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❙❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡s ❛♥❞ ❡①t❡♥s✐♦♥s ♦❢ ●▲ ❋❡❞♦r P❛❦❤♦♠♦✈ ✶ ❙t❡❦❧♦✈ ▼❛t❤❡♠❛t✐❝❛❧ ■♥st✐t✉t❡✱ ▼♦s❝♦✇ ♣❛❦❤❢❅♠✐✳r❛s✳r✉ ❆❞✈❛♥❝❡s ✐♥ ▼♦❞❛❧ ▲♦❣✐❝ ✷✵✶✻ ❙❡♣t❡♠❜❡r ✶ st ✷✵✶✻ ✶ ❘❡s❡❛r❝❤ s✉♣♣♦rt❡❞ ❜② t❤❡ ❘✉ss✐❛♥ ❙❝✐❡♥❝❡ ❋♦✉♥❞❛t✐♦♥ ✉♥❞❡r ❣r❛♥t ◆♦✳ ✶✹✕✺✵✕✵✵✵✵✺✳

❚❤❡♦r❡♠ ✭❙♦❧♦✈❛② ✶✾✼✻✮ ❋♦r ❡✈❡r② ♠♦❞❛❧ ❢♦r♠✉❧❛ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ●▲ ❀ ❢♦r ❡✈❡r② ✐♥t❡r♣r❡t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ Pr✈ P❆ ✇❡ ❤❛✈❡ P❆ ✳ ❆r✐t❤♠❡t✐❝❛❧ ❈♦♠♣❧❡t❡♥❡ss ♦❢ ●▲ Pr✈ P❆ ( x ) ✐s t❤❡ ●☎ ♦❞❡❧ ♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡ ❢♦r P❆✳ ( · ) ⋆ : ▲ � → ❆r✐t❤❙❡♥ ✐s ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ Pr✈ P❆ ( x ) ✐❢ ◮ ( � ϕ ) ⋆ ✐s ❡q✉❛❧ t♦ Pr✈ P❆ ( � ϕ ⋆ � ) ❀ ◮ ( · ) ⋆ ❝♦♠♠✉t❡s ✇✐t❤ ❛❧❧ ♥♦♥✲♠♦❞❛❧ ❝♦♥♥❡❝t✐✈❡❀ ◮ x ⋆ ❛r❡ ❛r❜✐tr❛r② ❢♦r ✈❛r✐❛❜❧❡s x ✳

❆r✐t❤♠❡t✐❝❛❧ ❈♦♠♣❧❡t❡♥❡ss ♦❢ ●▲ Pr✈ P❆ ( x ) ✐s t❤❡ ●☎ ♦❞❡❧ ♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡ ❢♦r P❆✳ ( · ) ⋆ : ▲ � → ❆r✐t❤❙❡♥ ✐s ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ Pr✈ P❆ ( x ) ✐❢ ◮ ( � ϕ ) ⋆ ✐s ❡q✉❛❧ t♦ Pr✈ P❆ ( � ϕ ⋆ � ) ❀ ◮ ( · ) ⋆ ❝♦♠♠✉t❡s ✇✐t❤ ❛❧❧ ♥♦♥✲♠♦❞❛❧ ❝♦♥♥❡❝t✐✈❡❀ ◮ x ⋆ ❛r❡ ❛r❜✐tr❛r② ❢♦r ✈❛r✐❛❜❧❡s x ✳ ❚❤❡♦r❡♠ ✭❙♦❧♦✈❛② ✶✾✼✻✮ ❋♦r ❡✈❡r② ♠♦❞❛❧ ❢♦r♠✉❧❛ ϕ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ◮ ●▲ ⊢ ϕ ❀ ◮ ❢♦r ❡✈❡r② ✐♥t❡r♣r❡t❛t✐♦♥ ( · ) ⋆ ✇✐t❤ r❡s♣❡❝t t♦ Pr✈ P❆ ( x ) ✇❡ ❤❛✈❡ P❆ ⊢ ϕ ⋆ ✳

■♥ t❤✐s ❝❛s❡ ✇❡ ❝❛❧❧ ❙Pr✈ s❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡✳ ❙Pr✈ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♠♦❞❛❧ ❢♦r♠✉❧❛s s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ✐♥t❡r♣❡t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❙Pr✈ ✇❡ ❤❛✈❡ ❚ ❙Pr✈ ✳ ❙❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡s ❙✉♣♣♦s❡ ❚ ✐♥t❡r♣r❡t❡s P❆✳ ♦❜ ❞❡r✐✈❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ❙Pr✈ ( x ) ✿ ❍✐❧❜❡rt✲❇❡r♥❛②s✲▲☎ ◮ ❚ ⊢ ϕ ⇒ ❚ ⊢ ❙Pr✈ ( � ϕ � ) ❀ ◮ ❚ ⊢ ❙Pr✈ ( � ϕ → ψ � ) → ( ❙Pr✈ ( � ϕ � ) → ❙Pr✈ ( � ψ � )) ❀ ◮ ❚ ⊢ ❙Pr✈ ( � ϕ � ) → ❙Pr✈ ( ❙Pr✈ ( � ϕ � )) ✳ ✭ ❙Pr✈ ( x ) ✏s❛t✐s✜❡s✑ ❑✹ ✮

❙Pr✈ ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♠♦❞❛❧ ❢♦r♠✉❧❛s s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ✐♥t❡r♣❡t❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❙Pr✈ ✇❡ ❤❛✈❡ ❚ ❙Pr✈ ✳ ❙❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡s ❙✉♣♣♦s❡ ❚ ✐♥t❡r♣r❡t❡s P❆✳ ♦❜ ❞❡r✐✈❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ❙Pr✈ ( x ) ✿ ❍✐❧❜❡rt✲❇❡r♥❛②s✲▲☎ ◮ ❚ ⊢ ϕ ⇒ ❚ ⊢ ❙Pr✈ ( � ϕ � ) ❀ ◮ ❚ ⊢ ❙Pr✈ ( � ϕ → ψ � ) → ( ❙Pr✈ ( � ϕ � ) → ❙Pr✈ ( � ψ � )) ❀ ◮ ❚ ⊢ ❙Pr✈ ( � ϕ � ) → ❙Pr✈ ( ❙Pr✈ ( � ϕ � )) ✳ ✭ ❙Pr✈ ( x ) ✏s❛t✐s✜❡s✑ ❑✹ ✮ ■♥ t❤✐s ❝❛s❡ ✇❡ ❝❛❧❧ ❙Pr✈ ( x ) s❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡✳

❙❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡s ❙✉♣♣♦s❡ ❚ ✐♥t❡r♣r❡t❡s P❆✳ ♦❜ ❞❡r✐✈❛❜✐❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ❙Pr✈ ( x ) ✿ ❍✐❧❜❡rt✲❇❡r♥❛②s✲▲☎ ◮ ❚ ⊢ ϕ ⇒ ❚ ⊢ ❙Pr✈ ( � ϕ � ) ❀ ◮ ❚ ⊢ ❙Pr✈ ( � ϕ → ψ � ) → ( ❙Pr✈ ( � ϕ � ) → ❙Pr✈ ( � ψ � )) ❀ ◮ ❚ ⊢ ❙Pr✈ ( � ϕ � ) → ❙Pr✈ ( ❙Pr✈ ( � ϕ � )) ✳ ✭ ❙Pr✈ ( x ) ✏s❛t✐s✜❡s✑ ❑✹ ✮ ■♥ t❤✐s ❝❛s❡ ✇❡ ❝❛❧❧ ❙Pr✈ ( x ) s❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡✳ L ( ❙Pr✈ ) ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♠♦❞❛❧ ❢♦r♠✉❧❛s ϕ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② ✐♥t❡r♣❡t❛t✐♦♥ ( · ) ⋆ ✇✐t❤ r❡s♣❡❝t t♦ ❙Pr✈ ✇❡ ❤❛✈❡ ❚ ⊢ ❙Pr✈ ( � ϕ ⋆ � ) ✳

■❢ ❙Pr✈ ✐s ✶ ❛♥❞ ❚ ✐s ✶ ✲s♦✉♥❞ t❤❡♥ ❙Pr✈ ●▲ ✭❜② ❙♦❧♦✈❛② t❡❝❤♥✐q✉❡✮✳ ❙♦❧♦✈❛② ❡①❛♠♣❧❡✿ ❙✉♣♣♦s❡ ❯♥✐✈ ✐s ❛ s❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡ ✐♥ ❩❋❈ s✉❝❤ t❤❛t ❯♥✐✈ ♠❡❛♥s t❤❛t ✐s tr✉❡ ✐♥ ❛❧❧ t❤❡ ♠♦❞❡❧s ❱ ✱ ✇❤❡r❡ ✐s ✐♥❛❝❝❡ss✐❜❧❡✳ ■❢ ✇❡ t❛❦❡ ❩❋❈ ✏t❤❡r❡ ❡①✐sts ✐♥✜♥✐t❡❧② ♠❛♥② ✐♥❛❝❝❡ss✐❜❧❡ ❝❛r❞✐♥❛❧s✑ ❛s ♦✉r ♠❡t❛t❤❡♦r② t❤❡♥✿ ❯♥✐✈ ●▲ ✸ ▲✐♠✐ts ♦❢ ❙♦❧♦✈❛② t❤❡♦r❡♠ ❋r♦♠ ▲☎ ♦❜✬s t❤❡♦r❡♠ ✐t ❢♦❧❧♦✇s t❤❛t L ( ❙Pr✈ ) ⊃ ●▲✱ ❢♦r ❛❧❧ ❙Pr✈ ( x )

❙♦❧♦✈❛② ❡①❛♠♣❧❡✿ ❙✉♣♣♦s❡ ❯♥✐✈ ✐s ❛ s❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡ ✐♥ ❩❋❈ s✉❝❤ t❤❛t ❯♥✐✈ ♠❡❛♥s t❤❛t ✐s tr✉❡ ✐♥ ❛❧❧ t❤❡ ♠♦❞❡❧s ❱ ✱ ✇❤❡r❡ ✐s ✐♥❛❝❝❡ss✐❜❧❡✳ ■❢ ✇❡ t❛❦❡ ❩❋❈ ✏t❤❡r❡ ❡①✐sts ✐♥✜♥✐t❡❧② ♠❛♥② ✐♥❛❝❝❡ss✐❜❧❡ ❝❛r❞✐♥❛❧s✑ ❛s ♦✉r ♠❡t❛t❤❡♦r② t❤❡♥✿ ❯♥✐✈ ●▲ ✸ ▲✐♠✐ts ♦❢ ❙♦❧♦✈❛② t❤❡♦r❡♠ ❋r♦♠ ▲☎ ♦❜✬s t❤❡♦r❡♠ ✐t ❢♦❧❧♦✇s t❤❛t L ( ❙Pr✈ ) ⊃ ●▲✱ ❢♦r ❛❧❧ ❙Pr✈ ( x ) ■❢ ❙Pr✈ ( x ) ✐s Σ ✶ ❛♥❞ ❚ ✐s Σ ✶ ✲s♦✉♥❞ t❤❡♥ L ( ❙Pr✈ ) = ●▲ ✭❜② ❙♦❧♦✈❛② t❡❝❤♥✐q✉❡✮✳

▲✐♠✐ts ♦❢ ❙♦❧♦✈❛② t❤❡♦r❡♠ ❋r♦♠ ▲☎ ♦❜✬s t❤❡♦r❡♠ ✐t ❢♦❧❧♦✇s t❤❛t L ( ❙Pr✈ ) ⊃ ●▲✱ ❢♦r ❛❧❧ ❙Pr✈ ( x ) ■❢ ❙Pr✈ ( x ) ✐s Σ ✶ ❛♥❞ ❚ ✐s Σ ✶ ✲s♦✉♥❞ t❤❡♥ L ( ❙Pr✈ ) = ●▲ ✭❜② ❙♦❧♦✈❛② t❡❝❤♥✐q✉❡✮✳ ❙♦❧♦✈❛② ❡①❛♠♣❧❡✿ ❙✉♣♣♦s❡ ❯♥✐✈ ( x ) ✐s ❛ s❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡ ✐♥ ❩❋❈ s✉❝❤ t❤❛t ❯♥✐✈ ( � ϕ � ) ♠❡❛♥s t❤❛t ϕ ✐s tr✉❡ ✐♥ ❛❧❧ t❤❡ ♠♦❞❡❧s ( ❱ κ , ∈ ) ✱ ✇❤❡r❡ κ ✐s ✐♥❛❝❝❡ss✐❜❧❡✳ ■❢ ✇❡ t❛❦❡ ❩❋❈ + ✏t❤❡r❡ ❡①✐sts ✐♥✜♥✐t❡❧② ♠❛♥② ✐♥❛❝❝❡ss✐❜❧❡ ❝❛r❞✐♥❛❧s✑ ❛s ♦✉r ♠❡t❛t❤❡♦r② t❤❡♥✿ L ( ❯♥✐✈ ) = ●▲ . ✸ .

❱ ❱ ❱ ❱ ✵ ✶ ❊①♣❧❛♥❛t✐♦♥ ♦❢ ❙♦❧♦✈❛② ❊①❛♠♣❧❡ κ α ✐s t❤❡ α ✲t❤ ✐♥❛❝❝❡ss✐❜❧❡✳ ❱ ❱ κ α . . . . . . ❱ κ ✵ ❱ κ ✶

❊①♣❧❛♥❛t✐♦♥ ♦❢ ❙♦❧♦✈❛② ❊①❛♠♣❧❡ κ α ✐s t❤❡ α ✲t❤ ✐♥❛❝❝❡ss✐❜❧❡✳ ❱ ❱ κ α . . . . . . ❱ κ ✵ ❱ κ ✶ . . . . . . ❱ κ ✵ ❱ κ ✶ ❱ κ α ❱

❑r✐♣❦❡ ❈♦♠♣❧❡t❡ ❙❡♠✐✲Pr♦✈❛❜✐❧✐t② Pr❡❞✐❝❛t❡s ❚❤❡♦r❡♠ ❙✉♣♣♦s❡ ▲ ✐s ❛ ❑r✐♣❦❡ ❝♦♠♣❧❡t❡ r❡❝✉rs✐✈❡❧② ❛①✐♦♠❛t✐③❛❜❧❡ ❡①t❡♥s✐♦♥ ♦❢ ●▲ ❛♥❞ ❩❋❈ ♣r♦✈❡s t❤❛t ▲ ✐s ❑r✐♣❦❡ ❝♦♠♣❧❡t❡✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❩❋❈ s❡♠✐✲♣r♦✈❛❜✐❧✐t② ♣r❡❞✐❝❛t❡ ❙Pr✈ ▲ ( x ) s✉❝❤ t❤❛t L ( ❙Pr✈ ▲ ) = ▲ ✳ ✭t❤❡♦r❡♠ ✐s ♣r♦✈❛❜❧❡ ✐♥ ❩❋❈ + ✏t❤❡r❡ ❡①✐sts ❛ ♣r♦♣❡r ❝❧❛ss ♦❢ ✐♥❛❝❝❡ss✐❜❧❡ ❝❛r❞✐♥❛❧s✑✮