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❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ■♥r✐❛ ❈◆❘❙✱ ▲❘■✱ ❯♥✐✈✳ P❛r✐s ❙✉❞✱ ❯♥✐✈❡rs✐té P❛r✐s✲❙❛❝❧❛② ❆♣r✐❧ ✽✱ ✷✵✶✾ Pr❛❣✉❡

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥ ❚❤✐s t❛❧❦ r❡❝❡♥t ✇♦r❦s✿ t✐♠❡ ❝r❡❞✐ts ❛✐♠✿ ♣r♦✈❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢ ❛ ♣r♦❣r❛♠ t❤✐s t❛❧❦✿ t✐♠❡ r❡❝❡✐♣ts ❛✐♠✿ ❛ss✉♠❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢ ❛ ♣r♦❣r❛♠ ❚❤❡s❡ ❛r❡ ❞✉❛❧ ♥♦t✐♦♥s✳ ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✶ ✴ ✶✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥ ❚❤✐s t❛❧❦ r❡❝❡♥t ✇♦r❦s✿ t✐♠❡ ❝r❡❞✐ts ❛✐♠✿ ♣r♦✈❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢ ❛ ♣r♦❣r❛♠ t❤✐s t❛❧❦✿ t✐♠❡ r❡❝❡✐♣ts ❛✐♠✿ ❛ss✉♠❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢ ❛ ♣r♦❣r❛♠ ❚❤❡s❡ ❛r❡ ❞✉❛❧ ♥♦t✐♦♥s✳ ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✶ ✴ ✶✼

✭✯ ✉♥s✐❣♥❡❞ ✻✹✲❜✐t ✐♥t❡❣❡r ✯✮ ✭✯ ♠❛② ♦✈❡r✢♦✇✦ ✯✮ ❙tr✐❝t❧② s♣❡❛❦✐♥❣✱ t❤✐s ❝♦❞❡ ✐s ♥♦t ❝♦rr❡❝t✳ ❲❡ st✐❧❧ ✇❛♥t t♦ ♣r♦✈❡ t❤❛t t❤✐s ❝♦❞❡ ✐s ✏❝♦rr❡❝t✑ ✐♥ s♦♠❡ s❡♥s❡✳ ■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥ ❊①❛♠♣❧❡✿ ❛ ✉♥✐q✉❡ s②♠❜♦❧ ❣❡♥❡r❛t♦r ❚❤❡ ❢✉♥❝t✐♦♥ genSym r❡t✉r♥s ❢r❡s❤ s②♠❜♦❧s✿ ❧❡t lastSym = r❡❢ ✵ ❧❡t genSym () = lastSym ✳ ✳ = ! lastSym + ✶ ; ! lastSym ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✷ ✴ ✶✼

❲❡ st✐❧❧ ✇❛♥t t♦ ♣r♦✈❡ t❤❛t t❤✐s ❝♦❞❡ ✐s ✏❝♦rr❡❝t✑ ✐♥ s♦♠❡ s❡♥s❡✳ ■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥ ❊①❛♠♣❧❡✿ ❛ ✉♥✐q✉❡ s②♠❜♦❧ ❣❡♥❡r❛t♦r ❚❤❡ ❢✉♥❝t✐♦♥ genSym r❡t✉r♥s ❢r❡s❤ s②♠❜♦❧s✿ ❧❡t lastSym = r❡❢ ✵ ✭✯ ✉♥s✐❣♥❡❞ ✻✹✲❜✐t ✐♥t❡❣❡r ✯✮ ❧❡t genSym () = lastSym ✳ ✳ = ! lastSym + ✶ ; ✭✯ ♠❛② ♦✈❡r✢♦✇✦ ✯✮ ! lastSym ❙tr✐❝t❧② s♣❡❛❦✐♥❣✱ t❤✐s ❝♦❞❡ ✐s ♥♦t ❝♦rr❡❝t✳ ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✷ ✴ ✶✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥ ❊①❛♠♣❧❡✿ ❛ ✉♥✐q✉❡ s②♠❜♦❧ ❣❡♥❡r❛t♦r ❚❤❡ ❢✉♥❝t✐♦♥ genSym r❡t✉r♥s ❢r❡s❤ s②♠❜♦❧s✿ ❧❡t lastSym = r❡❢ ✵ ✭✯ ✉♥s✐❣♥❡❞ ✻✹✲❜✐t ✐♥t❡❣❡r ✯✮ ❧❡t genSym () = lastSym ✳ ✳ = ! lastSym + ✶ ; ✭✯ ♠❛② ♦✈❡r✢♦✇✦ ✯✮ ! lastSym ❙tr✐❝t❧② s♣❡❛❦✐♥❣✱ t❤✐s ❝♦❞❡ ✐s ♥♦t ❝♦rr❡❝t✳ ❲❡ st✐❧❧ ✇❛♥t t♦ ♣r♦✈❡ t❤❛t t❤✐s ❝♦❞❡ ✐s ✏❝♦rr❡❝t✑ ✐♥ s♦♠❡ s❡♥s❡✳ ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✷ ✴ ✶✼

■♥ t❤✐s t❛❧❦✿ ❲❡ ❛♥s✇❡r t❤✐s q✉❡st✐♦♥ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts ✳ ❲❡ ♣r♦✈❡ t❤❛t ■r✐s✱ ❡①t❡♥❞❡❞ ✇✐t❤ t✐♠❡ r❡❝❡✐♣ts✱ ✐s s♦✉♥❞ ✳ ■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❇♦✉♥❞❡❞ ❚✐♠❡ ❍②♣♦t❤❡s✐s ❬❈❧♦❝❤❛r❞ ❡t ❛❧✳ ✱ ✷✵✶✺❪ ❈♦✉♥t✐♥❣ ❢r♦♠ ✵ t♦ ✷ ✻✹ t❛❦❡s ❝❡♥t✉r✐❡s ✇✐t❤ ❛ ♠♦❞❡r♥ ♣r♦❝❡ss♦r✳ ❚❤❡r❡❢♦r❡✱ t❤✐s ♦✈❡r✢♦✇ ✇♦♥✬t ❤❛♣♣❡♥ ✐♥ ❛ ❧✐❢❡t✐♠❡✳ ❍♦✇ t♦ ❡①♣r❡ss t❤✐s ✐♥❢♦r♠❛❧ ❛r❣✉♠❡♥t ✐♥ s❡♣❛r❛t✐♦♥ ❧♦❣✐❝❄ ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✸ ✴ ✶✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥ ❚❤❡ ❇♦✉♥❞❡❞ ❚✐♠❡ ❍②♣♦t❤❡s✐s ❬❈❧♦❝❤❛r❞ ❡t ❛❧✳ ✱ ✷✵✶✺❪ ❈♦✉♥t✐♥❣ ❢r♦♠ ✵ t♦ ✷ ✻✹ t❛❦❡s ❝❡♥t✉r✐❡s ✇✐t❤ ❛ ♠♦❞❡r♥ ♣r♦❝❡ss♦r✳ ❚❤❡r❡❢♦r❡✱ t❤✐s ♦✈❡r✢♦✇ ✇♦♥✬t ❤❛♣♣❡♥ ✐♥ ❛ ❧✐❢❡t✐♠❡✳ ❍♦✇ t♦ ❡①♣r❡ss t❤✐s ✐♥❢♦r♠❛❧ ❛r❣✉♠❡♥t ✐♥ s❡♣❛r❛t✐♦♥ ❧♦❣✐❝❄ ■♥ t❤✐s t❛❧❦✿ ❲❡ ❛♥s✇❡r t❤✐s q✉❡st✐♦♥ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts ✳ ❲❡ ♣r♦✈❡ t❤❛t ■r✐s✱ ❡①t❡♥❞❡❞ ✇✐t❤ t✐♠❡ r❡❝❡✐♣ts✱ ✐s s♦✉♥❞ ✳ ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✸ ✴ ✶✼

❆ ❝❧♦s❡r ❧♦♦❦ ❛t t❤❡ ♣r♦❜❧❡♠

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥ ❙♣❡❝✐✜❝❛t✐♦♥ ♦❢ ❣❡♥❙②♠ ❆ s♣❡❝✐✜❝❛t✐♦♥ ✭✐♥ s❡♣❛r❛t✐♦♥ ❧♦❣✐❝✮✿   { P S }   P ∅ ∗ ∀ S . genSym ()   { λ n . n / ∈ S ∗ P ( S ∪ { n } ) } ❢♦r s♦♠❡ ♣r♦♣♦s✐t✐♦♥ P S ✇❤✐❝❤ r❡♣r❡s❡♥ts✿ t❤❡ ♦✇♥❡rs❤✐♣ ♦❢ t❤❡ ❣❡♥❡r❛t♦r❀ t❤❡ ❢❛❝t t❤❛t S ✐s t❤❡ s❡t ♦❢ ❛❧❧ s②♠❜♦❧s r❡t✉r♥❡❞ s♦ ❢❛r✳ ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✹ ✴ ✶✼

■♥✈❛r✐❛♥t✿ ✵ ✶ ✷ ✻✹ ✶ ✶ ✷ ✻✹ ✷ ✻✹ ■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥ ❚❡♥t❛t✐✈❡ ♣r♦♦❢ ♦❢ ❣❡♥❙②♠ ❧❡t lastSym = r❡❢ ✵ ❧❡t genSym () = lastSym ✳ ✳ = ! lastSym + ✶ ; ! lastSym ●❧❡♥ ▼é✈❡❧ ✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✺ ✴ ✶✼