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  1. ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s ❙✐♠♦♥ ❈❛st❡❧❧❛♥ ✶ ✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t ✷ ✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥ ✸ ✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✸ ✶ ■♠♣❡r✐❛❧ ❈♦❧❧❡❣❡ ▲♦♥❞♦♥ ✷ ▲■P✱ ❊◆❙ ▲②♦♥ ✸ ❈♦♠♣✉t❡r ▲❛❜♦r❛t♦r②✱ ❯♥✐✈❡rs✐t② ♦❢ ❈❛♠❜r✐❞❣❡ ✶✻ ❛♣r✐❧ ✷✵✶✽ ❋♦❙❙❛❈❙ ✷✵✶✽

  2. ❆♣♣❧✐❝❛t✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ❜② ❝♦♠♣♦s✐t✐♦♥ ✿ ✐♥t❡r❛❝t✐♦♥ ✭❝♦♠♠✉♥✐❝❛t✐♦♥ ♦♥ ✮ ✭♦❢ ✐♥t❡r♥❛❧ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♦♥ ✮ ❤✐❞✐♥❣ ❍✐❞✐♥❣ ✐s ❦❡② t♦ ❝r✉❝✐❛❧ t♦ ❣❡t ✳ ●❛♠❡ s❡♠❛♥t✐❝s ❘❡♣r❡s❡♥ts ♣r♦❣r❛♠s ❜② t❤❡✐r ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ t❤❡ ❝♦♥t❡①t✿     � �  ⇒ ⇒  B B B B       q q     � �   λ x . ✐❢ x t❤❡♥ tr✉❡   � � , q q � � � = � ❡❧s❡ ⊥       tt ✛ � �� �       M       tt � �� � str❛t❡❣② = s❡t ♦❢ ♣❧❛②s ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✷ ✴ ✷✵

  3. ●❛♠❡ s❡♠❛♥t✐❝s ❘❡♣r❡s❡♥ts ♣r♦❣r❛♠s ❜② t❤❡✐r ✐♥t❡r❛❝t✐♦♥ ✇✐t❤ t❤❡ ❝♦♥t❡①t✿     � �  ⇒ ⇒  B B B B       q q     � �   λ x . ✐❢ x t❤❡♥ tr✉❡   � � , q q � � � = � ❡❧s❡ ⊥       tt ✛ � �� �       M       tt � �� � str❛t❡❣② = s❡t ♦❢ ♣❧❛②s ❆♣♣❧✐❝❛t✐♦♥ ( M : A → B ) ( N : A ) ✐s r❡♣r❡s❡♥t❡❞ ❜② ❝♦♠♣♦s✐t✐♦♥ ✿ ✐♥t❡r❛❝t✐♦♥ ( ⊛ ) ✭❝♦♠♠✉♥✐❝❛t✐♦♥ ♦♥ A ✮ + ❤✐❞✐♥❣ ✭♦❢ ✐♥t❡r♥❛❧ ❝♦♠♠✉♥✐❝❛t✐♦♥ ♦♥ A ✮ � ❍✐❞✐♥❣ ✐s ❦❡② t♦ ❝r✉❝✐❛❧ t♦ ❣❡t � ( λ x . M ) N � = � M [ N / x ] � ✳ ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✷ ✴ ✷✵

  4. q tt tt ❖♥❧② ❛❞❡q✉❛t❡ ❢♦r ❛♥❣❡❧✐❝ ♥♦♥❞❡t❡r♠✐♥✐s♠ ✭♠❛②✲❡q✉✐✈❛❧❡♥❝❡✮✳ ❚❤❡ ♣r♦❜❧❡♠ ✇✐t❤ ♥♦♥❞❡t❡r♠✐♥✐s♠ ❚r❛❞✐t✐♦♥❛❧ ❤✐❞✐♥❣ ♦❢ ❣❛♠❡ s❡♠❛♥t✐❝s ♦♥❧② ❦❡❡♣s ✈✐s✐❜❧❡ ❡✈❡♥ts✳ � M ❝❤♦✐❝❡ � = ❤✐❞❡ ( � M � ⊛ � ❝❤♦✐❝❡ � )   B B   q q       q , q = ❤✐❞❡       tt ✛     tt   B B     , = q q     tt ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✸ ✴ ✷✵

  5. ❚❤❡ ♣r♦❜❧❡♠ ✇✐t❤ ♥♦♥❞❡t❡r♠✐♥✐s♠ ❚r❛❞✐t✐♦♥❛❧ ❤✐❞✐♥❣ ♦❢ ❣❛♠❡ s❡♠❛♥t✐❝s ♦♥❧② ❦❡❡♣s ✈✐s✐❜❧❡ ❡✈❡♥ts✳ � M ❝❤♦✐❝❡ � = ❤✐❞❡ ( � M � ⊛ � ❝❤♦✐❝❡ � )   B B   q q       q , q = ❤✐❞❡       tt ✛     tt     B B B         , = q q  = q = � tt �          tt tt � ❖♥❧② ❛❞❡q✉❛t❡ ❢♦r ❛♥❣❡❧✐❝ ♥♦♥❞❡t❡r♠✐♥✐s♠ ✭♠❛②✲❡q✉✐✈❛❧❡♥❝❡✮✳ ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✸ ✴ ✷✵

  6. ❲❤❛t ✇❡ ❝❛♥ ❞♦ ❚✇♦ ❡①✐st✐♥❣ ❛♣♣r♦❛❝❤❡s✿ ◮ ❙t♦♣♣✐♥❣ tr❛❝❡s ♦❢ ❍❛r♠❡r ❛♥❞ ▼❝❈✉s❦❡r✿ � ❚❛✐❧♦r❡❞ ❢♦r ♠✉st✲❡q✉✐✈❛❧❡♥❝❡ ♦♥❧② ◮ P❧❛②❣r♦✉♥❞s ♦❢ ❍✐rs❝❤♦✇✐t③ ❡t✳ ❛❧ � ◆♦ ❝♦♠♣♦s✐t✐♦♥✳ ■♥ t❤✐s t❛❧❦✿ ◮ ❲❤❛t❄ ❖❜t❛✐♥ ♥♦♥✲❛♥❣❡❧✐❝ ♠♦❞❡❧s ✐♥ ❝♦♥❝✉rr❡♥t ❣❛♠❡s✳ � ❆❞❡q✉❛❝② ❢♦r ✇❡❛❦ ❜✐s✐♠✉❧❛t✐♦♥✱ ❛♥❞ ❛ ❝♦♠♣♦s✐t✐♦♥❛❧ st♦r②✳ ◮ ❍♦✇❄ ▼♦❞✐❢② ❤✐❞✐♥❣ t♦ r❡♠❡♠❜❡r ❤✐❞❞❡♥ ❞✐✈❡r❣❡♥❝❡s✳ � ❚❤❡② ❜❡❝♦♠❡ ✐♥t❡r♥❛❧ ❡✈❡♥ts ✳ q ∗ ∗ tt ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✹ ✴ ✷✵

  7. ◆♦ ❤✐❞✐♥❣ P❛rt✐❛❧ ❤✐❞✐♥❣ q q tt tt q q tt tt ❈● ♥♦ ❈● ♣❛rt✐❛❧ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✐♥ ❛ t❛❜❧❡ ❚♦t❛❧ ❤✐❞✐♥❣ ❬❘❲✶✶❪ q � N � tt q � ( λ x . x ) N � tt ∼ ❈● = t♦t❛❧ ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✺ ✴ ✷✵

  8. P❛rt✐❛❧ ❤✐❞✐♥❣ q tt q q tt tt ❈● ♥♦ ❈● ♣❛rt✐❛❧ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✐♥ ❛ t❛❜❧❡ ❚♦t❛❧ ❤✐❞✐♥❣ ◆♦ ❤✐❞✐♥❣ ❬❘❲✶✶❪ q q ∗ � N � ∗ ∗ tt tt q � ( λ x . x ) N � tt ∼ ❈● = t♦t❛❧ ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✺ ✴ ✷✵

  9. P❛rt✐❛❧ ❤✐❞✐♥❣ q tt q tt ❈● ♥♦ ❈● ♣❛rt✐❛❧ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✐♥ ❛ t❛❜❧❡ ❚♦t❛❧ ❤✐❞✐♥❣ ◆♦ ❤✐❞✐♥❣ ❬❘❲✶✶❪ q q ∗ � N � ∗ ∗ tt tt ∗ q q ∗ � ( λ x . x ) N � ∗ ∗ tt ∗ tt ∼ ❈● = t♦t❛❧ ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✺ ✴ ✷✵

  10. P❛rt✐❛❧ ❤✐❞✐♥❣ q tt q tt ❈● ♣❛rt✐❛❧ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✐♥ ❛ t❛❜❧❡ ❚♦t❛❧ ❤✐❞✐♥❣ ◆♦ ❤✐❞✐♥❣ ❬❘❲✶✶❪ q q ∗ � N � ∗ ∗ tt tt ∗ q q ∗ � ( λ x . x ) N � ∗ ∗ tt ∗ tt ∼ ❈● ❈● ≈ = ♥♦ t♦t❛❧ ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✺ ✴ ✷✵

  11. q tt q tt ❈● ♣❛rt✐❛❧ ❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦ ✐♥ ❛ t❛❜❧❡ ❚♦t❛❧ ❤✐❞✐♥❣ ◆♦ ❤✐❞✐♥❣ P❛rt✐❛❧ ❤✐❞✐♥❣ ❬❘❲✶✶❪ q q ∗ � N � ∗ ∗ tt tt ∗ q q ∗ � ( λ x . x ) N � ∗ ∗ tt ∗ tt ∼ ❈● ❈● ≈ = ♥♦ t♦t❛❧ ◆♦♥✲❛♥❣❡❧✐❝ ❝♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ❏♦♥❛t❤❛♥ ❍❛②♠❛♥✱ ●❧②♥♥ ❲✐♥s❦❡❧ ✺ ✴ ✷✵

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