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  1. ❋♦✉♥❞❛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 ✷✵✱ ✷✵✶✷ ✶✴✹✻

  2. ❋♦✉♥❞❛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 ✷✵✱ ✷✵✶✷ ✷✴✹✻

  3. ❋♦✉♥❞❛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 ✷✵✱ ✷✵✶✷ ✸✴✹✻

  4. ❋♦✉♥❞❛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 ✷✵✱ ✷✵✶✷ ✹✴✹✻

  5. ❋♦✉♥❞❛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 ❛ ♠❛♣♣✐♥❣ µ P ❢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 µ P ( I ) = HI ( lfp ( T P ∪ I | ext ( P ) )) | int ( P ) . ✭ I P reds ✳ ✳ ✳ r❡str✐❝t✐♦♥ ♦❢ I t♦ Preds ✮✳ ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✺✴✹✻

  6. ❋♦✉♥❞❛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 ✷✵✱ ✷✵✶✷ ✻✴✹✻

  7. ❋♦✉♥❞❛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 ✷✵✱ ✷✵✶✷ ✻✴✹✻

  8. ❋♦✉♥❞❛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 ✮✿ 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 T = lfp ( T P ) ❚❤♦♠❛s ❊✐t❡r ❛♥❞ ❘❡✐♥❤❛r❞ P✐❝❤❧❡r ❉❡❝❡♠❜❡r ✷✵✱ ✷✵✶✷ ✻✴✹✻

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