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  1. ����� � ���� ������ ������������������������ �������������������������� � � � � � � � � � � �������� ��� ����������� ���� ����� ��� �� ���� ���� �� ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts ❖♥ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❝❛rr✐❡❞ ❜② ♣r♦❣r❛♠s ❛❜♦✉t t❤❡ ♦❜❥❡❝ts t❤❡② ❝♦♠♣✉t❡ ▼❛t❤✐❡✉ ❍♦②r✉♣ ❛♥❞ ❈r✐stó❜❛❧ ❘♦❥❛s ▲❖❘■❆ ✲ ■♥r✐❛✱ ◆❛♥❝② ✭❋r❛♥❝❡✮

  2. ▼❛✐♥ q✉❡st✐♦♥s ❉♦❡s ✐t ♠❛❦❡ ❛ ❞✐✛❡r❡♥❝❡❄ ❈❛♥ t❤❡ t✇♦ ♠❛❝❤✐♥❡s ♣❡r❢♦r♠ t❤❡ s❛♠❡ t❛s❦s❄ ❉♦❡s t❤❡ ❝♦❞❡ ♦❢ ❛ ♣r♦❣r❛♠ ❣✐✈❡ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✇❤❛t ✐t ❝♦♠♣✉t❡s❄ ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts ❚❤❡ ♣r♦❜❧❡♠ ❚✇♦ ✇❛②s ♦❢ ♣r♦✈✐❞✐♥❣ ❛ ❝♦♠♣✉t❛❜❧❡ ❢✉♥❝t✐♦♥ f : N → N t♦ ❛ ♠❛❝❤✐♥❡✿ • ❱✐❛ t❤❡ ❣r❛♣❤ ♦❢ f ✭ ✐♥✜♥✐t❡ ♦❜❥❡❝t✮✱ • ❱✐❛ ❛ ♣r♦❣r❛♠ ❝♦♠♣✉t✐♥❣ f ✭ ✜♥✐t❡ ♦❜❥❡❝t✮✳

  3. ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts ❚❤❡ ♣r♦❜❧❡♠ ❚✇♦ ✇❛②s ♦❢ ♣r♦✈✐❞✐♥❣ ❛ ❝♦♠♣✉t❛❜❧❡ ❢✉♥❝t✐♦♥ f : N → N t♦ ❛ ♠❛❝❤✐♥❡✿ • ❱✐❛ t❤❡ ❣r❛♣❤ ♦❢ f ✭ ✐♥✜♥✐t❡ ♦❜❥❡❝t✮✱ • ❱✐❛ ❛ ♣r♦❣r❛♠ ❝♦♠♣✉t✐♥❣ f ✭ ✜♥✐t❡ ♦❜❥❡❝t✮✳ ▼❛✐♥ q✉❡st✐♦♥s • ❉♦❡s ✐t ♠❛❦❡ ❛ ❞✐✛❡r❡♥❝❡❄ • ❈❛♥ t❤❡ t✇♦ ♠❛❝❤✐♥❡s ♣❡r❢♦r♠ t❤❡ s❛♠❡ t❛s❦s❄ • ❉♦❡s t❤❡ ❝♦❞❡ ♦❢ ❛ ♣r♦❣r❛♠ ❣✐✈❡ ♠♦r❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✇❤❛t ✐t ❝♦♠♣✉t❡s❄

  4. ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts ❚❤❡ ♣r♦❜❧❡♠ ❚❤❡ ❛♥s✇❡r ❞❡♣❡♥❞s ♦♥✿ • ❲❤❡t❤❡r t❤❡ ❢✉♥❝t✐♦♥s f ❛r❡ ♣❛rt✐❛❧ ♦r t♦t❛❧ ✱ • ❚❤❡ t❛s❦ t♦ ❜❡ ♣❡r❢♦r♠❡❞ ❜② t❤❡ ♠❛❝❤✐♥❡ ✭❡✳❣✳ ❞❡❝✐❞❡ ♦r s❡♠✐✲❞❡❝✐❞❡ s♦♠❡t❤✐♥❣✮✳ ❉❡❝✐❞❛❜✐❧✐t② ❙❡♠✐✲❞❡❝✐❞❛❜✐❧✐t② P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ❚♦t❛❧ ❢✉♥❝t✐♦♥s

  5. ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts

  6. ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ❉❡❝✐❞❛❜✐❧✐t② ❙❡♠✐✲❞❡❝✐❞❛❜✐❧✐t② P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ? ❚♦t❛❧ ❢✉♥❝t✐♦♥s

  7. ❚❤❡♦r❡♠ ✭❚✉r✐♥❣✱ ✶✾✸✻✮ ●✐✈❡♥ ❛ ♣r♦❣r❛♠ ❢♦r ✱ ❛ ♠❛❝❤✐♥❡ ❝❛♥♥♦t ❞♦ ❜❡tt❡r✳ ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ❉❡❝✐❞❛❜✐❧✐t② ❙❡♠✐✲❞❡❝✐❞❛❜✐❧✐t② P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ? ❚♦t❛❧ ❢✉♥❝t✐♦♥s ●✐✈❡♥ ✭❛♥② ❡♥✉♠❡r❛t✐♦♥ ♦❢✮ t❤❡ ❣r❛♣❤ ♦❢ f ✱ ♦♥❡ ❝❛♥♥♦t ❞❡❝✐❞❡ ✇❤❡t❤❡r f (0) ✐s ❞❡✜♥❡❞✳

  8. ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ❉❡❝✐❞❛❜✐❧✐t② ❙❡♠✐✲❞❡❝✐❞❛❜✐❧✐t② P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ? ❚♦t❛❧ ❢✉♥❝t✐♦♥s ●✐✈❡♥ ✭❛♥② ❡♥✉♠❡r❛t✐♦♥ ♦❢✮ t❤❡ ❣r❛♣❤ ♦❢ f ✱ ♦♥❡ ❝❛♥♥♦t ❞❡❝✐❞❡ ✇❤❡t❤❡r f (0) ✐s ❞❡✜♥❡❞✳ ❚❤❡♦r❡♠ ✭❚✉r✐♥❣✱ ✶✾✸✻✮ ●✐✈❡♥ ❛ ♣r♦❣r❛♠ ❢♦r f ✱ ❛ ♠❛❝❤✐♥❡ ❝❛♥♥♦t ❞♦ ❜❡tt❡r✳

  9. ❚❤❡♦r❡♠ ✭❘✐❝❡✱ ✶✾✺✸✮ ●✐✈❡♥ ❛ ♣r♦❣r❛♠ ❢♦r ✱ ❛ ♠❛❝❤✐♥❡ ❝❛♥♥♦t ❞♦ ❜❡tt❡r✳ ❆♥s✇❡rs ●✐✈❡♥ t❤❡ ❣r❛♣❤ ♦❢ ✱ ♦♥❧② tr✐✈✐❛❧ ♣r♦♣❡rt✐❡s✿ t❤❡ ❞❡❝✐s✐♦♥ ❛❜♦✉t ❛♣♣❧✐❡s t♦ ❡✈❡r② ✳ ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ❉❡❝✐❞❛❜✐❧✐t② ❙❡♠✐✲❞❡❝✐❞❛❜✐❧✐t② P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ? ❚♦t❛❧ ❢✉♥❝t✐♦♥s ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❤❛t ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❛❜♦✉t f ❄

  10. ❚❤❡♦r❡♠ ✭❘✐❝❡✱ ✶✾✺✸✮ ●✐✈❡♥ ❛ ♣r♦❣r❛♠ ❢♦r ✱ ❛ ♠❛❝❤✐♥❡ ❝❛♥♥♦t ❞♦ ❜❡tt❡r✳ ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ❉❡❝✐❞❛❜✐❧✐t② ❙❡♠✐✲❞❡❝✐❞❛❜✐❧✐t② P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ? ❚♦t❛❧ ❢✉♥❝t✐♦♥s ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❤❛t ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❛❜♦✉t f ❄ ❆♥s✇❡rs ●✐✈❡♥ t❤❡ ❣r❛♣❤ ♦❢ f ✱ ♦♥❧② tr✐✈✐❛❧ ♣r♦♣❡rt✐❡s✿ t❤❡ ❞❡❝✐s✐♦♥ ❛❜♦✉t λx. ⊥ ❛♣♣❧✐❡s t♦ ❡✈❡r② f ✳

  11. ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ❉❡❝✐❞❛❜✐❧✐t② ❙❡♠✐✲❞❡❝✐❞❛❜✐❧✐t② P❛rt✐❛❧ ❢✉♥❝t✐♦♥s program ≡ graph ❚♦t❛❧ ❢✉♥❝t✐♦♥s ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❤❛t ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ❛❜♦✉t f ❄ ❆♥s✇❡rs ●✐✈❡♥ t❤❡ ❣r❛♣❤ ♦❢ f ✱ ♦♥❧② tr✐✈✐❛❧ ♣r♦♣❡rt✐❡s✿ t❤❡ ❞❡❝✐s✐♦♥ ❛❜♦✉t λx. ⊥ ❛♣♣❧✐❡s t♦ ❡✈❡r② f ✳ ❚❤❡♦r❡♠ ✭❘✐❝❡✱ ✶✾✺✸✮ ●✐✈❡♥ ❛ ♣r♦❣r❛♠ ❢♦r f ✱ ❛ ♠❛❝❤✐♥❡ ❝❛♥♥♦t ❞♦ ❜❡tt❡r✳

  12. ❚❤❡♦r❡♠ ✭❙❤❛♣✐r♦✱ ✶✾✺✻✮ ●✐✈❡♥ ❛ ♣r♦❣r❛♠ ❢♦r ✱ ❛ ♠❛❝❤✐♥❡ ❝❛♥♥♦t ❞♦ ❜❡tt❡r✳ ❆♥s✇❡rs ●✐✈❡♥ t❤❡ ❣r❛♣❤ ♦❢ ✱ ❡①❛❝t❧② t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦r♠✿ ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ❉❡❝✐❞❛❜✐❧✐t② ❙❡♠✐✲❞❡❝✐❞❛❜✐❧✐t② P❛rt✐❛❧ ❢✉♥❝t✐♦♥s program ≡ graph ? ❚♦t❛❧ ❢✉♥❝t✐♦♥s ❲❤❛t ❝❛♥ ❜❡ s❡♠✐✲❞❡❝✐❞❡❞ ❛❜♦✉t f ❄

  13. ❚❤❡♦r❡♠ ✭❙❤❛♣✐r♦✱ ✶✾✺✻✮ ●✐✈❡♥ ❛ ♣r♦❣r❛♠ ❢♦r ✱ ❛ ♠❛❝❤✐♥❡ ❝❛♥♥♦t ❞♦ ❜❡tt❡r✳ ❚❤❡ ♣r♦❜❧❡♠ ❍✐st♦r✐❝❛❧ r❡s✉❧ts ◆❡✇ r❡s✉❧ts ▲✐♠✐ts P❛rt✐❛❧ ❢✉♥❝t✐♦♥s ❉❡❝✐❞❛❜✐❧✐t② ❙❡♠✐✲❞❡❝✐❞❛❜✐❧✐t② P❛rt✐❛❧ ❢✉♥❝t✐♦♥s program ≡ graph ? ❚♦t❛❧ ❢✉♥❝t✐♦♥s ❲❤❛t ❝❛♥ ❜❡ s❡♠✐✲❞❡❝✐❞❡❞ ❛❜♦✉t f ❄ ❆♥s✇❡rs ●✐✈❡♥ t❤❡ ❣r❛♣❤ ♦❢ f ✱ ❡①❛❝t❧② t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❢♦r♠✿ ( f ( a 1 ) = u 1 ∧ . . . ∧ f ( a i ) = u i ) ∨ ( f ( b 1 ) = v 1 ∧ . . . ∧ f ( b j ) = v j ) ∨ ( f ( c 1 ) = w 1 ∧ . . . ∧ f ( c k ) = w k ) ∨ . . .

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