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❙tr♦♥❣ ◆♦r♠❛❧✐③❛❜✐❧✐t② ❛s ❛ ❋✐♥✐t❡♥❡ss ❙tr✉❝t✉r❡ ✈✐❛ t❤❡ ❚❛②❧♦r ❊①♣❛♥s✐♦♥ ♦❢ λ ✲t❡r♠s ▼✐❝❤❡❧❡ P❛❣❛♥✐ ◦ ✱ ❈❤r✐st✐♥❡ ❚❛ss♦♥ ◦ ❛♥❞ ▲✐♦♥❡❧ ❱❛✉① ∗ ◦ ■❘■❋✱ ❯♥✐✈❡rs✐té P❛r✐s ❉✐❞❡r♦t✱ ❋r❛♥❝❡ ∗ ■✷▼✱ ❆✐①✲▼❛rs❡✐❧❧❡ ❯♥✐✈❡rs✐té✱ ❋r❛♥❝❡ ❙✉♣♣♦rt❡❞ ❜② ❋r❡♥❝❤ ❆◆❘ Pr♦❥❡❝t ❈♦q✉❛s ✭❆◆❘ ✶✷ ❏❙✵✷ ✵✵✻ ✵✶✮✳ ❋♦❙❙❛❈❙ ❅ ❊❚❆P❙ ✷✵✶✻ ❊✐♥❞❤♦✈❡♥✱ ◆▲✱ ✹✲✼ ❆♣r✐❧ ✷✵✶✻ ✶ ✴ ✶✺

❊✈❡r②t❤✐♥❣ ✐s ✐♥ t❤❡ t✐t❧❡ ( ∀ M ∈ Λ + ) M ∈ SN ⇐ ⇒ T ( M ) ∈ F ❲❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ str♦♥❣ ♥♦r♠❛❧✐③❛❜✐❧✐t② ✭ SN ✮ ♦❢ ✭♥♦♥✲❞❡t❡r♠✐♥✐st✐❝✮ λ ✲t❡r♠s ✭ Λ + ✮ ❛s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ✭ F ✮ ✈✐❛ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✭ T ✮✳ ✷ ✴ ✶✺

❊✈❡r②t❤✐♥❣ ✐s ✐♥ t❤❡ t✐t❧❡ ( ∀ M ∈ Λ + ) M ∈ SN ⇐ ⇒ T ( M ) ∈ F ❲❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ str♦♥❣ ♥♦r♠❛❧✐③❛❜✐❧✐t② ✭ SN ✮ ♦❢ ✭♥♦♥✲❞❡t❡r♠✐♥✐st✐❝✮ λ ✲t❡r♠s ✭ Λ + ✮ ❛s ❛ ✜♥✐t❡♥❡ss str✉❝t✉r❡ ✭ F ✮ ✈✐❛ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✭ T ✮✳ ✷ ✴ ✶✺

❋✐♥✐t♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ❡❛r❧② ✷✵✵✵✬s✮ ❘❡❢♦r♠✉❧❛t❡ q✳s✳ ✐♥ ❛ ❧✐♥❡❛r ❧♦❣✐❝ s❡tt✐♥❣ ✉s✐♥❣ st❛♥❞❛r❞ ❛❧❣❡❜r❛✿ t②♣❡s ♣❛rt✐❝✉❧❛r t♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡s✿ ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ str✉❝t✉r❡ ❢✉♥❝t✐♦♥ t❡r♠s ♣♦✇❡r s❡r✐❡s ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ✲t❡r♠s ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r ✷✵✵✸✲✷✵✵✹✮ ❙♦ ✇❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ✲t❡r♠s✱ ❛♥❞ ❝♦♠♣✉t❡ t❤❡✐r ❚❛②❧♦r ❡①♣❛♥s✐♦♥✦ ❆♥❞ ♦♥❡ ❝❛♥ ♠✐♠✐❝❦ t❤❛t ✐♥ t❤❡ s②♥t❛①✿ ❞✐✛❡r❡♥t✐❛❧ ✲❝❛❧❝✉❧✉s ❛ ✜♥✐t❛r② ❢r❛❣♠❡♥t✿ r❡s♦✉r❝❡ ✲❝❛❧❝✉❧✉s ❂ t❤❡ t❛r❣❡t ♦❢ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s ❆ ♣r✐♠❡ ❛❣❡❞ ✐❞❡❛ ✭●✐r❛r❞✱ ✬✽✵s✱ ❜❡❢♦r❡ ▲▲✮ λ ✲t❡r♠s = ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s = ♣♦✇❡r s❡r✐❡s ❖r✐❣✐♥❛❧❧②✿ ❢♦r t❤❡ λ ✲❝❛❧❝✉❧✉s✱ ✐♥ ❛♥ ❛❜str❛❝t ❝❛t❡❣♦r✐❝❛❧ s❡tt✐♥❣ ✭❝♦❡✣❝✐❡♥ts ❛r❡ s❡ts✮ ✸ ✴ ✶✺

❉✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ ✲t❡r♠s ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r ✷✵✵✸✲✷✵✵✹✮ ❙♦ ✇❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ ✲t❡r♠s✱ ❛♥❞ ❝♦♠♣✉t❡ t❤❡✐r ❚❛②❧♦r ❡①♣❛♥s✐♦♥✦ ❆♥❞ ♦♥❡ ❝❛♥ ♠✐♠✐❝❦ t❤❛t ✐♥ t❤❡ s②♥t❛①✿ ❞✐✛❡r❡♥t✐❛❧ ✲❝❛❧❝✉❧✉s ❛ ✜♥✐t❛r② ❢r❛❣♠❡♥t✿ r❡s♦✉r❝❡ ✲❝❛❧❝✉❧✉s ❂ t❤❡ t❛r❣❡t ♦❢ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s ❆ ♣r✐♠❡ ❛❣❡❞ ✐❞❡❛ ✭●✐r❛r❞✱ ✬✽✵s✱ ❜❡❢♦r❡ ▲▲✮ λ ✲t❡r♠s = ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s = ♣♦✇❡r s❡r✐❡s ❖r✐❣✐♥❛❧❧②✿ ❢♦r t❤❡ λ ✲❝❛❧❝✉❧✉s✱ ✐♥ ❛♥ ❛❜str❛❝t ❝❛t❡❣♦r✐❝❛❧ s❡tt✐♥❣ ✭❝♦❡✣❝✐❡♥ts ❛r❡ s❡ts✮ ❋✐♥✐t♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ❡❛r❧② ✷✵✵✵✬s✮ ❘❡❢♦r♠✉❧❛t❡ q✳s✳ ✐♥ ❛ ❧✐♥❡❛r ❧♦❣✐❝ s❡tt✐♥❣ ✉s✐♥❣ st❛♥❞❛r❞ ❛❧❣❡❜r❛✿ ◮ t②♣❡s � ♣❛rt✐❝✉❧❛r t♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡s✿ � A � ⊆ k | A | ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ str✉❝t✉r❡ ◮ ❢✉♥❝t✐♦♥ t❡r♠s � ♣♦✇❡r s❡r✐❡s ✸ ✴ ✶✺

◗✉❛♥t✐t❛t✐✈❡ s❡♠❛♥t✐❝s ❆ ♣r✐♠❡ ❛❣❡❞ ✐❞❡❛ ✭●✐r❛r❞✱ ✬✽✵s✱ ❜❡❢♦r❡ ▲▲✮ λ ✲t❡r♠s = ❛♥❛❧②t✐❝ ❢✉♥❝t✐♦♥s = ♣♦✇❡r s❡r✐❡s ❖r✐❣✐♥❛❧❧②✿ ❢♦r t❤❡ λ ✲❝❛❧❝✉❧✉s✱ ✐♥ ❛♥ ❛❜str❛❝t ❝❛t❡❣♦r✐❝❛❧ s❡tt✐♥❣ ✭❝♦❡✣❝✐❡♥ts ❛r❡ s❡ts✮ ❋✐♥✐t♥❡ss s♣❛❝❡s ✭❊❤r❤❛r❞✱ ❡❛r❧② ✷✵✵✵✬s✮ ❘❡❢♦r♠✉❧❛t❡ q✳s✳ ✐♥ ❛ ❧✐♥❡❛r ❧♦❣✐❝ s❡tt✐♥❣ ✉s✐♥❣ st❛♥❞❛r❞ ❛❧❣❡❜r❛✿ ◮ t②♣❡s � ♣❛rt✐❝✉❧❛r t♦♣♦❧♦❣✐❝❛❧ ✈❡❝t♦r s♣❛❝❡s✿ � A � ⊆ k | A | ✰ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ str✉❝t✉r❡ ◮ ❢✉♥❝t✐♦♥ t❡r♠s � ♣♦✇❡r s❡r✐❡s ❉✐✛❡r❡♥t✐❛t✐♦♥ ♦❢ λ ✲t❡r♠s ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r ✷✵✵✸✲✷✵✵✹✮ ❙♦ ✇❡ ❝❛♥ ❞✐✛❡r❡♥t✐❛t❡ λ ✲t❡r♠s✱ ❛♥❞ ❝♦♠♣✉t❡ t❤❡✐r ❚❛②❧♦r ❡①♣❛♥s✐♦♥✦ ❆♥❞ ♦♥❡ ❝❛♥ ♠✐♠✐❝❦ t❤❛t ✐♥ t❤❡ s②♥t❛①✿ ◮ ❞✐✛❡r❡♥t✐❛❧ λ ✲❝❛❧❝✉❧✉s ◮ ❛ ✜♥✐t❛r② ❢r❛❣♠❡♥t✿ r❡s♦✉r❝❡ λ ✲❝❛❧❝✉❧✉s ❂ t❤❡ t❛r❣❡t ♦❢ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ✸ ✴ ✶✺

❧✐♥❡❛r✐t②✿ ✱ ✱ ❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥ ♣r❡s❡r✈❡s ❢r❡❡ ✈❛r✐❛❜❧❡s✱ ✐s s✐③❡✲❞❡❝r❡❛s✐♥❣✱ str♦♥❣❧② ❝♦♥✢✉❡♥t ❛♥❞ ♥♦r♠❛❧✐③✐♥❣✳ ❘❡s♦✉r❝❡ λ ✲❝❛❧❝✉❧✉s ❘❡s♦✉r❝❡ t❡r♠s ∆ ∋ s, t, . . . ::= x | λx.s | � s � t ∆ ! ∋ s, t, . . . ::= [ s 1 , . . . , s n ] ▼❡❛♥✐♥❣✿ � s � [ s 1 , . . . , s n ] = ( Ds ) 0 · ( s 1 , . . . , s n ) ❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥ � λx.s � t → ρ ∂ x s · t ✭❛♥②✇❤❡r❡✮ � � � � f ∈ S n s t f (1) , . . . , t f ( n ) /x 1 , . . . , x n ✐❢ deg x ( s ) = # t = n ∂ x s · t = 0 ♦t❤❡r✇✐s❡ ✹ ✴ ✶✺

❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥ ♣r❡s❡r✈❡s ❢r❡❡ ✈❛r✐❛❜❧❡s✱ ✐s s✐③❡✲❞❡❝r❡❛s✐♥❣✱ str♦♥❣❧② ❝♦♥✢✉❡♥t ❛♥❞ ♥♦r♠❛❧✐③✐♥❣✳ ❘❡s♦✉r❝❡ λ ✲❝❛❧❝✉❧✉s ❘❡s♦✉r❝❡ t❡r♠s ∆ ∋ s, t, . . . ::= x | λx.s | � s � t ∆ ! ∋ s, t, . . . ::= [ s 1 , . . . , s n ] ▼❡❛♥✐♥❣✿ � s � [ s 1 , . . . , s n ] = ( Ds ) 0 · ( s 1 , . . . , s n ) ❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥ � λx.s � t → ρ ∂ x s · t ✭❛♥②✇❤❡r❡✮ � � � � f ∈ S n s t f (1) , . . . , t f ( n ) /x 1 , . . . , x n ✐❢ deg x ( s ) = # t = n ∂ x s · t = 0 ♦t❤❡r✇✐s❡ ❧✐♥❡❛r✐t②✿ λx. 0 = 0 ✱ � s � [ t 1 + t 2 , u ] = � s � [ t 1 , u ] + � s � [ t 2 , u ] ✱ . . . ✹ ✴ ✶✺

❘❡s♦✉r❝❡ λ ✲❝❛❧❝✉❧✉s ❘❡s♦✉r❝❡ t❡r♠s ∆ ∋ s, t, . . . ::= x | λx.s | � s � t ∆ ! ∋ s, t, . . . ::= [ s 1 , . . . , s n ] ▼❡❛♥✐♥❣✿ � s � [ s 1 , . . . , s n ] = ( Ds ) 0 · ( s 1 , . . . , s n ) ❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥ � λx.s � t → ρ ∂ x s · t ✭❛♥②✇❤❡r❡✮ � � � � f ∈ S n s t f (1) , . . . , t f ( n ) /x 1 , . . . , x n ✐❢ deg x ( s ) = # t = n ∂ x s · t = 0 ♦t❤❡r✇✐s❡ ❧✐♥❡❛r✐t②✿ λx. 0 = 0 ✱ � s � [ t 1 + t 2 , u ] = � s � [ t 1 , u ] + � s � [ t 2 , u ] ✱ . . . ◮ ❘❡s♦✉r❝❡ r❡❞✉❝t✐♦♥ ♣r❡s❡r✈❡s ❢r❡❡ ✈❛r✐❛❜❧❡s✱ ✐s s✐③❡✲❞❡❝r❡❛s✐♥❣✱ str♦♥❣❧② ❝♦♥✢✉❡♥t ❛♥❞ ♥♦r♠❛❧✐③✐♥❣✳ ✹ ✴ ✶✺

❚❛②❧♦r ❡①♣❛♥s✐♦♥✿ ❚❤❡♦r❡♠ ✭❊❤r❤❛r❞✲❘❡❣♥✐❡r✱ ❈✐❊ ✷✵✵✻✮ ■❢ ✱ t❤❡♥ ♥♦r♠❛❧✐③❡s t♦ ✳ ▼♦r❛❧ ■♥ t❤❡ ♦r❞✐♥❛r② ✲❝❛❧❝✉❧✉s ✳ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ λ ✲t❡r♠s n ! � M � N n ✇❤❡r❡ N n = [ N, . . . , N ] ✳ 1 ❙❡♠❛♥t✐❝❛❧❧②✱ ( M ) N = � n ∈ N ✺ ✴ ✶✺