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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 ✶ ✴ ✶✼

■♥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 ✷ ✴ ✶✼

■♥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 ✷ ✴ ✶✼

■♥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 ✷ ✴ ✶✼

■♥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 ✸ ✴ ✶✼

■♥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()

{λn. 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 ✹ ✴ ✶✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥

❚❡♥t❛t✐✈❡ ♣r♦♦❢ ♦❢ ❣❡♥❙②♠

■♥✈❛r✐❛♥t✿ ❧❡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 ✺ ✴ ✶✼

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥

❚❡♥t❛t✐✈❡ ♣r♦♦❢ ♦❢ ❣❡♥❙②♠

■♥✈❛r✐❛♥t✿ {} ❧❡t lastSym = r❡❢ ✵ ✵ {P ∅} {P S} ❧❡t genSym() = lastSym ✳

✳= ! lastSym + ✶;

✶

✷✻✹

✶

✷✻✹

✶

✷✻✹

! lastSym {λn. n / ∈ S ∗ P(S ∪ {n})}

• ❧❡♥ ▼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✐♦♥

❚❡♥t❛t✐✈❡ ♣r♦♦❢ ♦❢ ❣❡♥❙②♠

■♥✈❛r✐❛♥t✿ P S lastSym → max S {} ❧❡t lastSym = r❡❢ ✵ ✵ {P ∅} {P S} ❧❡t genSym() = lastSym ✳

✳= ! lastSym + ✶;

✶

✷✻✹

✶

✷✻✹

✶

✷✻✹

! lastSym {λn. n / ∈ S ∗ P(S ∪ {n})}

• ❧❡♥ ▼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✐♦♥

❚❡♥t❛t✐✈❡ ♣r♦♦❢ ♦❢ ❣❡♥❙②♠

■♥✈❛r✐❛♥t✿ P S lastSym → max S {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹} ! lastSym {λn. n / ∈ S ∗ lastSym → n} {λn. n / ∈ S ∗ P(S ∪ {n})}

• ❧❡♥ ▼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✐♦♥

❚❡♥t❛t✐✈❡ ♣r♦♦❢ ♦❢ ❣❡♥❙②♠

■♥✈❛r✐❛♥t✿ P S lastSym → max S {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹} ! lastSym {λn. n / ∈ S ∗ lastSym → n} {λn. n / ∈ S ∗ P(S ∪ {n})} ❲r♦♥❣

• ❧❡♥ ▼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❦❛r♦✉♥❞✿ ♣❛t❝❤ t❤❡ s♣❡❝✐✜❝❛t✐♦♥

❲❡ ♠❛② ❛❞❞ ❛ ♣r❡❝♦♥❞✐t✐♦♥ t♦ ❡①❝❧✉❞❡ ❛♥② ❝❤❛♥❝❡ ♦❢ ♦✈❡r✢♦✇✿ P ∅ ∗ ∀S.    {P S ∗ |S| < ✷✻✹ − ✶} genSym()

{λn. n /

∈ S ∗ P(S ∪ {n})}    ❚❤✐s ♣♦❧❧✉t❡s ✉s❡r ♣r♦♦❢s ✇✐t❤ ❝✉♠❜❡rs♦♠❡ ♣r♦♦❢ ♦❜❧✐❣❛t✐♦♥s✳ ✳ ✳ ✇❤✐❝❤ ♠❛② ❡✈❡♥ ❜❡ ✉♥♣r♦✈❛❜❧❡✦

• ❧❡♥ ▼é✈❡❧✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r

❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✻ ✴ ✶✼

❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥

❚✐♠❡ r❡❝❡✐♣ts ✐♥ s❡♣❛r❛t✐♦♥ ❧♦❣✐❝

❚♦ ❝♦✉♥t ❡①❡❝✉t✐♦♥ st❡♣s✱ ✇❡ ✐♥tr♦❞✉❝❡ t✐♠❡ r❡❝❡✐♣ts✳ ❊❛❝❤ st❡♣ ♣r♦❞✉❝❡s ♦♥❡ t✐♠❡ r❡❝❡✐♣t✱ ❛♥❞ ♦♥❧② ♦♥❡✿

{❚r✉❡}

x + y

{λz. z = ⌊x + y⌋ ✷✻✹ ∗ ✶}

❚✐♠❡ r❡❝❡✐♣ts s✉♠ ✉♣✿

✶ ∗ . . . ∗ ✶

• n

≡

n

❇✉t t✐♠❡ r❡❝❡✐♣ts ❞♦ ♥♦t ❞✉♣❧✐❝❛t❡ ✭s❡♣❛r❛t✐♦♥ ❧♦❣✐❝✮✿

✶ −

∗ ✶ ∗ ✶ ❚❤❡r❡❢♦r❡✱ n ✐s ❛ ✇✐t♥❡ss t❤❛t ✭❛t ❧❡❛st✮ n st❡♣s ❤❛✈❡ ❜❡❡♥ 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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts

■♥✈❛r✐❛♥t✿ P S lastSym → max S {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹} ! lastSym {λn. n / ∈ S ∗ lastSym → n} {λn. n / ∈ S ∗ P(S ∪ {n})}

• ❧❡♥ ▼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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts

■♥✈❛r✐❛♥t✿ P S lastSym → max S ∗ (max S) {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹} ! lastSym {λn. n / ∈ S ∗ lastSym → n} {λn. n / ∈ S ∗ P(S ∪ {n})}

❲❡ ❦❡❡♣ tr❛❝❦ ♦❢ ❡❧❛♣s❡❞ 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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts

■♥✈❛r✐❛♥t✿ P S lastSym → max S ∗ (max S) {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵ ∗ ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S ∗ max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} ! lastSym {λn. n / ∈ S ∗ lastSym → n ∗ n} {λn. n / ∈ S ∗ P(S ∪ {n})}

• ❧❡♥ ▼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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts

■♥✈❛r✐❛♥t✿ P S lastSym → max S ∗ (max S) {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵ ∗ ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S ∗ max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} ! lastSym {λn. n / ∈ S ∗ lastSym → n ∗ n} {λn. n / ∈ S ∗ P(S ∪ {n})}

■♥✐t✐❛❧✐③❛t✐♦♥ ❲❡ ♦❜t❛✐♥ ✵ t✐♠❡ r❡❝❡✐♣ts ❢♦r ❢r❡❡✳

• ❧❡♥ ▼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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts

■♥✈❛r✐❛♥t✿ P S lastSym → max S ∗ (max S) {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵ ∗ ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S ∗ max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} ! lastSym {λn. n / ∈ S ∗ lastSym → n ∗ n} {λn. n / ∈ S ∗ P(S ∪ {n})}

❚✐❝❦ ❆❞❞✐t✐♦♥ ♣r♦❞✉❝❡s ♦♥❡ t✐♠❡ 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 ✇✐t❤ t✐♠❡ r❡❝❡✐♣ts

▲❡t N ❜❡ ❛♥ ❛r❜✐tr❛r② ✐♥t❡❣❡r✳ ❲❡ ♣♦s✐t t❤❡ ❇♦✉♥❞❡❞ ❚✐♠❡ ❍②♣♦t❤❡s✐s✿

N ⊢ ❋❛❧s❡

■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❛ss✉♠❡ t❤❛t ♥♦ ❡①❡❝✉t✐♦♥ ❧❛sts ❢♦r N st❡♣s✳ ❚❤❡ ❧❛r❣❡r N✱ t❤❡ ✇❡❛❦❡r t❤✐s ❛ss✉♠♣t✐♦♥✳ ❈♦♥s❡q✉❡♥❝❡✿

n ⊢ n < N

• ❧❡♥ ▼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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts ❛♥❞ t❤❡ ❇❚❍

■♥✈❛r✐❛♥t✿ P S lastSym → max S ∗ (max S) {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵ ∗ ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S ∗ max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ ( max S + ✶)} ! lastSym {λn. n / ∈ S ∗ lastSym → n ∗ n} {λn. n / ∈ S ∗ P(S ∪ {n})}

• ❧❡♥ ▼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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts ❛♥❞ t❤❡ ❇❚❍

■♥✈❛r✐❛♥t✿ P S lastSym → max S ∗ (max S) {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵ ∗ ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S ∗ max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ ( max S + ✶)} ! lastSym {λn. n / ∈ S ∗ lastSym → n ∗ n} {λn. n / ∈ S ∗ P(S ∪ {n})}

❇♦✉♥❞❡❞ ❚✐♠❡

(max S + ✶) ❡♥t❛✐❧s max S + ✶ < N✳

• ❧❡♥ ▼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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts ❛♥❞ t❤❡ ❇❚❍

■♥✈❛r✐❛♥t✿ P S lastSym → max S ∗ (max S) {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵ ∗ ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S ∗ max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} {⌊ max S + ✶⌋ ✷✻✹ / ∈ S ∗ lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ ( max S + ✶)} ! lastSym {λn. n / ∈ S ∗ lastSym → n ∗ n} {λn. n / ∈ S ∗ P(S ∪ {n})}

❇♦✉♥❞❡❞ ❚✐♠❡ ❲❡ ❢✉rt❤❡r r❡q✉✐r❡ N ≤ ✷✻✹✳

• ❧❡♥ ▼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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts ❛♥❞ t❤❡ ❇❚❍

■♥✈❛r✐❛♥t✿ P S lastSym → max S ∗ (max S) {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵ ∗ ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S ∗ max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} {max S + ✶ / ∈ S ∗ lastSym → max S + ✶ ∗ ( max S + ✶)} ! lastSym {λn. n / ∈ S ∗ lastSym → n ∗ n} {λn. n / ∈ S ∗ P(S ∪ {n})}

◆♦ ♦✈❡r✢♦✇ ❚❤❡♥✱ max 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✐♦♥

Pr♦♦❢ ♦❢ ❣❡♥❙②♠ ✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts ❛♥❞ t❤❡ ❇❚❍

■♥✈❛r✐❛♥t✿ P S lastSym → max S ∗ (max S) {} ❧❡t lastSym = r❡❢ ✵ {lastSym → ✵ ∗ ✵} {P ∅} {P S} ❧❡t genSym() = {lastSym → max S ∗ max S} lastSym ✳

✳= ! lastSym + ✶;

{lastSym → ⌊ max S + ✶⌋ ✷✻✹ ∗ (max S + ✶)} {max S + ✶ / ∈ S ∗ lastSym → max S + ✶ ∗ ( max S + ✶)} ! lastSym {λn. n / ∈ S ∗ lastSym → n ∗ n} {λn. n / ∈ S ∗ P(S ∪ {n})}

• ❧❡♥ ▼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✐♦♥

■r✐s

✱ ❛ ♣r♦❣r❛♠ ❧♦❣✐❝ ✇✐t❤ t✐♠❡ r❡❝❡✐♣ts

❚✐♠❡ r❡❝❡✐♣ts s❛t✐s❢② t❤❡ ❇♦✉♥❞❡❞ ❚✐♠❡ ❍②♣♦t❤❡s✐s✿

N ⊢ ❋❛❧s❡

❊❛❝❤ st❡♣ ♣r♦❞✉❝❡s ♦♥❡ t✐♠❡ r❡❝❡✐♣t❀ ❢♦r ✐♥st❛♥❝❡✿

{❚r✉❡}

x + y

{λz. z = ⌊x + y⌋ ✷✻✹ ∗ ✶}

❲❡ ❝❛♥ ♦❜t❛✐♥ ③❡r♦ t✐♠❡ r❡❝❡✐♣ts ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✿ ✵ ❚✐♠❡ r❡❝❡✐♣ts ❛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✐♦♥

■r✐s

✱ ❛ ♣r♦❣r❛♠ ❧♦❣✐❝ ✇✐t❤ t✐♠❡ r❡❝❡✐♣ts

❚✐♠❡ r❡❝❡✐♣ts s❛t✐s❢② t❤❡ ❇♦✉♥❞❡❞ ❚✐♠❡ ❍②♣♦t❤❡s✐s✿

N ⊢ ❋❛❧s❡

❊❛❝❤ st❡♣ ♣r♦❞✉❝❡s ♦♥❡ t✐♠❡ r❡❝❡✐♣t❀ ❢♦r ✐♥st❛♥❝❡✿

{❚r✉❡}

x + y

{λz. z = ⌊x + y⌋ ✷✻✹ ∗ ✶}

❲❡ ❝❛♥ ♦❜t❛✐♥ ③❡r♦ t✐♠❡ r❡❝❡✐♣ts ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✿ ⊢

✵

❚✐♠❡ r❡❝❡✐♣ts ❛r❡ ❛❞❞✐t✐✈❡✿

m ∗ n ≡ (m + n)

• ❧❡♥ ▼é✈❡❧✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r

❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✶✶ ✴ ✶✼

❙♦✉♥❞♥❡ss ♦❢ ■r✐s ✇✐t❤ t✐♠❡ r❡❝❡✐♣ts

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥

❙♦✉♥❞♥❡ss ♦❢ ■r✐s

• ❲❡ ✇❛♥t ♦✉r ♣r♦❣r❛♠ ❧♦❣✐❝ ■r✐s t♦ s❛t✐s❢② t❤✐s ♣r♦♣❡rt②✿

❚❤❡♦r❡♠ ✭❙♦✉♥❞♥❡ss ♦❢ ■r✐s✮ ■❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ■r✐s tr✐♣❧❡ ❤♦❧❞s✿

{❚r✉❡} e {❴}

t❤❡♥ e ❝❛♥♥♦t ❝r❛s❤ ✉♥t✐❧ N st❡♣s ❤❛✈❡ ❜❡❡♥ t❛❦❡♥✳ ❲❡ s❛② t❤❛t ✏e ✐s ✭N − ✶✮✲s❛❢❡✑✳ ❈r❛s❤✐♥❣ ♠❡❛♥s tr②✐♥❣ t♦ st❡♣ ✇❤✐❧❡ ✐♥ ❛ st✉❝❦ ❝♦♥✜❣✉r❛t✐♦♥❀ ❢♦r ❡①❛♠♣❧❡✱ ❞❡r❡❢❡r❡♥❝✐♥❣ ❛ ♥♦♥✲♣♦✐♥t❡r✳

• ❧❡♥ ▼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✐♦♥

Pr♦♦❢ s❦❡t❝❤ ♦❢ t❤❡ s♦✉♥❞♥❡ss t❤❡♦r❡♠

❲❡ ✉s❡ ■r✐s ❛s ❛ ♠♦❞❡❧ ♦❢ ■r✐s✳

{P} e {ϕ} {P}

e {ϕ} ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ · ✐♥s❡rts t✐❝❦s ✭s❡❡ ♥❡①t s❧✐❞❡s✮✳ ❚❤❡ ♣r♦♦❢ t❤❡♥ ✇♦r❦s ❛s ❢♦❧❧♦✇s✿

{❚r✉❡}

e {❴}

• ❙♦✉♥❞♥❡ss t❤❡♦r❡♠ ♦❢ ■r✐s ❬❏✉♥❣ ❡t ❛❧✳✱ ✷✵✶✺❪
• e

✐s s❛❢❡

• ❙✐♠✉❧❛t✐♦♥ ❧❡♠♠❛

e ✐s ✭N − ✶✮✲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✐♦♥

❚❤❡ ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥

❲❡ ❦❡❡♣ tr❛❝❦ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ st❡♣s ✉s✐♥❣ ❛ ❣❧♦❜❛❧ ❝♦✉♥t❡r c✱ ✐♥✐t✐❛❧✐③❡❞ ✇✐t❤ ✵✳ ❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐♥s❡rts ♦♥❡ t✐❝❦ ✐♥str✉❝t✐♦♥ ♣❡r ♦♣❡r❛t✐♦♥✳

• e✶ + e✷

t✐❝❦ ( e✶ + e✷ ) t✐❝❦ ✐♥❝r❡♠❡♥ts c✳ ❖♥ ✐ts Nt❤ ❡①❡❝✉t✐♦♥✱ ✐t ❞♦❡s ♥♦t r❡t✉r♥✳ ❧❡t t✐❝❦ x = ! c ✳

✳= ! c + ✶;

✐❢ ! c < N t❤❡♥ x ❡❧s❡ ❧♦♦♣ () ■❞❡❛✿ tr❛♥s❢♦r♠ ❛ ♣r♦❣r❛♠ t❤❛t r✉♥s ❢♦r t♦♦ ❧♦♥❣ ✐♥t♦ ❛ ♣r♦❣r❛♠ t❤❛t ♥❡✈❡r ❡♥❞s✱ ❤❡♥❝❡ ✐s 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✐♦♥ ❧❡♠♠❛

❚❤✐s ♣r♦❣r❛♠ tr❛♥s❢♦r♠❛t✐♦♥ ❞♦❡s s❛t✐s❢② t❤❡ ❞❡s✐r❡❞ ❧❡♠♠❛✿ ▲❡♠♠❛ ✭❙✐♠✉❧❛t✐♦♥✮ ■❢ e ✐s s❛❢❡ ✭✐✳❡✳ ✐t ❝❛♥♥♦t ❝r❛s❤✮✱ t❤❡♥ e ✐s ✭N − ✶✮✲s❛❢❡ ✭✐✳❡✳ ✐t ❝❛♥♥♦t ❝r❛s❤ ✉♥t✐❧ N st❡♣s ❤❛✈❡ ❜❡❡♥ 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✐♦♥

❚❤❡ ♠♦❞❡❧ ♦❢ t✐♠❡ r❡❝❡✐♣ts

❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♠❛✐♥t❛✐♥s t❤❡ ✐♥✈❛r✐❛♥t ! c < N✳

✶ ✐s ♠♦❞❡❧❡❞ ❛s ❛♥ ❡①❝❧✉s✐✈❡ ♣♦rt✐♦♥ ♦❢ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❝♦✉♥t❡r c

✭■r✐s ❢❡❛t✉r❡s ✉s❡❞✿ ❛✉t❤♦r✐t❛t✐✈❡ ♠♦♥♦✐❞❛❧ r❡s♦✉r❝❡✱ ✐♥✈❛r✐❛♥t✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ n ⊢ ! c ≥ n✳ ❍❡♥❝❡✱ N ⊢ ❋❛❧s❡✳ ❆❧❧ ♦t❤❡r ❛①✐♦♠s ♦❢ t✐♠❡ r❡❝❡✐♣ts ❛r❡ r❡❛❧✐s❡❞ ❛s ✇❡❧❧✳

• ❧❡♥ ▼é✈❡❧✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r

❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✶✻ ✴ ✶✼

❈♦♥❝❧✉s✐♦♥

■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠ ❚✐♠❡ r❡❝❡✐♣ts ✐♥ ❛❝t✐♦♥ ❙♦✉♥❞♥❡ss ❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

❈♦♥tr✐❜✉t✐♦♥s ✭♥❡✇✮✿ ❙♦✉♥❞♥❡ss ❆♣♣❧✐❝❛t✐♦♥ ❚✐♠❡ ❝r❡❞✐ts

• ❘❡❝♦♥str✉❝t✐♦♥ ♦❢ ❖❦❛s❛❦✐

❛♥❞ ❉❛♥✐❡❧ss♦♥✬s t❤✉♥❦s ✭❛♠♦rt✐③❡❞ ❛♥❛❧②s✐s✮ ❚✐♠❡ r❡❝❡✐♣ts ✭❡①❝❧✉s✐✈❡ ✴ ♣❡rs✐st❡♥t✮

• ❘❡❝♦♥str✉❝t✐♦♥ ♦❢ ❈❧♦❝❤❛r❞

❡t ❛❧✳✬s ♦✈❡r✢♦✇✲❢r❡❡ ✐♥t❡❣❡rs ❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts

• Pr♦♦❢ ♦❢ ❯♥✐♦♥✲❋✐♥❞✿

❝♦♠♣❧❡①✐t②✱ ❛❜s❡♥❝❡ ♦❢ ♦✈❡r✢♦✇ ✐♥ r❛♥❦s ❉❡✜♥❡❞ ✇✐t❤✐♥ ■r✐s✱ ♠❛❝❤✐♥❡✲❝❤❡❝❦❡❞ ✇✐t❤ ❈♦q ❖♣❡♥ q✉❡st✐♦♥✿ ❈❛♥ ✇❡ ♣r♦✈❡ ✉s❡❢✉❧ ❢❛❝ts ❛❜♦✉t ❝♦♥❝✉rr❡♥t ❝♦❞❡❄

• ❧❡♥ ▼é✈❡❧✱ ❏❛❝q✉❡s✲❍❡♥r✐ ❏♦✉r❞❛♥✱ ❋r❛♥ç♦✐s P♦tt✐❡r

❚✐♠❡ ❝r❡❞✐ts ❛♥❞ t✐♠❡ r❡❝❡✐♣ts ✐♥ ■r✐s ✶✼ ✴ ✶✼

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r t✐♠❡✳

✶✼

❲❤❛t ❛❜♦✉t ❝♦♥❝✉rr❡♥❝②❄

■r✐s ✐s ❛ ❝♦♥❝✉rr❡♥t s❡♣❛r❛t✐♦♥ ❧♦❣✐❝❀ t❤✉s✱ ♦✉r ♣r♦❣r❛♠ ❧♦❣✐❝s ❛❧r❡❛❞② s✉♣♣♦rt ❝♦♥❝✉rr❡♥❝②✿ t❤❡② ♠❡❛s✉r❡ t❤❡ ✇♦r❦ ✭t♦t❛❧ ♥✉♠❜❡r ♦❢ ♦♣❡r❛t✐♦♥s ✐♥ ❛❧❧ t❤r❡❛❞s✮✳ ❧❡t t✐❝❦ x = ✐❢ (❋❆❆ c ✶ < N − ✶) t❤❡♥ x ❡❧s❡ ❧♦♦♣ () ❲❤❛t ❛❜♦✉t ♠❡❛s✉r✐♥❣ t❤❡ s♣❛♥ ✭r✉♥♥✐♥❣ t✐♠❡ ♦❢ t❤❡ ❧♦♥❣❡st✲❧✐✈✐♥❣ t❤r❡❛❞✮❄ ❆ ♣❛t❤ t♦ ❡①♣❧♦r❡✿ ❛ s❡♣❛r❛t❡ ♥♦t✐♦♥ ♦❢ t✐♠❡ r❡❝❡✐♣t ❢♦r ❡❛❝❤ t❤r❡❛❞✱ ✇✐t❤ ❛ r✉❧❡ t♦ ❝❧♦♥❡ t✐♠❡ r❡❝❡✐♣ts ♦❢ t❤❡ ❝❛❧❧✐♥❣ t❤r❡❛❞ ✇❤❡♥ ❢♦r❦✐♥❣✳

❈♦♠♣✐❧✐♥❣ ❝♦❞❡ ❛♥❛❧②s❡❞ ✇✐t❤ t✐♠❡ r❡❝❡✐♣ts

❋♦r t✐♠❡ r❡❝❡✐♣t ♣r♦♦❢s t♦ ❜❡ ✈❛❧✐❞✱ ✇❡ ♥❡❡❞ t♦ ❢♦r❜✐❞ ♦♣t✐♠✐③❛t✐♦♥s✦ ❖t❤❡r✇✐s❡✱ ♣r♦❣r❛♠s ♠❛② ❝♦♠♣✉t❡ ❢❛st❡r t❤❛♥ ❡①♣❡❝t❡❞✳ ❋♦r ❡①❛♠♣❧❡✿ ❢♦r i ❢r♦♠ ✶ t♦ N ❞♦ () ❞♦♥❡; ✭✯ ❚❤✐s ♣♦✐♥t ✐s ❜❡②♦♥❞ t❤❡ s❝♦♣❡ ♦❢ ■r✐s✿ ✯ ❛♥②t❤✐♥❣ ❜❡❧♦✇ ♠❛② ❜❡ ✉♥s❛❢❡✱ ✯ ❜✉t ✐t s❤♦✉❧❞♥✬t ❜❡ r❡❛❝❤❡❞ ✐♥ ❛ ❧✐❢❡t✐♠❡✳ ✳ ✳ ✯✮ ❝r❛s❤ () ❆ ❝♦♠♣✐❧❡r ♠❛② ♦♣t✐♠✐③❡ ✐t t♦✿ ✭✯ ❚♦♦ ❜❛❞✦ ✯✮ ❝r❛s❤ () ❆ s♦❧✉t✐♦♥✿ ✐♥s❡rt ❛❝t✉❛❧ t✐❝❦ ♦♣❡r❛t✐♦♥s ❛♥❞ ♠❛❦❡ t❤❡♠ ♦♣❛q✉❡✳

❊①❛♠♣❧❡ ❛♣♣❧✐❝❛t✐♦♥✿ ❯♥✐♦♥✲❋✐♥❞

❲❡ ✐♠♣❧❡♠❡♥t t❤❡ ❯♥✐♦♥✲❋✐♥❞ ✇✐t❤ r❛♥❦s st♦r❡❞ ✐♥ ♠❛❝❤✐♥❡ ✇♦r❞s✳ ❲❤✐❧❡ ♣r♦✈✐♥❣ t❤❡ ❝♦rr❡❝t♥❡ss ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✱ ✇❡ ❛❧s♦ ♣r♦✈❡ ✐ts ❝♦♠♣❧❡①✐t② ✭✉s✐♥❣ t✐♠❡ ❝r❡❞✐ts✮ ❛♥❞ t❤❡ ❛❜s❡♥❝❡ ♦❢ ♦✈❡r✢♦✇s ❢♦r r❛♥❦s ✭✉s✐♥❣ t✐♠❡ r❡❝❡✐♣ts✮✳

• r❛♥t❡❞ t❤❛t x, y ∈ D ❛♥❞ log✷ log✷ N < ✇♦r❞❴s✐③❡ − ✶✱ ✇❡ s❤♦✇

t❤❡ ■r✐s$ tr✐♣❧❡✿

{✐s❯❋ D R V ∗ $(✹✹α(|D|) + ✶✺✷)}

✉♥✐♦♥ x y

{λz. ✐s❯❋ D R′ V ′ ∗ (z = R x ∨ z = R y)}$

❈♦♥s❡q✉❡♥❝❡s✿ t❤❡ ✭❛♠♦rt✐③❡❞✮ ❝♦♠♣❧❡①✐t② ✐s t❤❡ ✐♥✈❡rs❡ ❆❝❦❡r♠❛♥♥ ❢✉♥❝t✐♦♥❀ ✐❢ N = ✷✻✹✱ t❤❡♥ ✇♦r❞❴s✐③❡ ≥ ✽ ✐s ❡♥♦✉❣❤ t♦ ❛✈♦✐❞ ♦✈❡r✢♦✇s✳

❊①❛♠♣❧❡✿ ❛ ✉♥✐q✉❡ s②♠❜♦❧ ❣❡♥❡r❛t♦r ✭❢✉♥❝t✐♦♥❛❧ ✈❡rs✐♦♥✮

❈♦❞❡✿ ❧❡t makeGenSym() = ❧❡t lastSym = r❡❢ ✵ ✐♥ ✭✯ ✉♥s✐❣♥❡❞ ✻✹✲❜✐t ✐♥t❡❣❡r ✯✮ ❢✉♥ () → lastSym ✳

✳= ! lastSym + ✶;

✭✯ ♠❛② ♦✈❡r✢♦✇ ✯✮ ! lastSym ❙♣❡❝✐✜❝❛t✐♦♥ ✭✐♥ ❤✐❣❤❡r✲♦r❞❡r s❡♣❛r❛t✐♦♥ ❧♦❣✐❝✮✿

{❚r✉❡}

makeGenSym()

{

λ genSym. ∃P. P ∅ ∗ ∀S.   

{P S}

genSym()

{λn. n /

∈ S ∗ P(S ∪ {n})}   }

❆❧t❡r♥❛t✐✈❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ ♠❛❦❡●❡♥❙②♠

❙♣❡❝✐✜❝❛t✐♦♥ ✭✐♥ ■r✐s✮✿

{❚r✉❡}

makeGenSym()

{

λ genSym. ∃γ. ∀n.   

{❚r✉❡}

genSym() {λm. ❖✇♥❙②♠γ(m)}   } ❚❤❡ ♦✇♥❡rs❤✐♣ ♦❢ t❤❡ ❣❡♥❡r❛t♦r ✐s s❤❛r❡❞ t❤r♦✉❣❤ ❛♥ ✐♥✈❛r✐❛♥t✳ ❖✇♥❙②♠γ(m) ❛ss❡rts ✉♥✐q✉❡♥❡ss ♦❢ s②♠❜♦❧ m✿ ❖✇♥❙②♠γ(m✶) ∗ ❖✇♥❙②♠γ(m✷) − ∗ m✶ = m✷

❆ ♣r♦❣r❛♠ ❧♦❣✐❝ ✇✐t❤ t✐♠❡ ❝r❡❞✐ts

❊❛❝❤ st❡♣ ❝♦♥s✉♠❡s ♦♥❡ t✐♠❡ ❝r❡❞✐t❀ ❢♦r ✐♥st❛♥❝❡✿

{$✶}

x + y

{λz. z = ⌊x + y⌋ ✷✻✹}

❲❡ ❝❛♥ ♦❜t❛✐♥ ③❡r♦ t✐♠❡ ❝r❡❞✐ts ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✿ ⊢ $✵ ❚✐♠❡ ❝r❡❞✐ts ❛r❡ ❛❞❞✐t✐✈❡✿ $m ∗ $n ≡ $(m + n)

❆ ♣r♦❣r❛♠ ❧♦❣✐❝ ✇✐t❤ t✐♠❡ ❝r❡❞✐ts ✖ ❆❞❡q✉❛❝②

❖✉r ♣r♦❣r❛♠ ❧♦❣✐❝ ■r✐s$ s❛t✐s✜❡s t❤✐s ♣r♦♣❡rt②✿ ❚❤❡♦r❡♠ ✭❆❞❡q✉❛❝② ♦❢ ■r✐s$✮ ■❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ■r✐s tr✐♣❧❡ ❤♦❧❞s✿

{$n} e {ϕ}✩

t❤❡♥✿ e ❝❛♥♥♦t ❝r❛s❤❀ ✐❢ e ❝♦♠♣✉t❡s ❛ ✈❛❧✉❡ v✱ t❤❡♥ ϕ v ❤♦❧❞s❀ e ❝♦♠♣✉t❡s ❢♦r ❛t ♠♦st n st❡♣s✳

❆❞❡q✉❛❝② ♦❢ ■r✐s

• ❖✉r ♣r♦❣r❛♠ ❧♦❣✐❝ ■r✐s s❛t✐s✜❡s t❤✐s ♣r♦♣❡rt②✿

❚❤❡♦r❡♠ ✭❆❞❡q✉❛❝② ♦❢ ■r✐s✮ ■❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ■r✐s tr✐♣❧❡ ❤♦❧❞s✿

{❚r✉❡} e {ϕ}

t❤❡♥✿ e ❝❛♥♥♦t ❝r❛s❤ ✉♥t✐❧ N st❡♣s ❤❛✈❡ ❜❡❡♥ t❛❦❡♥❀ ✐❢ e ❝♦♠♣✉t❡s ❛ ✈❛❧✉❡ v ✐♥ ❧❡ss t❤❛♥ N st❡♣s✱ t❤❡♥ ϕ v ❤♦❧❞s✳

❆ ♣r♦❣r❛♠ ❧♦❣✐❝ ✇✐t❤ ❞✉♣❧✐❝❛❜❧❡ t✐♠❡ r❡❝❡✐♣ts

❉✉♣❧✐❝❛❜❧❡ t✐♠❡ r❡❝❡✐♣ts s❛t✐s❢② t❤❡ ❇♦✉♥❞❡❞ ❚✐♠❡ ❍②♣♦t❤❡s✐s✿

N ⊢ ❋❛❧s❡

❊❛❝❤ st❡♣ ✐♥❝r❡♠❡♥ts ❛ ❞✉♣❧✐❝❛❜❧❡ t✐♠❡ r❡❝❡✐♣t❀ ❢♦r ✐♥st❛♥❝❡✿

{ m}

x + y

{λz. z = ⌊x + y⌋ ✷✻✹ ∗ (m + ✶)}

❲❡ ❝❛♥ ♦❜t❛✐♥ ③❡r♦ ❞✉♣❧✐❝❛❜❧❡ t✐♠❡ r❡❝❡✐♣ts ✉♥❝♦♥❞✐t✐♦♥❛❧❧②✿ ⊢

✵

❉✉♣❧✐❝❛❜❧❡ t✐♠❡ r❡❝❡✐♣ts ♦❜❡② ♠❛①✐♠✉♠✿

m ∗ n ≡ max(m, n)

❉✉♣❧✐❝❛❜❧❡ t✐♠❡ r❡❝❡✐♣ts ❛r❡ ❞✉♣❧✐❝❛❜❧❡✿

m −

∗ m ∗ m ❘❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t✐♠❡ r❡❝❡✐♣ts ❛♥❞ ❞✉♣❧✐❝❛❜❧❡ t✐♠❡ r❡❝❡✐♣ts✿

m ⊢ m ∗ m

❖✈❡r✢♦✇✲❢r❡❡ ✐♥t❡❣❡rs ✭s✉♠♠❛❜❧❡✮

■s❈❧♦❝❦(v, n) ✵ ≤ n < ✷✻✹ ∗ v = n ∗ n ♥♦♥✲❞✉♣❧✐❝❛❜❧❡ s✉♣♣♦rts ❛❞❞✐t✐♦♥ ✭❝♦♥s✉♠❡s ✐ts ♦♣❡r❛♥❞s✮✿

{■s❈❧♦❝❦(v✶, n✶) ∗ ■s❈❧♦❝❦(v✷, n✷)}

v✶ + v✷

{λw. ■s❈❧♦❝❦(w, n✶ + n✷)}

♥♦ ♦✈❡r✢♦✇✦

❖✈❡r✢♦✇✲❢r❡❡ ✐♥t❡❣❡rs ✭✐♥❝r❡♠❡♥t❛❜❧❡✮

■s❙♥❛♣❈❧♦❝❦(v, n) ✵ ≤ n < ✷✻✹ ∗ v = n ∗ n ❞✉♣❧✐❝❛❜❧❡ s✉♣♣♦rts ✐♥❝r❡♠❡♥t❛t✐♦♥ ✭❞♦❡s ♥♦t ❝♦♥s✉♠❡ ✐ts ♦♣❡r❛♥❞✮✿

{■s❙♥❛♣❈❧♦❝❦(v, n)}

v + ✶

{λw. ■s❙♥❛♣❈❧♦❝❦(w, n + ✶)}

♥♦ ♦✈❡r✢♦✇✦

❍♦❛r❡ ❧♦❣✐❝ ♣r✐♠❡r

♣r❣♠ ✐s ❛ ♣r♦❣r❛♠ ✭s♦✉r❝❡ ❝♦❞❡✮✳ Pre ❛♥❞ Post ❛r❡ ❧♦❣✐❝❛❧ ❢♦r♠✉❧❛s✳

{Pre} ♣r❣♠ {Post}

❙♦✉♥❞♥❡ss✿ ✏■❢ Pre ❤♦❧❞s✱ t❤❡♥ ♣r❣♠ ✇♦♥✬t ❝r❛s❤✳✑ ✭P❛rt✐❛❧✮ ❝♦rr❡❝t♥❡ss✿ ✏■❢ Pre ❤♦❧❞s✱ t❤❡♥ ❛❢t❡r ♣r❣♠ ✐s r✉♥✱ Post ✇✐❧❧ ❤♦❧❞✳✑ ❚♦t❛❧ ❝♦rr❡❝t♥❡ss✿ ✏■❢ Pre ❤♦❧❞s✱ t❤❡♥ ♣r❣♠ t❡r♠✐♥❛t❡s ❛♥❞✱ ❛❢t❡r ♣r❣♠ ✐s r✉♥✱ Post ✇✐❧❧ ❤♦❧❞✳✑

❙❡♣❛r❛t✐♦♥ ❧♦❣✐❝ ♣r✐♠❡r

P ✐s ❛ r❡s♦✉r❝❡✳ x → v ✐s ❛♥ ❡①❝❧✉s✐✈❡ r❡s♦✉r❝❡✱ ✐ts ♦✇♥❡rs❤✐♣ ❝❛♥♥♦t ❜❡ s❤❛r❡❞✳ ❙t❛♥❞❛r❞ ❧♦❣✐❝✿ P ⇒ P ∧ P ❙❡♣❛r❛t✐♦♥ ❧♦❣✐❝✿ P − ∗ P ∗ P ✭r❡s♦✉r❝❡s ❛r❡ ♥♦t ❞✉♣❧✐❝❛❜❧❡✮ P ∗ Q ❛r❡ ❞✐s❥♦✐♥t r❡s♦✉r❝❡s✳ x → v ∗ x → v′ ✐s ❛❜s✉r❞✳ ❆✣♥❡ s❡♣✳ ❧♦❣✐❝✿ P ∗ Q − ∗ P ✭r❡s♦✉r❝❡s ❝❛♥ ❜❡ t❤r♦✇♥ ❛✇❛②✮

■r✐s ♣r✐♠❡r

■r✐s ✐s✿ ❛♥ ❛✣♥❡ s❡♣❛r❛t✐♦♥ ❧♦❣✐❝✱ ❤✐❣❤❡r✲♦r❞❡r✱ ❢✉❧❧✲❢❡❛t✉r❡❞ ✭✐♠♣r❡❞✐❝❛t✐✈❡ ✐♥✈❛r✐❛♥ts✱ ♠♦♥♦✐❞❛❧ r❡s♦✉r❝❡s✳ ✳ ✳ ✮✱ ✈❡r② ❡①t❡♥s✐❜❧❡✱ ❢♦r♠❛❧✐③❡❞ ✐♥ ❈♦q✳