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❋♦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 ✶ , . . . , c n ✱ ✇❤❡r❡ ❢♦r ❡❛❝❤ i ✱ c i → M c i + ✶ ✱ ✐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❡❥ , ⊲, R q , ✶ , R q , ✶ , R q q ′ q ′ , ✶ , L r❡❥ , ⊲, R r❡❥ , ✵ , S ❛❝❝ , ⊔ , 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 × Σ k ) → Q ∪ { ❛❝❝ , r❡❥ } × (Σ × { L , R , S } ) k ❙✐♠✐❧❛r❧②✱ ❛ ❝♦♥✜❣✉r❛t✐♦♥ ✐s ♦❢ t❤❡ ❢♦r♠✿ ( q , w ✶ , u ✶ , . . . , w k , u k ) ❆♥✉❥ ❉❛✇❛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 ✵✵ ) ( r❡❥ , u ✷ , w ✷ ) ( q ✶✵ , 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♥♦♠✐❛❧ ❚✐♠❡ ∞ � ❚■▼❊ ( n k ) P = k = ✶ ❚❤❡ ❝❧❛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 ✵✵ ) ( r❡❥ , u ✷ , w ✷ ) ( q ✶✵ , 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 ✳ ∞ � ◆❚■▼❊ ( n k ) ◆P = k = ✶ ❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

◆♦♥❞❡t❡r♠✐♥✐s♠ ( s , ⊲, x ) ( q ✷ , u ✷ , w ✷ ) ( q ✵ , u ✵ , w ✵ ) ( q ✶ , u ✶ , w ✶ ) ( q ✵✵ , u ✵✵ , w ✵✵ ) ( r❡❥ , u ✷ , w ✷ ) ( q ✶✵ , 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✐♠❡ n k ✳ ❲❡ ❞❡✜♥❡ t❤❡ ❞❡t❡r♠✐♥✐st✐❝ ❛❧❣♦r✐t❤♠ V ✇❤✐❝❤ ♦♥ ✐♥♣✉t ( x , c ) s✐♠✉❧❛t❡s M ♦♥ ✐♥♣✉t x ✳ ❆t t❤❡ i t❤ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❝❤♦✐❝❡ ♣♦✐♥t✱ V ❧♦♦❦s ❛t t❤❡ i t❤ ❝❤❛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 x V x ❣❡♥❡r❛t❡ ✈❡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❤✐♥ n k 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✿ ❢♦r ❡❛❝❤ i ≤ n k ❛♥❞ q ∈ Q S i , q ❢♦r ❡❛❝❤ i , j ≤ n k ❛♥❞ σ ∈ Σ T i , j ,σ ❢♦r ❡❛❝❤ i , j ≤ n k H i , j ❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

■♥t✉✐t✐✈❡❧②✱ t❤❡s❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐♥t❡♥❞❡❞ t♦ ♠❡❛♥✿ • S i , q ✕ t❤❡ st❛t❡ ♦❢ t❤❡ ♠❛❝❤✐♥❡ ❛t t✐♠❡ i ✐s q ✳ • T i , j ,σ ✕ ❛t t✐♠❡ i ✱ t❤❡ s②♠❜♦❧ ❛t ♣♦s✐t✐♦♥ j ♦❢ t❤❡ t❛♣❡ ✐s σ ✳ • H i , 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 ♦♥❝❡ � � � ( H i , j → ( ¬ H i , j ′ )) i j j ′ � = j ❚❤❡ ♠❛❝❤✐♥❡ ✐s ♥❡✈❡r ✐♥ t✇♦ st❛t❡s ❛t ♦♥❝❡ � � � ( S i , q → ( ¬ S i , q ′ )) q i q ′ � = q ❊❛❝❤ t❛♣❡ ❝❡❧❧ ❝♦♥t❛✐♥s ♦♥❧② ♦♥❡ s②♠❜♦❧ � � � � ( T i , j ,σ → ( ¬ T i , j ,σ ′ )) i j σ σ ′ � = σ ❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

❚❤❡ ✐♥✐t✐❛❧ t❛♣❡ ❝♦♥t❡♥ts ❛r❡ x � � T ✶ , j , x j ∧ T ✶ , j , ⊔ j ≤ n n < j ❚❤❡ t❛♣❡ ❞♦❡s ♥♦t ❝❤❛♥❣❡ ❡①❝❡♣t ✉♥❞❡r t❤❡ ❤❡❛❞ � � � � ( H i , j ∧ T i , j ′ ,σ ) → T i + ✶ , j ′ ,σ j ′ � = j σ i j ❊❛❝❤ st❡♣ ✐s ❛❝❝♦r❞✐♥❣ t♦ δ ✳ � � � � ( H i , j ∧ S i , q ∧ T i , j ,σ ) i j σ q � ( H i + ✶ , j ′ ∧ S i + ✶ , q ′ ∧ T i + ✶ , 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 ✐❢ D = S  j ′ = j − ✶ ✐❢ D = L j + ✶ ✐❢ D = R  ❋✐♥❛❧❧②✱ t❤❡ ❛❝❝❡♣t✐♥❣ st❛t❡ ✐s r❡❛❝❤❡❞ � S i , ❛❝❝ i ❆♥✉❥ ❉❛✇❛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 ✶ , . . . , v n ♦❢ V ❢♦r ✇❤✐❝❤ t❤❡ t♦t❛❧ ❝♦st✿ n − ✶ � c ( v n , v ✶ ) + c ( v i , v i + ✶ ) i = ✶ ✐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❡ � ✶ ✐❢ ( u , v ) ∈ E c ( u , v ) = ✷ otherwise ❛♥❞ 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✳ z v ✶ z v ✹ ¯ z v ✶ ¯ x v ✶ y v ✶ y v ✷ x v ✹ z v ✷ z v ✹ y v ✹ x v ✷ y v ✸ x v ✸ z v ✷ ¯ z v ✸ ¯ z v ✸ ❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②

■♥ ❛❞❞✐t✐♦♥✱ ❢♦r ❡✈❡r② ❝❧❛✉s❡ c ✱ ✇❡ ❤❛✈❡ t✇♦ ❡❧❡♠❡♥ts x c ❛♥❞ y c ✳ ■❢ t❤❡ ❧✐t❡r❛❧ v ♦❝❝✉rs ✐♥ c ✱ ✇❡ ✐♥❝❧✉❞❡ t❤❡ tr✐♣❧❡ ( x c , y c , z vc ) ✐♥ M ✳ ❙✐♠✐❧❛r❧②✱ ✐❢ ¬ v ♦❝❝✉rs ✐♥ c ✱ ✇❡ ✐♥❝❧✉❞❡ t❤❡ tr✐♣❧❡ ( x c , y c , ¯ z vc ) ✐♥ 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 ✶ , . . . , S m } ♦❢ 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 ✶ , . . . , S m } 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 ✈❛❧✉❡ v i ❛♥❞ ✇❡✐❣❤t w i ✳ ❲❡ ❛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 ✶ , . . . , S m } ✳ ❲❡ ♠❛♣ t❤✐s t♦ ❛♥ ✐♥st❛♥❝❡ ♦❢ ❑◆❆P❙❆❈❑ ✇✐t❤ m ❡❧❡♠❡♥ts ❡❛❝❤ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ♦♥❡ ♦❢ t❤❡ S i ✱ ❛♥❞ ❤❛✈✐♥❣ ✇❡✐❣❤t ❛♥❞ ✈❛❧✉❡ � ( m + ✶ ) j − ✶ j ∈ S i ❛♥❞ s❡t t❤❡ t❛r❣❡t ✇❡✐❣❤t ❛♥❞ ✈❛❧✉❡ ❜♦t❤ t♦ ✸ n − ✶ � ( m + ✶ ) j 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 ✱ p − ✶ ✶ < r < p ✱ s✉❝❤ t❤❛t r p − ✶ = ✶ ♠♦❞ p ❛♥❞ r � = ✶ ♠♦❞ p ❢♦r q ❛❧❧ ♣r✐♠❡ ❞✐✈✐s♦rs q ♦❢ p − ✶ ✳ ❆♥✉❥ ❉❛✇❛r ❈♦♠♣❧❡①✐t② ❚❤❡♦r②