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❙❡ss✐♦♥ ❚②♣❡s ❛♥❞ ●❛♠❡ ❙❡♠❛♥t✐❝s ❙②♥❝❤r♦♥② ❛♥❞ ❆s②♥❝❤r♦♥② ❙✐♠♦♥ ❈❛st❡❧❧❛♥ ✶ ✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t ✷ ✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✶ ✶ ■♠♣❡r✐❛❧ ❈♦❧❧❡❣❡ ▲♦♥❞♦♥ ✷ ▲■P✱ ❊◆❙ ▲②♦♥ ✶✹ ❛♣r✐❧ ✷✵✶✽ ●❛▲♦P ✷✵✶✽

❚❤❡ π ✲❝❛❧❝✉❧✉s ❬▼P❲✾✷❪ ❚❤❡ π ✲❝❛❧❝✉❧✉s ❞❡s❝r✐❜❡s ❛❣❡♥ts ❝♦♠♠✉♥✐❝❛t✐♥❣ t❤r♦✉❣❤ ❝❤❛♥♥❡❧s ✿ P , Q ::= ✵ | ( P | Q ) | ( ν ab ) P r❡str✐❝t✐♦♥ | ¯ a ! ℓ � u � . P ♦✉t♣✉t � | a ? ℓ ( x i ) . P i ✐♥♣✉t i ∈ I | P + Q ♥♦♥❞❡t✳ ❝❤♦✐❝❡ ❈♦♠♠✉♥✐❝❛t✐♦♥✿ ❞❛t❛ ✭ ℓ ✮ ❛♥❞ ❝❤❛♥♥❡❧s ✭ u ✮✳ ❙❤♦rt✲❤❛♥❞s ✿ ¯ a � u � := ¯ a ! ⋆ � � u � a ( x ) := a ? ⋆ ( x ) ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✷ ✴ ✷✹

❇❛s✐❝ ✐❞❡❛✿ ✐♥t❡r♣r❡t ❝❤❛♥♥❡❧s ❛s ❛♥ ❡✛❡❝t ❧✐❦❡ r❡❢❡r❡♥❝❡s✿ ❝❝ ●❛♠❡ s❡♠❛♥t✐❝s ❢♦r t❤❡ π ✲❝❛❧❝✉❧✉s ❊①✐st✐♥❣ ♠♦❞❡❧s✿ ◮ ▲❛✐r❞ ❬▲❛✐✵✺❪ r❡✜♥❡❞ ❜② ❚s✉❦❛❞❛ ✫ ❙❛❦❛②♦r✐ ❬❙❚✶✼❪ ✭❢♦r t❤❡ ❛s②♥❝❤r♦♥♦✉s ❢r❛❣♠❡♥t✮ ◮ ❍✐rs❝❤♦✇✐t③ ❡t✳ ❛❧✳ ❬❊❍❙✶✺❪ � ■♥ t❤✐s t❛❧❦✱ ❢♦❝✉s ♦♥ ❛♥❛❧②③✐♥❣ t❤❡ ✜rst ❧✐♥❡ ♦❢ ✐♥t❡r♣r❡t❛t✐♦♥✳ ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✸ ✴ ✷✹

●❛♠❡ s❡♠❛♥t✐❝s ❢♦r t❤❡ π ✲❝❛❧❝✉❧✉s ❊①✐st✐♥❣ ♠♦❞❡❧s✿ ◮ ▲❛✐r❞ ❬▲❛✐✵✺❪ r❡✜♥❡❞ ❜② ❚s✉❦❛❞❛ ✫ ❙❛❦❛②♦r✐ ❬❙❚✶✼❪ ✭❢♦r t❤❡ ❛s②♥❝❤r♦♥♦✉s ❢r❛❣♠❡♥t✮ ◮ ❍✐rs❝❤♦✇✐t③ ❡t✳ ❛❧✳ ❬❊❍❙✶✺❪ � ■♥ t❤✐s t❛❧❦✱ ❢♦❝✉s ♦♥ ❛♥❛❧②③✐♥❣ t❤❡ ✜rst ❧✐♥❡ ♦❢ ✐♥t❡r♣r❡t❛t✐♦♥✳ ❇❛s✐❝ ✐❞❡❛✿ ✐♥t❡r♣r❡t ❝❤❛♥♥❡❧s ❛s ❛♥ ❡✛❡❝t ❧✐❦❡ r❡❢❡r❡♥❝❡s✿ � ♯ T � = � T � ⊥ × � T � � ( ν a ) P � = � P � ⊙ ❝❝ ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✸ ✴ ✷✹

❆s②♥❝❤r♦♥②✿ ❣❛♠❡ s❡♠❛♥t✐❝s ❈♦♥❝✉rr❡♥t ❣❛♠❡ s❡♠❛♥t✐❝s ✐s tr❛❞t✐♦♥❛❧❧② ❛s②♥❝❤r♦♥♦✉s ✿ c A ⊙ σ = σ = = = ⇒ σ ❝♦✉rt❡♦✉s [ ▼▼✵✼ , ❘❲✶✶ ] c B B B B q q q q tt tt tt tt ✗ ✓ ❚❤✐s ❢♦r❝❡s s♦♠❡ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ♠♦❞❡❧✿ a � u � . ¯ b � v � . P � = � ¯ � ¯ b � v � . ¯ a � u � . P � � a ( x ) . b ( y ) . P � = � b ( y ) . a ( x ) . P � � ▲✐♠✐ts ❛❞❡q✉❛❝② r❡s✉❧ts ✳ ✳ ✳ ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✹ ✴ ✷✹

❍♦✇❡✈❡r✱ ♠✉st ◆♦ ❛❞❡q✉❛❝② ♣♦ss✐❜❧❡ ❢♦r ♥♦♥✲❛♥❣❡❧✐❝ t❡st✐♥❣ ❡q✉✐✈❛❧❡♥❝❡s ✳ ✳ ✳ ◆❡❡❞ t♦ t❛❦❡ s②♥❝❤r♦♥② s❡r✐♦✉s❧②✦ ❆s②♥❝❤r♦♥②✿ π ✲❝❛❧❝✉❧✉s ❬❍❚✾✶❪ ❆s②♥❝❤r♦♥② ✐♥ π ✲❝❛❧❝✉❧✉s✿ ♥♦ ❝♦♥t✐♥✉❛t✐♦♥ ❛❢t❡r s❡♥❞s✳ a � u � . ¯ � ¯ b � v ) ✐s ♥♦t ❛ t❡r♠✦ ▼♦r❡♦✈❡r✱ ✐♥ ❛s②♥❝❤r♦♥♦✉s π ✲❝❛❧❝✉❧✉s✿ a ( x ) . b ( y ) . P ≃ ♠❛② b ( y ) . a ( x ) . Q � ▼♦❞❡❧s ♦❢ ❬▲❛✐✵✺✱ ❙❚✶✼❪ ❛❞❡q✉❛t❡ ❢♦r ♠❛② ✳ ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✺ ✴ ✷✹

❆s②♥❝❤r♦♥②✿ π ✲❝❛❧❝✉❧✉s ❬❍❚✾✶❪ ❆s②♥❝❤r♦♥② ✐♥ π ✲❝❛❧❝✉❧✉s✿ ♥♦ ❝♦♥t✐♥✉❛t✐♦♥ ❛❢t❡r s❡♥❞s✳ a � u � . ¯ � ¯ b � v ) ✐s ♥♦t ❛ t❡r♠✦ ▼♦r❡♦✈❡r✱ ✐♥ ❛s②♥❝❤r♦♥♦✉s π ✲❝❛❧❝✉❧✉s✿ a ( x ) . b ( y ) . P ≃ ♠❛② b ( y ) . a ( x ) . Q � ▼♦❞❡❧s ♦❢ ❬▲❛✐✵✺✱ ❙❚✶✼❪ ❛❞❡q✉❛t❡ ❢♦r ♠❛② ✳ ❍♦✇❡✈❡r✱ a ( x ) . b ( y ) . P � ≃ ♠✉st b ( y ) . a ( x ) . Q ◆♦ ❛❞❡q✉❛❝② ♣♦ss✐❜❧❡ ❢♦r ♥♦♥✲❛♥❣❡❧✐❝ t❡st✐♥❣ ❡q✉✐✈❛❧❡♥❝❡s ✳ ✳ ✳ ⇒ ◆❡❡❞ t♦ t❛❦❡ s②♥❝❤r♦♥② s❡r✐♦✉s❧②✦ ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✺ ✴ ✷✹

❙❡ss✐♦♥ t②♣❡s ❬❍❱❑✾✽❪ ❚②♣✐♥❣ ❞✐s❝♣❧✐♥❡ ✇❤❡r❡ t②♣❡s ❛r❡ ♣r♦t♦❝♦❧s ✿ S , T ::= ❡♥❞ | ⊕ i ∈ I ℓ i ( S i ) . T i | & i ∈ I ℓ i ( S i ) . T i ❚②♣✐♥❣ ⊢ P :: a ✶ : S ✶ , . . . , a n : S n ❡♥s✉r❡s ♣r♦t♦❝♦❧ ♣r❡s❡r✈❛t✐♦♥ ✳ ⊢ P : a : T k , ∆ ⊢ a ! ℓ k � u � . P :: a : ⊕ i ∈ I ℓ i ( S i ) . T i , ∆ , u : S k ❉✉❛❧✐t② ❡①♣r❡ss❡s ❝♦♠♣❛t✐❜❧❡ ❡♥❞♣♦✐♥ts✿ ⊢ P :: ∆ , a : S , b : S ⊥ ⊢ ( ν ab ) P :: ∆ ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✻ ✴ ✷✹

❈♦✐♥❝✐❞❡♥t str❛t❡❣✐❡s ❈♦✉rt❡♦✉s Pr♦❝❡ss❡s ❚❤✐s t❛❧❦ ❙②♥❝❤r♦♥♦✉s Pr♦❝❡ss❡s ✭✶✮ ✐♥❛❞❡q✉❛t❡ ✐♥t❡r♣r❡t❛t✐♦♥ ❈♦♥❝✉rr❡♥t ❙tr❛t❡❣✐❡s ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✼ ✴ ✷✹

❈♦✐♥❝✐❞❡♥t str❛t❡❣✐❡s ❈♦✉rt❡♦✉s Pr♦❝❡ss❡s ❚❤✐s t❛❧❦ ❙②♥❝❤r♦♥♦✉s Pr♦❝❡ss❡s ✐♥❛❞❡q✉❛t❡ ✐♥t❡r♣r❡t❛t✐♦♥ ✜ ♥ ✐ t ❡ ❞ ❡ ❢ ✉ ♣ t ♦ ∼ = ❈♦♥❝✉rr❡♥t ❙tr❛t❡❣✐❡s ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✼ ✴ ✷✹

❈♦✐♥❝✐❞❡♥t str❛t❡❣✐❡s ❚❤✐s t❛❧❦ ❙②♥❝❤r♦♥♦✉s Pr♦❝❡ss❡s ✐♥❛❞❡q✉❛t❡ ✐♥t❡r♣r❡t❛t✐♦♥ ✜ ♥ ✐ t ❡ ❞ ❡ ❢ ✉ ♣ t ♦ ∼ = ✭ ✷ ✮ ❛ ❞ ❡ q ✉ ❛ t ❡ ✐ ♥ t ❡ r ♣ ✳ ❈♦✉rt❡♦✉s Pr♦❝❡ss❡s ❈♦♥❝✉rr❡♥t ❙tr❛t❡❣✐❡s ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✼ ✴ ✷✹

❚❤✐s t❛❧❦ ✭✸✮ ❛❞❡q✉❛t❡ ✐♥t❡r♣✳ ❙②♥❝❤r♦♥♦✉s Pr♦❝❡ss❡s ✭✸✮ ❈♦✐♥❝✐❞❡♥t str❛t❡❣✐❡s ✐ ♥ ❛ ❞ ❡ q ✉ ❛ t ❡ ✐ ♥ t ❡ ✜♥✐t❡ ❞❡❢ ✉♣ t♦ ∼ r ♣ r ❡ t ❛ t ✐ ♦ ♥ = ❛❞❡q✉❛t❡ ✐♥t❡r♣✳ ❈♦✉rt❡♦✉s Pr♦❝❡ss❡s ❈♦♥❝✉rr❡♥t ❙tr❛t❡❣✐❡s ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✼ ✴ ✷✹

❚❤✐s t❛❧❦ ❛❞❡q✉❛t❡ ✐♥t❡r♣✳ ❙②♥❝❤r♦♥♦✉s Pr♦❝❡ss❡s ❈♦✐♥❝✐❞❡♥t str❛t❡❣✐❡s ✐♥❛❞❡q✉❛t❡ ✐♥t❡r♣r❡t❛t✐♦♥ ✭✹✮ ❡♥❝♦❞✐♥❣ ✜ ♥ ✐ t ❡ ❞ ❡ ❢ ✉ ♣ t ♦ ∼ = ❛ ❞ ❡ q ✉ ❛ t ❡ ✐ ♥ t ❡ r ♣ ✳ ❈♦✉rt❡♦✉s Pr♦❝❡ss❡s ❈♦♥❝✉rr❡♥t ❙tr❛t❡❣✐❡s ✭❆✮s②♥❝❤r♦♥② ✐♥ ❣❛♠❡ s❡♠❛♥t✐❝s · ❈✳✱ P✐❡rr❡ ❈❧❛✐r❛♠❜❛✉❧t✱ ◆♦❜✉❦♦ ❨♦s❤✐❞❛ ✼ ✴ ✷✹