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❈❤❛♥♥❡❧ ✇✐t❤ ❚❡r♠✐♥❛t✐♦♥ ❉❡t❡❝t✐♦♥ ❙❡r✈✐❝❡

❙❤❛♥❦❛r ❆♣r✐❧ ✷✻✱ ✷✵✶✹

❖✈❡r✈✐❡✇

❉✐str✐❜✉t❡❞ t❡r♠✐♥❛t✐♦♥ ❞❡t❡❝t✐♦♥ ♣r♦❜❧❡♠ ❣✐✈❡♥ ❝♦♠♣✉t❛t✐♦♥ X ✇❤❡r❡ ❡❛❝❤ s②st❡♠ ✐s ❛❝t✐✈❡ ♦r ✐♥❛❝t✐✈❡ ❛❝t✐✈❡✿ s❡♥❞✴r❝✈ ♠s❣s✱ ❜❡❝♦♠❡ ✐♥❛❝t✐✈❡ ✐♥❛❝t✐✈❡✿ ❜❡❝♦♠❡ ❛❝t✐✈❡ ✉♣♦♥ r❝✈✐♥❣ ❛ ♠s❣ ♦❜t❛✐♥ ❝♦♠♣✉t❛t✐♦♥ Y t❤❛t ❞♦❡s ♥♦t ❞✐st✉r❜ X ❛♥❞ ✐♥❢♦r♠s ❛ ✏s✐♥❦✑ s②t❡♠ ✇❤❡♥✴✐❢ X ❤❛s t❡r♠✐♥❛t❡❞ t❡r♠✐♥❛t✐♦♥✿ ❛❧❧ s②st❡♠s ✐♥❛❝t✐✈❡✱ ♥♦ ♠s❣s ✐♥ tr❛♥s✐t ❋♦r♠❛❧✐③❡ ❛s ✜❢♦ ❝❤❛♥♥❡❧ ✇✐t❤ t❡r♠✐♥❛t✐♦♥ ❞❡t❡❝t✐♦♥ s❡r✈✐❝❡ s✉❜s❡t ♦❢ ❛❞❞r❡ss❡s ❛❝t✐✈❡ ✐♥✐t✐❛❧❧② ❛t ❡❛❝❤ ❛❞❞r✿ s❡♥❞ ♠s❣s❀ r❝✈ ♠s❣s❀ s✐❣♥❛❧ ✐♥❛❝t✐✈❡ ❛t ✏s✐♥❦✑ ❛❞❞r a0✿ ❣❡t t❡r♠✐♥❛t✐♦♥ st❛t✉s

❚❡r♠✐♥❛t✐♦♥ ❞❡t❡❝t✐♦♥ ❝❤❛♥♥❡❧✿ ♦✈❡r✈✐❡✇

❱❛rs✿

active ✢❛❣ ❢♦r ❡✈❡r② ❛❞❞r j txh ❛♥❞ rxh ❢♦r ❡✈❡r② j,k

✐♥♣✉t j.tx(k, msg) ❛❧❧♦✇❡❞ ♦♥❧② ✐❢ j ❛❝t✐✈❡❀ s❡♥❞ msg✱ ✉♣❞❛t❡ txh ✐♥♣✉t j.rx() r❝✈ ♠s❣ ❢r♦♠ ❛♥② k❀ s❡t j ❛❝t✐✈❡ ✐♥♣✉t j.inactive() ❛❧❧♦✇❡❞ ♦♥❧② ✐❢ j ❛❝t✐✈❡❀ s❡t j ✐♥❛❝t✐✈❡ ✐♥♣✉t a0.isTerminated() r❡t✉r♥ ♦♥❧② ✐❢ j ✐♥❛❝t✐✈❡ ❛♥❞ t①❤ ❡q✉❛❧s r①❤ ❢♦r ❡✈❡r② j,k

❚❡r♠✐♥❛t✐♦♥ ❞❡t❡❝t✐♦♥ ❝❤❛♥♥❡❧ ✕ ✶

❙❡r✈✐❝❡ TdChannel ( ADDR, a0, initActive )

initActive✿ ✐♥✐t✐❛❧❧② ❛❝t✐✈❡ ❛❞❞r❡ss❡s

▼❛✐♥

txhj,k ← []

✴✴ t① ❤✐st♦r② ❛t j t♦ k

rxhj,k ← []

✴✴ r① ❤✐st♦r② ❛t k ❢r♦♠ j

activej ← j ✐♥ initActive

✴✴ ❛❝t✐✈❡ st❛t✉s ❛t j

vj ← sid()

✴✴ ❛❝❝❡ss s②st❡♠ ❛t j

return {vj}

✴✴ s✐❞ ♠❛♣ ♦✈❡r ADDR ❍❡❧♣❡r ❢✉♥❝t✐♦♥ terminated():

forall(j: not activej)

❛♥❞

forall(j,k: txhj,k = rxhj,k)

❚❡r♠✐♥❛t✐♦♥ ❞❡t❡❝t✐♦♥ ❝❤❛♥♥❡❧ ✕ ✷

✐♥♣✉t vj.tx(k,msg) ✐❝ ④activej ❛♥❞ k = j ❛♥❞ ♥♦ ♦♥❣♦✐♥❣ vj.tx(.)⑥ ❛♣♣❡♥❞ msg t♦ txhj,k❀ ♦❝ ④true⑥ r❡t✉r♥ ✐♥♣✉t Seq vj.rx() ✐❝ ④♥♦ ♦♥❣♦✐♥❣ vj.rx()⑥ ♦✉t♣✉t msg, k ♦❝ {rxhk,j ◦ [msg] ♣r❡✜①✲♦❢ txhk,j⑥ ❛♣♣❡♥❞ msg t♦ rxhk,j

activej ← tr✉❡

r❡t✉r♥ msg

❚❡r♠✐♥❛t✐♦♥ ❞❡t❡❝t✐♦♥ ❝❤❛♥♥❡❧ ✕ ✸

✐♥♣✉t vj.inactive() ✐❝ ④activej ❛♥❞ ♥♦ ♦♥❣♦✐♥❣ vj.inactive()⑥

activej ← ❢❛❧s❡

♦❝ ④true⑥ r❡t✉r♥ ✐♥♣✉t va0.isTerminated() ✐❝ ④♥♦ ♦♥❣♦✐♥❣ va0.isTerminated()⑥ ♦❝ {terminated()} r❡t✉r♥

❚❡r♠✐♥❛t✐♦♥ ❞❡t❡❝t✐♦♥ ❝❤❛♥♥❡❧ ✕ ✹

❛t♦♠✐❝✐t② ❛ss✉♠♣t✐♦♥✿ ✐♥♣✉t ♣❛rts ❛♥❞ ♦✉t♣✉t ♣❛rts ♣r♦❣r❡ss ❛ss✉♠♣t✐♦♥ ♦♥❣♦✐♥❣ vj.tx ❧❡❛❞s✲t♦ ♥♦t ♦♥❣♦✐♥❣ vj.tx

txhk,j.size ≥ i

❧❡❛❞s✲t♦

rxhk,j.size ≥ i ♦r ♥♦t ♦♥❣♦✐♥❣ vj.rx

♦♥❣♦✐♥❣ vj.inactive ❧❡❛❞s✲t♦ ♥♦t ♦♥❣♦✐♥❣ vj.inactive()

(terminated() ❛♥❞ ♦♥❣♦✐♥❣ va0.isTerminated)

❧❡❛❞s✲t♦ ♥♦t ♦♥❣♦✐♥❣ va0.isTerminated