❡❞ ♣r❡❞✐❝❛t❡s✳ ▼❛❦❡ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ s♦❧✉t✐♦♥ ♥♦❞❡s ✭❣♦❛❧s✮ ❛♥❞ ❧♦♦❦✉♣ ♥♦❞❡s ✭❣♦❛❧s✮✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✸✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❖▲❉❚ ✕ ❇❛s✐❝ ❊❧❡♠❡♥ts

❉❡✜♥✐t✐♦♥ ✭❖▲❉❚✲str✉❝t✉r❡✮

❆♥ ❖▲❉❚✲str✉❝t✉r❡ (T, TS, TL) ❝♦♥s✐sts ♦❢ ❛♥ ❙▲❉✲tr❡❡ T✱ ❛ s♦❧✉t✐♦♥ t❛❜❧❡ TS✱ ✐✳❡✳✱ ❛ s❡t ♦❢ ♣❛✐rs (a, TS(a)) ✇❤❡r❡

• a ✐s ❛♥ ❛t♦♠ ❛♥❞
• TS(a) ✐s ❛ ❧✐st ♦❢ ✐♥st❛♥❝❡s ♦❢ a ❝❛❧❧❡❞ t❤❡ s♦❧✉t✐♦♥s ♦❢ a✱ ❛♥❞

❛ ❧♦♦❦✉♣ t❛❜❧❡ TL✱ ✐✳❡✳✱ ❛ s❡t ♦❢ ♣❛✐rs (a, TL(a)) ✇❤❡r❡ a ✐s ❛♥ ❛t♦♠ ❛♥❞ TL(a) ✐s ❛ ♣♦✐♥t❡r t♦ ❛♥ ❡❧❡♠❡♥t ♦❢ TS(a′) s✉❝❤ t❤❛t a ✐s ❛♥ ✐♥st❛♥❝❡ ♦❢ a′✳ TL ❝♦♥t❛✐♥s ♦♥❡ ♣❛✐r (a, TL(a)) ❢♦r ❛♥ ❛t♦♠ a ♦❝❝✉rr✐♥❣ ❛s ❛ ❧❡❢t♠♦st ❛t♦♠ ♦❢ ❛ ❣♦❛❧ ✐♥ T✳ ❚❤❡ ✐♥✐t✐❛❧ ❖▲❉❚✲str✉❝t✉r❡ ❤❛s ❛s T t❤❡ ❣♦❛❧ ❛♥❞ ✈♦✐❞ TS ❛♥❞ TL✳ ❚❤❡ ❖▲❉❚✲str✉❝t✉r❡ ✐s st❡♣✇✐s❡ ❡①t❡♥❞❡❞✱ ✉s✐♥❣ ❙▲❉ r❡s♦❧✉t✐♦♥ ❛♥❞ ❧♦♦❦✉♣✱ ❡♠♣❧♦②✐♥❣ ❛ ❧❡❢t✲t♦✲r✐❣❤t ❝♦♠♣✉t❛t✐♦♥ r✉❧❡✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✹✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❖▲❉❚✲❊①t❡♥s✐♦♥

❚❤❡ ❡①t❡♥s✐♦♥ ♦❢ ❛♥ ❖▲❉❚ str✉❝t✉r❡ (T, TS, TL) ❝♦♥s✐sts ♦❢ t❤r❡❡ st❡♣s✳

✶ r❡s♦❧✉t✐♦♥ st❡♣✿ ❛ ♥❡✇ ❣♦❛❧ G′ ✐s ❛❞❞❡❞ t♦ T✱ r❡s♦❧✈✐♥❣ s♦♠❡ ❣♦❛❧

G = ← a1, . . . , an ✐♥ T t❤❛t ✐s

✭✐✮ ❛ ♥♦♥✲t❛❜❧❡❞ ❣♦❛❧ ♦r ❛ s♦❧✉t✐♦♥ ❣♦❛❧✱ ✇✐t❤ ❛ ❝❧❛✉s❡ C✱ r❡s♣✳ ✭✐✐✮ ❛ ❧♦♦❦✉♣ ❣♦❛❧ ✇✐t❤ t❤❡ ❛t♦♠ a ❢r♦♠ TL(a1)✳

✷ ❝❧❛ss✐✜❝❛t✐♦♥ st❡♣✿ G′ ✐s

• ❛ ♥♦♥✲t❛❜❧❡ ❣♦❛❧✱ ✐❢ t❤❡ ❧❡❢t♠♦st ❛t♦♠ ♦❢ G′✱ a′

1✱ ❤❛s ♥♦t ❛ t❛❜❧❡ ♣r❡❞✐❝❛t❡✳

• ❛ t❛❜❧❡ ❣♦❛❧ ♦t❤❡r✇✐s❡✱ ❛♥❞ ✐s

❛ ❧♦♦❦✉♣ ❣♦❛❧✱ ✐❢ s♦♠❡ (a, TS(a)) ✐♥ TS ✇✐t❤ a ♠♦r❡ ❣❡♥❡r❛❧ t❤❛♥ a′

1 ❡①✐sts ✭a

✏❝♦✈❡rs✑ a′

1✮✳

❚❤❡♥✱ ❛❞❞ (a′

1, p) t♦ TL ✇❤❡r❡ p ♣♦✐♥ts t♦ t❤❡ ✜rst ❡❧❡♠❡♥t ♦❢ TS(a)✳

❛ s♦❧✉t✐♦♥ ♥♦❞❡✱ ✐❢ TS ❝♦♥t❛✐♥s ♥♦ (a, TS(a)) ✇❤❡r❡ a ✐s ♠♦r❡ ❣❡♥❡r❛❧ t❤❛♥ a′

1✳

■♥ t❤✐s ❝❛s❡✱ ❛❞❞ (a′

1, [ ]) t♦ TS✳

✸ t❛❜❧❡ ✉♣❞❛t❡ st❡♣✿ ❛❞❞ ♥❡✇ s♦❧✉t✐♦♥s t♦ TS✿

• ❙✉♣♣♦s❡ G′ = ← a2, . . . , an r❡s✉❧ts ❢r♦♠ s♦♠❡ t❛❜❧❡ ❣♦❛❧

G = ← a1, . . . , an ✐♥ T ❜② ❛♥ ❙▲❉ r❡s♦❧✉t✐♦♥ G0, G1, . . . , Gm ✇✐t❤ ϑ1, ϑ2, . . . , ϑm✳

• ❆❞❞ t❤❡ r❡str✐❝t✐♦♥ ϑ ♦❢ ϑ1 · · · ϑm t♦ t❤❡ ✈❛r✐❛❜❧❡s ♦❢ a1 ❛s ❛♥s✇❡r ❢♦r a1 t♦

TS(a1)✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✺✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❨✮←t✭❳✱❩✮✱❡✭❩✱❨✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳ ▲❡t t ❜❡ ❛ t❛❜❧❡ ♣r❡❞✐❝❛t❡

:- t(1,A) :- e(1,A) 1,{X/1,Y/A} :- 3, {A/2} :- t(1,Z),e(Z,A) 2,{X/1,Y/A} :- e(1,Z),e(Z,A) 1,{X/1,Y/Z} :- e(2,A) 3,{Z/2} :- 4,{A/1} :- t(1,Z'), e(Z',Z), e(Z,A) 2',{X'/1,Y'/Z} ... ....

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❨✮←t✭❳✱❩✮✱❡✭❩✱❨✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳ ▲❡t t ❜❡ ❛ t❛❜❧❡ ♣r❡❞✐❝❛t❡

:- t(1,A) :- e(1,A) 1,{X/1,Y/A} :- 3, {A/2} :- t(1,Z),e(Z,A) 2,{X/1,Y/A} :- e(1,Z),e(Z,A) 1,{X/1,Y/Z} :- e(2,A) 3,{Z/2} :- 4,{A/1} :- t(1,Z'), e(Z',Z), e(Z,A) 2',{X'/1,Y'/Z} ... ....

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❨✮←t✭❳✱❩✮✱❡✭❩✱❨✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳ ▲❡t t ❜❡ ❛ t❛❜❧❡ ♣r❡❞✐❝❛t❡

:- t(1,A) :- e(1,A) 1,{X/1,Y/A} :- 3, {A/2} :- t(1,Z),e(Z,A) 2,{X/1,Y/A} :- e(1,Z),e(Z,A) 1,{X/1,Y/Z} :- e(2,A) 3,{Z/2} :- 4,{A/1} :- t(1,Z'), e(Z',Z), e(Z,A) 2',{X'/1,Y'/Z} ... .... Lookup-Node Solution-Node

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❨✮←t✭❳✱❩✮✱❡✭❩✱❨✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳ ▲❡t t ❜❡ ❛ t❛❜❧❡ ♣r❡❞✐❝❛t❡

:- t(1,A) :- e(1,A) 1,{X/1,Y/A} :- 3, {A/2} :- t(1,Z),e(Z,A) 2,{X/1,Y/A} :- e(1,Z),e(Z,A) 1,{X/1,Y/Z} :- e(2,A) 3,{Z/2} :- 4,{A/1} :- t(1,Z'), e(Z',Z), e(Z,A) 2',{X'/1,Y'/Z} ... .... Lookup-Node Solution-Node

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❊①❛♠♣❧❡

✶✿ t✭❳✱❨✮←❡✭❳✱❨✮✳ ✷✿ t✭❳✱❨✮←t✭❳✱❩✮✱❡✭❩✱❨✮✳ ✸✿ ❡✭✶✱✷✮✳ ✹✿ ❡✭✷✱✶✮✳ ✺✿ ←t✭✶✱❆✮✳ ▲❡t t ❜❡ ❛ t❛❜❧❡ ♣r❡❞✐❝❛t❡

:- t(1,A) :- e(1,A) 1,{X/1,Y/A} :- 3, {A/2} :- t(1,Z),e(Z,A) 2,{X/1,Y/A} :- e(1,Z),e(Z,A) 1,{X/1,Y/Z} :- e(2,A) 3,{Z/2} :- 4,{A/1} :- t(1,Z'), e(Z',Z), e(Z,A) 2',{X'/1,Y'/Z} ... .... :- e(2,A) {Z/2} :- {A/1} Lookup-Node Solution-Node

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥

❈♦♠♣❧❡t❡♥❡ss

❖▲❉❚✲r❡s♦❧✉t✐♦♥ ✐s ♥♦t ❝♦♠♣❧❡t❡ ✐♥ ❣❡♥❡r❛❧

♣ ✭ ① ✮ ← q ✭ ① ✮ ✱ r ✳ q ✭ s ✭ ① ✮ ✮ ← q ✭ ① ✮ ✳ q ✭ ❛ ✮ ← ✳ r ← ✳ ← ♣ ✭ ① ✮ ✳

Pr♦❜❧❡♠ ❘❡❞✉❝t✐♦♥ st❡♣s ❛r❡ ♦♥❧② ❛♣♣❧✐❡❞ t♦ ❧♦♦❦✉♣ ❣♦❛❧ ← q(x′), r. ◆♦ s♦❧✉t✐♦♥s ❢♦r p(x) ✇✐❧❧ ❜❡ ♣r♦❞✉❝❡❞ ✐♥ ✜♥✐t❡ t✐♠❡✳ ❘❡♠❡❞② ❙♣❡❝✐❛❧ s❡❛r❝❤ str❛t❡❣② ✭♠✉❧t✐✲st❛❣❡ ❞❡♣t❤ ✜rst✱ ▼❙❉❋❙✮✿ ❖r❞❡r t❤❡ ♥♦❞❡s ✐♥ t❤❡ ❖▲❉❚ tr❡❡✱ ❛✈♦✐❞ r❡♣❡❛t✐♥❣ r❡❞✉❝t✐♦♥ ♦❢ ❛ ♥♦❞❡ ✐❢ ♦t❤❡r ♥♦❞❡s ❛r❡ ❛✈❛✐❧❛❜❧❡✳ ❆❜♦✈❡✿ ❛✈♦✐❞ r❡❞✉❝✐♥❣ t❤❡ ❧♦♦❦✉♣ ❣♦❛❧ ← q(x′), r. t✇✐❝❡✱ ❛♥❞ r❡❞✉❝❡ ← r✳ ❯♥❞❡r ▼❙❉❋❙✱ ❖▲❉❚✲r❡s♦❧✉t✐♦♥ ❜❡❝♦♠❡s ❝♦♠♣❧❡t❡✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✼✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✺ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❖✉t❧✐♥❡

✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥ ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠ ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥ ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥ ✻✳✺ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥ ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✽✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✺ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❯♥t✐❧ ♥♦✇✱ ✇❡ ❤❛✈❡ s❡❡♥✿

• ❢♦r✇❛r❞ ❝❤❛✐♥✐♥❣ ✭❞❛t❛ ❞r✐✈❡♥✮ ❡✈❛❧✉❛t✐♦♥ ♦❢ ▲P
• ❜❛❝❦✇❛r❞ ❝❤❛✐♥✐♥❣ ✭❣♦❛❧ ❞r✐✈❡♥✮ ❡✈❛❧✉❛t✐♦♥ ♦❢ ▲P
• ✐♠♣r♦✈❡♠❡♥t ♦❢ ❜❛❝❦✇❛r❞ ❝❤❛✐♥✐♥❣ ❜② t❛❜❧✐♥❣

■❞❡❛ ♦❢ t❤❡ ♠❛❣✐❝ t❡♠♣❧❛t❡s tr❛♥s❢♦r♠❛t✐♦♥✿

• t❛❦❡ t❤❡ ❜❡st ♦❢ ❜♦t❤ ✇♦r❧❞s✿

❊✣❝✐❡♥❝② ♦❢ ❣♦❛❧ ❞✐r❡❝t❡❞♥❡ss

• ♦♦❞ t❡r♠✐♥❛t✐♦♥ ♣r♦♣❡rt✐❡s ♦❢ ❢♦r✇❛r❞ ❝❤❛✐♥✐♥❣

❊❛s② ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛ ❢♦r✇❛r❞ ❝❤❛✐♥✐♥❣ r✉❧❡ ❡♥❣✐♥❡

❖r✐❣✐♥❛❧❧② ❞❡✈❡❧♦♣❡❞ ❢♦r ♣♦s✐t✐✈❡ ❞❛t❛❧♦❣ ♣r♦❣r❛♠s ❆❧s♦ ❢♦r ✉♥str❛t✐✜❡❞✴❞✐s❥✉♥❝t✐✈❡ ❧♦❣✐❝ ♣r♦❣r❛♠s ✉♥❞❡r ❛♥s✇❡r s❡t s❡♠❛♥t✐❝s ❬❋❛❜❡r ❡t ❛❧✳✱ ■❈❉❚ ✷✵✵✺✴❏❈❙❙ ✷✵✵✼❪✿ ❘❡❝❡♥t ✇♦r❦ ▼✳ ❆❧✈✐❛♥♦✿ ✏❉②♥❛♠✐❝ ▼❛❣✐❝ ❙❡ts✑ ❬P❤❞ ❚❤❡s✐s✱ ✷✵✶✵❪

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✷✾✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❖✉t❧✐♥❡

✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✶ ❙❡♠✐✲◆❛✐✈❡ ❊✈❛❧✉❛t✐♦♥ ✻✳✷ ❘❊❚❊ ❆❧❣♦r✐t❤♠ ✻✳✸ ❙▲❉ ❘❡s♦❧✉t✐♦♥ ✻✳✹ ❖▲❉❚ ❘❡s♦❧✉t✐♦♥ ✻✳✺ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥ ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✵✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s

❘❡❝❛❧❧

■❞❡❛✿ ❧❡❛✈❡ tr✉t❤ ✈❛❧✉❡ ✐♥❝❛s❡ ♦❢ ❝②❝❧✐❝ ♥❡❣❛t✐♦♥ ♦♣❡♥ ✭❡✳❣✳✱ p ← ¬p✮ ❯s❡ t❤r❡❡✲✈❛❧✉❡❞ ✐♥t❡r♣r❡t❛t✐♦♥s I ✭tr✉❡✱ ❢❛❧s❡✱ ✉♥❞❡✜♥❡❞✮✱ ✈✐❡✇❡❞ ❛s s❡ts ♦❢ ❣r♦✉♥❞ ❧✐t❡r❛❧s✳ ❊♠♣❧♦② ✉♥❢♦✉♥❞❡❞ s❡ts t♦ ♠❛❦❡ ❛t♦♠s ❞❡✜♥✐t❡❧② ❢❛❧s❡❀ ❛ ✉♥✐q✉❡ ♠❛①✐♠❛❧ ✭❂❣r❡❛t❡st✮ ✉♥❢♦✉♥❞❡❞ s❡t ❡①✐sts ❢♦r ❛♥② ✐♥t❡r♣r❡t❛t✐♦♥ I✳ ❉❡✜♥❡ ♠♦♥♦t♦♥✐❝ ♦♣❡r❛t♦rs T

S(I) ✭✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡s✮✱ U S

✭❣r❡❛t❡st ✉♥❢♦✉♥❞❡❞ s❡t✮✱ W

S ❂ T S ∪ U S

❚❤❡ ✇❡❧❧✲❢♦✉♥❞❡❞ ♠♦❞❡❧ ♦❢ ❛ s❡t ♦❢ ♥♦r♠❛❧ ❝❧❛✉s❡s S ✐s ❣✐✈❡♥ ❜② lfp(W

S)❀ ✐t

♠❛② ❜❡ ♣❛rt✐❛❧ ♦r t♦t❛❧

Pr♦❜❧❡♠

❈♦♠♣✉t✐♥❣ ✉♥❢♦✉♥❞❡❞ s❡t U

S ✭❣✉❡ss✐♥❣✮

❆ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t Pr♦❝❡❞✉r❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✶✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❚❤❡ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t Pr♦❝❡❞✉r❡

❈❡♥tr❛❧ ■❞❡❛

■t❡r❛t✐✈❡❧② ❜✉✐❧❞ ✉♣ ❛ s❡t ♦❢ ♥❡❣❛t✐✈❡ ❝♦♥❝❧✉s✐♦♥s ˜ A✱ ✇❤✐❝❤ ✉♥❞❡r❡st✐♠❛t❡s t❤❡ s❡t ♦❢ ❛t♦♠s t❤❛t ❛r❡ ❢❛❧s❡ ✐♥ ❲❋❙✳ ❚❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ ♣♦s✐t✐✈❡ ❝♦♥❝❧✉s✐♦♥s ❢r♦♠ t❤❡ ❡✈❡♥t✉❛❧ ˜ A str❛✐❣❤t❢♦r✇❛r❞✳

▼❡t❤♦❞✿ ❊❛❝❤ ✐t❡r❛t✐♦♥ ✐s ❛ t✇♦✲♣❤❛s❡ ♣r♦❝❡ss ❙✉♣♣♦s❡ ˜ I ✐s ❛♥ ✉♥❞❡r❡st✐♠❛t❡ ♦❢ t❤❡ ♥❡❣❛t✐✈❡ ❝♦♥❝❧✉s✐♦♥s ✉♥❞❡r ❲❋❙

✶ ❚r❛♥s❢♦r♠ ˜

I ✐♥t♦ ❛♥ ♦✈❡r❡st✐♠❛t❡ ❜② ˜ SP (˜ I) := lfp(T

P ˜

I ) := ¬ · (HB P − lfp(T

P ˜

I )),

✇❤❡r❡ P˜

I = P ∪ ˜

I✱ ✈✐❡✇✐♥❣ ♥❡❣❛t❡❞ ♣r❡❞✐❝❛t❡s ❛s ♥❡✇ ♣r❡❞✐❝❛t❡ s②♠❜♦❧s ✭HB P ✳ ✳ ✳ ❍❡r❜r❛♥❞ ❜❛s❡ ♦❢ P✮

✷ ❚r❛♥s❢♦r♠ t❤❡ ♦✈❡r❡st✐♠❛t❡ ❜❛❝❦ t♦ ❛♥ ✉♥❞❡r❡st✐♠❛t❡ ❜②

AP (˜ I) := ˜ SP (˜ SP (˜ I)) ❲❡ ❤❛✈❡ ˜ I ⊆ AP (˜ I) = ˜ S2

P (˜

I)❀ ✐♥✐t✐❛❧❧②✱ s❡t ˜ I = ∅✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t Pr♦❝❡❞✉r❡

˜ I ˜ W ✭♥❡❣❛t✐✈❡✮ W ? ✭✉♥❞❡✜♥❡❞✮ W + ✭♣♦s✐t✐✈❡✮ ✳ ✳ ✳ ˜ SP (˜ I) ˜ S2i+1

P

(˜ I) ✳ ✳ ✳ ˜ Ai

P (˜

I) = ˜ S2i

P (˜

I) ˜ S5

P (˜

I) ˜ A2

P (˜

I) = ˜ S4

P (˜

I) ˜ S3

P (˜

I) ˜ AP (˜ I) = ˜ S2

P (˜

I)

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✸✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✻✳ ❖♣❡r❛t✐♦♥❛❧ ❙❡♠❛♥t✐❝s ♦❢ ❘✉❧❡ ▲❛♥❣✉❛❣❡s ✻✳✻ ❲❡❧❧✲❋♦✉♥❞❡❞ ❙❡♠❛♥t✐❝s✿ ❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t

❆❧t❡r♥❛t✐♥❣ ❋✐①♣♦✐♥t Pr♦❝❡❞✉r❡✿ ❊①❛♠♣❧❡

a ← c, ¬b. b ← ¬a✳ c✳ p ← q, ¬s. p ← r, ¬s. p ← t. q ← p. r ← q. r ← ¬c✳

HB P = {a, b, c, p, q, r, s, t} ˜ I0 = ∅ lfp(T

P ∪˜ I0) = {c}

˜ I1 = ˜ SP (˜ I0) = ¬ · (HB P − lfp(T

P ∪˜ I0)) =

{¬a, ¬b, ¬p, ¬q, ¬r, ¬s, ¬t} lfp(T

P ∪˜ I1) = {c, a, b}

˜ I2 = ˜ SP (˜ I1) = ¬ · (HB P − lfp(T

P ∪˜ I1)) =

{¬p, ¬q, ¬r, ¬s, ¬t} ˜ I3 = ˜ I1 ❛♥❞ ˜ I4 = ˜ I2✳ ❋✐①♣♦✐♥t r❡❛❝❤❡❞✦ ❚❤❡ ✇❡❧❧✲❢♦✉♥❞❡❞ ♠♦❞❡❧ ✐s {c, ¬p, ¬q, ¬r, ¬s, ¬t}✳

◆♦t❡ ❋♦r ♣r♦♣♦s✐t✐♦♥❛❧ ♣r♦❣r❛♠s P✱ t❤❡ ❆❋P ❝♦♠♣✉t❛t✐♦♥ ✐s ♣♦❧②♥♦♠✐❛❧✳ ❲❤❡t❤❡r ❢♦r s✉❝❤ P t❤❡ ✇❡❧❧✲❢♦✉♥❞❡❞ ♠♦❞❡❧ ✐s ❝♦♠♣✉t❛❜❧❡ ✐♥ ❧✐♥❡❛r t✐♠❡ ✐s ✉♥❦♥♦✇♥✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✹✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❯♥t✐❧ ♥♦✇✱ ✇❡ ❤❛✈❡ s❡❡♥✿ ❢♦r✇❛r❞ ❝❤❛✐♥✐♥❣ ✭❞❛t❛ ❞r✐✈❡♥✮ ❡✈❛❧✉❛t✐♦♥ ♦❢ ▲P ❜❛❝❦✇❛r❞ ❝❤❛✐♥✐♥❣ ✭❣♦❛❧ ❞r✐✈❡♥✮ ❡✈❛❧✉❛t✐♦♥ ♦❢ ▲P ✐♠♣r♦✈❡♠❡♥t ♦❢ ❜❛❝❦✇❛r❞ ❝❤❛✐♥✐♥❣ ❜② t❛❜❧✐♥❣ ■❞❡❛ ♦❢ t❤❡ ♠❛❣✐❝ t❡♠♣❧❛t❡s tr❛♥s❢♦r♠❛t✐♦♥✿ t❛❦❡ t❤❡ ❜❡st ♦❢ ❜♦t❤ ✇♦r❧❞s✿

• ❊✣❝✐❡♥❝② ♦❢ ❣♦❛❧ ❞✐r❡❝t❡❞♥❡ss
• ●♦♦❞ t❡r♠✐♥❛t✐♦♥ ♣r♦♣❡rt✐❡s ♦❢ ❢♦r✇❛r❞ ❝❤❛✐♥✐♥❣
• ❊❛s② ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛ ❢♦r✇❛r❞ ❝❤❛✐♥✐♥❣ r✉❧❡ ❡♥❣✐♥❡

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✺✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❛❣✐❝ ❚❡♠♣❧❛t❡s ✕ ❊①❛♠♣❧❡

t✭❳✱❨✮ ← r✭❳✱❨✮ t✭❳✱❩✮ ← t✭❳✱❨✮✱ t✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

❇♦tt♦♠✲✉♣ ❡✈❛❧✉❛t✐♦♥ ♣r♦❞✉❝❡s ♠❛♥② ❢❛❝ts✿

t✭❛✱❜✮✱ t✭❛✱❝✮✱ t✭❛✱❞✮✱ t✭❜✱❝✮✱ t✭❜✱❞✮✱ t✭❝✱❞✮

❖♥❧② t✭❜✱❝✮✱ t✭❜✱❞✮ ❛r❡ r❡❧❡✈❛♥t ❢♦r q✉❡r② ❛♥s✇❡rs✳ ■❞❡❛✿

❯t✐❧✐③❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✇❤✐❝❤ ✈❛r✐❛❜❧❡s ✐♥ ❛t♦♠ ❛r❡ ❜♦✉♥❞ ♦r ❢r❡❡ ❢♦r ❡✈❛❧✉❛t✐♦♥✳ ❘❡✇r✐t❡ t❤❡ ♣r♦❣r❛♠ ✐♥t♦ ❛♥ ❛❞♦r♥❡❞ ♣r♦❣r❛♠✱ r❡s♣❡❝t✐♥❣ ❜✐♥❞✐♥❣ ♣❛tt❡r♥s✳ ❚r❛♥s❢♦r♠ t❤❡ ❛❞♦r♥❡❞ ♣r♦❣r❛♠ ✐♥t♦ ❛ s❡t ♦❢ r✉❧❡s t❤❛t ❝❛♥ ❜❡ ❡✣❝✐❡♥t❧② ❡✈❛❧✉❛t❡❞ ❜♦tt♦♠ ✉♣✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❛❣✐❝ ❚❡♠♣❧❛t❡s ✕ ❊①❛♠♣❧❡

t✭❳✱❨✮ ← r✭❳✱❨✮ t✭❳✱❩✮ ← t✭❳✱❨✮✱ t✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

❇♦tt♦♠✲✉♣ ❡✈❛❧✉❛t✐♦♥ ♣r♦❞✉❝❡s ♠❛♥② ❢❛❝ts✿

t✭❛✱❜✮✱ t✭❛✱❝✮✱ t✭❛✱❞✮✱ t✭❜✱❝✮✱ t✭❜✱❞✮✱ t✭❝✱❞✮

❖♥❧② t✭❜✱❝✮✱ t✭❜✱❞✮ ❛r❡ r❡❧❡✈❛♥t ❢♦r q✉❡r② ❛♥s✇❡rs✳ ■❞❡❛✿

❯t✐❧✐③❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✇❤✐❝❤ ✈❛r✐❛❜❧❡s ✐♥ ❛t♦♠ ❛r❡ ❜♦✉♥❞ ♦r ❢r❡❡ ❢♦r ❡✈❛❧✉❛t✐♦♥✳ ❘❡✇r✐t❡ t❤❡ ♣r♦❣r❛♠ ✐♥t♦ ❛♥ ❛❞♦r♥❡❞ ♣r♦❣r❛♠✱ r❡s♣❡❝t✐♥❣ ❜✐♥❞✐♥❣ ♣❛tt❡r♥s✳ ❚r❛♥s❢♦r♠ t❤❡ ❛❞♦r♥❡❞ ♣r♦❣r❛♠ ✐♥t♦ ❛ s❡t ♦❢ r✉❧❡s t❤❛t ❝❛♥ ❜❡ ❡✣❝✐❡♥t❧② ❡✈❛❧✉❛t❡❞ ❜♦tt♦♠ ✉♣✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❛❣✐❝ ❚❡♠♣❧❛t❡s ✕ ❊①❛♠♣❧❡

t✭❳✱❨✮ ← r✭❳✱❨✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

❇♦tt♦♠✲✉♣ ❡✈❛❧✉❛t✐♦♥ ♣r♦❞✉❝❡s ♠❛♥② ❢❛❝ts✿

t✭❛✱❜✮✱ t✭❛✱❝✮✱ t✭❛✱❞✮✱ t✭❜✱❝✮✱ t✭❜✱❞✮✱ t✭❝✱❞✮

❖♥❧② t✭❜✱❝✮✱ t✭❜✱❞✮ ❛r❡ r❡❧❡✈❛♥t ❢♦r q✉❡r② ❛♥s✇❡rs✳ ■❞❡❛✿

• ❯t✐❧✐③❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✇❤✐❝❤ ✈❛r✐❛❜❧❡s ✐♥ ❛t♦♠ ❛r❡ ❜♦✉♥❞ ♦r ❢r❡❡ ❢♦r

❡✈❛❧✉❛t✐♦♥✳

• ❘❡✇r✐t❡ t❤❡ ♣r♦❣r❛♠ ✐♥t♦ ❛♥ ❛❞♦r♥❡❞ ♣r♦❣r❛♠✱ r❡s♣❡❝t✐♥❣ ❜✐♥❞✐♥❣ ♣❛tt❡r♥s✳
• ❚r❛♥s❢♦r♠ t❤❡ ❛❞♦r♥❡❞ ♣r♦❣r❛♠ ✐♥t♦ ❛ s❡t ♦❢ r✉❧❡s t❤❛t ❝❛♥ ❜❡ ❡✣❝✐❡♥t❧②

❡✈❛❧✉❛t❡❞ ❜♦tt♦♠ ✉♣✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✻✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❆❞♦r♥♠❡♥t ♦❢ ❉❛t❛❧♦❣ ♣r♦❣r❛♠s

❙✐❞❡✇❛②s ■♥❢♦r♠❛t✐♦♥ P❛ss✐♥❣ ❙tr❛t❡❣②

❆ s✐❞❡✇❛②s ✐♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ str❛t❡❣② ✭❙■P❙✮ ❞❡t❡r♠✐♥❡s ❤♦✇ ✈❛r✐❛❜❧❡ ❜✐♥❞✐♥❣s ❣❛✐♥❡❞ ❢r♦♠ t❤❡ ✉♥✐✜❝❛t✐♦♥ ♦❢ ❛ r✉❧❡ ❤❡❛❞ ✇✐t❤ ❛ ❣♦❛❧ ♦r s✉❜✲❣♦❛❧ ❛r❡ ♣❛ss❡❞ t♦ t❤❡ ❜♦❞② ♦❢ t❤❡ r✉❧❡✱ ❛♥❞ ❤♦✇ t❤❡② ❛r❡ ♣❛ss❡❞ ❢r♦♠ ❛ s❡t ♦❢ ❧✐t❡r❛❧s ✐♥ t❤❡ ❜♦❞② t♦ ❛♥♦t❤❡r ❧✐t❡r❛❧✳ ❊✈❛❧✉❛t✐♦♥ ✐♥ Pr♦❧♦❣ ✐♠♣❧❡♠❡♥ts ❛ s♣❡❝✐❛❧ ❙■P❙ ✭❤❡❛❞✲t♦✲❜♦❞②✱ ❧❡❢t t♦ r✐❣❤t✮✳ ▼❛♥② ♦t❤❡r ❙■P❙ ♠✐❣❤t ❜❡ ❝♦♥✈❡♥✐❡♥t✳ ❲✳❧✳♦✳❣✳✱ t❤❡ q✉❡r② Q ✐s ♦❢ ❢♦r♠ ← q(t1, . . . , tn)✳

❇✐♥❞✐♥❣ P❛tt❡r♥

❆ ❜✐♥❞✐♥❣ ♣❛tt❡r♥ ❢♦r ❛♥ n✲❛r② ♣r❡❞✐❝❛t❡ ✐s ❛ str✐♥❣ x1 · · · xn✱ n ≥ 0✱ ✇❤❡r❡ ❡❛❝❤ xi ∈ {b, f} ✭✐♥t✉✐t✐✈❡❧②✱ b ♠❡❛♥s ✏❜♦✉♥❞✑ ❛♥❞ f ♠❡❛♥s ✏❢r❡❡✑✮✳ ❚❤❡ ❜✐♥❞✐♥❣ ♣❛tt❡r♥ ❢♦r t❤❡ q✉❡r② ❛t♦♠ q(t1, . . . , tn) ✐s x1 · · · xn s✉❝❤ t❤❛t xi = b ✐✛ ti ✐s ❛ ❝♦♥st❛♥t✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✼✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❘✉❧❡ ❆❞♦r♥♠❡♥t

• ✐✈❡♥ ❛ r✉❧❡

p(t1, . . . , tn) ← p1(t1,1, . . . , t1,n1), . . . pm(tm,1, . . . , tm,nm) ❛♥❞ ❛ ❜✐♥❞✐♥❣ ♣❛tt❡r♥ bp = x1 · · · xn ❢♦r p✱ t❤❡ r✉❧❡ ❛❞♦r♥❡❞ ✇✐t❤ bp✱ pbp(t1, . . . , tn) ← pa1

1 (t1,1, . . . , t1,n1), . . . pam m (tm,1, . . . , tm,nm),

✐s ❝♦♥str✉❝t❡❞ ❧❡❢t t♦ r✐❣❤t✱ ✇❤❡r❡ ❢♦r ❡①t❡♥s✐♦♥❛❧ pi✱ ai = ǫ ❛♥❞ ♦t❤❡r✇✐s❡ ✐♥ ai = xi,i1 · · · xi,ni ✇❡ ❤❛✈❡ xi,j = b ✐✛ ti,j ✐s ❡✐t❤❡r ❛ ❝♦♥st❛♥t ♦r ❡q✉❛❧ s♦♠❡ t′

j ♦r s♦♠❡ ti′,j′ ✇❤❡r❡ i′ < i✳

❙t❛rt✐♥❣ ✇✐t❤ t❤❡ ❜✐♥❞✐♥❣ ♣❛tt❡r♥ bp ❢♦r t❤❡ q✉❡r② ❛t♦♠ q(t1, . . . , tn)✱ ❛❧❧ r✉❧❡s ✇❤♦s❡ ❤❡❛❞ ✉♥✐✜❡s ✇✐t❤ t❤❡ q✉❡r② ❛t♦♠ q(t1, . . . , tn) ❛r❡ ❛❞♦r♥❡❞ ✇✐t❤ bp✳ ❘❡❝✉rs✐✈❡❧②✱ ❢♦r ❡❛❝❤ ❛❞♦r♥❡❞ ❛t♦♠ pai

i (ti,1, . . . , ti,ni)✱ ❛❧❧ r✉❧❡s ✇❤♦s❡ ❤❡❛❞

✉♥✐✜❡s ✇✐t❤ pi(ti,1, . . . , ti,ni) ❛r❡ ❛❞♦r♥❡❞ ✇✐t❤ ai✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✽✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

t✭❳✱❨✮ ← r✭❳✱❨✮ t✭❳✱❩✮ ← t✭❳✱❨✮✱ t✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

▲❛❜❡❧ ♦❝❝✉rr❡♥❝❡s ♦❢ t ❢♦r ❜❡tt❡r ❞✐st✐♥❝t✐♦♥ ■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t r✳ t t ✳ t t ✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✾✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

t✭❳✱❨✮ ← r✭❳✱❨✮ t1✭❳✱❩✮ ← t2✭❳✱❨✮✱ t3✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

▲❛❜❡❧ ♦❝❝✉rr❡♥❝❡s ♦❢ t ❢♦r ❜❡tt❡r ❞✐st✐♥❝t✐♦♥ ■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✾✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

tbf✭❳✱❨✮ ← r✭❳✱❨✮ t1bf✭❳✱❩✮ ← t2bf✭❳✱❨✮✱ t3bf✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

▲❛❜❡❧ ♦❝❝✉rr❡♥❝❡s ♦❢ t ❢♦r ❜❡tt❡r ❞✐st✐♥❝t✐♦♥ ■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✸✾✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

• ♦❛❧✲❉✐r❡❝t❡❞ ❘❡✇r✐t✐♥❣
• ✐✈❡♥ t❤❡ ❛❞♦r♥❡❞ ♣r♦❣r❛♠ P ad✱ tr❛♥s❢♦r♠ ✐t ✐♥t♦ ❛ ♣r♦❣r❛♠ P ad

m s✉❝❤ t❤❛t

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

m ✳

■♥t✉✐t✐♦♥✿ ♣r♦✈✐❞❡ ♣♦ss✐❜❧❡ ✈❛❧✉❡s ❢♦r t❤❡ ❜♦✉♥❞ ❛r❣✉♠❡♥ts ♦❢ ❛ ♣r❡❞✐❝❛t❡ ✭♠❛❣✐❝ s❡ts✮✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✵✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❡t❤♦❞✿

✶ ❋♦r ❡❛❝❤ ❛❞♦r♥❡❞ ♣r❡❞✐❝❛t❡ ♣a✱ ❝r❡❛t❡ ❛ ♣r❡❞✐❝❛t❡ ♠❛❣✐❝❴♣a ✇❤♦s❡ ❛r✐t② ✐s t❤❡

♥✉♠❜❡r ♦❢ b✬s ✐♥ a✳

✷ ❋♦r t❤❡ q✉❡r② ❛t♦♠ q(t1, . . . , tn) ✇✐t❤ ❜✐♥❞✐♥❣ ♣❛tt❡r♥ a✱ ❛❞❞ t♦ P ad ❛ ❢❛❝t

♠❛❣✐❝❴qa(c1, . . . , cm) ✇❤❡r❡ c1, . . . , cm ❛r❡ t❤❡ ❝♦♥st❛♥ts ❛♠♦♥❣ t1, . . . , tn ✭s❡❡❞✮✳

✸ ■♥tr♦❞✉❝❡ r✉❧❡s ❢♦r ❝♦♠♣✉t✐♥❣ s✉❜❣♦❛❧s r❡✢❡❝t✐♥❣ ❙■P✳

❋♦r pbp(t1, . . . , tn) ← pa1

1 (

t1), . . . , pam

m (

tm) ✭✶✮ ❛❞❞ t♦ P ad ❢♦r 0 ≤ j < m r✉❧❡s ♠❛❣✐❝❴♣

aj+1 j+1 (x1, . . . , xnj+1) ← ♠❛❣✐❝❴♣bp(t1, . . . , tn), p1(

t1), . . . , pj( tj) ✭✷✮ ✇❤❡r❡ pj+i ✐s ✐♥t❡♥s✐♦♥❛❧ ❛♥❞ x1, . . . , xnj+1 ❛r❡ t❤❡ ❜♦✉♥❞ ✈❛r✐❛❜❧❡s ❛♠♦♥❣ tj+1✳

✹ ❆❞❛♣t t❤❡ ♦r✐❣✐♥❛❧ r✉❧❡s ✭✶✮ ♦❢ P ad✳

❆❞❞ ✐♥ t❤❡ ❜♦❞②

• ♠❛❣✐❝❴♣bp(t1, . . . , tn)✱
• ♠❛❣✐❝❴♣

aj+1 j+1 (x1, . . . , xnj+1) ❢♦r ❡❛❝❤ ♠❛❣✐❝ r✉❧❡ ✭✷✮ ❛❜♦✈❡✱ ✉♥❧❡ss ❛❧❧ xi ❛r❡

❜♦✉♥❞ ❜② ❡①t❡♥s✐♦♥❛❧ ♣r❡❞✐❝❛t❡s✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✶✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

tbf✭❳✱❨✮ ← r✭❳✱❨✮ t1bf✭❳✱❩✮ ← t2bf✭❳✱❨✮✱ t3bf✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

tbf✭❳✱❨✮ ← r✭❳✱❨✮ t1bf✭❳✱❩✮ ← t2bf✭❳✱❨✮✱ t3bf✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮ ♠❛❣✐❝❴tbf✭❜✮✳ s❡❡❞

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

tbf✭❳✱❨✮ ← r✭❳✱❨✮ t1bf✭❳✱❩✮ ← t2bf✭❳✱❨✮✱ t3bf✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮ ♠❛❣✐❝❴tbf✭❜✮✳ ♠❛❣✐❝❴tbf✭❳✮ ← ♠❛❣✐❝❴tbf✭❳✮✳ s❡❡❞ ▼❛❣✐❝ ❘✉❧❡s

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

tbf✭❳✱❨✮ ← r✭❳✱❨✮ t1bf✭❳✱❩✮ ← t2bf✭❳✱❨✮✱ t3bf✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮ ♠❛❣✐❝❴tbf✭❜✮✳ ♠❛❣✐❝❴tbf✭❳✮ ← ♠❛❣✐❝❴tbf✭❳✮✳ ♠❛❣✐❝❴tbf✭❨✮ ← ♠❛❣✐❝❴tbf✭❳✮✱ t✭❳✱❨✮✳ s❡❡❞ ▼❛❣✐❝ ❘✉❧❡s

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

tbf✭❳✱❨✮ ← r✭❳✱❨✮ t1bf✭❳✱❩✮ ← t2bf✭❳✱❨✮✱ t3bf✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮ ♠❛❣✐❝❴tbf✭❜✮✳ ♠❛❣✐❝❴tbf✭❳✮ ← ♠❛❣✐❝❴tbf✭❳✮✳ ♠❛❣✐❝❴tbf✭❨✮ ← ♠❛❣✐❝❴tbf✭❳✮✱ t✭❳✱❨✮✳ t✭❳✱❨✮ ← ♠❛❣✐❝❴tbf✭❳✮✱ r✭❳✱❨✮✳ s❡❡❞ ▼❛❣✐❝ ❘✉❧❡s ❘❡✇r✐tt❡♥ ❘✉❧❡s

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

tbf✭❳✱❨✮ ← r✭❳✱❨✮ t1bf✭❳✱❩✮ ← t2bf✭❳✱❨✮✱ t3bf✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮ ♠❛❣✐❝❴tbf✭❜✮✳ ♠❛❣✐❝❴tbf✭❳✮ ← ♠❛❣✐❝❴tbf✭❳✮✳ ♠❛❣✐❝❴tbf✭❨✮ ← ♠❛❣✐❝❴tbf✭❳✮✱ t✭❳✱❨✮✳ t✭❳✱❨✮ ← ♠❛❣✐❝❴tbf✭❳✮✱ r✭❳✱❨✮✳ t✭❳✱❩✮ ← ♠❛❣✐❝❴tbf✭❳✮✱t✭❳✱❨✮✱ ♠❛❣✐❝❴tbf✭❨✮✱t✭❨✱❩✮✳ s❡❡❞ ▼❛❣✐❝ ❘✉❧❡s ❘❡✇r✐tt❡♥ ❘✉❧❡s

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

tbf✭❳✱❨✮ ← r✭❳✱❨✮ t1bf✭❳✱❩✮ ← t2bf✭❳✱❨✮✱ t3bf✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮ ♠❛❣✐❝❴tbf✭❜✮✳ ♠❛❣✐❝❴tbf✭❳✮ ← ♠❛❣✐❝❴tbf✭❳✮✳ ♠❛❣✐❝❴tbf✭❨✮ ← ♠❛❣✐❝❴tbf✭❳✮✱ t✭❳✱❨✮✳ t✭❳✱❨✮ ← ♠❛❣✐❝❴tbf✭❳✮✱ r✭❳✱❨✮✳ t✭❳✱❩✮ ← ♠❛❣✐❝❴tbf✭❳✮✱t✭❳✱❨✮✱ ♠❛❣✐❝❴tbf✭❨✮✱t✭❨✱❩✮✳ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ s❡❡❞ ▼❛❣✐❝ ❘✉❧❡s ❘❡✇r✐tt❡♥ ❘✉❧❡s ❊①t❡♥s✐♦♥❛❧ ❋❛❝ts

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

tbf✭❳✱❨✮ ← r✭❳✱❨✮ t1bf✭❳✱❩✮ ← t2bf✭❳✱❨✮✱ t3bf✭❨✱❩✮ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ← t✭❜✱ ❆♥s✇❡r✮✳

• ♦❛❧✲❞✐r❡❝t❡❞ ❡✈❛❧✉❛t✐♦♥

■♥❢♦r♠❛t✐♦♥ ♣❛ss✐♥❣ t ֒ →X r✳ t1 ֒ →X t2✳ t2 ֒ →Y t3✳ ❆❞♦r♥♠❡♥t ✭❜♦✉♥❞✱ ❢r❡❡✮ ♠❛❣✐❝❴tbf✭❜✮✳ ♠❛❣✐❝❴tbf✭❳✮ ← ♠❛❣✐❝❴tbf✭❳✮✳ ♠❛❣✐❝❴tbf✭❨✮ ← ♠❛❣✐❝❴tbf✭❳✮✱ t✭❳✱❨✮✳ t✭❳✱❨✮ ← ♠❛❣✐❝❴tbf✭❳✮✱ r✭❳✱❨✮✳ t✭❳✱❩✮ ← ♠❛❣✐❝❴tbf✭❳✮✱t✭❳✱❨✮✱ ♠❛❣✐❝❴tbf✭❨✮✱t✭❨✱❩✮✳ r✭❛✱❜✮✳ r✭❜✱❝✮✳ r✭❝✱❞✮✳ ❊✈❛❧✉❛t✐♦♥✿ ♠❛❣✐❝❴tbf✭❜✮✳ t✭❜✱❝✮✳ ♠❛❣✐❝❴tbf✭❝✮✳ t✭❝✱❞✮✳ ♠❛❣✐❝❴tbf✭❞✮✳ t✭❜✱❞✮✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✷✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❛❣✐❝ ❙❡t ❚r❛♥s❢♦r♠❛t✐♦♥ ✇✐t❤ ◆❡❣❛t✐♦♥

Pr♦❜❧❡♠ ✇✐t❤ ♥❡❣❛t✐♦♥

❊✈❡♥ ❢♦r str❛t✐✜❡❞ ♣r♦❣r❛♠s✱ t❤❡ ♠❛❣✐❝ s❡t tr❛♥s❢♦r♠❛t✐♦♥ ✭▼❙❚✮ ♠❛② ❤❛✈❡ ✉♥str❛t✐✜❡❞ ♦✉t❝♦♠❡✳

❈❛✉s❡s ❢♦r ✉♥str❛t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ▼❙❚

✶ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛ ❧✐t❡r❛❧ ✐♥ ❛ r✉❧❡ ❜♦❞② ✷ ♠✉❧t✐♣❧❡ ♥❡❣❛t✐✈❡ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛ ❧✐t❡r❛❧ ✐♥ ❛ r✉❧❡ ❜♦❞② ✸ ♥❡❣❛t✐✈❡ ❧✐t❡r❛❧ ✐♥ ❛ r❡❝✉rs✐✈❡ r✉❧❡

❙♦❧✉t✐♦♥ ✭❙♦✉r❝❡ ✶✮

❞✐st✐♥❝t✐♦♥ ♦❢ ❝♦♥t❡①ts ♦❢ ♣r♦❜❧❡♠❛t✐❝ ❛t♦♠s✿ ❧❛❜❡❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ♣r❡❞✐❝❛t❡ p t♦ p❴1✱ p❴2 ❡t❝ r❡♣❧✐❝❛t❡ ❡❛❝❤ r✉❧❡ ❞❡✜♥✐♥❣ p ✇✐t❤ p❴i ✭❛❧s♦ ✐♥ t❤❡ ❜♦❞②✮✱ ❢♦r ❛❧❧ p❴i

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✸✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❛❣✐❝ ❙❡t ❚r❛♥s❢♦r♠❛t✐♦♥ ✇✐t❤ ◆❡❣❛t✐♦♥

Pr♦❜❧❡♠ ✇✐t❤ ♥❡❣❛t✐♦♥

❊✈❡♥ ❢♦r str❛t✐✜❡❞ ♣r♦❣r❛♠s✱ t❤❡ ♠❛❣✐❝ s❡t tr❛♥s❢♦r♠❛t✐♦♥ ✭▼❙❚✮ ♠❛② ❤❛✈❡ ✉♥str❛t✐✜❡❞ ♦✉t❝♦♠❡✳

❈❛✉s❡s ❢♦r ✉♥str❛t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ▼❙❚

✶ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛ ❧✐t❡r❛❧ ✐♥ ❛ r✉❧❡ ❜♦❞② ✷ ♠✉❧t✐♣❧❡ ♥❡❣❛t✐✈❡ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛ ❧✐t❡r❛❧ ✐♥ ❛ r✉❧❡ ❜♦❞② ✸ ♥❡❣❛t✐✈❡ ❧✐t❡r❛❧ ✐♥ ❛ r❡❝✉rs✐✈❡ r✉❧❡

❙♦❧✉t✐♦♥ ✭❙♦✉r❝❡ ✶✮

❞✐st✐♥❝t✐♦♥ ♦❢ ❝♦♥t❡①ts ♦❢ ♣r♦❜❧❡♠❛t✐❝ ❛t♦♠s✿ ❧❛❜❡❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ♣r❡❞✐❝❛t❡ p t♦ p❴1✱ p❴2 ❡t❝ r❡♣❧✐❝❛t❡ ❡❛❝❤ r✉❧❡ ❞❡✜♥✐♥❣ p ✇✐t❤ p❴i ✭❛❧s♦ ✐♥ t❤❡ ❜♦❞②✮✱ ❢♦r ❛❧❧ p❴i

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✸✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❛❣✐❝ ❙❡t ❚r❛♥s❢♦r♠❛t✐♦♥ ✇✐t❤ ◆❡❣❛t✐♦♥

Pr♦❜❧❡♠ ✇✐t❤ ♥❡❣❛t✐♦♥

❊✈❡♥ ❢♦r str❛t✐✜❡❞ ♣r♦❣r❛♠s✱ t❤❡ ♠❛❣✐❝ s❡t tr❛♥s❢♦r♠❛t✐♦♥ ✭▼❙❚✮ ♠❛② ❤❛✈❡ ✉♥str❛t✐✜❡❞ ♦✉t❝♦♠❡✳

❈❛✉s❡s ❢♦r ✉♥str❛t✐✜❝❛t✐♦♥ ♦❢ t❤❡ ▼❙❚

✶ ♣♦s✐t✐✈❡ ❛♥❞ ♥❡❣❛t✐✈❡ ♦❝❝✉rr❡♥❝❡ ♦❢ ❛ ❧✐t❡r❛❧ ✐♥ ❛ r✉❧❡ ❜♦❞② ✷ ♠✉❧t✐♣❧❡ ♥❡❣❛t✐✈❡ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛ ❧✐t❡r❛❧ ✐♥ ❛ r✉❧❡ ❜♦❞② ✸ ♥❡❣❛t✐✈❡ ❧✐t❡r❛❧ ✐♥ ❛ r❡❝✉rs✐✈❡ r✉❧❡

❙♦❧✉t✐♦♥ ✭❙♦✉r❝❡ ✶✮

❞✐st✐♥❝t✐♦♥ ♦❢ ❝♦♥t❡①ts ♦❢ ♣r♦❜❧❡♠❛t✐❝ ❛t♦♠s✿ ❧❛❜❡❧ ♦❝❝✉rr❡♥❝❡s ♦❢ ♣r❡❞✐❝❛t❡ p t♦ p❴1✱ p❴2 ❡t❝ r❡♣❧✐❝❛t❡ ❡❛❝❤ r✉❧❡ ❞❡✜♥✐♥❣ p ✇✐t❤ p❴i ✭❛❧s♦ ✐♥ t❤❡ ❜♦❞②✮✱ ❢♦r ❛❧❧ p❴i

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✸✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❊①❛♠♣❧❡

❛✭①✮ ← ♥♦t ❜✭①✮✱ ❝✭①✱②✮✱ ❜✭②✮✳ ❜✭①✮ ← ❝✭①✱②✮✱ ❜✭②✮✳ ♠❛❣✐❝❴❛b✭✶✮✳ ♠❛❣✐❝❴❜b✭①✮ ← ♠❛❣✐❝❴❛b✭①✮ ♠❛❣✐❝❴❜b✭②✮ ← ♠❛❣✐❝❴❛b✭①✮✱ ♥♦t ❜✭①✮✱ ❝✭①✱②✮✳ ❛✭①✮ ← ♠❛❣✐❝❴❛b✭①✮✱ ♥♦t ❜✭①✮✱ ❝✭①✱②✮✱ ❜✭②✮✳ ♠❛❣✐❝❴❜b✭②✮ ← ♠❛❣✐❝❴❜b✭①✮✱ ❝✭①✱②✮✳ ❜✭①✮ ← ♠❛❣✐❝❴❜b✭①✮✱ ❝✭①✱②✮✱ ❜✭②✮✳ ❜ ♦❝❝✉rs ❜♦t❤ ♥❡❣❛t✐✈❡❧② ❛♥❞ ♣♦s✐t✐✈❡❧② ✐♥ t❤❡ ✜rst r✉❧❡✳

magic_ab magic_bb a b

❘❡s✉❧t✐♥❣ ♣r♦❣r❛♠ ✉♥str❛t✐✜❛❜❧❡✦

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✹✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

❛✭①✮ ← ♥♦t ❜❴✶✭①✮✱ ❝✭①✱②✮✱ ❜❴✷✭②✮✳ ❜❴✶✭①✮ ← ❝✭①✱②✮✱ ❜❴✶✭②✮✳ ❜❴✷✭①✮ ← ❝✭①✱②✮✱ ❜❴✷✭②✮✳ ♠❛❣✐❝❴❛b✭✶✮✳ ♠❛❣✐❝❴❜❴✶b✭①✮ ← ♠❛❣✐❝❴❛b✭①✮✳ ♠❛❣✐❝❴❜❴✷b✭②✮ ← ♠❛❣✐❝❴❛b✭①✮✱ ♥♦t ❜❴✶✭①✮✱ ❝✭①✱②✮✳ ♠❛❣✐❝❴❜b✭②✮ ← ♠❛❣✐❝❴❛b✭①✮✱ ♥♦t ❜✭①✮✱ ❝✭①✱②✮✳ ❛✭①✮ ← ♠❛❣✐❝❴❛b✭①✮✱ ♥♦t ❜✭①✮✱ ❝✭①✱②✮✱ ❜✭②✮✳ ♠❛❣✐❝❴❜❴✐b✭②✮ ← ♠❛❣✐❝❴❜❴✐b✭①✮✱ ❝✭①✱②✮✳ ❜❴✐✭①✮ ← ♠❛❣✐❝❴❜❴✐b✭①✮✱ ❝✭①✱②✮✱ ❜❴✐✭②✮✳ ✐❂✶✱✷ ❈♦♥t❡①t ❧❛❜❡❧✐♥❣ ♦❢ ♣r❡❞✐❝❛t❡s ❘✉❧❡ r❡♣❧✐❝❛t✐♦♥

magic_ab magic_b_1b a b_1 magic_b_2b b_2

❘❡s✉❧t ✐s str❛t✐✜❛❜❧❡✦ ❚❤❡ s❡❝♦♥❞ ❛♥❞ t❤✐r❞ s♦✉r❝❡ ♦❢ ✉♥str❛t✐✜❛❜✐❧✐t② ❝❛♥ ❜❡ ❡❧✐♠✐♥❛t❡❞ ♦♥ t❤❡ ❛❞♦r♥❡❞ r✉❧❡ s❡t ✭♣r❡♣r♦❝❡ss✐♥❣✮✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✺✴✹✻

❋♦✉♥❞❛t✐♦♥s ♦❢ ❉❑❙ ✼✳ ❆♣♣❡♥❞✐① ✼✳✶ ❆♣♣❡♥❞✐①✿ ▼❛❣✐❝ ❚❡♠♣❧❛t❡s ❚r❛♥s❢♦r♠❛t✐♦♥

▼❛❣✐❝ ❙❡ts ❢♦r ❯♥str❛t✐✜❡❞ Pr♦❣r❛♠s

❆❧s♦ ❢♦r ✉♥str❛t✐✜❡❞ ❧♦❣✐❝ ♣r♦❣r❛♠s ✉♥❞❡r st❛❜❧❡ ♠♦❞❡❧ s❡♠❛♥t✐❝s✱ ▼❙❚ ❝❛♥ ❜❡ ❞❡✈❡❧♦♣❡❞✳ ❊✳❣✳✱ ❬❋❛❜❡r ❡t ❛❧✳✱ ■❈❉❚ ✷✵✵✺✴❏❈❙❙ ✷✵✵✼❪✿

• ❡❛r❡❞ t♦✇❛r❞s q✉❡r② ❛♥s✇❡r✐♥❣✱ ❛ss✉♠✐♥❣ t❤❛t t❤❡ ♣r♦❣r❛♠ ❤❛s s♦♠❡

st❛❜❧❡ ♠♦❞❡❧✳ ❚❤❡② ✐♥tr♦❞✉❝❡❞ ❛ s✉✐t❛❜❧❡ ♥♦t✐♦♥ ♦❢ ♠♦❞✉❧❡ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t s❡t✱ t♦ ❢♦❝✉s ❝♦♠♣✉t❛t✐♦♥ ♦♥ ❛ s✉❜♣r♦❣r❛♠✳ ❚❤❡ ♠❡t❤♦❞ ♠❛❦❡s ❛❧s♦ ❜♦❞②✲t♦✲❤❡❛❞ ♣r♦♣❛❣❛t✐♦♥ ♦❢ ✈❛❧✉❡s✳ ❢r✉✐t❢✉❧ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♠❛❣✐❝ s❡ts ❡✳❣✳ ✐♥ t❤❡ ❛r❡❛ ♦❢ ❞❛t❛ ✐♥t❡❣r❛t✐♦♥ ✭■◆❋❖▼■❳ ♣r♦❥❡❝t✮✳

❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✹✻✴✹✻