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❇♦✉♥❞❡❞ ❇✉✛❡r ❙❤❛♥❦❛r ❙❡♣t❡♠❜❡r ✶✽✱ ✷✵✶✹

❖✈❡r✈✐❡✇ ❇♦✉♥❞❡❞✲❜✉✛❡r s❡r✈✐❝❡ ✐♥♣✉t ❢✉♥❝t✐♦♥s✿ void put(x) ✱ Val get() ✱ void end() ♥♦ ♦✉t♣✉t ❢✉♥❝t✐♦♥s ■♠♣❧❡♠❡♥t❛t✐♦♥s ✉s✐♥❣ st❛♥❞❛r❞ s②♥❝❤r♦♥✐③❛t✐♦♥ ❝♦♥str✉❝ts ❧♦❝❦s✱ ❝♦♥❞✐t✐♦♥ ✈❛r✐❛❜❧❡s s❡♠❛♣❤♦r❡s ■♠♣❧❡♠❡♥t❛t✐♦♥s ✉s✐♥❣ ❛✇❛✐t s②♥❝❤r♦♥✐③❛t✐♦♥ ❝♦♥str✉❝ts ♠♦r❡ ♣♦✇❡r❢✉❧✱ ❝♦♥✈❡♥✐❡♥t ♠❡❝❤❛♥✐❝❛❧ tr❛♥s❢♦r♠❛t✐♦♥ t♦ st❛♥❞❛r❞ s②♥❝❤ ❝♦♥str✉❝ts ❘❡❞✉❝❡ ❜❧♦❝❦✐♥❣ ✭✐♥❝r❡❛s❡ ♣❛r❛❧❧❡❧✐s♠✮ ✐♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❈❛♥❝❡❧❧✐♥❣ ❜❧♦❝❦❡❞ ❝❛❧❧s t♦ r❡♠♦t❡ s②st❡♠s ❛❧❧♦✇s ❛ ❝❛❧❧❡r ♥♦ ❧♦♥❣❡r ✐♥t❡r❡st❡❞ ✐♥ ❝❛❧❧ t♦ r❡tr✐❡✈❡ ✐ts❡❧❢

❖✉t❧✐♥❡ s❡r✈✐❝❡ ❇♦✉♥❞❡❞✲❇✉✛❡r ❙❡r✈✐❝❡ ❇♦✉♥❞❡❞✲❇✉✛❡r ❙❡r✈✐❝❡ ■♥✈❡rs❡ ❆✇❛✐ts ❇♦✉♥❞❡❞✲❇✉✛❡r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ❆✇❛✐ts ▲♦❝❦s ❛♥❞ ❈♦♥❞✐t✐♦♥ ❱❛r✐❛❜❧❡s ❇♦✉♥❞❡❞✲❇✉✛❡r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ▲♦❝❦s ❛♥❞ ❈♦♥❞✐t✐♦♥ ❱❛r✐❛❜❧❡s ❙❡♠❛♣❤♦r❡s ❇♦✉♥❞❡❞✲❇✉✛❡r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ❙❡♠❛♣❤♦r❡s ✴✴ ❙❊❊ ❚❊❳❚ ■♥❝r❡❛s✐♥❣ P❛r❛❧❧❡❧✐s♠ ❈❛♥❝❡❧✐♥❣ ❇❧♦❝❦❡❞ ❈❛❧❧s ✴✴ ❙❊❊ ❚❊❳❚

❇♦✉♥❞❡❞✲❇✉✛❡r ❙❡r✈✐❝❡ ✕ ✶ s❡r✈✐❝❡ ❋✐❢♦ ❜♦✉♥❞❡❞ ❜✉✛❡r ♦❢ s✐③❡ N ✐♥♣✉t ❢✉♥❝t✐♦♥s✿ put(x) ✱ get() ✱ end() ♥♦ ♦✉t♣✉t ❢✉♥❝t✐♦♥s ▼❛✐♥ buff ✿ s❡q✉❡♥❝❡ ♦❢ ✐t❡♠s ✐♥ ❜✉✛❡r ending ✿ tr✉❡ ✐✛ end() ❤❛s ❜❡❡♥ ❝❛❧❧❡❞ putBusy ✿ tr✉❡ ✐✛ put ❝❛❧❧ ♦♥❣♦✐♥❣ getBusy ✿ tr✉❡ ✐✛ get ❝❛❧❧ ♦♥❣♦✐♥❣ void mysid.put(x) ✐❝ ④♥♦t ❡♥❞✐♥❣ ❛♥❞ ♥♦ ♦♥❣♦✐♥❣ put ❝❛❧❧⑥ ♦❝ ④ buff ❤❛s s♣❛❝❡⑥ ❛♣♣❡♥❞ x t♦ buff ❀ r❡t✉r♥

❇♦✉♥❞❡❞✲❇✉✛❡r ❙❡r✈✐❝❡ ✕ ✷ s❡r✈✐❝❡ Val mysid.get() ✐❝ ④♥♦t ❡♥❞✐♥❣ ❛♥❞ ♥♦ ♦♥❣♦✐♥❣ get ❝❛❧❧⑥ ♦✉t♣✉t rval ♦❝ ④ buff ❤❛s ✐t❡♠✱ rval ✐s buff.head ⑥ ❜❡❤❡❛❞ buff ❀ r❡t✉r♥ rval void mysid.end() ✐❝ ④♥♦t ❡♥❞✐♥❣⑥ s❡t ❡♥❞✐♥❣ ♦❝ ④tr✉❡⑥ r❡t✉r♥ Pr♦❣r❡ss ❛ss✉♠♣t✐♦♥ put ❝❛❧❧ r❡t✉r♥s ✐❢ buff ❤❛s s♣❛❝❡ ✴✴ ✉s❡s putBusy get ❝❛❧❧ r❡t✉r♥s ✐❢ buff ❤❛s ✐t❡♠ ✴✴ ✉s❡s getBusy end ❝❛❧❧ r❡t✉r♥s ✴✴ ✉s❡s ✏t❤r❡❛❞ ✐♥ mysid.end ✑

BoundedBuffer(int N) ✕ ✶ s❡r✈✐❝❡ // main input Val mysid.get() ic {N ≥ 1} ic {not ending and buff ← []; not getBusy} ending ← false; getBusy ← true; putBusy ← false; output(Val rval) getBusy ← false; oc {buff.size > 0 return mysid; and rval = buff[0]} buff.remove(); input void mysid.put(Val x) getBusy ← false; ic {not ending and return rval; not putBusy} putBusy ← true; input void mysid.end() oc {buff.size < N} ic {not ending} buff.append(x); ending ← true; putBusy ← false; oc {true} return; return;

BoundedBuffer(int N) ✕ ✷ s❡r✈✐❝❡ atomicity assumption {input and output parts} progress assumption { // thread in put returns if buffer has space (putBusy and buff.size < N) ❧❡❛❞s✲t♦ not putBusy; // thread in get returns if buffer has an item (getBusy and buff.size > 0) ❧❡❛❞s✲t♦ not getBusy; // thread in end returns (thread u in mysid.end) ❧❡❛❞s✲t♦ (not u in mysid.end); } }

❖✉t❧✐♥❡ s❡r✈✐❝❡ ✐♥✈❡rs❡ ❇♦✉♥❞❡❞✲❇✉✛❡r ❙❡r✈✐❝❡ ❇♦✉♥❞❡❞✲❇✉✛❡r ❙❡r✈✐❝❡ ■♥✈❡rs❡ ❆✇❛✐ts ❇♦✉♥❞❡❞✲❇✉✛❡r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ❆✇❛✐ts ▲♦❝❦s ❛♥❞ ❈♦♥❞✐t✐♦♥ ❱❛r✐❛❜❧❡s ❇♦✉♥❞❡❞✲❇✉✛❡r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ▲♦❝❦s ❛♥❞ ❈♦♥❞✐t✐♦♥ ❱❛r✐❛❜❧❡s ❙❡♠❛♣❤♦r❡s ❇♦✉♥❞❡❞✲❇✉✛❡r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ❙❡♠❛♣❤♦r❡s ✴✴ ❙❊❊ ❚❊❳❚ ■♥❝r❡❛s✐♥❣ P❛r❛❧❧❡❧✐s♠ ❈❛♥❝❡❧✐♥❣ ❇❧♦❝❦❡❞ ❈❛❧❧s ✴✴ ❙❊❊ ❚❊❳❚

❙❡r✈✐❝❡ ■♥✈❡rs❡ ❖✈❡r✈✐❡✇ ✕ ✶ s❡r✈✐❝❡ ✐♥✈❡rs❡ BoundedBufferInverse(N, bb) ✴✴ bb ✿ s✐❞ ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♠❛✐♥✿ buff ✱ ending ✱ putBusy ✱ getBusy ♦✉t♣✉t ❢✉♥❝t✐♦♥s✿ doPut(x) ✱ doGet() ✱ doEnd() doPut(x) ♦❝ ④♥♦t ❡♥❞✐♥❣ ❛♥❞ ♥♦ ♦♥❣♦✐♥❣ put ❝❛❧❧⑥ bb.put(x) ✐❝ ④ buff ❤❛s s♣❛❝❡⑥ ❛♣♣❡♥❞ x t♦ buff doGet() ♦❝ ④♥♦t ❡♥❞✐♥❣ ❛♥❞ ♥♦ ♦♥❣♦✐♥❣ get ❝❛❧❧⑥ rval ← bb.get() ✐❝ ④ buff ❤❛s ✐t❡♠✱ rval ✐s buff.head ⑥ ❜❡❤❡❛❞ buff

❙❡r✈✐❝❡ ■♥✈❡rs❡ ❖✈❡r✈✐❡✇ ✕ ✷ s❡r✈✐❝❡ ✐♥✈❡rs❡ doEnd() ♦❝ ④♥♦t ❡♥❞✐♥❣⑥ s❡t ❡♥❞✐♥❣❀ bb.end() ✐❝ ④tr✉❡⑥ r❡t✉r♥ ♣r♦❣r❡ss ❝♦♥❞✐t✐♦♥ put ❝❛❧❧ r❡t✉r♥s ✐❢ buff ❤❛s s♣❛❝❡ ✴✴ ✉s❡s putBusy get ❝❛❧❧ r❡t✉r♥s ✐❢ buff ❤❛s ✐t❡♠ ✴✴ ✉s❡s getBusy end ❝❛❧❧ r❡t✉r♥s ✴✴ ✉s❡s ✏t❤r❡❛❞ ✐♥ bb.end ✑

BoundedBufferInverse(N, bb) s❡r✈✐❝❡ ✐♥✈❡rs❡ // main output doGet() { ic {N ≥ 1} oc {not ending and not getBusy} buff ← []; getBusy ← true; putBusy ← false; Val x ← bb.get(x); getBusy ← false; ic {buff.size > 0 and x = buff[0]} return mysid; buff.remove(); getBusy ← false; output doPut(Val x) { oc {not end- output doEnd() { ing and oc {not ending} not putBusy} ending ← true; putBusy ← true; lck.end(); bb.put(x); ic {true} ic {buff.size < N} buff.append(x); progress condition {... mysid bb...} putBusy ← false;

❖✉t❧✐♥❡ ❛✇❛✐t ❇♦✉♥❞❡❞✲❇✉✛❡r ❙❡r✈✐❝❡ ❇♦✉♥❞❡❞✲❇✉✛❡r ❙❡r✈✐❝❡ ■♥✈❡rs❡ ❆✇❛✐ts ❇♦✉♥❞❡❞✲❇✉✛❡r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ❆✇❛✐ts ▲♦❝❦s ❛♥❞ ❈♦♥❞✐t✐♦♥ ❱❛r✐❛❜❧❡s ❇♦✉♥❞❡❞✲❇✉✛❡r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ▲♦❝❦s ❛♥❞ ❈♦♥❞✐t✐♦♥ ❱❛r✐❛❜❧❡s ❙❡♠❛♣❤♦r❡s ❇♦✉♥❞❡❞✲❇✉✛❡r ■♠♣❧❡♠❡♥t❛t✐♦♥ ✉s✐♥❣ ❙❡♠❛♣❤♦r❡s ✴✴ ❙❊❊ ❚❊❳❚ ■♥❝r❡❛s✐♥❣ P❛r❛❧❧❡❧✐s♠ ❈❛♥❝❡❧✐♥❣ ❇❧♦❝❦❡❞ ❈❛❧❧s ✴✴ ❙❊❊ ❚❊❳❚

❆✇❛✐t ❝♦♥str✉❝t ❛✇❛✐t ❆✇❛✐t✿ ♣♦✇❡r❢✉❧ s②♥❝❤r♦♥✐③❛t✐♦♥ ❝♦♥str✉❝t ♥♦t ♣r♦✈✐❞❡❞ ❜② P▲s ❝♦♥✈❡♥✐❡♥t ❢♦r ✇r✐t✐♥❣ ♣r♦❣❛♠s ✐♠♣❧❡♠❡♥t❛❜❧❡ ❜② st❛♥❞❛r❞ s②♥❝❤r♦♥✐③❛t✐♦♥ ❝♦♥str✉❝ts await (B) S B ✐s ♣r❡❞✐❝❛t❡✱ S ✐s ♥♦♥✲❜❧♦❝❦✐♥❣ ❝♦❞❡ ❛t♦♠✐❝❛❧❧② ❡①❡❝✉t❡ S ♦♥❧② ✐❢ B ❤♦❧❞s✱ ♦t❤❡r✇✐s❡ ✇❛✐t ✇❡❛❦ ♦r str♦♥❣ ❢❛✐r♥❡ss await (B) S ✿ ♠♦r❡ ❣❡♥❡r❛❧ t❤❛♥ oc {B} S S ❝❛♥ ♠❛❦❡ ✭♥♦♥✲❜❧♦❝❦✐♥❣✮ ♦✉t♣✉t ❝❛❧❧s✱ ✉s❡ r❡t✉r♥ ✈❛❧✉❡s atomic S ✿ s❤♦rt ❢♦r await (true) S