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❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❆♥✉❥ ❉❛✇❛r

❚❡①ts

❚❤❡ ♠❛✐♥ t❡①ts ❢♦r t❤❡ ❝♦✉rs❡ ❛r❡✿ ❈♦♠♣✉t❛t✐♦♥❛❧ ❈♦♠♣❧❡①✐t②✳ ❈❤r✐st♦s ❍✳ P❛♣❛❞✐♠✐tr✐♦✉✳ ■♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ❚❤❡♦r② ♦❢ ❈♦♠♣✉t❛t✐♦♥✳ ▼✐❝❤❛❡❧ ❙✐♣s❡r✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡❢❡r❡♥❝❡s

❖t❤❡r ✉s❡❢✉❧ r❡❢❡r❡♥❝❡s ✐♥❝❧✉❞❡✿ ❈♦♠♣✉t❡rs ❛♥❞ ■♥tr❛❝t❛❜✐❧✐t②✿ ❆ ❣✉✐❞❡ t♦ t❤❡ t❤❡♦r② ♦❢ ◆P✲❝♦♠♣❧❡t❡♥❡ss✳ ▼✐❝❤❛❡❧ ❘✳ ●❛r❡② ❛♥❞ ❉❛✈✐❞ ❙✳ ❏♦❤♥s♦♥✳ P✱ ◆P ❛♥❞ ◆P✲❝♦♠♣❧❡t❡♥❡ss✳ ❖❞❡❞ ●♦❧❞r❡✐❝❤✳ ❈♦♠♣✉t❛❜✐❧✐t② ❛♥❞ ❈♦♠♣❧❡①✐t② ❢r♦♠ ❛ Pr♦❣r❛♠♠✐♥❣ P❡rs♣❡❝t✐✈❡✳ ◆❡✐❧ ❏♦♥❡s✳ ❈♦♠♣✉t❛t✐♦♥❛❧ ❈♦♠♣❧❡①✐t② ✲ ❆ ▼♦❞❡r♥ ❆♣♣r♦❛❝❤✳ ❙❛♥❥❡❡✈ ❆r♦r❛ ❛♥❞ ❇♦❛③ ❇❛r❛❦✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❖✉t❧✐♥❡

❆ r♦✉❣❤ ❧❡❝t✉r❡✲❜②✲❧❡❝t✉r❡ ❣✉✐❞❡✱ ✇✐t❤ r❡❧❡✈❛♥t s❡❝t✐♦♥s ❢r♦♠ t❤❡ t❡①t ❜② P❛♣❛❞✐♠✐tr✐♦✉ ✭♦r ❙✐♣s❡r✱ ✇❤❡r❡ ♠❛r❦❡❞ ✇✐t❤ ❛♥ ❙✮✳

• ❆❧❣♦r✐t❤♠s ❛♥❞ ♣r♦❜❧❡♠s✳ ✶✳✶✕✶✳✸✳
• ❚✐♠❡ ❛♥❞ s♣❛❝❡✳ ✷✳✶✕✷✳✺✱ ✷✳✼✳
• ❚✐♠❡ ❈♦♠♣❧❡①✐t② ❝❧❛ss❡s✳ ✼✳✶✱ ❙✼✳✷✳
• ◆♦♥❞❡t❡r♠✐♥✐s♠✳ ✷✳✼✱ ✾✳✶✱ ❙✼✳✸✳
• ◆P✲❝♦♠♣❧❡t❡♥❡ss✳ ✽✳✶✕✽✳✷✱ ✾✳✷✳
• ●r❛♣❤✲t❤❡♦r❡t✐❝ ♣r♦❜❧❡♠s✳ ✾✳✸

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❖✉t❧✐♥❡ ✲ ❝♦♥t❞✳

• ❙❡ts✱ ♥✉♠❜❡rs ❛♥❞ s❝❤❡❞✉❧✐♥❣✳ ✾✳✹
• ❝♦◆P✳ ✶✵✳✶✕✶✵✳✷✳
• ❈r②♣t♦❣r❛♣❤✐❝ ❝♦♠♣❧❡①✐t②✳ ✶✷✳✶✕✶✷✳✷✳
• ❙♣❛❝❡ ❈♦♠♣❧❡①✐t② ✼✳✶✱ ✼✳✸✱ ❙✽✳✶✳
• ❍✐❡r❛r❝❤② ✼✳✷✱ ❙✾✳✶✳
• ❉❡s❝r✐♣t✐✈❡ ❈♦♠♣❧❡①✐t② ✺✳✼✱ ✽✳✸✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❆❧❣♦r✐t❤♠s ❛♥❞ Pr♦❜❧❡♠s

■♥s❡rt✐♦♥ ❙♦rt r✉♥s ✐♥ t✐♠❡ O(n✷)✱ ✇❤✐❧❡ ▼❡r❣❡ ❙♦rt ✐s ❛♥ O(n ❧♦❣ n) ❛❧❣♦r✐t❤♠✳ ❚❤❡ ✜rst ❤❛❧❢ ♦❢ t❤✐s st❛t❡♠❡♥t ✐s s❤♦rt ❢♦r✿ ■❢ ✇❡ ❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢ st❡♣s ♣❡r❢♦r♠❡❞ ❜② t❤❡ ■♥s❡rt✐♦♥ ❙♦rt ❛❧❣♦r✐t❤♠ ♦♥ ❛♥ ✐♥♣✉t ♦❢ s✐③❡ n✱ t❛❦✐♥❣ t❤❡ ❧❛r❣❡st s✉❝❤ ♥✉♠❜❡r✱ ❢r♦♠ ❛♠♦♥❣ ❛❧❧ ✐♥♣✉ts ♦❢ t❤❛t s✐③❡✱ t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ ♦❢ n s♦ ❞❡✜♥❡❞ ✐s ❡✈❡♥t✉❛❧❧② ❜♦✉♥❞❡❞ ❜② ❛ ❝♦♥st❛♥t ♠✉❧t✐♣❧❡ ♦❢ n✷✳ ■t ♠❛❦❡s s❡♥s❡ t♦ ❝♦♠♣❛r❡ t❤❡ t✇♦ ❛❧❣♦r✐t❤♠s✱ ❜❡❝❛✉s❡ t❤❡② s❡❡❦ t♦ s♦❧✈❡ t❤❡ s❛♠❡ ♣r♦❜❧❡♠✳ ❇✉t✱ ✇❤❛t ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ s♦rt✐♥❣ ♣r♦❜❧❡♠❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

▲♦✇❡r ❛♥❞ ❯♣♣❡r ❇♦✉♥❞s

❲❤❛t ✐s t❤❡ r✉♥♥✐♥❣ t✐♠❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ❢❛st❡st ❛❧❣♦r✐t❤♠ t❤❛t s♦rts ❛ ❧✐st❄ ❇② t❤❡ ❛♥❛❧②s✐s ♦❢ t❤❡ ▼❡r❣❡ ❙♦rt ❛❧❣♦r✐t❤♠✱ ✇❡ ❦♥♦✇ t❤❛t t❤✐s ✐s ♥♦ ✇♦rs❡ t❤❛♥ O(n ❧♦❣ n)✳ ❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❛ ♣❛rt✐❝✉❧❛r ❛❧❣♦r✐t❤♠ ❡st❛❜❧✐s❤❡s ❛♥ ✉♣♣❡r ❜♦✉♥❞ ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ t❤❡ ♣r♦❜❧❡♠✳ ❚♦ ❡st❛❜❧✐s❤ ❛ ❧♦✇❡r ❜♦✉♥❞✱ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t ♥♦ ♣♦ss✐❜❧❡ ❛❧❣♦r✐t❤♠✱ ✐♥❝❧✉❞✐♥❣ t❤♦s❡ ❛s ②❡t ✉♥❞r❡❛♠❡❞ ♦❢✱ ❝❛♥ ❞♦ ❜❡tt❡r✳ ■♥ t❤❡ ❝❛s❡ ♦❢ s♦rt✐♥❣✱ ✇❡ ❝❛♥ ❡st❛❜❧✐s❤ ❛ ❧♦✇❡r ❜♦✉♥❞ ♦❢ Ω(n ❧♦❣ n)✱ s❤♦✇✐♥❣ t❤❛t ▼❡r❣❡ ❙♦rt ✐s ❛s②♠♣t♦t✐❝❛❧❧② ♦♣t✐♠❛❧✳ ❙♦rt✐♥❣ ✐s ❛ r❛r❡ ❡①❛♠♣❧❡ ✇❤❡r❡ ❦♥♦✇♥ ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❜♦✉♥❞s ♠❛t❝❤✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡✈✐❡✇

❚❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❛♥ ❛❧❣♦r✐t❤♠ ✭✇❤❡t❤❡r ♠❡❛s✉r✐♥❣ ♥✉♠❜❡r ♦❢ st❡♣s✱ ♦r ❛♠♦✉♥t ♦❢ ♠❡♠♦r②✮ ✐s ✉s✉❛❧❧② ❞❡s❝r✐❜❡❞ ❛s②♠♣t♦t✐❝❛❧❧②✿ ❉❡✜♥✐t✐♦♥ ❋♦r ❢✉♥❝t✐♦♥s f : ■ ◆ → ■ ◆ ❛♥❞ g : ■ ◆ → ■ ◆✱ ✇❡ s❛② t❤❛t✿

• f = O(g)✱ ✐❢ t❤❡r❡ ✐s ❛♥ n✵ ∈ ■

◆ ❛♥❞ ❛ ❝♦♥st❛♥t c s✉❝❤ t❤❛t ❢♦r ❛❧❧ n > n✵✱ f (n) ≤ cg(n)❀

• f = Ω(g)✱ ✐❢ t❤❡r❡ ✐s ❛♥ n✵ ∈ ■

◆ ❛♥❞ ❛ ❝♦♥st❛♥t c s✉❝❤ t❤❛t ❢♦r ❛❧❧ n > n✵✱ f (n) ≥ cg(n)✳

• f = θ(g) ✐❢ f = O(g) ❛♥❞ f = Ω(g)✳

❯s✉❛❧❧②✱ O ✐s ✉s❡❞ ❢♦r ✉♣♣❡r ❜♦✉♥❞s ❛♥❞ Ω ❢♦r ❧♦✇❡r ❜♦✉♥❞s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

▲♦✇❡r ❇♦✉♥❞ ♦♥ ❙♦rt✐♥❣

❆♥ ❛❧❣♦r✐t❤♠ A s♦rt✐♥❣ ❛ ❧✐st ♦❢ n ❞✐st✐♥❝t ♥✉♠❜❡rs a✶, . . . , an✳

❞♦♥❡ ❞♦♥❡ ❞♦♥❡ ❞♦♥❡ ❞♦♥❡ ai < aj❄ ak < al❄ ap < aq❄ ar < as❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❚♦ ✇♦r❦ ❢♦r ❛❧❧ ♣❡r♠✉t❛t✐♦♥s ♦❢ t❤❡ ✐♥♣✉t ❧✐st✱ t❤❡ tr❡❡ ♠✉st ❤❛✈❡ ❛t ❧❡❛st n! ❧❡❛✈❡s ❛♥❞ t❤❡r❡❢♦r❡ ❤❡✐❣❤t ❛t ❧❡❛st ❧♦❣✷(n!) = θ(n ❧♦❣ n)✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❚r❛✈❡❧❧✐♥❣ ❙❛❧❡s♠❛♥

• ✐✈❡♥
• V ✖ ❛ s❡t ♦❢ ♥♦❞❡s✳
• c : V × V → ■

◆ ✖ ❛ ❝♦st ♠❛tr✐①✳ ❋✐♥❞ ❛♥ ♦r❞❡r✐♥❣ v✶, . . . , vn ♦❢ V ❢♦r ✇❤✐❝❤ t❤❡ t♦t❛❧ ❝♦st✿ c(vn, v✶) +

n−✶

• i=✶

c(vi, vi+✶) ✐s t❤❡ s♠❛❧❧❡st ♣♦ss✐❜❧❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣❧❡①✐t② ♦❢ ❚❙P

❖❜✈✐♦✉s ❛❧❣♦r✐t❤♠✿ ❚r② ❛❧❧ ♣♦ss✐❜❧❡ ♦r❞❡r✐♥❣s ♦❢ V ❛♥❞ ✜♥❞ t❤❡ ♦♥❡ ✇✐t❤ ❧♦✇❡st ❝♦st✳ ❚❤❡ ✇♦rst ❝❛s❡ r✉♥♥✐♥❣ t✐♠❡ ✐s θ(n!)✳ ▲♦✇❡r ❜♦✉♥❞✿ ❆♥ ❛♥❛❧②s✐s ❧✐❦❡ t❤❛t ❢♦r s♦rt✐♥❣ s❤♦✇s ❛ ❧♦✇❡r ❜♦✉♥❞ ♦❢ Ω(n ❧♦❣ n)✳ ❯♣♣❡r ❜♦✉♥❞✿ ❚❤❡ ❝✉rr❡♥t❧② ❢❛st❡st ❦♥♦✇♥ ❛❧❣♦r✐t❤♠ ❤❛s ❛ r✉♥♥✐♥❣ t✐♠❡ ♦❢ O(n✷✷n)✳ ❇❡t✇❡❡♥ t❤❡s❡ t✇♦ ✐s t❤❡ ❝❤❛s♠ ♦❢ ♦✉r ✐❣♥♦r❛♥❝❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❋♦r♠❛❧✐s✐♥❣ ❆❧❣♦r✐t❤♠s

❚♦ ♣r♦✈❡ ❛ ❧♦✇❡r ❜♦✉♥❞ ♦♥ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❛ ♣r♦❜❧❡♠✱ r❛t❤❡r t❤❛♥ ❛ s♣❡❝✐✜❝ ❛❧❣♦r✐t❤♠✱ ✇❡ ♥❡❡❞ t♦ ♣r♦✈❡ ❛ st❛t❡♠❡♥t ❛❜♦✉t ❛❧❧ ❛❧❣♦r✐t❤♠s ❢♦r s♦❧✈✐♥❣ ✐t✳ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ ❢❛❝ts ❛❜♦✉t ❛❧❧ ❛❧❣♦r✐t❤♠s✱ ✇❡ ♥❡❡❞ ❛ ♠❛t❤❡♠❛t✐❝❛❧❧② ♣r❡❝✐s❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛❧❣♦r✐t❤♠✳ ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ❚✉r✐♥❣ ♠❛❝❤✐♥❡✳ ❚❤❡ s✐♠♣❧✐❝✐t② ♦❢ t❤❡ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ♠❡❛♥s ✐t✬s ♥♦t ✉s❡❢✉❧ ❢♦r ❛❝t✉❛❧❧② ❡①♣r❡ss✐♥❣ ❛❧❣♦r✐t❤♠s✱ ❜✉t ✈❡r② ✇❡❧❧ s✉✐t❡❞ ❢♦r ♣r♦♦❢s ❛❜♦✉t ❛❧❧ ❛❧❣♦r✐t❤♠s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❚✉r✐♥❣ ▼❛❝❤✐♥❡s

❋♦r ♦✉r ♣✉r♣♦s❡s✱ ❛ ❚✉r✐♥❣ ▼❛❝❤✐♥❡ ❝♦♥s✐sts ♦❢✿

• Q ✖ ❛ ✜♥✐t❡ s❡t ♦❢ st❛t❡s❀
• Σ ✖ ❛ ✜♥✐t❡ s❡t ♦❢ s②♠❜♦❧s✱ ✐♥❝❧✉❞✐♥❣ ⊔ ❛♥❞ ⊲✳
• s ∈ Q ✖ ❛♥ ✐♥✐t✐❛❧ st❛t❡❀
• δ : (Q × Σ) → (Q ∪ {❛❝❝, r❡❥}) × Σ × {L, R, S}

❆ tr❛♥s✐t✐♦♥ ❢✉♥❝t✐♦♥ t❤❛t s♣❡❝✐✜❡s✱ ❢♦r ❡❛❝❤ st❛t❡ ❛♥❞ s②♠❜♦❧ ❛ ♥❡①t st❛t❡ ✭♦r ❛❝❝❡♣t ❛❝❝ ♦r r❡❥❡❝t r❡❥✮✱ ❛ s②♠❜♦❧ t♦ ♦✈❡r✇r✐t❡ t❤❡ ❝✉rr❡♥t s②♠❜♦❧✱ ❛♥❞ ❛ ❞✐r❡❝t✐♦♥ ❢♦r t❤❡ t❛♣❡ ❤❡❛❞ t♦ ♠♦✈❡ ✭L ✕ ❧❡❢t✱ R ✕ r✐❣❤t✱ ♦r S ✲ st❛t✐♦♥❛r②✮

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♥✜❣✉r❛t✐♦♥s

❆ ❝♦♠♣❧❡t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❛ ♠❛❝❤✐♥❡ ❝❛♥ ❜❡ ❣✐✈❡♥ ✐❢ ✇❡ ❦♥♦✇ ✇❤❛t st❛t❡ ✐t ✐s ✐♥✱ ✇❤❛t ❛r❡ t❤❡ ❝♦♥t❡♥ts ♦❢ ✐ts t❛♣❡✱ ❛♥❞ ✇❤❛t ✐s t❤❡ ♣♦s✐t✐♦♥ ♦❢ ✐ts ❤❡❛❞✳ ❚❤✐s ❝❛♥ ❜❡ s✉♠♠❡❞ ✉♣ ✐♥ ❛ s✐♠♣❧❡ tr✐♣❧❡✿ ❉❡✜♥✐t✐♦♥ ❆ ❝♦♥✜❣✉r❛t✐♦♥ ✐s ❛ tr✐♣❧❡ (q, w, u)✱ ✇❤❡r❡ q ∈ Q ❛♥❞ w, u ∈ Σ⋆ ❚❤❡ ✐♥t✉✐t✐♦♥ ✐s t❤❛t (q, w, u) r❡♣r❡s❡♥ts ❛ ♠❛❝❤✐♥❡ ✐♥ st❛t❡ q ✇✐t❤ t❤❡ str✐♥❣ wu ♦♥ ✐ts t❛♣❡✱ ❛♥❞ t❤❡ ❤❡❛❞ ♣♦✐♥t✐♥❣ ❛t t❤❡ ❧❛st s②♠❜♦❧ ✐♥ w✳ ❚❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❛ ♠❛❝❤✐♥❡ ❝♦♠♣❧❡t❡❧② ❞❡t❡r♠✐♥❡s t❤❡ ❢✉t✉r❡ ❜❡❤❛✈✐♦✉r ♦❢ t❤❡ ♠❛❝❤✐♥❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣✉t❛t✐♦♥s

• ✐✈❡♥ ❛ ♠❛❝❤✐♥❡ M = (Q, Σ, s, δ) ✇❡ s❛② t❤❛t ❛ ❝♦♥✜❣✉r❛t✐♦♥ (q, w, u)

②✐❡❧❞s ✐♥ ♦♥❡ st❡♣ (q′, w ′, u′)✱ ✇r✐tt❡♥ (q, w, u) →M (q′, w ′, u′) ✐❢

• w = va ❀
• δ(q, a) = (q′, b, D)❀ ❛♥❞
• ❡✐t❤❡r D = L ❛♥❞ w ′ = v ❛♥❞ u′ = bu

♦r D = S ❛♥❞ w ′ = vb ❛♥❞ u′ = u ♦r D = R ❛♥❞ w ′ = vbc ❛♥❞ u′ = x✱ ✇❤❡r❡ u = cx✳ ■❢ u ✐s ❡♠♣t②✱ t❤❡♥ w ′ = vb⊔ ❛♥❞ u′ ✐s ❡♠♣t②✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣✉t❛t✐♦♥s

❚❤❡ r❡❧❛t✐♦♥ →⋆

M ✐s t❤❡ r❡✢❡①✐✈❡ ❛♥❞ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ♦❢ →M✳

❆ s❡q✉❡♥❝❡ ♦❢ ❝♦♥✜❣✉r❛t✐♦♥s c✶, . . . , cn✱ ✇❤❡r❡ ❢♦r ❡❛❝❤ i✱ ci →M ci+✶✱ ✐s ❝❛❧❧❡❞ ❛ ❝♦♠♣✉t❛t✐♦♥ ♦❢ M✳ ❚❤❡ ❧❛♥❣✉❛❣❡ L(M) ⊆ Σ⋆ ❛❝❝❡♣t❡❞ ❜② t❤❡ ♠❛❝❤✐♥❡ M ✐s t❤❡ s❡t ♦❢ str✐♥❣s {x | (s, ⊲, x) →⋆

M (❛❝❝, w, u) ❢♦r s♦♠❡ w ❛♥❞ u}

❆ ♠❛❝❤✐♥❡ M ✐s s❛✐❞ t♦ ❤❛❧t ♦♥ ✐♥♣✉t x ✐❢ ❢♦r s♦♠❡ w ❛♥❞ u✱ ❡✐t❤❡r (s, ⊲, x) →⋆

M (❛❝❝, w, u) ♦r (s, ⊲, x) →⋆ M (r❡❥, w, u)

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❉❡❝✐❞❛❜✐❧✐t②

❆ ❧❛♥❣✉❛❣❡ L ⊆ Σ⋆ ✐s r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡ ✐❢ ✐t ✐s L(M) ❢♦r s♦♠❡ M✳ ❆ ❧❛♥❣✉❛❣❡ L ✐s ❞❡❝✐❞❛❜❧❡ ✐❢ ✐t ✐s L(M) ❢♦r s♦♠❡ ♠❛❝❤✐♥❡ M ✇❤✐❝❤ ❤❛❧ts ♦♥ ❡✈❡r② ✐♥♣✉t✳ ❆ ❧❛♥❣✉❛❣❡ L ✐s s❡♠✐✲❞❡❝✐❞❛❜❧❡ ✐❢ ✐t ✐s r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡✳ ❆ ❢✉♥❝t✐♦♥ f : Σ⋆ → Σ⋆ ✐s ❝♦♠♣✉t❛❜❧❡✱ ✐❢ t❤❡r❡ ✐s ❛ ♠❛❝❤✐♥❡ M✱ s✉❝❤ t❤❛t ❢♦r ❛❧❧ x✱ (s, ⊲, x) →⋆

M (❛❝❝, ⊲f (x), ε)

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ ♠❛❝❤✐♥❡ ✇✐t❤ δ ❣✐✈❡♥ ❜②✿ ⊲ ✵ ✶ ⊔ s s, ⊲, R r❡❥, ✵, S r❡❥, ✶, S q, ⊔, R q r❡❥, ⊲, R q, ✶, R q, ✶, R q′, ✵, R q′ r❡❥, ⊲, R r❡❥, ✵, S q′, ✶, L ❛❝❝, ⊔, S ❚❤✐s ♠❛❝❤✐♥❡✱ ✇❤❡♥ st❛rt❡❞ ✐♥ ❝♦♥✜❣✉r❛t✐♦♥ (s, ⊲, ⊔✶n✵) ❡✈❡♥t✉❛❧❧② ❤❛❧ts ✐♥ ❝♦♥✜❣✉r❛t✐♦♥ (❛❝❝, ⊲ ⊔ ✶n+✶✵⊔, ε)✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

▼✉❧t✐✲❚❛♣❡ ▼❛❝❤✐♥❡s

❚❤❡ ❢♦r♠❛❧✐s❛t✐♦♥ ♦❢ ❚✉r✐♥❣ ♠❛❝❤✐♥❡s ❡①t❡♥❞s ✐♥ ❛ ♥❛t✉r❛❧ ✇❛② t♦ ♠✉❧t✐✲t❛♣❡ ♠❛❝❤✐♥❡s✳ ❋♦r ✐♥st❛♥❝❡ ❛ ♠❛❝❤✐♥❡ ✇✐t❤ k t❛♣❡s ✐s s♣❡❝✐✜❡❞ ❜②✿

• Q✱ Σ✱ s❀ ❛♥❞
• δ : (Q × Σk) → Q ∪ {❛❝❝, r❡❥} × (Σ × {L, R, S})k

❙✐♠✐❧❛r❧②✱ ❛ ❝♦♥✜❣✉r❛t✐♦♥ ✐s ♦❢ t❤❡ ❢♦r♠✿ (q, w✶, u✶, . . . , wk, uk)

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘✉♥♥✐♥❣ ❚✐♠❡

❲✐t❤ ❛♥② ❚✉r✐♥❣ ♠❛❝❤✐♥❡ M✱ ✇❡ ❛ss♦❝✐❛t❡ ❛ ❢✉♥❝t✐♦♥ r : ■ ◆ → ■ ◆ ❝❛❧❧❡❞ t❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢ M✳ r(n) ✐s ❞❡✜♥❡❞ t♦ ❜❡ t❤❡ ❧❛r❣❡st ✈❛❧✉❡ R s✉❝❤ t❤❛t t❤❡r❡ ✐s ❛ str✐♥❣ x ♦❢ ❧❡♥❣t❤ n s♦ t❤❛t t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ M st❛rt✐♥❣ ✇✐t❤ ❝♦♥✜❣✉r❛t✐♦♥ (s, ⊲, x) ✐s ♦❢ ❧❡♥❣t❤ R ✭✐✳❡✳ ❤❛s R s✉❝❝❡ss✐✈❡ ❝♦♥✜❣✉r❛t✐♦♥s ✐♥ ✐t✮ ❛♥❞ ❡♥❞s ✇✐t❤ ❛♥ ❛❝❝❡♣t✐♥❣ ❝♦♥✜❣✉r❛t✐♦♥✳ ■♥ s❤♦rt✱ r(n) ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❧♦♥❣❡st ❛❝❝❡♣t✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢ M ♦♥ ❛♥ ✐♥♣✉t ♦❢ ❧❡♥❣t❤ n✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣❧❡①✐t②

❋♦r ❛♥② ❢✉♥❝t✐♦♥ f : ■ ◆ → ■ ◆✱ ✇❡ s❛② t❤❛t ❛ ❧❛♥❣✉❛❣❡ L ✐s ✐♥ ❚■▼❊(f ) ✐❢ t❤❡r❡ ✐s ❛ ♠❛❝❤✐♥❡ M = (Q, Σ, s, δ)✱ s✉❝❤ t❤❛t✿

• L = L(M)❀ ❛♥❞
• ❚❤❡ r✉♥♥✐♥❣ t✐♠❡ ♦❢ M ✐s O(f )✳

❙✐♠✐❧❛r❧②✱ ✇❡ ❞❡✜♥❡ ❙P❆❈❊✭f ✮ t♦ ❜❡ t❤❡ ❧❛♥❣✉❛❣❡s ❛❝❝❡♣t❡❞ ❜② ❛ ♠❛❝❤✐♥❡ ✇❤✐❝❤ ✉s❡s O(f (n)) t❛♣❡ ❝❡❧❧s ♦♥ ✐♥♣✉ts ♦❢ ❧❡♥❣t❤ n✳ ■♥ ❞❡✜♥✐♥❣ s♣❛❝❡ ❝♦♠♣❧❡①✐t②✱ ✇❡ ❛ss✉♠❡ ❛ ♠❛❝❤✐♥❡ M✱ ✇❤✐❝❤ ❤❛s ❛ r❡❛❞✲♦♥❧② ✐♥♣✉t t❛♣❡✱ ❛♥❞ ❛ s❡♣❛r❛t❡ ✇♦r❦ t❛♣❡✳ ❲❡ ♦♥❧② ❝♦✉♥t ❝❡❧❧s ♦♥ t❤❡ ✇♦r❦ t❛♣❡ t♦✇❛r❞s t❤❡ ❝♦♠♣❧❡①✐t②✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❉❡❝✐❞❛❜✐❧✐t② ❛♥❞ ❈♦♠♣❧❡①✐t②

❋♦r ❡✈❡r② ❞❡❝✐❞❛❜❧❡ ❧❛♥❣✉❛❣❡ L✱ t❤❡r❡ ✐s ❛ ❝♦♠♣✉t❛❜❧❡ ❢✉♥❝t✐♦♥ f s✉❝❤ t❤❛t L ∈ ❚■▼❊(f ) ■❢ L ✐s ❛ s❡♠✐✲❞❡❝✐❞❛❜❧❡ ✭❜✉t ♥♦t ❞❡❝✐❞❛❜❧❡✮ ❧❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ ❜② M✱ t❤❡♥ t❤❡r❡ ✐s ♥♦ ❝♦♠♣✉t❛❜❧❡ ❢✉♥❝t✐♦♥ f s✉❝❤ t❤❛t ❡✈❡r② ❛❝❝❡♣t✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢ M✱ ♦♥ ✐♥♣✉t ♦❢ ❧❡♥❣t❤ n ✐s ♦❢ ❧❡♥❣t❤ ❛t ♠♦st f (n)✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆♦♥❞❡t❡r♠✐♥✐s♠

■❢✱ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❚✉r✐♥❣ ♠❛❝❤✐♥❡✱ ✇❡ r❡❧❛① t❤❡ ❝♦♥❞✐t✐♦♥ ♦♥ δ ❜❡✐♥❣ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✐♥st❡❛❞ ❛❧❧♦✇ ❛♥ ❛r❜✐tr❛r② r❡❧❛t✐♦♥✱ ✇❡ ♦❜t❛✐♥ ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡✳ δ ⊆ (Q × Σ) × (Q ∪ {❛❝❝, r❡❥} × Σ × {R, L, S}). ❚❤❡ ②✐❡❧❞s r❡❧❛t✐♦♥ →M ✐s ❛❧s♦ ♥♦ ❧♦♥❣❡r ❢✉♥❝t✐♦♥❛❧✳ ❲❡ st✐❧❧ ❞❡✜♥❡ t❤❡ ❧❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ ❜② M ❜②✿ {x | (s, ⊲, x) →⋆

M (❛❝❝, w, u) ❢♦r s♦♠❡ w ❛♥❞ u}

t❤♦✉❣❤✱ ❢♦r s♦♠❡ x✱ t❤❡r❡ ♠❛② ❜❡ ❝♦♠♣✉t❛t✐♦♥s ❧❡❛❞✐♥❣ t♦ ❛❝❝❡♣t✐♥❣ ❛s ✇❡❧❧ ❛s r❡❥❡❝t✐♥❣ st❛t❡s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣✉t❛t✐♦♥ ❚r❡❡s

❲✐t❤ ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡✱ ❡❛❝❤ ❝♦♥✜❣✉r❛t✐♦♥ ❣✐✈❡s r✐s❡ t♦ ❛ tr❡❡ ♦❢ s✉❝❝❡ss✐✈❡ ❝♦♥✜❣✉r❛t✐♦♥s✳

(s, ⊲, x) (q✵, u✵, w✵) (q✶, u✶, w✶) (q✷, u✷, w✷) (q✵✵, u✵✵, w✵✵) (q✶✶, u✶✶, w✶✶) ✳ ✳ ✳ ✳ ✳ ✳ (r❡❥, u✷, w✷) (❛❝❝, . . .) (q✶✵, u✶✵, w✶✵)

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣❧❡①✐t② ❈❧❛ss❡s

❆ ❝♦♠♣❧❡①✐t② ❝❧❛ss ✐s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❧❛♥❣✉❛❣❡s ❞❡t❡r♠✐♥❡❞ ❜② t❤r❡❡ t❤✐♥❣s✿

• ❆ ♠♦❞❡❧ ♦❢ ❝♦♠♣✉t❛t✐♦♥ ✭s✉❝❤ ❛s ❛ ❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡✱ ♦r

❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❚▼✱ ♦r ❛ ♣❛r❛❧❧❡❧ ❘❛♥❞♦♠ ❆❝❝❡ss ▼❛❝❤✐♥❡✮✳

• ❆ r❡s♦✉r❝❡ ✭s✉❝❤ ❛s t✐♠❡✱ s♣❛❝❡ ♦r ♥✉♠❜❡r ♦❢ ♣r♦❝❡ss♦rs✮✳
• ❆ s❡t ♦❢ ❜♦✉♥❞s✳ ❚❤✐s ✐s ❛ s❡t ♦❢ ❢✉♥❝t✐♦♥s t❤❛t ❛r❡ ✉s❡❞ t♦ ❜♦✉♥❞

t❤❡ ❛♠♦✉♥t ♦❢ r❡s♦✉r❝❡ ✇❡ ❝❛♥ ✉s❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

P♦❧②♥♦♠✐❛❧ ❇♦✉♥❞s

❇② ♠❛❦✐♥❣ t❤❡ ❜♦✉♥❞s ❜r♦❛❞ ❡♥♦✉❣❤✱ ✇❡ ❝❛♥ ♠❛❦❡ ♦✉r ❞❡✜♥✐t✐♦♥s ❢❛✐r❧② ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ♠♦❞❡❧ ♦❢ ❝♦♠♣✉t❛t✐♦♥✳ ❚❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❧❛♥❣✉❛❣❡s r❡❝♦❣♥✐s❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐s t❤❡ s❛♠❡ ✇❤❡t❤❡r ✇❡ ❝♦♥s✐❞❡r ❚✉r✐♥❣ ♠❛❝❤✐♥❡s✱ r❡❣✐st❡r ♠❛❝❤✐♥❡s✱ ♦r ❛♥② ♦t❤❡r ❞❡t❡r♠✐♥✐st✐❝ ♠♦❞❡❧ ♦❢ ❝♦♠♣✉t❛t✐♦♥✳ ❚❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❧❛♥❣✉❛❣❡s r❡❝♦❣♥✐s❡❞ ✐♥ ❧✐♥❡❛r t✐♠❡✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐s ❞✐✛❡r❡♥t ♦♥ ❛ ♦♥❡✲t❛♣❡ ❛♥❞ ❛ t✇♦✲t❛♣❡ ❚✉r✐♥❣ ♠❛❝❤✐♥❡✳ ❲❡ ❝❛♥ s❛② t❤❛t ❜❡✐♥❣ r❡❝♦❣♥✐s❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ✐s ❛ ♣r♦♣❡rt② ♦❢ t❤❡ ❧❛♥❣✉❛❣❡✱ ✇❤✐❧❡ ❜❡✐♥❣ r❡❝♦❣♥✐s❛❜❧❡ ✐♥ ❧✐♥❡❛r t✐♠❡ ✐s s❡♥s✐t✐✈❡ t♦ t❤❡ ♠♦❞❡❧ ♦❢ ❝♦♠♣✉t❛t✐♦♥✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

P♦❧②♥♦♠✐❛❧ ❚✐♠❡

P =

∞

• k=✶

❚■▼❊(nk)

❚❤❡ ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s ❞❡❝✐❞❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❚❤❡ ❝♦♠♣❧❡①✐t② ❝❧❛ss P ♣❧❛②s ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ♦✉r t❤❡♦r②✳

• ■t ✐s r♦❜✉st✱ ❛s ❡①♣❧❛✐♥❡❞✳
• ■t s❡r✈❡s ❛s ♦✉r ❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ✇❤❛t ✐s ❢❡❛s✐❜❧② ❝♦♠♣✉t❛❜❧❡

❖♥❡ ❝♦✉❧❞ ❛r❣✉❡ ✇❤❡t❤❡r ❛♥ ❛❧❣♦r✐t❤♠ r✉♥♥✐♥❣ ✐♥ t✐♠❡ θ(n✶✵✵) ✐s ❢❡❛s✐❜❧❡✱ ❜✉t ✐t ✇✐❧❧ ❡✈❡♥t✉❛❧❧② r✉♥ ❢❛st❡r t❤❛♥ ♦♥❡ t❤❛t t❛❦❡s t✐♠❡ θ(✷n)✳ ▼❛❦✐♥❣ t❤❡ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ r❡s✉❧ts ✐♥ ❛ ✉s❡❢✉❧ ❛♥❞ ❡❧❡❣❛♥t t❤❡♦r②✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊①❛♠♣❧❡✿ ❘❡❛❝❤❛❜✐❧✐t②

❚❤❡ ❘❡❛❝❤❛❜✐❧✐t② ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✐s✱ ❣✐✈❡♥ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ G = (V , E) ❛♥❞ t✇♦ ♥♦❞❡s a, b ∈ V ✱ t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡r❡ ✐s ❛ ♣❛t❤ ❢r♦♠ a t♦ b ✐♥ G✳ ❆ s✐♠♣❧❡ s❡❛r❝❤ ❛❧❣♦r✐t❤♠ ❛s ❢♦❧❧♦✇s s♦❧✈❡s ✐t✿ ✶✳ ♠❛r❦ ♥♦❞❡ a✱ ❧❡❛✈✐♥❣ ♦t❤❡r ♥♦❞❡s ✉♥♠❛r❦❡❞✱ ❛♥❞ ✐♥✐t✐❛❧✐s❡ s❡t S t♦ {a}❀ ✷✳ ✇❤✐❧❡ S ✐s ♥♦t ❡♠♣t②✱ ❝❤♦♦s❡ ♥♦❞❡ i ✐♥ S✿ r❡♠♦✈❡ i ❢r♦♠ S ❛♥❞ ❢♦r ❛❧❧ j s✉❝❤ t❤❛t t❤❡r❡ ✐s ❛♥ ❡❞❣❡ (i, j) ❛♥❞ j ✐s ✉♥♠❛r❦❡❞✱ ♠❛r❦ j ❛♥❞ ❛❞❞ j t♦ S❀ ✸✳ ✐❢ b ✐s ♠❛r❦❡❞✱ ❛❝❝❡♣t ❡❧s❡ r❡❥❡❝t✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❆♥❛❧②s✐s

❚❤✐s ❛❧❣♦r✐t❤♠ r❡q✉✐r❡s O(n✷) t✐♠❡ ❛♥❞ O(n) s♣❛❝❡✳ ❚❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✇♦✉❧❞ ❤❛✈❡ t♦ ❜❡ r❡✜♥❡❞ ❢♦r ❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦♥ ❛ ❚✉r✐♥❣ ♠❛❝❤✐♥❡✱ ❜✉t ✐t ✐s ❡❛s② ❡♥♦✉❣❤ t♦ s❤♦✇ t❤❛t✿ ❘❡❛❝❤❛❜✐❧✐t② ∈ P ❚♦ ❢♦r♠❛❧❧② ❞❡✜♥❡ ❘❡❛❝❤❛❜✐❧✐t② ❛s ❛ ❧❛♥❣✉❛❣❡✱ ✇❡ ✇♦✉❧❞ ❤❛✈❡ t♦ ❛❧s♦ ❝❤♦♦s❡ ❛ ✇❛② ♦❢ r❡♣r❡s❡♥t✐♥❣ t❤❡ ✐♥♣✉t (V , E, a, b) ❛s ❛ str✐♥❣✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊①❛♠♣❧❡✿ ❊✉❝❧✐❞✬s ❆❧❣♦r✐t❤♠

❈♦♥s✐❞❡r t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✭♦r ❧❛♥❣✉❛❣❡✮ ❘❡❧Pr✐♠❡ ❞❡✜♥❡❞ ❜②✿ {(x, y) | ❣❝❞(x, y) = ✶} ❚❤❡ st❛♥❞❛r❞ ❛❧❣♦r✐t❤♠ ❢♦r s♦❧✈✐♥❣ ✐t ✐s ❞✉❡ t♦ ❊✉❝❧✐❞✿ ✶✳ ■♥♣✉t (x, y)✳ ✷✳ ❘❡♣❡❛t ✉♥t✐❧ y = ✵✿ x ← x ♠♦❞ y❀ ❙✇❛♣ x ❛♥❞ y ✸✳ ■❢ x = ✶ t❤❡♥ ❛❝❝❡♣t ❡❧s❡ r❡❥❡❝t✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❆♥❛❧②s✐s

❚❤❡ ♥✉♠❜❡r ♦❢ r❡♣❡t✐t✐♦♥s ❛t st❡♣ ✷ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ✐s ❛t ♠♦st O(❧♦❣ x)✳ ✇❤②❄ ❚❤✐s ✐♠♣❧✐❡s t❤❛t ❘❡❧Pr✐♠❡ ✐s ✐♥ P✳ ■❢ t❤❡ ❛❧❣♦r✐t❤♠ t♦♦❦ θ(x) st❡♣s t♦ t❡r♠✐♥❛t❡✱ ✐t ✇♦✉❧❞ ♥♦t ❜❡ ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✱ ❛s x ✐s ♥♦t ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥♣✉t✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

Pr✐♠❛❧✐t②

❈♦♥s✐❞❡r t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✭♦r ❧❛♥❣✉❛❣❡✮ Pr✐♠❡ ❞❡✜♥❡❞ ❜②✿ {x | x ✐s ♣r✐♠❡} ❚❤❡ ♦❜✈✐♦✉s ❛❧❣♦r✐t❤♠✿ ❋♦r ❛❧❧ y ✇✐t❤ ✶ < y ≤ √x ❝❤❡❝❦ ✇❤❡t❤❡r y|x✳ r❡q✉✐r❡s Ω(√x) st❡♣s ❛♥❞ ✐s t❤❡r❡❢♦r❡ ♥♦t ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥♣✉t✳ ■s Pr✐♠❡ ∈ P❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❇♦♦❧❡❛♥ ❊①♣r❡ss✐♦♥s

❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥s ❛r❡ ❜✉✐❧t ✉♣ ❢r♦♠ ❛♥ ✐♥✜♥✐t❡ s❡t ♦❢ ✈❛r✐❛❜❧❡s X = {x✶, x✷, . . .} ❛♥❞ t❤❡ t✇♦ ❝♦♥st❛♥ts tr✉❡ ❛♥❞ ❢❛❧s❡ ❜② t❤❡ r✉❧❡s✿

• ❛ ❝♦♥st❛♥t ♦r ✈❛r✐❛❜❧❡ ❜② ✐ts❡❧❢ ✐s ❛♥ ❡①♣r❡ss✐♦♥❀
• ✐❢ φ ✐s ❛ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥✱ t❤❡♥ s♦ ✐s (¬φ)❀
• ✐❢ φ ❛♥❞ ψ ❛r❡ ❜♦t❤ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥s✱ t❤❡♥ s♦ ❛r❡ (φ ∧ ψ) ❛♥❞

(φ ∨ ψ)✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊✈❛❧✉❛t✐♦♥

■❢ ❛♥ ❡①♣r❡ss✐♦♥ ❝♦♥t❛✐♥s ♥♦ ✈❛r✐❛❜❧❡s✱ t❤❡♥ ✐t ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ t♦ ❡✐t❤❡r tr✉❡ ♦r ❢❛❧s❡✳ ❖t❤❡r✇✐s❡✱ ✐t ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞✱ ❣✐✈❡♥ ❛ tr✉t❤ ❛ss✐❣♥♠❡♥t t♦ ✐ts ✈❛r✐❛❜❧❡s✳ ❊①❛♠♣❧❡s✿ (tr✉❡ ∨ ❢❛❧s❡) ∧ (¬❢❛❧s❡) (x✶ ∨ ❢❛❧s❡) ∧ ((¬x✶) ∨ x✷) (x✶ ∨ ❢❛❧s❡) ∧ (¬x✶) (x✶ ∨ (¬x✶)) ∧ tr✉❡

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❇♦♦❧❡❛♥ ❊✈❛❧✉❛t✐♦♥

❚❤❡r❡ ✐s ❛ ❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡✱ ✇❤✐❝❤ ❣✐✈❡♥ ❛ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ ✇✐t❤♦✉t ✈❛r✐❛❜❧❡s ♦❢ ❧❡♥❣t❤ n ✇✐❧❧ ❞❡t❡r♠✐♥❡✱ ✐♥ t✐♠❡ O(n✷) ✇❤❡t❤❡r t❤❡ ❡①♣r❡ss✐♦♥ ❡✈❛❧✉❛t❡s t♦ tr✉❡✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✇♦r❦s ❜② s❝❛♥♥✐♥❣ t❤❡ ✐♥♣✉t✱ r❡✇r✐t✐♥❣ ❢♦r♠✉❧❛s ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ r✉❧❡s✿

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘✉❧❡s

• (tr✉❡ ∨ φ) ⇒ tr✉❡
• (φ ∨ tr✉❡) ⇒ tr✉❡
• (❢❛❧s❡ ∨ φ) ⇒ φ
• (❢❛❧s❡ ∧ φ) ⇒ ❢❛❧s❡
• (φ ∧ ❢❛❧s❡) ⇒ ❢❛❧s❡
• (tr✉❡ ∧ φ) ⇒ φ
• (¬tr✉❡) ⇒ ❢❛❧s❡
• (¬❢❛❧s❡) ⇒ tr✉❡

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❆♥❛❧②s✐s

❊❛❝❤ s❝❛♥ ♦❢ t❤❡ ✐♥♣✉t ✭O(n) st❡♣s✮ ♠✉st ✜♥❞ ❛t ❧❡❛st ♦♥❡ s✉❜❡①♣r❡ss✐♦♥ ♠❛t❝❤✐♥❣ ♦♥❡ ♦❢ t❤❡ r✉❧❡ ♣❛tt❡r♥s✳ ❆♣♣❧②✐♥❣ ❛ r✉❧❡ ❛❧✇❛②s ❡❧✐♠✐♥❛t❡s ❛t ❧❡❛st ♦♥❡ s②♠❜♦❧ ❢r♦♠ t❤❡ ❢♦r♠✉❧❛✳ ❚❤✉s✱ t❤❡r❡ ❛r❡ ❛t ♠♦st O(n) s❝❛♥s r❡q✉✐r❡❞✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✇♦r❦s ✐♥ O(n✷) st❡♣s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙❛t✐s✜❛❜✐❧✐t②

❋♦r ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥s φ t❤❛t ❝♦♥t❛✐♥ ✈❛r✐❛❜❧❡s✱ ✇❡ ❝❛♥ ❛s❦ ■s t❤❡r❡ ❛♥ ❛ss✐❣♥♠❡♥t ♦❢ tr✉t❤ ✈❛❧✉❡s t♦ t❤❡ ✈❛r✐❛❜❧❡s ✇❤✐❝❤ ✇♦✉❧❞ ♠❛❦❡ t❤❡ ❢♦r♠✉❧❛ ❡✈❛❧✉❛t❡ t♦ tr✉❡❄ ❚❤❡ s❡t ♦❢ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥s ❢♦r ✇❤✐❝❤ t❤✐s ✐s tr✉❡ ✐s t❤❡ ❧❛♥❣✉❛❣❡ ❙❆❚ ♦❢ s❛t✐s✜❛❜❧❡ ❡①♣r❡ss✐♦♥s✳ ❚❤✐s ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❜② ❛ ❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✐♥ t✐♠❡ O(n✷✷n)✳ ❆♥ ❡①♣r❡ss✐♦♥ ♦❢ ❧❡♥❣t❤ n ❝❛♥ ❝♦♥t❛✐♥ ❛t ♠♦st n ✈❛r✐❛❜❧❡s✳ ❋♦r ❡❛❝❤ ♦❢ t❤❡ ✷n ♣♦ss✐❜❧❡ tr✉t❤ ❛ss✐❣♥♠❡♥ts t♦ t❤❡s❡ ✈❛r✐❛❜❧❡s✱ ✇❡ ❝❤❡❝❦ ✇❤❡t❤❡r ✐t r❡s✉❧ts ✐♥ ❛ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ t❤❛t ❡✈❛❧✉❛t❡s t♦ tr✉❡✳ ■s ❙❆❚ ∈ P❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈✐r❝✉✐ts

❆ ❝✐r❝✉✐t ✐s ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ G = (V , E)✱ ✇✐t❤ V = {✶, . . . , n} t♦❣❡t❤❡r ✇✐t❤ ❛ ❧❛❜❡❧✐♥❣✿ l : V → {tr✉❡, ❢❛❧s❡, ∧, ∨, ¬}✱ s❛t✐s❢②✐♥❣✿

• ■❢ t❤❡r❡ ✐s ❛♥ ❡❞❣❡ (i, j)✱ t❤❡♥ i < j❀
• ❊✈❡r② ♥♦❞❡ ✐♥ V ❤❛s ✐♥❞❡❣r❡❡ ❛t ♠♦st ✷✳
• ❆ ♥♦❞❡ v ❤❛s

✐♥❞❡❣r❡❡ ✵ ✐✛ l(v) ∈ {tr✉❡, ❢❛❧s❡}❀ ✐♥❞❡❣r❡❡ ✶ ✐✛ l(v) = ¬❀ ✐♥❞❡❣r❡❡ ✷ ✐✛ l(v) ∈ {∨, ∧} ❚❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❡①♣r❡ss✐♦♥ ✐s ❣✐✈❡♥ ❜② t❤❡ ✈❛❧✉❡ ❛t ♥♦❞❡ n✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈❱P

❆ ❝✐r❝✉✐t ✐s ❛ ♠♦r❡ ❝♦♠♣❛❝t ✇❛② ♦❢ r❡♣r❡s❡♥t✐♥❣ ❛ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥✳ ■❞❡♥t✐❝❛❧ s✉❜❡①♣r❡ss✐♦♥s ♥❡❡❞ ♥♦t ❜❡ r❡♣❡❛t❡❞✳ ❈❱P ✲ t❤❡ ❝✐r❝✉✐t ✈❛❧✉❡ ♣r♦❜❧❡♠ ✐s✱ ❣✐✈❡♥ ❛ ❝✐r❝✉✐t✱ ❞❡t❡r♠✐♥❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ r❡s✉❧t ♥♦❞❡ n✳ ❈❱P ✐s s♦❧✈❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✱ ❜② t❤❡ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❡①❛♠✐♥❡s t❤❡ ♥♦❞❡s ✐♥ ✐♥❝r❡❛s✐♥❣ ♦r❞❡r✱ ❛ss✐❣♥✐♥❣ ❛ ✈❛❧✉❡ tr✉❡ ♦r ❢❛❧s❡ t♦ ❡❛❝❤ ♥♦❞❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣♦s✐t❡s

❈♦♥s✐❞❡r t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✭♦r ❧❛♥❣✉❛❣❡✮ ❈♦♠♣♦s✐t❡ ❞❡✜♥❡❞ ❜②✿ {x | x ✐s ♥♦t ♣r✐♠❡} ❚❤✐s ✐s t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ t❤❡ ❧❛♥❣✉❛❣❡ Pr✐♠❡✳ ■s ❈♦♠♣♦s✐t❡ ∈ P❄ ❈❧❡❛r❧②✱ t❤❡ ❛♥s✇❡r ✐s ②❡s ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ Pr✐♠❡ ∈ P✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❍❛♠✐❧t♦♥✐❛♥ ●r❛♣❤s

• ✐✈❡♥ ❛ ❣r❛♣❤ G = (V , E)✱ ❛ ❍❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡ ✐♥ G ✐s ❛ ♣❛t❤ ✐♥ t❤❡

❣r❛♣❤✱ st❛rt✐♥❣ ❛♥❞ ❡♥❞✐♥❣ ❛t t❤❡ s❛♠❡ ♥♦❞❡✱ s✉❝❤ t❤❛t ❡✈❡r② ♥♦❞❡ ✐♥ V ❛♣♣❡❛rs ♦♥ t❤❡ ❝②❝❧❡ ❡①❛❝t❧② ♦♥❝❡✳ ❆ ❣r❛♣❤ ✐s ❝❛❧❧❡❞ ❍❛♠✐❧t♦♥✐❛♥ ✐❢ ✐t ❝♦♥t❛✐♥s ❛ ❍❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡✳ ❚❤❡ ❧❛♥❣✉❛❣❡ ❍❆▼ ✐s t❤❡ s❡t ♦❢ ❡♥❝♦❞✐♥❣s ♦❢ ❍❛♠✐❧t♦♥✐❛♥ ❣r❛♣❤s✳ ■s ❍❆▼ ∈ P❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊①❛♠♣❧❡s

❚❤❡ ✜rst ♦❢ t❤❡s❡ ❣r❛♣❤s ✐s ♥♦t ❍❛♠✐❧t♦♥✐❛♥✱ ❜✉t t❤❡ s❡❝♦♥❞ ♦♥❡ ✐s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

• r❛♣❤ ■s♦♠♦r♣❤✐s♠
• ✐✈❡♥ t✇♦ ❣r❛♣❤s G✶ = (V✶, E✶) ❛♥❞ G✷ = (V✷, E✷)✱ ✐s t❤❡r❡ ❛ ❜✐❥❡❝t✐♦♥

ι : V✶ → V✷ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② u, v ∈ V✶✱ (u, v) ∈ E✶ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ (ι(u), ι(v)) ∈ E✷. ■s ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ∈ P❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

P♦❧②♥♦♠✐❛❧ ❱❡r✐✜❝❛t✐♦♥

❚❤❡ ♣r♦❜❧❡♠s ❈♦♠♣♦s✐t❡✱ ❙❆❚✱ ❍❆▼ ❛♥❞ ●r❛♣❤ ■s♦♠♦r♣❤✐s♠ ❤❛✈❡ s♦♠❡t❤✐♥❣ ✐♥ ❝♦♠♠♦♥✳ ■♥ ❡❛❝❤ ❝❛s❡✱ t❤❡r❡ ✐s ❛ s❡❛r❝❤ s♣❛❝❡ ♦❢ ♣♦ss✐❜❧❡ s♦❧✉t✐♦♥s✳ t❤❡ ♥✉♠❜❡rs ❧❡ss t❤❛♥ x❀ tr✉t❤ ❛ss✐❣♥♠❡♥ts t♦ t❤❡ ✈❛r✐❛❜❧❡s ♦❢ φ❀ ❧✐sts ♦❢ t❤❡ ✈❡rt✐❝❡s ♦❢ G❀ ❛ ❜✐❥❡❝t✐♦♥ ❜❡t✇❡❡♥ V✶ ❛♥❞ V✷✳ ❚❤❡ s✐③❡ ♦❢ t❤❡ s❡❛r❝❤ ✐s ❡①♣♦♥❡♥t✐❛❧ ✐♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ✐♥♣✉t✳

• ✐✈❡♥ ❛ ♣♦t❡♥t✐❛❧ s♦❧✉t✐♦♥ ✐♥ t❤❡ s❡❛r❝❤ s♣❛❝❡✱ ✐t ✐s ❡❛s② t♦ ❝❤❡❝❦ ✇❤❡t❤❡r

♦r ♥♦t ✐t ✐s ❛ s♦❧✉t✐♦♥✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❱❡r✐✜❡rs

❆ ✈❡r✐✜❡r V ❢♦r ❛ ❧❛♥❣✉❛❣❡ L ✐s ❛♥ ❛❧❣♦r✐t❤♠ s✉❝❤ t❤❛t L = {x | (x, c) ✐s ❛❝❝❡♣t❡❞ ❜② V ❢♦r s♦♠❡ c} ■❢ V r✉♥s ✐♥ t✐♠❡ ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ ❧❡♥❣t❤ ♦❢ x✱ t❤❡♥ ✇❡ s❛② t❤❛t L ✐s ♣♦❧②♥♦♠✐❛❧❧② ✈❡r✐✜❛❜❧❡✳ ▼❛♥② ♥❛t✉r❛❧ ❡①❛♠♣❧❡s ❛r✐s❡✱ ✇❤❡♥❡✈❡r ✇❡ ❤❛✈❡ t♦ ❝♦♥str✉❝t ❛ s♦❧✉t✐♦♥ t♦ s♦♠❡ ❞❡s✐❣♥ ❝♦♥str❛✐♥ts ♦r s♣❡❝✐✜❝❛t✐♦♥s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆♦♥❞❡t❡r♠✐♥✐s♠

■❢✱ ✐♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❚✉r✐♥❣ ♠❛❝❤✐♥❡✱ ✇❡ r❡❧❛① t❤❡ ❝♦♥❞✐t✐♦♥ ♦♥ δ ❜❡✐♥❣ ❛ ❢✉♥❝t✐♦♥ ❛♥❞ ✐♥st❡❛❞ ❛❧❧♦✇ ❛♥ ❛r❜✐tr❛r② r❡❧❛t✐♦♥✱ ✇❡ ♦❜t❛✐♥ ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡✳ δ ⊆ (Q × Σ) × (Q ∪ {❛❝❝, r❡❥} × Σ × {R, L, S}). ❚❤❡ ②✐❡❧❞s r❡❧❛t✐♦♥ →M ✐s ❛❧s♦ ♥♦ ❧♦♥❣❡r ❢✉♥❝t✐♦♥❛❧✳ ❲❡ st✐❧❧ ❞❡✜♥❡ t❤❡ ❧❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ ❜② M ❜②✿ {x | (s, ⊲, x) →⋆

M (❛❝❝, w, u) ❢♦r s♦♠❡ w ❛♥❞ u}

t❤♦✉❣❤✱ ❢♦r s♦♠❡ x✱ t❤❡r❡ ♠❛② ❜❡ ❝♦♠♣✉t❛t✐♦♥s ❧❡❛❞✐♥❣ t♦ ❛❝❝❡♣t✐♥❣ ❛s ✇❡❧❧ ❛s r❡❥❡❝t✐♥❣ st❛t❡s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣✉t❛t✐♦♥ ❚r❡❡s

❲✐t❤ ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡✱ ❡❛❝❤ ❝♦♥✜❣✉r❛t✐♦♥ ❣✐✈❡s r✐s❡ t♦ ❛ tr❡❡ ♦❢ s✉❝❝❡ss✐✈❡ ❝♦♥✜❣✉r❛t✐♦♥s✳

(s, ⊲, x) (q✵, u✵, w✵) (q✶, u✶, w✶) (q✷, u✷, w✷) (q✵✵, u✵✵, w✵✵) (q✶✶, u✶✶, w✶✶) ✳ ✳ ✳ ✳ ✳ ✳ (r❡❥, u✷, w✷) (❛❝❝, . . .) (q✶✵, u✶✵, w✶✵)

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆♦♥❞❡t❡r♠✐♥✐st✐❝ ❈♦♠♣❧❡①✐t② ❈❧❛ss❡s

❲❡ ❤❛✈❡ ❛❧r❡❛❞② ❞❡✜♥❡❞ ❚■▼❊(f ) ❛♥❞ ❙P❆❈❊(f )✳ ◆❚■▼❊(f ) ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❝❧❛ss ♦❢ t❤♦s❡ ❧❛♥❣✉❛❣❡s L ✇❤✐❝❤ ❛r❡ ❛❝❝❡♣t❡❞ ❜② ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ M✱ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x ∈ L✱ t❤❡r❡ ✐s ❛♥ ❛❝❝❡♣t✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢ M ♦♥ x ♦❢ ❧❡♥❣t❤ ❛t ♠♦st f (n)✱ ✇❤❡r❡ n ✐s t❤❡ ❧❡♥❣t❤ ♦❢ x✳ ◆P =

∞

• k=✶

◆❚■▼❊(nk)

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆♦♥❞❡t❡r♠✐♥✐s♠

(s, ⊲, x) (q✵, u✵, w✵) (q✶, u✶, w✶) (q✷, u✷, w✷) (q✵✵, u✵✵, w✵✵) (q✶✶, u✶✶, w✶✶) ✳ ✳ ✳ ✳ ✳ ✳ (r❡❥, u✷, w✷) (❛❝❝, . . .) (q✶✵, u✶✵, w✶✵)

❋♦r ❛ ❧❛♥❣✉❛❣❡ ✐♥ ◆❚■▼❊(f )✱ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ tr❡❡ ❝❛♥ ❜❡ ❜♦✉♥❞❡❞ ❜② f (n) ✇❤❡♥ t❤❡ ✐♥♣✉t ✐s ♦❢ ❧❡♥❣t❤ n✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆P

❆ ❧❛♥❣✉❛❣❡ L ✐s ♣♦❧②♥♦♠✐❛❧❧② ✈❡r✐✜❛❜❧❡ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ✐t ✐s ✐♥ ◆P✳ ❚♦ ♣r♦✈❡ t❤✐s✱ s✉♣♣♦s❡ L ✐s ❛ ❧❛♥❣✉❛❣❡✱ ✇❤✐❝❤ ❤❛s ❛ ✈❡r✐✜❡r V ✱ ✇❤✐❝❤ r✉♥s ✐♥ t✐♠❡ p(n)✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡s❝r✐❜❡s ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ t❤❛t ❛❝❝❡♣ts L ✶✳ ✐♥♣✉t x ♦❢ ❧❡♥❣t❤ n ✷✳ ♥♦♥❞❡t❡r♠✐♥✐st✐❝❛❧❧② ❣✉❡ss c ♦❢ ❧❡♥❣t❤ ≤ p(n) ✸✳ r✉♥ V ♦♥ (x, c)

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆P

■♥ t❤❡ ♦t❤❡r ❞✐r❡❝t✐♦♥✱ s✉♣♣♦s❡ M ✐s ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡ t❤❛t ❛❝❝❡♣ts ❛ ❧❛♥❣✉❛❣❡ L ✐♥ t✐♠❡ nk✳ ❲❡ ❞❡✜♥❡ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ V ✇❤✐❝❤ ♦♥ ✐♥♣✉t (x, c) s✐♠✉❧❛t❡s M ♦♥ ✐♥♣✉t x✳ ❆t t❤❡ it❤ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡ ♣♦✐♥t✱ V ❧♦♦❦s ❛t t❤❡ it❤ ❝❤❛r❛❝t❡r ✐♥ c t♦ ❞❡❝✐❞❡ ✇❤✐❝❤ ❜r❛♥❝❤ t♦ ❢♦❧❧♦✇✳ ■❢ M ❛❝❝❡♣ts t❤❡♥ V ❛❝❝❡♣ts✱ ♦t❤❡r✇✐s❡ ✐t r❡❥❡❝ts✳ V ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ✈❡r✐✜❡r ❢♦r L✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

• ❡♥❡r❛t❡ ❛♥❞ ❚❡st

❲❡ ❝❛♥ t❤✐♥❦ ♦❢ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠s ✐♥ t❤❡ ❣❡♥❡r❛t❡✲❛♥❞ t❡st ♣❛r❛❞✐❣♠✿

②❡s ♥♦ ❣❡♥❡r❛t❡ x Vx ✈❡r✐❢②

❲❤❡r❡ t❤❡ ❣❡♥❡r❛t❡ ❝♦♠♣♦♥❡♥t ✐s ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❛♥❞ t❤❡ ✈❡r✐❢② ❝♦♠♣♦♥❡♥t ✐s ❞❡t❡r♠✐♥✐st✐❝✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡❞✉❝t✐♦♥s

• ✐✈❡♥ t✇♦ ❧❛♥❣✉❛❣❡s L✶ ⊆ Σ⋆

✶✱ ❛♥❞ L✷ ⊆ Σ⋆ ✷✱

❆ r❡❞✉❝t✐♦♥ ♦❢ L✶ t♦ L✷ ✐s ❛ ❝♦♠♣✉t❛❜❧❡ ❢✉♥❝t✐♦♥ f : Σ⋆

✶ → Σ⋆ ✷

s✉❝❤ t❤❛t ❢♦r ❡✈❡r② str✐♥❣ x ∈ Σ⋆

✶✱

f (x) ∈ L✷ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ x ∈ L✶

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡s♦✉r❝❡ ❇♦✉♥❞❡❞ ❘❡❞✉❝t✐♦♥s

■❢ f ✐s ❝♦♠♣✉t❛❜❧❡ ❜② ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠✱ ✇❡ s❛② t❤❛t L✶ ✐s ♣♦❧②♥♦♠✐❛❧ t✐♠❡ r❡❞✉❝✐❜❧❡ t♦ L✷✳ L✶ ≤P L✷ ■❢ f ✐s ❛❧s♦ ❝♦♠♣✉t❛❜❧❡ ✐♥ ❙P❆❈❊(❧♦❣ n)✱ ✇❡ ✇r✐t❡ L✶ ≤L L✷

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡❞✉❝t✐♦♥s ✷

■❢ L✶ ≤P L✷ ✇❡ ✉♥❞❡rst❛♥❞ t❤❛t L✶ ✐s ♥♦ ♠♦r❡ ❞✐✣❝✉❧t t♦ s♦❧✈❡ t❤❛♥ L✷✱ ❛t ❧❡❛st ❛s ❢❛r ❛s ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ❝♦♥❝❡r♥❡❞✳ ❚❤❛t ✐s t♦ s❛②✱ ■❢ L✶ ≤P L✷ ❛♥❞ L✷ ∈ P✱ t❤❡♥ L✶ ∈ P ❲❡ ❝❛♥ ❣❡t ❛♥ ❛❧❣♦r✐t❤♠ t♦ ❞❡❝✐❞❡ L✶ ❜② ✜rst ❝♦♠♣✉t✐♥❣ f ✱ ❛♥❞ t❤❡♥ ✉s✐♥❣ t❤❡ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r L✷✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣❧❡t❡♥❡ss

❚❤❡ ✉s❡❢✉❧♥❡ss ♦❢ r❡❞✉❝t✐♦♥s ✐s t❤❛t t❤❡② ❛❧❧♦✇ ✉s t♦ ❡st❛❜❧✐s❤ t❤❡ r❡❧❛t✐✈❡ ❝♦♠♣❧❡①✐t② ♦❢ ♣r♦❜❧❡♠s✱ ❡✈❡♥ ✇❤❡♥ ✇❡ ❝❛♥♥♦t ♣r♦✈❡ ❛❜s♦❧✉t❡ ❧♦✇❡r ❜♦✉♥❞s✳ ❈♦♦❦ ✭✶✾✼✷✮ ✜rst s❤♦✇❡❞ t❤❛t t❤❡r❡ ❛r❡ ♣r♦❜❧❡♠s ✐♥ ◆P t❤❛t ❛r❡ ♠❛①✐♠❛❧❧② ❞✐✣❝✉❧t✳ ❆ ❧❛♥❣✉❛❣❡ L ✐s s❛✐❞ t♦ ❜❡ ◆P✲❤❛r❞ ✐❢ ❢♦r ❡✈❡r② ❧❛♥❣✉❛❣❡ A ∈ ◆P✱ A ≤P L✳ ❆ ❧❛♥❣✉❛❣❡ L ✐s ◆P✲❝♦♠♣❧❡t❡ ✐❢ ✐t ✐s ✐♥ ◆P ❛♥❞ ✐t ✐s ◆P✲❤❛r❞✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙❆❚ ✐s ◆P✲❝♦♠♣❧❡t❡

❈♦♦❦ s❤♦✇❡❞ t❤❛t t❤❡ ❧❛♥❣✉❛❣❡ ❙❆❚ ♦❢ s❛t✐s✜❛❜❧❡ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥s ✐s ◆P✲❝♦♠♣❧❡t❡✳ ❚♦ ❡st❛❜❧✐s❤ t❤✐s✱ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t ❢♦r ❡✈❡r② ❧❛♥❣✉❛❣❡ L ✐♥ ◆P✱ t❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ r❡❞✉❝t✐♦♥ ❢r♦♠ L t♦ ❙❆❚✳ ❙✐♥❝❡ L ✐s ✐♥ ◆P✱ t❤❡r❡ ✐s ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ M = (Q, Σ, s, δ) ❛♥❞ ❛ ❜♦✉♥❞ k s✉❝❤ t❤❛t ❛ str✐♥❣ x ♦❢ ❧❡♥❣t❤ n ✐s ✐♥ L ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ✐t ✐s ❛❝❝❡♣t❡❞ ❜② M ✇✐t❤✐♥ nk st❡♣s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❇♦♦❧❡❛♥ ❋♦r♠✉❧❛

❲❡ ♥❡❡❞ t♦ ❣✐✈❡✱ ❢♦r ❡❛❝❤ x ∈ Σ⋆✱ ❛ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ f (x) ✇❤✐❝❤ ✐s s❛t✐s✜❛❜❧❡ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ t❤❡r❡ ✐s ❛♥ ❛❝❝❡♣t✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢ M ♦♥ ✐♥♣✉t x✳ f (x) ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛r✐❛❜❧❡s✿ Si,q ❢♦r ❡❛❝❤ i ≤ nk ❛♥❞ q ∈ Q Ti,j,σ ❢♦r ❡❛❝❤ i, j ≤ nk ❛♥❞ σ ∈ Σ Hi,j ❢♦r ❡❛❝❤ i, j ≤ nk

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

■♥t✉✐t✐✈❡❧②✱ t❤❡s❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥t❡♥❞❡❞ t♦ ♠❡❛♥✿

• Si,q ✕ t❤❡ st❛t❡ ♦❢ t❤❡ ♠❛❝❤✐♥❡ ❛t t✐♠❡ i ✐s q✳
• Ti,j,σ ✕ ❛t t✐♠❡ i✱ t❤❡ s②♠❜♦❧ ❛t ♣♦s✐t✐♦♥ j ♦❢ t❤❡ t❛♣❡ ✐s σ✳
• Hi,j ✕ ❛t t✐♠❡ i✱ t❤❡ t❛♣❡ ❤❡❛❞ ✐s ♣♦✐♥t✐♥❣ ❛t t❛♣❡ ❝❡❧❧ j✳

❲❡ ♥♦✇ ❤❛✈❡ t♦ s❡❡ ❤♦✇ t♦ ✇r✐t❡ t❤❡ ❢♦r♠✉❧❛ f (x)✱ s♦ t❤❛t ✐t ❡♥❢♦r❝❡s t❤❡s❡ ♠❡❛♥✐♥❣s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

■♥✐t✐❛❧ st❛t❡ ✐s s ❛♥❞ t❤❡ ❤❡❛❞ ✐s ✐♥✐t✐❛❧❧② ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ t❛♣❡✳ S✶,s ∧ H✶,✶ ❚❤❡ ❤❡❛❞ ✐s ♥❡✈❡r ✐♥ t✇♦ ♣❧❛❝❡s ❛t ♦♥❝❡

• i
• j

(Hi,j →

• j′=j

(¬Hi,j′)) ❚❤❡ ♠❛❝❤✐♥❡ ✐s ♥❡✈❡r ✐♥ t✇♦ st❛t❡s ❛t ♦♥❝❡

• q
• i

(Si,q →

• q′=q

(¬Si,q′)) ❊❛❝❤ t❛♣❡ ❝❡❧❧ ❝♦♥t❛✐♥s ♦♥❧② ♦♥❡ s②♠❜♦❧

• i
• j
• σ

(Ti,j,σ →

• σ′=σ

(¬Ti,j,σ′))

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❚❤❡ ✐♥✐t✐❛❧ t❛♣❡ ❝♦♥t❡♥ts ❛r❡ x

• j≤n

T✶,j,xj ∧

• n<j

T✶,j,⊔ ❚❤❡ t❛♣❡ ❞♦❡s ♥♦t ❝❤❛♥❣❡ ❡①❝❡♣t ✉♥❞❡r t❤❡ ❤❡❛❞

• i
• j
• j′=j
• σ

(Hi,j ∧ Ti,j′,σ) → Ti+✶,j′,σ ❊❛❝❤ st❡♣ ✐s ❛❝❝♦r❞✐♥❣ t♦ δ✳

• i
• j
• σ
• q

(Hi,j ∧ Si,q ∧ Ti,j,σ) →

• ∆

(Hi+✶,j′ ∧ Si+✶,q′ ∧ Ti+✶,j,σ′)

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

✇❤❡r❡ ∆ ✐s t❤❡ s❡t ♦❢ ❛❧❧ tr✐♣❧❡s (q′, σ′, D) s✉❝❤ t❤❛t ((q, σ), (q′, σ′, D)) ∈ δ ❛♥❞ j′ =    j ✐❢ D = S j − ✶ ✐❢ D = L j + ✶ ✐❢ D = R ❋✐♥❛❧❧②✱ t❤❡ ❛❝❝❡♣t✐♥❣ st❛t❡ ✐s r❡❛❝❤❡❞

• i

Si,❛❝❝

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈◆❋

❆ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ ✐s ✐♥ ❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠ ✐❢ ✐t ✐s t❤❡ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ ❛ s❡t ♦❢ ❝❧❛✉s❡s✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ✐s t❤❡ ❞✐s❥✉♥❝t✐♦♥ ♦❢ ❛ s❡t ♦❢ ❧✐t❡r❛❧s✱ ❡❛❝❤ ♦❢ t❤❡s❡ ❜❡✐♥❣ ❡✐t❤❡r ❛ ✈❛r✐❛❜❧❡ ♦r t❤❡ ♥❡❣❛t✐♦♥ ♦❢ ❛ ✈❛r✐❛❜❧❡✳ ❋♦r ❛♥② ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ φ✱ t❤❡r❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥t ❡①♣r❡ss✐♦♥ ψ ✐♥ ❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠✳ ψ ❝❛♥ ❜❡ ❡①♣♦♥❡♥t✐❛❧❧② ❧♦♥❣❡r t❤❛♥ φ✳ ❍♦✇❡✈❡r✱ ❈◆❋✲❙❆❚✱ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ s❛t✐s✜❛❜❧❡ ❈◆❋ ❡①♣r❡ss✐♦♥s✱ ✐s ◆P✲❝♦♠♣❧❡t❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

✸❙❆❚

❆ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ ✐s ✐♥ ✸❈◆❋ ✐❢ ✐t ✐s ✐♥ ❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠ ❛♥❞ ❡❛❝❤ ❝❧❛✉s❡ ❝♦♥t❛✐♥s ❛t ♠♦st ✸ ❧✐t❡r❛❧s✳ ✸❙❆❚ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❧❛♥❣✉❛❣❡ ❝♦♥s✐st✐♥❣ ♦❢ t❤♦s❡ ❡①♣r❡ss✐♦♥s ✐♥ ✸❈◆❋ t❤❛t ❛r❡ s❛t✐s✜❛❜❧❡✳ ✸❙❆❚ ✐s ◆P✲❝♦♠♣❧❡t❡✱ ❛s t❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ r❡❞✉❝t✐♦♥ ❢r♦♠ ❈◆❋✲❙❆❚ t♦ ✸❙❆❚✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣♦s✐♥❣ ❘❡❞✉❝t✐♦♥s

P♦❧②♥♦♠✐❛❧ t✐♠❡ r❡❞✉❝t✐♦♥s ❛r❡ ❝❧❡❛r❧② ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣♦s✐t✐♦♥✳ ❙♦✱ ✐❢ L✶ ≤P L✷ ❛♥❞ L✷ ≤P L✸✱ t❤❡♥ ✇❡ ❛❧s♦ ❤❛✈❡ L✶ ≤P L✸✳ ■❢ ✇❡ s❤♦✇✱ ❢♦r s♦♠❡ ♣r♦❜❧❡♠ A ✐♥ ◆P t❤❛t ❙❆❚ ≤P A ♦r ✸❙❆❚ ≤P A ✐t ❢♦❧❧♦✇s t❤❛t A ✐s ❛❧s♦ ◆P✲❝♦♠♣❧❡t❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

■♥❞❡♣❡♥❞❡♥t ❙❡t

• ✐✈❡♥ ❛ ❣r❛♣❤ G = (V , E)✱ ❛ s✉❜s❡t X ⊆ V ♦❢ t❤❡ ✈❡rt✐❝❡s ✐s s❛✐❞ t♦ ❜❡

❛♥ ✐♥❞❡♣❡♥❞❡♥t s❡t✱ ✐❢ t❤❡r❡ ❛r❡ ♥♦ ❡❞❣❡s (u, v) ❢♦r u, v ∈ X✳ ❚❤❡ ♥❛t✉r❛❧ ❛❧❣♦r✐t❤♠✐❝ ♣r♦❜❧❡♠ ✐s✱ ❣✐✈❡♥ ❛ ❣r❛♣❤✱ ✜♥❞ t❤❡ ❧❛r❣❡st ✐♥❞❡♣❡♥❞❡♥t s❡t✳ ❚♦ t✉r♥ t❤✐s ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠ ✐♥t♦ ❛ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠✱ ✇❡ ❞❡✜♥❡ ■◆❉ ❛s✿ ❚❤❡ s❡t ♦❢ ♣❛✐rs (G, K)✱ ✇❤❡r❡ G ✐s ❛ ❣r❛♣❤✱ ❛♥❞ K ✐s ❛♥ ✐♥t❡❣❡r✱ s✉❝❤ t❤❛t G ❝♦♥t❛✐♥s ❛♥ ✐♥❞❡♣❡♥❞❡♥t s❡t ✇✐t❤ K ♦r ♠♦r❡ ✈❡rt✐❝❡s✳ ■◆❉ ✐s ❝❧❡❛r❧② ✐♥ ◆P✳ ❲❡ ♥♦✇ s❤♦✇ ✐t ✐s ◆P✲❝♦♠♣❧❡t❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡❞✉❝t✐♦♥

❲❡ ❝❛♥ ❝♦♥str✉❝t ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ ✸❙❆❚ t♦ ■◆❉✳ ❆ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ φ ✐♥ ✸❈◆❋ ✇✐t❤ m ❝❧❛✉s❡s ✐s ♠❛♣♣❡❞ ❜② t❤❡ r❡❞✉❝t✐♦♥ t♦ t❤❡ ♣❛✐r (G, m)✱ ✇❤❡r❡ G ✐s t❤❡ ❣r❛♣❤ ♦❜t❛✐♥❡❞ ❢r♦♠ φ ❛s ❢♦❧❧♦✇s✿ G ❝♦♥t❛✐♥s m tr✐❛♥❣❧❡s✱ ♦♥❡ ❢♦r ❡❛❝❤ ❝❧❛✉s❡ ♦❢ φ✱ ✇✐t❤ ❡❛❝❤ ♥♦❞❡ r❡♣r❡s❡♥t✐♥❣ ♦♥❡ ♦❢ t❤❡ ❧✐t❡r❛❧s ✐♥ t❤❡ ❝❧❛✉s❡✳ ❆❞❞✐t✐♦♥❛❧❧②✱ t❤❡r❡ ✐s ❛♥ ❡❞❣❡ ❜❡t✇❡❡♥ t✇♦ ♥♦❞❡s ✐♥ ❞✐✛❡r❡♥t tr✐❛♥❣❧❡s ✐❢ t❤❡② r❡♣r❡s❡♥t ❧✐t❡r❛❧s ✇❤❡r❡ ♦♥❡ ✐s t❤❡ ♥❡❣❛t✐♦♥ ♦❢ t❤❡ ♦t❤❡r✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊①❛♠♣❧❡

(x✶ ∨ x✷ ∨ ¬x✸) ∧ (x✸ ∨ ¬x✷ ∨ ¬x✶)

x✶ x✷ ¬x✸ ¬x✶ ¬x✷ x✸

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈❧✐q✉❡

• ✐✈❡♥ ❛ ❣r❛♣❤ G = (V , E)✱ ❛ s✉❜s❡t X ⊆ V ♦❢ t❤❡ ✈❡rt✐❝❡s ✐s ❝❛❧❧❡❞ ❛

❝❧✐q✉❡✱ ✐❢ ❢♦r ❡✈❡r② u, v ∈ X✱ (u, v) ✐s ❛♥ ❡❞❣❡✳ ❆s ✇✐t❤ ■◆❉✱ ✇❡ ❝❛♥ ❞❡✜♥❡ ❛ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠✿ ❈▲■◗❯❊ ✐s ❞❡✜♥❡❞ ❛s✿ ❚❤❡ s❡t ♦❢ ♣❛✐rs (G, K)✱ ✇❤❡r❡ G ✐s ❛ ❣r❛♣❤✱ ❛♥❞ K ✐s ❛♥ ✐♥t❡❣❡r✱ s✉❝❤ t❤❛t G ❝♦♥t❛✐♥s ❛ ❝❧✐q✉❡ ✇✐t❤ K ♦r ♠♦r❡ ✈❡rt✐❝❡s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈❧✐q✉❡ ✷

❈▲■◗❯❊ ✐s ✐♥ ◆P ❜② t❤❡ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❣✉❡ss❡s ❛ ❝❧✐q✉❡ ❛♥❞ t❤❡♥ ✈❡r✐✜❡s ✐t✳ ❈▲■◗❯❊ ✐s ◆P✲❝♦♠♣❧❡t❡✱ s✐♥❝❡ ■◆❉ ≤P ❈▲■◗❯❊ ❜② t❤❡ r❡❞✉❝t✐♦♥ t❤❛t ♠❛♣s t❤❡ ♣❛✐r (G, K) t♦ ( ¯ G, K)✱ ✇❤❡r❡ ¯ G ✐s t❤❡ ❝♦♠♣❧❡♠❡♥t ❣r❛♣❤ ♦❢ G✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

k✲❈♦❧♦✉r❛❜✐❧✐t②

❆ ❣r❛♣❤ G = (V , E) ✐s k✲❝♦❧♦✉r❛❜❧❡✱ ✐❢ t❤❡r❡ ✐s ❛ ❢✉♥❝t✐♦♥ χ : V → {✶, . . . , k} s✉❝❤ t❤❛t✱ ❢♦r ❡❛❝❤ u, v ∈ V ✱ ✐❢ (u, v) ∈ E✱ χ(u) = χ(v) ❚❤✐s ❣✐✈❡s r✐s❡ t♦ ❛ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❡❛❝❤ k✳ ✷✲❝♦❧♦✉r❛❜✐❧✐t② ✐s ✐♥ P✳ ❋♦r ❛❧❧ k > ✷✱ k✲❝♦❧♦✉r❛❜✐❧✐t② ✐s ◆P✲❝♦♠♣❧❡t❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

✸✲❈♦❧♦✉r❛❜✐❧✐t②

✸✲❈♦❧♦✉r❛❜✐❧✐t② ✐s ✐♥ ◆P✱ ❛s ✇❡ ❝❛♥ ❣✉❡ss ❛ ❝♦❧♦✉r✐♥❣ ❛♥❞ ✈❡r✐❢② ✐t✳ ❚♦ s❤♦✇ ◆P✲❝♦♠♣❧❡t❡♥❡ss✱ ✇❡ ❝❛♥ ❝♦♥str✉❝t ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ ✸❙❆❚ t♦ ✸✲❈♦❧♦✉r❛❜✐❧✐t②✳ ❋♦r ❡❛❝❤ ✈❛r✐❛❜❧❡ x✱ ✇❡ ❤❛✈❡ t✇♦ ✈❡rt✐❝❡s x✱ ¯ x ✇❤✐❝❤ ❛r❡ ❝♦♥♥❡❝t❡❞ ✐♥ ❛ tr✐❛♥❣❧❡ ✇✐t❤ t❤❡ ✈❡rt❡① a ✭❝♦♠♠♦♥ t♦ ❛❧❧ ✈❛r✐❛❜❧❡s✮✳ ■♥ ❛❞❞✐t✐♦♥✱ ❢♦r ❡❛❝❤ ❝❧❛✉s❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ ❧✐t❡r❛❧s l✶✱ l✷ ❛♥❞ l✸ ✇❡ ❤❛✈❡ ❛ ❣❛❞❣❡t✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

• ❛❞❣❡t

l✶ l✷ l✸ b

❲✐t❤ ❛ ❢✉rt❤❡r ❡❞❣❡ ❢r♦♠ a t♦ b✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❍❛♠✐❧t♦♥✐❛♥ ●r❛♣❤s

❘❡❝❛❧❧ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❍❆▼✖t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ❍❛♠✐❧t♦♥✐❛♥ ❣r❛♣❤s✳

• ✐✈❡♥ ❛ ❣r❛♣❤ G = (V , E)✱ ❛ ❍❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡ ✐♥ G ✐s ❛ ♣❛t❤ ✐♥ t❤❡

❣r❛♣❤✱ st❛rt✐♥❣ ❛♥❞ ❡♥❞✐♥❣ ❛t t❤❡ s❛♠❡ ♥♦❞❡✱ s✉❝❤ t❤❛t ❡✈❡r② ♥♦❞❡ ✐♥ V ❛♣♣❡❛rs ♦♥ t❤❡ ❝②❝❧❡ ❡①❛❝t❧② ♦♥❝❡✳ ❆ ❣r❛♣❤ ✐s ❝❛❧❧❡❞ ❍❛♠✐❧t♦♥✐❛♥ ✐❢ ✐t ❝♦♥t❛✐♥s ❛ ❍❛♠✐❧t♦♥✐❛♥ ❝②❝❧❡✳ ❚❤❡ ❧❛♥❣✉❛❣❡ ❍❆▼ ✐s t❤❡ s❡t ♦❢ ❡♥❝♦❞✐♥❣s ♦❢ ❍❛♠✐❧t♦♥✐❛♥ ❣r❛♣❤s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❍❛♠✐❧t♦♥✐❛♥ ❈②❝❧❡

❲❡ ❝❛♥ ❝♦♥str✉❝t ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ ✸❙❆❚ t♦ ❍❆▼ ❊ss❡♥t✐❛❧❧②✱ t❤✐s ✐♥✈♦❧✈❡s ❝♦❞✐♥❣ ✉♣ ❛ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ ❛s ❛ ❣r❛♣❤✱ s♦ t❤❛t ❡✈❡r② s❛t✐s❢②✐♥❣ tr✉t❤ ❛ss✐❣♥♠❡♥t t♦ t❤❡ ❡①♣r❡ss✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❍❛♠✐❧t♦♥✐❛♥ ❝✐r❝✉✐t ♦❢ t❤❡ ❣r❛♣❤✳ ❚❤✐s r❡❞✉❝t✐♦♥ ✐s ♠✉❝❤ ♠♦r❡ ✐♥tr✐❝❛t❡ t❤❛♥ t❤❡ ♦♥❡ ❢♦r ■◆❉✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❚r❛✈❡❧❧✐♥❣ ❙❛❧❡s♠❛♥

❘❡❝❛❧❧ t❤❡ tr❛✈❡❧❧✐♥❣ s❛❧❡s♠❛♥ ♣r♦❜❧❡♠

• ✐✈❡♥
• V ✖ ❛ s❡t ♦❢ ♥♦❞❡s✳
• c : V × V → ■

◆ ✖ ❛ ❝♦st ♠❛tr✐①✳ ❋✐♥❞ ❛♥ ♦r❞❡r✐♥❣ v✶, . . . , vn ♦❢ V ❢♦r ✇❤✐❝❤ t❤❡ t♦t❛❧ ❝♦st✿ c(vn, v✶) +

n−✶

• i=✶

c(vi, vi+✶) ✐s t❤❡ s♠❛❧❧❡st ♣♦ss✐❜❧❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❚r❛✈❡❧❧✐♥❣ ❙❛❧❡s♠❛♥

❆s ✇✐t❤ ♦t❤❡r ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠s✱ ✇❡ ❝❛♥ ♠❛❦❡ ❛ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✈❡rs✐♦♥ ♦❢ t❤❡ ❚r❛✈❡❧❧✐♥❣ ❙❛❧❡s♠❛♥ ♣r♦❜❧❡♠✳ ❚❤❡ ♣r♦❜❧❡♠ ❚❙P ❝♦♥s✐sts ♦❢ t❤❡ s❡t ♦❢ tr✐♣❧❡s (V , c : V × V → ■ ◆, t) s✉❝❤ t❤❛t t❤❡r❡ ✐s ❛ t♦✉r ♦❢ t❤❡ s❡t ♦❢ ✈❡rt✐❝❡s V ✱ ✇❤✐❝❤ ✉♥❞❡r t❤❡ ❝♦st ♠❛tr✐① c✱ ❤❛s ❝♦st t ♦r ❧❡ss✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡❞✉❝t✐♦♥

❚❤❡r❡ ✐s ❛ s✐♠♣❧❡ r❡❞✉❝t✐♦♥ ❢r♦♠ ❍❆▼ t♦ ❚❙P✱ ♠❛♣♣✐♥❣ ❛ ❣r❛♣❤ (V , E) t♦ t❤❡ tr✐♣❧❡ (V , c : V × V → ■ ◆, n)✱ ✇❤❡r❡ c(u, v) =

• ✶

✐❢ (u, v) ∈ E ✷

• therwise

❛♥❞ n ✐s t❤❡ s✐③❡ ♦❢ V ✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙❡ts✱ ◆✉♠❜❡rs ❛♥❞ ❙❝❤❡❞✉❧✐♥❣

■t ✐s ♥♦t ❥✉st ♣r♦❜❧❡♠s ❛❜♦✉t ❢♦r♠✉❧❛s ❛♥❞ ❣r❛♣❤s t❤❛t t✉r♥ ♦✉t t♦ ❜❡ ◆P✲❝♦♠♣❧❡t❡✳ ▲✐t❡r❛❧❧② ❤✉♥❞r❡❞s ♦❢ ♥❛t✉r❛❧❧② ❛r✐s✐♥❣ ♣r♦❜❧❡♠s ❤❛✈❡ ❜❡❡♥ ♣r♦✈❡❞ ◆P✲❝♦♠♣❧❡t❡✱ ✐♥ ❛r❡❛s ✐♥✈♦❧✈✐♥❣ ♥❡t✇♦r❦ ❞❡s✐❣♥✱ s❝❤❡❞✉❧✐♥❣✱ ♦♣t✐♠✐s❛t✐♦♥✱ ❞❛t❛ st♦r❛❣❡ ❛♥❞ r❡tr✐❡✈❛❧✱ ❛rt✐✜❝✐❛❧ ✐♥t❡❧❧✐❣❡♥❝❡ ❛♥❞ ♠❛♥② ♦t❤❡rs✳ ❙✉❝❤ ♣r♦❜❧❡♠s ❛r✐s❡ ♥❛t✉r❛❧❧② ✇❤❡♥❡✈❡r ✇❡ ❤❛✈❡ t♦ ❝♦♥str✉❝t ❛ s♦❧✉t✐♦♥ ✇✐t❤✐♥ ❝♦♥str❛✐♥ts✱ ❛♥❞ t❤❡ ♠♦st ❡✛❡❝t✐✈❡ ✇❛② ❛♣♣❡❛rs t♦ ❜❡ ❛♥ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ♦❢ ❛♥ ❡①♣♦♥❡♥t✐❛❧ s♦❧✉t✐♦♥ s♣❛❝❡✳ ❲❡ ♥♦✇ ❡①❛♠✐♥❡ t❤r❡❡ ♠♦r❡ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s✱ ✇❤♦s❡ s✐❣♥✐✜❝❛♥❝❡ ❧✐❡s ✐♥ t❤❛t t❤❡② ❤❛✈❡ ❜❡❡♥ ✉s❡❞ t♦ ♣r♦✈❡ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ♦t❤❡r ♣r♦❜❧❡♠s ◆P✲❝♦♠♣❧❡t❡✱ t❤r♦✉❣❤ r❡❞✉❝t✐♦♥s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

✸❉ ▼❛t❝❤✐♥❣

❚❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ♦❢ ✸❉ ▼❛t❝❤✐♥❣ ✐s ❞❡✜♥❡❞ ❛s✿

• ✐✈❡♥ t❤r❡❡ ❞✐s❥♦✐♥t s❡ts X✱ Y ❛♥❞ Z✱ ❛♥❞ ❛ s❡t ♦❢ tr✐♣❧❡s

M ⊆ X × Y × Z✱ ❞♦❡s M ❝♦♥t❛✐♥ ❛ ♠❛t❝❤✐♥❣❄ ■✳❡✳ ✐s t❤❡r❡ ❛ s✉❜s❡t M′ ⊆ M✱ s✉❝❤ t❤❛t ❡❛❝❤ ❡❧❡♠❡♥t ♦❢ X✱ Y ❛♥❞ Z ❛♣♣❡❛rs ✐♥ ❡①❛❝t❧② ♦♥❡ tr✐♣❧❡ ♦❢ M′❄ ❲❡ ❝❛♥ s❤♦✇ t❤❛t ✸❉▼ ✐s ◆P✲❝♦♠♣❧❡t❡ ❜② ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ ✸❙❆❚✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡❞✉❝t✐♦♥

■❢ ❛ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ φ ✐♥ ✸❈◆❋ ❤❛s n ✈❛r✐❛❜❧❡s✱ ❛♥❞ m ❝❧❛✉s❡s✱ ✇❡ ❝♦♥str✉❝t ❢♦r ❡❛❝❤ ✈❛r✐❛❜❧❡ v t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❛❞❣❡t✳

zv✶ zv✷ zv✸ zv✹ xv✶ yv✶ ¯ zv✶ ¯ zv✷ yv✷ xv✷ yv✸ xv✸ yv✹ xv✹ ¯ zv✸ ¯ zv✹

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

■♥ ❛❞❞✐t✐♦♥✱ ❢♦r ❡✈❡r② ❝❧❛✉s❡ c✱ ✇❡ ❤❛✈❡ t✇♦ ❡❧❡♠❡♥ts xc ❛♥❞ yc✳ ■❢ t❤❡ ❧✐t❡r❛❧ v ♦❝❝✉rs ✐♥ c✱ ✇❡ ✐♥❝❧✉❞❡ t❤❡ tr✐♣❧❡ (xc, yc, zvc) ✐♥ M✳ ❙✐♠✐❧❛r❧②✱ ✐❢ ¬v ♦❝❝✉rs ✐♥ c✱ ✇❡ ✐♥❝❧✉❞❡ t❤❡ tr✐♣❧❡ (xc, yc, ¯ zvc) ✐♥ M✳ ❋✐♥❛❧❧②✱ ✇❡ ✐♥❝❧✉❞❡ ❡①tr❛ ❞✉♠♠② ❡❧❡♠❡♥ts ✐♥ X ❛♥❞ Y t♦ ♠❛❦❡ t❤❡ ♥✉♠❜❡rs ♠❛t❝❤ ✉♣✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊①❛❝t ❙❡t ❈♦✈❡r✐♥❣

❚✇♦ ♦t❤❡r ✇❡❧❧ ❦♥♦✇♥ ♣r♦❜❧❡♠s ❛r❡ ♣r♦✈❡❞ ◆P✲❝♦♠♣❧❡t❡ ❜② ✐♠♠❡❞✐❛t❡ r❡❞✉❝t✐♦♥ ❢r♦♠ ✸❉▼✳ ❊①❛❝t ❈♦✈❡r ❜② ✸✲❙❡ts ✐s ❞❡✜♥❡❞ ❜②✿

• ✐✈❡♥ ❛ s❡t U ✇✐t❤ ✸n ❡❧❡♠❡♥ts✱ ❛♥❞ ❛ ❝♦❧❧❡❝t✐♦♥

S = {S✶, . . . , Sm} ♦❢ t❤r❡❡✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ U✱ ✐s t❤❡r❡ ❛ s✉❜✲❝♦❧❧❡❝t✐♦♥ ❝♦♥t❛✐♥✐♥❣ ❡①❛❝t❧② n ♦❢ t❤❡s❡ s❡ts ✇❤♦s❡ ✉♥✐♦♥ ✐s ❛❧❧ ♦❢ U❄ ❚❤❡ r❡❞✉❝t✐♦♥ ❢r♦♠ ✸❉▼ s✐♠♣❧② t❛❦❡s U = X ∪ Y ∪ Z✱ ❛♥❞ S t♦ ❜❡ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ t❤r❡❡✲❡❧❡♠❡♥t s✉❜s❡ts r❡s✉❧t✐♥❣ ❢r♦♠ M✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙❡t ❈♦✈❡r✐♥❣

▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❡ ❤❛✈❡ t❤❡ ❙❡t ❈♦✈❡r✐♥❣ ♣r♦❜❧❡♠✿

• ✐✈❡♥ ❛ s❡t U✱ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ S = {S✶, . . . , Sm} s✉❜s❡ts ♦❢ U

❛♥❞ ❛♥ ✐♥t❡❣❡r ❜✉❞❣❡t B✱ ✐s t❤❡r❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ B s❡ts ✐♥ S ✇❤♦s❡ ✉♥✐♦♥ ✐s U❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❑♥❛♣s❛❝❦

❑◆❆P❙❆❈❑ ✐s ❛ ♣r♦❜❧❡♠ ✇❤✐❝❤ ❣❡♥❡r❛❧✐s❡s ♠❛♥② ♥❛t✉r❛❧ s❝❤❡❞✉❧✐♥❣ ❛♥❞ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠s✱ ❛♥❞ t❤r♦✉❣❤ r❡❞✉❝t✐♦♥s ❤❛s ❜❡❡♥ ✉s❡❞ t♦ s❤♦✇ ♠❛♥② s✉❝❤ ♣r♦❜❧❡♠s ◆P✲❝♦♠♣❧❡t❡✳ ■♥ t❤❡ ♣r♦❜❧❡♠✱ ✇❡ ❛r❡ ❣✐✈❡♥ n ✐t❡♠s✱ ❡❛❝❤ ✇✐t❤ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r ✈❛❧✉❡ vi ❛♥❞ ✇❡✐❣❤t wi✳ ❲❡ ❛r❡ ❛❧s♦ ❣✐✈❡♥ ❛ ♠❛①✐♠✉♠ t♦t❛❧ ✇❡✐❣❤t W ✱ ❛♥❞ ❛ ♠✐♥✐♠✉♠ t♦t❛❧ ✈❛❧✉❡ V ✳ ❈❛♥ ✇❡ s❡❧❡❝t ❛ s✉❜s❡t ♦❢ t❤❡ ✐t❡♠s ✇❤♦s❡ t♦t❛❧ ✇❡✐❣❤t ❞♦❡s ♥♦t ❡①❝❡❡❞ W ✱ ❛♥❞ ✇❤♦s❡ t♦t❛❧ ✈❛❧✉❡ ❡①❝❡❡❞s V ❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡❞✉❝t✐♦♥

❚❤❡ ♣r♦♦❢ t❤❛t ❑◆❆P❙❆❈❑ ✐s ◆P✲❝♦♠♣❧❡t❡ ✐s ❜② ❛ r❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❊①❛❝t ❈♦✈❡r ❜② ✸✲❙❡ts✳

• ✐✈❡♥ ❛ s❡t U = {✶, . . . , ✸n} ❛♥❞ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✸✲❡❧❡♠❡♥t s✉❜s❡ts ♦❢ U✱

S = {S✶, . . . , Sm}✳ ❲❡ ♠❛♣ t❤✐s t♦ ❛♥ ✐♥st❛♥❝❡ ♦❢ ❑◆❆P❙❆❈❑ ✇✐t❤ m ❡❧❡♠❡♥ts ❡❛❝❤ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♦♥❡ ♦❢ t❤❡ Si✱ ❛♥❞ ❤❛✈✐♥❣ ✇❡✐❣❤t ❛♥❞ ✈❛❧✉❡

• j∈Si

(m + ✶)j−✶ ❛♥❞ s❡t t❤❡ t❛r❣❡t ✇❡✐❣❤t ❛♥❞ ✈❛❧✉❡ ❜♦t❤ t♦

✸n−✶

• j=✵

(m + ✶)j

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙❝❤❡❞✉❧✐♥❣

❙♦♠❡ ❡①❛♠♣❧❡s ♦❢ t❤❡ ❦✐♥❞s ♦❢ s❝❤❡❞✉❧✐♥❣ t❛s❦s t❤❛t ❤❛✈❡ ❜❡❡♥ ♣r♦✈❡❞ ◆P✲❝♦♠♣❧❡t❡ ✐♥❝❧✉❞❡✿ ❚✐♠❡t❛❜❧❡ ❉❡s✐❣♥

• ✐✈❡♥ ❛ s❡t H ♦❢ ✇♦r❦ ♣❡r✐♦❞s✱ ❛ s❡t W ♦❢ ✇♦r❦❡rs ❡❛❝❤ ✇✐t❤ ❛♥

❛ss♦❝✐❛t❡❞ s✉❜s❡t ♦❢ H ✭❛✈❛✐❧❛❜❧❡ ♣❡r✐♦❞s✮✱ ❛ s❡t T ♦❢ t❛s❦s ❛♥❞ ❛♥ ❛ss✐❣♥♠❡♥t r : W × T → ■ ◆ ♦❢ r❡q✉✐r❡❞ ✇♦r❦✱ ✐s t❤❡r❡ ❛ ♠❛♣♣✐♥❣ f : W × T × H → {✵, ✶} ✇❤✐❝❤ ❝♦♠♣❧❡t❡s ❛❧❧ t❛s❦s❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙❝❤❡❞✉❧✐♥❣

❙❡q✉❡♥❝✐♥❣ ✇✐t❤ ❉❡❛❞❧✐♥❡s

• ✐✈❡♥ ❛ s❡t T ♦❢ t❛s❦s ❛♥❞ ❢♦r ❡❛❝❤ t❛s❦ ❛ ❧❡♥❣t❤ l ∈ ■

◆✱ ❛ r❡❧❡❛s❡ t✐♠❡ r ∈ ■ ◆ ❛♥❞ ❛ ❞❡❛❞❧✐♥❡ d ∈ ■ ◆✱ ✐s t❤❡r❡ ❛ ✇♦r❦ s❝❤❡❞✉❧❡ ✇❤✐❝❤ ❝♦♠♣❧❡t❡s ❡❛❝❤ t❛s❦ ❜❡t✇❡❡♥ ✐ts r❡❧❡❛s❡ t✐♠❡ ❛♥❞ ✐ts ❞❡❛❞❧✐♥❡❄ ❏♦❜ ❙❝❤❡❞✉❧✐♥❣

• ✐✈❡♥ ❛ s❡t T ♦❢ t❛s❦s✱ ❛ ♥✉♠❜❡r m ∈ ■

◆ ♦❢ ♣r♦❝❡ss♦rs ❛ ❧❡♥❣t❤ l ∈ ■ ◆ ❢♦r ❡❛❝❤ t❛s❦✱ ❛♥❞ ❛♥ ♦✈❡r❛❧❧ ❞❡❛❞❧✐♥❡ D ∈ ■ ◆✱ ✐s t❤❡r❡ ❛ ♠✉❧t✐✲♣r♦❝❡ss♦r s❝❤❡❞✉❧❡ ✇❤✐❝❤ ❝♦♠♣❧❡t❡s ❛❧❧ t❛s❦s ❜② t❤❡ ❞❡❛❞❧✐♥❡❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡s♣♦♥s❡s t♦ ◆P✲❈♦♠♣❧❡t❡♥❡ss

❈♦♥❢r♦♥t❡❞ ❜② ❛♥ ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠✱ s❛② ❝♦♥str✉❝t✐♥❣ ❛ t✐♠❡t❛❜❧❡✱ ✇❤❛t ❝❛♥ ♦♥❡ ❞♦❄

• ■t✬s ❛ s✐♥❣❧❡ ✐♥st❛♥❝❡✱ ❞♦❡s ❛s②♠♣t♦t✐❝ ❝♦♠♣❧❡①✐t② ♠❛tt❡r❄
• ❲❤❛t✬s t❤❡ ❝r✐t✐❝❛❧ s✐③❡❄ ■s s❝❛❧❛❜✐❧✐t② ✐♠♣♦rt❛♥t❄
• ❆r❡ t❤❡r❡ ❣✉❛r❛♥t❡❡❞ r❡str✐❝t✐♦♥s ♦♥ t❤❡ ✐♥♣✉t❄ ❲✐❧❧ ❛ s♣❡❝✐❛❧

♣✉r♣♦s❡ ❛❧❣♦r✐t❤♠ s✉✣❝❡❄

• ❲✐❧❧ ❛♥ ❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ s✉✣❝❡❄ ❆r❡ ♣❡r❢♦r♠❛♥❝❡ ❣✉❛r❛♥t❡❡s

r❡q✉✐r❡❞❄

• ❆r❡ t❤❡r❡ ✉s❡❢✉❧ ❤❡✉r✐st✐❝s t❤❛t ❝❛♥ ❝♦♥str❛✐♥ ❛ s❡❛r❝❤❄ ❲❛②s ♦❢

♦r❞❡r✐♥❣ ❝❤♦✐❝❡s t♦ ❝♦♥tr♦❧ ❜❛❝❦tr❛❝❦✐♥❣❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❱❛❧✐❞✐t②

❲❡ ❞❡✜♥❡ ❱❆▲✖t❤❡ s❡t ♦❢ ✈❛❧✐❞ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥s✖t♦ ❜❡ t❤♦s❡ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥s ❢♦r ✇❤✐❝❤ ❡✈❡r② ❛ss✐❣♥♠❡♥t ♦❢ tr✉t❤ ✈❛❧✉❡s t♦ ✈❛r✐❛❜❧❡s ②✐❡❧❞s ❛♥ ❡①♣r❡ss✐♦♥ ❡q✉✐✈❛❧❡♥t t♦ tr✉❡✳ φ ∈ ❱❆▲ ⇔ ¬φ ∈ ❙❆❚ ❇② ❛♥ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ❛❧❣♦r✐t❤♠ s✐♠✐❧❛r t♦ t❤❡ ♦♥❡ ❢♦r ❙❆❚✱ ❱❆▲ ✐s ✐♥ ❚■▼❊(n✷✷n)✳ ■s ❱❆▲ ∈ ◆P❄

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❱❛❧✐❞✐t②

❱❆▲ = {φ | φ ∈ ❱❆▲}✖t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ ❱❆▲ ✐s ✐♥ ◆P✳

• ✉❡ss ❛ ❢❛❧s✐❢②✐♥❣ tr✉t❤ ❛ss✐❣♥♠❡♥t ❛♥❞ ✈❡r✐❢② ✐t✳

❙✉❝❤ ❛♥ ❛❧❣♦r✐t❤♠ ❞♦❡s ♥♦t ✇♦r❦ ❢♦r ❱❆▲✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡ ❤❛✈❡ t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r ❡✈❡r② tr✉t❤ ❛ss✐❣♥♠❡♥t r❡s✉❧ts ✐♥ tr✉❡✖❛ r❡q✉✐r❡♠❡♥t t❤❛t ❞♦❡s ♥♦t s✐t ❛s ✇❡❧❧ ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛❝❝❡♣t❛♥❝❡ ❜② ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣❧❡♠❡♥t❛t✐♦♥

■❢ ✇❡ ✐♥t❡r❝❤❛♥❣❡ ❛❝❝❡♣t✐♥❣ ❛♥❞ r❡❥❡❝t✐♥❣ st❛t❡s ✐♥ ❛ ❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡ t❤❛t ❛❝❝❡♣ts t❤❡ ❧❛♥❣✉❛❣❡ L✱ ✇❡ ❣❡t ♦♥❡ t❤❛t ❛❝❝❡♣ts L✳ ■❢ ❛ ❧❛♥❣✉❛❣❡ L ∈ P✱ t❤❡♥ ❛❧s♦ L ∈ P✳ ❈♦♠♣❧❡①✐t② ❝❧❛ss❡s ❞❡✜♥❡❞ ✐♥ t❡r♠s ♦❢ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡ ♠♦❞❡❧s ❛r❡ ♥♦t ♥❡❝❡ss❛r✐❧② ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❧❛♥❣✉❛❣❡s✳ ❉❡✜♥❡✱ ❝♦✲◆P ✕ t❤❡ ❧❛♥❣✉❛❣❡s ✇❤♦s❡ ❝♦♠♣❧❡♠❡♥ts ❛r❡ ✐♥ ◆P✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙✉❝❝✐♥❝t ❈❡rt✐✜❝❛t❡s

❚❤❡ ❝♦♠♣❧❡①✐t② ❝❧❛ss ◆P ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐s❡❞ ❛s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❧❛♥❣✉❛❣❡s ♦❢ t❤❡ ❢♦r♠✿ L = {x | ∃yR(x, y)} ❲❤❡r❡ R ✐s ❛ r❡❧❛t✐♦♥ ♦♥ str✐♥❣s s❛t✐s❢②✐♥❣ t✇♦ ❦❡② ❝♦♥❞✐t✐♦♥s ✶✳ R ✐s ❞❡❝✐❞❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ✷✳ R ✐s ♣♦❧②♥♦♠✐❛❧❧② ❜❛❧❛♥❝❡❞✳ ❚❤❛t ✐s✱ t❤❡r❡ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ p s✉❝❤ t❤❛t ✐❢ R(x, y) ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ x ✐s n✱ t❤❡♥ t❤❡ ❧❡♥❣t❤ ♦❢ y ✐s ♥♦ ♠♦r❡ t❤❛♥ p(n)✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙✉❝❝✐♥❝t ❈❡rt✐✜❝❛t❡s

y ✐s ❛ ❝❡rt✐✜❝❛t❡ ❢♦r t❤❡ ♠❡♠❜❡rs❤✐♣ ♦❢ x ✐♥ L✳ ❊①❛♠♣❧❡✿ ■❢ L ✐s ❙❆❚✱ t❤❡♥ ❢♦r ❛ s❛t✐s✜❛❜❧❡ ❡①♣r❡ss✐♦♥ x✱ ❛ ❝❡rt✐✜❝❛t❡ ✇♦✉❧❞ ❜❡ ❛ s❛t✐s❢②✐♥❣ tr✉t❤ ❛ss✐❣♥♠❡♥t✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❝♦✲◆P

❆s ❝♦✲◆P ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❝♦♠♣❧❡♠❡♥ts ♦❢ ❧❛♥❣✉❛❣❡s ✐♥ ◆P✱ ❛♥❞ P ✐s ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥✱ ❝♦✲◆P ❝❛♥ ❛❧s♦ ❜❡ ❝❤❛r❛❝t❡r✐s❡❞ ❛s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❧❛♥❣✉❛❣❡s ♦❢ t❤❡ ❢♦r♠✿ L = {x | ∀y |y| < p(|x|) → R′(x, y)} ◆P ✕ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❧❛♥❣✉❛❣❡s ✇✐t❤ s✉❝❝✐♥❝t ❝❡rt✐✜❝❛t❡s ♦❢ ♠❡♠❜❡rs❤✐♣✳ ❝♦✲◆P ✕ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❧❛♥❣✉❛❣❡s ✇✐t❤ s✉❝❝✐♥❝t ❝❡rt✐✜❝❛t❡s ♦❢ ❞✐sq✉❛❧✐✜❝❛t✐♦♥✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆P ❝♦✲◆P P ❆♥② ♦❢ t❤❡ s✐t✉❛t✐♦♥s ✐s ❝♦♥s✐st❡♥t ✇✐t❤ ♦✉r ♣r❡s❡♥t st❛t❡ ♦❢ ❦♥♦✇❧❡❞❣❡✿

• P = ◆P = ❝♦✲◆P
• P = ◆P ∩ ❝♦✲◆P = ◆P = ❝♦✲◆P
• P = ◆P ∩ ❝♦✲◆P = ◆P = ❝♦✲◆P
• P = ◆P ∩ ❝♦✲◆P = ◆P = ❝♦✲◆P

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❝♦✲◆P✲❝♦♠♣❧❡t❡

❱❆▲ ✕ t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥s t❤❛t ❛r❡ ✈❛❧✐❞ ✐s ❝♦✲◆P✲❝♦♠♣❧❡t❡✳ ❆♥② ❧❛♥❣✉❛❣❡ L t❤❛t ✐s t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ ❛♥ ◆P✲❝♦♠♣❧❡t❡ ❧❛♥❣✉❛❣❡ ✐s ❝♦✲◆P✲❝♦♠♣❧❡t❡✳ ❆♥② r❡❞✉❝t✐♦♥ ♦❢ ❛ ❧❛♥❣✉❛❣❡ L✶ t♦ L✷ ✐s ❛❧s♦ ❛ r❡❞✉❝t✐♦♥ ♦❢ ¯ L✶✕t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ L✶✕t♦ ¯ L✷✕t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ L✷✳ ❚❤❡r❡ ✐s ❛♥ ❡❛s② r❡❞✉❝t✐♦♥ ❢r♦♠ t❤❡ ❝♦♠♣❧❡♠❡♥t ♦❢ ❙❆❚ t♦ ❱❆▲✱ ♥❛♠❡❧② t❤❡ ♠❛♣ t❤❛t t❛❦❡s ❛♥ ❡①♣r❡ss✐♦♥ t♦ ✐ts ♥❡❣❛t✐♦♥✳ ❱❆▲ ∈ P ⇒ P = ◆P = ❝♦✲◆P ❱❆▲ ∈ ◆P ⇒ ◆P = ❝♦✲◆P

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

Pr✐♠❡ ◆✉♠❜❡rs

❈♦♥s✐❞❡r t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ P❘■▼❊✿

• ✐✈❡♥ ❛ ♥✉♠❜❡r x✱ ✐s ✐t ♣r✐♠❡❄

❚❤✐s ♣r♦❜❧❡♠ ✐s ✐♥ ❝♦✲◆P✳ ∀y(y < x → (y = ✶ ∨ ¬(❞✐✈(y, x)))) ◆♦t❡ ❛❣❛✐♥✱ t❤❡ ❛❧❣♦r✐t❤♠ t❤❛t ❝❤❡❝❦s ❢♦r ❛❧❧ ♥✉♠❜❡rs ✉♣ t♦ √n ✇❤❡t❤❡r ❛♥② ♦❢ t❤❡♠ ❞✐✈✐❞❡s n✱ ✐s ♥♦t ♣♦❧②♥♦♠✐❛❧✱ ❛s √n ✐s ♥♦t ♣♦❧②♥♦♠✐❛❧ ✐♥ t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥♣✉t str✐♥❣✱ ✇❤✐❝❤ ✐s ❧♦❣ n✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

Pr✐♠❛❧✐t②

❆♥♦t❤❡r ✇❛② ♦❢ ♣✉tt✐♥❣ t❤✐s ✐s t❤❛t ❈♦♠♣♦s✐t❡ ✐s ✐♥ ◆P✳ Pr❛tt ✭✶✾✼✻✮ s❤♦✇❡❞ t❤❛t P❘■▼❊ ✐s ✐♥ ◆P✱ ❜② ❡①❤✐❜✐t✐♥❣ s✉❝❝✐♥❝t ❝❡rt✐✜❝❛t❡s ♦❢ ♣r✐♠❛❧✐t② ❜❛s❡❞ ♦♥✿ ❆ ♥✉♠❜❡r p > ✷ ✐s ♣r✐♠❡ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ t❤❡r❡ ✐s ❛ ♥✉♠❜❡r r✱ ✶ < r < p✱ s✉❝❤ t❤❛t r p−✶ = ✶ ♠♦❞ p ❛♥❞ r

p−✶ q

= ✶ ♠♦❞ p ❢♦r ❛❧❧ ♣r✐♠❡ ❞✐✈✐s♦rs q ♦❢ p − ✶✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

Pr✐♠❛❧✐t②

■♥ ✷✵✵✷✱ ❆❣r❛✇❛❧✱ ❑❛②❛❧ ❛♥❞ ❙❛①❡♥❛ s❤♦✇❡❞ t❤❛t P❘■▼❊ ✐s ✐♥ P✳ ■❢ a ✐s ❝♦✲♣r✐♠❡ t♦ p✱ (x − a)p ≡ (xp − a) (♠♦❞ p) ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ p ✐s ❛ ♣r✐♠❡✳ ❈❤❡❝❦✐♥❣ t❤✐s ❡q✉✐✈❛❧❡♥❝❡ ✇♦✉❧❞ t❛❦❡ t♦ ❧♦♥❣✳ ■♥st❡❛❞✱ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ✐s ❝❤❡❝❦❡❞ ♠♦❞✉❧♦ ❛ ♣♦❧②♥♦♠✐❛❧ xr − ✶✱ ❢♦r ✏s✉✐t❛❜❧❡✑ r✳ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ s✉✐t❛❜❧❡ s♠❛❧❧ r r❡❧✐❡s ♦♥ ❞❡❡♣ r❡s✉❧ts ✐♥ ♥✉♠❜❡r t❤❡♦r②✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❋❛❝t♦rs

❈♦♥s✐❞❡r t❤❡ ❧❛♥❣✉❛❣❡ ❋❛❝t♦r {(x, k) | x ❤❛s ❛ ❢❛❝t♦r y ✇✐t❤ ✶ < y < k} ❋❛❝t♦r ∈ ◆P ∩ ❝♦✲◆P ❈❡rt✐✜❝❛t❡ ♦❢ ♠❡♠❜❡rs❤✐♣✖❛ ❢❛❝t♦r ♦❢ x ❧❡ss t❤❛♥ k✳ ❈❡rt✐✜❝❛t❡ ♦❢ ❞✐sq✉❛❧✐✜❝❛t✐♦♥✖t❤❡ ♣r✐♠❡ ❢❛❝t♦r✐s❛t✐♦♥ ♦❢ x✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

• r❛♣❤ ■s♦♠♦r♣❤✐s♠
• ✐✈❡♥ t✇♦ ❣r❛♣❤s G✶ = (V✶, E✶) ❛♥❞ G✷ = (V✷, E✷)✱ ✐s t❤❡r❡ ❛ ❜✐❥❡❝t✐♦♥

ι : V✶ → V✷ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② u, v ∈ V✶✱ (u, v) ∈ E✶ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ (ι(u), ι(v)) ∈ E✷.

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

• r❛♣❤ ■s♦♠♦r♣❤✐s♠
• r❛♣❤ ■s♦♠♦r♣❤✐s♠ ✐s
• ✐♥ ◆P
• ♥♦t ❦♥♦✇♥ t♦ ❜❡ ✐♥ P
• ♥♦t ❦♥♦✇♥ t♦ ❜❡ ✐♥ ❝♦✲◆P
• ♥♦t ❦♥♦✇♥ ✭♦r ❡①♣❡❝t❡❞✮ t♦ ❜❡ ◆P✲❝♦♠♣❧❡t❡
• r❡❝❡♥t❧② s❤♦✇♥ t♦ ❜❡ ✐♥ q✉❛s✐✲♣♦❧②♥♦♠✐❛❧ t✐♠❡✱ ✐✳❡✳ ✐♥

❚■▼❊(n(❧♦❣ n)k) ❢♦r ❛ ❝♦♥st❛♥t k✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❖♣t✐♠✐s❛t✐♦♥

❚❤❡ ❚r❛✈❡❧❧✐♥❣ ❙❛❧❡s♠❛♥ Pr♦❜❧❡♠ ✇❛s ♦r✐❣✐♥❛❧❧② ❝♦♥❝❡✐✈❡❞ ♦❢ ❛s ❛♥ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠ t♦ ✜♥❞ ❛ ♠✐♥✐♠✉♠ ❝♦st t♦✉r✳ ❲❡ ❢♦r❝❡❞ ✐t ✐♥t♦ t❤❡ ♠♦✉❧❞ ♦❢ ❛ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✕ ❚❙P ✕ ✐♥ ♦r❞❡r t♦ ✜t ✐t ✐♥t♦ ♦✉r t❤❡♦r② ♦❢ ◆P✲❝♦♠♣❧❡t❡♥❡ss✳ ❙✐♠✐❧❛r ❛r❣✉♠❡♥ts ❝❛♥ ❜❡ ♠❛❞❡ ❛❜♦✉t t❤❡ ♣r♦❜❧❡♠s ❈▲■◗❯❊ ❛♥❞ ■◆❉✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❚❤✐s ✐s st✐❧❧ r❡❛s♦♥❛❜❧❡✱ ❛s ✇❡ ❛r❡ ❡st❛❜❧✐s❤✐♥❣ t❤❡ ❞✐✣❝✉❧t② ♦❢ t❤❡ ♣r♦❜❧❡♠s✳ ❆ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ s♦❧✉t✐♦♥ t♦ t❤❡ ♦♣t✐♠✐s❛t✐♦♥ ✈❡rs✐♦♥ ✇♦✉❧❞ ❣✐✈❡ ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠✳ ❆❧s♦✱ ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ s♦❧✉t✐♦♥ t♦ t❤❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✇♦✉❧❞ ❛❧❧♦✇ ❛ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r ✜♥❞✐♥❣ t❤❡ ♦♣t✐♠❛❧ ✈❛❧✉❡✱ ✉s✐♥❣ ❜✐♥❛r② s❡❛r❝❤✱ ✐❢ ♥❡❝❡ss❛r②✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❋✉♥❝t✐♦♥ Pr♦❜❧❡♠s

❙t✐❧❧✱ t❤❡r❡ ✐s s♦♠❡t❤✐♥❣ ✐♥t❡r❡st✐♥❣ t♦ ❜❡ s❛✐❞ ❢♦r ❢✉♥❝t✐♦♥ ♣r♦❜❧❡♠s ❛r✐s✐♥❣ ❢r♦♠ ◆P ♣r♦❜❧❡♠s✳ ❙✉♣♣♦s❡ L = {x | ∃yR(x, y)} ✇❤❡r❡ R ✐s ❛ ♣♦❧②♥♦♠✐❛❧❧②✲❜❛❧❛♥❝❡❞✱ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❞❡❝✐❞❛❜❧❡ r❡❧❛t✐♦♥✳ ❆ ✇✐t♥❡ss ❢✉♥❝t✐♦♥ ❢♦r L ✐s ❛♥② ❢✉♥❝t✐♦♥ f s✉❝❤ t❤❛t✿

• ✐❢ x ∈ L✱ t❤❡♥ f (x) = y ❢♦r s♦♠❡ y s✉❝❤ t❤❛t R(x, y)❀
• f (x) = ✏♥♦✑ ♦t❤❡r✇✐s❡✳

❚❤❡ ❝❧❛ss ❋◆P ✐s ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✇✐t♥❡ss ❢✉♥❝t✐♦♥s ❢♦r ❧❛♥❣✉❛❣❡s ✐♥ ◆P✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❋◆P ❛♥❞ ❋P

❆ ❢✉♥❝t✐♦♥ ✇❤✐❝❤✱ ❢♦r ❛♥② ❣✐✈❡♥ ❇♦♦❧❡❛♥ ❡①♣r❡ss✐♦♥ φ✱ ❣✐✈❡s ❛ s❛t✐s❢②✐♥❣ tr✉t❤ ❛ss✐❣♥♠❡♥t ✐❢ φ ✐s s❛t✐s✜❛❜❧❡✱ ❛♥❞ r❡t✉r♥s ✏♥♦✑ ♦t❤❡r✇✐s❡✱ ✐s ❛ ✇✐t♥❡ss ❢✉♥❝t✐♦♥ ❢♦r ❙❆❚✳ ■❢ ❛♥② ✇✐t♥❡ss ❢✉♥❝t✐♦♥ ❢♦r ❙❆❚ ✐s ❝♦♠♣✉t❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✱ t❤❡♥ P = ◆P✳ ■❢ P = ◆P✱ t❤❡♥ ❢♦r ❡✈❡r② ❧❛♥❣✉❛❣❡ ✐♥ ◆P✱ s♦♠❡ ✇✐t♥❡ss ❢✉♥❝t✐♦♥ ✐s ❝♦♠♣✉t❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✱ ❜② ❛ ❜✐♥❛r② s❡❛r❝❤ ❛❧❣♦r✐t❤♠✳ ❯♥❞❡r ❛ s✉✐t❛❜❧❡ ❞❡✜♥✐t✐♦♥ ♦❢ r❡❞✉❝t✐♦♥✱ t❤❡ ✇✐t♥❡ss ❢✉♥❝t✐♦♥s ❢♦r ❙❆❚ ❛r❡ ❋◆P✲❝♦♠♣❧❡t❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❋❛❝t♦r✐s❛t✐♦♥

❚❤❡ ❢❛❝t♦r✐s❛t✐♦♥ ❢✉♥❝t✐♦♥ ♠❛♣s ❛ ♥✉♠❜❡r n t♦ ✐ts ♣r✐♠❡ ❢❛❝t♦r✐s❛t✐♦♥✿ ✷k✶✸k✷ · · · pkm

m .

❚❤✐s ❢✉♥❝t✐♦♥ ✐s ✐♥ ❋◆P✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✭❢♦r ✇❤✐❝❤ ✐t ✐s ❛ ✇✐t♥❡ss ❢✉♥❝t✐♦♥✮ ✐s tr✐✈✐❛❧ ✲ ✐t ✐s t❤❡ s❡t ♦❢ ❛❧❧ ♥✉♠❜❡rs✳ ❙t✐❧❧✱ ✐t ✐s ♥♦t ❦♥♦✇♥ ✇❤❡t❤❡r t❤✐s ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈r②♣t♦❣r❛♣❤②

❆❧✐❝❡ ❇♦❜ ❊✈❡

❆❧✐❝❡ ✇✐s❤❡s t♦ ❝♦♠♠✉♥✐❝❛t❡ ✇✐t❤ ❇♦❜ ✇✐t❤♦✉t ❊✈❡ ❡❛✈❡s❞r♦♣♣✐♥❣✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

Pr✐✈❛t❡ ❑❡②

■♥ ❛ ♣r✐✈❛t❡ ❦❡② s②st❡♠✱ t❤❡r❡ ❛r❡ t✇♦ s❡❝r❡t ❦❡②s e ✕ t❤❡ ❡♥❝r②♣t✐♦♥ ❦❡② d ✕ t❤❡ ❞❡❝r②♣t✐♦♥ ❦❡② ❛♥❞ t✇♦ ❢✉♥❝t✐♦♥s D ❛♥❞ E s✉❝❤ t❤❛t✿ ❢♦r ❛♥② x✱ D(E(x, e), d) = x ❋♦r ✐♥st❛♥❝❡✱ t❛❦✐♥❣ d = e ❛♥❞ ❜♦t❤ D ❛♥❞ E ❛s ❡①❝❧✉s✐✈❡ ♦r✱ ✇❡ ❤❛✈❡ t❤❡ ♦♥❡ t✐♠❡ ♣❛❞✿ (x ⊕ e) ⊕ e = x

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❖♥❡ ❚✐♠❡ P❛❞

❚❤❡ ♦♥❡ t✐♠❡ ♣❛❞ ✐s ♣r♦✈❛❜❧② s❡❝✉r❡✱ ✐♥ t❤❛t t❤❡ ♦♥❧② ✇❛② ❊✈❡ ❝❛♥ ❞❡❝♦❞❡ ❛ ♠❡ss❛❣❡ ✐s ❜② ❦♥♦✇✐♥❣ t❤❡ ❦❡②✳ ■❢ t❤❡ ♦r✐❣✐♥❛❧ ♠❡ss❛❣❡ x ❛♥❞ t❤❡ ❡♥❝r②♣t❡❞ ♠❡ss❛❣❡ y ❛r❡ ❦♥♦✇♥✱ t❤❡♥ s♦ ✐s t❤❡ ❦❡②✿ e = x ⊕ y

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

P✉❜❧✐❝ ❑❡②

■♥ ♣✉❜❧✐❝ ❦❡② ❝r②♣t♦❣r❛♣❤②✱ t❤❡ ❡♥❝r②♣t✐♦♥ ❦❡② e ✐s ♣✉❜❧✐❝✱ ❛♥❞ t❤❡ ❞❡❝r②♣t✐♦♥ ❦❡② d ✐s ♣r✐✈❛t❡✳ ❲❡ st✐❧❧ ❤❛✈❡✱ ❢♦r ❛♥② x✱ D(E(x, e), d) = x ■❢ E ✐s ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❝♦♠♣✉t❛❜❧❡ ✭❛♥❞ ✐t ♠✉st ❜❡ ✐❢ ❝♦♠♠✉♥✐❝❛t✐♦♥ ✐s ♥♦t t♦ ❜❡ ♣❛✐♥❢✉❧❧② s❧♦✇✮✱ t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ t❤❛t t❛❦❡s y = E(x, e) t♦ x ✭✇✐t❤♦✉t ❦♥♦✇✐♥❣ d✮✱ ♠✉st ❜❡ ✐♥ ❋◆P✳ ❚❤✉s✱ ♣✉❜❧✐❝ ❦❡② ❝r②♣t♦❣r❛♣❤② ✐s ♥♦t ♣r♦✈❛❜❧② s❡❝✉r❡ ✐♥ t❤❡ ✇❛② t❤❛t t❤❡ ♦♥❡ t✐♠❡ ♣❛❞ ✐s✳ ■t r❡❧✐❡s ♦♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❢✉♥❝t✐♦♥s ✐♥ ❋◆P − ❋P✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❖♥❡ ❲❛② ❋✉♥❝t✐♦♥s

❆ ❢✉♥❝t✐♦♥ f ✐s ❝❛❧❧❡❞ ❛ ♦♥❡ ✇❛② ❢✉♥❝t✐♦♥ ✐❢ ✐t s❛t✐s✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ ✶✳ f ✐s ♦♥❡✲t♦✲♦♥❡✳ ✷✳ ❢♦r ❡❛❝❤ x✱ |x|✶/k ≤ |f (x)| ≤ |x|k ❢♦r s♦♠❡ k✳ ✸✳ f ∈ ❋P✳ ✹✳ f −✶ ∈ ❋P✳ ❲❡ ❝❛♥♥♦t ❤♦♣❡ t♦ ♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥s ✇✐t❤♦✉t ❛t t❤❡ s❛♠❡ t✐♠❡ ♣r♦✈✐♥❣ P = ◆P✳ ■t ✐s str♦♥❣❧② ❜❡❧✐❡✈❡❞ t❤❛t t❤❡ ❘❙❆ ❢✉♥❝t✐♦♥✿ f (x, e, p, q) = (xe ♠♦❞ pq, pq, e) ✐s ❛ ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❯P

❚❤♦✉❣❤ ♦♥❡ ❝❛♥♥♦t ❤♦♣❡ t♦ ♣r♦✈❡ t❤❛t t❤❡ ❘❙❆ ❢✉♥❝t✐♦♥ ✐s ♦♥❡✲✇❛② ✇✐t❤♦✉t s❡♣❛r❛t✐♥❣ P ❛♥❞ ◆P✱ ✇❡ ♠✐❣❤t ❤♦♣❡ t♦ ♠❛❦❡ ✐t ❛s s❡❝✉r❡ ❛s ❛ ♣r♦♦❢ ♦❢ ◆P✲❝♦♠♣❧❡t❡♥❡ss✳ ❉❡✜♥✐t✐♦♥ ❆ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡ ✐s ✉♥❛♠❜✐❣✉♦✉s ✐❢✱ ❢♦r ❛♥② ✐♥♣✉t x✱ t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡ ❛❝❝❡♣t✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ♠❛❝❤✐♥❡✳ ❯P ✐s t❤❡ ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s ❛❝❝❡♣t❡❞ ❜② ✉♥❛♠❜✐❣✉♦✉s ♠❛❝❤✐♥❡s ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❯P

❊q✉✐✈❛❧❡♥t❧②✱ ❯P ✐s t❤❡ ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s ♦❢ t❤❡ ❢♦r♠ {x | ∃yR(x, y)} ❲❤❡r❡ R ✐s ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❝♦♠♣✉t❛❜❧❡✱ ♣♦❧②♥♦♠✐❛❧❧② ❜❛❧❛♥❝❡❞✱ ❛♥❞ ❢♦r ❡❛❝❤ x✱ t❤❡r❡ ✐s ❛t ♠♦st ♦♥❡ y s✉❝❤ t❤❛t R(x, y)✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❯P ❖♥❡✲✇❛② ❋✉♥❝t✐♦♥s

❲❡ ❤❛✈❡ P ⊆ ❯P ⊆ ◆P ■t s❡❡♠s ✉♥❧✐❦❡❧② t❤❛t t❤❡r❡ ❛r❡ ❛♥② ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s ✐♥ ❯P✳ ❖♥❡✲✇❛② ❢✉♥❝t✐♦♥s ❡①✐st ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ P = ❯P✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❖♥❡✲❲❛② ❋✉♥❝t✐♦♥s ■♠♣❧② P = ❯P

❙✉♣♣♦s❡ f ✐s ❛ ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥✳ ❉❡✜♥❡ t❤❡ ❧❛♥❣✉❛❣❡ Lf ❜② Lf = {(x, y) | ∃z(z ≤ x ❛♥❞ f (z) = y)}. ❲❡ ❝❛♥ s❤♦✇ t❤❛t Lf ✐s ✐♥ ❯P ❜✉t ♥♦t ✐♥ P✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

P = ❯P ■♠♣❧✐❡s ❖♥❡✲❲❛② ❋✉♥❝t✐♦♥s ❊①✐st

❙✉♣♣♦s❡ t❤❛t L ✐s ❛ ❧❛♥❣✉❛❣❡ t❤❛t ✐s ✐♥ ❯P ❜✉t ♥♦t ✐♥ P✳ ▲❡t U ❜❡ ❛♥ ✉♥❛♠❜✐❣✉♦✉s ♠❛❝❤✐♥❡ t❤❛t ❛❝❝❡♣ts L✳ ❉❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ fU ❜② ✐❢ x ✐s ❛ str✐♥❣ t❤❛t ❡♥❝♦❞❡s ❛♥ ❛❝❝❡♣t✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢ U✱ t❤❡♥ fU(x) = ✶y ✇❤❡r❡ y ✐s t❤❡ ✐♥♣✉t str✐♥❣ ❛❝❝❡♣t❡❞ ❜② t❤✐s ❝♦♠♣✉t❛t✐♦♥✳ fU(x) = ✵x ♦t❤❡r✇✐s❡✳ ❲❡ ❝❛♥ ♣r♦✈❡ t❤❛t fU ✐s ❛ ♦♥❡✲✇❛② ❢✉♥❝t✐♦♥✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙♣❛❝❡ ❈♦♠♣❧❡①✐t②

❲❡✬✈❡ ❛❧r❡❛❞② s❡❡♥ t❤❡ ❞❡✜♥✐t✐♦♥ ❙P❆❈❊✭f ✮✿ t❤❡ ❧❛♥❣✉❛❣❡s ❛❝❝❡♣t❡❞ ❜② ❛ ♠❛❝❤✐♥❡ ✇❤✐❝❤ ✉s❡s O(f (n)) t❛♣❡ ❝❡❧❧s ♦♥ ✐♥♣✉ts ♦❢ ❧❡♥❣t❤ n✳ ❈♦✉♥t✐♥❣ ♦♥❧② ✇♦r❦ s♣❛❝❡✳ ◆❙P❆❈❊(f ) ✐s t❤❡ ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s ❛❝❝❡♣t❡❞ ❜② ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✉s✐♥❣ ❛t ♠♦st O(f (n)) ✇♦r❦ s♣❛❝❡✳ ❆s ✇❡ ❛r❡ ♦♥❧② ❝♦✉♥t✐♥❣ ✇♦r❦ s♣❛❝❡✱ ✐t ♠❛❦❡s s❡♥s❡ t♦ ❝♦♥s✐❞❡r ❜♦✉♥❞✐♥❣ ❢✉♥❝t✐♦♥s f t❤❛t ❛r❡ ❧❡ss t❤❛♥ ❧✐♥❡❛r✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈❧❛ss❡s

▲ = ❙P❆❈❊(❧♦❣ n) ◆▲ = ◆❙P❆❈❊(❧♦❣ n) P❙P❆❈❊ = ∞

k=✶ ❙P❆❈❊(nk)

❚❤❡ ❝❧❛ss ♦❢ ❧❛♥❣✉❛❣❡s ❞❡❝✐❞❛❜❧❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ s♣❛❝❡✳ ◆P❙P❆❈❊ = ∞

k=✶ ◆❙P❆❈❊(nk)

❆❧s♦✱ ❞❡✜♥❡✿ ❝♦✲◆▲ ✕ t❤❡ ❧❛♥❣✉❛❣❡s ✇❤♦s❡ ❝♦♠♣❧❡♠❡♥ts ❛r❡ ✐♥ ◆▲✳ ❝♦✲◆P❙P❆❈❊ ✕ t❤❡ ❧❛♥❣✉❛❣❡s ✇❤♦s❡ ❝♦♠♣❧❡♠❡♥ts ❛r❡ ✐♥ ◆P❙P❆❈❊✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

■♥❝❧✉s✐♦♥s

❲❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❝❧✉s✐♦♥s✿ ▲ ⊆ ◆▲ ⊆ P ⊆ ◆P ⊆ P❙P❆❈❊ ⊆ ◆P❙P❆❈❊ ⊆ ❊❳P ✇❤❡r❡ ❊❳P = ∞

k=✶ ❚■▼❊(✷nk)

▼♦r❡♦✈❡r✱ ▲ ⊆ ◆▲ ∩ ❝♦✲◆▲ P ⊆ ◆P ∩ ❝♦✲◆P P❙P❆❈❊ ⊆ ◆P❙P❆❈❊ ∩ ❝♦✲◆P❙P❆❈❊

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♥str✉❝t✐❜❧❡ ❋✉♥❝t✐♦♥s

❆ ❝♦♠♣❧❡①✐t② ❝❧❛ss s✉❝❤ ❛s ❚■▼❊(f ) ❝❛♥ ❜❡ ✈❡r② ✉♥♥❛t✉r❛❧✱ ✐❢ f ✐s✳ ❲❡ r❡str✐❝t ♦✉r ❜♦✉♥❞✐♥❣ ❢✉♥❝t✐♦♥s f t♦ ❜❡ ♣r♦♣❡r ❢✉♥❝t✐♦♥s✿ ❉❡✜♥✐t✐♦♥ ❆ ❢✉♥❝t✐♦♥ f : ■ ◆ → ■ ◆ ✐s ❝♦♥str✉❝t✐❜❧❡ ✐❢✿

• f ✐s ♥♦♥✲❞❡❝r❡❛s✐♥❣✱ ✐✳❡✳ f (n + ✶) ≥ f (n) ❢♦r ❛❧❧ n❀ ❛♥❞
• t❤❡r❡ ✐s ❛ ❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡ M ✇❤✐❝❤✱ ♦♥ ❛♥② ✐♥♣✉t ♦❢ ❧❡♥❣t❤ n✱

r❡♣❧❛❝❡s t❤❡ ✐♥♣✉t ✇✐t❤ t❤❡ str✐♥❣ ✵f (n)✱ ❛♥❞ M r✉♥s ✐♥ t✐♠❡ O(n + f (n)) ❛♥❞ ✉s❡s O(f (n)) ✇♦r❦ s♣❛❝❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊①❛♠♣❧❡s

❆❧❧ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢✉♥❝t✐♦♥s ❛r❡ ❝♦♥str✉❝t✐❜❧❡✿

• ⌈❧♦❣ n⌉❀
• n✷❀
• n❀
• ✷n✳

■❢ f ❛♥❞ g ❛r❡ ❝♦♥str✉❝t✐❜❧❡ ❢✉♥❝t✐♦♥s✱ t❤❡♥ s♦ ❛r❡ f + g✱ f · g✱ ✷f ❛♥❞ f (g) ✭t❤✐s ❧❛st✱ ♣r♦✈✐❞❡❞ t❤❛t f (n) > n✮✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❯s✐♥❣ ❈♦♥str✉❝t✐❜❧❡ ❋✉♥❝t✐♦♥s

◆❚■▼❊(f ) ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❛s t❤❡ ❝❧❛ss ♦❢ t❤♦s❡ ❧❛♥❣✉❛❣❡s L ❛❝❝❡♣t❡❞ ❜② ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ M✱ s✉❝❤ t❤❛t ❢♦r ❡✈❡r② x ∈ L✱ t❤❡r❡ ✐s ❛♥ ❛❝❝❡♣t✐♥❣ ❝♦♠♣✉t❛t✐♦♥ ♦❢ M ♦♥ x ♦❢ ❧❡♥❣t❤ ❛t ♠♦st O(f (n))✳ ■❢ f ✐s ❛ ❝♦♥str✉❝t✐❜❧❡ ❢✉♥❝t✐♦♥ t❤❡♥ ❛♥② ❧❛♥❣✉❛❣❡ ✐♥ ◆❚■▼❊(f ) ✐s ❛❝❝❡♣t❡❞ ❜② ❛ ♠❛❝❤✐♥❡ ❢♦r ✇❤✐❝❤ ❛❧❧ ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ♦❢ ❧❡♥❣t❤ ❛t ♠♦st O(f (n))✳ ❆❧s♦✱ ❣✐✈❡♥ ❛ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ M ❛♥❞ ❛ ❝♦♥str✉❝t✐❜❧❡ ❢✉♥❝t✐♦♥ f ✱ ✇❡ ❝❛♥ ❞❡✜♥❡ ❛ ♠❛❝❤✐♥❡ t❤❛t s✐♠✉❧❛t❡s M ❢♦r f (n) st❡♣s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊st❛❜❧✐s❤✐♥❣ ■♥❝❧✉s✐♦♥s

❚♦ ❡st❛❜❧✐s❤ t❤❡ ❦♥♦✇♥ ✐♥❝❧✉s✐♦♥s ❜❡t✇❡❡♥ t❤❡ ♠❛✐♥ ❝♦♠♣❧❡①✐t② ❝❧❛ss❡s✱ ✇❡ ♣r♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ❢♦r ❛♥② ❝♦♥str✉❝t✐❜❧❡ f ✳

• ❙P❆❈❊(f (n)) ⊆ ◆❙P❆❈❊(f (n))❀
• ❚■▼❊(f (n)) ⊆ ◆❚■▼❊(f (n))❀
• ◆❚■▼❊(f (n)) ⊆ ❙P❆❈❊(f (n))❀
• ◆❙P❆❈❊(f (n)) ⊆ ❚■▼❊(k❧♦❣ n+f (n))❀

❚❤❡ ✜rst t✇♦ ❛r❡ str❛✐❣❤t❢♦r✇❛r❞ ❢r♦♠ ❞❡✜♥✐t✐♦♥s✳ ❚❤❡ t❤✐r❞ ✐s ❛♥ ❡❛s② s✐♠✉❧❛t✐♦♥✳ ❚❤❡ ❧❛st r❡q✉✐r❡s s♦♠❡ ♠♦r❡ ✇♦r❦✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡❛❝❤❛❜✐❧✐t②

❘❡❝❛❧❧ t❤❡ ❘❡❛❝❤❛❜✐❧✐t② ♣r♦❜❧❡♠✿ ❣✐✈❡♥ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ G = (V , E) ❛♥❞ t✇♦ ♥♦❞❡s a, b ∈ V ✱ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡r❡ ✐s ❛ ♣❛t❤ ❢r♦♠ a t♦ b ✐♥ G✳ ❆ s✐♠♣❧❡ s❡❛r❝❤ ❛❧❣♦r✐t❤♠ s♦❧✈❡s ✐t✿ ✶✳ ♠❛r❦ ♥♦❞❡ a✱ ❧❡❛✈✐♥❣ ♦t❤❡r ♥♦❞❡s ✉♥♠❛r❦❡❞✱ ❛♥❞ ✐♥✐t✐❛❧✐s❡ s❡t S t♦ {a}❀ ✷✳ ✇❤✐❧❡ S ✐s ♥♦t ❡♠♣t②✱ ❝❤♦♦s❡ ♥♦❞❡ i ✐♥ S✿ r❡♠♦✈❡ i ❢r♦♠ S ❛♥❞ ❢♦r ❛❧❧ j s✉❝❤ t❤❛t t❤❡r❡ ✐s ❛♥ ❡❞❣❡ (i, j) ❛♥❞ j ✐s ✉♥♠❛r❦❡❞✱ ♠❛r❦ j ❛♥❞ ❛❞❞ j t♦ S❀ ✸✳ ✐❢ b ✐s ♠❛r❦❡❞✱ ❛❝❝❡♣t ❡❧s❡ r❡❥❡❝t✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❲❡ ❝❛♥ ✉s❡ t❤❡ O(n✷) ❛❧❣♦r✐t❤♠ ❢♦r ❘❡❛❝❤❛❜✐❧✐t② t♦ s❤♦✇ t❤❛t✿ ◆❙P❆❈❊(f (n)) ⊆ ❚■▼❊(k❧♦❣ n+f (n)) ❢♦r s♦♠❡ ❝♦♥st❛♥t k✳ ▲❡t M ❜❡ ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡ ✇♦r❦✐♥❣ ✐♥ s♣❛❝❡ ❜♦✉♥❞s f (n)✳ ❋♦r ❛♥② ✐♥♣✉t x ♦❢ ❧❡♥❣t❤ n✱ t❤❡r❡ ✐s ❛ ❝♦♥st❛♥t c ✭❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ st❛t❡s ❛♥❞ ❛❧♣❤❛❜❡t ♦❢ M✮ s✉❝❤ t❤❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ ❝♦♥✜❣✉r❛t✐♦♥s ♦❢ M ✇✐t❤✐♥ s♣❛❝❡ ❜♦✉♥❞s f (n) ✐s ❜♦✉♥❞❡❞ ❜② n · cf (n)✳ ❍❡r❡✱ cf (n) r❡♣r❡s❡♥ts t❤❡ ♥✉♠❜❡r ♦❢ ❞✐✛❡r❡♥t ♣♦ss✐❜❧❡ ❝♦♥t❡♥ts ♦❢ t❤❡ ✇♦r❦ s♣❛❝❡✱ ❛♥❞ n ❞✐✛❡r❡♥t ❤❡❛❞ ♣♦s✐t✐♦♥s ♦♥ t❤❡ ✐♥♣✉t✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♥✜❣✉r❛t✐♦♥ ●r❛♣❤

❉❡✜♥❡ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ❣r❛♣❤ ♦❢ M, x t♦ ❜❡ t❤❡ ❣r❛♣❤ ✇❤♦s❡ ♥♦❞❡s ❛r❡ t❤❡ ♣♦ss✐❜❧❡ ❝♦♥✜❣✉r❛t✐♦♥s✱ ❛♥❞ t❤❡r❡ ✐s ❛♥ ❡❞❣❡ ❢r♦♠ i t♦ j ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ i →M j✳ ❚❤❡♥✱ M ❛❝❝❡♣ts x ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ s♦♠❡ ❛❝❝❡♣t✐♥❣ ❝♦♥✜❣✉r❛t✐♦♥ ✐s r❡❛❝❤❛❜❧❡ ❢r♦♠ t❤❡ st❛rt✐♥❣ ❝♦♥✜❣✉r❛t✐♦♥ (s, ⊲, x, ⊲, ε) ✐♥ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ❣r❛♣❤ ♦❢ M, x✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❯s✐♥❣ t❤❡ O(n✷) ❛❧❣♦r✐t❤♠ ❢♦r ❘❡❛❝❤❛❜✐❧✐t②✱ ✇❡ ❣❡t t❤❛t L(M)✖t❤❡ ❧❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ ❜② M✖❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❜② ❛ ❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡ ♦♣❡r❛t✐♥❣ ✐♥ t✐♠❡ c′(ncf (n))✷ ∼ c′c✷(❧♦❣ n+f (n)) ∼ k(❧♦❣ n+f (n)) ■♥ ♣❛rt✐❝✉❧❛r✱ t❤✐s ❡st❛❜❧✐s❤❡s t❤❛t ◆▲ ⊆ P ❛♥❞ ◆P❙P❆❈❊ ⊆ ❊❳P✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆▲ ❘❡❛❝❤❛❜✐❧✐t②

❲❡ ❝❛♥ ❝♦♥str✉❝t ❛♥ ❛❧❣♦r✐t❤♠ t♦ s❤♦✇ t❤❛t t❤❡ ❘❡❛❝❤❛❜✐❧✐t② ♣r♦❜❧❡♠ ✐s ✐♥ ◆▲✿ ✶✳ ✇r✐t❡ t❤❡ ✐♥❞❡① ♦❢ ♥♦❞❡ a ✐♥ t❤❡ ✇♦r❦ s♣❛❝❡❀ ✷✳ ✐❢ i ✐s t❤❡ ✐♥❞❡① ❝✉rr❡♥t❧② ✇r✐tt❡♥ ♦♥ t❤❡ ✇♦r❦ s♣❛❝❡✿

✷✳✶ ✐❢ i = b t❤❡♥ ❛❝❝❡♣t✱ ❡❧s❡ ❣✉❡ss ❛♥ ✐♥❞❡① j ✭❧♦❣ n ❜✐ts✮ ❛♥❞ ✇r✐t❡ ✐t ♦♥ t❤❡ ✇♦r❦ s♣❛❝❡✳ ✷✳✷ ✐❢ (i, j) ✐s ♥♦t ❛♥ ❡❞❣❡✱ r❡❥❡❝t✱ ❡❧s❡ r❡♣❧❛❝❡ i ❜② j ❛♥❞ r❡t✉r♥ t♦ ✭✷✮✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙❛✈✐t❝❤✬s ❚❤❡♦r❡♠

❋✉rt❤❡r s✐♠✉❧❛t✐♦♥ r❡s✉❧ts ❢♦r ♥♦♥❞❡t❡r♠✐♥✐st✐❝ s♣❛❝❡ ❛r❡ ♦❜t❛✐♥❡❞ ❜② ♦t❤❡r ❛❧❣♦r✐t❤♠s ❢♦r ❘❡❛❝❤❛❜✐❧✐t②✳ ❲❡ ❝❛♥ s❤♦✇ t❤❛t ❘❡❛❝❤❛❜✐❧✐t② ❝❛♥ ❜❡ s♦❧✈❡❞ ❜② ❛ ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ ✐♥ O((❧♦❣ n)✷) s♣❛❝❡✳ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❝✉rs✐✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ❞❡t❡r♠✐♥✐♥❣ ✇❤❡t❤❡r t❤❡r❡ ✐s ❛ ♣❛t❤ ❢r♦♠ a t♦ b ♦❢ ❧❡♥❣t❤ ❛t ♠♦st i ✭❢♦r i ❛ ♣♦✇❡r ♦❢ ✷✮✿

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

O((❧♦❣ n)✷) s♣❛❝❡ ❘❡❛❝❤❛❜✐❧✐t② ❛❧❣♦r✐t❤♠✿ P❛t❤(a, b, i) ✐❢ i = ✶ ❛♥❞ a = b ❛♥❞ (a, b) ✐s ♥♦t ❛♥ ❡❞❣❡ r❡❥❡❝t ❡❧s❡ ✐❢ (a, b) ✐s ❛♥ ❡❞❣❡ ♦r a = b ❛❝❝❡♣t ❡❧s❡✱ ❢♦r ❡❛❝❤ ♥♦❞❡ x✱ ❝❤❡❝❦✿ ✶✳ ✐s t❤❡r❡ ❛ ♣❛t❤ a − x ♦❢ ❧❡♥❣t❤ i/✷❀ ❛♥❞ ✷✳ ✐s t❤❡r❡ ❛ ♣❛t❤ x − b ♦❢ ❧❡♥❣t❤ i/✷❄ ✐❢ s✉❝❤ ❛♥ x ✐s ❢♦✉♥❞✱ t❤❡♥ ❛❝❝❡♣t✱ ❡❧s❡ r❡❥❡❝t✳ ❚❤❡ ♠❛①✐♠✉♠ ❞❡♣t❤ ♦❢ r❡❝✉rs✐♦♥ ✐s ❧♦❣ n✱ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❜✐ts ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❦❡♣t ❛t ❡❛❝❤ st❛❣❡ ✐s ✸ ❧♦❣ n✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙❛✈✐t❝❤✬s ❚❤❡♦r❡♠

❚❤❡ s♣❛❝❡ ❡✣❝✐❡♥t ❛❧❣♦r✐t❤♠ ❢♦r r❡❛❝❤❛❜✐❧✐t② ✉s❡❞ ♦♥ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ❣r❛♣❤ ♦❢ ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡ s❤♦✇s✿ ◆❙P❆❈❊(f ) ⊆ ❙P❆❈❊(f ✷) ❢♦r f (n) ≥ ❧♦❣ n✳ ❚❤✐s ②✐❡❧❞s P❙P❆❈❊ = ◆P❙P❆❈❊ = ❝♦✲◆P❙P❆❈❊.

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣❧❡♠❡♥t❛t✐♦♥

❆ st✐❧❧ ♠♦r❡ ❝❧❡✈❡r ❛❧❣♦r✐t❤♠ ❢♦r ❘❡❛❝❤❛❜✐❧✐t② ❤❛s ❜❡❡♥ ✉s❡❞ t♦ s❤♦✇ t❤❛t ♥♦♥❞❡t❡r♠✐♥✐st✐❝ s♣❛❝❡ ❝❧❛ss❡s ❛r❡ ❝❧♦s❡❞ ✉♥❞❡r ❝♦♠♣❧❡♠❡♥t❛t✐♦♥✿ ■❢ f (n) ≥ ❧♦❣ n✱ t❤❡♥ ◆❙P❆❈❊(f ) = ❝♦✲◆❙P❆❈❊(f ) ■♥ ♣❛rt✐❝✉❧❛r ◆▲ = ❝♦✲◆▲.

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

▲♦❣❛r✐t❤♠✐❝ ❙♣❛❝❡ ❘❡❞✉❝t✐♦♥s

❲❡ ✇r✐t❡ A ≤L B ✐❢ t❤❡r❡ ✐s ❛ r❡❞✉❝t✐♦♥ f ♦❢ A t♦ B t❤❛t ✐s ❝♦♠♣✉t❛❜❧❡ ❜② ❛ ❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✉s✐♥❣ O(❧♦❣ n) ✇♦r❦s♣❛❝❡ ✭✇✐t❤ ❛ r❡❛❞✲♦♥❧② ✐♥♣✉t t❛♣❡ ❛♥❞ ✇r✐t❡✲♦♥❧② ♦✉t♣✉t t❛♣❡✮✳ ◆♦t❡✿ ❲❡ ❝❛♥ ❝♦♠♣♦s❡ ≤L r❡❞✉❝t✐♦♥s✳ ❙♦✱ ✐❢ A ≤L B ❛♥❞ B ≤L C t❤❡♥ A ≤L C

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆P✲❝♦♠♣❧❡t❡ Pr♦❜❧❡♠s

❆♥❛❧②s✐♥❣ ❝❛r❡❢✉❧❧② t❤❡ r❡❞✉❝t✐♦♥s ✇❡ ❝♦♥str✉❝t❡❞ ✐♥ ♦✉r ♣r♦♦❢s ♦❢ ◆P✲❝♦♠♣❧❡t❡♥❡ss✱ ✇❡ ❝❛♥ s❡❡ t❤❛t ❙❆❚ ❛♥❞ t❤❡ ✈❛r✐♦✉s ♦t❤❡r ◆P✲❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s ❛r❡ ❛❝t✉❛❧❧② ❝♦♠♣❧❡t❡ ✉♥❞❡r ≤L r❡❞✉❝t✐♦♥s✳ ❚❤✉s✱ ✐❢ ❙❆❚ ≤L A ❢♦r s♦♠❡ ♣r♦❜❧❡♠ A ✐♥ ▲ t❤❡♥ ♥♦t ♦♥❧② P = ◆P ❜✉t ❛❧s♦ ▲ = ◆P✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

P✲❝♦♠♣❧❡t❡ Pr♦❜❧❡♠s

■t ♠❛❦❡s ❧✐tt❧❡ s❡♥s❡ t♦ t❛❧❦ ♦❢ ❝♦♠♣❧❡t❡ ♣r♦❜❧❡♠s ❢♦r t❤❡ ❝❧❛ss P ✇✐t❤ r❡s♣❡❝t t♦ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ r❡❞✉❝✐❜✐❧✐t② ≤P✳ ❚❤❡r❡ ❛r❡ ♣r♦❜❧❡♠s t❤❛t ❛r❡ ❝♦♠♣❧❡t❡ ❢♦r P ✇✐t❤ r❡s♣❡❝t t♦ ❧♦❣❛r✐t❤♠✐❝ s♣❛❝❡ r❡❞✉❝t✐♦♥s ≤L✳ ❖♥❡ ❡①❛♠♣❧❡ ✐s ❈❱P✖t❤❡ ❝✐r❝✉✐t ✈❛❧✉❡ ♣r♦❜❧❡♠✳ ❚❤❛t ✐s✱ ❢♦r ❡✈❡r② ❧❛♥❣✉❛❣❡ A ✐♥ P✱ A ≤L ❈❱P

• ■❢ ❈❱P ∈ ▲ t❤❡♥ ▲ = P✳
• ■❢ ❈❱P ∈ ◆▲ t❤❡♥ ◆▲ = P✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❘❡❛❝❤❛❜✐❧✐t②

❙✐♠✐❧❛r❧②✱ ✐t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ❘❡❛❝❤❛❜✐❧✐t② ✐s✱ ✐♥ ❢❛❝t✱ ◆▲✲❝♦♠♣❧❡t❡✳ ❋♦r ❛♥② ❧❛♥❣✉❛❣❡ A ∈ ◆▲✱ ✇❡ ❤❛✈❡ A ≤L ❘❡❛❝❤❛❜✐❧✐t② ▲ = ◆▲ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ❘❡❛❝❤❛❜✐❧✐t② ∈ ▲ ◆♦t❡✿ ✐t ✐s ❦♥♦✇♥ t❤❛t t❤❡ r❡❛❝❤❛❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r ✉♥❞✐r❡❝t❡❞ ❣r❛♣❤s ✐s ✐♥ ▲✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

Pr♦✈❛❜❧❡ ■♥tr❛❝t❛❜✐❧✐t②

❖✉r ❛✐♠ ♥♦✇ ✐s t♦ s❤♦✇ t❤❛t t❤❡r❡ ❛r❡ ❧❛♥❣✉❛❣❡s ✭♦r✱ ❡q✉✐✈❛❧❡♥t❧②✱ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s✮ t❤❛t ✇❡ ❝❛♥ ♣r♦✈❡ ❛r❡ ♥♦t ✐♥ P✳ ❚❤✐s ✐s ❞♦♥❡ ❜② s❤♦✇✐♥❣ t❤❛t✱ ❢♦r ❡✈❡r② r❡❛s♦♥❛❜❧❡ ❢✉♥❝t✐♦♥ f ✱ t❤❡r❡ ✐s ❛ ❧❛♥❣✉❛❣❡ t❤❛t ✐s ♥♦t ✐♥ ❚■▼❊(f )✳ ❚❤❡ ♣r♦♦❢ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❞✐❛❣♦♥❛❧ ♠❡t❤♦❞✱ ❛s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ t❤❡ ✉♥❞❡❝✐❞❛❜✐❧✐t② ♦❢ t❤❡ ❤❛❧t✐♥❣ ♣r♦❜❧❡♠✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❚✐♠❡ ❍✐❡r❛r❝❤② ❚❤❡♦r❡♠

❋♦r ❛♥② ❝♦♥str✉❝t✐❜❧❡ ❢✉♥❝t✐♦♥ f ✱ ✇✐t❤ f (n) ≥ n✱ ❞❡✜♥❡ t❤❡ f ✲❜♦✉♥❞❡❞ ❤❛❧t✐♥❣ ❧❛♥❣✉❛❣❡ t♦ ❜❡✿ Hf = {[M], x | M ❛❝❝❡♣ts x ✐♥ f (|x|) st❡♣s} ✇❤❡r❡ [M] ✐s ❛ ❞❡s❝r✐♣t✐♦♥ ♦❢ M ✐♥ s♦♠❡ ✜①❡❞ ❡♥❝♦❞✐♥❣ s❝❤❡♠❡✳ ❚❤❡♥✱ ✇❡ ❝❛♥ s❤♦✇ Hf ∈ ❚■▼❊(f (n)✷) ❛♥❞ Hf ∈ ❚■▼❊(f (⌊n/✷⌋)) ❚✐♠❡ ❍✐❡r❛r❝❤② ❚❤❡♦r❡♠ ❋♦r ❛♥② ❝♦♥str✉❝t✐❜❧❡ ❢✉♥❝t✐♦♥ f (n) ≥ n✱ ❚■▼❊(f (n)) ✐s ♣r♦♣❡r❧② ❝♦♥t❛✐♥❡❞ ✐♥ ❚■▼❊(f (✷n + ✶)✷)✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙tr♦♥❣ ❍✐❡r❛r❝❤② ❚❤❡♦r❡♠s

❋♦r ❛♥② ❝♦♥str✉❝t✐❜❧❡ ❢✉♥❝t✐♦♥ f (n) ≥ n✱ ❚■▼❊(f (n)) ✐s ♣r♦♣❡r❧② ❝♦♥t❛✐♥❡❞ ✐♥ ❚■▼❊(f (n)(❧♦❣ f (n)))✳ ❙♣❛❝❡ ❍✐❡r❛r❝❤② ❚❤❡♦r❡♠ ❋♦r ❛♥② ♣❛✐r ♦❢ ❝♦♥str✉❝t✐❜❧❡ ❢✉♥❝t✐♦♥s f ❛♥❞ g✱ ✇✐t❤ f = O(g) ❛♥❞ g = O(f )✱ t❤❡r❡ ✐s ❛ ❧❛♥❣✉❛❣❡ ✐♥ ❙P❆❈❊(g(n)) t❤❛t ✐s ♥♦t ✐♥ ❙P❆❈❊(f (n))✳ ❙✐♠✐❧❛r r❡s✉❧ts ❝❛♥ ❜❡ ❡st❛❜❧✐s❤❡❞ ❢♦r ♥♦♥❞❡t❡r♠✐♥✐st✐❝ t✐♠❡ ❛♥❞ s♣❛❝❡ ❝❧❛ss❡s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♥s❡q✉❡♥❝❡s

• ❋♦r ❡❛❝❤ k✱ ❚■▼❊(nk) = P✳
• P = ❊❳P✳
• ▲ = P❙P❆❈❊✳
• ❆♥② ❧❛♥❣✉❛❣❡ t❤❛t ✐s ❊❳P✲❝♦♠♣❧❡t❡ ✐s ♥♦t ✐♥ P✳
• ❚❤❡r❡ ❛r❡ ♥♦ ♣r♦❜❧❡♠s ✐♥ P t❤❛t ❛r❡ ❝♦♠♣❧❡t❡ ✉♥❞❡r ❧✐♥❡❛r t✐♠❡

r❡❞✉❝t✐♦♥s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❉❡s❝r✐♣t✐✈❡ ❈♦♠♣❧❡①✐t②

❉❡s❝r✐♣t✐✈❡ ❈♦♠♣❧❡①✐t② ✐s ❛♥ ❛tt❡♠♣t t♦ st✉❞② t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ♣r♦❜❧❡♠s ❛♥❞ ❝❧❛ss✐❢② t❤❡♠✱ ♥♦t ♦♥ t❤❡ ❜❛s✐s ♦❢ ❤♦✇ ❞✐✣❝✉❧t ✐t ✐s t♦ ❝♦♠♣✉t❡ s♦❧✉t✐♦♥s✱ ❜✉t ♦♥ t❤❡ ❜❛s✐s ♦❢ ❤♦✇ ❞✐✣❝✉❧t ✐t ✐s t♦ ❞❡s❝r✐❜❡ t❤❡ ♣r♦❜❧❡♠✳ ❚❤✐s ❣✐✈❡s ❛♥ ❛❧t❡r♥❛t✐✈❡ ✇❛② t♦ st✉❞② ❝♦♠♣❧❡①✐t②✱ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ♣❛rt✐❝✉❧❛r ♠❛❝❤✐♥❡ ♠♦❞❡❧s✳ ❇❛s❡❞ ♦♥ ❞❡✜♥❛❜✐❧✐t② ✐♥ ❧♦❣✐❝✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

• r❛♣❤ Pr♦♣❡rt✐❡s

❆s ❛♥ ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ♦♥ ❣r❛♣❤s✳ ✶✳ ●✐✈❡♥ ❛ ❣r❛♣❤ G = (V , E) ❞♦❡s ✐t ❝♦♥t❛✐♥ ❛ tr✐❛♥❣❧❡❄ ✷✳ ●✐✈❡♥ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ G = (V , E) ❛♥❞ t✇♦ ♦❢ ✐ts ✈❡rt✐❝❡s a, b ∈ V ✱ ❞♦❡s G ❝♦♥t❛✐♥ ❛ ♣❛t❤ ❢r♦♠ a t♦ b❄ ✸✳ ●✐✈❡♥ ❛ ❣r❛♣❤ G = (V , E) ✐s ✐t ✸✲❝♦❧♦✉r❛❜❧❡❄ ❚❤❛t ✐s✱ ✐s t❤❡r❡ ❛ ❢✉♥❝t✐♦♥ χ : V → {✶, ✷, ✸} s♦ t❤❛t ✇❤❡♥❡✈❡r (u, v) ∈ E✱ χ(u) = χ(v)✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

• r❛♣❤ Pr♦♣❡rt✐❡s

✶✳ ❈❤❡❝❦✐♥❣ ✐❢ G ❝♦♥t❛✐♥s ❛ tr✐❛♥❣❧❡ ❝❛♥ ❜❡ s♦❧✈❡❞ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛♥❞ ❧♦❣❛r✐t❤♠✐❝ s♣❛❝❡✳ ✷✳ ❈❤❡❝❦✐♥❣ ✐❢ G ❝♦♥t❛✐♥s ❛ ♣❛t❤ ❢r♦♠ a t♦ b ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❈❛♥ ✐t ❜❡ ❞♦♥❡ ✐♥ ❧♦❣❛r✐t❤♠✐❝ s♣❛❝❡❄ ❯♥❧✐❦❡❧②✳ ■t ✐s ◆▲✲❝♦♠♣❧❡t❡✳ ✸✳ ❈❤❡❝❦✐♥❣ ✐❢ G ✐s ✸✲❝♦❧♦✉r❛❜❧❡ ❝❛♥ ❜❡ ❞♦♥❡ ✐♥ ❡①♣♦♥❡♥t✐❛❧ t✐♠❡ ❛♥❞ ♣♦❧②♥♦♠✐❛❧ s♣❛❝❡✳ ❈❛♥ ✐t ❜❡ ❞♦♥❡ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡❄ ❯♥❧✐❦❡❧②✳ ■t ✐s ◆P✲❝♦♠♣❧❡t❡✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

▲♦❣✐❝❛❧ ❉❡✜♥❛❜✐❧✐t②

■♥ ✇❤❛t ❦✐♥❞ ♦❢ ❢♦r♠❛❧ ❧❛♥❣✉❛❣❡ ❝❛♥ t❤❡s❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ❜❡ s♣❡❝✐✜❡❞ ♦r ❞❡✜♥❡❞❄ ❚❤❡ ❣r❛♣❤ G = (V , E) ❝♦♥t❛✐♥s ❛ tr✐❛♥❣❧❡✳ ∃x, y, z ∈ V (x = y ∧ y = z ∧ x = z ∧ E(x, y) ∧ E(x, z) ∧ E(y, z)) ❚❤❡ ♦t❤❡r t✇♦ ♣r♦♣❡rt✐❡s ❛r❡ ♣r♦✈❛❜❧② ♥♦t ❞❡✜♥❛❜❧❡ ✇✐t❤ ♦♥❧② ✜rst✲♦r❞❡r q✉❛♥t✐✜❝❛t✐♦♥ ♦✈❡r ✈❡rt✐❝❡s✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❋✐rst✲❖r❞❡r ▲♦❣✐❝

❈♦♥s✐❞❡r ✜rst✲♦r❞❡r ♣r❡❞✐❝❛t❡ ❧♦❣✐❝✳ ❆ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s x, y, . . .✱ ❛♥❞ ❢♦r♠✉❧❛s✿ E(x, y) | φ ∧ ψ | φ ∨ ψ | ¬φ | ∃xφ | ∀xφ ❆♥② ♣r♦♣❡rt② ♦❢ ❣r❛♣❤s t❤❛t ✐s ❡①♣r❡ss✐❜❧❡ ✐♥ ✜rst✲♦r❞❡r ❧♦❣✐❝ ✐s ✐♥ ▲✳ ❚❤❡ ♣r♦❜❧❡♠ ♦❢ ❞❡❝✐❞✐♥❣ ✇❤❡t❤❡r G | = φ ❢♦r ❛ ✜rst✲♦r❞❡r φ ✐s ✐♥ t✐♠❡ O(lnm) ❛♥❞ O(m ❧♦❣ n) s♣❛❝❡✳ ✇❤❡r❡✱ l ✐s t❤❡ ❧❡♥❣t❤ ♦❢ φ ❛♥❞ n t❤❡ ♦r❞❡r ♦❢ G ❛♥❞ m ✐s t❤❡ ♥❡st✐♥❣ ❞❡♣t❤ ♦❢ q✉❛♥t✐✜❡rs ✐♥ φ✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❈♦♠♣❧❡①✐t② ♦❢ ❋✐rst✲❖r❞❡r ▲♦❣✐❝

❚❤❡ str❛✐❣❤t❢♦r✇❛r❞ ❛❧❣♦r✐t❤♠ ♣r♦❝❡❡❞s r❡❝✉rs✐✈❡❧② ♦♥ t❤❡ str✉❝t✉r❡ ♦❢ φ✿

• ❆t♦♠✐❝ ❢♦r♠✉❧❛s ❜② ❞✐r❡❝t ❧♦♦❦✉♣✳
• ❇♦♦❧❡❛♥ ❝♦♥♥❡❝t✐✈❡s ❛r❡ ❡❛s②✳
• ■❢ φ ≡ ∃x ψ t❤❡♥ ❢♦r ❡❛❝❤ v ✐♥ G ❝❤❡❝❦ ✇❤❡t❤❡r

(G, x → v) | = ψ.

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❙❡❝♦♥❞✲❖r❞❡r ◗✉❛♥t✐✜❡rs

✸✲❈♦❧♦✉r❛❜✐❧✐t② ❛♥❞ ❘❡❛❝❤❛❜✐❧✐t② ❝❛♥ ❜❡ ❞❡✜♥❡❞ ✇✐t❤ q✉❛♥t✐✜❝❛t✐♦♥ ♦✈❡r s❡ts ♦❢ ✈❡rt✐❝❡s✳ ∃R ⊆ V ∃B ⊆ V ∃G ⊆ V ∀x(Rx ∨ Bx ∨ Gx)∧ ∀x(¬(Rx ∧ Bx) ∧ ¬(Bx ∧ Gx) ∧ ¬(Rx ∧ Gx))∧ ∀x∀y(Exy → (¬(Rx ∧ Ry)∧ ¬(Bx ∧ By)∧ ¬(Gx ∧ Gy))) ∀S ⊆ V (a ∈ S ∧ ∀x∀y((x ∈ S ∧ E(x, y)) → y ∈ S) → b ∈ S)

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❊①✐st❡♥t✐❛❧ ❙❡❝♦♥❞✲❖r❞❡r ▲♦❣✐❝

❙❡❝♦♥❞✲♦r❞❡r ❧♦❣✐❝ ✐s ♦❜t❛✐♥❡❞ ❜② ❛❞❞✐♥❣ t♦ t❤❡ ❞❡✜♥✐♥❣ r✉❧❡s ♦❢ ✜rst✲♦r❞❡r ❧♦❣✐❝ t✇♦ ❢✉rt❤❡r ❝❧❛✉s❡s✿ ❛t♦♠✐❝ ❢♦r♠✉❧❛❡ ✕ X(t✶, . . . , ta)✱ ✇❤❡r❡ X ✐s ❛ s❡❝♦♥❞✲♦r❞❡r ✈❛r✐❛❜❧❡ s❡❝♦♥❞✲♦r❞❡r q✉❛♥t✐✜❡rs ✕ ∃Xφ✱ ∀Xφ ❊①✐st❡♥t✐❛❧ ❙❡❝♦♥❞✲❖r❞❡r ▲♦❣✐❝ ✭❊❙❖✮ ❝♦♥s✐sts ♦❢ ❢♦r♠✉❧❛s ♦❢ t❤❡ ❢♦r♠ ∃X✶ · · · ∃Xkφ ✇❤❡r❡ φ ✐s ✜rst✲♦r❞❡r

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❋❛❣✐♥✬s ❚❤❡♦r❡♠

❚❤❡♦r❡♠ ✭❋❛❣✐♥✮ ❆ ❝❧❛ss ♦❢ ❣r❛♣❤s ✐s ❞❡✜♥❛❜❧❡ ❜② ❛ ❢♦r♠✉❧❛ ♦❢ ❡①✐st❡♥t✐❛❧ s❡❝♦♥❞✲♦r❞❡r ❧♦❣✐❝ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ✐t ✐s ❞❡❝✐❞❛❜❧❡ ❜② ❛ ♥♦♥❞❡t❡r♠✐♥✐s✐t✐❝ ♠❛❝❤✐♥❡ r✉♥♥✐♥❣ ✐♥ ♣♦❧②♥♦♠✐❛❧ t✐♠❡✳ ❊❙❖ = ◆P ❖♥❡ ❞✐r❡❝t✐♦♥ ✐s ❡❛s②✿ ●✐✈❡♥ G ❛♥❞ ∃X✶ . . . ∃Xkφ✳ ❛ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ♠❛❝❤✐♥❡ ❝❛♥ ❣✉❡ss ❛♥ ✐♥t❡r♣r❡t❛t✐♦♥ ❢♦r X✶, . . . , Xk ❛♥❞ t❤❡♥ ✈❡r✐❢② φ✳ ❚❤❡ ♦t❤❡r ❞✐r❡❝t✐♦♥ r❡q✉✐r❡s ❛ ♣r♦♦❢ s✐♠✐❧❛r t♦ ❈♦♦❦✬s t❤❡♦r❡♠✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❆ ▲♦❣✐❝ ❢♦r P❄

■s t❤❡r❡ ❛ ❧♦❣✐❝✱ ✐♥t❡r♠❡❞✐❛t❡ ❜❡t✇❡❡♥ ✜rst ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ❧♦❣✐❝ t❤❛t ❡①♣r❡ss❡s ❡①❛❝t❧② ❣r❛♣❤ ♣r♦♣❡rt✐❡s ✐♥ P❄ ❚❤✐s ✐s ❛♥ ♦♣❡♥ q✉❡st✐♦♥✱ st✐❧❧ t❤❡ s✉❜❥❡❝t ♦❢ ❛❝t✐✈❡ r❡s❡❛r❝❤✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❚❤❡ ❊♥❞

P❧❡❛s❡ ♣r♦✈✐❞❡ ❢❡❡❞❜❛❝❦✱ ✉s✐♥❣ t❤❡ ❧✐♥❦ s❡♥t t♦ ②♦✉ ❜② ❡✲♠❛✐❧✳

❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②