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❆ ✉♥✐❢♦r♠ ❢r❛♠❡✇♦r❦ ❢♦r ❞❡❧✐♠✐t❡❞ ❝♦♥t✐♥✉❛t✐♦♥s ❊✛❡❝ts ✐♥ ❚②♣❡ ❚❤❡♦r② ❚❛❧❧✐♥♥ ❉❡❝❡♠❜❡r ✷✵✵✼ ✶

❲❤❛t ✐s ❞❡❧✐♠✐t❡❞ ❝♦♥tr♦❧❄ ❉❡❧✐♠✐t❡❞ ❝♦♥tr♦❧ ❂ ❝❛❧❧✲❝❝ ✰ ❛❜♦rt ✰ ❛ ❝♦♥t✐♥✉❛t✐♦♥ ❞❡❧✐♠✐t❡r ❏♦❤♥ss♦♥✱ ❏♦❤♥ss♦♥ ❛♥❞ ❉✉❣❣❛♥ ✶✾✽✼ ●▲✿ ❛ ❧❛♥❣✉❛❣❡ ✇✐t❤ ❞❡❧✐♠✐t❡❞ ❝♦♥tr♦❧ ❋❡❧❧❡✐s❡♥ ✶✾✽✽ ❝♦♥tr♦❧ ❛♥❞ ♣r♦♠♣t ✭✇r✐tt❡♥ F ❛♥❞ # ✮ ❉❛♥✈② ❛♥❞ ❋✐❧✐♥s❦✐ ✶✾✽✾ s❤✐❢t ❛♥❞ r❡s❡t ✭✇r✐tt❡♥ S ❛♥❞ � ❴ � ✮ ❋✐❧✐♥s❦✐ ✶✾✾✹ s❤✐❢t ✴ r❡s❡t ✐s ❝♦♠♣❧❡t❡ ❢♦r r❡♣r❡s❡♥t✐♥❣ ♠♦♥❛❞s ✷

❘❡♣r❡s❡♥t❛t✐♥❣ ♠♦♥❛❞s ▲❡t ( T, η, ∗ ) ❜❡ ❛ ❝♦♥❝r❡t❡ ♠♦♥❛❞ ✐♥ s♦♠❡ ❣✐✈❡♥ ❝❛❧❝✉❧✉s✳ ❖♥❡ ❝❛♥ ✉s❡ t❤❡ ♠♦♥❛❞ ✐♥ ✏❞✐r❡❝t st②❧❡✑ ✐❢ ♦♥❡ ❤❛s ❼ ❛ r❡✢❡❝t✐♦♥ ❝♦♥str✉❝t♦r µ ( t ) : X ❢♦r t : T ( X ) ✱ ❼ ❛ r❡✐✜❝❛t✐♦♥ ❝♦♥str✉❝t♦r [ t ] : T ( X ) ❢♦r t : X t❤❛t ❛❧❧♦✇ t♦ ❧♦❝❛❧❧② s✇✐t❝❤ t♦ t❤❡ ✉♥❞❡r❧②✐♥❣ r✐❝❤❡r ♠♦♥❛❞✐❝ str✉❝t✉r❡✳ ❋♦r ✐♥st❛♥❝❡✱ ❢♦r t❤❡ st❛t❡ ♠♦♥❛❞ ❛♥❞ ❢♦r t❤❡ ❡①❝❡♣t✐♦♥ ♠♦♥❛❞✿ T ( X ) � S → X × S T ( X ) � A + exn � λs. ( x, s ) � inl x η ( x ) η ( x ) � λx.λs. let ( x ′ , s ′ ) = x s in f x ′ s ′ f ∗ f ∗ � λx. case x of inl y → f y | inr z → inr z t❤❡ ❡①✐st❡♥❝❡ ♦❢ µ ❛♥❞ [ ] ❛❧❧♦✇s t♦ ❞❡✜♥❡✿ � µ ( λs. ( s, s )) � µ ( inr x ) : unit → S read () raise x write s � µ ( λ ❴ . (() , s )) : S → unit try t with z → u � case [ t ] of inl y → y | inr z → u ❚❤❡♦r❡♠ ✭ ❋✐❧✐♥s❦✐ ✮ ●✐✈❡♥ η ❛♥❞ ∗ ✱ µ ❛♥❞ [ ] ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❢r♦♠ S ❛♥❞ � � ✳ ❈♦r♦❧❧❛r② ✭ ❋✐❧✐♥s❦✐ ✮ λ ✲❝❛❧❝✉❧✉s ✇✐t❤ s❤✐❢t ✴ r❡s❡t ❝❛♥ s✐♠✉❧❛t❡ ❛❧❧ ♠♦♥❛❞✐❝ ❡✛❡❝ts ✐♥ ❞✐r❡❝t st②❧❡✳ ✸

▲❡t✬s ♥♦✇ ❢♦❝✉s ♦♥ ♣✉r❡ ❞❡❧✐♠✐t❡❞ ❝♦♥tr♦❧✳✳✳ ✹

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ❆ ❢♦✉♥❞❛t✐♦♥ ❢♦r ❝♦♥tr♦❧✿ λµ t♣✲❝❛❧❝✉❧✉s ✭t♣ ❛ ❝♦♥t✐♥✉❛t✐♦♥ ❝♦♥st❛♥t✮ ❆ ❢♦✉♥❞❛t✐♦♥ ❢♦r ❞❡❧✐♠✐t❡❞ ❝♦♥tr♦❧✿ λµ � t♣ ✲❝❛❧❝✉❧✉s ✭ � t♣ ❛ ❞②♥❛♠✐❝ ❝♦♥t✐♥✉❛t✐♦♥ ✈❛r✐❛❜❧❡✮ ✲ ❈❇❱ λµ � t♣ ✲❝❛❧❝✉❧✉s ≃ s❤✐❢t ✴ r❡s❡t ❝❛❧❝✉❧✉s ֒ → ❝♦♠♣❧❡t❡ ❢♦r r❡♣r❡s❡♥t✐♥❣ s②♥t❛❝t✐❝ ♠♦♥❛❞s ✭❡①❝❡♣t✐♦♥s✱ r❡❢❡r❡♥❝❡s✱ ✳✳✳✮ ✲ ❈❇◆ λµ � t♣ ✲❝❛❧❝✉❧✉s ≃ ❞❡ ●r♦♦t❡ ❛♥❞ ❙❛✉r✐♥✬s λµ ✲❝❛❧❝✉❧✉s → ♦❜s❡r✈❛t✐♦♥❛❧❧② ✭❇ö❤♠✮ ❝♦♠♣❧❡t❡ ֒ ✲ ❚❤❡ t✇♦ ✏❧❛③②✑ ✈❛r✐❛♥ts ♦❢ ❈❇◆ ❛♥❞ ❈❇❱ ✺

❚❤❡ ✇❡❛❦♥❡ss❡s ♦❢ ❋❡❧❧❡✐s❡♥ ❡t ❛❧ ✬s λ C ✲❝❛❧❝✉❧✉s ✲ ❝❛♥♥♦t ❢❛✐t❤❢✉❧❧② ❡①♣r❡ss t❤❡ s❡♠❛♥t✐❝s ♦❢ ❝❛❧❧❝❝ ✲ ❝❛♥♥♦t ❢❛✐t❤❢✉❧❧② ❡①♣r❡ss ✐ts ♦✇♥ ♦♣❡r❛t✐♦♥❛❧ s❡♠❛♥t✐❝s ✲ ♠❛② ✐♥tr♦❞✉❝❡ s♣❛❝❡ ❧❡❛❦s ✐♥ ❝♦♠♣✉t❛t✐♦♥ ֒ → ❛❧❧ ♦❢ t❤❡s❡ ❞✉❡ t♦ r❡✐✜❝❛t✐♦♥ ♦❢ ❡✈❛❧✉❛t✐♦♥ ❝♦♥t❡①ts ❛s ♦r❞✐♥❛r② ❢✉♥❝t✐♦♥s E [ C ( λk... k t ... )] ւ ց . . . ( λx.E [ x ]) t . . . . . . E [ t ] . . . � � s✉❜st✐t✉t✐♦♥ ♦❢ t❤❡ ( str✉❝t✉r❛❧ s✉❜st✐t✉t✐♦♥ ) r❡✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❝♦♥t❡①t ( λ C − st②❧❡ ) ( λµ − st②❧❡ ) ✻

❚❤❡ ✇❡❛❦♥❡ss ♦❢ ♣✉r❡ λµ ✲❝❛❧❝✉❧✉s P❛r✐❣♦t✬s λµ ✲❝❛❧❝✉❧✉s ❝❛♥♥♦t ❡①♣r❡ss abort ✱ ❤❡♥❝❡ ♥♦t t❤❡ s❡♠❛♥t✐❝s ♦❢ ❝❛❧❧❝❝ ❡✐t❤❡r✳✳✳ ❛ ❞❡♥♦t❛t✐♦♥ ❢♦r t❤❡ t♦♣❧❡✈❡❧ ✐s ♠✐ss✐♥❣✳ ✼

❚❤❡ r♦❧❡ ♦❢ t❤❡ t♦♣❧❡✈❡❧ ❝♦♥t✐♥✉❛t✐♦♥ ✐♥ ❡①♣r❡ss✐♥❣ t❤❡ s❡♠❛♥t✐❝s ♦❢ ❝❛❧❧❝❝ ★ ❧❡t ② ❂ ❝❛❧❧❝❝ ✭❢✉♥ ❦ ✲❃ ❢✉♥ ① ✲❃ t❤r♦✇ ❦ ✭❢✉♥ ② ✲❃ ①✰②✮✮❀❀ ✈❛❧ ② ✿ ✐♥t ✲❃ ✐♥t ❂ ❁❢✉♥❃ ★ ② ✸❀❀ ✈❛❧ ② ✿ ✐♥t ✲❃ ✐♥t ❂ ❁❢✉♥❃ ✭✯ ✦✦✦✦ ✯✮ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ y ✐s ✏♣r✐♥t t❤❡ ✈❛❧✉❡ ♦❢ y ✑✳ ▲❡t✬s ❝❛❧❧ ✐t k 0 ✳ ❚❤❡ ✈❛❧✉❡ ♦❢ y ✐s ❢✉♥ ① ✲❃ t❤r♦✇ k 0 ✭❢✉♥ ② ✲❃ ①✰②✮ ✳ ❲❤❡♥ y ✐s ❛♣♣❧✐❡❞✱ t❤r♦✇ ✐s ❝❛❧❧❡❞ ❛♥❞ ✐t r❡t✉r♥s t♦ k 0 ✳ ❲❡ ❝❛♥♥♦t ❡①♣r❡ss t❤❡ s❡♠❛♥t✐❝s ♦❢ callcc ✇✐t❤♦✉t r❡❢❡rr✐♥❣ t♦ t❤❡ t♦♣❧❡✈❡❧ ❝♦♥t✐♥✉❛t✐♦♥✳ ❈♦♥✈❡♥t✐♦♥❛❧❧②✱ t❤✐s s❡♠❛♥t✐❝s ✐s ❡①♣r❡ss❡❞ ✉s✐♥❣ ❛♥ ❛❜♦rt✐♦♥ ♦♣❡r❛t♦r A ✇❤✐❝❤ ✐ts❡❧❢ ❤✐❞❡s ❛ ❝❛❧❧ t♦ t❤❡ t♦♣❧❡✈❡❧ ❝♦♥t✐♥✉❛t✐♦♥✳ E [ callcc ( λk.M )] �→ E [ M [ λx. A ( E [ x ]) /k ]] ✽

❆ ❢♦✉♥❞❛t✐♦♥ ❢♦r ❝♦♥tr♦❧✿ λµ t♣✲❝❛❧❝✉❧✉s t ::= V | t t | µα.c ✭t❡r♠s✮ ::= [ β ] t | [ t♣ ] t c ✭❝♦♠♠❛♥❞s ♦r st❛t❡s✮ ::= x | λx.t V ✭✈❛❧✉❡s✮ λµ t♣✲❝❛❧❝✉❧✉s s❛t✐s✜❡s✿ ✲ ❋❛✐t❤❢✉❧ s✐♠✉❧❛t✐♦♥ ♦❢ ❝❛❧❧❝❝ ✱ A ✱ C ✱ ✳✳✳ ✲ ❖❜s❡r✈❛t✐♦♥❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ λ C ✭❜✉t ♥♦t ♦♣❡r❛t✐♦♥❛❧❧② ❡q✉✐✈❛❧❡♥t✮ ✲ ❙t❛♥❞❛r❞ ❡①♣❡❝t❡❞ ♣r♦♣❡rt✐❡s ✲ ■♥t❡r♥❛❧ ♥♦t✐♦♥ ♦❢ st❛t❡ ■♥t❡r❡st✐♥❣❧②✱ λµ t♣✲❝❛❧❝✉❧✉s s❤♦✇s t❤❛t t❤❡ ♠♦st ❣❡♥❡r❛❧ t②♣❡ ❢♦r C ✐s ♥♦t ¬¬ A → A ❜✉t (( A → B ) → T ) → A ✇❤❡r❡ T ✐s t❤❡ t♦♣❧❡✈❡❧ t②♣❡✳ ✾

❆ ❢♦✉♥❞❛t✐♦♥ ❢♦r ❞❡❧✐♠✐t❡❞ ❝♦♥tr♦❧✿ λµ � t♣ ✲❝❛❧❝✉❧✉s ::= V | t t | µα.c | µ � t t♣ .c ✭t❡r♠s✮ ::= [ β ] t | [ � t♣ ] t ✭❝♦♠♠❛♥❞s ♦r st❛t❡s✮ c V ::= x | λx.t ✭✈❛❧✉❡s✮ ❚❤❡ ♥❡✇ ♦♣❡r❛t♦r µ � t♣ .c ❞❡❧✐♠✐ts ❛ ❧♦❝❛❧ t♦♣❧❡✈❡❧✿ ✐t ❜✐♥❞s � t♣ ❞②♥❛♠✐❝❛❧❧② ✭ � t♣ ✐s ♥♦t r❡♥❛♠❡❞ ✇❤❡♥ s✉❜st✐t✉t❡❞ ❜❡❧♦✇ ❛ µ � t♣ ✮✳ ✶✵

❊①♣r❡ss✐✈❡♥❡ss ♦❢ λµ � t♣ ✲❝❛❧❝✉❧✉s � µ � t♣ . [ � � t � t♣ ] t � µ ❴ . [ � A t t♣ ] t � µα. [ � t♣ ]( t [ λx.µ � S ( λk.t ) t♣ . [ α ] x/k ]) � µα. [ � C ( λk.t ) t♣ ]( t [ λx.µ ❴ . [ α ] x/k ]) � µα. [ α ]( t [ λx.µ ❴ . [ α ] x/k ]) callcc ( λk.t ) � µ ❴ . [ � raise t t♣ ] t try t with ♣❛tt❡r♥s � ❝❛s❡ µ � t♣ . [ � t♣ ]( ❱❛❧ t ) ♦❢ | ❱❛❧ x ⇒ x | ♣❛tt❡r♥s | x ⇒ µ ❴ . [ � t♣ ] x t♣ . [ α ] x ) ∗ t ) � µα. [ � t♣ ](( λx.µ � µ ( t ) � µ � t♣ . [ � [ t ] t♣ ]( η t ) � λ () .µα. [ � t♣ ] λs. (( µ � t♣ . [ α ] s ) s ) read � λs.µα. [ � t♣ ] λ ❴ . (( µ � t♣ . [ α ]()) s ) write ✶✶

▲❡t✬s ❢♦❝✉s ♦♥ ❝❛❧❧✲❜②✲♥❛♠❡ ❛♥❞ ♦❜s❡r✈❛t✐♦♥❛❧ ✭❇ö❤♠✮ ❝♦♠♣❧❡t❡♥❡ss ✐♥ λµ ✲❝❛❧❝✉❧✉s✳✳✳ ✶✷