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❆♥ ❡♥✉♠❡r❛t✐♦♥✲❢r❡❡ ♣r♦♦❢ ♦❢ ●ö❞❡❧✬s ❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠ ✇✐t❤ s✐❞❡ ❡✛❡❝ts ✭✇♦r❦ ✐♥ ♣r♦❣r❡ss✮ ❍✉❣♦ ❍❡r❜❡❧✐♥ ✸ ❋❡❜r✉❛r② ✷✵✶✷ ❋❆❚P❆ ✷✵✶✷ ❇❡❧❣r❛❞❡ ✭r❡✈✐s❡❞ ✺✴✷✴✶✷✮ ✶

❚❤❡ ♣r♦♦❢✲❛s✲♣r♦❣r❛♠s ❝♦rr❡s♣♦♥❞❡♥❝❡ ❆♥ ✐♥t✉✐t✐♦♥✐st✐❝ ♣r♦♦❢ ✐s ❛ ♣✉r❡❧② ❢✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠ ❬❈✉rr②✱ ✶✾✺✽✱ ❍♦✇❛r❞✱ ✶✾✻✾❪ ✲ ❛ ♣r♦♦❢ ♦❢ A → B ✐s ❛ ♣r♦❣r❛♠ ♦❢ t❤❡ ❢♦r♠ x = > p ✇❤❡r❡ x ❤❛s t②♣❡ A ❛♥❞ p ❤❛s t②♣❡ B ✲ ❛ ♣r♦♦❢ ♦❢ A ∧ B ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♣❛✐r ♦❢ ❛♥ ❡❧❡♠❡♥t ♦❢ A ❛♥❞ ♦❢ ❛♥ ❡❧❡♠❡♥t ♦❢ B ✲ ❞❡r✐✈✐♥❣ B ❢r♦♠ A → B ❛♥❞ A ✐s t❤❡ s❛♠❡ ❛s ❛♣♣❧②✐♥❣ t❤❡ ❣✐✈❡♥ ♣r♦♦❢✲❢✉♥❝t✐♦♥ ♦❢ t②♣❡ A → B t♦ t❤❡ ❣✐✈❡♥ ❡❧❡♠❡♥t ♦❢ t②♣❡ A ✲ ❡t❝ ✷

❚❤❡ ♣r♦♦❢✲❛s✲♣r♦❣r❛♠ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ❆ ❝❧❛ss✐❝❛❧ ♣r♦♦❢ ✐s ❛ ♣r♦❣r❛♠ ✇✐t❤ ❝♦♥tr♦❧ ❬●r✐✣♥✱ ✶✾✾✵❪ ❝❛❧❧❝❝ ✴ t❤r♦✇ ✭▼▲ ❧❛♥❣✉❛❣❡s✮ ♦r ❝❛❧❧✲✇✐t❤✲❝✉rr❡♥t✲❝♦♥t✐♥✉❛t✐♦♥ ✭❙❝❤❡♠❡✮ ❝❛♥ ❜❡ ✉s❡❞ t♦ ♣r♦✈❡ ¬¬ A → A ❊①❝❡♣t✐♦♥ ♠❡❝❤❛♥✐s♠s ✭▼▲ ❧❛♥❣✉❛❣❡s✱ ❏❛✈❛✱ ✳✳✳✮✱ ♦r ❡✈❡♥ s❡t❥✉♠♣ ✴ ❧♦♥❣❥✉♠♣ ✭✐♥ ❈✮ ♣r♦✈✐❞❡ ❛ ✇❡❛❦ ❢♦r♠ ♦❢ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✿ t❤❡② ❝❛♥ ❜❡ ✉s❡❞ t♦ ♣r♦✈❡ ¬¬ A → A ✇❤❡♥ A ✐s ❛ ❢♦r♠✉❧❛ ✇✐t❤♦✉t ✐♠♣❧✐❝❛t✐♦♥ ♥♦r ✉♥✐✈❡rs❛❧ q✉❛♥t✐✜❝❛t✐♦♥ t②♣❡ ✭t❤✐s ❧❛tt❡r ♣r✐♥❝✐♣❧❡ ❝❛♥ ❜❡ s❡❡♥ ❛s ❛ ✈❛r✐❛♥t ♦❢ ▼❛r❦♦✈✬s ♣r✐♥❝✐♣❧❡ ❢♦r ♣r❡❞✐❝❛t❡ ❧♦❣✐❝✮ ❬❍❍✱ ✷✵✶✵❪ ✸

■s ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ❛ s✐❞❡ ❡✛❡❝t❄ ❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ ♣✉r❡ ❢✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠♠✐♥❣ ✭❍❛s❦❡❧❧✱ ♦r t❤❡ ❝♦r❡ ♦❢ ❋★✮✱ ❝♦♥tr♦❧ ♦♣❡r❛t♦rs ✐♠♣❧❡♠❡♥t s✐❞❡ ❡✛❡❝ts ❚♦ ❞❡❛❧ ✇✐t❤ s✐❞❡ ❡✛❡❝ts ✐♥ ❍❛s❦❡❧❧✱ t❤❡ ❝♦♠♠♦♥ ❛♣♣r♦❛❝❤ ✐s t♦ r❡❛s♦♥ ✐♥ ❛ ✏♠♦♥❛❞✑ ❛♥❞ t♦ ❡①tr❛❝t ❛❢t❡r✇❛r❞s ♣✉r❡ ❝♦♥t❡♥ts ❜② ✏r✉♥♥✐♥❣✑ t❤❡ ♠♦♥❛❞ ❈❛♥ ✇❡ ✐♠❛❣✐♥❡ ❛❞❞✐♥❣ ♦t❤❡r s✐❞❡ ❡✛❡❝ts t♦ ❧♦❣✐❝ t❤❛t ❝♦✉❧❞ ❜❡ ❝❧❡❛r❡❞ ❜② ✏r✉♥♥✐♥❣✑ t❤❡ ♠♦♥❛❞❄ ✹

❊①❛♠♣❧❡ ♦❢ ♠♦♥❛❞✐❝ ♣r♦❣r❛♠♠✐♥❣✿ t❤❡ st❛t❡ ♠♦♥❛❞ ❚♦ s✐♠✉❧❛t❡ t❤❡ ❛❜✐❧✐t② t♦ r❡❛❞ ❛♥❞ ✇r✐t❡ ❛ ❣❧♦❜❛❧ ♠❡♠♦r② ♦❢ t②♣❡ S ✇✐t❤♦✉t ✉s✐♥❣ ❛♥② ♠❡♠♦r② ❛t ❛❧❧✱ ♦♥❡ s❡ts T st ( A ) � S → S × A ❛♥❞✱ ✇❤❡♥❡✈❡r ♦♥❡ ✇❛♥ts t♦ ✇r✐t❡ ❛ ♣r♦❣r❛♠ ♦❢ t②♣❡ A ✱ ✇❡ ❛❝t✉❛❧❧② ✇r✐t❡ ❛ ♣r♦❣r❛♠ ♦❢ t②♣❡ T st ( A ) ❚♦ ❜❡ ❛❜❧❡ t♦ ✇r✐t❡ ❛♥② ❢✉♥❝t✐♦♥❛❧ ♣r♦❣r❛♠ ✐♥ ❛ ✏♠♦♥❛❞✑ T ✱ ✇❡ ♥❡❡❞ t✇♦ ♦♣❡r❛t✐♦♥s ✶ : A → T ( A ) η t♦ ❡♥t❡r t❤❡ ♠♦♥❛❞ ∗ : ( A → T ( B )) → T ( A ) → T ( B ) t♦ ❛♣♣❧② ❢✉♥❝t✐♦♥s run : T ( base ) → base t♦ ✏r✉♥✑ t❤❡ ♠♦♥❛❞ ♦♥ ❜❛s❡ t②♣❡s ✇❤✐❝❤✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ st❛t❡ ♠♦♥❛❞✱ ❛r❡ ✐♠♣❧❡♠❡♥t❡❞ ❜②✿ � λs. ( s, x ) η x f ∗ x � λs. let ( s ′ , x ′ ) = xs in f x ′ s ′ run x � let ( ❴ , x ′ ) = xs 0 in x ′ ✇❤❡r❡ s 0 ✐s t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♦❢ t❤❡ ♠❡♠♦r②✳ ✶ ❚❤❡s❡ ♦♣❡r❛t✐♦♥s ❤❛✈❡ t♦ s❛t✐s❢② t❤❡ ❡q✉❛t✐♦♥s η ∗ x = x ✱ f ∗ ( ηx ) = fx ✱ g ∗ ( f ∗ x ) = ( g ∗ ◦ f ) ∗ x ✳ ✺

❲❤❛t ❞✐❞ ✇❡ ❣❛✐♥❄ ❲❤❛t ✇❡ ❣❛✐♥❡❞ ✐s t❤❛t ✐♥ t❤❡ st❛t❡ ♠♦♥❛❞✱ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ♥❡✇ ♦♣❡r❛t✐♦♥s ❛r❡ ❛✈❛✐❧❛❜❧❡ ❢♦r ❢r❡❡✦ : unit → T st ( S ) read write : S → T st ( unit ) ✇❤✐❝❤ ❛r❡ ✐♠♣❧❡♠❡♥t❡❞ ❜②✿ � λs. ( s, s ) read () write s ′ � λ ❴ . ( s ′ , ()) ✻

▼♦♥❛❞✐❝✲st②❧❡ ✈s ❞✐r❡❝t✲st②❧❡✿ t❤❡ ❝❛s❡ ♦❢ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ❘❡♠❛r❦✿ ❞♦✉❜❧❡✲♥❡❣❛t✐♦♥ tr❛♥s❧❛t✐♦♥ T cont ( A ) � ¬¬ A ✐s t❤❡ ✏❝♦♥t✐♥✉❛t✐♦♥✑ ♠♦♥❛❞ t❤❛t ❛❧❧♦✇s t♦ r❡❛s♦♥ ❝❧❛ss✐❝❛❧❧② ✭✐✳❡✳ ✉s✐♥❣ ¬¬ A → A ✮ ✐♥s✐❞❡ ✐♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝✳ ❈♦♥✈❡rs❡❧②✱ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ❝❛♥ ❜❡ s❡❡♥ ❛s r❡❛s♦♥✐♥❣ ✐♥t✉✐t✐♦♥✐st✐❝❛❧❧② ✇✐t❤✐♥ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♠♦♥❛❞✱ ❜✉t ❞♦✐♥❣ ✐t ✐♥ ❞✐r❡❝t✲st②❧❡ ✳ ✼

▼♦♥❛❞✐❝✲st②❧❡ ✈s ❞✐r❡❝t✲st②❧❡✿ t❤❡ ❝❛s❡ ♦❢ ▼❛r❦♦✈✬s ♣r✐♥❝✐♣❧❡ ❚❤❡r❡ ✐s ❛ ✇❡❛❦ ❢♦r♠ ♦❢ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ♠♦♥❛❞✱ ♥❛♠❡❧② t❤❡ ✏❡①❝❡♣t✐♦♥✑ ♠♦♥❛❞ T exc ( A ) � A ∨ E ❢♦r s✉♣♣♦rt✐♥❣ ❤❛♥❞✐♥❣ ❛♥❞ r❛✐s✐♥❣ ❡①❝❡♣t✐♦♥s ✐♥ s♦♠❡ t②♣❡ E ✳ ❚❤✐s ✐s ❡♥♦✉❣❤ t♦ ♣r♦✈✐❞❡ ❛ ♣r❡❞✐❝❛t❡ ❧♦❣✐❝ ✈❛r✐❛♥t ♦❢ ▼❛r❦♦✈✬s ♣r✐♥❝✐♣❧❡ ✭✐✳❡✳ ¬¬ A → A ❢♦r A ✇✐t❤♦✉t ✐♠♣❧✐❝❛t✐♦♥ ♥♦r ✉♥✐✈❡rs❛❧ q✉❛♥t✐✜❝❛t✐♦♥✮✳ ✽

▼♦♥❛❞✐❝✲st②❧❡ ✈s ❞✐r❡❝t✲st②❧❡ ■t t✉r♥s ♦✉t t❤❛t s♦♠❡ ❢♦r♠ ♦❢ ♠❡♠♦r② ❛ss✐❣♥♠❡♥t ✐s ✉s❡❞ ✐♥ ❧♦❣✐❝✿ ❜♦t❤ ♦❢ ✲ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❜❛s❡❞ tr❛♥s❧❛t✐♦♥s ✲ ❈♦❤❡♥✬s ❢♦r❝✐♥❣ tr❛♥s❧❛t✐♦♥ ❝♦rr❡s♣♦♥❞ t♦ ♣r♦✈✐❞✐♥❣ ♠♦♥♦t♦♥✐❝❛❧❧② ♠❡♠♦r② ❛ss✐❣♥♠❡♥t ✐♥ ♠♦♥❛❞✐❝ st②❧❡ ✳ ❑r✐♣❦❡ s❡♠❛♥t✐❝s ❜❛s❡❞ tr❛♥s❧❛t✐♦♥s ✭✐✳❡✳ T kr ( A )( x ) � ∀ x ′ ≥ x A ( x ′ ) ❢♦r x r❛♥❣✐♥❣ ♦✈❡r W ✮ ✐s ❛ ❞❡♣❡♥❞❡♥t ❢♦r♠ ♦❢ t❤❡ ❡♥✈✐r♦♥♠❡♥t ♠♦♥❛❞ ✭ T env ( A ) � W → A ✮ ❦♥♦✇♥ t♦ ♣r♦✈✐❞❡ ▲✐s♣✲st②❧❡ ❞②♥❛♠✐❝ ❜✐♥❞✐♥❣s✳ ❈♦❤❡♥✬s ❢♦r❝✐♥❣ tr❛♥s❧❛t✐♦♥ ✭ T forc ( A )( x ) � ∀ x ′ ≥ x ∃ x ′′ ≥ x ′ A ( x ′′ ) ❢♦r x r❛♥❣✐♥❣ ♦✈❡r s♦♠❡ ❞♦♠❛✐♥ S ✮ ✐s ❛ ❞❡♣❡♥❞❡♥t ❢♦r♠ ♦❢ t❤❡ st❛t❡ ♠♦♥❛❞ ✭ T st ( A ) � S → S × A ✮✳ ❲❤❛t ✐❢ ✇❡ tr② t♦ ❞❡s✐❣♥ ❛ ❧♦❣✐❝❛❧ s②st❡♠ t❤❛t ♣r♦✈✐❞❡s s✉❝❤ ❦✐♥❞s ♦❢ ♠❡♠♦r② ❛ss✐❣♥♠❡♥t ✐♥ ❞✐r❡❝t✲st②❧❡ ❄ ✾