t rrt r tr trs - - PowerPoint PPT Presentation

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦ ▲✉❝❛ ❙❛✐✉ ❤tt♣✿✴✴✇✇✇✲❧✐♣♥✳✉♥✐✈✲♣❛r✐s✶✸✳❢r✴⑦❧♦❞❞♦ ❤tt♣✿✴✴✇✇✇✲❧✐♣♥✳✉♥✐✈✲♣❛r✐s✶✸✳❢r✴⑦s❛✐✉

▲■P◆✱ ❯♥✐✈❡rs✐té P❛r✐s ✶✸

P❛r✐s✱ ❏✉❧② ✺✱ ✷✵✶✵

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ ⊕✲⊙ ❋♦r♠❛❧✐③❛t✐♦♥ ❛s ❛ r❡✇r✐t❡ s②st❡♠ ✴ s♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s

▼♦t✐✈❛t✐♦♥

▼❛♥② ♣r♦❜❧❡♠s ❝❛♥ ❜❡ s♦❧✈❡❞ ✇✐t❤ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ s❡❛r❝❤ ♦♥ ❛ s♣❛❝❡ ✇❤❡r❡ s♦♠❡ ♠❡tr✐❝ ✐s ♦♣t✐♠✐③❡❞✳

✳ ✳

❚r♦♣✐❝❛❧ ❣❛♠❡s ❛r❡ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♠✐♥✲♠❛① ❣❛♠❡s ❚r♦♣✐❝❛❧ ❣❛♠❡s s✉♣♣♦rt tr♦♣✐❝❛❧ ✲♣r✉♥✐♥❣

tr♦♣✐❝❛❧ ❣❛♠❡s ✇❤♦s❡ ❞✉❛❧ ✐s ❛❧s♦ tr♦♣✐❝❛❧ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✇✐t❤ ✲ ✭♥♦t ♣r♦✈❡❞ ❢♦r♠❛❧❧②✱ ❜✉t ✐♥t✉✐t✐✈❡✮

❆♣♣r♦①✐♠❛t❡❞ ♣❛rs✐♥❣ ✐s ❛ tr♦♣✐❝❛❧ ❣❛♠❡

✲♣r✉♥✐♥❣ s❡❡♠s t♦ ❜❡ ❡✛❡❝t✐✈❡ ✳✳✳✇❡ s✉s♣❡❝t ♠❛♥② ♠♦r❡ ♣r♦❜❧❡♠s t♦ ❜❡ tr♦♣✐❝❛❧ ❣❛♠❡s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ ⊕✲⊙ ❋♦r♠❛❧✐③❛t✐♦♥ ❛s ❛ r❡✇r✐t❡ s②st❡♠ ✴ s♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s

▼♦t✐✈❛t✐♦♥

▼❛♥② ♣r♦❜❧❡♠s ❝❛♥ ❜❡ s♦❧✈❡❞ ✇✐t❤ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ s❡❛r❝❤ ♦♥ ❛ s♣❛❝❡ ✇❤❡r❡ s♦♠❡ ♠❡tr✐❝ ✐s ♦♣t✐♠✐③❡❞✳

✳ ✳

❚r♦♣✐❝❛❧ ❣❛♠❡s ❛r❡ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♠✐♥✲♠❛① ❣❛♠❡s ❚r♦♣✐❝❛❧ ❣❛♠❡s s✉♣♣♦rt tr♦♣✐❝❛❧ ✲♣r✉♥✐♥❣

tr♦♣✐❝❛❧ ❣❛♠❡s ✇❤♦s❡ ❞✉❛❧ ✐s ❛❧s♦ tr♦♣✐❝❛❧ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✇✐t❤ ✲ ✭♥♦t ♣r♦✈❡❞ ❢♦r♠❛❧❧②✱ ❜✉t ✐♥t✉✐t✐✈❡✮

❆♣♣r♦①✐♠❛t❡❞ ♣❛rs✐♥❣ ✐s ❛ tr♦♣✐❝❛❧ ❣❛♠❡

✲♣r✉♥✐♥❣ s❡❡♠s t♦ ❜❡ ❡✛❡❝t✐✈❡ ✳✳✳✇❡ s✉s♣❡❝t ♠❛♥② ♠♦r❡ ♣r♦❜❧❡♠s t♦ ❜❡ tr♦♣✐❝❛❧ ❣❛♠❡s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ ⊕✲⊙ ❋♦r♠❛❧✐③❛t✐♦♥ ❛s ❛ r❡✇r✐t❡ s②st❡♠ ✴ s♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s

▼♦t✐✈❛t✐♦♥

▼❛♥② ♣r♦❜❧❡♠s ❝❛♥ ❜❡ s♦❧✈❡❞ ✇✐t❤ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ s❡❛r❝❤ ♦♥ ❛ s♣❛❝❡ ✇❤❡r❡ s♦♠❡ ♠❡tr✐❝ ✐s ♦♣t✐♠✐③❡❞✳

✳ ✳

❚r♦♣✐❝❛❧ ❣❛♠❡s ❛r❡ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♠✐♥✲♠❛① ❣❛♠❡s ❚r♦♣✐❝❛❧ ❣❛♠❡s s✉♣♣♦rt tr♦♣✐❝❛❧ α✲♣r✉♥✐♥❣

tr♦♣✐❝❛❧ ❣❛♠❡s ✇❤♦s❡ ❞✉❛❧ ✐s ❛❧s♦ tr♦♣✐❝❛❧ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✇✐t❤ α✲β ✭♥♦t ♣r♦✈❡❞ ❢♦r♠❛❧❧②✱ ❜✉t ✐♥t✉✐t✐✈❡✮

❆♣♣r♦①✐♠❛t❡❞ ♣❛rs✐♥❣ ✐s ❛ tr♦♣✐❝❛❧ ❣❛♠❡

✲♣r✉♥✐♥❣ s❡❡♠s t♦ ❜❡ ❡✛❡❝t✐✈❡ ✳✳✳✇❡ s✉s♣❡❝t ♠❛♥② ♠♦r❡ ♣r♦❜❧❡♠s t♦ ❜❡ tr♦♣✐❝❛❧ ❣❛♠❡s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ ⊕✲⊙ ❋♦r♠❛❧✐③❛t✐♦♥ ❛s ❛ r❡✇r✐t❡ s②st❡♠ ✴ s♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s

▼♦t✐✈❛t✐♦♥

▼❛♥② ♣r♦❜❧❡♠s ❝❛♥ ❜❡ s♦❧✈❡❞ ✇✐t❤ ❛ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ s❡❛r❝❤ ♦♥ ❛ s♣❛❝❡ ✇❤❡r❡ s♦♠❡ ♠❡tr✐❝ ✐s ♦♣t✐♠✐③❡❞✳

✳ ✳

❚r♦♣✐❝❛❧ ❣❛♠❡s ❛r❡ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ♠✐♥✲♠❛① ❣❛♠❡s ❚r♦♣✐❝❛❧ ❣❛♠❡s s✉♣♣♦rt tr♦♣✐❝❛❧ α✲♣r✉♥✐♥❣

tr♦♣✐❝❛❧ ❣❛♠❡s ✇❤♦s❡ ❞✉❛❧ ✐s ❛❧s♦ tr♦♣✐❝❛❧ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✇✐t❤ α✲β ✭♥♦t ♣r♦✈❡❞ ❢♦r♠❛❧❧②✱ ❜✉t ✐♥t✉✐t✐✈❡✮

❆♣♣r♦①✐♠❛t❡❞ ♣❛rs✐♥❣ ✐s ❛ tr♦♣✐❝❛❧ ❣❛♠❡

α✲♣r✉♥✐♥❣ s❡❡♠s t♦ ❜❡ ❡✛❡❝t✐✈❡ ✳✳✳✇❡ s✉s♣❡❝t ♠❛♥② ♠♦r❡ ♣r♦❜❧❡♠s t♦ ❜❡ tr♦♣✐❝❛❧ ❣❛♠❡s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ ⊕✲⊙ ❋♦r♠❛❧✐③❛t✐♦♥ ❛s ❛ r❡✇r✐t❡ s②st❡♠ ✴ s♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s ✭✶✮✿ ♠✐♥✲♠❛①

▲❡t ❛ ❣❛♠❡ ❛r❡♥❛ ❙ = (P, λ, s✉❝❝) ❛♥❞ ❛ ♣❛②♦✛ ❢✉♥❝t✐♦♥ ♣ : P❚ → Z ∪ {+∞, −∞} ❜❡ ❣✐✈❡♥✳

✳ ✳

❚❤❡ ❣❛♠❡ ✈❛❧✉❡ ♦❢ ❛ ♣♦s✐t✐♦♥ π ∈ P ✐s tr❛❞✐t✐♦♥❛❧❧② ❞❡✜♥❡❞ ❛s✿ ✈♣(π) =      ♣(π), π ∈ P❚ ♠✐♥♥

✐=✶ ✈♣(π✐),

s✉❝❝(π) = π✶...π♥, λ(π) = P ♠❛①♥

✐=✶ ✈♣(π✐),

s✉❝❝(π) = π✶...π♥, λ(π) = O

✳

❲❤❛t ✐❢ ✇❡ ❣❡♥❡r❛❧✐③❡❄ ❧❡t✬s t❛❦❡ ❛ ❣❡♥❡r✐❝ ✐♥st❡❛❞ ♦❢ ♠✐♥ ♠❛①

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ ⊕✲⊙ ❋♦r♠❛❧✐③❛t✐♦♥ ❛s ❛ r❡✇r✐t❡ s②st❡♠ ✴ s♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s ✭✶✮✿ ♠✐♥✲♠❛①

▲❡t ❛ ❣❛♠❡ ❛r❡♥❛ ❙ = (P, λ, s✉❝❝) ❛♥❞ ❛ ♣❛②♦✛ ❢✉♥❝t✐♦♥ ♣ : P❚ → Z ∪ {+∞, −∞} ❜❡ ❣✐✈❡♥✳

✳ ✳

❚❤❡ ❣❛♠❡ ✈❛❧✉❡ ♦❢ ❛ ♣♦s✐t✐♦♥ π ∈ P ✐s tr❛❞✐t✐♦♥❛❧❧② ❞❡✜♥❡❞ ❛s✿ ✈♣(π) =      ♣(π), π ∈ P❚ ♠✐♥♥

✐=✶ ✈♣(π✐),

s✉❝❝(π) = π✶...π♥, λ(π) = P ♠❛①♥

✐=✶ ✈♣(π✐),

s✉❝❝(π) = π✶...π♥, λ(π) = O

✳

❲❤❛t ✐❢ ✇❡ ❣❡♥❡r❛❧✐③❡❄ ❧❡t✬s t❛❦❡ ❛ ❣❡♥❡r✐❝ A = (U, ⊕, ⊙) ✐♥st❡❛❞ ♦❢ (Z ∪ {+∞, −∞}, ♠✐♥, ♠❛①)

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ ⊕✲⊙ ❋♦r♠❛❧✐③❛t✐♦♥ ❛s ❛ r❡✇r✐t❡ s②st❡♠ ✴ s♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s ✭✷✮✿ ⊕✲⊙

▲❡t ❛ ❣❛♠❡ ❛r❡♥❛ ❙ = (P, λ, s✉❝❝) ❛♥❞ ❛ ♣❛②♦✛ ❢✉♥❝t✐♦♥ ♣ : P❚ → U ❜❡ ❣✐✈❡♥✳

✳ ✳

❚❤❡ ❣❛♠❡ ✈❛❧✉❡ ♦❢ ❛ ♣♦s✐t✐♦♥ π ∈ P ✐s ❞❡✜♥❡❞ ❛s✿ ✈♣(π) =                ♣(π), π ∈ P❚

♥

• ✐=✶

✈♣(π✐), s✉❝❝(π) = π✶...π♥, λ(π) = P

♥

• ✐=✶

✈♣(π✐), s✉❝❝(π) = π✶...π♥, λ(π) = O

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ ⊕✲⊙ ❋♦r♠❛❧✐③❛t✐♦♥ ❛s ❛ r❡✇r✐t❡ s②st❡♠ ✴ s♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s

❙♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s ■

π ∈ P❚ ♣(π) = ✈ ❬P❛②♦✛❪ π → ✈ s✉❝❝(π) = t λ(π) = P ❬P✲❡①♣❛♥❞❪ # t ≥ ✶ π → t s✉❝❝(π) = t λ(π) = O ❬O✲❡①♣❛♥❞❪ # t ≥ ✶ π → t ❤❛❝❦ ❬P✲r❡❞✉❝❡❪ ✈✶ ⊕ ✈✷ = ✈

• t✈✶ ✈✷

③ → t✈ ③

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ ■♥tr♦❞✉❝t✐♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ ⊕✲⊙ ❋♦r♠❛❧✐③❛t✐♦♥ ❛s ❛ r❡✇r✐t❡ s②st❡♠ ✴ s♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s

❙♠❛❧❧✲st❡♣ s❡♠❛♥t✐❝s ■■

❤❛❝❦ ❬O✲r❡❞✉❝❡❪ ✈✶ ⊙ ✈✷ = ✈

• t✈✶ ✈✷

③ → t✈ ③ ❤❛❝❦ ❬❘❡t✉r♥❪ Λ✈ → ✈ t → t′ ❬❈♦♥t❡①t❪ ❢♦r ❛❧❧ ❝♦♥t❡①ts ❈ ❈[t] →❝ ❈[t′] ❚❤❡ r❡✢❡①✐✈❡✲tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ♦❢ →❝ ❞❡✜♥❡s r❡✇r✐t❡s✳

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

❚r♦♣✐❝❛❧ ❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥ ✭❚r♦♣✐❝❛❧ ❛❧❣❡❜r❛✮ A = (U, ⊕, ⊙) ✐s ❛ tr♦♣✐❝❛❧ ❛❧❣❡❜r❛ ✐❢ ⊕ ❛♥❞ ⊙ ❛r❡ ❛ss♦❝✐❛t✐✈❡ ⊙ ❞✐str✐❜✉t❡s ♦✈❡r ⊕✿ ❢♦r ❛❧❧ ❛, ❜, ❝ ∈ U ❛ ⊙ (❜ ⊕ ❝) = (❛ ⊙ ❜) ⊕ (❛ ⊙ ❝) ❛♥❞ (❛ ⊕ ❜) ⊙ ❝ = (❛ ⊙ ❝) ⊕ (❜ ⊙ ❝)

❙❧✐❣❤t❧② s✐♠♣❧✐✜❡❞✿ s❡❡ t❤❡ ♣❛♣❡r✳ ✳

❚❤❡ ♠✐♥✱✰ ❛❧❣❡❜r❛ ♠✐♥ ✐s ❛♥ ✐♠♣♦rt❛♥t ❡①❛♠♣❧❡ ♦❢ tr♦♣✐❝❛❧ ❛❧❣❡❜r❛✳

✳ ✳ ❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

❚r♦♣✐❝❛❧ ❛❧❣❡❜r❛s

❉❡✜♥✐t✐♦♥ ✭❚r♦♣✐❝❛❧ ❛❧❣❡❜r❛✮ A = (U, ⊕, ⊙) ✐s ❛ tr♦♣✐❝❛❧ ❛❧❣❡❜r❛ ✐❢ ⊕ ❛♥❞ ⊙ ❛r❡ ❛ss♦❝✐❛t✐✈❡ ⊙ ❞✐str✐❜✉t❡s ♦✈❡r ⊕✿ ❢♦r ❛❧❧ ❛, ❜, ❝ ∈ U ❛ ⊙ (❜ ⊕ ❝) = (❛ ⊙ ❜) ⊕ (❛ ⊙ ❝) ❛♥❞ (❛ ⊕ ❜) ⊙ ❝ = (❛ ⊙ ❝) ⊕ (❜ ⊙ ❝)

❙❧✐❣❤t❧② s✐♠♣❧✐✜❡❞✿ s❡❡ t❤❡ ♣❛♣❡r✳ ✳

❚❤❡ ♠✐♥✱✰ ❛❧❣❡❜r❛ (Z ∪ {+∞}, ♠✐♥, +) ✐s ❛♥ ✐♠♣♦rt❛♥t ❡①❛♠♣❧❡ ♦❢ tr♦♣✐❝❛❧ ❛❧❣❡❜r❛✳

✳ ✳ ❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

❚❤❡ ❘❛t✐♦♥❛❧✐t② ❤②♣♦t❤❡s✐s

❋♦r ❛♥② α, β ✐t✬s ❡❛s② t♦ s❡❡ t❤❛t✿ α ≤ β + α ❖r✱ ✉s✐♥❣ ♦♥❧② t❤❡ ♦♣❡r❛t✐♦♥s ✐♥ ♠✐♥ ✿ ♠✐♥

• ❡♥❡r❛❧✐③✐♥❣ t♦

✇❡ ♦❜t❛✐♥✿ ❉❡✜♥✐t✐♦♥ ✭❘❛t✐♦♥❛❧✐t②✮ ✐s r❛t✐♦♥❛❧ ✐✛ ❢♦r ❛♥② ✇❡ ❤❛✈❡ t❤❛t

❙❧✐❣❤t❧② s✐♠♣❧✐✜❡❞✿ s❡❡ t❤❡ ♣❛♣❡r✳

✏❚❤❡ ♣❧❛②❡r ♣r❡❢❡rs ♦✈❡r ✇♦rs❡♥❡❞ ❜② ✑ ✲♣r✉♥✐♥❣ ❞❡♣❡♥❞s ♦♥ tr♦♣✐❝❛❧✐t② ❛♥❞ r❛t✐♦♥❛❧✐t②

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

❚❤❡ ❘❛t✐♦♥❛❧✐t② ❤②♣♦t❤❡s✐s

❋♦r ❛♥② α, β ✐t✬s ❡❛s② t♦ s❡❡ t❤❛t✿ α ≤ β + α ❖r✱ ✉s✐♥❣ ♦♥❧② t❤❡ ♦♣❡r❛t✐♦♥s ✐♥ (Z ∪ {+∞}, ♠✐♥, +)✿ α = ♠✐♥{α, β + α}

• ❡♥❡r❛❧✐③✐♥❣ t♦

✇❡ ♦❜t❛✐♥✿ ❉❡✜♥✐t✐♦♥ ✭❘❛t✐♦♥❛❧✐t②✮ ✐s r❛t✐♦♥❛❧ ✐✛ ❢♦r ❛♥② ✇❡ ❤❛✈❡ t❤❛t

❙❧✐❣❤t❧② s✐♠♣❧✐✜❡❞✿ s❡❡ t❤❡ ♣❛♣❡r✳

✏❚❤❡ ♣❧❛②❡r ♣r❡❢❡rs ♦✈❡r ✇♦rs❡♥❡❞ ❜② ✑ ✲♣r✉♥✐♥❣ ❞❡♣❡♥❞s ♦♥ tr♦♣✐❝❛❧✐t② ❛♥❞ r❛t✐♦♥❛❧✐t②

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

❚❤❡ ❘❛t✐♦♥❛❧✐t② ❤②♣♦t❤❡s✐s

❋♦r ❛♥② α, β ✐t✬s ❡❛s② t♦ s❡❡ t❤❛t✿ α ≤ β + α ❖r✱ ✉s✐♥❣ ♦♥❧② t❤❡ ♦♣❡r❛t✐♦♥s ✐♥ (Z ∪ {+∞}, ♠✐♥, +)✿ α = ♠✐♥{α, β + α}

• ❡♥❡r❛❧✐③✐♥❣ t♦ (U, ⊕, ⊙) ✇❡ ♦❜t❛✐♥✿

❉❡✜♥✐t✐♦♥ ✭❘❛t✐♦♥❛❧✐t②✮ A = (U, ⊕, ⊙) ✐s r❛t✐♦♥❛❧ ✐✛ ❢♦r ❛♥② α, β ∈ U ✇❡ ❤❛✈❡ t❤❛t α = α ⊕ (β ⊙ α)

❙❧✐❣❤t❧② s✐♠♣❧✐✜❡❞✿ s❡❡ t❤❡ ♣❛♣❡r✳

✏❚❤❡ ♣❧❛②❡r ♣r❡❢❡rs ♦✈❡r ✇♦rs❡♥❡❞ ❜② ✑ ✲♣r✉♥✐♥❣ ❞❡♣❡♥❞s ♦♥ tr♦♣✐❝❛❧✐t② ❛♥❞ r❛t✐♦♥❛❧✐t②

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

❚❤❡ ❘❛t✐♦♥❛❧✐t② ❤②♣♦t❤❡s✐s

❋♦r ❛♥② α, β ✐t✬s ❡❛s② t♦ s❡❡ t❤❛t✿ α ≤ β + α ❖r✱ ✉s✐♥❣ ♦♥❧② t❤❡ ♦♣❡r❛t✐♦♥s ✐♥ (Z ∪ {+∞}, ♠✐♥, +)✿ α = ♠✐♥{α, β + α}

• ❡♥❡r❛❧✐③✐♥❣ t♦ (U, ⊕, ⊙) ✇❡ ♦❜t❛✐♥✿

❉❡✜♥✐t✐♦♥ ✭❘❛t✐♦♥❛❧✐t②✮ A = (U, ⊕, ⊙) ✐s r❛t✐♦♥❛❧ ✐✛ ❢♦r ❛♥② α, β ∈ U ✇❡ ❤❛✈❡ t❤❛t α = α ⊕ (β ⊙ α)

❙❧✐❣❤t❧② s✐♠♣❧✐✜❡❞✿ s❡❡ t❤❡ ♣❛♣❡r✳

✏❚❤❡ ♣❧❛②❡r P ♣r❡❢❡rs α ♦✈❡r α ✇♦rs❡♥❡❞ ❜② β✑ ✲♣r✉♥✐♥❣ ❞❡♣❡♥❞s ♦♥ tr♦♣✐❝❛❧✐t② ❛♥❞ r❛t✐♦♥❛❧✐t②

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

❚❤❡ ❘❛t✐♦♥❛❧✐t② ❤②♣♦t❤❡s✐s

❋♦r ❛♥② α, β ✐t✬s ❡❛s② t♦ s❡❡ t❤❛t✿ α ≤ β + α ❖r✱ ✉s✐♥❣ ♦♥❧② t❤❡ ♦♣❡r❛t✐♦♥s ✐♥ (Z ∪ {+∞}, ♠✐♥, +)✿ α = ♠✐♥{α, β + α}

• ❡♥❡r❛❧✐③✐♥❣ t♦ (U, ⊕, ⊙) ✇❡ ♦❜t❛✐♥✿

❉❡✜♥✐t✐♦♥ ✭❘❛t✐♦♥❛❧✐t②✮ A = (U, ⊕, ⊙) ✐s r❛t✐♦♥❛❧ ✐✛ ❢♦r ❛♥② α, β ∈ U ✇❡ ❤❛✈❡ t❤❛t α = α ⊕ (β ⊙ α)

❙❧✐❣❤t❧② s✐♠♣❧✐✜❡❞✿ s❡❡ t❤❡ ♣❛♣❡r✳

✏❚❤❡ ♣❧❛②❡r P ♣r❡❢❡rs α ♦✈❡r α ✇♦rs❡♥❡❞ ❜② β✑ α✲♣r✉♥✐♥❣ ❞❡♣❡♥❞s ♦♥ tr♦♣✐❝❛❧✐t② ❛♥❞ r❛t✐♦♥❛❧✐t②

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

P✲❝✉t

■♥t✉✐t✐♦♥✿ t❤✐♥❦ ♦❢ ⊕ ❛s ♠✐♥ ❛♥❞ ⊙ ❛s +✳ ❋♦r ❛♥② ✈✵✱ ✈✶✱ t✵, t✶, α, β ✐❢ α ⊕ β = α t❤❡♥ t ≤ t′ ✭✏t′ s✐♠✉❧❛t❡s t✑✮✿

t α β v0 v1 t′ α v1

✏❚❤❡ ♣❧❛②❡r P ♣r❡❢❡rs α t♦ β✑ s♦ ❝♦♠♣✉t✐♥❣ ✈✵ ✐s ❛ ✇❛st❡ ♦❢ t✐♠❡ ✐❢ ♠✐♥ t❤❡♥ ♠✐♥ ✈✵ ✈✶ ♠✐♥ ✈✶ ✐❢ t❤❡♥ ✈✵ ✈✶ ✈✶

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

P✲❝✉t

■♥t✉✐t✐♦♥✿ t❤✐♥❦ ♦❢ ⊕ ❛s ♠✐♥ ❛♥❞ ⊙ ❛s +✳ ❋♦r ❛♥② ✈✵✱ ✈✶✱ t✵, t✶, α, β ✐❢ α ⊕ β = α t❤❡♥ t ≤ t′ ✭✏t′ s✐♠✉❧❛t❡s t✑✮✿

t α β v0 v1 t′ α v1

✏❚❤❡ ♣❧❛②❡r P ♣r❡❢❡rs α t♦ β✑ s♦ ❝♦♠♣✉t✐♥❣ ✈✵ ✐s ❛ ✇❛st❡ ♦❢ t✐♠❡ ✐❢ ♠✐♥ t❤❡♥ ♠✐♥ ✈✵ ✈✶ ♠✐♥ ✈✶ ✐❢ t❤❡♥ ✈✵ ✈✶ ✈✶

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

P✲❝✉t

■♥t✉✐t✐♦♥✿ t❤✐♥❦ ♦❢ ⊕ ❛s ♠✐♥ ❛♥❞ ⊙ ❛s +✳ ❋♦r ❛♥② ✈✵✱ ✈✶✱ t✵, t✶, α, β ✐❢ α ⊕ β = α t❤❡♥ t ≤ t′ ✭✏t′ s✐♠✉❧❛t❡s t✑✮✿

t α β v0 v1 t′ α v1

✏❚❤❡ ♣❧❛②❡r P ♣r❡❢❡rs α t♦ β✑ s♦ ❝♦♠♣✉t✐♥❣ ✈✵ ✐s ❛ ✇❛st❡ ♦❢ t✐♠❡ ✐❢ ♠✐♥{α, β} = α t❤❡♥ ♠✐♥{α, (β + ✈✵), ✈✶} = ♠✐♥{α, ✈✶} ✐❢ t❤❡♥ ✈✵ ✈✶ ✈✶

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

P✲❝✉t

■♥t✉✐t✐♦♥✿ t❤✐♥❦ ♦❢ ⊕ ❛s ♠✐♥ ❛♥❞ ⊙ ❛s +✳ ❋♦r ❛♥② ✈✵✱ ✈✶✱ t✵, t✶, α, β ✐❢ α ⊕ β = α t❤❡♥ t ≤ t′ ✭✏t′ s✐♠✉❧❛t❡s t✑✮✿

t α β v0 v1 t′ α v1

✏❚❤❡ ♣❧❛②❡r P ♣r❡❢❡rs α t♦ β✑ s♦ ❝♦♠♣✉t✐♥❣ ✈✵ ✐s ❛ ✇❛st❡ ♦❢ t✐♠❡ ✐❢ ♠✐♥{α, β} = α t❤❡♥ ♠✐♥{α, (β + ✈✵), ✈✶} = ♠✐♥{α, ✈✶} ✐❢ α ⊕ β = α t❤❡♥ α ⊕ (β ⊙ ✈✵) ⊕ ✈✶ = α ⊕ ✈✶

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

P✲✇✐❧❧✿ ❢♦r ❛♥② t✵, t✶, t✷, α, β ✇❡ ❤❛✈❡ t ≤ t′

t α β v0 v1 v2 t′ α β α v0 v1 v2

✏❚❤❡ ♣❧❛②❡r P ❝❛♥ ❜❡q✉❡❛t❤ α t♦ ❤❡r ❣r❛♥❞❝❤✐❧❞✑ ✭✇❤✐❝❤ ❛❧❧♦✇s t♦ ♣r✉♥❡ ♠♦r❡✮ ♠✐♥ ✈✵ ✈✶ ✈✷ ♠✐♥ ♠✐♥ ✈✵ ✈✶ ✈✷ ✈✵ ✈✶ ✈✷ ✈✵ ✈✶ ✈✷

✳ ✳

④❚❤✐s ✐s ❞✉❡ t♦ r❛t✐♦♥❛❧✐t② ❛♥❞ t♦ ♣r♦♣❡rt✐❡s ♦❢ tr♦♣✐❝❛❧ ❛❧❣❡❜r❛s⑥

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

P✲✇✐❧❧✿ ❢♦r ❛♥② t✵, t✶, t✷, α, β ✇❡ ❤❛✈❡ t ≤ t′

t α β v0 v1 v2 t′ α β α v0 v1 v2

✏❚❤❡ ♣❧❛②❡r P ❝❛♥ ❜❡q✉❡❛t❤ α t♦ ❤❡r ❣r❛♥❞❝❤✐❧❞✑ ✭✇❤✐❝❤ ❛❧❧♦✇s t♦ ♣r✉♥❡ ♠♦r❡✮ ♠✐♥{α, (β + ✈✵ + ✈✶), ✈✷} = ♠✐♥{α, (β + ♠✐♥{α, ✈✵} + ✈✶), ✈✷} ✈✵ ✈✶ ✈✷ ✈✵ ✈✶ ✈✷

✳ ✳

④❚❤✐s ✐s ❞✉❡ t♦ r❛t✐♦♥❛❧✐t② ❛♥❞ t♦ ♣r♦♣❡rt✐❡s ♦❢ tr♦♣✐❝❛❧ ❛❧❣❡❜r❛s⑥

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

P✲✇✐❧❧✿ ❢♦r ❛♥② t✵, t✶, t✷, α, β ✇❡ ❤❛✈❡ t ≤ t′

t α β v0 v1 v2 t′ α β α v0 v1 v2

✏❚❤❡ ♣❧❛②❡r P ❝❛♥ ❜❡q✉❡❛t❤ α t♦ ❤❡r ❣r❛♥❞❝❤✐❧❞✑ ✭✇❤✐❝❤ ❛❧❧♦✇s t♦ ♣r✉♥❡ ♠♦r❡✮ ♠✐♥{α, (β + ✈✵ + ✈✶), ✈✷} = ♠✐♥{α, (β + ♠✐♥{α, ✈✵} + ✈✶), ✈✷} α ⊕ (β ⊙ ✈✵ ⊙ ✈✶) ⊕ ✈✷ = α ⊕ (β ⊙ (α ⊕ ✈✵) ⊙ ✈✶) ⊕ ✈✷ ✳

✳

④❚❤✐s ✐s ❞✉❡ t♦ r❛t✐♦♥❛❧✐t② ❛♥❞ t♦ ♣r♦♣❡rt✐❡s ♦❢ tr♦♣✐❝❛❧ ❛❧❣❡❜r❛s⑥

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

P✲✇✐❧❧✿ ❢♦r ❛♥② t✵, t✶, t✷, α, β ✇❡ ❤❛✈❡ t ≤ t′

t α β v0 v1 v2 t′ α β α v0 v1 v2

✏❚❤❡ ♣❧❛②❡r P ❝❛♥ ❜❡q✉❡❛t❤ α t♦ ❤❡r ❣r❛♥❞❝❤✐❧❞✑ ✭✇❤✐❝❤ ❛❧❧♦✇s t♦ ♣r✉♥❡ ♠♦r❡✮ ♠✐♥{α, (β + ✈✵ + ✈✶), ✈✷} = ♠✐♥{α, (β + ♠✐♥{α, ✈✵} + ✈✶), ✈✷} α ⊕ (β ⊙ ✈✵ ⊙ ✈✶) ⊕ ✈✷ = α ⊕ (β ⊙ (α ⊕ ✈✵) ⊙ ✈✶) ⊕ ✈✷ ✳

✳

④❚❤✐s ✐s ❞✉❡ t♦ r❛t✐♦♥❛❧✐t② ❛♥❞ t♦ ♣r♦♣❡rt✐❡s ♦❢ tr♦♣✐❝❛❧ ❛❧❣❡❜r❛s⑥

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

Pr✉♥✐♥❣ r✉❧❡s ✭❢♦r ❜✐✲tr♦♣✐❝❛❧ ❣❛♠❡s✮

❤❛❝❦ ❬P✲✇✐❧❧❪ α [β ( t✶) t✷] t✸ → α [β (α t✶) t✷] t✸ ❤❛❝❦ ❬O✲✇✐❧❧❪ β [α ( t✶) t✷] t✸ → β [α (β t✶) t✷] t✸ α ⊕ β = α ❬P✲❝✉t❪ α (β t✶) t✷ → α t✷ β ⊙ α = β ❬O✲❝✉t❪ β (α t✶) t✷ → β t✷

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

α✲β✲♣r✉♥✐♥❣ ✭✶✮ ❬α ❜❡st t✐❧❧ ♥♦✇ ❢♦r P✴♠✐♥❀ β ❜❡st t✐❧❧ ♥♦✇ ❢♦r O✴♠❛①❪

✶ ❢✉♥❝t✐♦♥ ❛❧♣❤❛❴❜❡t❛✭π : P; α, β : Z ✮ ✿ Z ✷ ✐ ❢ π ∈ P❚ t❤❡♥ ✸ r❡t✉r♥ ♣(π) ✹ π✶...π♥ := s✉❝❝(π) # ♥ ≥ ✶ ✺ ✐ ❢ λ(π) ❂ P t❤❡♥ ✻ ✈ := α ★ ❇❡st ❢♦r P t✐❧❧ ♥♦✇ ✼ ❢♦r ✐ ❢r♦♠ ✶ t♦ ♥ ✽ ❛♥❞ ✇❤✐❧❡ β <Z ✈ ❞♦ ✾ ✈ := ♠✐♥{✈, ❛❧♣❤❛❴❜❡t❛(π✐, ✈, β)} ✶✵ ❡❧s❡ ★ λ(π) ❂ O ✶✶ ✈ := β ✶✷ ❢♦r ✐ ❢r♦♠ ✶ t♦ ♥ ✶✸ ❛♥❞ ✇❤✐❧❡ ✈ <Z α ❞♦ ✶✹ ✈ := ♠❛①{✈, ❛❧♣❤❛❴❜❡t❛(π✐, α, ✈)} ✶✺ r❡t✉r♥ ✈

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

α✲β✲♣r✉♥✐♥❣ ✭✷✮ ❬α ❜❡st t✐❧❧ ♥♦✇ ❢♦r P✴♠✐♥❀ β ❜❡st t✐❧❧ ♥♦✇ ❢♦r O✴♠❛①❪

✶ ❢✉♥❝t✐♦♥ ❛❧♣❤❛❴❜❡t❛✭π : P; α, β : Z ✮ ✿ Z ✷ ✐ ❢ π ∈ P❚ t❤❡♥ ✸ r❡t✉r♥ ♣(π) ✹ π✶...π♥ := s✉❝❝(π) # ♥ ≥ ✶ ✺ ✐ ❢ λ(π) ❂ P t❤❡♥ ✻ ✈ := α ★ ❇❡st ❢♦r P t✐❧❧ ♥♦✇ ✼ ❢♦r ✐ ❢r♦♠ ✶ t♦ ♥ ✽ ❛♥❞ ✇❤✐❧❡ β <Z ✈ ❞♦ ★ ❚❤❡r❡✬s ♥♦ β ✾ ✈ := ♠✐♥{✈, ❛❧♣❤❛❴❜❡t❛(π✐, ✈, β)} ★ ❚❤❡r❡✬s ♥♦ β ✶✵ ❡❧s❡ ★ λ(π) ❂ O ✶✶ ✈ := β ★ ❚❤❡r❡✬s ♥♦ β ✶✷ ❢♦r ✐ ❢r♦♠ ✶ t♦ ♥ ✶✸ ❛♥❞ ✇❤✐❧❡ ✈ <Z α ❞♦ ✶✹ ✈ := ♠❛①{✈, ❛❧♣❤❛❴❜❡t❛(π✐, α, ✈)} ✶✺ r❡t✉r♥ ✈

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣ Pr✉♥✐♥❣ tr❡❡s ❘❡✇r✐t❡ r✉❧❡s ❢♦r ♣r✉♥✐♥❣ ■♠♣❡r❛t✐✈❡✲st②❧❡ ❛❧❣♦r✐t❤♠

❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❬α ❝✉rr❡♥t ❜❡st ❢♦r P✴⊕❪

✶ ❢✉♥❝t✐♦♥ tr♦♣✐❝❛❧✭π : P; α : U ✮ ✿ U ✷ ✐ ❢ π ∈ P❚ t❤❡♥ ✸ r❡t✉r♥ ♣(π) ✹ π✶...π♥ := s✉❝❝(π) # ♥ ≥ ✶ ✺ ✐ ❢ λ(π) ❂ P t❤❡♥ ✻ ✈ := α ★ ❇❡st ❢♦r P t✐❧❧ ♥♦✇ ✼ ❢♦r ✐ ❢r♦♠ ✶ t♦ ♥ ❞♦ ✽ # ❞♦ ♥♦t ♣r✉♥❡ ❛t P✬s ❧❡✈❡❧ ✾ ✈ := ✈ ⊕ tr♦♣✐❝❛❧(π✐, ✈) ✶✵ ❡❧s❡ ★ λ(π) ❂ O ✶✶ ✈ := tr♦♣✐❝❛❧(π✶, α) ★ ◆♦ ✶U ✶✷ ❢♦r ✐ ❢r♦♠ ✷ t♦ ♥ ✶✸ ❛♥❞ ✇❤✐❧❡ α ⊕ ✈ = α ❞♦ ✶✹ ✈ := ✈ ⊙ tr♦♣✐❝❛❧(π✐, α) ✶✺ r❡t✉r♥ ✈

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r

❚r❛❞✐t✐♦♥❛❧ ❉✫❈ ❛❧❣♦r✐t❤♠s ❝♦♥s✐st ✐♥✿ ❉✐✈✐❞❡ ❛ ♣r♦❜❧❡♠ ✐♥t♦ s✉❜✲♣r♦❜❧❡♠s ❬❞❡t❡r♠✐♥✐st✐❝❛❧❧②❪ ❈♦♠❜✐♥❡ s♦❧✉t✐♦♥s

✳ ✳

❋♦r ❡①❛♠♣❧❡✱ ♠❡r❣❡✲s♦rt✳

✳ ✳

▲❡t✬s ❣❡♥❡r❛❧✐③❡ t❤✐s t♦ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❞✐✈✐s✐♦♥s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r

❚r❛❞✐t✐♦♥❛❧ ❉✫❈ ❛❧❣♦r✐t❤♠s ❝♦♥s✐st ✐♥✿ ❉✐✈✐❞❡ ❛ ♣r♦❜❧❡♠ ✐♥t♦ s✉❜✲♣r♦❜❧❡♠s ❬❞❡t❡r♠✐♥✐st✐❝❛❧❧②❪ ❈♦♠❜✐♥❡ s♦❧✉t✐♦♥s

✳ ✳

❋♦r ❡①❛♠♣❧❡✱ ♠❡r❣❡✲s♦rt✳

✳ ✳

▲❡t✬s ❣❡♥❡r❛❧✐③❡ t❤✐s t♦ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❞✐✈✐s✐♦♥s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r

♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ✐♥ ❛ s♦❧✉t✐♦♥ s♣❛❝❡ ❝❤♦♦s❡ ❛ ❞✐✈✐s✐♦♥ ✭ ✱ ✮✿ ❢♦r ❡①❛♠♣❧❡ ♠✐♥ s♦♠❡ ✇❛② ♦❢ ✏❝♦♠❜✐♥✐♥❣✑ s✉❜✲s♦❧✉t✐♦♥s✳ ✏❛❝❝✉♠✉❧❛t✐♥❣ ✈❛❧✉❡s✑ ✭ ✱ ✮✿ ❢♦r ❡①❛♠♣❧❡

✳ ✳

■❢ ✐s tr♦♣✐❝❛❧ ✇❡ ❝❛♥ ✉s❡ ✲♣r✉♥✐♥❣

✳ ✳

r❡♣r❡s❡♥ts ❝❤♦✐❝❡ ❛❝❝♦r❞✐♥❣ t♦ ❛ q✉❛❧✐t② ❝r✐t❡r✐♦♥ ✏❛❝❝✉♠✉❧❛t❡s✑

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r

♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ✐♥ ❛ s♦❧✉t✐♦♥ s♣❛❝❡ ❝❤♦♦s❡ ❛ ❞✐✈✐s✐♦♥ ✭P✱ ⊕✮✿ ❢♦r ❡①❛♠♣❧❡ ♠✐♥ s♦♠❡ ✇❛② ♦❢ ✏❝♦♠❜✐♥✐♥❣✑ s✉❜✲s♦❧✉t✐♦♥s✳ ✏❛❝❝✉♠✉❧❛t✐♥❣ ✈❛❧✉❡s✑ ✭ ✱ ✮✿ ❢♦r ❡①❛♠♣❧❡

✳ ✳

■❢ ✐s tr♦♣✐❝❛❧ ✇❡ ❝❛♥ ✉s❡ ✲♣r✉♥✐♥❣

✳ ✳

r❡♣r❡s❡♥ts ❝❤♦✐❝❡ ❛❝❝♦r❞✐♥❣ t♦ ❛ q✉❛❧✐t② ❝r✐t❡r✐♦♥ ✏❛❝❝✉♠✉❧❛t❡s✑

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r

♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ✐♥ ❛ s♦❧✉t✐♦♥ s♣❛❝❡ ❝❤♦♦s❡ ❛ ❞✐✈✐s✐♦♥ ✭P✱ ⊕✮✿ ❢♦r ❡①❛♠♣❧❡ ♠✐♥ s♦♠❡ ✇❛② ♦❢ ✏❝♦♠❜✐♥✐♥❣✑ s✉❜✲s♦❧✉t✐♦♥s✳ ✏❛❝❝✉♠✉❧❛t✐♥❣ ✈❛❧✉❡s✑ ✭O✱ ⊙✮✿ ❢♦r ❡①❛♠♣❧❡ +

✳ ✳

■❢ ✐s tr♦♣✐❝❛❧ ✇❡ ❝❛♥ ✉s❡ ✲♣r✉♥✐♥❣

✳ ✳

r❡♣r❡s❡♥ts ❝❤♦✐❝❡ ❛❝❝♦r❞✐♥❣ t♦ ❛ q✉❛❧✐t② ❝r✐t❡r✐♦♥ ✏❛❝❝✉♠✉❧❛t❡s✑

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r

♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ✐♥ ❛ s♦❧✉t✐♦♥ s♣❛❝❡ ❝❤♦♦s❡ ❛ ❞✐✈✐s✐♦♥ ✭P✱ ⊕✮✿ ❢♦r ❡①❛♠♣❧❡ ♠✐♥ s♦♠❡ ✇❛② ♦❢ ✏❝♦♠❜✐♥✐♥❣✑ s✉❜✲s♦❧✉t✐♦♥s✳ ✏❛❝❝✉♠✉❧❛t✐♥❣ ✈❛❧✉❡s✑ ✭O✱ ⊙✮✿ ❢♦r ❡①❛♠♣❧❡ +

✳ ✳

■❢ A = (U, ⊕, ⊙) ✐s tr♦♣✐❝❛❧ ✇❡ ❝❛♥ ✉s❡ α✲♣r✉♥✐♥❣ ✳

✳

r❡♣r❡s❡♥ts ❝❤♦✐❝❡ ❛❝❝♦r❞✐♥❣ t♦ ❛ q✉❛❧✐t② ❝r✐t❡r✐♦♥ ✏❛❝❝✉♠✉❧❛t❡s✑

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r

♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡s ✐♥ ❛ s♦❧✉t✐♦♥ s♣❛❝❡ ❝❤♦♦s❡ ❛ ❞✐✈✐s✐♦♥ ✭P✱ ⊕✮✿ ❢♦r ❡①❛♠♣❧❡ ♠✐♥ s♦♠❡ ✇❛② ♦❢ ✏❝♦♠❜✐♥✐♥❣✑ s✉❜✲s♦❧✉t✐♦♥s✳ ✏❛❝❝✉♠✉❧❛t✐♥❣ ✈❛❧✉❡s✑ ✭O✱ ⊙✮✿ ❢♦r ❡①❛♠♣❧❡ +

✳ ✳

■❢ A = (U, ⊕, ⊙) ✐s tr♦♣✐❝❛❧ ✇❡ ❝❛♥ ✉s❡ α✲♣r✉♥✐♥❣ ✳

✳

⊕ r❡♣r❡s❡♥ts ❝❤♦✐❝❡ ❛❝❝♦r❞✐♥❣ t♦ ❛ q✉❛❧✐t② ❝r✐t❡r✐♦♥ ⊙ ✏❛❝❝✉♠✉❧❛t❡s✑

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡

▲❡t ❛ ❝♦♥t❡①t✲❢r❡❡ ❣r❛♠♠❛r ❜❡ ❣✐✈❡♥✳

✳ ✳

❋✐♥❞ t❤❡ ✏❧❡❛st✲✇r♦♥❣✑ ♣❛rs❡✱ ✇❤❡r❡ ❡❛❝❤ ♣❛rs❡ ♠❛② ❝♦♥t❛✐♥ ✉♥♠❛t❝❤❡❞ t❡r♠✐♥❛❧s ▼✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❡rr♦rs ▼✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ✉♥♠❛t❝❤❡❞ ❝❤❛r❛❝t❡rs

✳ ✳ ❍②♣♦t❤❡s❡s t♦ ♠❛❦❡ ❡①♣♦s✐t✐♦♥ s✐♠♣❧❡r ✭❧✐❢t❛❜❧❡✮✿ ◆♦ ✲♣r♦❞✉❝t✐♦♥s ◆♦ t✇♦ ❛❞❥❛❝❡♥t ♥♦♥✲t❡r♠✐♥❛❧s ◆♦ ♥♦♥✲t❡r♠✐♥❛❧ ❛❧♦♥❡ ♦♥ ❛ r✐❣❤t s✐❞❡ ❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡

▲❡t ❛ ❝♦♥t❡①t✲❢r❡❡ ❣r❛♠♠❛r ❜❡ ❣✐✈❡♥✳

✳ ✳

❋✐♥❞ t❤❡ ✏❧❡❛st✲✇r♦♥❣✑ ♣❛rs❡✱ ✇❤❡r❡ ❡❛❝❤ ♣❛rs❡ ♠❛② ❝♦♥t❛✐♥ ✉♥♠❛t❝❤❡❞ t❡r♠✐♥❛❧s ▼✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❡rr♦rs ▼✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ✉♥♠❛t❝❤❡❞ ❝❤❛r❛❝t❡rs

✳ ✳ ❍②♣♦t❤❡s❡s t♦ ♠❛❦❡ ❡①♣♦s✐t✐♦♥ s✐♠♣❧❡r ✭❧✐❢t❛❜❧❡✮✿ ◆♦ ✲♣r♦❞✉❝t✐♦♥s ◆♦ t✇♦ ❛❞❥❛❝❡♥t ♥♦♥✲t❡r♠✐♥❛❧s ◆♦ ♥♦♥✲t❡r♠✐♥❛❧ ❛❧♦♥❡ ♦♥ ❛ r✐❣❤t s✐❞❡ ❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡

▲❡t ❛ ❝♦♥t❡①t✲❢r❡❡ ❣r❛♠♠❛r ❜❡ ❣✐✈❡♥✳

✳ ✳

❋✐♥❞ t❤❡ ✏❧❡❛st✲✇r♦♥❣✑ ♣❛rs❡✱ ✇❤❡r❡ ❡❛❝❤ ♣❛rs❡ ♠❛② ❝♦♥t❛✐♥ ✉♥♠❛t❝❤❡❞ t❡r♠✐♥❛❧s ▼✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ❡rr♦rs ▼✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ✉♥♠❛t❝❤❡❞ ❝❤❛r❛❝t❡rs

✳ ✳ ❍②♣♦t❤❡s❡s t♦ ♠❛❦❡ ❡①♣♦s✐t✐♦♥ s✐♠♣❧❡r ✭❧✐❢t❛❜❧❡✮✿ ◆♦ ǫ✲♣r♦❞✉❝t✐♦♥s ◆♦ t✇♦ ❛❞❥❛❝❡♥t ♥♦♥✲t❡r♠✐♥❛❧s ◆♦ ♥♦♥✲t❡r♠✐♥❛❧ ❛❧♦♥❡ ♦♥ ❛ r✐❣❤t s✐❞❡ ❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡✿ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♣❛rs✐♥❣

❲❡ ❤❛✈❡ ❛ s✉❜str✐♥❣ t♦ ♣❛rs❡ ✇✐t❤ ❛ ♥♦♥t❡r♠✐♥❛❧✱ ❢♦r ❡①❛♠♣❧❡ ✧✶ ✰ ✷ ✰ ✸✧ ✇✐t❤ ❊✳ ✿ ❈❤♦♦s❡ ❛ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ❛ ❞✐✈✐s✐♦♥ ✭♣✐✈♦t ❝❤❛r❛❝t❡rs✮ ✿ P❛rs❡ ♥♦♥t❡r♠✐♥❛❧s ✏✐♥ ❜❡t✇❡❡♥✑ ♣✐✈♦ts

✳ ✳

✶ ✷ ✸ ❊ ✶ ❊ ✷ ✸ ❊

✳ ✳

♠✐♥

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡✿ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♣❛rs✐♥❣

❲❡ ❤❛✈❡ ❛ s✉❜str✐♥❣ t♦ ♣❛rs❡ ✇✐t❤ ❛ ♥♦♥t❡r♠✐♥❛❧✱ ❢♦r ❡①❛♠♣❧❡ ✧✶ ✰ ✷ ✰ ✸✧ ✇✐t❤ ❊✳ P✿ ❈❤♦♦s❡ ❛ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ❛ ❞✐✈✐s✐♦♥ ✭♣✐✈♦t ❝❤❛r❛❝t❡rs✮ ✿ P❛rs❡ ♥♦♥t❡r♠✐♥❛❧s ✏✐♥ ❜❡t✇❡❡♥✑ ♣✐✈♦ts

✳ ✳

✶ ✷ ✸ ❊ ✶ ❊ ✷ ✸ ❊

✳ ✳

♠✐♥

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡✿ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♣❛rs✐♥❣

❲❡ ❤❛✈❡ ❛ s✉❜str✐♥❣ t♦ ♣❛rs❡ ✇✐t❤ ❛ ♥♦♥t❡r♠✐♥❛❧✱ ❢♦r ❡①❛♠♣❧❡ ✧✶ ✰ ✷ ✰ ✸✧ ✇✐t❤ ❊✳ P✿ ❈❤♦♦s❡ ❛ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ❛ ❞✐✈✐s✐♦♥ ✭♣✐✈♦t ❝❤❛r❛❝t❡rs✮ O✿ P❛rs❡ ♥♦♥t❡r♠✐♥❛❧s ✏✐♥ ❜❡t✇❡❡♥✑ ♣✐✈♦ts

✳ ✳

✶ ✷ ✸ ❊ ✶ ❊ ✷ ✸ ❊

✳ ✳

♠✐♥

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡✿ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♣❛rs✐♥❣

❲❡ ❤❛✈❡ ❛ s✉❜str✐♥❣ t♦ ♣❛rs❡ ✇✐t❤ ❛ ♥♦♥t❡r♠✐♥❛❧✱ ❢♦r ❡①❛♠♣❧❡ ✧✶ ✰ ✷ ✰ ✸✧ ✇✐t❤ ❊✳ P✿ ❈❤♦♦s❡ ❛ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ❛ ❞✐✈✐s✐♦♥ ✭♣✐✈♦t ❝❤❛r❛❝t❡rs✮ O✿ P❛rs❡ ♥♦♥t❡r♠✐♥❛❧s ✏✐♥ ❜❡t✇❡❡♥✑ ♣✐✈♦ts

✳ ✳

πP = (”✶ + ✷ + ✸”, ❊) ✶ ❊ ✷ ✸ ❊

✳ ✳

♠✐♥

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡✿ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♣❛rs✐♥❣

❲❡ ❤❛✈❡ ❛ s✉❜str✐♥❣ t♦ ♣❛rs❡ ✇✐t❤ ❛ ♥♦♥t❡r♠✐♥❛❧✱ ❢♦r ❡①❛♠♣❧❡ ✧✶ ✰ ✷ ✰ ✸✧ ✇✐t❤ ❊✳ P✿ ❈❤♦♦s❡ ❛ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ❛ ❞✐✈✐s✐♦♥ ✭♣✐✈♦t ❝❤❛r❛❝t❡rs✮ O✿ P❛rs❡ ♥♦♥t❡r♠✐♥❛❧s ✏✐♥ ❜❡t✇❡❡♥✑ ♣✐✈♦ts

✳ ✳

πP = (”✶ + ✷ + ✸”, ❊) πO = (”✶”, ❊), (”✷ + ✸”, ❊) ✳

✳

♠✐♥

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡✿ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♣❛rs✐♥❣

❲❡ ❤❛✈❡ ❛ s✉❜str✐♥❣ t♦ ♣❛rs❡ ✇✐t❤ ❛ ♥♦♥t❡r♠✐♥❛❧✱ ❢♦r ❡①❛♠♣❧❡ ✧✶ ✰ ✷ ✰ ✸✧ ✇✐t❤ ❊✳ P✿ ❈❤♦♦s❡ ❛ ♣r♦❞✉❝t✐♦♥ ❛♥❞ ❛ ❞✐✈✐s✐♦♥ ✭♣✐✈♦t ❝❤❛r❛❝t❡rs✮ O✿ P❛rs❡ ♥♦♥t❡r♠✐♥❛❧s ✏✐♥ ❜❡t✇❡❡♥✑ ♣✐✈♦ts

✳ ✳

πP = (”✶ + ✷ + ✸”, ❊) πO = (”✶”, ❊), (”✷ + ✸”, ❊) ✳

✳

A = (Z ∪ {+∞}, ♠✐♥, +)

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡✿ ❡①❛♠♣❧❡

P❛rs✐♥❣ ✧✶ ✰ ✷ ✰ ✸✧✿ ❊ ::= ♥ ❊ ::= ✈ ❊ ::= ✭ ❊ ✮ ❊ ::= ❧❡t ✈ ❂ ❊ ✐♥ ❊ ❊ ::= ✐❢ ❊ t❤❡♥ ❊ ❡❧s❡ ❊ ❊ ::= ❊ ❂ ❊ ❊ ::= ❊ + ❊ ❊ ::= ❊ ∗ ❊

(”1 + 2 + 3”, E) (”1”, E) (”2 + 3”, E) (”1 + 2”, E) (”3”, E)

✳ ✳

❲❤❛t ✐❢ ✇❡ ❤❛❞ ❧❡t ✐♥st❡❛❞ ♦❢ ✸❄ ❆♥ ❡rr♦r✳✳✳

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

P❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡✿ ❡①❛♠♣❧❡

P❛rs✐♥❣ ✧✶ ✰ ✷ ✰ ✸✧✿ ❊ ::= ♥ ❊ ::= ✈ ❊ ::= ✭ ❊ ✮ ❊ ::= ❧❡t ✈ ❂ ❊ ✐♥ ❊ ❊ ::= ✐❢ ❊ t❤❡♥ ❊ ❡❧s❡ ❊ ❊ ::= ❊ ❂ ❊ ❊ ::= ❊ + ❊ ❊ ::= ❊ ∗ ❊

(”1 + 2 + 3”, E) (”1”, E) (”2 + 3”, E) (”1 + 2”, E) (”3”, E)

✳ ✳

❲❤❛t ✐❢ ✇❡ ❤❛❞ ❧❡t ✐♥st❡❛❞ ♦❢ ✸❄ ❆♥ ❡rr♦r✳✳✳

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

▼❡♠♦✐③❛t✐♦♥

▼❡♠♦✐③❛t✐♦♥✿ ❛✈♦✐❞✐♥❣ t♦ r❡✲❡①❛♠✐♥❡❞ r❡♣❡❛t❡❞ ♣♦s✐t✐♦♥s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

Pr❡❧✐♠✐♥❛r② ♣r❛❝t✐❝❛❧ t❡st

■s α✲♣r✉♥✐♥❣ ✉s❡❢✉❧❄

✳ ✳

■s ♠❡♠♦✐③❛t✐♦♥ ✉s❡❢✉❧❄

✳ ✳ ✳ ✳

❬❧✐tt❧❡ ❞❡♠♦❪

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❖t❤❡r α✲♣r✉♥✐♥❣ ❡①❛♠♣❧❡s

❙❡❛r❝❤✐♥❣ t❤❡ ♠♦st ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐♥ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ❬▲♦❞❞♦✲❉✐❈♦s♠♦✱ ▲P❆❘ ✷✵✵✵❪ ❬▲♦❞❞♦ P❤❉ ✷✵✵✷❪

♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥✿ ✏❧❡ss✲✐♥st❛♥t✐❛t❡❞ t❡r♠s✑✳ ✐♥❢ s✉♣ ❜✐✲tr♦♣✐❝❛❧

❉r❛✇ ❛ ♣❧❛♥❛r ❣r❛♣❤✳✳✳ ✳✳✳♠✐♥✐♠✐③✐♥❣✿

t❤❡ ♦❝❝✉♣✐❡❞ ❛r❡❛ t❤❡ ♥✉♠❜❡r ♦❢ ❝r♦ss✐♥❣s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❖t❤❡r α✲♣r✉♥✐♥❣ ❡①❛♠♣❧❡s

❙❡❛r❝❤✐♥❣ t❤❡ ♠♦st ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐♥ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ❬▲♦❞❞♦✲❉✐❈♦s♠♦✱ ▲P❆❘ ✷✵✵✵❪ ❬▲♦❞❞♦ P❤❉ ✷✵✵✷❪

♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥✿ ✏❧❡ss✲✐♥st❛♥t✐❛t❡❞ t❡r♠s✑✳ ✐♥❢ , s✉♣ ❜✐✲tr♦♣✐❝❛❧

❉r❛✇ ❛ ♣❧❛♥❛r ❣r❛♣❤✳✳✳ ✳✳✳♠✐♥✐♠✐③✐♥❣✿

t❤❡ ♦❝❝✉♣✐❡❞ ❛r❡❛ t❤❡ ♥✉♠❜❡r ♦❢ ❝r♦ss✐♥❣s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❖t❤❡r α✲♣r✉♥✐♥❣ ❡①❛♠♣❧❡s

❙❡❛r❝❤✐♥❣ t❤❡ ♠♦st ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐♥ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ❬▲♦❞❞♦✲❉✐❈♦s♠♦✱ ▲P❆❘ ✷✵✵✵❪ ❬▲♦❞❞♦ P❤❉ ✷✵✵✷❪

♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥✿ ✏❧❡ss✲✐♥st❛♥t✐❛t❡❞ t❡r♠s✑✳ ✐♥❢ , s✉♣ ❜✐✲tr♦♣✐❝❛❧

❉r❛✇ ❛ ♣❧❛♥❛r ❣r❛♣❤✳✳✳ ✳✳✳♠✐♥✐♠✐③✐♥❣✿

t❤❡ ♦❝❝✉♣✐❡❞ ❛r❡❛ t❤❡ ♥✉♠❜❡r ♦❢ ❝r♦ss✐♥❣s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❖t❤❡r α✲♣r✉♥✐♥❣ ❡①❛♠♣❧❡s

❙❡❛r❝❤✐♥❣ t❤❡ ♠♦st ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐♥ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ❬▲♦❞❞♦✲❉✐❈♦s♠♦✱ ▲P❆❘ ✷✵✵✵❪ ❬▲♦❞❞♦ P❤❉ ✷✵✵✷❪

♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥✿ ✏❧❡ss✲✐♥st❛♥t✐❛t❡❞ t❡r♠s✑✳ ✐♥❢ , s✉♣ ❜✐✲tr♦♣✐❝❛❧

❉r❛✇ ❛ ♣❧❛♥❛r ❣r❛♣❤✳✳✳ ✳✳✳♠✐♥✐♠✐③✐♥❣✿

t❤❡ ♦❝❝✉♣✐❡❞ ❛r❡❛ t❤❡ ♥✉♠❜❡r ♦❢ ❝r♦ss✐♥❣s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❖t❤❡r α✲♣r✉♥✐♥❣ ❡①❛♠♣❧❡s

❙❡❛r❝❤✐♥❣ t❤❡ ♠♦st ❣❡♥❡r❛❧ s♦❧✉t✐♦♥ ✐♥ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ❬▲♦❞❞♦✲❉✐❈♦s♠♦✱ ▲P❆❘ ✷✵✵✵❪ ❬▲♦❞❞♦ P❤❉ ✷✵✵✷❪

♦♣t✐♠✐③❛t✐♦♥ ❝r✐t❡r✐♦♥✿ ✏❧❡ss✲✐♥st❛♥t✐❛t❡❞ t❡r♠s✑✳ ✐♥❢ , s✉♣ ❜✐✲tr♦♣✐❝❛❧

❉r❛✇ ❛ ♣❧❛♥❛r ❣r❛♣❤✳✳✳ ✳✳✳♠✐♥✐♠✐③✐♥❣✿

t❤❡ ♦❝❝✉♣✐❡❞ ❛r❡❛ t❤❡ ♥✉♠❜❡r ♦❢ ❝r♦ss✐♥❣s

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❈♦♥❝❧✉s✐♦♥s

❲❡✳✳✳ ✳✳✳❞❡✜♥❡❞ ❛♥❞ ♣r♦✈❡❞ ❝♦rr❡❝t tr♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ✳✳✳❢♦r♠❛❧✐③❡❞ ♠✐♥✲♠❛① ❣❛♠❡s ❛s ♣❛rt✐❝✉❧❛r ✭❜✐✲✮tr♦♣✐❝❛❧ ❣❛♠❡s ✳✳✳♠♦❞❡❧❡❞ ❛♣♣r♦①✐♠❛t❡❞ ♣❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡

α✲♣r✉♥✐♥❣ s❡❡♠s t♦ ❜❡ ❡✛❡❝t✐✈❡

❲❡ ✇♦✉❧❞ ❧✐❦❡✳✳✳ ✳✳✳t♦ ✜♥❞ ♠♦r❡ ❛♣♣❧✐❝❛t✐♦♥s ✳✳✳♠♦r❡ r❡❛❧✐st✐❝ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

❚❤❛♥❦s✦

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s

❈♦♠❜✐♥❛t♦r✐❛❧ ❣❛♠❡s✿ ❢r♦♠ ♠✐♥✲♠❛① t♦ tr♦♣✐❝❛❧ ❚r♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ❈❤♦♦s❡✲❍♦✇✲❚♦✲❉✐✈✐❞❡ ❛♥❞ ❈♦♥q✉❡r ❆ tr♦♣✐❝❛❧ ❣❛♠❡ ❡①❛♠♣❧❡✿ ♣❛rs✐♥❣

❈♦♥❝❧✉s✐♦♥s

❲❡✳✳✳ ✳✳✳❞❡✜♥❡❞ ❛♥❞ ♣r♦✈❡❞ ❝♦rr❡❝t tr♦♣✐❝❛❧ α✲♣r✉♥✐♥❣ ✳✳✳❢♦r♠❛❧✐③❡❞ ♠✐♥✲♠❛① ❣❛♠❡s ❛s ♣❛rt✐❝✉❧❛r ✭❜✐✲✮tr♦♣✐❝❛❧ ❣❛♠❡s ✳✳✳♠♦❞❡❧❡❞ ❛♣♣r♦①✐♠❛t❡❞ ♣❛rs✐♥❣ ❛s ❛ tr♦♣✐❝❛❧ ❣❛♠❡

α✲♣r✉♥✐♥❣ s❡❡♠s t♦ ❜❡ ❡✛❡❝t✐✈❡

❲❡ ✇♦✉❧❞ ❧✐❦❡✳✳✳ ✳✳✳t♦ ✜♥❞ ♠♦r❡ ❛♣♣❧✐❝❛t✐♦♥s ✳✳✳♠♦r❡ r❡❛❧✐st✐❝ ✐♠♣❧❡♠❡♥t❛t✐♦♥s

❚❤❛♥❦s✦

❏❡❛♥✲❱✐♥❝❡♥t ▲♦❞❞♦✱ ▲✉❝❛ ❙❛✐✉ ❍♦✇ t♦ ❝♦rr❡❝t❧② ♣r✉♥❡ tr♦♣✐❝❛❧ tr❡❡s