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❖❜❥❡❝t ❚r❛♥s❢❡r ✉s✐♥❣ P❛t❤ ❘❡✈❡rs❛❧✿ ❉✐str✐❜✉t❡❞ P❛t❤✲❘❡✈❡rs❛❧ ❆❧❣♦r✐t❤♠ ❙❤❛♥❦❛r ❙❡♣t❡♠❜❡r ✶✽✱ ✷✵✶✹

❖✉t❧✐♥❡ ♣r ❛❧❣ P❛t❤ r❡✈❡rs❛❧✿ ❛❧❣♦r✐t❤♠ P❛t❤ r❡✈❡rs❛❧✿ s❛❢❡t② ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ ♣r♦❣r❡ss ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ s❡r✐❛❧✐③❛❜✐❧✐t② ❛♥❛❧②s✐s

P❛t❤ r❡✈❡rs❛❧ ❛❧❣♦r✐t❤♠ ♣r ❛❧❣ ❙②st❡♠s ❛tt❛❝❤❡❞ t♦ ❛ ✜❢♦ ❝❤❛♥♥❡❧❀ ♦❜❥ ✐♥✐t✐❛❧❧② ❛t a0 ▼❡ss❛❣❡s [REQ,j] ✿ r❡q✉❡st ♠s❣ // j ✐s ✐ss✉❡r ✭♥♦t ❢♦r✇❛r❞❡r✮ [OBJ] ✿ ♦❜❥❡❝t✲❝❛rr②✐♥❣ ♠s❣ // ✐❣♥♦r❡ ✈❛❧✉❡ ❢♦r ♥♦✇ ❙②st❡♠ j ✿ ❡❛t✐♥❣ ✭❤❛s ♦❜❥✮✱ ❤✉♥❣r② ✭✇❛♥ts ♦❜❥✮✱ t❤✐♥❦✐♥❣ ✭♦✴✇✮ ❙②st❡♠ j ❤❛s ❛ ✏❧❛st✑ ♣♦✐♥t❡r ❛❞❞r ✐♥ t❤❡ ❧❛st r❡q ♠s❣ r❝✈❞ ❜② j ❛❢t❡r ❧❛st ❜❡❝♦♠✐♥❣ ❤✉♥❣r②✳ ♥✐❧ ✐❢ ♥♦ s✉❝❤ ♠s❣ ✐♥✐t✐❛❧❧② nil ❛t a0 ✱ ❛♥❞ a0 ❡❧s❡✇❤❡r❡ ❙②st❡♠ j ❤❛s ❛ ✏♥❡①t✑ ♣♦✐♥t❡r ♥✐❧ ✐❢ j t❤✐♥❦✐♥❣ ♦r ♥♦t r❝✈❞ r❡q s✐♥❝❡ ♥♦♥✲t❤✐♥❦✐♥❣ ♦✴✇ ❡q✉❛❧s ❛❞❞r ✐♥ t❤❡ ✜rst r❡q ♠s❣ r❝✈❞ s✐♥❝❡ ♥♦♥✲t❤✐♥❦✐♥❣ ✐♥✐t✐❛❧❧② ♥✐❧

❙②st❡♠ j r✉❧❡s ♣r ❛❧❣ ❜❡❝♦♠❡ ❤✉♥❣r② ♦♥❧② ✐❢ t❤✐♥❦✐♥❣✿ ✴✴ H ( j ) s❡♥❞ [REQ,j] t♦ lst s❡t lst t♦ ♥✐❧ r❝✈ [OBJ] ✿ ✴✴ E ( j ) ❜❡❝♦♠❡ ❡❛t✐♥❣ ❜❡❝♦♠❡ t❤✐♥❦✐♥❣ ♦♥❧② ✐❢ ❡❛t✐♥❣ ❛♥❞ nxt ♥♦♥✲♥✐❧✿ ✴✴ T ( j ) s❡♥❞ ♦❜❥❡❝t t♦ nxt s❡t nxt t♦ ♥✐❧ r❝✈ [REQ,k] ✿ ✴✴ R ( j,k ) ✐❢ lst ♥♦t ♥✐❧ s❡♥❞ [REQ,k] t♦ lst s❡t lst t♦ k ❡❧s❡ s❡t nxt ❛♥❞ lst t♦ k

❈♦♥✈❡♥t✐♦♥s ♣r ❛❧❣ j ✕ k ✐s ❛ ❧❛st ❡❞❣❡✿ j.lst ✐s ♥♦t ♥✐❧ ❛♥❞ ❡q✉❛❧s k j ✕ k ✐s ❛ ♥❡①t ❡❞❣❡✿ j.nxt ✐s ♥♦t ♥✐❧ ❛♥❞ ❡q✉❛❧s k j ✕ k ✐s ❛ r❡q✉❡st ❡❞❣❡✿ ♠❡ss❛❣❡ [REQ,j] ✐s ✐♥ tr❛♥s✐t t♦ k ❞✐❣r❛♣❤✿ ❞✐r❡❝t❡❞ ♠✉❧t✐✲❣r❛♣❤ LNR ✿ ❞✐❣r❛♣❤ ❬ ❛❞❞r❡ss❡s❀ ❧❛st✴♥❡①t✴r❡q✉❡st ❡❞❣❡s ❪ L ✿ ❞✐❣r❛♣❤ ❬ ❛❞❞r❡ss❡s❀ ❧❛st ❡❞❣❡s ❪ LR ✿ ❞✐❣r❛♣❤ ❬ ❛❞❞r❡ss❡s❀ ❧❛st✴r❡q✉❡st ❡❞❣❡s ❪ ❉r❛✇✐♥❣ ❝♦♥✈❡♥t✐♦♥s ❧❛st ❡❞❣❡s✿ − − − − − ♥❡①t ❡❞❣❡s✿ ✲ ✲ ✲ ✲ ✲ r❡q✉❡st ❡❞❣❡s✿ · · · · · ·

❙❡r✐❛❧ ❡✈♦❧✉t✐♦♥✿ ❛t ♠♦st ✶ ❤✉♥❣r② ❛t ❛ t✐♠❡ ♣r ❛❧❣ 1 1 1 1 1 1 2 2 2 2 2 3 2 3 3 3 3 3 4 4 4 4 5 4 5 5 5 4 5 5 6 6 6 6 6 6 initially 4 turns hungry, 2 forwards 1 rcvs 1 turns thinking 4 rcvs object, 1 eating sends [REQ,4] to 2 [REQ,4] to 1 [REQ,4] sends obj to 4 turns eating L ✐s ❛♥ ✐♥✲tr❡❡ ✇❤❡♥ ♥♦ r❡q ♠s❣ ✐♥ tr❛♥s✐t ❡❛❝❤ r❡q✉❡st ❡✛❡❝ts ❛ ♣❛t❤ r❡✈❡rs❛❧ j ✬s r❡q tr❛✈❡❧s ❢r♦♠ j t♦ r♦♦t ❛❧❧ ♥♦❞❡s ♦♥ ♣❛t❤ ♥♦✇ ♣♦✐♥t t♦ j ❛♠♦rt✐③❡❞ ❝♦st ♦❢ ❧♦❣ N

❙❡r✐❛❧ ❡✈♦❧✉t✐♦♥✿ ♠♦r❡ t❤❛♥ ✶ ❤✉♥❣r② ❛t ❛ t✐♠❡ ♣r ❛❧❣ 1 1 1 1 1 2 2 3 2 3 3 2 2 3 3 4 5 4 5 4 5 4 5 5 4 initially 3 turns hungry, 1 rcvs [REQ,3]. 1 frwrds [REQ,2]. 1 turns thinking, sends [REQ,3]. 2 turns hungry, 3 rcvs [REQ,2] sends object. 1 eating sends [REQ,2] 3 turns eating L ❡✈♦❧✈❡s ❛s ❜❡❢♦r❡✱ s♦ ❛♠♦rt✐③❡❞ ❝♦st s❛♠❡ ♥❡①t ♣trs ❢♦r♠ q✉❡✉❡

◆♦♥✲s❡r✐❛❧ ❡✈♦❧✉t✐♦♥ ♣r ❛❧❣ 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 5 4 5 4 5 4 5 4 5 4 5 initially 2, 4 turn 2 rcvs 3 turns hungry. 1 forwards 3 rcvs [REQ,2]. 1 eating hungry [REQ,4] 1 rcvs [REQ,3] [REQ,2]. 1 turns thinking. 3 turns eating L ♠❛② ♥❡✈❡r ❜❡ ❛♥ ✐♥✲tr❡❡ ❝❛♥ ❜❡ s❡✈❡r❛❧ ♥❡①t✲♣tr q✉❡✉❡s ♣r♦❣r❡ss❄ ❛♠♦rt✐③❡❞ ❝♦st❄

❉♦❡s ❤✉♥❣r② j ❡✈❡♥t✉❛❧❧② ❡❛t❄ ♣r ❛❧❣ P♦ss✐❜✐❧✐t✐❡s ✇❤❡♥ j ✬s r❡q✉❡st ♠s❣ ❞✐❡s r p p q k k k k p k j j j j j (a) (b) (c) (d) (e) ❆❜♦✈❡ ❤❛s ✐♠♣❧✐❝✐t ❛ss✉♠♣t✐♦♥s L r❡♠❛✐♥s ❛❝②❝❧✐❝ j.nxt ♥❡✈❡r ♣♦✐♥ts t♦ j · · · ◆♦✇ t♦ ♠❛❦❡ ❛r❣✉♠❡♥t r✐❣♦r♦✉s

❖✉t❧✐♥❡ ♣r s❛❢❡t② P❛t❤ r❡✈❡rs❛❧✿ ❛❧❣♦r✐t❤♠ P❛t❤ r❡✈❡rs❛❧✿ s❛❢❡t② ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ ♣r♦❣r❡ss ❛♥❛❧②s✐s P❛t❤ r❡✈❡rs❛❧✿ s❡r✐❛❧✐③❛❜✐❧✐t② ❛♥❛❧②s✐s

❇❛s✐❝ s❛❢❡t② ♣r♦♣❡rt✐❡s ✕ ✶ ♣r s❛❢❡t② ■♥✈ A ✶ ✕ A ✸ ❤♦❧❞s // ✈✐❛ ✐♥✈ r✉❧❡ A ✶ : ❢♦r♦♥❡✭ j ✿ ❡✐t❤❡r ✭ j ❡❛t✐♥❣✮ ♦r ✭♦❜❥ ✐♥ tr❛♥s✐t t♦ j ✮✮ A ✷ : j.nxt � = ♥✐❧ ⇒ ✭ j.lst � = ♥✐❧ ❛♥❞ ✭ j ♥♦t t❤✐♥❦✐♥❣✮✮ A ✸ : ✭ j t❤✐♥❦✐♥❣✮ ⇒ j.lst � = ♥✐❧ ■♥✈ B ✶ ✕ B ✷ ❤♦❧❞s // ✈✐❛ ✐♥✈ r✉❧❡ ❛ss✉♠✐♥❣ ■♥✈ A ✶ ✕ A ✸ B ✶ : LR ❤❛s ❡①❛❝t❧② ✶ ✉♥❞✐r❡❝t❡❞ ♣❛t❤ ❜❡t✇❡❡♥ ❡✈❡r② t✇♦ ♥♦❞❡s B ✷ : ❢♦r❛❧❧✭ j: j.lst � = j ✮

❇❛s✐❝ s❛❢❡t② ♣r♦♣❡rt✐❡s ✕ ✷ ♣r s❛❢❡t② ■♥✈ B ✸ ✕ B ✺ ❤♦❧❞s // ✈✐❛ ✐♥✈ r✉❧❡ ❛ss✉♠✐♥❣ ■♥✈ A ✶ ✕ A ✸ , B ✶ ✕ B ✸ B ✸ : ❢♦r❛❧❧ j ✿ ❡①❛❝t❧② ✶ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s j t❤✐♥❦✐♥❣ ♦r [REQ,j] ✐♥ tr❛♥s✐t ♦r ❢♦rs♦♠❡ ✭ k ✿ k.nxt = j ✮ ♦r [OBJ] ✐♥ tr❛♥s✐t t♦ j ♦r j ❡❛t✐♥❣ B ✹ : ❢♦r❛❧❧ ✭ j ✿ ❛t ♠♦st ♦♥❡ [REQ,j] ✐♥ tr❛♥s✐t✮ B ✺ : ❢♦r❛❧❧ ✭ j ✿ j.nxt � = j ❛♥❞ ♥✉♠✭ k ✿ k.nxt = j ✮ ≤ ✶✮

Pr✐♦r✐t② ❞✐❣r❛♣❤ Pr ♣r s❛❢❡t② ❲❛♥t ❛ ❞✐❣r❛♣❤ Pr t❤❛t ❝❛♣t✉r❡s r❡❧❛t✐✈❡ ♣r✐♦r✐t✐❡s ♦❢ ♥♦❞❡s ❲❛♥t j ✕ k ✐♥ Pr t♦ ♠❡❛♥ j ❤❛s ❧♦✇❡r ♣r✐♦r✐t② t❤❛♥ k j ✕ k ✐s ❛ ♣r✲♥❡①t ❡❞❣❡✿ k ✕ j ✐s ❛ ♥❡①t ❡❞❣❡ j ✕ k ✐s ❛ ♣r✲❧❛st ❡❞❣❡✿ j ✕ k ✐s ❛ ❧❛st ❡❞❣❡ ❛♥❞ j t❤✐♥❦✐♥❣ j ✕ k ✐s ❛ ♣r✲r❡q✉❡st ❡❞❣❡✿ j ✕ k ✐s ❛ r❡q✉❡st ❡❞❣❡ Pr ✿ ❞✐❣r❛♣❤ ❬ ❛❞❞r❡ss❡s❀ r❡q✴♣r✲♥❡①t✴♣r✲❧❛st ❡❞❣❡s ❪ ❉❡✜♥❡ ♣r✲♣❛t❤✿ ❞✐r❡❝t❡❞ ♣❛t❤ ✐♥ Pr j ♣r✲r❡❛❝❤❛❜❧❡ ❢r♦♠ k ✿ ♣r✲♣❛t❤ ❢r♦♠ k t♦ j ❧r✲♣❛t❤✿ ✉♥❞✐r❡❝t❡❞ ♣❛t❤ ✐♥ LR

❙♦♠❡ s❛❢❡t② ♣r♦♣❡rt✐❡s ♦❢ Pr ♣r s❛❢❡t② ■♥✈ C ✶ ✕ C ✸ // ✈✐❛ ✐♥✈ r✉❧❡ ❛ss✉♠✐♥❣ ■♥✈ A ✶ ✕ A ✸ , B ✶ ✕ B ✺ C ✶ : ✭ Pr ✐♥✲tr❡❡✮ ❛♥❞ ✭ Pr ✬s r♦♦t ❡❛t✐♥❣ ♦r ♦❜❥ ✐♥ tr❛♥s✐t t♦ ✐t✮ C ✷ : ♣r✲♣❛t❤ ❢r♦♠ k t♦ j ⇒ ❢♦r❛❧❧ x ♦♥ t❤❡ ❧r✲♣❛t❤ ❜❡t✇❡❡♥ j ❛♥❞ k ✿ ♣r✲♣❛t❤ ❢r♦♠ x t♦ j C ✸ : ✭ j ♥♦t t❤✐♥❦✐♥❣✮ ❛♥❞ j.lst = k � = nil ⇒ ✭✭♣r✲♣❛t❤ ❢r♦♠ k t♦ j ✮ ❛♥❞ ✭ k ❤✉♥❣r②✮✮