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  1. ❊♥❝♦❞✐♥❣ ◗✉❛♥t✐✜❡❞ ❈❙Ps ❛s ◗✉❛♥t✐✜❡❞ ❇♦♦❧❡❛♥ ❋♦r♠✉❧❛❡ ■❛♥ P✳ ●❡♥t✱ P❡t❡r ◆✐❣❤t✐♥❣❛❧❡ ❛♥❞ ❆♥❞r❡✇ ❘♦✇❧❡② ✶

  2. ❋✐♥✐t❡ ❞♦♠❛✐♥ ❝♦♥str❛✐♥t s❛t✐s❢❛❝t✐♦♥ ♣r♦❜❧❡♠ ✭❈❙P✮ • ❱❛r✐❛❜❧❡s ✇✐t❤ ❛ ✜♥✐t❡ ❞♦♠❛✐♥ ✕ ❡✳❣✳ A ∈ { 2 , 3 } , B ∈ { 1 , 2 , 4 } • ❈♦♥str❛✐♥ts ♣❧❛❝❡❞ ♦♥ ✈❛r✐❛❜❧❡s ✕ A � = B ✱ A + B = 4 • ❆ s♦❧✉t✐♦♥ ✐s ❛ ✈❛❧✐❞ ❛ss✐❣♥♠❡♥t t♦ ❛❧❧ ✈❛r✐❛❜❧❡s ✕ A = 3 ✱ B = 1 • ◆P✲❝♦♠♣❧❡t❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✷

  3. ■♥tr♦❞✉❝✐♥❣ q✉❛♥t✐✜❡rs ✭◗❈❙P✮ • ❊①✐st❡♥t✐❛❧ ✭ ∃ ✮ ❛♥❞ ✉♥✐✈❡rs❛❧ ✭ ∀ ✮ q✉❛♥t✐✜❡rs • A ∈ { 2 , 3 } , B ∈ { 1 , 2 , 4 } ✱ ∃ A ∃ B ✱ A � = B ✱ A + B = 4 • ∀ A ∃ B ✱ A + B = 4 ✕ ❙♦❧✉t✐♦♥ tr❡❡ ✭str❛t❡❣②✮ ✸

  4. ■♥tr♦❞✉❝✐♥❣ q✉❛♥t✐✜❡rs ✭◗❈❙P✮ • ◗✉❛♥t✐✜❝❛t✐♦♥ ♦r❞❡r ✐s s✐❣♥✐✜❝❛♥t ✕ ∀ A ∃ B ✱ A + B = 4 ✕ ∃ B ∀ A ✱ A + B = 4 • P❙P❆❈❊✲❝♦♠♣❧❡t❡ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✕ P❙P❆❈❊ ❛❧❣♦r✐t❤♠ tr❛✈❡rs❡s s♦❧✉t✐♦♥ tr❡❡ • ❊①♣♦♥❡♥t✐❛❧ s♣❛❝❡ t♦ ♣r♦✈✐❞❡ ❛ s♦❧✉t✐♦♥ ✹

  5. ❚❤❡ ❣❛♠❡ ♦❢ ◗❈❙P • ◗❈❙P ❝❛♥ ❜❡ t❤♦✉❣❤t ♦❢ ❛s ❛ ❣❛♠❡ • P❧❛②❡rs ❛r❡ ❡①✐st❡♥t✐❛❧ ❛♥❞ ✉♥✐✈❡rs❛❧ • ❙♦♠❡ ❣❛♠❡s ♠❛♣ ✐♥t♦ ◗❈❙P ✕ ❈♦♥♥❡❝t✲✹ ✭●❡♥t ❛♥❞ ❘♦✇❧❡②✮ ✕ ❆ ✈❛r✐❛♥t ♦❢ ●♦ ✭▲✐❝❤t❡♥st❡✐♥ ❛♥❞ ❙✐♣s❡r✮ ✕ ❖t❤❡❧❧♦ ✭■✇❛t❛ ❛♥❞ ❑❛s❛✐✮ ✺

  6. ◆♦✉❣❤ts ❛♥❞ ❝r♦ss❡s ✻

  7. ❲❤② ❝♦♥s✐❞❡r ◗❈❙P❄ • ◆❛t✉r❛❧ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❈❙P • Pr♦❜❧❡♠ s♦❧✈✐♥❣ ✇✐t❤ ✉♥❝❡rt❛✐♥t② ✕ ❯♥❝❡rt❛✐♥ ❞❛t❛ ❛t s♦❧✉t✐♦♥ t✐♠❡ ❡✳❣✳ ❞❡❧✐✈❡r② t✐♠❡ ✶✵ ❛♠ ± ✶ ❤♦✉r ∗ ✭▼✐♥✐♠❛❧✮ ❈♦✈❡r✐♥❣ s❡t ♦❢ s♦❧✉t✐♦♥s ✭❨♦r❦❡✲❙♠✐t❤ ❛♥❞ ●❡r✈❡t✮ ✕ ❯♥❝❡rt❛✐♥t② r❡s♦❧✈❡❞ ❞✉r✐♥❣ ❡①❡❝✉t✐♦♥ ♦❢ ♣❧❛♥ ∗ ●❛♠❡ ❛❣❛✐♥st t❤❡ ❡♥✈✐r♦♥♠❡♥t ✼

  8. ◗✉❛♥t✐✜❡❞ ❇♦♦❧❡❛♥ ❋♦r♠✉❧❛❡ ✭◗❇❋✮ • ❙✉❜s❡t ♦❢ ◗❈❙P ✭❛❧s♦ P❙P❆❈❊✲❝♦♠♣❧❡t❡✮ • ❲❡ ❝♦♥s✐❞❡r ❝♦♥❥✉♥❝t✐✈❡ ♥♦r♠❛❧ ❢♦r♠ ◗❇❋ ✐♥ ♣r❡♥❡① ❢♦r♠ ∀ a, b ∃ c, ( a ∨ ¬ c ) ∧ ( ¬ a ∨ ¬ b ∨ ¬ c ) • ❯♥✐t ♣r♦♣❛❣❛t✐♦♥ r✉❧❡s s✐♠✐❧❛r t♦ ❙❆❚ ✖ s❧✐❣❤t❧② str♦♥❣❡r ✽

  9. ❲❤② ❡♥❝♦❞❡❄ • ◗❇❋ ✐s t❤❡ s✉❜❥❡❝t ♦❢ r❡❝❡♥t r❡s❡❛r❝❤ ✕ ❇❛s✐❝ ❝♦♠♣❧❡t❡ ❛❧❣♦r✐t❤♠ ❜❛s❡❞ ♦♥ ❉❛✈✐s P✉t♥❛♠ ▲♦❣❡♠❛♥♥ ▲♦✈❡❧❛♥❞ ❛❧❣♦r✐t❤♠ ✕ ❈♦♥✢✐❝t ❛♥❞ s♦❧✉t✐♦♥ ❞✐r❡❝t❡❞ ❜❛❝❦❥✉♠♣✐♥❣ ✭●✉✐♥❝❤✐❣❧✐❛✱ ◆❛r✐③✲ ③❛♥♦ ❛♥❞ ❚❛❝❝❤❡❧❧❛✮ ✕ ❊✣❝✐❡♥t ✇❛t❝❤❡❞ ❞❛t❛ str✉❝t✉r❡s ✭●❡♥t✱ ●✉✐♥❝❤✐❣❧✐❛✱ ◆❛r✐③③❛♥♦✱ ❘♦✇❧❡② ❛♥❞ ❚❛❝❝❤❡❧❧❛✮ • ❚❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❢❛st ◗❇❋ s♦❧✈❡rs ❢♦r ◗❈❙P ✾

  10. ❉✐r❡❝t ❡♥❝♦❞✐♥❣ • ❲❡ ❝♦♥s✐❞❡r ❜✐♥❛r② ◗❈❙P ❢♦r t❤✐s ✇♦r❦ • ❊♥❝♦❞❡ ❈❙P ✈❛r✐❛❜❧❡ v ✇✐t❤ ❙❆❚ ✈❛r✐❛❜❧❡s x v i ❢♦r ❡❛❝❤ ✈❛❧✉❡ i • ❆t✲❧❡❛st✲♦♥❡ ❝❧❛✉s❡ ( � d i =1 x v i ) ✭ v t❛❦❡s ❛t ❧❡❛st ♦♥❡ ✈❛❧✉❡✮ • ❆t✲♠♦st✲♦♥❡ ❝❧❛✉s❡s � d � d j = i +1 ( ¬ x v i ∨ ¬ x v j ) i =1 • ❈♦♥✢✐❝t ❝❧❛✉s❡s ( ¬ x v i ∨ ¬ x w j ) ✶✵

  11. ●❧♦❜❛❧ ❆❝❝❡♣t❛❜✐❧✐t② ❊♥❝♦❞✐♥❣ ❢♦r ◗❈❙P • ❈♦♥s✐❞❡r❛❜❧② ♠♦r❡ ✐♥✈♦❧✈❡❞ t❤❛♥ ❞✐r❡❝t ❡♥❝♦❞✐♥❣ • ❆❝❝❡♣t❛❜❧❡ ❛ss✐❣♥♠❡♥t t♦ t❤❡ ❡♥❝♦❞❡❞ ◗❇❋ ❝♦rr❡s♣♦♥❞s t♦ ◗❈❙P ❛ss✐❣♥♠❡♥t • ❚❤❡ ❢♦r♠✉❧❛ ✐s r❡q✉✐r❡❞ t♦ ❜❡ tr✉❡ ❢♦r s♦♠❡ ✉♥❛❝❝❡♣t❛❜❧❡ ❛ss✐❣♥✲ ♠❡♥ts ✖ ✇❤❡r❡ ✉♥✐✈❡rs❛❧ ✈❛r✐❛❜❧❡s t❛❦❡ � = 1 ✈❛❧✉❡s • ❆❞❞✐t✐♦♥❛❧ ❧✐t❡r❛❧ z ✐♥ ♠♦st ❝❧❛✉s❡s • ❈♦♥✢✐❝t ❝❧❛✉s❡s ( ¬ x v i ∨ ¬ x w j ∨ z ) • Pr❡✈❡♥ts ✉♥✐t ♣r♦♣❛❣❛t✐♦♥ ✉♥t✐❧ ✐♥♥❡r♠♦st ✉♥✐✈❡rs❛❧ ✈❛r✐❛❜❧❡ ✐s s❡t ✶✶

  12. ▲♦❝❛❧ ❆❝❝❡♣t❛❜✐❧✐t② ❊♥❝♦❞✐♥❣ ✭r❡✜♥❡♠❡♥t ♦❢ ❛❜♦✈❡✮ • ▲♦❝❛❧ z u ✈❛r✐❛❜❧❡s ❛r❡ s❡t ❡❛r❧✐❡r t❤❛♥ z ❛♥❞ ❛❧❧♦✇ ✉♥✐t ♣r♦♣❛❣❛t✐♦♥ • . . . ∀ x v i . . . ∀ x w j . . . ( ¬ x v i ∨ ¬ x w j ∨ z w ) • . . . ∀ x u h . . . ∃ x v i . . . ∃ x w j . . . ( ¬ x v i ∨ ¬ x w j ∨ z u ) • ❙✐♠✉❧❛t❡s ❢♦r✇❛r❞ ❝❤❡❝❦✐♥❣ ✭▼❛♠♦✉❧✐s ❛♥❞ ❙t❡r❣✐♦✉✮ • ▲❛r❣❡ ♥✉♠❜❡r ♦❢ ✉♥❛❝❝❡♣t❛❜❧❡ ❛ss✐❣♥♠❡♥ts ✶✷

  13. ❆❞❛♣t❡❞ ▲♦❣ ❊♥❝♦❞✐♥❣ ✭❢✉rt❤❡r r❡✜♥❡♠❡♥t✮ • ❯♥❛r② ❡♥❝♦❞✐♥❣ ♦❢ ✉♥✐✈❡rs❛❧ ✈❛r✐❛❜❧❡s ❤❛s O (2 d ) ✉♥❛❝❝❡♣t❛❜❧❡ ❛s✲ s✐❣♥♠❡♥ts ✖ ▲♦❣ ❡♥❝♦❞✐♥❣ ❤❛s O ( d ) ✉♥❛❝❝❡♣t❛❜❧❡ ❛ss✐❣♥♠❡♥ts • Pr♦✈❡♥ ❝♦rr❡❝t • ❈❤❛♥♥❡❧ ❧♦❣ ❡♥❝♦❞✐♥❣ t♦ ✉♥❛r② ❡♥❝♦❞✐♥❣ ( z u ∨ x v 1 ∨ b v 2 ∨ b v 1 ∨ b v 0 ) ( z u ∨ x v 2 ∨ b v 2 ∨ b v 1 ∨ ¬ b v 0 ) ( z u ∨ x v 3 ∨ b v 2 ∨ ¬ b v 1 ∨ b v 0 ) ( z u ∨ x v 4 ∨ b v 2 ∨ ¬ b v 1 ∨ ¬ b v 0 ) ( z u ∨ x v 5 ∨ ¬ b v 2 ∨ b v 1 ∨ b v 0 ) • ❖♥❡✲✇❛② ❝❤❛♥♥❡❧❧✐♥❣ ♣r❡s❡r✈❡s ♣✉r❡ ❧✐t❡r❛❧ ♣r♦♣❛❣❛t✐♦♥ ✶✸

  14. ❉✐r❡❝t s♦❧✉t✐♦♥ ✈s✳ ❡♥❝♦❞✐♥❣ ✶✹

  15. ❋❧❛✇s ✐♥ ◗❈❙Ps • ❙♦♠❡ ✐♥st❛♥❝❡s tr✐✈✐❛❧❧② ❢❛❧s❡ • ❯♥✐✈❡rs❛❧s u 1 . . . u 7 ❢♦❧❧♦✇❡❞ ❜② ❡①✐st❡♥t✐❛❧ e • ❊❛❝❤ ✈❛❧✉❡ ♦❢ e ❝♦♥✢✐❝ts ✇✐t❤ s♦♠❡ ✈❛❧✉❡ ♦❢ s♦♠❡ u i • ❆rt✐✜❝✐❛❧❧② s❤✐❢ts ♣❤❛s❡ tr❛♥s✐t✐♦♥ • ❘❡❝❡♥t ✇♦r❦ ♦♥ ❝♦♥tr♦❧❧✐♥❣ ♣❛r❛♠❡t❡rs t♦ ❛✈♦✐❞ t❤✐s ✶✺

  16. ❈♦♥❝❧✉s✐♦♥s • ❊♥❝♦❞✐♥❣ ♦✉t♣❡r❢♦r♠s ❞✐r❡❝t s♦❧✉t✐♦♥ ♦♥ s♦♠❡ ♣r♦❜❧❡♠s ✕ ❙♦♠❡t✐♠❡s ❜② ♦r❞❡rs ♦❢ ♠❛❣♥✐t✉❞❡ • ▲♦✇ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❡✛♦rt • ❙✉♣♣♦rt ❡♥❝♦❞✐♥❣ r❡♠❛✐♥s ♦♣❡♥ • ●♦♦❞ ❜❡♥❝❤♠❛r❦ ♣r♦❜❧❡♠s r❡q✉✐r❡❞ ✶✻

  17. ❚❤❛♥❦ ②♦✉ ✶✼

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