◆❚P ✷ ❆rt❡♠ ❈❤❡r♥✐❦♦✈ ❯♥✐✈❡rs✐t② ▲②♦♥ ✶✱ ❋r❛♥❝❡ ✶✺t❤ ▲❛t✐♥ ❆♠❡r✐❝❛♥ ❙②♠♣♦s✐✉♠ ♦♥ ▼❛t❤❡♠❛t✐❝❛❧ ▲♦❣✐❝ ❇♦❣♦t❛✱ ❏✉♥❡ ✽ ✷✵✶✷
●❡♥❡r❛❧✐③❛t✐♦♥s ♦❢ st❛❜✐❧✐t②
◆❚P ✷ ✿ ❉❡✜♥✐t✐♦♥ ❉❡✜♥✐t✐♦♥ ✭❙❤❡❧❛❤✮ ❆ ❢♦r♠✉❧❛ φ ( ① , ② ) ❤❛s ❚P ✷ ✐❢ t❤❡r❡ ❛r❡ ( ❛ ✐ , ❥ ) ✐ , ❥ ∈ ω ❛♥❞ ❦ ∈ ω s✉❝❤ t❤❛t✿ ◮ { φ ( ① , ❛ ✐ , ❥ ) } ❥ ∈ ω ✐s ❦ ✲✐♥❝♦♥s✐st❡♥t ❢♦r ❡✈❡r② ✐ ∈ ω ✱ ◮ � � φ ( ① , ❛ ✐ , ❢ ( ✐ ) ) ✐ ∈ ω ✐s ❝♦♥s✐st❡♥t ❢♦r ❡✈❡r② ❢ : ω → ω ✳ ❚ ✐s ❝❛❧❧❡❞ ◆❚P ✷ ✐❢ ♥♦ ❢♦r♠✉❧❛ ❤❛s ❚P ✷ ✳
◆❚P ✷ ✿ ❊①❛♠♣❧❡s ◮ ❊✈❡r② s✐♠♣❧❡ ♦r ◆■P t❤❡♦r② ✐s ◆❚P ✷ ✳ ◮ ▲❡t ❚ ❜❡ ❛ ♠♦❞❡❧ ❝♦♠♣❧❡t❡ ❣❡♦♠❡tr✐❝ t❤❡♦r② ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ▲ ✭✐✳❡✳ ✐t ❡❧✐♠✐♥❛t❡s ∃ ∞ ❛♥❞ t❤❡ ♠♦❞❡❧ t❤❡♦r❡t✐❝ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡ s❛t✐s✜❡s ❡①❝❤❛♥❣❡✮✳ ▲❡t ▲ ′ ❜❡ ❛♥ ❡①♣❛♥s✐♦♥ ♦❢ ▲ ❜② ❛ ♥❡✇ ✉♥❛r② ♣r❡❞✐❝❛t❡ P ( ① ) ✳ ❚❤❡♥ ❚ ❤❛s ❛ ♠♦❞❡❧ ❝♦♠♣❛♥✐♦♥ ❚ ′ ✐♥ ▲ ′ ❛♥❞ t❤✐s ♠♦❞❡❧ ❝♦♠♣❛♥✐♦♥ ✐s ◆❚P ✷ ✭❣❡♥❡r❛❧✐③✐♥❣ ❈❤❛t③✐❞❛❦✐s✲P✐❧❧❛②✮✳ ◮ ❊✳❣✳ ❢✉s✐♥❣ ❛ ❞❡♥s❡ ❧✐♥❡❛r ♦r❞❡r ✇✐t❤ ❛ r❛♥❞♦♠ ❣r❛♣❤ ❣✐✈❡s ❛♥ ◆❚P ✷ t❤❡♦r②✳
❚P ✷ ✐s ✇✐t♥❡ss❡❞ ✐♥ ♦♥❡ ✈❛r✐❛❜❧❡ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s q✉✐t❡ ✉s❡❢✉❧ ❢♦r ❝❤❡❝❦✐♥❣ t❤❛t ❛ ♣❛rt✐❝✉❧❛r str✉❝t✉r❡ ✐s ◆❚P ✷ ✳ ❚❤❡♦r❡♠ ✭❈❤✳✮ ❚ ✐s ◆❚P ✷ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② ❢♦r♠✉❧❛ ϕ ( ① , ② ) ✇✐t❤ ① s✐♥❣❧❡t♦♥ ✐s ◆❚P ✷ ✳ ✭■♥ ❢❛❝t✱ t❤✐s ❢♦❧❧♦✇s ❢r♦♠ ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ♦♥ s✉❜✲♠✉❧t✐♣❧✐❝❛t✐✈✐t② ♦❢ ❜✉r❞❡♥ ✐♥ ❛r❜✐tr❛r② t❤❡♦r✐❡s ❛♥❞ ❛♥s✇❡rs ❛ q✉❡st✐♦♥ ♦❢ ❙❤❡❧❛❤✮✳
◆❚P ✷ ✿ ❱❛❧✉❡❞ ✜❡❧❞s ◮ ❈♦♥s✐❞❡r t❤❡ ✈❛❧✉❡❞ ✜❡❧❞ ❑ = � ♣ ♣r✐♠❡ Q ♣ / U ✱ ✇❤❡r❡ U ✐s ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ✉❧tr❛✜❧t❡r✳ ◮ ❚❤❡ t❤❡♦r② ♦❢ ❑ ✐s ♥♦t s✐♠♣❧❡✿ ❜❡❝❛✉s❡ t❤❡ ✈❛❧✉❡ ❣r♦✉♣ ✐s ❧✐♥❡❛r❧② ♦r❞❡r❡❞✳ ◮ ❚❤❡ t❤❡♦r② ♦❢ ❑ ✐s ♥♦t ◆■P✿ t❤❡ r❡s✐❞✉❡ ✜❡❧❞ ✐s ♣s❡✉❞♦✲✜♥✐t❡✱ t❤✉s ❤❛s t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♣r♦♣❡rt② ❜② ❛ r❡s✉❧t ♦❢ ❉✉r❡t✳ ◮ ❊✈❡♥ ✐♥ t❤❡ ♣✉r❡ ✜❡❧❞ ❧❛♥❣✉❛❣❡✱ ❛s t❤❡ ✈❛❧✉❛t✐♦♥ r✐♥❣ ✐s ❞❡✜♥❛❜❧❡ ✉♥✐❢♦r♠❧② ✐♥ ♣ ✭❆①✮✳
◆❚P ✷ ✿ ❱❛❧✉❡❞ ✜❡❧❞s ❍♦✇❡✈❡r✱ ❑ ✐s ◆❚P ✷ ✭❛♥❞ ❡✈❡♥ str♦♥❣✱ ♦❢ ✜♥✐t❡ ❜✉r❞❡♥✮ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✭❈❤✳✮ ▲❡t ❑ = ( ❑ , ❦ , Γ) ❜❡ ❛ ❤❡♥s❡❧✐❛♥ ✈❛❧✉❡❞ ✜❡❧❞ ♦❢ ❡q✉✐❝❤❛r❛❝t❡r✐st✐❝ ✵ ✱ ✐♥ t❤❡ ❉❡♥❡❢✲P❛s ❧❛♥❣✉❛❣❡✳ ❆ss✉♠❡ t❤❛t ❦ ✐s ◆❚P ✷ ✳ ❚❤❡♥ ❑ ✐s ◆❚P ✷ ✳ ❆♥❛❧♦❣♦✉s t♦ t❤❡ t❤❡♦r❡♠ ♦❢ ❉❡❧♦♥ ❢♦r ◆■P✳
❑✐❦②♦✲❙❤❡❧❛❤✿ ■t ❚ ❤❛s t❤❡ ❙tr✐❝t ❖r❞❡r Pr♦♣❡rt② ✭✇❤✐❝❤ ✐s t❤❡ ❝❛s❡ ✇✐t❤ ✈❛❧✉❡❞ ✜❡❧❞s✮✱ t❤❡♥ t❤❡ ♠♦❞❡❧ ❝♦♠♣❛♥✐♦♥ ♦❢ ❚ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ❞♦❡s ♥♦t ❡①✐st✳ ❍r✉s❤♦✈s❦✐✴❆③❣✐♥✿ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ ✐♠♣♦s❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t ✐s ❝♦♥tr❛❝t✐✈❡ ✭✐✳❡✳ ✈ ① ♥ ✈ ① ❢♦r ❛❧❧ ♥ ✮✱ t❤❡♥ t❤❡ ♠♦❞❡❧ ❝♦♠♣❛♥✐♦♥ ❱❋❆ ✵ ❡①✐sts✳ ■t ✐s ❛①✐♦♠❛t✐③❡❞ ❜② s❛②✐♥❣ t❤❛t ❦ ✐s ❛ ♠♦❞❡❧ ♦❢ ❆❈❋❆ ✵ ✱ ✐s ❛ ❞✐✈✐s✐❜❧❡ ♦r❞❡r❡❞ ♠♦❞✉❧❡ ❛♥❞ ❑ ✐s ✲❤❡♥s❡❧✐❛♥✳ ❆ ♥❛t✉r❛❧ ♠♦❞❡❧ ♦❢ ❱❋❆ ✵ ✿ ❛ ♥♦♥✲st❛♥❞❛r❞ ❋r♦❜❡♥✐✉s ❛❝t✐♥❣ ♦♥ ❛♥ ❛❧❣❡❜r❛✐❝❛❧❧② ❝❧♦s❡❞ ✈❛❧✉❡❞ ✜❡❧❞ ♦❢ ❝❤❛r ✵✳ ❆❣❛✐♥ ♥❡✐t❤❡r s✐♠♣❧❡ ♥♦r ◆■P✳ ◆❚P ✷ ✿ ❱❛❧✉❡❞ ❞✐✛❡r❡♥❝❡ ✜❡❧❞s ◮ ❲❡ ❝♦♥s✐❞❡r ✈❛❧✉❡❞ ❞✐✛❡r❡♥❝❡ ✜❡❧❞s ❑ = ( ❑ , ❦ , Γ , σ ) ♦❢ ❡q✉✐❝❤❛r❛❝t❡r✐st✐❝ ✵ ✭✐✳❡✳ σ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❑ ♣r❡s❡r✈✐♥❣ t❤❡ ✈❛❧✉❛t✐♦♥ r✐♥❣✮✳
◆❚P ✷ ✿ ❱❛❧✉❡❞ ❞✐✛❡r❡♥❝❡ ✜❡❧❞s ◮ ❲❡ ❝♦♥s✐❞❡r ✈❛❧✉❡❞ ❞✐✛❡r❡♥❝❡ ✜❡❧❞s ❑ = ( ❑ , ❦ , Γ , σ ) ♦❢ ❡q✉✐❝❤❛r❛❝t❡r✐st✐❝ ✵ ✭✐✳❡✳ σ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ❑ ♣r❡s❡r✈✐♥❣ t❤❡ ✈❛❧✉❛t✐♦♥ r✐♥❣✮✳ ◮ ❑✐❦②♦✲❙❤❡❧❛❤✿ ■t ❚ ❤❛s t❤❡ ❙tr✐❝t ❖r❞❡r Pr♦♣❡rt② ✭✇❤✐❝❤ ✐s t❤❡ ❝❛s❡ ✇✐t❤ ✈❛❧✉❡❞ ✜❡❧❞s✮✱ t❤❡♥ t❤❡ ♠♦❞❡❧ ❝♦♠♣❛♥✐♦♥ ♦❢ ❚ ∪ { σ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ } ❞♦❡s ♥♦t ❡①✐st✳ ◮ ❍r✉s❤♦✈s❦✐✴❆③❣✐♥✿ ◮ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ ✐♠♣♦s❡ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t σ ✐s ❝♦♥tr❛❝t✐✈❡ ✭✐✳❡✳ ✈ ( σ ( ① )) > ♥ · ✈ ( ① ) ❢♦r ❛❧❧ ♥ ∈ ω ✮✱ t❤❡♥ t❤❡ ♠♦❞❡❧ ❝♦♠♣❛♥✐♦♥ ❱❋❆ ✵ ❡①✐sts✳ ■t ✐s ❛①✐♦♠❛t✐③❡❞ ❜② s❛②✐♥❣ t❤❛t ( ❦ , σ ) ✐s ❛ ♠♦❞❡❧ ♦❢ ❆❈❋❆ ✵ ✱ (Γ , σ ) ✐s ❛ ❞✐✈✐s✐❜❧❡ ♦r❞❡r❡❞ Z [ σ ] ♠♦❞✉❧❡ ❛♥❞ ❑ ✐s σ ✲❤❡♥s❡❧✐❛♥✳ ◮ ❆ ♥❛t✉r❛❧ ♠♦❞❡❧ ♦❢ ❱❋❆ ✵ ✿ ❛ ♥♦♥✲st❛♥❞❛r❞ ❋r♦❜❡♥✐✉s ❛❝t✐♥❣ ♦♥ ❛♥ ❛❧❣❡❜r❛✐❝❛❧❧② ❝❧♦s❡❞ ✈❛❧✉❡❞ ✜❡❧❞ ♦❢ ❝❤❛r ✵✳ ◮ ❆❣❛✐♥ ♥❡✐t❤❡r s✐♠♣❧❡ ♥♦r ◆■P✳
◆❚P ✷ ✿ ❱❛❧✉❡❞ ❞✐✛❡r❡♥❝❡ ✜❡❧❞s ❚❤❡♦r❡♠ ✭❈❤✳✱ ❍✐❧s✮ ▲❡t ❑ = ( ❑ , ❦ , Γ , σ ) ❜❡ ❛ σ ✲❤❡♥s❡❧✐❛♥ ❝♦♥tr❛❝t✐✈❡ ✈❛❧✉❡❞ ❞✐✛❡r❡♥❝❡ ✜❡❧❞ ♦❢ ❡q✉✐❝❤❛r❛❝t❡r✐st✐❝ ✵ ✳ ❆ss✉♠❡ t❤❛t ❜♦t❤ ( ❦ , σ ) ❛♥❞ (Γ , σ ) ❛r❡ ◆❚P ✷ ✳ ❚❤❡♥ ❑ ✐s ◆❚P ✷ ✳
❆ ♥♦♥✲❡①❛♠♣❧❡ ▲❡t ❚ ❜❡ t❤❡ t❤❡♦r② ♦❢ ❛ tr✐❛♥❣❧❡✲❢r❡❡ r❛♥❞♦♠ ❣r❛♣❤✳ ▲❡t ϕ ( ① , ② ✶ ② ✷ ) = ①❘② ✶ ∧ ①❘② ✷ ✳ ❚❤❡♥ ✐t ❤❛s ❚P ✷ ✿
Recommend
More recommend