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t r r ts - PowerPoint PPT Presentation

Jul 11, 2023 •5 likes •1.55k views

t r r ts tts r t rs r tt t rsts

  1. ❋♩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

  2. ❚✉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❡❀ • ÎŽ : ( 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

  3. ❈♊♥✜❣✉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

  4. ❈♊♠♣✉t❛t✐♊♥s ●✐✈❡♥ ❛ ♠❛❝❀✐♥❡ M = ( Q , Σ , s , ÎŽ ) ✇❡ 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 ❀ • ÎŽ ( 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

  5. ❈♊♠♣✉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

  6. ❉❡❝✐❞❛❜✐❧✐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

  7. ❊①❛♠♣❧❡ ❈♊♥s✐❞❡r t❀❡ ♠❛❝❀✐♥❡ ✇✐t❀ ÎŽ ❣✐✈❡♥ ❜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

  8. ▌✉❧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 ❀ ❛♥❞ • ÎŽ : ( 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

  9. ❘✉♥♥✐♥❣ ❚✐♠❡ ❲✐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

  10. ❈♊♠♣❧❡①✐t② ❋♩r ❛♥② ❢✉♥❝t✐♊♥ f : ■ ◆ → ■ ◆ ✱ ✇❡ s❛② t❀❛t ❛ ❧❛♥❣✉❛❣❡ L ✐s ✐♥ ❚■▌❊ ( f ) ✐❢ t❀❡r❡ ✐s ❛ ♠❛❝❀✐♥❡ M = ( Q , Σ , s , ÎŽ ) ✱ 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

  11. ❉❡❝✐❞❛❜✐❧✐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

  12. ◆♊♥❞❡t❡r♠✐♥✐s♠ ■❢✱ ✐♥ t❀❡ ❞❡✜♥✐t✐♊♥ ♊❢ ❛ ❚✉r✐♥❣ ♠❛❝❀✐♥❡✱ ✇❡ r❡❧❛① t❀❡ ❝♊♥❞✐t✐♊♥ ♊♥ ÎŽ ❜❡✐♥❣ ❛ ❢✉♥❝t✐♊♥ ❛♥❞ ✐♥st❡❛❞ ❛❧❧♩✇ ❛♥ ❛r❜✐tr❛r② r❡❧❛t✐♊♥✱ ✇❡ ♩❜t❛✐♥ ❛ ♥♊♥❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❀✐♥❡ ✳ ÎŽ ⊆ ( 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

  13. ❈♊♠♣✉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

  14. ❈♊♠♣❧❡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

  15. 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

  16. P♊❧②♥♊♠✐❛❧ ❚✐♠❡ ∞ ï¿œ ❚■▌❊ ( 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✐♠❡ Ξ ( n ✶✵✵ ) ✐s ❢❡❛s✐❜❧❡✱ ❜✉t ✐t ✇✐❧❧ ❡✈❡♥t✉❛❧❧② r✉♥ ❢❛st❡r t❀❛♥ ♊♥❡ t❀❛t t❛❊❡s t✐♠❡ Ξ ( ✷ n ) ✳ ▌❛❊✐♥❣ t❀❡ ❞✐st✐♥❝t✐♊♥ ❜❡t✇❡❡♥ ♣♊❧②♥♊♠✐❛❧ ❛♥❞ ❡①♣♊♥❡♥t✐❛❧ r❡s✉❧ts ✐♥ ❛ ✉s❡❢✉❧ ❛♥❞ ❡❧❡❣❛♥t t❀❡♊r②✳ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  17. ❊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

  18. ❆♥❛❧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

  19. ❊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

  20. ❆♥❛❧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♩♩❩ Ξ ( 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

  21. 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 ℩( √ x ) st❡♣s ❛♥❞ ✐s t❀❡r❡❢♊r❡ ♥♊t ♣♊❧②♥♊♠✐❛❧ ✐♥ t❀❡ ❧❡♥❣t❀ ♊❢ t❀❡ ✐♥♣✉t✳ ■s Pr✐♠❡ ∈ P❄ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  22. ❇♊♊❧❡❛♥ ❊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

  23. ❊✈❛❧✉❛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

  24. ❇♊♊❧❡❛♥ ❊✈❛❧✉❛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

  25. ❘✉❧❡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

  26. ❆♥❛❧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

  27. ❙❛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

  28. ❈✐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

  29. ❈❱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

  30. ❈♊♠♣♊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

  31. ❍❛♠✐❧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

  32. ❊①❛♠♣❧❡s ❚❀❡ ✜rst ♊❢ t❀❡s❡ ❣r❛♣❀s ✐s ♥♊t ❍❛♠✐❧t♊♥✐❛♥✱ ❜✉t t❀❡ s❡❝♊♥❞ ♊♥❡ ✐s✳ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  33. ●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

  34. 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

  35. ❱❡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

  36. ◆♊♥❞❡t❡r♠✐♥✐s♠ ■❢✱ ✐♥ t❀❡ ❞❡✜♥✐t✐♊♥ ♊❢ ❛ ❚✉r✐♥❣ ♠❛❝❀✐♥❡✱ ✇❡ r❡❧❛① t❀❡ ❝♊♥❞✐t✐♊♥ ♊♥ ÎŽ ❜❡✐♥❣ ❛ ❢✉♥❝t✐♊♥ ❛♥❞ ✐♥st❡❛❞ ❛❧❧♩✇ ❛♥ ❛r❜✐tr❛r② r❡❧❛t✐♊♥✱ ✇❡ ♩❜t❛✐♥ ❛ ♥♊♥❞❡t❡r♠✐♥✐st✐❝ ❚✉r✐♥❣ ♠❛❝❀✐♥❡ ✳ ÎŽ ⊆ ( 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

  37. ❈♊♠♣✉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

  38. ◆♊♥❞❡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 ✳ ∞ ï¿œ ◆❚■▌❊ ( n k ) ◆P = k = ✶ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  39. ◆♊♥❞❡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

  40. ◆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

  41. ◆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

  42. ●❡♥❡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

  43. ❘❡❞✉❝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

  44. ❘❡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

  45. ❘❡❞✉❝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

  46. ❈♊♠♣❧❡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

  47. ❙❆❚ ✐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 , ÎŽ ) ❛♥❞ ❛ ❜♊✉♥❞ 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

  48. ❇♊♊❧❡❛♥ ❋♩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

  49. ■♥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

  50. ■♥✐t✐❛❧ st❛t❡ ✐s s ❛♥❞ t❀❡ ❀❡❛❞ ✐s ✐♥✐t✐❛❧❧② ❛t t❀❡ ❜❡❣✐♥♥✐♥❣ ♊❢ t❀❡ t❛♣❡✳ S ✶ , s ∧ H ✶ , ✶ ❚❀❡ ❀❡❛❞ ✐s ♥❡✈❡r ✐♥ t✇♩ ♣❧❛❝❡s ❛t ♊♥❝❡ ï¿œ ï¿œ ï¿œ ( H i , j → ( ¬ H i , j ′ )) i j j ′ ï¿œ = j ❚❀❡ ♠❛❝❀✐♥❡ ✐s ♥❡✈❡r ✐♥ t✇♩ st❛t❡s ❛t ♊♥❝❡ ï¿œ ï¿œ ï¿œ ( S i , q → ( ¬ S i , q ′ )) q i q ′ ï¿œ = q ❊❛❝❀ t❛♣❡ ❝❡❧❧ ❝♊♥t❛✐♥s ♊♥❧② ♊♥❡ s②♠❜♊❧ ï¿œ ï¿œ ï¿œ ï¿œ ( T i , j ,σ → ( ¬ T i , j ,σ ′ )) i j σ σ ′ ï¿œ = σ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  51. ❚❀❡ ✐♥✐t✐❛❧ t❛♣❡ ❝♊♥t❡♥ts ❛r❡ x ï¿œ ï¿œ T ✶ , j , x j ∧ T ✶ , j , ⊔ j ≀ n n < j ❚❀❡ t❛♣❡ ❞♊❡s ♥♊t ❝❀❛♥❣❡ ❡①❝❡♣t ✉♥❞❡r t❀❡ ❀❡❛❞ ï¿œ ï¿œ ï¿œ ï¿œ ( H i , j ∧ T i , j ′ ,σ ) → T i + ✶ , j ′ ,σ j ′ ï¿œ = j σ i j ❊❛❝❀ st❡♣ ✐s ❛❝❝♩r❞✐♥❣ t♩ ÎŽ ✳ ï¿œ ï¿œ ï¿œ ï¿œ ( H i , j ∧ S i , q ∧ T i , j ,σ ) i j σ q ï¿œ ( H i + ✶ , j ′ ∧ S i + ✶ , q ′ ∧ T i + ✶ , j ,σ ′ ) → ∆ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  52. ✇❀❡r❡ ∆ ✐s t❀❡ s❡t ♊❢ ❛❧❧ tr✐♣❧❡s ( q ′ , σ ′ , D ) s✉❝❀ t❀❛t (( q , σ ) , ( q ′ , σ ′ , D )) ∈ ÎŽ ❛♥❞  j ✐❢ D = S  j ′ = j − ✶ ✐❢ D = L j + ✶ ✐❢ D = R  ❋✐♥❛❧❧②✱ t❀❡ ❛❝❝❡♣t✐♥❣ st❛t❡ ✐s r❡❛❝❀❡❞ ï¿œ S i , ❛❝❝ i ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  53. ❈◆❋ ❆ ❇♊♊❧❡❛♥ ❡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

  54. ✾❙❆❚ ❆ ❇♊♊❧❡❛♥ ❡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

  55. ❈♊♠♣♊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

  56. ■♥❞❡♣❡♥❞❡♥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

  57. ❘❡❞✉❝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

  58. ❊①❛♠♣❧❡ ( x ✶ √ x ✷ √ ¬ x ✾ ) ∧ ( x ✾ √ ¬ x ✷ √ ¬ x ✶ ) x ✶ ¬ x ✾ x ✷ ¬ x ✶ ¬ x ✷ x ✾ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  59. ❈❧✐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

  60. ❈❧✐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

  61. 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 ) ï¿œ = 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

  62. ✞✲❈♊❧♊✉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

  63. ●❛❞❣❡t l ✶ l ✷ l ✾ b ❲✐t❀ ❛ ❢✉rt❀❡r ❡❞❣❡ ❢r♩♠ a t♩ b ✳ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  64. ❍❛♠✐❧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

  65. ❍❛♠✐❧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

  66. ❚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 − ✶ ï¿œ c ( v n , v ✶ ) + c ( v i , v i + ✶ ) i = ✶ ✐s t❀❡ s♠❛❧❧❡st ♣♊ss✐❜❧❡✳ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  67. ❚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

  68. ❘❡❞✉❝t✐♊♥ ❚❀❡r❡ ✐s ❛ s✐♠♣❧❡ r❡❞✉❝t✐♊♥ ❢r♩♠ ❍❆▌ t♩ ❚❙P✱ ♠❛♣♣✐♥❣ ❛ ❣r❛♣❀ ( V , E ) t♩ t❀❡ tr✐♣❧❡ ( V , c : V × V → ■ ◆ , n ) ✱ ✇❀❡r❡ ï¿œ ✶ ✐❢ ( u , v ) ∈ E c ( u , v ) = ✷ otherwise ❛♥❞ n ✐s t❀❡ s✐③❡ ♊❢ V ✳ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  69. ❙❡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

  70. ✾❉ ▌❛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

  71. ❘❡❞✉❝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

  72. ■♥ ❛❞❞✐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

  73. ❊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

  74. ❙❡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

  75. ❑♥❛♣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

  76. ❘❡❞✉❝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 ❛♥❞ ✈❛❧✉❡ ï¿œ ( m + ✶ ) j − ✶ j ∈ S i ❛♥❞ s❡t t❀❡ t❛r❣❡t ✇❡✐❣❀t ❛♥❞ ✈❛❧✉❡ ❜♩t❀ t♩ ✾ n − ✶ ï¿œ ( m + ✶ ) j j = ✵ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  77. ❙❝❀❡❞✉❧✐♥❣ ❙♊♠❡ ❡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

  78. ❙❝❀❡❞✉❧✐♥❣ ❙❡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

  79. ❘❡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

  80. ❱❛❧✐❞✐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 ￜ∈ ❙❆❚ ❇② ❛♥ ❡①❀❛✉st✐✈❡ s❡❛r❝❀ ❛❧❣♊r✐t❀♠ s✐♠✐❧❛r t♩ t❀❡ ♊♥❡ ❢♊r ❙❆❚✱ ❱❆▲ ✐s ✐♥ ❚■▌❊ ( n ✷ ✷ n ) ✳ ■s ❱❆▲ ∈ ◆P❄ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  81. ❱❛❧✐❞✐t② ❱❆▲ = { φ | φ ￜ∈ ❱❆▲ } ✖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

  82. ❈♊♠♣❧❡♠❡♥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

  83. ❙✉❝❝✐♥❝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

  84. ❙✉❝❝✐♥❝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

  85. ❝♊✲◆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

  86. ◆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 ï¿œ = ◆P ï¿œ = ❝♊✲◆P • P ï¿œ = ◆P ∩ ❝♊✲◆P = ◆P = ❝♊✲◆P • P ï¿œ = ◆P ∩ ❝♊✲◆P ï¿œ = ◆P ï¿œ = ❝♊✲◆P ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

  87. ❝♊✲◆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

  88. 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

  89. 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 ï¿œ = ✶ ♠♩❞ p ❢♊r q ❛❧❧ ♣r✐♠❡ ❞✐✈✐s♩rs q ♊❢ p − ✶ ✳ ❆♥✉❥ ❉❛✇❛r ❈♊♠♣❧❡①✐t② ❚❀❡♊r②

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